analyze guarantee run efficient vertex selection technique include parallelization efficiency social various scale accelerate server distribute machine hundred inference formulation partitioning also present brief overview structure instance solve measure fitting regularizer correlate exhibit sparsity many word attribute ad pattern graphical joint summation clique graph inference undirecte interact clique inference set represent vertex vertex neighbor computationally dependencie problem vertex illustrate risk consist sample consist zero working dependency natural edge clique add edge clique example challenge inference store machine divide exploit decompose solve simplify develop efficient algorithms variant execute server framework computational node divide worker globally server worker server two way parameter recent parameter dependency block loss generality consider server server aggregate without bipartite graph part assign server assign illustrate want worker server goal implement graph balance load computational load incur roughly small parameter keep ram cost memory communication minimize goal cost worker communication show server never maintain cost communication server worker request server minimize machine partitioning attract interest scientific scaling database social previous cut recently vertex cut different work partitioning give partition art several large quality objective note np complete rather task partition intuitively assign balance minimize minimize tb partition iteration improvement probability fail evenly complete several solve key important algorithm proceed follow add increment fu fu I detail quality algorithm partitioning generate partitioning refer iteration assume maximize fu u note monotonicity fu fu fu finally balance happen chernoff terminate increment hence finally fu server neighbor iv program totally constraint solve optimization index server maintain record whether integer exploit constraint consequence integral relax convex optimization single find locally optimal one variable due optima neighbor perform complexity could infeasible first server describe neighbor efficiently tb initial neighbor partition pick assign remove remove tb expensive problem strategy vertex find vertex load balance assign balancing improve naive calculate find fraction undesirable remain impractical graph vertex accelerate cost compute store obtain new adding add change vertex vertice sparsity cost small build datum store cost illustrate th th entry doubly link array increase low vertex update doubly list preserve portion cost small equal degree store small array head location cost jump accelerate list cover input partition union vertex update address sort integer bound degree update evaluate vertex doubly link list access vertex due sequential storing array find vertex list link keep cost bad head average fast partition balance assign additional address balance optimal remain computation randomly construct subgraph neighbor subgraph sequentially denote subgraphs feed union subgraph begin strategy subgraph advantage link efficiency place parallelization soon keep current partition graph size large memory quality efficiency extreme consuming though reduce remove quality single parallelization cpu time subgraph node server node issue partition progress maintain request worker partition subgraph worker read file subgraph subgraph use set vertex well potentially assign assignment improve result use subgraph subgraph old vertex old cost parallel partitioning start subgraph result initialization way want already partition old use initialization use neighbor set partitioning worth impose maximal delay
reformulate reject sample reject rejection quality leave rejection vary reject fraction initial reject fraction operating region accuracy respectively apply classification rejection data performance classifier supervise arise enforce accord improvement achieve rejection rejection rejection ht cccc derive minimization classification highly truth rejection bottom row color derive split learn design sample classifier learn classification parameter accuracy rejection misclassifie classify parameter vary hardness degree overlap gaussian ability cope hard table class accuracy evaluate classification comparison rejection measure reject assess usefulness electrical computer superior university center pa edu electrical engineering lx electrical computer engineering pa systems application three system rejection measure ability correctly classify sample ability measure relative incorrectly classified compare classify applicability rejection loss function automate classification application consequence critical introduce great advantageous classify diagnosis retrieval furthermore cope set noise classifier rejection sample reject fraction nontrivial classification analyze error trade allow determination incorporate reject option embed achieve risk minimize one embed option design framework assess rejection variant base rejection conceptual point feasible classifier reject combine prohibitive rejection world system characteristic roc surface obtain false positive belong positive outlier belong volume roc suffer ratio accuracy gain small measure evaluate performance regard reject evolution rejection relate accuracy accuracy insight rejection unbounded rejection classification allow assessment reject allow three embed reject option label reject correspond derive classifier rejection potential classifier measure show performance completely describe application conclude system see couple classification map dimensional nr represent whether rgb rgb cycle cycle cycle rgb rgb rgb rgb rgb rgb rgb rgb rejection derive system reconstruct probabilistic general rejection element reject small element thus reject incorrect classification classification function separation induce problem index reject pose separation define reject represent support norm subscript present approach lose correct classification element incorrectly incorrectly classify classified ratio incorrectly correctly classify evaluate compare concentration concentration note objective zero classify reject sample classify everything reject rejection fast option add variation express reject length variation allow estimate measure fundamental obtain give support define rejected rejected reject accuracy regardless separates reject correctly newly reject classified sample since express combine give eq quality decision absolute difference rejection q incorrectly since consider classification rejection couple classification obtain reject ideally incorrectly sample reject quality become rejection become rate fraction ratio classify sample triplet relate triplet confusion knowledge confusion binary pair mean triplet reconstruction confusion system sufficient describe behavior classifier q denote classified fraction incorrectly classify reject classified reject incorrectly classify reject give binary classification confusion rejection computation quality knowledge reject mutually exclusive reject unable reformulate computation measure rejection accuracy reject fraction quality accuracy reject accuracy entire quality comparison among classification measure
genetic marker specific mutation incorporate marker poisson allele profile develop mcmc origin genome come proportion population allele frequency highlight existence rare european highlight future development relax incorporate population article devote infer method automatically must careful setup generally sensible correspond common structure update remain manuscript manuscript iteration population restaurant update new link segment chinese restaurant customer ik customer simple parametric observation marker beta al iteration I auxiliary b r walk around parametric hence leave form snp section carlo nonparametric infer extend linkage allow infer infer population origin region assume mutation present mcmc simulation genetic consider variation frequency lead rare modelling snp data mcmc refer presence systematic genetic marker allele population population central history gene map useful gene event nucleotide snp year marker assess population investigate relationship focus particular occur isolated result population american broadly detection sample iv population population population proportion vi identify genetic segment individual propose two analysis estimation pca population decade low individual reflect genetic note principal capture may reflect linkage aim historical event explicitly association assign possibly membership influential early structure assume allele jointly simple extended structure determine genome comes model independently profile markov monte extension address dna mixture dna group pass along day contain original population shorter segment reflect long event occur segment recent linkage necessary thin link correlation quality et improve contiguous share proportion grain process allele model introduce profile segmentation profile rich dependence population important statistical concern model determination use selection probability bayesian though specification population population nonparametric offer unbounded size apply use nonparametric counterpart model linkage model design scalable bayesian dirichlet hdp uncertain costly simplify event identity scale nonparametric slice truncation describe gene section allele individual denote proportion individual individual work take population account employ split contiguous genetic denote follow split segment et complete specify l nucleotide snp distribution use proportion assume equal asymmetric population genetic material thompson control asymmetric allow nonparametric population limit lead mathematically symmetric hierarchical stick break representation diversity population large uniform proportion dirichlet extend specifically constructive follow assume infinite number population particular random number population use population subsection detail complete dirichlet allele frequency case beta allele population concentration independent reason specify fairly denote marker genetic available proxy physical measure interpret issue population range truncation allow resource propose subsection discuss hierarchical prior break population impose population higher undesirable modelling induce order artificial undesirable switch inference address abstract formalism base construction hierarchical dirichlet dirichlet accord dirichlet parameter consist positive concentration parameter base measure constructive process review nonparametric generalize dirichlet surely independent base atom stick break modelling correspond population allele random introduce variable next condition population fall select state individual update finitely achieve simulate stick breaking fall threshold forward backward slice backward variable slice dependency latent filtering instead ignore cause slice algorithm metropolis hasting proposal proportion slice threshold population index proposal slice population indicator forward dynamic eq starts proposal q denote way expression account effect conditioning proportion current backward population computationally mcmc since technology straightforwardly extend assume individual individual chain allele frequency follow straightforward correlate allele allele frequency another likely due specify allele population moment extra physical e sample individual source modify require specification measure hdp perspective employ generalization dp stick break chinese restaurant cluster size decay possible recover population simulation bi genetic marker mutation per population history particular iii population recover number follow population I specification number sequence marker increase spurious dirichlet nevertheless cover I prior specification difficult agreement value aim capture arguably implement parametric linkage model describe allele al value likelihood latter harmonic example seem log software drawback criterion interpret result give suggestion improvement demonstrate human wide study american uniquely advantageous history extensive occur year hard recent american branch separate drift shape genetic landscape cause drift east eventually become allele common snp rs allele european alternative allele snp human straight shape rs snps include strength absence snp rs informative snp act specify carry variation range effect mean selection individual snps available population cell panel publish american population west south european population run sampler iteration every iteration precision hdp mean beta observe frequency prior evidence mcmc minimize alternative major I occurrence use panel cardinality european origin cluster respectively confirm look frequent assign major verify firstly
concave conjugate machine reduce solve result saddle widely saddle exhibit say likewise keep notation learn convex concave saddle explicit form separable convex gradient smooth subgradient smooth strong convexity convex minimization erm eq convex function predictor convex regularization fall regularize erm formulation regularize detail erm employ eq lead sep compare general consider composite solve alternate multiplier admm problem particularly erm zhang propose dual descent descent primal dual dual stepsize unnormalized primal solving carefully exploit subproblem propose adaptive stepsize summarize primal elaborate theoretical comparison erm superiority conclude first primal could linear achieve method stochastic dual coordinate iteration intensive variant batch method handle however conservative constant stepsize primal convergence reality observation exploit propose adaptive solve configuration optimize alternatively update dual principled separable iteration variable follow coordinate block exploit erm couple block norm small block primal couple large help variable update proximal also incremental variable intermediate convergence adaptively contrary tb block initialize pick coordinate block configuration update update use whole procedure sep notable characteristic compare size whole bring independent parallel modern help use available possible convergence assume strongly n technical coordinate e practical implementation measure performance term w r pass firstly problem zhang point set diagonal optimization ridge employ conjugate ridge sep ridge regression dual closed objective measure entire ill conditioning observe substantially condition stepsize time sampling dataset number ccc protein ccc dataset protein compare real set benchmark list table collection obtain dataset take term aim minimize regularize method hinge smoothing conjugate hinge ridge necessity whose dual convex initial newton sag stepsize sag try stepsize result theoretical stepsize sag loss algorithm smooth hinge compare method task early epoch result epoch cause stepsize choice primal iteration explicitly couple block primal theoretically superiority erm immediate size theoretically optimization also acknowledgment support china thank discussion proof present firstly characterize semi partition consider configuration matrix definite firstly q cauchy schwarz fact view inequality obvious exist far simplify consider zero expand positive firstly variable th let I strongly minimize saddle happen round consequently mn dual strongly obtain side
similar classic correlation propose software package usa confirm solid thin parameterization compression wavelet evident wavelet match signal address wavelet signal suggest methodology wavelet version wavelet science engineering research development pt cr tag tag university electrical engineering I electrical role diagnosis low cost clinical conduct although extract information utilize recently advanced wavelet I wavelet offer frequency ii perform characterize global design wavelet match detection wavelet match signal systematically numerous issue generally regard represented wavelet wavelet often select research wavelet compression detect minimize compressed wavelet compress signal minimal wavelet consider original signal analysis wavelet aim six seven nine medical usa place projection eight channel internal present process internal visual verify normal diagram depict cc perform signal low analog use frequency analog solution bc configuration hour four body index sd one utilize university raw contaminate potential iv signal appear channel visually g identify discard practice recommend obtain reliable subsequent subject interval hand recursive iterate procedure iterate coefficient observe perform similar recursive analysis ii level compression discard criterion base large absolute coefficient retain eq original compressed analysis introduce reconstruction evaluation effect compression commonly percent root square utilize study compression signal subject cycle human scale period magnitude fourier consequently minimize difference decomposition wavelet wavelet wavelet human haar set optimize choice wavelet abundance wavelet prohibitive result wavelet introduce know wavelet finite impulse filter length great wavelet find haar wavelet one six utilize wavelet wavelet wavelet parameterization define every scheme generate parameterization minima surface wavelet
iteration rbf kernel distance neighbor far fine classification rbf rbf svm classification three baseline original kernel rbf figure fourier dimensionality grow accuracy rbf fact approximate use section mse show dataset achieve compact kernel compare fourier figure comparable marginally well lead well classification kernel map advantage train scalable nonlinear computational complexity advantage achieve observation transform dft denote element dft dft dft efficiently store element dft kernel nonlinear feature although fourier procedure element trick convolution nonlinear performance storage tend performance feature recent allow large map shift invariant substantially performance attribute mostly simultaneous along parameter neural nonlinearity bridge stream high dimensional impose capture deep compact li david discussion kernel via feature machine challenge method approximation pick nonlinear map prediction propose achieve kernel joint achieve map function definite induce could fortunately one utilize rich despite popularity high complexity vary prohibitive million tend growth svms application since space utilize method expressive reasoning nonlinear become popular machine formally find nonlinear even approximate design approximate kernel result work propose optimize directly definite shift learn approximate achieve predefine map compare baseline method propose structured projection computational number nonlinear input seminal map mapping form rbf additive skewed multiplicative monte build random fourier besides base matrix significant effort good kernel mkl optimize directly optimize joint kernel type machine decomposition limit vector scale truly datum alternative apply machine partitioning relate begin approximate shift positive fourier characterization definite borel transform interpret expectation carlo fouri use type kernel include despite popularity feature notable issue matter influence kernel approximate approximation technique try lead kernel work fourier lead feature function kernel shift invariance show map positive shift multi training optimize map dependent hinge initialize output challenge number optimize problem nonconvex propose find sgd traditional eq sgd mini point write set step optimize initialize random fourier sgd optimize algorithm classification also kernel mse note classification former compact
patch follow patch whole pose patch learn sum input output project forest suitable multiclass classification parametric tune parallel robust processing patch spatio forest tree tune depth laplace calculate pose construct drawback pose pose real image back follow training recover might choose sample test viewpoint regard labeling write appear pose pose assume view trend reduce range multiply smoothed step turn trend turn clean rf rf patch rf promise clean background real background conditional automatically rf clean set truth indicate global although achieve drop learn patch importance verify patch hence overfitte show test test pose misclassifie pose like square front often misclassifie estimation learn image experiment verify effectiveness future foreground background utilize image assumption patch generalize height title date em project aim pose crucial know pose object besides also drive good pose estimation driving system pose benefit field pose implement dataset multi camera however create extremely consume limited dataset research specific poor work utilize power shape specific balanced precisely million thousand image easily pose base predict category tight although choose probability vector pose give grid score combine whole set pose feature diverse project focus transmission method iteratively classical problem research line rely diagnostic object view category view invariant group part feature svm classifier pose latent discrete convolutional propose base simultaneous gain structural object like model base pose estimation represent obtain rough object annotate viewpoint motion limitation utilize solve collect model annotate thousand object category vision utilize evenly accordingly image first pixel divide patch patch preprocesse extract whole evaluate build increase test
dimensional significantly curse canonical design set variable instance complexity certain cca view regression cluster etc theoretically interpret cca probabilistic foundation practical cca view utilize application extract kind texture popular visual base video typical cca view correlation view drawback information obtained ignore tackle develop cca generalize view yet natural particular analyze view approximate covariance tensor investigate alternate square adopt measure covariance view represent feature whereas tensor illustrative subspace feature dimension useful extensive experiment challenge task internet cca approach confirm summarize work introduction extension extension view extensive experiment paper dimension find dimensional feature former subset variable original transform datum reduction e laplacian e lda amount label view attract multi view graphic multi view family weight focus remove irrelevant reduce multiple leverage dependency coherence view conditionally independent thus shared exploiting correlation exploit method multi dimension among view consensus share adaptively view pairwise constraint dimension incorporate use margin latent subspace al formulation share conditional constraint correlation analysis originally find basis variable project maximally pattern recognition svm view prove rademacher svms cca correlation conditionally simple subspace base view show weak condition addition cca concentrate tensor vector cca without extension multilinear cca consider study volume tensor flexibility obtained term cca difference multiple vector view closely work concerned cca cca cca least cca perform combination canonical representation costly svd train drawback reformulate cca couple ls seek pair cca efficient adaptively disadvantage cca ls namely pairwise exploit correlation view ignore framework introduce correlation generalization several column cca find call canonical correlation canonical maximize problem matrix stack cca constraint cca cca view cca canonical variable avoid solution adaptive reformulate impose l cca cca pairwise tensor multi reduction high view diagram kind color histogram wavelet texture feature extract feature intuitive illustration generality different data covariance subsequently sum r feature low projected briefly concept multilinear tensor denote I denote mapping associate multiplication store express series kronecker c p finally frobenius tensor give q instance view two variable tp appendix far add become identity nonnegative trade base instance include web annotation near neighbor feature nn view long cca formulation view accord web term average voting term cca view base ls unsupervised dimension induce cca empirically evaluate supervise dataset secondary sequence instance randomly unlabeled instance performance three cca method cca cca ls amount unlabele follow find evaluated set since directly learn datum large needs solve thus position divide attribute context attribute base current position middle position view attribute dimension c cat cca cca l compare relation dimension table concatenation comparable strategy single cat baseline dataset cca view utilize former accuracy cca increase avg l cca good dimension cca cca l significantly cca ls cca reason seek canonical factor decomposition tend uniformly factor discover explore kind explore information ad internet uci repository whether dataset instance instance utilize unlabeled sample use attribute represent absence l feature view view term site view feature cat cca cca avg ls show compare observation concatenation strategy cat cat high label much steady underlying correlation compare unlabele utilize order correlation reason effectiveness natural image conduct subset cat image distinguish similar cat label utilize unlabeled color auto wavelet texture represent annotation improve number l cca avg peak cca cca avg l accuracy avg cca ls decrease satisfactory even method label cat cca avg cca web task non linear problem dimension infinite linear visual histogram achieve nn averaging use formulation view setup tensor parameter optimize unlabele utilize separability trick strategy comparable slightly well c avg empirically computational conduct matlab computer ram cost result high decomposition adopt paper could satisfactory accuracy efficient demonstrate superiority unsupervised cca deal multi ignore correlation feature resolve tensor cca discover analyze view application conclude subspace view feature especially high examining may utilize outperform high cca traditional main disadvantage could utilize accelerate tensor accord element denote th additionally accord
label show dual kernel rank fraction maintain representation align manifold sort unfold form image synthesis remarkably distortion manifold unit author address issue machine proposition property laboratory de alignment match correspond property generalize align complexity unfold plus alignment cope multimodal align dimensionality robust transfer address issue reduce computational principle exhibit synthetic example recognition constitute interest pathway characteristic depend availability family attempt adaptation canonical cca source meet seek target discrepancy mmd geodesic datum suppose feature idea geodesic distance linear subspace intermediate flow subspace path dimensionality pca cca gm information projection discriminative label source find feature partial pls another know domain coherence plan source target try align feature move decision hyperplane know matching generally use appealing method specify amount pair semi alignment project belong become geometry preserve perform deal multiple cope nonlinear introduce generalization property allow dimensional property cope extract rotation manifold structure unfold simultaneous domain domain align manifold kernel numerical inversion permit alignment meaningful propose summarize align manifold kernel nonlinear cca align pair may useful align lead pair demand counterpart must stress laplacian take effort compute efficiently solve resort reduce remainder review section introduce formulation practical present linear method toy real visual conclude section classification belong close apart entity lead component class otherwise class similarity represent radial basis rbf near compute three entity joint contain column serve common project inverting domain adaptation synthesis hilbert replace reproduce kernel dimension ii block row hilbert thank sum hilbert theory resort representation theorem express linear map replace dot product contain become instead dual advantageous operate numerical computational current fisher deep latent space map define eq feature eigenvalue confirm use benchmark among candidate kernel tight graph artificial distortion mis alignment visual database recognition first series toy domain column fig scaling experiment hold rotation line third dim illustrate classification source align basically rotation manifold allow sort unfold experiment even alignment projection resolve provide picture green linearly projection trend experiment provide ccc cc cc space exp domain exp exp linear extract feature second right accuracy rbf kernel project randomly label resort operation simultaneously unfold full actually h c adapt svm domain show inversion direct linear kernel shown label keep fig reconstruction capable invert accurate basically significantly domain rbf achieve target rotation exp class domain domain class c exp plot average run consider amazon four domain amazon feature extract normalize histogram visual word obtained subset amazon dataset feature adaptation proposal supervise adaptation geodesic eigenvector pls use pls eigenvector source domain constrain source project correct decision source domain method label pixel domain sample alignment ordinary nn classifier label pls use target sensible kernel problem histogram intersection k j du nn top bottom report outperform compete improve supervised method provide art handling domain dimensionality classifier align
partially ac ac may satisfy acc ax ac different subset unconditional association set bc ac x cf ac z always hold pairwise ac z ac represent separate give separate reduce model explain separation criterion imply away association hold ac bc ac bc acc thus association connect graphical etc conditioning association figure ac notice appendix z condition bc one six ac bc ac bc also hold bc hold may six distinct rule supplement hand four draw dag consequently bc exclude ac ac ac ii bit conditioning weak square seem ac bc illustrate vc acc ac bc theorem corollary relationship ac vc acc example argue ac bc dag relation replace covariance comparison ac bc x ac ac z ac ac ac ac qualitatively ac iff theorems implication vertex compare square parent behaviour nature path illustrative theorem move drop move path illustrate drop ac z factor numerator denominator thus find theorem possibly whenever graphical association tree vertex definition connect vertex say subset give separate separate separate hand unique give relation set describe separation criterion satisfy independence formal operation triple define satisfy ac ac ac satisfy ac ac figure ie ac ac ac substitute direct skeleton tree path connect connect vertex ie path k ac ac ac ac ba z ac bc ac ac b require iff independent none find present none direction water point consideration measurement water level bx ie water neither however also ac ac none point ad stream ie ie ie section relationship square qualitatively compare well qualitative condition relationship reduce counter unless ie conditional graph law clearly keep plot compare relaxed z necessity figure edge respectively dash regression set plot title plot coefficient coefficient unconditional ac ac solid edge ie qualitatively ac ac ac z violate concerned square partial correlation condition violate condition square qualitatively compare either theorem ac z figure satisfy violate qualitatively correlation square qualitatively relaxed qualitative comparison sufficient involve mutual information matrix square comparison qualitatively rule sign comparison correlation regression develop definite e trivial constant k either zero denominator sign numerator algebraic positivity c correlation replace unless assume abuse still correlation ax cx ax cx ac ac ax cx ac denote inequality inequality ac ac ax cx ac denote bb ac ac bm ac ac ac ac enough ii hold ie ac bc bb b cx notice eq bb substitution proof ie notice zero none bb bb z ac follow ac result condition denote simplification ac give connect clearly connect connect clearly connect convenience square need ac ac result separate imply ac z I q show consider subgraph vertex c ig ix ib I ac I ib triple ig ac ac n conditioning suppose separate ac z ac z ac ac ac z follow assumption decomposition ac intersect g q v vertex arrange l k x result represent consideration ac z acc ac furthermore show prove direct acyclic vertex exist path non connection possibly connect say set empty separate separate dag whenever separate relevant vertex ac z ba ba clearly converse path however ac possible vertex imply least cv ac would ac thus separation cc give get ac path ac ac ac ac ac z ac b ac ac give ac ac fact three supplement closely refer treatment mixed graph contain three terminology describe relation graph define say vertex pt mixed condition sp separation mixed vertex path ie path vertex graph connect set every connect separate pair disjoint set say relation represent disjoint
box object view get fan pathway two objective fine tuning label multimodal initialize bfgs mnist pathway likelihood optimize fine tune pathway deep evaluate joint test joint analyzing image digit mnist randomly draw digit noisy learn dnn mnist digits default model dnn classifier test test clean testing noisy digit db train triplet clean digit test multi view svm indicate helpful multimodal svm fan multiple boost well achieve goal want output source boost matter objective fan deep multimodal powerful lr error multimodal fan share representation multimodal powerful branch deep function different step initialization tuning stage cd maximize joint fine stage define label object recognition answer leverage source boost experimental fan structure fan easily fan extend branch share node branch lstm joint multiclass object recognition jointly prediction experimental demonstrate baseline multimodal year approach propose multimodal joint rbm gaussian unit et al low image additional svm demonstrate various object recognition multimodal deep attract attention use autoencoder speech composition unimodal undirected pathway pathway large unlabeled potentially multimodal allow naturally effectively handle specifically paper unified fan learn compose lstm generalizing model initialize cd fine parameter optimize leverage multiple handle different visual recognition visual denoise autoencoder extend corrupt boltzmann introduce recognition denoise shape indicate ignore handle effectiveness regular structural multi propose joint method detection boost competitive baseline multimodal svms share handle modality clarity deep fan explain boltzmann machines rbms branch boltzmann fine deep pathway part rbms introduce rbm hide unit parametric form likelihood calculate cd ignore bias observation layer update bias restrict boltzmann stack rbms high activity hide layer energy configuration represent visible rbms real likelihood wise basically rbms cd rbm treat next rbm stack multiple deep kind couple stochastic binary unit hierarchical via rbms clarity way suppose input layer unit clarity modeling layer visible rbms bias visible clarity analogously get likelihood formula fan additional top right multi modal layer ignore input cd effectively tune parameter initialization joint representation modality discriminative multimodal update via multimodal deep basically stack rbms layer learn rbm treat next stack v x manner mention model approximate expectation mcmc procedure expect sufficient statistic learn latent replace update log h entropy approximate posterior naive factorize x field dependent item sample mcmc parameter cd eq fine determine visual labeling deep different visual labeling corrupt way clarity loss specify indicate share triplet l sigmoid case branch simplicity clarity note branch structure keep class object assume input object object belong purpose answer view image specify ignore clarity initialization cd tune w r via backpropagation bfgs visual label two even multimodal cd view recognition leverage joint learn joint output input latter leverage source bi feed consist encoder recover share feed multiple model adjust visual cd instead recover autoencoder multimodal unimodal undirected pathway completely unsupervised fan multimodal fan lstm cnn multimodal multimodal discriminative modal feed forward neural share handle learn together representation multiple
create density image plane translation add snr synthetic optimization perform seven traditional sgd accelerated optimization particle rescale deviation parameter radius minibatch size use except iteration minibatch minibatch trade small result size slow base tune training accord base rate momentum cm online hessian epoch online increase log remain initialize density result versus gradient online minibatch projection match density cross correlation search reconstructed iterate similar publicly approach run method evaluation five fail converge particularly approach approximately formulate cast result stochastic array seven range sgd quasi synthetic method little exist make epoch minima notably require per hessian free simply take iteration method problem challenge highlight exist manual amount manual every would new need automatically department determine structure biological key biology heavily paper structure latent model seven method recover less epoch initialization find slowly simple method converge optima method fast protein traditional ray limitation target grow impossible biological determination raise attempt thin pass measure produce image visible relate capture interference image nature biological keep lead extremely around imaging image illustrate particle image ct ct projection direction corruption density coarse shape visible fine presence practically explore em density introduce image formation em posteriori estimation stochastic optimization speed gain ability compute estimate stochastic less initialization previous bad initialization able quickly initialization compare em mixed result suggest serve optimization algorithm paper real benchmark aim refinement initial estimate project image consider orientation state slice origin projection direction particle fundamentally ideal circumstance impossible error refine poor initialization structure clearly wrong appears result publication incorrectly crucially use method orientation image orientation parameter analytically intractable numerically far perform marginalization originally image alignment marginalization originally perform find individual particle marginalization raw stochastic progress make quickly optimization recently fundamental sgd popular surprisingly momentum method sgd iteration natural manifold fisher accounting attempt approximate curvature hyper attempt operating gradient still limited strong scope interested reader compare evaluate performance good converge optimal simple complex costly formulate observe unknown seek projection density direction corrupt primary interference model frequency spectrum typical zero information zero vary setting condition every refer reader noise arise exposure noise model iid formalize density denote cubic grid orientation represent transformation project image denote image plane center represent denote consideration evaluate make slice eq fourier transform image fouri diagonal interpolation operator fourier transform speed frequency fourier specify provide shift however unknown cope shift double analytically resort quadrature quadrature direction account rotation quadrature shift quadrature weight quadrature scheme consequently frequency image combination exponential encourage density specifically prior possible promise directly marginalization
fundamental tucker tucker tensor q matrix tensor tucker unchanged size remove rotation encode abstract space eq invariance minima isolated isolate consequently systematic decomposition unconstraine interpret search endow hessian order riemannian metric hessian costly simplified index candidate riemannian metric simplicity consider element note convex convex individually td reasonable novel riemannian space early square propose tangent line abstract object total run total riemannian riemannian key riemannian conceptually transform unconstrained briefly development cost first equivalence blue color manifold tangent induce equivalence realize vertical space tangent equivalence sense metric horizontal abstract abstract tangent riemannian g x along equivalently riemannian well pose horizontal x tangent vector transformation straightforward endow riemannian principle concrete abstract g start dimension need tangent operation extract ambient normal matrix dd matrix characterization eq q efficiently matlab routine vertical vertical characterization dd horizontal projection lyapunov skew couple lyapunov routine combine horizontal local search manifold I depend representation horizontal abstract tangent manifold smooth mapping tangent tangent generalize concept manifold lift xt depend riemannian completion choice development use conjugate smooth complete conjugate remain cost end riemannian guess concrete formula total n n r n r dd x lyapunov order r f auxiliary euclidean partial derivative respect due partial scaling riemannian lift subsequently lyapunov routine numerical cost derivative initial guess square cost degree direction f algorithm state tucker decomposition nuclear intel machine gb ram matlab handle select entry dimension os create stop either mse iteration exceed five deviation comparison os asymmetric os euclidean benefit conventional symmetry natural randomly os simplicity compare descent backtracking give conventional consider os b fast error scale tensor size rank os outperform influence instance os complete superior decrease ill conditioning instance case additionally impose exponentially decay number cn influence evaluate property noise noise train os asymmetric rank along different case consider tensor size consider rank hyperspectral hyperspectral five random pixel os rank r adopt randomly split validation algorithm stop correspond ratio propose corresponding day wide bin size completion dataset reveal validation partition iteration rank second l os mse problem stem riemannian exploit fundamental uniqueness tucker decomposition riemannian enable riemannian concrete expression work superior benchmark future research direction look update rank tensor tensor tucker acknowledgment thank van present dynamical science office research national height tensor supplementary mm width concrete r tucker transformation riemannian product tangent manifold tangent x inner extract tangent characterization hold tangent space characterization skew matrix generality n characterize horizontal tangent remove along vertical vertical linearization dd linearization orthogonal vertical characterization space orthogonal relationship vertical characterization trace trace trace trace mode unfolding since skew extract component ambient definition tangent satisfy equation solve matlab routine tangent h result dd note tensor r plugging equivalent r matrix skew lyapunov equation routine combined gauss auxiliary unfold specific scale metric derivative scale consequently relationship lift requirement tangent space mode dd lyapunov equation square lyapunov solve routine representative comparison numerical span synthetic instance tensor os figure show different manuscript mean figure os figure fast consistently competitive especially ratio tensor rank os algorithm show algorithm instance result sample complete rank convergence os propose five dimension figure superior propose outperform additionally
fail hold interpretation differential natural tradeoff privacy release database achieve easy goal aggregate interest conversely complexity privacy complexity differential setting less recently show optimal privacy guarantee meaningful complete privacy give private choice smoothly approximate differential algorithm compute extremely fundamental database marginal row say sense md meaning e md differentially private marginal average privacy combine result must multiplicative answer family query additive necessary laplace differential low mechanism average case error guarantee guarantee mechanism matching surprisingly degradation answer marginal sample laplace mechanism factor widely technique query bad error efficient differentially private guarantee laplace sample differential cc c lower n code originally digital recent bound differential al code construct demonstrate mechanism accurately marginal private give database construct answer attack privacy accurately break private release specifically state algorithm differentially private private meaningful privacy guarantee size reduction start row uses differentially apply attack value low quantify pure sample proportional md correspond add squared turn well tail bound add noise differential privacy different union chernoff show namely ensure error marginal trick marginal rather row row binary database differ single denote replace another privacy differentially private adjacent database know differential privacy generalize smoothly say database differential privacy differentially private adjacent marginal privacy exist differentially private exist differentially private every accuracy bound namely differentially private main differentially private sufficiently large introduction rearrange convenient introduction code tailor originally dd code subsequent adapted existence code bind differentially choose create copy entry output privacy differentially private mechanism nk contradiction sound code completeness differentially contradiction ensures require k contradiction seem natural one way case error rather average bound mechanism distribution sample distribution function formally measurable firstly clearly verify shift distribution give bind obtain infinity surface precisely gamma q inequality eq require verify first gamma r I give circuit use privacy powerful take input database query differentially assume simply differentially define mechanism private differentially private differentially private require q guarantee hold q failure failure dominate work differentially simple private suppose special hoeffding hoeffding copy let give e dd row eq q rearrange require combine sample uniformly sample entry sample independently n let verify construction
latent latent version complete coordinate descent minimization constraint hence duality strong factorize case slack variable latent model product linearity expectation fact qx ix kl primal strong duality maximize constraint minimize kl minimal value fully variable assignment q fall slack relaxed remainder slack form slack equivalent rgb rgb rgb rgb rgb rgb rgb rgb rgb cs berkeley edu
vector real hyperplane side sum along direction consider shift hyperplane decision determine anomalous shift hyperplane store anomaly network design single dimensional weight regression hyperplane directly eqs hyperplane anomalous non anomalous distance fall derive threshold point form center correlate center class hyperplane anomalous identify anomalous vector activation identify anomalous point cm cm theorem corollary height em expect value anomaly give identification class anomaly identify neural describe decision separate offline recall class measure datum throughout probability measure weighted lee anomaly detection traffic case measure identifie measure accurately entropy responsible formation particular anomalous identify attribute sample assume subset bound area define partitioning size maintain partition maintain graph base anomaly composite score scoring structure asymptotically optimal false sample increase normal produce distribute maintain disjoint score anomaly center respectively hyperplane measure center hyperplane anomaly datum point regression greater bind anomalous likewise within hyperplane great anomalous shift show activation neural obtaining point value hypercube unit area dataset unit square plane give minimum sample region region radial circle q circle decrease probability contain within one contiguous otherwise order exceed maintain radial size less disjoint thm lemma circle radius partition copy contiguous classification remain new know member class member candidate outside new linear respective hyperplane question center whereby hyperplane linear hyperplane suppose regression give otherwise micro anomaly detection anomalous radius center hyperplane anomalous anomalous class triangle definition anomalous section use obtain dx dl anomalous anomalous dx show exist anomalous
flat fix representation second qualitatively show able automatically proper conduct extensive empirical superiority previous fed ensemble length word column distribute vocabulary vector pool vector insensitive order word length sentence recurrent network connection hide form nature recurrent sequential generation task model machine step time recurrent bias non recurrent summarize past composition recurrent htb build parse initialize word non child composition recursive network hide parent parse child connection recurrent sentence tree recursive parse recursive neural neural composition recurrent neural network chain recursive composition pyramid locally combine reach top pyramid global sentence refer build differ neural try form decide illustrate acyclic graph show input sequence pyramid let level define j tt th j view phrase tt unit consecutive phrases original rd two level whole sentence enter pyramid apply word phrase rich pyramid matrix embed u v tailor factorization help parameter composition pyramid range recurrent way length linear representation fundamental behind encode semantic phrase composition two phrase express hand hope phrase variation decide parametrize decide non forward future composition mechanism adopt mlp output system multinomial pyramid define apply pyramid worth note composition implicitly form layer along phrase recurrent net recursive net htb pyramid build phrase original sentence illustrate rd pyramid straightforward short phrase high focus part sentence interested review goal phrase opinion set question mr cr cv cv list class measure split train fold list autoencoder word gradually phrase along tree sentence apply generalize pool paragraph unsupervise public top paragraph continuous pool phrase sentence recurrent recurrent neural network network convolutional pyramid use pyramid difficulty train recurrent largely due vanishing discuss dag vanish still recurrent show application propagation problem frobenius norm recurrent act since computationally exact value direct formulate coefficient typical minibatch optimize word train wikipedia embedding performance composition relate softmax implement try mlp implement improve htb mr cr nb rnn compare consistently rnn margin comparable number code phrase dependency hard help phrase nearby word consistently also outperform set well fail length encoder machine quite task think due encode characteristic surprising datum average limit window mr cr rnn rnn also v rnn table time hyper initialization report run consistently outperform set htb study consensus pre visualize belief score vary sentence set train adaptively appropriate hierarchy give give concrete mr prediction level score fig row show ty incorrectly high representation first belief correct multiscale combined allow correspond belief automatically component st row implicitly property explicitly objective achieve sequence explore new direction sequence instead length continuous effectiveness short represent acknowledgment author technology helpful support china national cb conjecture com ability accurately word phrase self sentence hierarchy phrase composition adjacent segment mechanism network particular task effectively vanish persistent qualitative quantitative analysis automatically suitable task yield task cast semantic parsing describe sentence representation representation quite translation matching perhaps simple continuous bag word sentence max pooling word word unsupervised fashion effective
result compare perfect seem show affect show accurate design work consider several perfect assumption suggest type compare theoretical result rank long cdf order th rank resample denote result sample result bootstrap compare base literature liu functions empirical methodology powerful design comprehensive likelihood balanced assumption liu al likelihood method data cycle liu et al use overcome balanced figure plot algorithm liu third htb contain birth month weight research university well rank quantile month weight intensive active nature activity birth treat record month perfect seven month month weight process result weight seven month time summary histogram seven month birth seven seven month birth observe median month weight birth different design pt bootstrap et df show satisfactory perfect propose nonparametric cumulative et estimator exhibit order property balanced applicable result application et df size acknowledge constructive comment associate support national institute health grant gm department statistics usa mb usa nonparametric df base set et use unbalanced rank bootstrap estimator df asymptotically exhibit finite scheme use estimator estimator keyword rank powerful collection technique collect representative small fairly accurately order take actual costly unit find application environmental sciences chen rank balanced unbalanced unbalanced rank rank order statistic quantify size unit rank quantification unit order pre suppose measurement th underlie population denote n identically df reduce balanced show chen easily see equal df underlie property balanced chen use estimate practitioner standard statistic make inference characteristic interesting develop efficient technique exact obtain chen al estimator use resample underlie desirable outline consider validity method problem describe finite sampling parametric resample population mean consider five design propose liu testing problem real consist seven month provide conclude remark likelihood powerful nonparametric use subject et estimation spatial estimation calibration among nx px minimize dp I multipli testing p lead nonparametric distance q size estimator element draw retain generate bootstrap repeat bootstrap easily bootstrappe purpose unknown parameter balanced modification sample underlie desirable similar et bootstrap sample mx underlie simplicity write test incorporate estimator introduce lagrange coefficient exp nan testing bootstrap mx function estimator proceed select continue retain repeat step bootstrap interest resample perform use b experiment also summarize example design per denote design distribution proceed proposition df interest row represent population converge distribution normal diag f df approximately refer rest work consider approximation degree freedom bootstrap use nan resample calculate proportion estimate testing sample normal logistic perform see unknown let x calculate bootstrap nominal
feature approximation diameter grow preserve derivative analyze uniform convergence propose motivated feature map enable task derivative consider kx ph tu ph tu ph tu j impose constant square expense slightly q case p p analogue obtain assumption boundedness impose certain moment expense bernstein net union extend differentiable z dd conv h continuity z supplement continuously clear p kx qx theorem comparison handle unbounded function detailed feature context improve machine approximation compact analyze approximation reduce equality change lemma bind rademacher bind combine x follow jx ix I theorem conditioning bind imply covering guarantee center combine bind arbitrary point quantity center lipschitz uniform optimize p cover center net eq cauchy schwarz get imply l z linearity get net center use propagate center thereby condition notice bernstein give union substitute jensen dc matching prove boundedness imply let random suppose yield bernstein satisfy theorem college house ar uk kernel powerful tool tackle capability relation good intensive scalability require operation limitation construction literature fouri construction kernel popularity theoretically provide norm derivative quality success several fundamental range extraction estimation discovery hypothesis capability relation possibly heart product kernel kx flexibility operate raise serious dealing order resolve numerous solution design additive construct shift paper yet efficient rely appeal kernel low dimensional map empirically x explicit low dimensional feature primal fast solver thereby enable scale algorithm degradation approach low rank approximation approximate online applicability area privacy causal surprisingly literature theoretical insight pr number fourier feature involve empirically statistic systematic entire characteristic decay diameter guarantee asymptotic nature finite sample optimal optimal almost sure convergence apart kx ml traditionally involve kernel involve kernel gram consist derivative address numerous task supervise multi infinite derivative elegant method mention quantify quality summary derivative term rate preserve match growth various along brief section material map definition continuous f finite borel denote banach integrable lebesgue measure rl df ix da r gx p nb nr gamma dr dr continuous translation invariant kernel kx fourier e symmetric kx tx replace measure write inner algorithm primal thereby well complexity solve mr precise dd differentiable tail compact optimal order convergence excess order significance grow analogue rate dependence section introduce practical theoretically understand theoretical optimality section see improve logarithm discuss consequence next suppose kx interest hoeffde inequality refine apply provide show consistent topology convergence convergence sure lemma instead discrete absolutely lebesgue density interesting study diameter therefore mention rate write
office building day usage business twice investigation business warm day probably figure present usage pm business day type system day turn middle day day special spike energy consumption spike model understand exploit differ treat day replicate modelling usage arising give usage give usage observe directly infer form general arise unobserved function main latent single switching among process sequence estimate derive curve replicate replicate switching among author single realization realization paper discuss detail paper contain replicate principle unlike work focus generalize change control parameter maximum penalize realization consider replicate replicate replicate simplicity across replicate hide unobserve consider identically follow fix replicate nk nz jx jx intercept ik depend identity matrix variance variable govern distribute length transition estimate standard measure replicate user depend since expectation maximization maximize em generate arise section overview propose intercept cubic entry find criterion propose cv smoothing simplicity application simulation choose r joint complete application replicate write p r entry eq hide state application calculate supplementary function fix smoothing guarantee depend dependence via summarize f maximize parameter obtain maximize obtain present update show form k ii pz ik see dx intercept probability diagonal step still change however easy except algorithm ik eq py nk ik obtain propose hold maximize fix eq initial work test datum discard treat set follow replicate cross maximizer pf convergence computationally intensive fortunately replicate estimate present use process restrict ease explanation use derive matrix plug formula yield second ik process obtain simultaneously supplementary section q r find supplementary datum replicate usage give give power assume covariance intercept smoothing describe detail figure fit nk fit curve give assume parameter estimate pz ik report fit give usage assume smoother estimate follow k z nk also estimate curve replicate transition transition actually happen gradually incorrectly replicate come get fit switching nonparametric regression observation within replicate correlate describe analysis assume follow table present obtain observe agree table show figure power usage usage consider upper seem building intercept variance therefore current follow markov generate replicate replicate locate type type simulate datum markov study vector plotted figure space true simulation obtained generate distribution repeat step time replicate repeat step obtain calculate simulation assume markov figure study replicate value try converge take long choose fit curve simulate simulation quality via produce deviation supplementary plot simulation improvement could way adjust freedom estimate parameter process error desire coverage interval form quantile coverage close supplementary box plot observe estimate close true follow maximizer maximizer rewrite positive
detector operate reliable regime adjacency size community characterize adjacency matrix entry bernoulli overall stochastic two connect parameterize probability matrix adjacency corrupt adjacency graph connection community restrict increase plant clique detection restrict block study critical size network study external edge community dimensional one matrix noiseless graph laplacian community adjacency two obtain eq leave multiply since entry let th singular value large rectangular prove concentration mean n I prove square inner almost surely almost recall show transition exceed certain case grow sign two term psd surely eq transition whereas asymptotic low substitute asymptotic value stochastic matrix random prove c c c p limit community detection independent generate node correctly transition empirically almost perfect low derive opposite fig threshold empirically empirically reliability denote network community detection classify three category network network intermediate region american political books amazon estimate political book book frequently determine perform separate book book neutral label investigate sensitivity detection detection mostly reliable community mostly indicate fact may detect extremely phase transition network corrupt perfectly phase estimator reliability community say empirical network model transition theory community spectral corollary remark department electrical university usa community detection subject random transition community connection edge two arbitrarily connect community external specifically almost inter community edge connection critical low transition transition threshold noisy processing node consider regular disjoint community external edge total number topology characterize otherwise community graph partitioning correctly separate community cluster specifie cut dimensional define ni semidefinite small know connectivity algebraic laplacian vector entry
unbiased frequency component experimentally sample random convergence strategy enyi star simulation analysis briefly discrete graph foundation adjacency graph shift undirected underlie th dependency filter comparison normalize shift write norm normalize basis graph signal inverse may orthonormal stability constant review class sample work previous graph coefficient node signal measure smoothness graph shift graph signal lead graph graph graph frequency k note smoothness recovery graph theory requirement restrictive requirement real world approximately control contribution frequency speed decaying frequency component ellipsoid frequency flexibility subset follow sample signal denote index call noiseless recover either mean index clear subset experimentally propose experimentally mainly concentrate frequency concentrated frequency perfectly frequency provide unbiased consider follow node estimate frequency component reconstruct fouri transform bandwidth result appealing graph disadvantage energy first component recover stability frobenius due compare rate two roughly evenly element example graph discrete space unweighte theorem rate bound set bias upper see experimentally design nk k bandwidth bandwidth achieve type bias term experimentally exhibit much random sampling f definition asymptotic real graph bandwidth definition recover globally recovery focus domain optimum close frequency signal experimentally noisy signal biased frequency signal band project onto component sense recover low frequency need reliable ccc nearest neighbor enyi graph star blue mse compare graph enyi star generate frequency component nearest near eigenvector discrete evenly definition base connect near neighbor graph enyi since eigenvector enyi graph simulation connect simulation star expect algorithm star performance test represent red represent algorithm black dot linear lemma enyi corollary class two base sampling frequency design graph star simulation give bandwidth mm signal versus experimentally sample study recovery graph sample smooth experimentally design sampling use
branch metric vs sentence corpus nine much sentence approach propose annotated training rely sense sentence correct incorporate posterior complete deterministic transform text corpus wikipedia estimate principle small number hand linguistic process widely parse generate candidate pick successfully parse parse approach close parse e sentence result train parse manually annotate good train iteration unsupervise dependency search variant move represent hypothesis expectation whole complexity complexity increase rapidly reduce sampling difficulty search base orient concrete parsing unsupervised parse automatically set raw sentence search among objective measure search indeed greedy structure iterate firstly result generate list candidate good candidate number bootstrapping employ bring benefit allow expressive generative propose similar machine learning increase call dependency parse intuitive sentence head difficult long syntactic category would much easy length look usually name adopt iterate framework model set sentence traditional start iterate phase continue use start exception supervise employ extremely sparsity outside employ generative intuitively root generate generate root dependent generative continue current rooted generate respectively plus child token generate le context generate contain context allow traditional impractical thus estimate neural topology make rnns inner content phrase cover representation context allow flow rnn make tool estimate parse generative model first inside depict head approximation plausible dominate inner word vector initially pos tag dependent conditioning context context g generate clearly context represent context full head inner representation indeed le q equation generative ir third party dependency parse party generate list phase corpus moderate report provide merge sentence evaluate english dataset portion corpus corpus phase section worth note phase iterate search define list shape thus fitness far away proportional fitness fitness exist diversity set create constraint control point determine diversity want search distant area maximal inner expressive feed network universal capable perfectly avoid early system contain sentence length whereas phase force phase run parse sentence corpus training testing unsupervise parse sub appear remain tree head rule gold pos tag accuracy take occur label every digit dim word embedding word embedding learn wikipedia train rate recent system length analyse aspect examine lexical start contribute phase phase contain sentence e contain sentence length within phase explore area reason short rich difference wrong system less long role semantics role lexical semantic embedding sentence without word embedding use accuracy drop lexical semantic performance system generative phase vs suggest phase capable dependency conjecture capture claim examine experiment employ extension framework harmonic harmonic training sentence performance iterate start phase remarkable iterate phase find parse start experiment certainly order lexical semantic exploit iterate result end phase head improve except correspond increase attribute improvement context capture correct pos tag ir pos tag g jj ir conjunction cc modal md modal show expressive order generative context share avoid rare system distant node informative conditioning addition free recursive neural net able unseen map close exploit lexical semantic learn turn exploit lexical limited tag distinguish word mean similarity mean close table annotated ir thus relate ir framework option use parse explore experimental iterate disadvantage compare system harmonic innovation capable exist expressive external expect
solve solves write equivalently relax positive constraint solve parametrize constrain simplified framework going arise signal exploit structure part q weighted application appears signal diag diag therefore generality assume every linearly formulate elliptical complex elliptical apply go efficient function surrogate achieve element inequality equality know achieve hand find eq new require single tp j toeplitz toeplitz structure constrain apply idea embed toeplitz toeplitz size parametrize row semidefinite false restrict feasible fourier toeplitz eq take j bar stand wise satisfied toeplitz base noisy augment toeplitz structure construct l impose toeplitz toeplitz stationary satisfie correlation increase embed subsection formulate compactly define real relate toeplitz toeplitz structure positive w w algorithm q apply assumption limit generate conclusion continuity assumption arbitrarily apply adapt solve section handle go structure tractable applying refer differ essentially principle application follow inner reduce propose estimator mean eq normalized expect carlo follow four estimator namely constrain toeplitz parameter indicate small addition embed toeplitz toeplitz sequential estimation error toeplitz second estimator bandwidth choose semidefinite consumption run scale computational increase reflect different toeplitz second constrain simulation toeplitz fast decay slow decay impose toeplitz structure vary estimator fig either bandwidth covariance regularize impose toeplitz structure bandwidth examine robustness direction arrival additive ideal element mean direction uncorrelated e signal direction distribute sensor music locate interval stepsize arrival correctly angle denote denote fig constrain constrain accurately music constrain plot see fast unlike arrival music estimation estimator generate spike varied set obtain measure estimator structure error plot smoothly gauss latter double former loop fig kronecker toeplitz impose help reduce kronecker kronecker structural constraint kronecker toeplitz problem information log constraint minimization community although thm consider elliptical mean covariance structure account beneficial propose incorporate cost structural convex finding base tailor special wide range process relate sum toeplitz toeplitz structure addition kronecker derive mm show estimator low cost constraint minimization arise wireless communication financial engineering notice possesse special exploiting imply beneficial improve various type toeplitz structure application model sparsity inverse matrix covariance closely problem structure previously application follow attempt covariance realize normally poorly cause set heavy lead way address aforementioned find matrix perform seek worst precise belong uncertainty asymptotic estimator uncertainty contamination structure constrain distribution give independent elliptical estimator minimax distribution sense obtain investigate focus group symmetry cost symmetry global study generalize type numerical semidefinite prove suffer drawback either grow formulate generalize consider large convexity focus convex mm case application exploit structure load discuss end convex turn kronecker theoretically prove converge structure unique initialization convex structure consider efficiency tractable present vi consider number elliptical proportional characterize estimator random angular gaussian fitting word normal independent symmetric notice possess structure take account motivated idea focus include improve accuracy formulate characterize cone proceed minimizer throughout assumption exclude algorithm hereafter assumption continuous boundary sufficient assumption constraint tackle possess convexity instead try find appear reason point capable mm derive briefly completeness set successively simple point th eq surrogate function satisfy stand directional point problem assume level x stand partition mn update x h ordinary block update derive form surrogate detailed characterization fine move minimax corrupted belong kolmogorov class estimator define elliptical completely differ develop constraint semidefinite matrix convex minimizer
persistent generate improve adopt consuming since normally linear constant cd unlike cd adopted consider document novel heuristic develop uniform cd treat partition parameter computable ml setting model proxy document document assume sample gradient sgd per document distribution result indicate might satisfactory alternative part dd remain computation simply word advance derive readily remain assume length towards incomplete load fortunately also store fast computation cd implement length ratio accommodate probability ability necessarily integer choose value weight weight define thus derive cp combining testing instance label well sentiment sentiment consist movie divide positive select set advance count slightly improve initialize learn partition separately nearly surprisingly constant without enhanced force value implement cd gpu fair cd share learn dictionary size range computation cd variant htbp minibatch time slow cd reasonable evaluate performance learn distinguishing like feature generative fair reason evaluated model indirect precision curve htbp retrieve recall p mean map evaluation check testing regularize logistic train label rate sentiment several baseline dataset compare cd naturally without posterior unit use train cd show size greatly cd retrieval slightly find retrieval count document greatly ccccc cd dataset htb model full cd sentiment dataset sentiment clear learn cd outperform gram input also arguably reach extend consider syntactic dependency relationship achieve well outperform full achieve well however extremely explain drop dependent large nonetheless systematic strategy explore work undirecte efficiently allow size estimator adapt length learn softmax powerful extract representation novel speed length benchmark high powerful analyze document categorization mainly like vast development great typical boltzmann machine rbms learn rbms cd really rbms softmax great inside especially vocabulary require hundred thousand thus limit typical undirected bag word
come attack group student student act two introduce student exercise score score observe might bias supervise scenario score true depend whether actual use compare score induce ranking agreement inversion ranking equality set estimate median assumption exercise inherent certain intuition tendency give strict whereas account normally prior fit observed purpose give result elaborate procedure confirm find true score artificial tb unsupervise course learn learn reliability vary work ordinal bt suggest estimating might purpose assume rank two score induce ranking use truth follow naive baseline exercise ta bias compute ta solution set accord task another ta put ta automatically towards student ta tb actually raw able performance rescale exercise lie rescale lead get interesting surprisingly self tend look shift version ta histogram contrary shape distribute first obvious skewed exercise many student mark score multi modal big solve part tendency generally ta evident receive moderate look reveal mistake mark describe meet runtime template solution unlikely full lower tb simulate abstract reveal error due suggest probabilistic serious problematic get whether really justify solution mistake get score realistic abundance analyze simply optimize ta histogram shift example unsupervised model ta score might bias shift measure control strength regularization little accuracy reliability student reliability artificial datum sensitive choice actually em equal one believe improvement fitting explain ta different ground none outperform study standard ta result considerably outperform mean step look actual ta consistency record performance compare student seen leave amongst bias amongst student next histogram figure little amongst histogram looks consistently believe use ta major probabilistic mid bias student report quite half student least roughly gain value student mean would improvement correct confirm dominate strict serious lack understanding material come structure less used report whether student find scenario perform mean job bias room improvement improvement compare totally wrong student none job student try effort much however look reveal portion student figure picture change compare l error calculate value student place supervision seem effect evaluate structure positive optimistic ta size reasonably inform picture reason use competitive elaborate model helpful understand acceptable student complicate model final machine test mean enough much well job variance among rather lack solution student mention successful generate publicly material contain couple report constraint ask ordinal conduct currently release platform group course acknowledgment partly education pl universit universit author responsible content publication lemma exercise theorem evaluate literature aggregate come solution ordinal ranking none improve student work co scoring become increasingly student crowd hope student perfect fair accurate aggregate many challenge good aggregation take suggestion literature bottom science algorithm structure ad course student exercise apply see publicly aggregate contrary researcher none satisfactory baseline reason course department computer student active participant traditional two student exercise
consistency classifier projection margin sample set maximize margin margin perfectly project maximum small classifier particular many whether classifier able arbitrary model characteristic call consistency class minimum going search maximize cross estimate project analogy approach take minimization start notation accuracy eq unbalanced might make important despite lead balanced weight ignore distribution small measure accuracy linear tf bad whole obviously make classify minimum greatest average follow section density approximate grow infinity consequence result true limit case use classifier give begin simple perfect distinguish regularize consistent linearly separable perfectly linear opt integrate function zero integral result non regularize attain radial regularize consistent radial gaussians variance projection normal projection give maximize al minimize maximize regularize unfortunately neither consistent linear start integral integral function integrable schwarz prove main paper minimal linearly separable projection narrow q connect minimize potential also optimize log density hold integrable another dot denote solution confirm claim simple numerical life evaluation analyze possible compare hinge case non classifier behave radial gaussian distribution place dataset embed positive dataset notice hinge optima fourth optimum core local however term locate near sufficient problem loss convex distant underlie comparable away fourth dataset gauss gauss mixed mnist loss non regularize classifier truly simple class hinge loss truly lead nearly grow many machine minimization additive loss perceptron machine meaning sense grow infinity answer classic error directly translate function define risk even overcome classifier
vary level volume spherical middle ccc separation middle compare parametric select data hyperparameter hyperparameter great eigenvalue table approximate marginal likelihood volume poorly equal general good mixture provide respectively volume mixture separation separation separate c c marginal likelihood separate mixture c estimate poorly mixture ht c c log obtained propose well separate mixture ht log mixture situation situation table see select provide separate parsimonious misclassification error thing observe select cluster term misclassification model one situation volume obtain propose situation show evidence majority select competitive table bad mixture volume evidence evidence go almost substantial evidence especially cluster volume model volume highlight ht vs vs vs competitive situation respectively vs vs bayes select denote situation table estimate across mixture equal volume good spherical also figure structure mixture ccc see spherical diagonal different cluster equal actual cluster hyperparameter mixture estimation two parsimonious sample component gaussian follow covariance volume figure partition two datum respect hyperparameter hyperparameter degree equal variate control control consider four situation four situation model diagonal log value see situation correct cluster c propose model competitive accord bayes propose confirm stability variation partition cccc partition confirm available old diabetes summarize model conduct lack four datum ht l diabetes description comprise old national usa comprise minute long alternative vary report propose c c log marginal old parsimonious except number vary parsimonious model spherical stable marginal orientation notice term strong compare competitive likely high ccc old partition posterior component comprise observation describe length colour collect classified vary report value see good provide actual see provide estimate cluster hand cluster addition cluster four provide bayes set evidence value performance confirm rand index misclassification parsimonious one high rand lowest rand index show partition precise posterior close first principal axis actual leave optimal middle diabetes consist curve area area steady group chemical diabetes diabetes alternative report likelihood obtain correctly parsimonious k c c diabetes datum diabete compare term competitive k rand index parsimonious model k indicate error figure optimal quite rate ccc diabetes set area partition empirical posterior component set cover three specie feature propose propose high value partition solution approach four marginal datum width middle component propose rand evidence four significant show log report accord table diabetes latter three make old diabetes vs vs parsimonious base infinite eigenvalue matrix chinese restaurant derive flexible avoid encounter maximum automatically simulate highlight represent finite partition cluster use bayesian parsimonious potential future may concern parsimonious desirable extend simultaneously formulation figure plot sensitivity figure parsimonious show cluster likelihood parsimonious represent nonparametric parsimonious gaussian parsimonious parsimonious technique posteriori map framework bayes factor parsimonious mixture obtain parsimonious alternative parsimonious model mixture dirichlet bayesian selection parametric model weighted multivariate gmm focus mixture datum parsimonious gmm exploit gmm wide flexible clustering demonstrate cluster analysis likelihood framework maximization em gaussian mle first normal mixture fail bayesian estimation intensive dealing encounter allow replace estimator achieve introduce assume uniform mle perform parsimonious mixture mixture carlo mixture decomposition parsimonious gibbs usually mixture fold pre establish approach penalize bayesian likelihood via compute posterior factor parsimonious mixture indeed natural criterion etc represent extension analyse component death refer fully bayesian mixture jointly bayesian one parametric unknown model go take mixture capability advance namely method dirichlet chinese restaurant process crp represent principled mixture cluster offer principle jointly infer parameter rather form grow represent formulation mixture group flexibility approach parsimonious covariance structure dirichlet parsimonious parsimonious dirichlet overcome issue automatically infer structure simple one complex maximum posteriori model factor simultaneously parsimonious flexible modeling cluster infer organize discuss based evaluate devoted discussion conclude remark label point possibly unknown base base mix proportion mean vector matrix gmm generative process mixture mixture multinomial give proportion component correspond parameter mle maximize em extension mle mixture fail avoid describe estimation gmm conjugate proportion inverse wishart normal view datum gmm mix conjugate mean component multivariate inverse wishart summarize follow step wishart gmm wishart mixture maximize posterior em namely prior prior gmm gibbs gmm cluster extend parsimonious exploiting eigenvalue matrix provide range flexible parsimonious decompose determinant term previously unseen assigning possibly unseen cluster observe chinese restaurant crp customer table customer social th proportional customer previously new proportional positive real number crp crp customer correspond crp mixture complete cluster gmm one inverse customer choose crp crp density crp covariance common wishart dp parsimonious crp parsimonious matrix flexible volume crp table decomposition mixture cluster parametric clustering investigate eight parsimonious covering family family parsimonious spherical full summarize gaussian parameter ht l l type diagonal diag diagonal diag structure denote inverse mix proportion multivariate wishart parsimonious dirichlet parsimonious parsimonious mixture model mixture perform miss dirichlet label posterior mt complete prior mixture sampler generate posterior multivariate normal wishart gamma depend markov stationary therefore n approximately distribute couple give hyperparameter z z detail material also component hyperparameter dirichlet sample arbitrary consist sampling prior introduce conditionally cluster beta n hyperparameter pseudo ht nk nz I cluster get distribution output retain solution correspond frequently sampling strategy number parsimonious bayes general modeling
year extensively computer movement fusion modality dynamic movement device computer interface hardware like computer active mobile sensor much device form environment device active continuously study four representative modality usage behavior device modality due relatively consumption remainder four modality error measure false far false reject collect author subject mobile device day kind literature duration absence restriction usage factor mobile device participant good representation close world organization device organization decision modality serial strength level additional classifier add without classifier system multimodal characterize fuse decision fuse aspect particular portfolio behavioral mobile device extent temporal multimodal mobile device four behavioral analyze modality fuse system user datum get motivate recent multimodal verification human computer interaction approach min combination classifier classifier fuse initialization utilize multimodal system knowledge subject characterize overall significantly achieve fusion combination available already design allow contribute drastically decision active mobile device year rich ultimately one contribute fusion verification achieve subject incorporate modality behavioral achieve user knowledge subject duration day portfolio modality linguistic verification thorough domain along traditionally machine prove rate impractical mobile device come modality context block text location extensively aware web study purpose understanding web source identification far computer mit reality portfolio usage location usage position low gps trace utilize work behavioral subject collection carry author requirement study user subject version table device device track device period track device modality visit location gps characteristic duration participant area place usage device os long duration study could modality tracking power include front tracking face recognition gps frequency device text web location refer character soft refer new device location show modality aggregate instance modality single soft website take rather gps table order three remove period device consider minute minute minute date time change divide dataset reflect event compression period cross utilize window active duration hour active order htbp home facebook phone message home sense home gps united world characteristic location city vary day day human location modality analysis soft visit physical gps modality extract raw modality produce train describe detail classifier take event tx tt current classifier score fig mobile device varied activity majority facebook event therefore final record associate record actual principle well representation linguistic style capture analyze classifier construct count user domain number time training example com domain www com refer domain entity across user entity visit user quantity normalize frequency value visit user valid user maximum entity machine radial rbf kernel function score form regression svm score additional validation learn library htbp box fire event associate modality multiple decision modality fuse sensor describe parallel architecture describe comprise local detector fusion center detector make favor local detector order decide favor favor detector assume observation detector use set couple equation word impractical detector condition fusion center performance decision suboptimal since detector scalable design parallel fusion scheme observe make decision decision form fusion center combine optimum priori represent optimum eq practice determine false alarm interaction mobile device divide fold three fold fold phase purpose individually fold test individual fusion three phase characterization relate fold fold characterization fold fold characterization fold fold characterization fold fold fold characterization fold characterization common fusion measure datum fold characterization testing far phase classify window system decrease increase show window bar compute characterization result use fusion center indicate activity produce metric thought decision window size older cover consideration window add text even average hour show gps correctly verify four modality hour mark fire character small list h modality different text gps web show within window distribution period window decrease long asynchronous modality event fusion utilize operate characteristic roc far decision fusion drop window window utilize modality contribution global decision paper verification evaluating error rate portfolio system relative contribution window minute contribute contribute window explanation significantly short
odd expert remain measure might suboptimal regret note yet play role algorithm divergence begin vector call sequence adaptively could special recover eq lemma convention banach function respect step optimistic mirror yield z tight learner advance align learner regret tuning lipschitz lipschitz bregman case divergence uniform stay case constant develop optimistic mirror incorporate step trick monotonically trick monotone sense necessary variation throughout game gradually horizon take adopt strategy worst describe algorithm banach optimistic descent statement automatically cast regularity far round satisfied accumulate variation regularity note importantly non monotone initialize check tx tm td c optimistic mirror descent epoch number instance epoch technical shall become lemma theorem assume enjoy follow regret summarize v follow gradient regime allow divide batch play smooth batch gradient recognize mirror prox period far regret mapping dynamic correspond history mapping index prescribed strategy payoff player lk player arbitrary switch convergence constant static sequence propose vary dynamic regret fully adaptive environment derive dynamic regret capture sequence sequence interestingly consider rate minimax acknowledgement acknowledge decentralized online nsf grant dms proof primal dual pair cd entail combine return eq simple bind summing q complete define choice sequence belong last entail divide batch use single batch horizon let r ib tu tt b otherwise precede upper complete sake clarity presentation stick epoch recall bar refer quantity tune pair identity fails suffer simple function sum fact batch use precede prescribed correspond optimistic strongly correspondingly get I tt notice choice guarantee specify splitting like payoff get denote appear last player denote regularity combine appear account identity bind would like player regardless adopt player drop negative well entail statement pay average recent develop adaptive take observation retain guarantee direction method perform benchmark direction benchmark regret guarantee scale notably measure apply zero player game action play nature action nature reveal learner aim regret numerous static minimax algorithm direction non benchmark hard guarantee advantage sequence non adversarial quality move distinct investigation develop regret show respect benchmark dynamic generally dynamic regret bad obtain dynamic regret furthermore propose dynamical potential forward idea generic sequence learner achieve variation type adversarial capture intuitive sequence version reveal online get regret regret valid begin full setting receive gradient trivially obtain simply play round online study knowledge dynamic full online regret variant optimistic mirror noiseless priori automatically adapt length term second technical derive apply monotone investigate provide player play zero game player play drift guarantee sequence vary slowly generalization action
support weather phenomenon click decision history web theoretical formally world record imply rather degenerate single could entirely score probabilistic typically state world pose problem generalization value observation possible correspond model outcome like low expectation eq since hold square however immediately truth evaluate however algebra true assign outcome kl divergence shannon entropy divergence g difference sample want mean pair sign test test mean option trick theory score thorough develop algebra logarithmic scoring get entropy term prediction theoretically comparison hold outcome pair determine score indicate pairwise mean score rule less unobserved distribution model rule logarithmic distance kl former interest quick outcome report wrong often free add parameter report undesirable popularity connection proper use scoring cosine similarity correspond rule general bregman compare probabilistic proper rule score reward report effect nonetheless theoretically preferable un comparison score accuracy probabilistic
overfitte hour provide dataset roughly year period price load load load second load unit measurement variable additional source competition total load load skew decide natural log variable much target calculate variable date extract day week integer day year integer day month week integer hour month month integer early hour early hour difference z z strong hour show select shift autocorrelation window hour autocorrelation price hour shift day see autocorrelation value much hour mean shift surprisingly hour boost apart find relatively importance hour suggest within lr hour month fairly result use validation research competition mark par decide window day train substantial drawback leave day month period testing leave day total assess forecast compare provide mae root mean rmse boost mae price day table statistic daily day rmse median low average day filter represent error confirm difference two mae mae rmse count mean competition forecast probabilistic worth investigate energy effort focus develop forecast help fairly approach capable achieve competition price surprisingly conventional forecasting observe wider cover future competition filter greatly improve forecast hard task series variable gradient competition year forecast experiment reveal perform auto correlation box real forecasting price task participant inside outside market ahead forecast unique inform short term forecast beneficial business ahead market forecast help operator ahead methodology current last team position approach forecasting competition establish competition put forecasting forecasting track forecasting hypothesis series candidate consider linear follow denote combination auto around zero parameter element weight usually acceptable error suitable minimize error boost responsible
bayes always surely imply martingale one tp tp h let integer decrease h tp infinity tp consider generate lemma desire trivial exact rest devote one stop bt monotone convergence use recurrence inequality markov process thompson decompose tp tp tp recurrence previous choice tp p tb tp tp kp h finally small h get notation let value later decompose use use lemma lebesgue monotone convergence recurrence regret thompson tp tp tp tp fact p establish recurrence inequality tp notation see decompose devoted bound lebesgue monotone establish desire recurrence inequality lemma process thompson follow tp tp ta q two b rearrange newly get p tp desire recurrence observe q use p axiom microsoft successful thompson bandit property encode exploitation prior bad dependence fully dependence yet case result true generating low discovery thompson appear well know tradeoff repeatedly action agent receive action reward potentially randomness make reward generate countable underlie reward random draw know underlying yield highest measure incur always select optimal action precisely frequentist selection reward take randomness impose bayes regret discretize abstract bandit arm coefficient reward reward useful result still unbounded probably paper take thompson action accord equivalently thompson I thompson thompson gain lot largely furthermore often analyse strategy classic arm comparable obtain bandit prove provide insight assume informative prior thompson regret bound unfortunately thompson bind another prior thompson always dependent prior thompson thompson unclear good perspective bayes regret extreme trivially knowledge work frequentist thompson informative thompson sensitivity prior important question useful implication thompson thompson important characterize bad yet meaningful highly nontrivial main summarize statement section case mild loss generality let thompson bound furthermore factor exist instance respectively low bound low bound countable model exist instance thompson show true p bound remove logarithmic far small open thompson exponentially expert specify assign expert sake simplicity thus exp thompson partly bad advantage efficiently impose prior remove core difficulty analyze state upper rely proposition proof proposition differ limitation sketch proof proposition proof proposition thompson precede problem specific smoothness thompson fairly problem notation obviously simplify notation tp decrease notation function tp complete therefore word time lebesgue lemma regret thompson tp tp decrease recurrence assumption q hold sketch reward define upper lemma stop dominate lemma next thompson tp tp use decrease get recurrence rearrange newly inequality lemma frequentist thompson consider thompson problem recall moreover lebesgue dominate monotone aspect thompson bandit focus representative case fully bad prior good matching upper extend quantify inherent sensitivity poor bandit version true strong frequentist still sensitivity insight thompson bad thompson range underlie negative functional process tp carry computation p kl p p step follow p hand jensen definition absolutely measure eq complete count
sample conditional cluster shown likelihood introduce component hyperparameter equip form dataset finite exhaustive enumeration clustering practice implement greedy search space successive application operator chain considerable model heuristic advance follow simplified speed cluster specify advance first exhaustive enumeration special correspond heuristic success simplify largely database find condition equation recent year generation theoretic possess desirable strong mathematical foundation similarity conduct conclude preferred purpose denote clustering co occur marginal derivation joint occur cluster cc ns describe item formulate clustering reduces also measure sense model cell truth retrieve value typically give general truth subset one include exclude disease respectively measure take successively retrieve experiment number cluster initially select top low gene gene experiment emphasize stage find minor simplified suggest search heuristic range end al et ultimately specific initially choose six gene complete linkage cl euclidean pearson correlation correlation cosine cosine fix gene result performance show clustering result well proceed evaluate approach previously model approach clustering dataset maximum posteriori find cluster restrict restriction trivially retrieval evaluating distance evaluate marginal query recently marginal average keep comparable discussion third differentially conduct specific suggest differential expression profile obtain condition potential achieve retrieval performance hand assume preprocesse suggest default gene use correlation instead two result expression retrieval model expression formulate comparison retrieval show retrieval scheme clearly outperform result far indicate retrieval quite robust allow vary assume fix mean conclude proxy conclusion generalize beyond scope b approach surprisingly gene essence predict learn therefore instead query may necessarily aspect experiment task relevance aspect current setup ground truth cell type disease match require modelling evaluate ground truth require match class ground truth require relevance increase retrieval capturing agreement figure comparable figure ground retrieve query match b b note phenotype g age disease retrieval idea next model retrieval retrieve experiment multiply element clustering experiment match type combination result type retrieve assume retrieval ranking paper general probabilistic model al likelihood query learn query compare argue reduce nuisance characteristic relevant retain comparative simulation inferior encounter real scenario see outperform counterpart contrary approach family seem somewhat restrictive potential individual store arise result model fairly standardized repository assumption belong gene experiment cluster turn good purpose preprocesse prior experiment yield retrieval combine keyword would like thank useful centre coin public relevant improve search annotation retrieval retrieve express profile retrieve query criterion well retrieval general model separately expression model induce cluster gene pattern empirically fast mean suggest clustering scalable construct purpose use distance package molecular continue ever amount biological store retrieve experimental relevance make experiment current rely search
survey answer question sake privacy survey recommender miss entry recommendation netflix service facebook prediction problem miss entry demand clean lead also noisy datum component focus sparse miss completion element account know imagine miss sparse stage propose unified yield performance low sparsity datum optimization simultaneously learn underlying take alternate learn ambient performance compete conclude learn regressor miss prediction particularly sample see formalize notion index n entry index label regressor unseen predict label parameter statistical motivated even though ambient large close regressor predict miss exploit incomplete regressor regression inherent regressor relevant network sensor would entry sensor exploit increase miss definition mp mi pls assumption handle missing introduce dictionary dimensionality dimensionality projection central independence dependency fully practical investigate investigate maximum work different regressor ambient explicitly miss exploit sparse relate task pose problem form concatenation author approach exploit statistical test finally procedure perform pca dataset follow regression exploit limitation stream ensure algorithm thus track variation track quickly algorithm inherently unsupervised directly approach exploit coefficient aim incomplete product entry identification alternate minimization alternate case problem perform learn regressor sparse two problem utilize structure linear model algorithm label information inherently inefficient exploit reflect potential basis subspace know rotation well solve hard necessary purpose row correspond zero matter inefficient particularly deal subspace ct slice procedure propose joint simultaneously learn relevant solve optimization formulation measure regressor low third penalty encourage type formulation prediction unlabeled datum first project onto span column projection output depend shall detail jointly minimization alternatively call pass order update six detail initialize matrix miss singular vector value matrix completion literature project subspace initialize initialize update responsible replace quadratic system equation step mp routine estimating problem current mp solve optimization start implement recursive mp routine available rectangular perform solve orthogonal subject construction subspace perform gradient require form regressor achieves necessarily regressor modified term skip update linear take parallel row take time orthogonal numerical miss rough span attempt span need amount computational past etc similar guarantee sgd base drive use alternate minimization describe convergence far r x perform regression surely replacement n fp lf fp r tv j p j j connect rademacher lemma calculation technical convenience allow state possible trade classical rich class small error generalization error term think class imply richer bind potentially wish surface empirical predict many value still error easy hence small column onto dimension know discussion complexity fraction training sufficiently yet regressor regressor ambient generate orthonormal dataset entry standard separate validation generating entry resp simulate retain ease choose analysis behave similarly report five different generate choose hold search parameter mse sensitive hence coarse perform stochastic lasso allow decay try good rate noiseless figure dataset red break figure mse impact noiseless study impact gradually noiseless substantially small examine zero dataset increase increase plot seem noise increase like increase comparison discussion provide slice task prediction ct scan ct cancer paper clear noisy advantage gain utilize bad ct slice ct
ie sdp dimensional instead linear determine rearrange identity eq rearrange yield formula since feasible sdp goal sdp event theorem characterize acceptable fail motivate constraint dual start characterization determine may write formulate task clustering determine determine symmetric find analyze eigenvalue eigenvector lie span write finally eigenvector column correspond eigenvector want pick thereby impose strong remain impose satisfy since choice satisfie immediately satisfie also necessarily implicitly require division desire check suffice eq imply summarize cluster let sdp relaxation bind take cluster whose column center first triangle separately combine first whose entry eq triangle inequality equality verify point recall stochastic prove ball surely lift hoeffding notice sum hoeffding stochastic expand result sum explain definition schwarz inequalities eq finish argument apply subtract ball add expand get q triangle give occur cauchy schwarz last cauchy probability union triangle iid whose outcome determine hoeffding equality linearity q probability combine leverage assumption probability conclude combine combine result column eq pass frobenius term may second union q suffice rearrange q acknowledgment nsf dms reflect united air conjecture problem sdp mean problem model ball radius draw common probability prove recover two explicit ball cluster task machine k dissimilarity usually choose mind common clustering criterion set centroid c I solve calculate centroid may objective furthermore output far slow preferable convex relaxation attack hard combinatorial know round paradigm region set seek approximation guarantee framework relate relaxation particular rounding happen find feasible original phenomenon know optimality thank focus problem geometric question appear provide work separation main sdp recover cluster strategy sdp ball sdp relaxation vertical exact derive prof show high hard many solve entry observe give sdp relaxation ax give dual semidefinite need interpret remain exploit express
cover class interest amazon characterize south locate forest constitute background post consist collection type user construct portion begin change configuration partition series sample pixel year profile datum miss four batch minimum propose method threshold correspond metric recall predictive value precision accuracy classify truth configuration q denominator accuracy mention threshold furthermore achieve across substantial amount overall stem tendency belief proceed change region apply pixel locate amazon several natural include substantial area convert production area build ice class bar change aim expect change pixel quality call monitor amazon produce national pixel particular segmentation note regard year panel panel ground truth higher estimate capture lower infer estimate year procedure pixel panel probability change j pattern bottom panel agreement reference middle highlight panel across concentrated cluster cc change characterize top illustrate result pixel region panel probability change w dark gray bar truth gray background represent plot outline infer year dash represent see profile datum reasonably project profile close bottom cover probability change spatially study region see change gray highlight change v right region pixel colored class infer case panel contain pattern panel propose inferential routine art mainly band combine statistic finally capture perform world study interestingly seem follow spatial pattern going operate spatial pattern high whole region pixel localize transition background cluster large pixel major ground percentage year pixel profile profile forest background forest reasonable profile summarize year year sub pixel leave figure respectively top plot correspond classified localize cover isolate correspond degradation area region detect cover crucial use resource management modeling effort sense rich inference change broad unfortunately dataset hierarchical account miss site extensive characterize dimensional forest analyze pixel detect distributional dataset point posterior use informally change change series flexible change suitably define methodology propose recovery characterize essential sense hyper carefully region general em successfully cover change accommodate situation characterize surface estimate post detect kind change pixel effectiveness method art ground change infer localization overall consider change serie contain formally change define devise computationally efficient g g section filter pixel change derive update identify depend change update need jointly pixel value miss entry evaluate vx jk vx situation involve datum thorough history miss exploitation series remain image literature period experience change estimation dimensionality change amazon maintain insight change occur forest enhance monitoring must maintain area affect human depend extent modern resource management observation surface enable especially human activity decade advanced series name moderate imaging balance moderate high capability observation measurement error contamination geometry challenge develop bi temporal forest series grow exploitation e however nature optical cloud contamination pre great address structure change proceed neighbor interpolation polynomial interpolation imputation thorough handle statistical analysis handle imputation change statistical assess tailor series large characterize reasonably homogeneous background cover classify detect understand nature well assess forest convert cover specify change aim consider apply issue product due view geometry spectral band visible pixel seven band exclude proportion treat year unit effect subset keep enough year profile avoid band miss one band discard year since happen make hard change evident year hard attribute minor plot year plot highlight possibility profile seem background change class carefully establish international percent cover green never green percent almost green without green dominate height consist period forest dominate tree percent consist community leaf mix dominate relate spatial resolution step temporal band dimension partitioning variation spectral distinguish class code cover equivalently q datum employ kronecker temporal note cover covariance cover class dimensionality profile temporal variability nature dimensionality large transformation approximate k x opt pre jointly end approach simplify burden simplify notation remainder parametric require miss pursuit point pixel exist devise account miss model procedure allow segment post change segment background lack change represent configuration pixel segment segment follow multivariate mean flexibility pixel set prior pixel segment affect smoothed spirit interest parameter em pixel recovery pixel weakly informative probability change occur specify change equally one recovery two give
accurately like two costly intensive formation explicit retain generative capability original world effectively capture several implementation parameter runtime number mixed number specify base preference validation may number give hold choice another reduction form signature guide signature merge consideration use cluster suitable control take iterative begin size increase significant portion cluster keep proportion portion assign remain runtime assign center affect cluster determine total cluster result fast convex shape become normalize axis suffer slow density estimator retain choose subsample compute density result computationally pa usa unsupervised spatio without conceptually spatio process property regular discrete seek joint spatio temporal process causal system propagate concept formally eq cone set future event affect light joint product likelihood pdf given seek equivalence light similarity discover give introduce method predictive state reconstruction follow model extension cone forecast mixture predictive require light predictive considerably large introduce reconstruction spatio temporal scalable consist reduction instance require maximization density density differ cluster proof describe spatio lastly discuss result principle mix reconstruct soft light state light cone unlike retain benefit soft likelihood forecast appendix parameter arise algorithm htbp light successive reduce object light final cluster output predictive state nonparametric step density assume consequence neighborhood point assign effectively density coverage avoid formation cluster g cluster use density density affect remain point signature family state signature final predictive assign predictive form nonparametric decompose spatio light tuple density compute condition merge reduce final htbp simplify step light cluster consequence space difference state minimal new map step spatio temporal mean become user scalar measurement since resemble video prediction frame pixel light exclude cone extract result simple compare take value use prediction current consistency low error light directly future cone regularize implement learn package change remainder near regressor light cone near light cone output learn default setting experiment light density estimation well original method gaussian density parametric unable delta set error mse ground truth distributional per avg negative truth distributional test well compact apply likelihood show three respectively frame model remain percentage maximum range actual prediction predict frame pixel qualitatively predict capture much frame give smoothed extreme htbp like knn linear material regression low confidence pearson lastly one proof highest low overall one relatively spatio three accurately forecast
publicly achieve speedup merely center view imagenet pre excellent accuracy challenge object segmentation exploit accelerate fast r speedup degradation benchmark manuscript conference manuscript extend initial version acceleration deep among deep investigate important imagenet evidence share inferior discovery architecture acceleration cnn I decomposition reduce former attention directly address essential decomposition equally fine good several investigate nonlinearity neuron influential accuracy present whole present decomposition separate filter dimension scheme reconstruction filter minimize conjugate solve filter reconstruction demonstrate character imagenet evaluate unclear adopt decomposition imagenet single acceleration report fail sgd tuning suggest nontrivial optimize layer imagenet preliminary acceleration open research layer stream improve testing particularly hand also thin besides reduce run stream decomposition close svd nonlinear simplicity asymmetric reconstruction deep response pixel lie low decomposition find rank minimize response convolutional filter channel filter volume denote volume entry number filter rank response expand filter complexity eqn eqn complexity illustrate layer filter correspond convolutional spatial randomly note arbitrary dd approximate solve svd actually good response convolutional imagenet layer covariance plot large eigenvalue substantial portion eigenvector conv contribute energy original filter rank work adopt low input local volume investigate unit nonlinear focus relu drive eqn reconstruction nonlinear r nonlinear due nonlinearity feasible relax auxiliary variable penalty alternate solver involve similar eqn form svd reduce rank regression belong broad category problem let dd decomposition z problem follow consider ij applicable dimensional nonlinear least problem warm solver run gradually infinity find iterative solver increase run find matlab much fast approximated accumulate deep propose asymmetric layer deep layer map previous current layer term layer incorporate c channel complexity pool conv pool conv conv conv conv layer follow relu conv spatial pyramid totally bin fed layer fc follow another fc softmax column response sparsity proper used uniform layer solution approximation classification accuracy energy empirically reduce energy degradation classification linear pca reduce roughly product pca th layer approximate approximated whole optimize layer speedup ratio maximize accumulate greedy initialize lc small large iterate achieve channel operate channel firstly easily control rank enable svd secondly optimize exactly decompose close solution subset operate use speedup might combine spatial thank asymmetric reconstruction effectively accumulate architecture decompose conv rank filter output channel speedup ratio original speedup contribute decompose determined reconstruction adopt optimize layer reconstruction eqn layer without spatial reconstruction important accumulate asymmetric speedup ratio complexity may tune imagenet datum asymmetric version speedup layer conv conv conv conv filter rate compare asymmetric approximate case approximate conv approximate conv comparison multi layer involve deep asymmetric previous approximated version try symmetric asymmetric asymmetric layer approximate speedup effectively layer approximate simultaneously rate drastically result acceleration solver layer rank selection conv selection consistently outperform counterpart rank advantage observe often choose rank rank selection conv assign conv explain conv less concentrated high rank prominent diversity art acceleration rarely address ratio whole rate top view cascade focus evaluate single network imagenet filter reconstruction speedup increase report conv speedup evaluate another pt speedup ft discuss backpropagation reconstruction find nontrivial work imagenet observe carefully independently start converge initialization range imagenet optima report single imagenet filter investigate deep involve whole speedup speedup sequentially conv speedup conv decompose speedup speedup conv conv accelerate large ratio ratio asymmetric speedup speedup increase speedup ratio get version decomposition strategy small speedup solver extensively easy speedup completeness asymmetric drop accuracy speedup also imagenet dataset underlie architecture acceleration train much deep model layer adopt initialization method otherwise common comparison train worse accelerate effectively model redundancy increase c filter complexity conv conv conv conv pool conv conv pool conv conv pool conv conv conv conv fc complexity show relative number total convolutional portion c cc speedup c c c speedup ratio convolutional accelerate column filter rate compare table comparison performance also fast accelerate gpu ignore view one report top accelerate top fine accelerate per intel ghz cpu version actual speedup ratio speedup ratio overhead come fc speedup accelerate easy parallelism gpu actual recognition image believe practical significance accelerate view speedup ft top pt speedup view ft value single accelerate speedup speedup ratio firstly important without rank selection rank selection increase repeatedly evenly feature size besides conv increase rank table conv conv filter trade compactly maintain evaluate imagenet evaluate challenge conv layer absolute somewhat fine fine speedup previous suffer greatly accumulate fine tuning increase view error speedup degradation deep model redundant increase cpu speedup report tuning suggest fine whole decomposition current object method exploit model evaluate accelerate default publicly cnn evaluate average imagenet task approximate asymmetric unlike layer dominate cnn detection conv speedup conv result model detection speedup degradation believe speed advance fast feature extraction considerable conv speedup detector fast acceleration speedup ratio accumulate nonlinear asymmetric demonstrate complex imagenet c microsoft com aim accelerate convolutional neural cnns cnns substantially vision unlike approximate
formally strictly speak write limit write define regularity condition mean combine turning intuition datum set generate outcome fig mean increase partition two cluster high context appear yield region interior within square diameter represent diameter seek find clustering say reverse split cluster splitting give tend split consequence decrease assume item follow split write enough q repeat proposition establish part next linkage sl sl merge sl question remain two first give set point distance take approximately order program percentile linkage dissimilarity also give study well give perform unable suggest result give worse well suppose closure disjoint disjoint large strictly suffice regular two less ensure one perfectly sl point together point subset component metric close proved theorem dendrogram sl separate perfectly enough perfectly sl use limit convexity regard perfect representation key force component thin closure satisfy always find work well seem satisfy regard sufficient examine technique find outperform interpretable merely enough th much sensitive value cluster influential point valid point affect representative influence cluster merge versus presence preferred indeed extreme linkage random start randomness cluster point regard ensemble represent pool clustering analog play linkage group clustering result final algorithm dendrogram linkage desire case dendrogram must back grow dendrogram little may split remove outlier small result therefore course merge however merge use final representative place root design advantage linkage merge become remain justify see dendrogram define tendency usually put near leave dendrogram reasonable place well separate homogeneous dissimilarity small refinement separation among remove k table place place early despite poor average suggest spectral cluster fig good seem broadly size technique assessment ct compare seven half depict show clustering panel little theoretic well separate randomness include panel merely panel fourth panel cluster respectively overall inference h ct consider nonconvex run generally use preferred htp six indicate ct show six clustering density whether ct give poor merge half top see low three nonzero ambiguity portion half bottom ambiguity versus recognize ambiguity indicate panel ct fu dna website set method well none fail completely greatly contrast outlier effect ct put cluster indicate low htp six ct generate evaluate successful strategy projection onto however systematically difficulty dimension present table separate less likely dimension mean perform well observation give replication obviously ct two provide ct work data expression seven class expect seven missing profile present ct red seven find find white attribute case could said omit see poorly example example well method h datum example ct rarely perform aggregation never likewise never perform really normal design difference occur outperform lead challenge something know datum mathematically estimate cluster instance statistic implement package addition cluster different sub deviation take integer though identify indeed worse bad percentile good sd seem guide poor sd well aggregation differ similarity single linkage usual euclidean percentile establish formal geometric percentile sometimes euclidean performance test variety qualitatively clustering simulated clustering clustering component separate suggest lead clustering satisfactory course complicated shape little separation lead equal outperform hybrid eight form always yield robust acknowledge nsf nr example department statistics usa usa propose non convex three stage first stage use produce series clustering select cluster linkage stage dendrogram stage dendrogram variant argument justify step stage involve real keyword hybrid mean linkage unsupervised technique dataset cluster wide list reference recent centroid variant come agglomerative come scope limitation strength map precisely combine centroid agglomerative careful treatment influential convexity quantity dissimilarity cluster principle correct give rarely particular outperform clustering convex formal clustering ensure corollary method case give result ensure condition simple knowledge except establish iid variable need effectively assume draw variety size create linkage sl clustering size clustering clustering pool zero one ham sl choose grow dendrogram cut dendrogram similarity similarity cluster merge ignore cluster possibly merge merge small cutoff sl usual short one effective generate disjoint closure find distance far describe distinct close pool point minus linkage final evidence accumulation pool clustering way range hence membership hamming ensemble co third use grow prune dendrogram tuning seem technique conceptually hereafter separate centroid ct key use place ct linkage average linkage technique stage fourth grow unlike unclear ct technique conceptually pass first partitioning divide small agglomerative theoretic first simply closeness cluster boundary fourth look ahead contrast cluster mean enable technique propose hybrid clustering way estimate combine modification follow sec cluster unable provide interpretation present concluding remark sec begin generation five input large reasonable hybrid technique number serve cutoff cluster work reasonably find require right give start initial agglomerative merge clusters bm bm construct similarity ix I I I sd sl vertical dendrogram leave correspond namely maximum branch dendrogram length line dendrogram cluster final cluster write clustering exist adjust cluster sl submatrix use sl give final size brevity refer use dissimilarity
crowd recommendation preference collection aim order good preference grow recover order inconsistent preference acquire challenge explore existence ground parametric item assign preference restaurant etc compare preference number repeat comparison snr reveal item recover ground rank ideally rank partial well aforementioned adopt accordance popular paradigm arguably mle inherent convexity comparison efficient another within produce estimate nearly minimax square centrality finding parametric consider therein rank square reliable realistic scenario receive rank fall ensure top item accurate identification term question minimum comparison affect preference score address question algorithmic minimax optimal contribution two begin characterize fundamental three number comparison preference perspective emphasize separation quantify preference minimal evaluation reflect separation rate propose nearly identification soon exceed limit constant careful score sense rank iteratively pointwise comparison design primarily estimate minimal optimal accuracy furthermore numerical mle centrality ranking receive considerable item draw distribution underlie one observe identification model adaptive term rank multi value preference preference scheme exploration tradeoff perfect characterize complexity sampling relative provide item admit dimensional embedding explore basically approximately ranking accurate ordering approach accommodate query noisy item motivate generalize often top assume preference collect manner apart centrality mle variety aggregation guarantee convergence sample centrality mle nevertheless total ordering selection justification derive work principle existence permutation generalize involve ranking ranking another work distance rank broadly scope remainder key top main fundamental nearly linear summarize present detail spectral treatment direction proof rank mle e defer appendix respectively provide brief notation denote norm respectively independently besides mean constant formalize present performance metric item understand rank outcome numerous existence item depend item without throughout otherwise pair outcome denote indicate throughout ease presentation statistic acquire item snr comparison comparison observe item assume throughout dynamic score irrespective positive away regime grows readily translate separate vanish e voting like pairwise identifiable index denote characterize reliable ranking rank perspective scheme bad challenging aggregation distinguishing near decision acquire finite measure see play determine identification employ comparison absolutely preference would main finding tight condition identifiability state exactly assume throughout fed identifiable identification plausible detail control entry report identifiable behave choose worst compatible impose comparison necessary reliable boundary around sample complexity preference separation another barrier away remark dominant fraction prior focus latent score infimum almost identical potential pointwise achieve minimax pointwise present bottleneck top two base control estimation identifiable region separation item arise specify fine coarse readily minimax separation case fine item specifically minimax separation consecutive many hardness ordering item item necessarily easily unless impose fairly snr requirement comparison snr snr could increasingly passive requirement reliable eq employ preference upper q challenge dominant consecutive suggest rank outperform ranking separate active pair evaluation rank initialization around ground truth sense via spectral pointwise manner consist coordinate wise operate splitting within describe detail throughout preference particularly discover preference incur enable desirable fortunately method serve ideal initial guess seed average outcome large comparison I ji j distribution centrality word centrality proceed markov chain return distribution lead transition completeness centrality reasonably probability analysis mle e mle I rather graph model utilize mle iterate log coordinate method mle far apart contraction pointwise accordance formal summarize analytical slightly splitting recommend leave justification spectral recent arrive rapidly attract towards optimum restrict loss kind gap mle minimax optimal successive characterize interval role outli guess sense entry ground outlier mle computational initialization step rank centrality accuracy instance likelihood sum term find one program refinement stage accomplish mle provide np basically succeed separate high object long additionally cycle require stage spectral achieve linear sequel like interpretation estimate heuristic since present calculation suggest around control locally existence low leibler calculation precise truth rely surrogate result plug fortunately incur employ sense make I iw iw I truth dominate surrogate pointwise loss exceed procedure solution low guess converge rapid heuristic suppose constant simultaneously appropriate wise expect q replacement choose outlier one refinement stage another obeys recognize sequence recurrence point f bf b specialized obey sufficiently apart truth careful reader spectral centrality indeed analyse lead initialization estimate reasonably spectral mle converge one naturally seed refinement stage order mle desire via configuration establish analytical bottleneck bias tradeoff accounting randomness random general independent randomness go avoid nevertheless case tradeoff acquire comparison applicability important report calculate trial pair comparison centrality illustrate tradeoff repeat comparison sparsity spectral mle outperform centrality resolution comparison e next identification varies impose rank accuracy centrality situation interestingly mle relative apparent seem capable achieve randomized simulate future aggregation develop aware perspective return item accordance combine wise identifiability preference separation come develop mle investigate remain characterize choice pair comparison draw model collaborative rank pool user ranking guarantee mle ranking subject theorem heavily reverse coordinate likelihood resp vector resp np np coordinate lemma later concern separate fix eq derive call cover within evident one produce cover cardinality n l occur n recognize lipschitz result pick sure cardinality cover sufficiently large putting suggest truth use constant simplicity wise assume generate clearly obey calculate taking reveal kl use p say coordinate sense mle ground truth fortunately surrogate likelihood true coordinate likelihood brevity depend consequence gap iw gap obey two substitution give notably j develop
monitor spectrum mobile suit behavior consist link topology measurement one neighbor typically perform adaptation stage lead communication intermediate aspect network way essential role network diffusion classical optimally different combination rule laplacian neither account operate snr result performance snr vary across scheme adjust optimize performance circumstance regime mechanism switch consider compose homogeneous filter circumstance nod advantageous tracking capability previous approach adjust alternative deal case alternative drawback firstly local neighbor feed back present separation adaptation implement rule suit track scenario addition asynchronous extend combination weight mean illustrative propose analyze diffusion derive close steady square deviation new adjust similar introduce include stationary tracking theoretically analyze performance l adapt network derive close present main conclusion possibility vector letter transpose context length equal one summarize notation cardinality exclude index belong index unknown node node local combination weight assign node neighbor exclude connected topology depict share information neighbor instant minimize mse instant measurement length realization unknown length linear denote measurement power across possibly parameter c c kn k diffusion solve estimation manner iterate phase however differently diffusion purely combine combined neighboring straightforwardly write local vector typical projection characterize keep combine constrain coefficient explain section impose negativity constraint kn update recursively dash highlight adaptation phase combination phase adaptation even affect clear accommodate compose adaptation influence selection weight suboptimal adaptation phase affect deal adaptation independent node neighbor delay slow detail strategy steady present full consider compose adaptation equation consider diffusion algorithm prevent division fig steady steady deviation mean network convergence slope conclusion extract steady network heterogeneous optimal convergence reach optimal consequently select section state environment thank energy directly steady several difficulty encounter assume static although straightforwardly adaptation introduce variation independent distribute autocorrelation tn kn kn kn commonly adaptive throughout stationary input regressor u tn assumption widely analysis diffusion realistic many application assume white spatially n kn nk condition sufficiently independence similar behavior stand filter diffusion strategy local estimate notational convenience product across represent obtain stack entry ergodicity regressor filter series govern radius eq choose impose negativity limit scheme converge radius simulation converge much fast solution steady often diffusion scheme iterate limit analytical steady network replace matrix equal triangular arrive theoretical steady coefficient u kn u tn un equivalent apply average principle tn form delay line factor assumption adaptive filter analytical approximation q steady use iii crucial node operate instance favor fast favor steady strategy learn suitable kn kn tn kn kn stand optimize square bind emphasize note application order negativity constraint guarantee field remain subsection stochastic mse stack define kn tn kn collect rewrite newton control autocorrelation avoid division identity matrix average regularize kn kn kn kn l attractive implementation node invoke inversion inversion constitute adapt subsection replace cost temporal represent combine time obtain equation read symmetric th neighbor could condition filter interpret autocorrelation see correlation window weight factor towards rectangular convergence steady instability affine window rectangular window efficient term outperform state simulation rule stationary tracking simulate fig employ adaptation step size input multidimensional unless observation node variance length vector uniformly equation set aim validate compare state art carry node scheme stationary scenario expect objective subsection predict steady well although metropolis rule check would analysis plot variance scenario c last part study situation fig plot steady match good particular slow medium db stationary fig learning rule trade steady influence parameter couple factor correctly steady dramatically instability degradation art tu also table simulation b l provide combination consequently seem adaptive homogeneous network noise explain gap regard gain scheme surprising term clear network tr analyze track steady state keep rate conclude steady fast tracking diffusion rule conjunction rectangular low complexity number compare vector cost reduce regard equivalent paper novel scheme meaningful
train binary notice first subsequently draw mnist example fig illustrate draw sequence attention depict fig whereas attention attention like person motivation large look nonetheless image draw able capture colour composition architecture demonstrate highly house mnist generation dimensional attention embed beneficial generation paper ba draw image mechanism sequential auto encoding substantially improve model mnist view house distinguish ask fashion modification rough precise picture generation aim generative typically single possibility iterative fundamentally architecture represent towards create independently successively successive stage generation area precision width recurrent real combine family recently deep variational lead significant advance differ rather generate single iteratively construct accumulation modification part scene ignore result year capture sequential attention reinforcement policy gradient backpropagation sense resemble read neural machine present selective attention read modify mnist house generate conclude lastly like direct reader read generate variational determine salient information input network receive key decoder encoder decoder previous decoder secondly decoder successively ultimately oppose dynamically restrict encoder decoder well feedforward encoder pass feedforward decoder passed encoder produce compute px pass decoder step result pass rnn rnns encoder encoder encoder output may implement use architecture record handle paper notation encoder receive decoder form operation latent experiment diagonal bernoulli gaussians latent great propagate gradient pass decoder operation ultimately reconstruct specify advance h bias compute concatenation single logistic latent show fig pass omit binary natural define kullback leibler draw latent simple standard total expectation reconstruction loss interpret sample reconstruct total compression decoder generate draw iteratively pick latent decoder repetition generate operation eqs one selective one simple draw image operation create modify vector provide selective attention crucial generation draw attention selective without benefit training mechanism machine aforementione array filter smoothly vary configuration resemble computer base autoencoder image centre location indicate patch green indicate boundary patch digit middle patch whole image patch illustrate filter specify centre filter patch large attention patch grid filter column fully specify intensity filter attention dynamically determine ensure positivity ensure initial decoder horizontal vertical define attention patch constant ensure extract reconstruct filter gaussian filter display leave last bottom patch attention intensity concatenation error error write operation colour input read write rgb triple reading writing channel realistic three visual house cifar network always indistinguishable cifar image natural preliminary exercise module mnist classification bernoulli reconstruction cifar green colour emission intensity model approach work training cost image optimisation algorithm example sequence video efficacy attention aid image performance translate mnist like digit
every consider dnn well supervise dnn performance shape pre dnn cdf value see dnn compare non dnn confirm training deep many much explain enough second good training layer difficult final vanishing may auto error favor final training stack auto supervise criterion supervise learned intermediate straightforward output pre perform capability structural dependency face shape output space output space output difficulty input pre layer link dnn c experiment exhibit dependency consist dataset pre shape fig face model configuration learn shape difficulty dnn large opposite present variation pose well cdf curve dnn dnn emphasize test dnn configuration truth htbp htbp article generic incorporate neural base pre deal two initialization well hide deal structured train pre validate test two challenging outperform demonstrate output structured capability perform detection raw give application trick show supervise future plan help supervise partly support project cl france pre deep architecture vanish issue paper characterize internal dependency pixel label problem generally model fully architecture learn strongly evaluate building system generalize single mapping focus constitute sequence string tree graph discover unknown statistical language application parse output iii part tag bioinformatic model tree speech processing speech speech structure category discriminative category discover latter dependency output unconditional distribution kde functions space add reconstruction output back approach output task classification scheme support learn space inverse need graphical structured capability capture hmm output suppose many world relation conditional fields crf thank output widely deal crf propose random provide crf signals diagrams crf signal segmentation hmm crf cost graphical model structure generic unified incorporate regression task deep dnn training make dependency output apply structure real world detection deal structure add constraint output structure output structure structure discover output complexity help result generic incorporate output dependency final output input dependency framework formulate learn space apply input dependency part datum firstly cost function unsupervise pre dependency layer layer supervise back allow part input layer consist mlp new describe refine mapping input input stack stack mapping eq function input replace reconstruction x keep initialize output reconstruction optimize initialize link link respectively supervise dnn describe experimental implementation present version library role recognition study year task remain complex pose expression point image application face dependent task face widely capture constrain face shape shape image many face also match carefully convolutional field whereas consist define propose face discover consist training unconstrained variation illumination dataset dataset truth divide sample similar resolution collect box truth dataset image normalize face deep hide
intuition plain paper existence framework facilitate degree explore problem rough coefficient fast possible follow pde discretization mesh example symmetric fast problem type affect lack progress development robust method rough hierarchical resolution wavelet method wavelet application wavelet arbitrarily preserve classical wavelet away property rigorously prove sum harmonic provable rough coefficient pre operation reformulate quantification solve automate paper possible pre support fast direct orthogonal nearly bound condition surprising achieve wave operator essential playing miss find possibly strategy play game fast completely find estimator decision sample analogous theory generalization quantification model require analogous formulation guide discovery identity difficulty generalize concept rough lie priori accurate adapt idea concept harmonic coordinate essential rough coefficient assumption find ergodicity fine sequence lack robustness rough basis must provably localize identification element numerical replace try give rough spline spline discover numerical instance information optimal method optimal strategy min game choose finite incomplete player minimize remarkable von strategie deterministic randomized strategy although information decision theory sufficient compact game purely deterministic priori connection player place distribution candidate strategy b place candidate prior although employ player player distribution bayesian prior appear due min determine player employ prior linearity calculation restrict prior linearity investigation algebraic framework linear systems b optimal accuracy apply discovery elementary gamble characterize exponential enable localize high section game must nest measurement coarse fine resolution approximation form martingale conditioning martingale hierarchy interpolation orthogonal system number elementary gamble nested orthogonal scalar product norm solution compute condition enable computation complexity identify equivalent algebraic set recover approximate linear equation let element purpose eq measurement know vector definite matrix example instance recover choose b lead purpose result select accordingly preserve linearity efficiency restrict b step mixed player q matrix follow whose minimum define eq simple calculation nest define write write subspace write projection scalar rectangular nan follow problem unique norm respect product matrix note zero conversely belong dimension conclude particular z z belong complement observe k solution v k right calculation equation control entry power radial basis krige assume observe energy norm significance quantify simplify energy estimate choice ax z approximation approximation interesting knowledge corollary remark motivate problem good measurement use quantify recovery write write orthonormal form eigenvector measurement correspond span small eigenvalue associate observe minimal matrix nearly randomization sense accuracy multiplicative factor log law entry application derive see indeed conclude observe p measurement value difficulty associate small one problematic randomization play game modify game player decision player randomization strategy game computationally measurement positive symmetric solution ab symmetric constant measurement identify positive error factor pde discretization laplace pde piecewise mesh norm resp solution resp resp span row resp span right side generalize continue analysis design interpolation interpolation formulation choose approximate linearly express q interpolation condition acting subsection continuous covariance function pde algebra interpolation condition measurement since test kronecker delta formula expansion discussion solution admit formula formula variational property conditioning intuition precision inverse constraint optimal recovery admit unique minimizer define unique subject furthermore obtain proof define measurement dy follow respect extend green observe fy dy imply fy dy fy dx dy h directly see allow solution partition closure sufficient write simplify modify equal one elsewhere exist construction additional equal use localization subsection require solution convexity constant via present similarly use construction clarity degree via constrain problem energy local energy rough show property beyond span lie exponentially support localize diameter element write center exponential decay basis eq aa euler let contain closure domain contain closure define q exist jx dy monotonicity green jx dx dy dy x start ball center contain diameter follow j dx dy sum inequality constant simplification integration eq follow let naturally outside localization hold need unit center diameter piece direct piecewise apply let union union restriction zero w lead h combine decay deduce combine conclude proof simplification solution note localize I furthermore v square contrast need slightly change avoid technical without generality scale u r u lemma preserve localization localize numerical section building level resolution hierarchical nest decomposition game say resolution diameter diameter least constant depth tuple j ks say regularity resolution regularity h j hierarchical nest measurement mix mixed player value measurement iy space nest hold measurement element hierarchy computed question formulation player express replace mixed dy follow measurement e investigate game hierarchical manner coarse game value theorem realization martingale increment vx form martingale increment gaussian measurement martingale martingale property martingale time cost towards martingale l condition gaussian covariance q direct martingale increment operator complement within write direct element hold belong orthogonal u u k nest restriction transpose interpolation follow restriction interpolation player mixed strategy player information l k restriction true equality observe I imply coefficient observe k identity onto restriction onto vx symmetric matrix invertible admit symmetric positive restriction operator define representation definition imply j j consequence formula provide restriction interpolation nest follow transpose furthermore r k nested imply take theorem lead orthogonal decomposition q sense basis see wavelet orthogonal rather adapt space pde subsection induce decomposition condition uniformly hold q furthermore l analogously h simple consequence k scalar direct consequence lead good integral player rectangle rectangle level basis involve subsection basis unconstraine element k theoretic interpretation figure observe let k k ic I quadratic q direct therefore inversion effect decomposition system uniformly dimensional define furthermore eq consequence u discuss subsection system solution mesh regularity conjugate cg guess yield approximation arithmetic writing prove size k extra underlie localize remain error fine localize check define definite element fine give allow localize element unconstraine localize ai k k z control reverse induction hold constraint writing hold eq equality l domain I proposition constant aa imply finish proof solution b constant aa ax need symmetric let b aa j bound number k imply aa eq low imply conclusion replace theorem aa solution follow
parent eq derivative directly calculate transpose derivative gate accordingly omit check approximated structure complicated consider simply overall solve semantic piece understand attempt sentiment phrase within stanford bank sentiment early factorize consider small component phrase phrase recent start principled formation semantic enhance composition node lstm stanford sentiment bank evaluate benchmark work datum annotation tree comprehensive lstm stanford sentiment bank review discuss stanford phrase manually annotate sentiment split training predict sentiment root sentence phrase sentence sentiment sentence phrase sentiment phrase classification mention minimize setting phrase regularize th element multinomial iterate regularization tune data split structure conduct conduct accuracy stanford bank result sentence column root machine correspond confusion merge vector interaction nb svm lstm table lstm batch size hyper fine weight leave word unit word initialization word depict converge root phrase fast phrase task start minute efficiently lstm experimental first keep lstm depict name stand gold sentence circumstance phrase node phrase however comment sentiment bank bank enable study change keep annotation tree sentiment available cover vocabulary concern sentiment setting lstm margin lstm obtain compare label lstm improvements internal learn parameter hand ability model lstm root leaf lstm performance figure length unbalanced trend show advantage deep semantic lstm effort attempt utilize structure prefer chain recurrent neural implicitly linear compare first word lstm short right lstm read right left phrase correspond sentiment bank annotation version experiment root include annotation lr lstm lstm root lstm leaf lstm leaf lstm root parsing help improve label leave recursive lstm recursive lstm inferior use gold gap sentiment dictionary structure gap conventional structured short propose reflect memory provide principled structure learn sentiment text replace enhance lstm show useful lstm research community contain line attempt representation utilize believe recurrent actually structure implicitly give empirical toward answer input high root semantic structure short lstm wide speech recognition translation tree reflect recursive principled consider interaction language understand mean text art recursive model layer lstm helpful achieve well year long lstm demonstrate translation code hierarchical modality semantic language merely concatenation instead sentence yield art performance task scene segmentation lstm learn reflect multiple call neural lstm potential avoid hence interaction tree deep lstm together lstm instead structure lstm mean piece text representation text understand human language stanford tree bank determine sentiment favorable benchmark much annotation enable explore experimentally lstm art recursive composition lstm memory structure consider recursion modality recent year demonstrate achieve performance semantic analysis segmentation recursive tree recursive leaf node combine backpropagation effort neural include amongst leverage syntactic parse recursive neural subject vanish result difficulty compare claim simply result performance recognition previous lstm model utilize achieve ignore priori lstm structures child hence multiple cell hierarchical show blue figure line indicate often sigmoid later lstm principle respectively soft compose lstm specific gate gate forget child datum forget rather denote hadamard sign hide child right gate resource child forget gate forget gate weight combine formula regular child forget
factorize perspective arise application trick objective interpretation ignore dependency activation may retain dropout form row multiply dropout capture multiplicative noise univariate meet training uniform prior weight format store p interpretation put kl dropout prior possibility respect approximate discuss detail analytically approximate eq use dropout rate low learn separate dropout per layer neuron separate although specification beneficial set maximize variance rate optima objective cause variational bias introduce good recurrent estimator variable trick unbiased stochastic variational inference focus variable extensive parameter report application long history use infer probability type show dirac posterior variation dropout similar focus approximations monte compare binary dropout type pre name correspond name include type noise introduce type weight write choose fully neural hidden follow recommendation rate early method epoch empirically different estimator describe epoch epoch variational dropout independent gradient full gradient rather advance encourage result format number format type value number exponent closely format receiver approximately double common specification kl uniform equal transform interpretation control digits dropout column vector result previous input expect variance rate dependency dropout q gaussian dropout parameterize treat weight variational optimize w marginal wish optimize kl divergence p prior approximation straight bernoulli sign approximate consist kl divergence kl divergence conditional scale kl bit evaluate weight divergence part transform uniform log putting involve cdf entropy define q divide family gaussian determinant variable rewrite term depend kl posterior log prior consistent additional term numerically use rd dropout alternatively improper prior allow indistinguishable correspond rate approximated term vanish claim draw stochastic minibatch draw random weight hand q decompose identical trick vanish uniquely compare model variational noise dropout correlate generally regularization htb top bottom stochastic gradient estimator epoch epoch epoch sample var university california research drastically efficiency rely parameter minibatch variance minibatch drastically upon uncertainty global minibatch local trivially variance minibatch fast convergence dropout dropout parameterize posterior specifically propose parameterize posterior generalization million minibatch gradient due high neural capacity wide diversity nonlinear pattern lead spurious happen train various controlling overfitte currently popular effective binary dropout gaussian approximation call identical regularization much marginal extend exploit greatly direct generalization parameterize bayesian posterior neural network overfitte computationally design markov mcmc inference asymptotic network alternative framework modern variant variational infer neural show modern deep neural much dropout variational datum simple regularization trick drastically gradient uncertainty minibatch flexible popular relationship dataset tuple standard observe belief posterior rule pp p involve integral approximation necessary optimize parameterize leibl practice likelihood plus w maximize minimize variational exist minibatch base especially basic trick new minibatch likelihood minibatch random draw gradient correct proceed perform stochastic tell asymptotically local weight crucially depend fail make objective monte calculate approximated indicator minibatch shorthand rewrite give inequality arise nn variance minibatch however random entire minibatch variance moderately variance intermediate weight translate form independent global translate yield computationally statistically trick generally applicable explain contain consist neuron receive feature layer multiply nonlinearity specify factorize minibatch need million number layer neural would hard originally perform simple turn importantly device optimize basic algebra happen architecture library deal activation directly factorize eq rather gaussian result activation activation j million
decomposition copula allow gradient copula introduce scalable augment copula easy later I mean augment factorization maximize fix copula field special running employ learn outline alternate set iterative procedure alternate share objective function well fix px describe copula automatically learn structure copula family selection among family supplement preliminary copula change requirement variational mean arbitrary copula add everywhere thus augmentation naturally address augment repeatedly inverse cdf individual copula efficiently bad dimension distribution follow copula calculation conditional cdf tree copula condition loop copula copula separate log therefore field change mean augment space use analogous gradient pre copula gradient set marginal gradient simplify copula contain gradient copula gradient copula sum pair conditioning condition copula copula require copula gradient arbitrary family convenience augmentation easily incorporate efficacy datum use model feasibility case dependency arbitrarily framework mini variational combine idea adopt nesterov accelerate velocity iteration momentum look update allow change quickly adopt al rao replace expectation lagrangian meet sample normal specifie assign copula augment factorization pair compare response variational bayes technique perturbation argument estimate mf truth set display simulation effective indicate display diagonal mf variance well also mf copula optima mf parameter posterior hand mf use set handwritten digits membership example report mean model latent draw bernoulli independent factor mean field copula hold field minibatch take average take minute take minute mf fit copula perform mf require minute already inversion well fast apply inversion outperform drastically upon convergence runtime hamiltonian five variational mean iteration comparison field either field copula iteration already field upon copula preserve dependency propose principled perform field copula alternate scalable manner stochastic mean mean easily add bias form approximation capability variational achieving acknowledgement foundation discussion assume structure copula family black box inference tree copula family synthetic calculate possibility intractable grow exist sequentially dependency sequential subset fix require copula preliminary find outline copula family certain conditional bivariate copula among family maximize sequential package easy also family experiment frank close family correlation include version copulas frank copula tail theorem general family structure copula copula allow dependency variational distribution approximation divergence augment stochastic straightforwardly original mean reduce sensitivity local hyperparameter help characterize interpret dependency latent keyword bayesian inference copulas network efficient approach approximate distribution applicability complex tractable make either field original variational order preserve mean monotonically kl budget demonstrate fit bivariate addition copula field structure approach knowledge fall dependency augment variational calculate easily place copula example mixture consistently parameter reduce optima implication feasibility restrict inference make write variational preserve variational inference study solution class model differ inference explicit denote cdf bivariate pearson copula copula specify tractable much focus two dimensional copula student frank copula multivariate lack flexibility accurately model dependency successful bivariate specifie factorization copula conditional bivariate copula also copula specify
answer representation together cnn outperform image need answer multimodal convolution propose image question question cnn multimodal concatenation prediction concatenation complicate relationship multimodal input multimodal interaction answer multimodal convolution word lstm answer reason meet word exploit lstm without modal convolution question compose reliable high semantic pool reach cnn possess language high representation examine whether reliable randomly question significantly compare language language natural representation content answer question drop generate representation greatly demonstrate moreover lstm question learn lstm question well introduce lstm performance future representation paper propose cnn neural cnn model architecture representation answer public dataset demonstrate outperform com li com propose question end composition inter modal generation answer image cnn cnn question layer multimodal joint candidate efficacy recently answer substantially multimodal language specifically rapid sentence retrieval far explore complete answer answer image content produce building block vision language processing however understanding pose challenge regard automatic image ai multimodal learn image produce image condition related human computer question understand instead example question successful well represent pay multimodal input question employ convolutional cnn image triplet consist question answer cnn learn like question content cnn learn answer question extensive result dataset state architecture encode conditioning recently sentence automatic image multimodal require make use widely recognition cnn successfully language multimodal relation sentence retrieval term lstm sentence image representation generate answer question et al binary image answer al question answer visual parse question answer neural research formulate conditioning question compare image solely lstm concatenation sentence cnn however question question answer tend denote color neural inspire rnn name visual semantic embed image question multimodal correct lstm treat question learn treat individual word exploit handle drawback cnn employ learn inter relationship prediction construct cnn well interaction make answer cnn related input figure cnn one cnn high semantic convolution layer representation softmax generate answer multimodal answer train reliable answer question firstly multimodal exploit sentence individually multimodal input many cnn representation achieve image recognition work encode content activation sigmoid relu take softmax layer relu cnn mapping dimension provide benefit firstly meaningful meaningful composition pool composition convolution max summarize component convolution max pooling question scale whole layer max generate max last pooling representation representation sentence multimodal input generate answer multimodal cnn treat semantic consecutive semantic question interaction multimodal input multimodal convolution multimodal similarly paper treat image representation treat generation word far exploit vanish time step begin perform manner interact closely question question well exploit question firstly semantic question interaction demonstrate feed show softmax layer answer question introduce cnn train evaluation measurement employ convolution cnn accommodate length question length choose embedding obtain gram cnn top softmax relu map new eq dimension image accommodate convolution multimodal cnn sentence joint softmax input cnn train sgd tune image multimodal softmax prevent dropout construct however publicly cnn public database testing answer image image type specifically type object color comprise training testing constrain generation answer dataset large color answer evaluation correct testing question besides wu base subsequence require use respectively image multi employ semantic answer develop compare specifically neural cnn interaction world human answer word language multiple multiple single word guess lstm performance
vary reinforcement admissible decay reinforcement configuration mdps hierarchical reinforcement basis markov explore temporal aspect task subproblem agent mean observe agent action move rewrite give reward function respectively one framework option way call create idea hierarchy option option option call mdp define choose environment option terminate probability notice semi policy I next option entire terminate continue accord rewrite q probability terminate update eq propose human expert still expert manually define behavior specify behavior tool compact maintain scale expressive power rl agent situation adapt change configuration reinforcement together reinforcement modular composite action example different kind position case could robot configuration arm reward return value must represent child child return reward unique child rest behavior would child health could one formal exploit option approach use policy option execute past state reward begin option generate probability argue model option reinforcement option reinforcement primitive type trial room next room moment room lose agent room figure branch highest low represent learn use receive action node receive show activation behavior expect activation behavior expert behavior use behavior baseline save use room action intensity give room intensity specify perform take complete notice wrong room show agent learn one learn node receive wrong save room root child try leave room show behavior difference learn effective behavior create character game quick receive development prove tree hybrid dynamical work root acyclic graph dag implementation level modeling level prefer extension core description horizontal due sort et propose look reason tree reinforcement behavior option similarity reinforcement abstract general hierarchical reinforcement area manual behavior problem part manual behavior view expert reinforcement agent physical agent constraint tree composite local node work hierarchical ensure nest intra option validate expert knowledge confirm nest node action expand use tree expect capability us agent entity scientific inf use behavior human agent execute desirable adapt human environment discard due framework node behavior address capability agent option ensure show affect execution must right moment goal minimum ideally zero chance consider human robot operate carefully neither human bt plan make video appeal human expert control execution action agent reliability agent behavior build maintain behavior also good behavior differently event bt entirely manual create behavior variation behavior agent agent human agent adapt environment interact bring rl agent adapt environment real online discard agent bring problem stability agent bring robot character human generalize poorly converge add capability reinforcement behavior rl minimize risk behavior reinforcement overall tree agent option framework reinforcement remainder present overview tool reinforcement reinforcement learn validate empirically control relation discuss behavior representation make agent create game character character control hierarchical state machine code rule effort tolerance system notation formalize behavior tree controller relation dynamical coherent behavior tree current minor please behavior provide transition especially collaborative bt define explicit box design independently another add modify remove necessary piece model bt decompose graphical bt node project although another parallelization possible easy worker contain parallel behavior tree root direct incoming child child leave subtree single leaf node propagate branch reach return call immediately root sometimes back type stop reach tree category present category constraint category commonly refer propagate composite return computation necessary symbol change behavior return state child repeat execution child represent action node leave propagate instead environment internal robot involve play sound turn spatial transformation play etc action instead signal meet commonly criterion condition agent visible low represent reason could return still never handle differently return receive core action five composite node node controller specific sequentially child return return receive sequence child return node selector child sequentially child stop
batch mini size ms gd compare good performance ms gd mini effective pass count effective mini batch gd comparable well gd parallelism ms gd parallelism ideal speedup parallelism achievable could evaluate efficiently ex descent give effective pass give thresholding experiment version average analyze well instead gradient descent apply give good gd safe ignore give demonstrate superiority gd algorithms ms gd propose batch variance nonsmooth unconstrained processing ms enjoy former admit parallelism comparison parallelism potential q proximal collection vector apply iy k q q convenience iterate x v ty x divide side inequality notation analysis put decrease obtain eq available eq summing multiply side right combine strong convexity combine h define statement relation recursively expectation eq value desire strong also choice verify equivalent side positive trivial thing need verify denominator need apply operator give update apply follow proximal operator calculate separable mini coordinate list let k k g k eq conclude summarize liu text bind evaluation need predefined accuracy iteration speedup mini simplify n essentially always indeed resource outer translate case reach speedup assume condition prove fix set outer minibatch ms gd perhaps explain behaviour need simplify assume big long processor approximate b convexity also convergence epoch accuracy epoch desire expect epoch enforce rest choose small highlight limitation efficient speedup processor probably gpu architectures parallelism loop follow note evaluation remark exercise question plot liu mathematics university unite department university usa gd incorporate improve complexity semi stochastic gd sum nonsmooth convex perform computation follow become introduction mini compute gradient base benefit effect reach predefine mini scheme admit parallel parallelization method empirical risk descent reduction separable nonsmooth number smooth function convex lipschitz constant e q parameter allow bound combine develop ms gd mini proximal gd enjoy parallel speedup environment attain mini formalize predict speedup mini batch loop one gd intensive past year accelerate nesterov fista scale big iteration issue randomize closely mini variant gd motivated accelerate proximal batch acc acc prox largely mini stochastic update limitation sgd inherently parallelism mini gradient via assumption whenever procedure equivalently write follow scale proximal stochastic eq estimate gd old gradient already prox point reference outer index new counter instead notation outer loop square variance ultimately extremely semi complexity complexity clear semi outperform regime fista achieve momentum gd max stochastic stepsize start mini batch store ix kt size estimate ij k iy h tx loop index epoch counter loop epoch start compute statement predefine particular speedup target follow demonstrate base logarithm e gd stepsize gd translate speedup mini threshold prove outer minibatch ms gd related mini acc prox acc incorporate mini nesterov acceleration claim define pn acceleration batch acc prox gd theoretical acc prox ms gd numerically total component evaluation compare condition theory mini gd advantageous acc prox acceleration acc prox illustrate ms gd ms ms gd ms gd set regularizer eq conduct q set training perform mini batch parallelism regression lipschitz result evaluate ill give four sparsity proportion nonzero element constant h sparsity might ask ms gd sparse sgd nonzero test operation ms gd fully
patient diagnosis diagnosis along vector recover separate selector various grateful anonymous helpful comment author wang matlab approximate selector use direction theorem example iterative selector two stage formulation selector stage construct selector direction simulation alternate numerical fast selector operator consider linear predictor parametric among square great deal variable selector selector sparsity involve selector receive amount attention selector fit minimization technique selector strictly guarantee ensure uniqueness selector right censor outcome importance selector demonstrate work cast program interior interior large scale problem cast solve alternate direction find selector show usually cpu problem rewrite iterative iterate subproblem successively subproblem solution gradient subproblem linearize selector world datum proximity optimization via solve primal one implement achieve comparable consume outline follow section numerical efficiency propose propose first simulated patient rest let absolute th otherwise give vector hadamard denote denote whose th natural product df fx fx develop proximity solve optimization exist alternate direction method linearize propose reformulate augment lagrangian lagrange multipli penalty optimization elementary equivalently subproblem gradient scheme k subproblem complete square constant form subproblem efficiently iterate selector base matrix appear rewrite thank norm characterize proximity operator review definition map solution vector conversely satisfy solution proof straightforwardly chain aa previous problem comment proximity appear soft operator cube length far b check find amount couple iterative upon equation dual dual exactly scheme apply word introduce iterative scheme sequence iterative initial seed particular limit sequence selector often nonzero correct bias assume generate iterative step step submatrix extract coordinate eq set parameter parameter compute I terminate reach stationary terminate stop meet change successive tolerance stationary successive iteration fix stage complexity comparison loop relate subproblem approximate subproblem terminate stage complexity next indicate short situation require iteration follow proximity approach method present selector matlab center advantage typical utilize equipped intel core cpu ghz gb matlab pc intel ghz processor gb windows simulation generate recovered combination algorithm stop priori noise well approximated exist method event speed affect accuracy suffer gaussian support size select uniformly identically sample normal collection selector collection zero standard simulation experiment selector measure square ideal estimator approximate noise simulation standard ii illustrated selector magnitude nonzero component estimate nonzero curve represent vertical away
appropriately optimal order iterate learn main goal compare implicit update suppose approximately term limit dominant stability work implicit define explicit loss low initial prefer discount explicit term discount reflect misspecification cause instability initial decay convergence implicit rate explicit conduct standard benchmark simulate real dataset display behavior scale challenge alpha xx gradient descent xx xx stochastic iterate replace proximal reduction proximal gradient sag establish minimizer mle stochastic scaling adaptively second sensitive default version descent aware importance equivalent interpretation optimality normal regression separate normal n ny xx plot full xx optimality follow xx follow xx let step proximal method pass single pass achieve indicate pass converge perform varying affect parameter perform increasingly bad stable achieve classify cover type leave validation datum use use logistic set regularization method exclude rate specify setting paper hyperparameter misclassification pass hinge figure intensive hyperparameter easy way investigate change iteration ht digit indicate misclassification pass respectively xx supplement hinge ht accuracy term proximal implicit update size average iterate er rao performance significantly par explicit theoretically aforementioned robust convexity effectively learn stability come model simplicity comparison proximal tuning calculation dataset storage information understand strongly convex objective grain analysis average corollary university university university become unstable statistically inefficient information term combine proximal implicit iterate er non stability robustness learn respect function demonstrate state averaging method utilize update simple consider wish expectation typically loss wide mean cast approximate approximation seminal maximum posteriori learn mle usually incremental procedure realization stream subgradient respect realize combine idea implicit iterate implicit update hand operator point generalize splitting comprehensive implicit derive efficiently implicit proximal method geometry proximal idea replace stochastic gradient update average average accord aforementioned finite thus rate estimator well difference regime incremental simplifie employ keep average periodic several whereas storage averaging iterate analyze optimality application convergent average work analyze update superiority substitute convexity simple certain aspect f expect asymptotic iterate imply non averaging experiment several task confirm suggest combine method definite theoretical eq differentiable surely differentiable decompose yx dy convex sequence random surely hessian almost surely twice convexity convergence assume believe implicit vector see probabilistic implicit efficient restrictive include variety logistic linear time series constraint notable exception form well regularization confirm either subject use average study proof
identity testing closeness address mild logarithmic low indeed logarithmic section proof follow sp section motivate special highlight two aspect test distinguish distinguish testing later subroutine simple case testing determine belong suffice thus chernoff suffice set next element pick determine probability increase experiment several uniformity illustrate difficult find element distinguish distinguish observe differ argument method near proceed testing argument chi capture dependence complexity example consider distinguish suffice capture test exist chi two n else sample correctly hence give find select however consider way find element quantify idea pick element definition generality heavy pair symbol heavy distinguish easily auxiliary heavy property algorithm achieve good trade tuple useful achieve near independent tuple tuple return heavy lemma element apply pi find distance furthermore increase find element heavy yet belong higher precise trade complexity recall find distinguish near distinguish easy meta subroutine want use yield help complexity return tuple run previous tuple output recall distinguish distinguish element distinguish candidate combination element know set form ideally scenario arise constitute partition ss combination possible scenario find tuple case randomly entire test yet scenario output probability otherwise step propose identity testing test calculate complexity tuple g h g h test return tuple closeness unknown identity closeness testing identity part find distinguish distinguish algorithm closeness testing extend identity closeness distinguish order probability decrease distinguish closeness find organize identify distinguish formalize distinguish element one frequency distinguish closeness require additional testing order subset distinguish closeness test outline finding serve symmetric compare simplicity rest come find eq indicator variable precise argument remove ps ps pi efficiently show state distinguish convert calculation analysis expectation seem analyze success event take fair amount conditional generate probability show use need element pi pg heavy although probability tuple ir use sample know multiplicative factor later furthermore I I threshold ensure guess value output I search fast recall problem approximately know guess instead find assume closeness least remove pruning element probability step tuple tuple return main binary set underlie result n sn nj ks ss ni else return obtain set heavy whenever discuss sufficient closeness firstly remove element big number indistinguishable probability probability concentration concentrate many time address pick ensure none element consider find perform closeness use distinguish run run distinguish main repeat majority success close none probability complexity run otherwise tuple I p return thank cl distribute align universal suppose poisson variance term chebyshev inequality chernoff p qp bn chebyshev eq chernoff bind argument hence n tuple since count prove thus proof pi upon expand furthermore similarly hence lemma return th overall chi square numerical simplification lemma error use theorem result output probability return element let correspond induced output probability pg h pg x x g x g pg ab pg pg py py proof chi substitute minimize pick condition event discuss probability fall three complexity use sum tuple yield complexity tuple p g sample summing use define section interested element end notion element show property reduce index go sum suffice eq inequality side sign different sign sign find distribution draw sample output tuple py generality technique algorithm element large far element get pi pi good element satisfy heavy exist element appear j first state follow independently distribution find run underlie good tuple tuple else find remove heavy two small tuple run initialize set element p r set sn nj obtain else remove remove element chernoff less n remove remove call identity let element sample th heavy eq suffice call proof remove element q time substitute rhs simplify follow hence iteration ii remove element heavy pruning pruning rhs inside ir simplify union lemma since time union bind divide proof return notational return call first showing heavy never remove remove element two part event I p conclusion lemma show r time clearly probability bind contain j rs rs chi eq return event happen set pg pg happen output union element prune unchanged prune pruning convexity fact hence q r eq
surrogate problem bound derive use surrogate minimization address issue approximate projection outer experiment synthetic efficiency machine svm well machine hinge loss surrogate classical majority class poorly biological constitute diagnosis pearson risk consist type type ii risk minimize dealing framework empirical risk propose optimization empirical subject empirical constraint present deal propose guarantee solve pearson problem section deal art pearson classification finally experiment synthetic rna seq cancer henceforth underlie classifier mapping sign predict occur classifier e must shall nonconvex convex fig surrogate problem x x I classical surrogate al quantitative relationship risk use show excess calibrate risk type eq pearson introduce let denote mx unbalanced unbalanced alternative sensitive lagrangian svm svm lagrangian cross propose pearson calibrate classification empirical type risk property classifier classifier mm eq suppose surrogate ii restrict attention calibrate surrogate loss convex everywhere lipschitz twice calibrate posterior directly value connection assumption differentiable vision normalize high loss calibrate hinge boost satisfy solve pearson algorithm backward splitting method introduction processing offer guarantee reliability suppose satisfie reasonably note lipschitz split separately proximal forward backward algorithm implementation projection operator assume rewrite derive convergence iterate let except computation lower set onto fortunately compute perform sufficient efficient design level algorithm proceed successive outer subgradient projection dd eq terminate principle compute half space serve outer outer onto express explicitly describe onto onto magnitude perform q fp fp fp kp kp fp k p iteration contain low fp fp kp projection half need sequence onto dp k fp kp dp fp fp dp dp fp n k computable inner take risk clinical set work dna empirical recent review selection detail alternate pearson classification experimental surrogate half figure risk typical throughput sequencing rna seq microarray preprocesse library jx total number read library sequencing expression propose transformation optimum eq transformation last negligible count patient patient gene sequence generate variable modelling measurement randomly change value decrease impact binomial transformation whether artificial patient class unbalanced sample
drop ill unbounded irrespective geometrically r measure noiseless analog constrain least long allow could relate much class asymptotic wide pointed square logarithmic achieve regularize estimator nuclear regularize achieve error arrive example upper imply appear sufficient slow vector slow exist definite ball scaling small number simple satisfied interesting design matrix appropriate tail moment know concentrate hence weak broad finite specialized statement indicate constant eq begin quantity square hence feasible condition later high henceforth eigenvalue orthogonal z indicate noiseless case regression model yet quantity appearance bind possible improve place dependent nuclear quantity square instance standard much indeed fail trivial entail albeit convex problem block practical reach numerically find apart regime without result particular face stock different employ rip boundedness involve problem fix standard random wishart behave form probability moment order scale replication phase specifically turn quantile associate triple summary may mask observation identify quantile drop model concern reasonable give rise noiseless yield solid line curves fit exceed empirically relative wishart expect behaviour design replication wishart I parameter oracle error negligible see color grid validation minimized note ensure pick nuclear minimized nuclear minimization chen et specific choice regularize assess impose constraint add constraint yield parameter grid specify worse report conclude case differ regularization oracle seem present ir z eq factor random motivate connection explain popularity model straightforward available constrain square estimator take consist image covariance turn obtain price stock technology begin year total retrieve correlation precede wishart difficult observation point replacement range replication replication approximate measurement perturb reasonable albeit extreme million achieve reason stop picture use full hand fast ex f c ex paper investigate trace symmetric semidefinite excellent employ nuclear side usefulness finding recover li partially nsf dms fa invariance suffice consider orthonormal canonical basis operator minimizer coincide projection law zero follow cone z r optimization symmetric proposition contain dimension conclude problem lagrangian dual proposition follow remainder establish dual obtain kkt optimal pair obey take inner complementary substitute feasible choosing ingredient give write last eq equip inequality lemma h maximum case consequently obtain follow easily see substitute finish recall bind back v yield desire concentration extreme eigenvalue let eq expand theorem obtain two assertion sequel respectively minimizer satisfie contain sphere consequently conversely satisfied otherwise divide proof analog suppose inside lower expand back yield collect obtain ex theorem section theorem proposition definition lemma sketch l trace positive computer department science nj usa past year receive considerable completion quantum estimation notably nuclear great popularity argue long positive condition situation approach entail knowledge estimation come trace interest estimate measurement matrix attract focused set sense phase retrieval work nuclear amenable modern technique lasso arise regularization less clear incorporate present semidefinite denote cone interest gram method rank kernel estimation measurement employ nuclear norm regularization proper parameter interesting practice choose finding negative dimensional paper certain design achieve regularize generalizing noiseless paper good paper compress sense goal complement finding summarize terminology throughout matrix inner dm dm usual number b linear adjoint consider error convenience cover tail symmetric projection orthogonal complement sequel estimation refer
denote information contour say frame mass induce independent evidence use form contour contour function contour pl pl algorithm maximum missing cause measure estimation density express become know uncertain information belief contour uncertain observe likelihood contour respect eq regard contour observation therefore eq em e observe small threshold special incomplete datum censor denote de two observe censor let datum mix knowledge experience partial simulate label draw randomly change probability uncertain label em censor em run estimation bias commonly equal exact estimation c degradation noisy soft suffer label supervise learn estimation unsupervise learn uncertain traditional indicate follow experiment censor censor class label appear moreover maximum e censor datum e algorithm estimation available belief improve project failure device stage continue regard evaluation perform priori frank com frank er evaluate em compute maximum estimation especially uncertain datum e due censor uncertain derive base knowledge integrate life censor right censor term censor drawback removal terminal censor possesse become popular year censor kind reliability evaluation uncertainty ever tendency take uncertain account last decade uncertainty latter lack information restriction cause compose expect datum carry uncertainty censor e uncertain use analysis censor consider special simultaneously censor merge uncertain unlabeled value hide occur early believe therefore belief prior pseudo prior method maximize show label algorithm censoring attract recent year flexibility removal terminal theory belief first later give brief ii censor pc describe identical
neutral frequent remove stop distribution distribute top frequent neutral feature pool unbalanced positive select pool axis unbalanced dataset remove unbalanced knowledge classify obtain balanced movie unbalanced remove document unbalanced dataset maximum improvement obvious neutral entropy since assume neutral distribute suggest gradually bias label incorporate kl well balanced unbalanced compare feature balanced label feature pool conduct movie positive unbalanced randomly positive balanced little reason label unbalanced maximum neutral bad approach pool axis unbalanced construct remove unbalanced manually movie dataset distribution positive original balanced movie balance unbalance pool unbalanced result show unbalanced become remarkable neutral decrease significantly remarkably divergence guide label unbalanced kl robust model al highly unsupervise assignment al constraint instances newly propose several dataset unlabele cross constraint self et objective al framework project distribution al explore distribution reference propose incorporate explore monotonicity chen try leverage incorporate class distribution objective discuss perspective address paper try prior et instance feature et propose active learning problem model propose regularization term expectation experimental considerably improve comparative knowledge work leverage may present detailed discussion incorporate neutral feature simple require modification common neutral unbalanced entropy regularization controlling extra nothing corpus assumption violate unbalanced reason kl utilize assumption fact suggest additional kl knowledge sometimes fortunately insensitive rough possibility perform reality distribution domain china cn many approach knowledge robust justify experimental propose remarkable improvement robust baseline language processing task text categorization indicator sentiment leverage guide nlp previous study address problem line leverage encode commonly see knowledge variable dependency last knowledge latent variable crucial knowledge fan provide word less heavy handle undesirable investigate aim reveal factor practical regularization formalize output easy neutral namely indicator reveal neutral boost remarkably make manual annotation neutral regularization maximum class regularization simply use neutral neutral uniformly contribution regularization term neutral distribution kl outperform briefly justify survey robustness method label knowledge indicator manually example label sentiment criterion provide preferred guide parameter preference expectation give express et label label parameter indicate word otherwise number softmax bfgs framework term constraint indicator correspond elsewhere load annotation well number label often neutral feature feature frequent preference neutral uniform neutral prevent dominate neutral class term manual neutral take neutral work successfully way prevent desire unlabele take maximum principle predict x p x number label entropy empirical objective already distribution roughly labeling kl predict objective preference follow
reveal ensemble anneal inverse temperature heat peak indicate temperature dash critical swap rate schedule heat gray curve anneal literature possibility anneal near path histogram temperature anneal construction schedule energy histogram successive overlap schedule logarithmic grow phase construct schedule e approach temperature automatically avoid worker anneal decade algorithm find temperature schedule entropy production temperature root heat rule change proportional heat capacity entropy increment inverse temperature increment generate schedule annealing show proportional anneal computation put system anneal seed right panel exchange anneal drop close heat peak section multiple second order relative term vanish view definite define parameterized special temperature information therein switch state therefore optimal geodesic manifold equip metric present relative follow anneal successful relative entropy accumulate approximate resource ensemble canonical ensemble temperature temperature anneal intermediate ensemble multiple parameter accumulate discrete geodesic generally reliably entire connect generate geodesic dash hand curve energy peak dash true line major powerful simulate canonical ensemble algorithms bridge motivation due tail overlap exchange difference intermediate boltzmann confirm ise model correct canonical whereas correspond canonical however advantage reason proportional solve temperature canonical peak temperature temperature ensemble multiple run fashion control set ise peak around peak anneal systematic produce schedule phase separation less consequently boltzmann also energy difference large produce boltzmann anneal algorithmic nest inference bayesian normalization essentially inference nest view special annealing zero temperature relative entropy ensemble constant cumulative contour reduce nest anneal ensemble result ensemble implement build nest principle guide nest utilize truncate truncation fact therefore volume energy energy among result nest ensemble achieve nn next energy also anneal produce ise also estimate example run speed three order magnitude anneal anneal energy instantaneous nested ensemble anneal histogram contour histogram slowly protein computation nest anneal ise geometrically canonical compression rewrite anneal canonical ensemble fast anneal ensemble ise fig canonical sampling reason maximum nest many bridge anneal carlo step ensemble generate along also density manner maintain constant entropy variety canonical family close fact annealing aim implement compression reliable nest tie whereas anneal hybrid monte carlo canonical ensemble worker anneal rich find ensemble simulate difficult mean contribute ensemble ensemble anneal grant configuration also family protocol intermediate canonical configuration factor temperature obviously method parallel boltzmann ensemble prior function cutoff temperature ensemble configuration energy great nested configuration system potential plus state denote delta function energy choose intermediate bridge successive ensemble kullback equality kullback leibler divergence distance ensemble contrast distance broader contain support quantify member integral energy th relative energy article relative distance ensemble overlap ensemble might useful exchange ensemble jensen shannon control mainly ensemble article explore measure ensemble infinitely many annealing reach ensemble intermediate fix entropy amount speed ensemble need relative average difference problem outline normalization constant evaluate state integral reduce integral anneal histogram interact whereas anneal energy visit configuration entire simulation update ensemble relation histogram free energy histogram update partition start previous set state histogram estimate ensemble reliably configuration previously ensemble ensemble anneal iteration schedule infinitely slow annealing accord entropy lead schedule shift direction depend relative next ensemble impose close canonical anneal illustrative sufficient monte consist draw uniform relative configuration criterion temperature annealing localize accurately anneal decrease represent boltzmann could decrease number thereby resource explore apply anneal simulation lattice particle relative monte randomly lattice try flip spin initial start spin cover proceed continue energy range accurate produce accuracy
make implementation next short delay big sampling factor nature front regime stage subsample use front end guarantee frequency sparse bi successfully singleton high stage subsample front architecture brief reader regime use r front end period achieve operate less regime decrease function follow ok f front noiseless delay bin contrary additive need path role structure front architecture detail observation b w b refer bin measurement bin sequel vector generally observation vector bin stage architecture per per chain j bin sampling period delay circular shift r sub front sample dft singleton singleton signal singleton try bin somewhat similar compressed sense incoherence restrict isometry rip widely recovery sparse although sufficient incoherence rip bin measurement stable propose style q property measurement restrict positive rip characterize preserve capability matrix operate good vector since bound away point discussion reliability stable circular front mutual incoherence bin least easy offline incoherence unit mutual incoherence eq circular delay chain front column bin frequency circular shift front uniformly shift matrix make slow consist detect reliable circular shift front combination randomness structure enable consist delay delay circular shift shift measurement exploit structure intra singleton e end decoder recover big dft output front routine singleton singleton exploit bin bin determine dft connect explain operation focus bin square observation estimator observation estimate correspond potential bin estimate justify bin observation satisfactory appropriately choose bin cluster measurement mmse coefficient output boolean singleton singleton dft coefficient dft coefficient pt singleton false set I bt I process colored estimate thus estimate successively refine continuous slight abuse multiply two frequency length fold result increase h r architecture circular shift stage sample frequency red fig reconstruct fully frequency fig acquisition propose manner reconstruct brain image acquire reconstruct image fourier reconstruct differential vertical operation dft operation creates approximately reconstruct access fouri sample brain reconstruct chain stage architecture r differential brain image inversion fully frequency reconstruct brain operation divide dft center fourier non center total fouri reconstruct show column bin lemma circular shift term I support hoeffding inequality apply summation thus choice circular shift least consider circle eigen eigen absolute entry matrix equal provide absolute diagonal consist part first show r dft second decoder denote fail reconstruct dft bin fails correctly classify bin singleton bin bin identify dft coefficient event entire decode bin wrong decision put piece bin process entire perform noiseless front construction vector bin number iteration dft get reliability role event threshold arbitrary complex noise corrupt value cn please use bin cn bin bin multi bin bin possibility consider bin identify zero bin zero dft bin let bin bin zero inequality use singleton inequality carefully dft coefficient discussion bin let process classified bin bin cn bin bin dft compute compute dft binomial need end pre b l k c l constant r front end constant stage bin dft sample presence moreover decoder bin constant proposition bin process cn get pr pr corrupt periodic interval observation singleton bin proposition frequency estimator true singleton bin sample overall singleton assertion theorem conjecture edu fast fourier transform fourier dft arbitrary length dft signal question fast transform induce code chinese theorem exploit devise iterative compute dft computation applicable whenever attractive adapt corrupt particular compute dft particularly implementation feasible randomized measurement permit flexibility choose variant r present dft corrupt signal dft white like mr spectrum dft coefficient fast way compute dft arbitrary complexity signal signal fourier assumption noise spectrum dft compute length dft arithmetic computation million million relevant big gain highlight conceptual framework underlie paper adapt noiseless robustness modify detail front stage stage call delay chain identical shift shift shift precisely asymptotically delay shift random choice circular shift measurement good mutual incoherence property enable motivating show domain minimize fourier reconstruct image brain acquire mr fourier sample elaborate art technique promise direction demonstrate dft practice uniform typical however compute dft corrupt hardware desirable even sample process set dft noise variant present useful application flexibility randomize measurement system elaborate dft emphasize assume secondly sub noise rest provide signal overview literature key provide review result propose recovery generic description validate complex transform signal corrupt cn signal dft zero dft remark dft signal ratio front dft operation linear sample flexibility dft computation r algorithm dft property computational recover dft please dft perfectly corrupted assume reconstruction dft long zero dft coefficient arbitrary value dft applicability complex dft successful recover perfectly also successfully transform rich processing compressive estimation study decomposition music approach tool theory theory many issue manner sense literature random pursuit standard isometry rip characterize matrix unitary sparse like exhibit rip scale practice dft good knowledge characterization rip consist dft sub scale contrast bad fourier sampling innovation level though key difference dft indeed compressive sensing inspire work compute dft high sub require bound noise dft compute dft applicable signal sub regime domain process end sub front architecture decoder later front signal dft furth dft length input form sub front decrease path period delay stage input signal delay chain shift illustrative processing corrupt stage sampling stage furth signal delay output front end far grouping bin decoder big dft short bin bin obtain node bi dft node bin sequel vector edge connect dft contribute bin e bin e observation dft identity let bin bi dft coefficient represent bin connect dft coefficient contribute bin sub dft contribute bin stage bin stage computing dft transform decode support bi I decode bi bin contribution dft signal dft coefficient noise vector contribution non dft bin stage verify stage non dft g bin corrupt observation sample front bin stage height estimate dft denote decoder threshold choose appendix stage bin singleton singleton v multi r decoder function singleton exploit high addition determine dft select right graph remove neighboring contribution check decode successful remove decoder successful decode coefficient decoder bipartite sub
study sophisticated estimation fast pdf common pdf belong poisson parameter advantage often back environmental estimate nice actual lie occur potentially priori pdf estimation option maximize grow add penalty unique maxima compute typically lose combination scale kernel fall decade choose appropriate process additionally slow parametric square approach search discuss wherein maximum likelihood estimator lipschitz pdfs property efficiently computable form present ml computable pdf pdf band bl bl think however propose preserve property estimator consistency efficiency bl infinite pdfs case test grid outperform bl ij x j support si c solution locate local value global maximum maxima however exhaustive entail solution compare programming theory additional knowledge know bl strictly positive estimate theorem test remarkable test prove construct plug result divergence si consider fourier transform frequency follow cutoff data j multidimensional dimensional result si next step select solves computationally state si lie bl bl strictly estimator simplicity computational kde estimate equation solve bl quick briefly bins pmf pmf bandwidth pmf make true pdf small small bin bin reduce f lx c lie np toolbox improve nearby hamming equal neighbor nearby computationally surrogate pdfs several surrogate pdfs bl panels use surrogate strictly pdfs panel plot size pdfs theory whereas marginally computationally expensive remainder pdf strictly strictly pdf cut assume calculate compare estimator fast kde nd order kde plot adaptive bl non bl pdf respectively bl band pdf bl respectively alternate hz position spike sort accomplish manual mit care activity grid cell peak fire grid spike histogram position generate spike dot trajectory inside blue neuron factor x x allow nonparametric spike estimate kde smoother capture sharp spike kde cut frequency combination frequency fit low ks circular glm spike fit rescale kolmogorov ks compute cdf estimate quantify ks plot close ks statistic ks estimator figure estimate ks remain inside ks ks glm glm glm nd activity glm structure glm certainly neuron covariate glm glm neurons know kde show mark co position spike ks kde nd glm glm bl develop three quick presented generate estimate strictly remarkably estimator even parametric apply mechanic development si wave mechanic absolute density mechanic wave momentum wave wave position bl versa occurrence single observe think experiment box wave finite momentum wave bl pdfs bl macro phenomenon bl double set bl bl convolution pdfs bl phenomenon level phenomenon bl macro phenomenon bl pdf macro observe pdfs process almost bl cutoff lie finite impossible distinguish bl pdf logarithm exist method select estimate fit ml pdf band converge pdf band increase frequency cut frequency infinity e sophisticated likelihood cut frequency figure n ij x increase infer complete analysis leave present approach numerical technique incorporate idea study prove estimate clear normal study normality first
location useful model could applied stock transformation model median improve modelling log transform residual allow require model raw stock comparison enable come recommendation assess course assess question interest ask availability system available cost comparison several criterion know need fitting applying may add required model validation good know p yield model performing regard specific comparing since context study carry advanced approach covariate krige consistently improve increase one involve multiple tool study predictor turn bring additional instance rank stock drive factor recommendation recommendation france diversity make recommendation information contain relationship stock flexible prove map national care dataset model national systematic scheme check spatial autocorrelation residual simple fail national residual provide accurate stock prediction highlight thereby guide research model acknowledgement analyse scientific environment management institute research institute european author thank involve address technical bank handle publication result correction mechanism reflect document publication volume page play major global source improve stock national monitor study recent first consider several increasingly tree convenient multiple perform network procedure limited predictor prediction significantly improve modelling prediction adequate care allow contain behaviour source increase temporal pool distribution international reliability suitable content density comprehensive may parameter dynamic interestingly regard use validity scale precision extent mapping rule relate stock use model adapt tree regression study study extent km predictor extent study approach national mapping france decade specific stock location reach stock unbiased bias ensure stock national potentially room especially way improve recently relate environmental method design outlier nevertheless currently qualitative variable nonlinear automate share robustness problem autocorrelation stock consider aim stock france quality network use model useful modelling advantage stock france france stock site monitor base km locate grid cell possible site km center site site locate locate position cell site individual take cm within individual composite composite south measurement stock horizon layer density mass account reach stock variable variable depict available mapping national scale site management biological forest forest median ph per national testing estimate database adapt content european european link possibly type surface unit rank also include instance occur available application production combine matter concentration month potential month temperature km average observational variable estimate site spatial join grid map primary site content possible site correspond lastly moderate resolution image primary get site development mapping mapping tool software stock present area south west part belong model onto apply base learner final combination weight learner boost algorithm algorithm stagewise base learner specialized regression gradient boost algorithm rely base produce draw replacement besides determine base minimum terminal available stop internal avoiding thank stochastic aim minimize risk overfitte predictive handle linear among predictor qualitative variable assess contribution predictor partial assess predictor thorough guide use fit predict stock refer model represent lot know site stock additionally content predict complex occurrence site month mm month addition matter leave value recommendation investigate represent et different class spatially three method stock contain covariate observation transform due skewness stock vector residual prediction transform transform response assume spatial model observation identify extreme reduce effect spurious exclude predict close observation confirm stock dataset represent log valid left inequality observe exceed possible measurement depend verify check leave validation valid property square standardized error note cross median procedure validity aim estimate spatial counterpart variation model model spatially variation effect model use fit due anomaly ordinary krige consist ordinary krige transforming predict stock prediction lagrange multiplier krige variance lagrange multiplier unbiased ordinary krige six refer counterpart validate cross procedure involve stock commonly suggest mean root square error coefficient determination strength value distance inter whole well model error root calculate hereafter name provide picture skew use monte fold enable external prefer preferred leave dataset model dataset krige fit krige step spatial spatial validate present section check perform validation procedure provide one prediction counterpart indicate performance metric external model validation fitting krige distinguished krige fitting fitting result distribution performance use adjustment name repetition validation sp residual result dependence residual dependence indicate simple variability deterministic spatial model value obtain site stock location extreme appear evenly distribute yield valid spatial validation repetition f spatial counterpart model express result improve bias important appear root square error fig skewed prediction median bias skewness result compare median spatial resulted improvement significant variance spatial express km smoothness three plot horizontal bar cross repetition line diagram f map instance give minu indicate size dot absolute error c whole dataset improvement model add term reveal south west areas central area area bias part west france site error area prediction prediction strongly note strongly model model yield significant degradation spatial south west improvement limit site improve difference indicate spatial model yield spatial ht residual decrease control factor content include range lie km give correlation controlling factor stock look km content decrease residual around km handle ph parent material however include control spatial km many functional spatially component another explanation high derive difficult draw conclusion study deal density stock stock might great estimate accounting associate error wrong bias
distance shorter hold shorter predict standard great hold shorter long supplementary iv vi qualitatively remain predictor supplement predictor run finding confirm validity find ii offer parsimonious science major power component find iii novel record reason suppose governed law empirical correctness extent individual determine explanation broken power record law record explain component distinct individual hold record unchanged gender age number summary vi universal three predictive validation imply finding number law iv second describe great negative third middle distance vs separate fall cluster middle distance provide capture social attempt influence statement cause conjecture datum clinical leverage uk predictor middle improvement rank version low middle maximum duration check verify level oppose relative predict world km unlikely provide complexity performance long short may implication finding length primarily vs conjecture methodology difference whether capability age cross standard decrease shift et al possible longitudinal bias older amenable quantitative validation number attempt notably cluster long resp shorter strong high example half good performance yet produce even also quantification em plan especially event ability accurately difference achieve drop prediction summary description immediate assessment plan potential scoring population prediction precise fair run resp capable conjecture validate power collective acknowledgment provide code norm use thank regard implementation local high rank helpful comment early manuscript computation carry fellowship author prediction acquire jointly methodology working algorithm rank concrete performance jointly analyse carry jointly paper responsible raw process matlab upon request analysis obtain link www achieve event event country event obtain database automate www ten track road track road exclude reason attempt country datum consist table field birth contain record attempt event field date second set available request database error removal record range old see gender database record date birth miss due record eight record road road half record record record leave recorded attempt set preprocessing table index event contain index column index entry date store mode yield ff proceed find percentile year good event period miss mode table ff event table select mode ensure close fitness performance fitness high attempt per influence chance use depend fitness narrow specific summary attempt performance wise event percentile miss preferred geometric distance preferred corresponding event mostly train characteristic depend reference affect removal matrix obtain high score sub gender certain amount per sub text table ff row gender age well event discard refer retain score take measure squared rmse mae rmse sub sample obtain validation entry entry scenario question performance event predict predict causality preserve lie predict third iii make extensive task due bias old attempt many recent attempt argue absence technical ii iii performance matter opinion contrary influence history due occur scientific statistical viewpoint outcome present pass variant column column complete natural logarithm second complete event column indicate mae order base measure unless otherwise learn experiment level repetition intersect summary good relative rmse report fair describe fair bar event event fair short event typically certain distance case whereby predict lie curve vs distance fair repeat sampling point standard analysis series experiment term mae result mae qualitatively similar measure prediction rmse mae log time c predictor high great middle unit event rank perform predict w predict event performance prediction prediction run fast model proxy real singular recover appropriate group singular unchanged prefer optimal notable prefer shorter old prefer transition right panel distance anti correlation distance short three positively long mae table report method mae compare rmse mae indicate presence qualitatively report mae bias towards long qualitatively log mathematically rmse mae rmse mae accuracy rank predict time top year improvement rank tend short distance accordance iv indicate individual good descriptor stability learn predict learn influence prediction log learn calibration goodness time good error bootstrap significantly prediction time normalize overall prediction preferable stability effect predictor year compute close point formula show predict distance predict year stability improve power predictor five choose point main prediction experiment display table different residual case incorporation lead aggregation sign demonstrate event may datum formula performance affect attempt predict performance make table report year performance random prediction qualitatively use prediction time comparison matrix completion single generate describe repeat nuclear pre cross display figure nuclear matrix available attempt rank ii synthetic assumption generative synthetic performance synthetic datum number three summary independently gaussian summary estimate component top event model distance miss uniformly miss per entry non nuclear norm repeat result display figure robust norm affected rmse approach minimization compare plausible model pattern identical top plausible entry singular svd real datum confidence display component missing entry rank identical observe component recover almost exactly slightly sub methodology component old year percentile range consider respective display estimate law unchanged component reduction explain consider display compare iii summary prefer summary display individual exponent score score correlation correlation approximation prefer number vs vs vs score optimal distance predictor performance distance vs top event compute achieve determine percentile refer prefer attempt event shorter old long explain phenomenon train shorter regardless old c iv phase transition closely phenomenon discuss consider set sequentially predict predict km triple consecutive distance exclude study perform triple rank performance distance way compare correspond red term distance perturb perturb calculate perturb prediction distance short km slow predict predictor find distance great km shorter long middle axis equivalent performance triple line middle decrease increase keep evaluation event mean individual law power nuclear em log time log cccc cccc cc nuclear em speed normalize log good cccc cccc cc event data nn individual nuclear log well good cccc k nuclear mean error cccc cccc c event type individual cccc rank leave equally cc cccc law law corollary conjecture scientific level individual performance low dominate individual law evaluate individual quantitative scheme break law basis prediction contribution modeling implication focus collective uk individual law explain law record individual training performance prediction accurate date minute mae table performance leverage insight record improve human prediction briefly law model performance dependence know describe extensively run law practitioner formula fix performance performance parsimonious international association distance comparable present performance performance may score forecasting performance science speed max direct prediction clinical appeal none prediction parsimonious power accurate scoring interpretable may one clinical measurement interpretable usually predict explain present desirable b avoid parsimonious empirically database uk yield descriptor basis prediction explanatory database step parsimonious performance discover individual explain range summary relate number great interpret exponent hold remarkably average describe linear correction law correction record three allow assessment completion record distance distinct approximate law power display performance world record exact straight align closely straight line notable record straight line break world individual variation optimally break law record individual explain variation scenario base top green predict green blue account predict green blue account event red distance blue green whose remain supplementary phenomenon present throughout quantitative simplest display panel predict green pattern performance exactly red event explanation green mathematically demand sum red blue I mathematically blue red green blue vanish equation multiple pattern triple average minimize model optimal prediction scheme instance apply performance event completion matrix predict behaviour recommender system predict cope finding supplement see appendix detail performance parsimonious explain consideration translate power demand depend since coefficient remarkably find due miss data analysis analyse online www contain removal individual range old comprise km km contain th percentile performance main analysis supplementary request full link acceptance manuscript state practice validate subset square absolute mae performance correct merely report data validation setup supplementary finding uk accuracy evaluate include include near representative art quantitative formula exponent power predictor exponent exponent point completion expectation maximization b minimization completion method performance event performance
variance ni use simple fact assumption denote asymptotic eigenvector motivate eigenvector natural insight normality regime relationship understand notation denote entry generic investigate behavior denote th normality distribution theorem strong weak asymptotically unbiased estimate conclusion equal eigenvalue matrix reveal control divide part element jk furthermore normality eigenvector noise radius addition jk j k factor dimensional small covariance start fact eigenvector spike understand important high remark datum note create distribute project eigenvector detail factor secondly strong generalize distribution part invariant general regime component compare eigenvalue limit eigenvalue normality except bias cause high estimation motivate sparse insight show eigenvector consistently estimate eigenvector name shrinkage principal thresholding time result management portfolio false discovery proportion finance observe stock e expression latent factor loading uncorrelated index repeat simplicity respectively load matrix condition covariance away covariance exhibit particular nonzero element decompose matrix spike separately correspond equivalent entry threshold thresholde determined section measure relative error device weak meaningful scale discussion drawback empirical inconsistent significantly dominate non part see recent eigenvalue drawback assumption p addition bound specific impose avoid sophisticated condition need drawback shrinkage inspire shrinkage first simply thresholding constant three common factor loading ready abuse svd decomposition obviously thus separately notice error eigenvalue reflect control estimating rate apply relative norm norm different incoherence space come care error space separately eigenvector correct dominate dimensional setting see study shrinkage estimator reason recommend practice subsection assume factor achievable perform eigenvalue risk management finance volatility risk portfolio allocation covariance underlie portfolio although curse dimensionality basic bound risk early similar exposure portfolio mathematically make error large mostly drive regardless converge portfolio make management model limit exposure position application nd st rd nd rd st nd eigenvector rd short match theoretical asymptotic effectiveness I eigenvalue much generate model simulated multivariate row standard multivariate matter shrinkage necessary maintain comparable even decrease however increase order result spike serve benchmark compare get b right degree indicate accuracy scatter plot true basically meaning signal perform benchmark proof asymptotic weighted wishart variance classical hold bound weight omit treat lemma column assumption independent note row normalize factor therefore independent easily spike pn contribute limiting eigenvalue b bn sub therefore involve I idea technical invariance prove normality q sub identity eigenvalue eigenvector j b n c leave multiply equation employ define derive right show element complete proof r mn b r distribute unit orthogonal remain r j easy since r c p conclude r distribute distribute derivation prove norm need random sphere pc e n pn pc could notice inequality therefore claim prove hence indeed say proof lemma second fact pc n pn pc bound asymptotically n suffice vector element characteristic expansion easily hand side mean clearly pn element must lie element empirical eigenvector p establish list e u mt tc u jt introduce mix let coefficient independence u error lemma sequence pa adaptive estimator apply get p assumption together fact pc pc lemma claim pt tp one last rate max q simplify schwarz ii theorem ready build prove write nc eigenvector section imply right iii ii p pp already thus term pc q denominator bound risk rate come follow eigenvalue corollary assumption support nsf grant dms dms grant gm gm wang eigenvectors unify spike dimensionality play principal analysis device low set new insight reveal bias eigenvalue covariance shrinkage principal estimation risk discovery proportion statistic study principal component factor relative management widely reduction visualization eigenvalue regime substantial amount effort understand empirical eigen structure early establish normality sample substantial eigen recent behavior relate development weak asymptotic bound grow factor grow dimensionality consistently question asymptotic eigenvalue question arise convergence empirical eigenvalue dimensional eigenvalue around theoretical counterpart covariance quantity role determine asymptotic eigen eigenvalue theoretical pca three angle contribute whereas serve low component pursuit incoherence rate sparse assume eigenvalue correspondingly reduce possibility accumulation effort spike size covariance almost scale lie scale threshold regular investigate part eigenvector corresponds normally scale random regime study principal literature regime allow spike size lead lead perspective regime bound eigenvalue bound ratio offset assume signal factor
image represent category instance image training generate category label mahalanobis mahalanobis entire dataset select fold training run use different initialization select log approach perform cholesky frobenius log covariance treat element space triangular cholesky bregman affine distance make k highly distance riemannian geodesic cluster log frobenius learn original cholesky decomposition compare logarithm domain riemannian metric geodesic distance show geodesic learn mahalanobis face supervised categorization clearly learn geodesic well c cholesky decomposition frobenius gain gain frobenius reference x n riemannian riemannian manifold f descriptor detection dictionary v log calculus diffusion tensor learn label face recognition environment analyze contour object categorization similarity search divergence wang invariant tensor value segmentation covariance pt pt considerable propose compare matrix affine invariant induce riemannian focus riemannian geometry propose drive approach riemannian metric distance learn face denote product pt pt frobenius triangular cholesky exp logarithm vision involve underlie metric feature shape rotation matrix etc lie riemannian need develop inference technique structure manifold considerable vision diffusion tensor medical imaging diffusion tensor water tensor characterize optical motion often employ encode descriptor texture track recognition correlate feature represent cross sample filter feature histogram covariance rotation invariance different various measure literature comparison log log euclidean reason metric euclidean riemannian euclidean riemannian ns equip metric space geodesic distance remarkable community learn log riemannian metric correspond geodesic distance distance like near neighbor distance explore idea metric geodesic mahalanobis brief overview literature technique euclidean riemannian log geodesic distance experimental conclude section distance statistical consideration linearity riemannian property frobenius one information similarity dissimilarity constraint let point point point aim learn mahalanobis parametrize similar pair give threshold mahalanobi denote index component similarity dissimilarity capture control bregman publicly result n ns right geodesic distance map identity equal usual space denote vector form operation inner uniquely characterize inner product give define unique product simply extend lie multiplication euclidean geodesic give directly say mahalanobis vector geodesic correspond riemannian uniquely mahalanobis hence riemannian metric geodesic datum mahalanobis geodesic mahalanobis learn cm mahalanobis mahalanobis section propose riemannian application face label face ii dataset experiment pair image correspond person dataset design face dataset face development pair pair test image pair image person consist image pair pair randomly development subset final image subset convert coordinate represent standard experimental protocol development set pair perform split train split parameter face matching learn compare performance
flow q round need polynomial state introduction equilibrium game function canonical game thick color text blue right bend bend bend node edge bend bend right let exclude equilibrium consider equilibrium hard player edge experience half player total edge plus see observe matter player path edge plus edge plus go whereas go every total exactly thus since eq lagrangian play define game zero sum approximate equilibrium follow fix maximization game good response game write equality approximately hence find minimax equilibrium lagrangian pair strategy action p equilibrium concave know argument use round regret select particular good response round average approximate equilibrium equilibrium lagrangian game induced gradient observation well game lagrangian description algorithm present descent close convex gradient regret dynamic lagrangian gradient bound q let average approximate equilibrium plug approximate base guarantee plug recover call therefore observation q plug constant extend version theorem theorem edu play play payoff game capture choose maximize goal price production know utility price bundle efficiently computation complexity utility reveal preference access maximization natural choose utility observe bundle would like price road unknown determine minimize round quite share important feature choose function understand unknown maximize minimized concave unfortunately pose pose maximize resp minimize concave resp maximization instance non concave price price unit unfortunately concave generic high dimension efficiently maximize also convexity generally play utility objective minimize traditionally game assume know utility utility several natural equilibrium efficiently preference function clarity detail maximization reveal preference optimally class unknown apply general include mention version technical reveal preference simply game rd problem main challenge class many write price setting face price bundle induce concave continue arbitrary cost meaningful well family thus bandit maximize unfortunately get bundle bundle set bundle price reduce simple maximize price suffice give access bundle set next ingredient efficiently find approximately function strongly feasible specifically bundle price query precede induce strongly convex induce target interest subproblem procedure flow detailed optimize follow strongly action objective concave objective action finally third demonstrate tolerance simple production stochastic procedure similar variable solution variable survey substantially certain game np ignore computational efficiency assume knowledge utility game learn query pure continuous result problem learn security learn optimal strategy polynomial game pure strategy algorithm np neither despite give polynomial maximization recent pricing reveal special quite extend game also work find atomic function ellipsoid form degree induce target game game increase ellipsoid subroutine approximately induce induce flow exactly strictly comparable motivation recent line design game direct denote key ingredient ability minimize concave function use descent radius projection onto subgradient descent start algorithm descent q alternatively within essential every strongly extremely let concave xx useful noisy say df ff noisy optimize use specified membership concave maximize reveal problem want bundle assume allow price good bundle price maximize price bundle maximize utility period choose price induce bundle would design nearly mild convex closed contain unit least empty bundle lastly cost differentiable decrease important technical assumption satisfied class concave bundle concave price bundle iteratively price price price allow simulate query use query feedback iteratively quickly bundle carry demonstrate function bundle bundle might induce set induce bundle observe vector price characterize feasible vector bundle maximize vx since maximizer thus concave know ascent direction contradict desire function close vx cx bundle concave problem meaningful pose rx vx pose reveal preference without specify bundle sensible often formalize homogeneous homogeneous unit scale preferred bundle concavity differentiable rx vx prove claim invoke euler euler homogeneous continuous homogeneous conclude interest strongly differentiable concavity follow induce bundle price vector bundle actually x p concavity actual bundle close quantitative lipschitz norm vx cx x vx vx bundle accuracy initialize restrict update descent bundle analyze define whose bundle solution vx constraint price price induce bundle subgradient restrict price unchanged even primal program primal compact let strategy function guarantee later restrict play minimax strategy state sum minimax equilibrium observe fix response write eq equality choice pair minimax induce reduce equilibrium lagrangian game pair equilibrium induce satisfie definition fix concave function play descent good dynamic define action player average play equilibrium simulate dynamic observe induced gradient lagrangian compute recall lagrangian sum mean end average description restrict update action regret lagrangian average form equilibrium plug induce action ready utility write must play allow utility p quite often achieve reveal preference operate interior trivially satisfied whenever induce game application evaluation function value induce action approximation guarantee find optimizer iteration present initialize td fx tx px observation q approximate satisfie end guarantee plug guarantee require total value bind constant introduction find induce approximately flow atomic unknown graph represent specify interested agent infinitely agent aggregate decision induce flow ff flow equilibrium game lemma state whenever decrease associated equilibrium compute whenever call social equilibrium power edge induce flow approximately game player flow player l tool problem begin assumption function match induce flow implement cost flow induce flow potential flow condition guarantee potential variable imply assumption implement flow require sufficient guarantee vector flow fix game satisfy flow need without last solve induce use form additive manner edge ellipsoid achieve rate induce processing step maximal polytope span transform body round transform need behavior exactly run maximization contract produce produce quality stochastically map work effort agent dimensional problem agent dimension effort agent know stochastically map abstract away effort loss generality contribution strongly produce contribution contract however stochastically principal want optimize response realize agent optimize minus agent contract attempt contribute utility utility ap cx expect principal contribution realize price utility agent utility merely version version adapt assumption feasible contribution hypercube attempt contribute unit dimension nothing lastly agent learn approximately observe hold generality principal lagrangian dual induce contribution satisfie optimize contribution perturbation response subgradient price principal observe realize contribution unbiased subgradient gradient realize contract price descent satisfie target price realize agent theorem w consider induced satisfie agent contribution zero agent form approximate minimax equilibrium induce contribution simulate regret run gradient realize price take player dynamic recall form descent contribution need
multiplication multiplication require multiplicative reach theoretical complexity transform layer layer cope column combine order addition multiplication finally one multiplication besides two multiplication post multiplication minimum multiplication multiplication another finite combination depend field transform order second pre addition layer cope respectively combine column layer pre multiplication q multiplication multiplication multiplication multiplication column pre fast algorithm fast split example dft multiplicative regard roughly hadamard decomposition short low popular ft attractive implement high dedicated acknowledgment support proposition tag cr tag email de mail de finite transform code interesting field inverse promise transform concern digital access system spread transform existence transform ft transform operation short decomposition hadamard decomposition discrete hadamard transform multiplicative dft implement scheme processor high speed dedicated hardware compute transform gaussian comment minimal multiplicative ft additive implementation plus introduce pair assume computed observe combine hadamard reduce first column make q therefore procedure addition q multiplication let
correlate several numerical approximation still expectation propagation unstable specific reason guarantee need principle convex set objective multivariate truncate integration truncation inequality provide absolutely continuous respect entropy vb disadvantage vb general come disadvantage include base convenient subspace vb bind original integral idea able truncate denote negative energy univariate normal truncate zero variational q bind could iteratively solve round find parameter c vb ep vs ep vs property integration integration dimension consider ground pseudo main validity various truncate provide integral vb minimize bfgs matlab precision draw varied vary compare accuracy moment moment compute euclidean ep give value integral consistently seem interesting vb correlation unable use correlation family compose truncate moment give result also become respect vb ep correlation handle generalization old inequality binary recover method minimize multiple variational approximation minimization great practical advantage vb practitioner integration could express special unique approximation maintain heavy tail far behave posterior optimize variety speedup vb ep decade focus type integral common problem glm prior potentially discrete design dedicated graphical upper know old conjecture express approach span symbol measurable scalar assume inequality expand right pair old exponent result prove equation lead symmetry proof proposition q college old vb involve maximization respect minimization bind problem literature integrate ep art integration many involve integral approximation technique explore involve likelihood criterion slow require empirical boltzmann simple ml estimator function partition soon model include distribution graphical non need set design decade including performance field approximation algorithm provable guarantee guarantee hard one variational expectation ep reweighte classical scheme base basically information apply bind tend variance example interestingly inequality suffer avoid vb lack guarantee connection introduce old tractable possibly previous work focus product potential parameter tool compare ep contribution show well approximation highlight effectively bayes variational properly variational unconstrained minimization optimize find intractable computed fashion approximate amount seminal minimized exponent discrete far bind continuous make old illustrate previous define univariate lebesgue assume want common k regression sparse univariate number observation integrate remain
assumption improvement burden iterative framework report concern screen efficiency screening method projection new assumption focus computation concern design compute article lie aspect screen consistency unified method strong insight assess screening method relate sufficient sign another arbitrarily flexibility hold even ic carefully choose equivalence screen consistency comment relationship commonly illustrate study screening measure signal evaluate compare design assume ic article follow basic screening provide condition relationship condition design probability task learn coefficient impose portion coordinate split phase recover phase point dimensionality regularization raise computationally suggest find eq define hope comparable step usually involve screening exist comprehensive put definition article estimator strong eq definition much usual study weak see relaxation sign reduce screen choice article property take sure screening projection screening computationally theoretically ordinary least square estimator although sufficient consistent fix define contain large coordinate terminology help establish theory dominant symmetric restrict dominant notice dominant screening noiseless consistent dominant dominant hold prove sign notice necessity material noiseless good point intuitively preserve coefficient need dominant diagonal dominate combination theorem need change accommodate certain estimator screen tailor term addition current tight condition necessary consistency rule example satisfy sure sis condition ic standardize zero letting give ic represent matrix ic verify screening matrix dominant explicit ic illustrate dominant satisfie ic restrict dominant demonstrate ic however requirement ic impose make violate predictor contrast ic impose flexibility equivalently ic satisfie correlate weak sis screening matrix make ic follow dominant ic screen consistency sis imply illustrate necessary guarantee lasso avoid give advantage computational screening consistency commonly screen common high strong sub row contrast next estimator satisfy row tail screen random broad distribution elliptical focus gaussian screening essential magnitude chi square states matrix sis screen sis asymptotically screen lemma indicate necessary usually condition inspire sis I sis dominant plug solve inequality notice correlation rely heavily material sketch leave material define angular central bound projection decompose choose central gaussian conditional te te desire distribution te h screening illustrate lemma diagonal satisfy exist provide supplementary material combine assume satisfie restrict imply c union matrix entry satisfied provide precise condition observation expression suggest screening evaluate closely screen question study establish necessary condition verify design relationship ic sis arbitrarily predictor see compressed technique marginal selection proof theorem section dominant similarly consistency coefficient notice screen fix screening sign j argument choice notice consistency proof prove cover complete proof row except ic without assume become sign sign either q sign check value first argument restrict dominant choose q sis proof part provide degree q eq iid proposition diagonal prove lemma dominant plug solve notice section section proposition need establish result reader reference orthogonal manifold manifold mathematically call haar orthogonal left probability suppose positive definite orientation angular right become angular unit sphere let invariant decompose distribute property entry probability essentially small eigenvalue eq diagonal belong quantity interested coordinate divide second part take care diagonal q distribute distribute begin evaluate part vector denote although still due decomposition term decompose haar point possess identity determinant easy result definition simplify angular relate quantity
leverage theoretical connection field different toy datum follow imply express derivative gaussian combination quickly acquire potentially locate acquisition new datum e tuple toy region within ten correct approximate evidence region improve full quickly perform intractable simulation several magnitude toy optimization two major difficulty free discrepancy parameter small former give discrepancy region science generating likelihood form inference infer yield datum difficulty choice discrepancy difficulty discrepancy tackle difficulty play role statistic joint see interest realization stage sample implicitly define via datum compute computationally well possibility simulate likelihood indirect economic bayesian process aforementione highlighted measurement simulate discriminate similar generate parameter solve free inference identify chance sample normal curve simulate mean green density discriminant lda yield green dash curve example curve drop curve
relu respectively layer layer c acc acc layers acc acc ar c present show representation learn measure component ar show high compare layer ar contract layer due reason activation hide compute set ar acc acc lc acc acc lc lc extend select image category ar select change select per training rest six setting unit change maintain unit system improve utilize abstraction table classification accuracy improve deep good image construct stacking module module low layer representation spatial pyramid classification experimental public database partially support national science distinguish chinese education cb fundamental research central program team sparse code classification however computationally expensive though signal discriminative dictionary sparse simplified module avoid inference module module module stack stacking evaluate four database extend ar outperform reach practical learn code promising digit sparse code sparse sparse expensive researcher use signal learn expensive algorithm train dictionary avoid expensive fortunately simplify module train dictionary fast linearly map layer sigmoid function map ability output label infer calculate multiplication stack module far stack stack network increase speech classification retrieval additionally batch offer potential problem amount despite classification limitation conventional sigmoid nonlinear layer literature slow solution suffer sigmoid recent relu train play key role image noise high representation lead promise image classification evidence reasonable representation module technique generally zero unit unit dependency unit unit observe module connection dependency divide group capture local dependency among hidden dependency module exploit stack image stacking module relu regularization hide modular advantage code dictionary lead compare extract retain ar experiment get particular scene originally yu deep stack layer module perceptron mathematically describe follow I ji ci ji di connect layer connect hide form eq square derive gradient module element one convex basic module stack deep low feature module replace hide representation bilinear stacking paradigm module expand module modular different sigmoid relu sparse penalty add unit upper hidden follow activation relu activation unit group g objective upper hide unit impose sparse norm enforce sparsity representation neural advantageous representation dependency dependency divide within force hidden unit group norm regularization conduct modular architecture matrix belong solve layer fix gradient square objective wise division relu activation non optimization unit tb parameter epoch initialize random repeat fast find deterministic plug square e simplify derive gradient eq define process outline tb describe layer architecture module output label decompose three module square generate module module iterate construct summarize implement hide unit activation module advantage parallelism four database extend database scene database face original image normalize pixel ar database database color people person take image variation include illumination standard
b conversely prefer commonly kronecker kernel anti use symmetric learn preference successful theoretical learning eigenvalue eigenfunction obtain learn depend introduction several recent therein determination another tool analysis universal intuitively enforce experimental previous thus far rigorous theoretical enforce property literature anti pairwise kernel kronecker universal result approximate arbitrarily anti function anti expressive concern provide guarantee anti pairwise symmetric anti regularization symmetric anti symmetric set feature literature conversely write type mapping simplify consideration couple input bound function generate write equivalence unique known kernel kx kk reproduce mapping often define rkh hand change norm composition diagram hilbert schmidt indicate require consideration operator suppose compact adjoint operator consist notation advantage eigen basis negative eigenfunction eigenfunction yield self adjoint eigen system adjoint next concept cone monotonically hilbert infinite space doubly doubly denote banach operator operation operator hilbert hilbert hilbert iff exist doubly kernel write space kernel type inner product joint pair define pairwise immediate forward permutation permutation invariant invariant equal permutation give projection define know kernel anti symmetric kernel symmetric pairwise q analogously anti anti projection connection symmetric anti permutation moreover anti form projection invariant obtain definition express sum anti form invariant anti integral operator arbitrary anti integral permutation q projection anti look consider divide preference projection eigenvalue function zero anti symmetric risk consideration eq anti error counterpart hypothesis anti whether restriction aim anti discrepancy literature split error part cause bias cause draw input caused briefly consider follow subsection discuss roughly kernel value eq show eigenvalue operator prove know infinite length non segment end dimension inequality straightforwardly draw affect effective limited error guarantee enough approximate may universal regression anti restrict design equivalence omit due lack formalize concept definition space close universal rkhs universal approximate property rkh universal rkhs real accordingly kernel arbitrarily rkh armed definition next characterize anti kernel arbitrary q anti symmetric determined section theorem generalization anti kronecker anti analyze pairwise rank correspond interpret anti second thus formalize rkh approximate see completeness somewhat square eq k proof regularization symmetric anti symmetric knowledge regression regularization may type symmetric decrease bias bias v anti cause kernel moreover cause operator anti observe eq integral anti symmetric integral analogously mapping stack operator check kernel reproduce kernel pi k p together rv rv pairs function belong rkhs prove claim anti kernel start form operator bias product function inequality due regularization anti
ascent dual separate last analyze analysis minimize nonsmooth obtain free asynchronous parallel sg describe sublinear asynchronous successfully server deep randomly training sample index k star shape master serve master exchange master simultaneously basically step select master compute master master basically aggregate master source collect predefine master perform atomic network especially server asynchronous sg serial parallel sg update compute instead serial sg asynchronous parallel overhead delay value gradient vanish asymptotically section parallel implementation parameter master example value denote delay evaluate th th asynchronous sg shape summarize short read mean inconsistent read worth star structure asynchronous cyclic delayed architecture delay architecture independence delay worker might important pointed asynchronous implementation old intuitively age idea assume commonly asynchronous note roughly worker q ergodic convergence iterate take optimization nonsmooth ergodic totally think roughly theorem properly assume ergodic corollary basically stochastic gradient worker speedup serial sg optimization sg nonconvex observation long roughly compare serial sg speedup achievable compare analysis consistent result upper worker ensure speedup consider widely asynchronous cut completion always involve machine platform sharing randomly select index randomly platform exactly software asynchronous implementation sg preferred basically consider implementation share worker read modify simultaneously read read share compute training stochastic share worker share cause inconsistent read read share memory share memory eq close look properly hold iteration exceed consistent result argue numerator count factor comparison rate essentially read big difference result sg consider analysis consider inconsistent assume impractical consistent read absolutely speedup achievable maximal worker speedup long worker result strictly gradient k convergence sparsity reader validate property computer following validate speedup interested speedup speedup speedup exactly speedup count level achieve hardware running time speedup speedup affect hardware generally bad speedup deep evaluate package convolution max find website network mnist dataset cifar initialize server worker server worker handle gradient worker core default tune serial sg use chosen default well draw report table observe iteration speedup sense problem table speedup cifar drop bandwidth parameter full require communication threshold dramatically htp l speedup speedup image minibatch conv fc cifar full x rgb intel core datum totally total number gb generate core mini batch choose choose base sg draw speedup report speedup computer delay usually htp machine leave c c speedup study popular asynchronous system sublinear prove consistent result achievable improve early share proof proof assumption expectation q unbiased stochastic next equality inequality next use equality last inequality side substitute full optimization complete globally apply theorem equality complete theorem q expectation side due take take upper come complete lower bind far relax show satisfie bind acquire substitute thm counter thm counter thm counter counter com asynchronous implementation stochastic broadly neural practice explain speedup mainly asynchronous mechanism gap provide support implementation sg memory establish prove speedup achievable generalize asynchronous learn asynchronous parallelism largely overhead parallelism asynchronous parallelism worker synchronization asynchronous parallelism speedup many art stochastic coordinate ascent randomize asynchronous parallel optimization mainly research effort speedup people exist explain excellent practice due deep asynchronous mechanism people nonconvex use widely share memory system fill paper try first nonconvex smooth necessarily specification consider asynchronous originally memory architecture diversity key computer naturally efficiently read share system unable usually ensure read write coordinate implementation sg gradient consistent inconsistent asynchronous share memory platform theoretical establish rate size minibatch achievable number speedup property highlight theoretical many knowledge first offer support early particularly maximal worker ensure speedup accurate free strictly dominate apply scenario solution th natural take paper make item
title empty empty format empty title title ed ed format emphasize emphasize volume empty ed ed swap connect series emphasize volume number check empty mid sentence connect empty function multi page global format page page page multi page page connect page format journal empty number empty format page page page empty chapter chapter connect page format ed emphasize format emphasize format format thesis empty technical format article format name et format format volume nan author series format format format name format format ff jj format format name format name author author empty function empty key author key organization key key organization label empty organization author label empty key name empty author label manual author organization type label author organization organization full label swap label output write function output format check format output check title title check format format format page output note format annotation format format either skip year format check format new format new sentence format function format author format output year check format title new block entry author format author author check skip year format title format format chapter page check new format series format chapter page chapter page format book note format format author check author format title title check format ed output format format chapter page sentence format output format annotation author author format output year block format title miss ed format format series format page output organization organization format output page annotation conference manual organization organization organization format author format new format title output organization block address organization organization check block entry format format check format title new master thesis thesis school school check block entry format annotation write format format key year title format block entry format annotation author author format format title ph thesis format thesis school school address entry annotation organization organization format format output check format title format output output format annotation check author block format title title format tr check block entry format format year check block title title output format annotation default macro macro macro macro macro sep macro macro macro survey macro macro intelligence macro communication macro journal macro journal macro transaction engineering macro transaction computer macro transaction computer integrate circuit macro letter macro journal macro journal computer macro science macro journal macro transaction computer macro transaction system macro graphic macro software macro transaction office read integer sort format name name skip jj format name sort format title word author empty sort need sort author sort author empty sort sort name sort author key sort author key author sort format organization sort empty sort organization key format sort manual organization sort sort sort label title format max sort iterate string label extra extra last extra label year extra year nan max last label reverse extra skip year label sort secondly condition computationally infeasible methodology jump employ strategy accelerate rejection condition jump light develop enable framework establish adaptive simulating condition extend simulate condition framework represent mean simulate dimensional principle comprise definition principle outline skeleton diffusion approximation mean computation determine path constrain path process partition path simulate conditional proposal rv skeleton simulate impose coefficient sufficiently regular ensure existence unique continuity drift coefficient sufficient allow volatility time interval univariate transform let apply jump f v induce transform point measure induce condition jump diffusion constrain end jump diffusion volatility compound jump coefficient jump distribute compound poisson exist compact set order tx tx density condition form paper directly boundedness bound set suppose lx outline simulate diffusion transform represent sde simulate rao less section direct extension term serve consider bridge path algorithm sampler operate diffusion introduce simulate measure brownian sample proceeding sampling draw accept exact rhs diffusion path infinite unbiased construct entire path simulate alg inf skeleton compose else return principle simulate skeleton require path employ construct simulate disjoint layer proposal belong aid precisely ax approach make computationally efficient strategy reject conditional reject acceptance simulate additional event alternate graph critical acceptance formally implement rx exact algorithm ii letting reject return simulate skeleton per accept return computational link graph naturally want choose graph alg perform simulate exponential set order interval refine upper accelerate rejection essence find conduct remainder simulation suit simulate condition long infeasible path efficient simulate extent interval mid mid simulate equal acceptance three associate consider evaluation next computation conditional accept accelerate rejection begin evaluate computationally respect sub new sample path tighter coincide iterate notation comprise evaluate acceptance estimate comprise regard time interval bound note h simulate l layer information return return satisfy principle augmentation illustrative accept path two path trajectory skeleton sample trajectory skeleton methodology represent sde denote construct algorithm develop upon skeleton collection principle employ rejection computational rejection employ condition jump proposal measure key contribution alternate construct point compound simulated ensure bridge condition component consider superposition compound sample path start end induce sde measure proceed follow exact simply draw accept x I consider acceptance simulate dimensional evaluate without simulate sample leave skeleton construct denote law acceptance decompose acceptance accept rejection strategy acceptance compound jump simulate leave form jump path brownian methodology recall compute require acceptance condition jump exact acceptance let simulate finite suggest omit incorporate idea illustrative accept skeleton simulate compound poisson per x reject reject return simulate l layer information l reject return set skeleton x acknowledgment mp thank work k author base intractable new application along interest lie methodology smc email ac uk ty empty ty empty e option without option argument option without option def def def def def def def def def def def sp large rest j cause page inconsistent pt height ne em sp plain plain pt height stream stream stream stream stream start file percent
z lattice g lattice obtained reconstruct self energy three estimate solely polynomial learn fig typical weak different use example worst predict large study rigorous learning material detail median example different show size around predictive interesting choose scenario lattice function chemical ml reconstruct lattice g use result shift prediction problematic ml prediction function ml good job database fig ard ard global finally analyse totally database homogeneous database choose actual previous overfitte influence approach well predict database choose full loose predictive machine really use body physic show predict solution function output function apply change solver cluster theory self way ml material accuracy largely enough problem important might cost account approach adapt department er numerous discussion implement j department university national solve equation dynamical dimensional technical issue map distinguish validity machine full particle indicate development attractive computational efficient option prediction system body contain entity decade difficulty long monte generic exponentially property law approximate phenomenon investigate leverage exist solution generic quantum physics essence use matter physics context molecular density functional context weakly formation energy material physics particle quantum ml energy scalar equilibrium body problem solve many issue arise application include infer quantum body relate build involve function formalism capable solve question relation position exchange correlation approximate interact quantum quantum body physics local green consistency band material question parametrize self consistency member test machine neural forests accuracy problem critical transition outperform decide kernel ridge detail follow determine process detail explain later text first implement database condition span range possibility database site strength range density ed discuss implement descriptor output datum naturally formulate lp green energy section material function parameter denote scalar interaction strength chemical ml concerned database energy different
heterogeneity represent effect represent estimation class dyadic longitudinal dyadic illustrate package analysis keyword dyadic latent factor mcmc social individual measure dyadic relational particularly variable dyadic population individual node direction quantification person international trade log country country ir include package specifically analyze trade rgb ir ir ir na na na dyadic exhibit dependency row heterogeneity mean heterogeneity popularity evaluate heterogeneity around overall additive row heterogeneity mean normal heterogeneity equal normal r response df sum pr heterogeneity row comparison estimate column na usa close square maximum straightforward implement classical fundamental characteristic dyadic relation refer node additive effect evaluate correlate evaluate popular additionally pair node outcome correlation effect dyadic model volume highly volume country analyze node describe variability row follow conditional row specific heterogeneity heterogeneity column summarize describe equivalently row mean variability beyond capture effect covariance element covariance evaluation tool progress display via sequence plot store vc intercept include default intercept store style fit beta vb complete estimate give goodness summary deviation row deviation within dependence histogram represent histogram generally speak histogram model lack datum respect discrepancy surprising dependency variance often dyadic dyadic model vector characteristic receiver dyadic covariate characteristic refer sometimes student success popularity like fit dyadic log number membership ir ir ir ir dyadic store array dyadic covariate value psd col col variance psd vb divide mean posterior deviation standard normal exceed calculation appear association population share positively country assume I residual package model column dyadic false fit psd row row col col col variance psd vb deviation almost fit explanation precision via exhibit dyadic regard fail dependence plot similarity individual associate relationship suppose node indicator organization indicator co organization may anti measure dyadic covariate multiplication see dyadic cluster people prefer form tie lot triple triple link one explanation link occur must would tie node multiple link triangle visual network person relate person characteristic set covariate product element dyadic length regression account describe type network pattern equivalence group way related equivalence estimate multiplicative trade share dyadic fit effect rgb predictive statistic include show raise exist attribute multiplicative case characteristic unobserve factor characteristic describe mean depend extent vector magnitude type network dyadic essentially dyadic aspect model package option letter stand factor fit model provide adequate dependence statistic estimate share latent psd col col shared variance psd way factor blue magnitude indicate trade regression additive row effect example identify trade volume dyadic outcome case trade datum transform binary transformation binary friend friend neutral discrete event amount people phone pair population accommodate dyadic follow ordinal meaningful include discrete outcome binary indicator count order outcome medium high ordinal ordinal probit simplest ordinal dyadic variable binary whether dyadic indicate interaction include social tie member dyadic several display office location status office age practice fit without explanatory describe observe latent specify model probit contain simple goodness compare fail fit fail term common description positively estimate large illustrate fitting age age coefficient psd age age age col col result positive effect age older dominate dyadic effect dominate summary intercept intercept identifiable parameter estimate intercept term part transformation nuisance human social fix people friend national health ask school student five member friend five friend scheme ordinal friend ordinal also censor complicated ask five friend five people person people five person censor absence modeling develop dyadic treat outcome continuous letting coding indicate rank rank positive correspond relation people could consider person censor person rank implement fit base use study ask rank variety rgb na specify rgb psd intercept psd goodness plot heterogeneity simulate satisfy impose amount dyadic design ask participant friend dyadic censor way datum survey person less approach analyze censor option dyadic outcome ordinal case treat heterogeneity tie across rank outcome ordinal dyadic impose unobserved option dyadic happen cost pair node partially dyadic datum distinguished pair dyadic missing mcmc iteratively simulate along value value way approximate speak miss procedure specifically study popular dyadic node randomly sample ask tie tie friend participant friend friend friend illustrate analysis college describe record record code ai already design share na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na datum bin fit ess beta intercept col ess intercept row col estimate similar second output fitting predict imputation dyadic dataset example relation obtain dataset na goodness respectively indicate reasonable obtain variability sample probit illustrate histogram illustrative exercise decrease sample concentration dyadic dyadic base accommodate dyadic model allow dyadic point word seem dyadic allow possibility certain parameter longitudinal relation college student person graph seven period surprisingly graph include student program member effect model indicator model include indicator product dyadic regressor dyadic program include dyadic possibility function array dimension dyadic array array datum analysis dim n dim dim using previously summarize fit bin psd intercept col psd vb indicate correlation status program evidence bit note whereas consider effect regressor might vary depend lag measurement possibility term regressor vary interval create dyadic covariate binary measurement regressor dim dim w vb ar vb psd intercept row col w col col
distribution concern three unknown benchmark useful procedure known analyzing rare detecting phenotype large rw although mh abc obtain locate finally proper allow implement assume sequel type explain analyze aim mutation occur dna site mutation mutation summary generate independent mean mean n ns exp w ns marginal poisson employ report result pilot run top calculate jacobian bottom simulate top leave jacobian grid pilot bottom credible posterior mean choose summary statistic site may every course statistic abc method approximate jacobian nearly relation calculate relative figure respect rather discrepancy prior posterior quantile parametric abc variance sample deviation normal n parameter linear illustrate residual output former posterior center contour posterior vertical line constant report scale variance move variance differ quantile stochastic representation available hence difficult focus stochastic z skewness abc quantiles range transformation describe pilot show var report residual variance abc rw mcmc four density posterior posterior chain assess dataset expect abc error mse automatic abc fp abc compatible observable summary one application dna phenotype dna single nucleotide observe million fast need question snps mainly disease usually collect open open nonetheless certain genetic snps disease come I human collect genetic world method develop genetic population compose family snp snp white affected red snp configuration phenotype snp observed phenotype level usual independent instance fisher test association snp former individual affect snp status treat highly genetic variant transmission relate phenotype configuration snps constitute relate logistic also volume disease genetic inside determine individual later observe similar disease snps level inside separately overall analyse snps logistic log odd snp transmission snp configuration transmission assume usual law segregation individual configuration summary odd affect among individual configuration specifically number occurrence individual numerator denominator constitute limitation perform pilot point grid pilot depend observe phenotype effort pilot see hypothesis genetic illustrate logarithm respect algorithm conditional conditional simulate snp term make distance tend match configuration obtain acceptance rw mcmc figure chain rw b dot factor along snp exhibit large posterior posterior snp skew center around also reflect logarithm bayes snp risk snp analysis snp rs bayes factor first precise signal summary seem vary grid suitable abc proposal rw fashion rw constant definition quasi likelihood proposal analogously scalar summary moreover happen part happen mutation properly discuss another lie argument perform quite notion rw mh focus constant assumption find lead discuss proposal reflect costly likelihood use abc di di computation abc bayesian practice basic may inefficient presence discrepancy elaborate monte abc difficult automatic proposal likelihood model value statistic sampling pilot establish conditional construct extended variance value many application biology involve computationally rapidly literature lead set leading later become area sample index prior aim n observable g quantile etc author suggest say accept assumption moreover sufficient agreement improve abc drawback inefficient easy issue monte carlo method abc analyze lee monte carlo smc attempt require analyst major literature concern suggestion current propose posterior mean pilot specific composite focus mcmc method building proposal proposal model account adopt approach function indirect distribution arise tractable context kernel transformation scalar typically pilot run regardless sample thus serve routine analysis appeal application genome fact end available asymptotically target follow throughout propose abc formally discuss illustrate propose conclusion remark conclusion profile receive wang summary whose observe assume convenience summary lie real line suitable specific replace pilot run state sequel abc purpose input density convergence rp smooth monotone differentiable spline possible convenience analyst diagram goodness estimate depend resource achieve make wide increasing curse large observe gain precision monotonicity abc relation g monotonicity automatically recognize proposal essentially simulate fix quantile include approximate jacobian reduce interpolation spline mcmc jacobian estimation effort desire proposal regular taylor around mode monotone possibly conditional covariance
section want exploit sake present respect argument suitable change expansion denote decrease non eigenfunction moreover pc admit j j decay exponentially tend decay decay super decay exponentially hyper hilbert pca basis carry orthonormal basis provide pca present decay variance devote define functional result subsection illustrate deal discriminant define I maximization approach identify parameter mixture highlight carry information orient tool detect latent proposal group surrogate large surface assign group consistently proximity look pc assign mean nn modal q empirical version discuss estimate projection span eigenfunction semi attain case b times bound integrable support follow thus justification univariate smooth scale whose sense specify concern spurious select mode jj use estimate play coefficient identification mode graphic visualization system software prototype estimate pc belong upper datum proposition avoid curse dimensionality estimate non explain large practice solution external criterion index depend combination accordingly validation criterion choice lead criterion find reference therein index pre extent class datum cluster proportion proportion calculate cluster clearly range obtain choose trace estimate matrix select discriminant differently presence establish model aim new incoming group structure typical one assign class correspond equivalently known simplifie follow argument apply straightforwardly setting consider classification assign eventually tend hard thank whenever pc group asymptotic parameter possibly different dimension simplify mixture decay start represent straight min operator order eigenvalue least eigenvalue much concentrate concentrate decay exponentially simplify similarly surrogate th group tend equivalently one density parametric introduce interesting parallelism conditional density assume mixture theoretically finite subspace slowly ensure projective discrimination full probability context gd eigenfunction occur balanced concern span eigenfunction pool discussion coefficient bandwidth particular behave converge tend subsection dedicate control cluster dataset dedicate simulation quantitative comparison goodness detect measure misclassification error exercise cluster noise point detect keep mind simulation exercise expansion element control mean shape coefficient beta scale coordinate spherical g limited semi unitary whose center choose un easily identifiable concern choose avoid noise direction pc replicate set proposition equally exponentially correspond great suggest generate sake depict plot algorithm htb middle right set mode space monte return misclassification mean mixture first pc code summary misclassification error quantile gm combine bic misclassification whenever configuration correctly recognize misclassification equal gm whenever bic cluster ccc st km dataset aim bring phenomenon domain analysis consumption estimate composition light decade widely explore kind cluster near nm grid water chemical original avoid calibration presence shift since represent way chemical composition chemical available chemical correlation linear particular water equal content protein positive water chemical composition first explain kernel pc figure reduce look local minima whose present three modal well pca concentrate first pc explain variability use reach internal external criterion computing couple accord summarize couple possibility reproduce distribution chemical measure curve heat centralized location entire system scheduling generate line demand flow mainly demand load aspect weather forecasting load demand application cluster heat consumption west centre produce generation previously regression consumption year privacy figure display behaviour demand intra daily due demand aggregate behaviour differently demand difference appear dataset way period daily load discretized mesh figure display procedure perform functional spectrum pc sufficient limit provide contribution exploit mean curve plus suitable see weather highlight difference demand heat less counter pose three systematically day cluster algorithm use choice reflect daily level demand peak moderate peak effect element cluster represent load daily modal curve plot grey moreover box plot daily label multi modal external cluster exercise pattern mid forecasting performance predict activity condition central acquire essential contribution activity procedure neuron sort detect thought correspond single neuron experiment reach target virtual detailed description neural activity channel versus discretized analysis also perform spectrum explain variability observe appear good build index lead admissible correspond cluster combine produce maximal level procedure curves briefly simulate consist misclassification fold validation evaluate estimate remain glm basis functional nonparametric discrimination classic see computation setting training set balanced translation variability around two small medium high cc cccc ccc glm nn discrimination exercise pc obtain misclassification summary deviation error comparable due spherical data glm result performance dataset belong domain dataset come website curve discretization relate berkeley growth data fit discretized datum aim discriminate curve base gender detail
impact output specifically address three type consist likelihood estimate characterize resort expectation attain iterate experiment reconstruct impulse compare base scheme account initial regression long history tool reduce mean regressor compare square novel method identification get estimate impulse class recently kernel spline estimation decay kernel identification see instance spline two estimate effective rely bayes argument exploit interpretation regularization impulse response model estimate marginal output impulse among impulse compute situation preferable record five reason standard variance mse g time time ignore rest experiment discard depend preferable initial condition context first incorporate unknown assume autoregressive average stationary estimate initial minimum initial marginal exploit problem iterative method technique method blind identification close involve grid search organize follow review estimation relate system algorithm discuss error figure output impulse convenience delay capture dynamic corrupt zero mean interested impulse q contain problem instance discard collect however considerable rest improve present interpretation impulse hyperparameter structure determine velocity impulse response introduce follow write follow posterior estimator sense hyperparameter quantity need datum consist compute hyperparameter quantity estimate computing variance residual identification approach ml integrate hyperparameter effectiveness bayes approach serve new aim initial straightforward quantity impulse maximization condition nonconvex possibly dimensional impulse response devise maximization method iterate suppose calculate impulse well impulse response introduce impulse toeplitz relation toeplitz impulse response definition iterative solve hyperparameter guess initial convergence sequence global marginal impulse limit use information available miss rational spectrum realization white noise namely probability initial write joint probabilistic available size hyperparameter amount solve problem unknown hyperparameter estimator guess update estimate impulse exploit miss less joint statistical propose mixed map highlight act put agree case find consider start guess initial hyperparameter incorporate information covariance general conditional case estimator conversely set obtain conditional yield degenerate iteration rely monte number carlo system radius plane impulse filter unit order filter filter carlo noise noiseless hold correspond priori avoid initial discard conditional estimator present base hyperparameter score impulse response test ccccc kb ic zeros kb kb ic kb kb kb ic oracle percent impulse response information discard kb kb zero suffer effect wrong system estimate impulse response record performance kb ic mean estimator kb ic perform oracle initial
measure code ab days total ab million ab constraint support constraint ab patient contain medical prior ab ab instance lift great non cause square ignore medical record ab record ab association risk ab reaction drug record time month never month association rule generate rule patient minimum confidence constraint patient contain medical record one association b association lift cause square value risk adjust failure severe occur month month rarely record instance refine adjusted show refine pair pair evaluation may show refinement able read unlikely able refinement c failure due read code rule may generate rarely record read code limitation c probably reason ab record prescribed commonly database medical ab consequence interestingly rule identify cause lead support refine consider work short medical history prior require item newly unlikely item record medical medical consider read prescribe thing read record frequently age patient year birth record rule set support minimum confidence tune base common rare whereas increase rule tend suggest poor apply rule rare common small contain lift account restrict rule three mining help refine improve adjusted filter signal occur still amount concept refinement record history instance cause drug efficiently refine signal require tuning confidence suggestion efficiently implement distribute enable contain investigate age signal cause go thank side prescribe common occurrence effect automatically leave likely essential refined correctly majority correspond paper filter patient medical require parameter patient aim improve health unfortunately majority induce drug clinical view positive often researcher identifying generate drug refinement severe death thorough generation database present additionally occur decrease take conclusion recently refinement thin database uk million record thin birth gender decide record novel medical remain use method develop subset patient thin million patient thin event via structure read code medical specificity medical diagnosis laboratory read consist alphabet dot level read define level medical parent read direct parent parent child read medical parent read code correspond child correspond code cause drug calculate implement thin issue still main issue move home within thin database medical new incorrect patient medical event find record date start prevent newly patient drug exclude identify drug record month thin database exclude identify drug prevent reporting mining frequently transaction read frequently occurrence event occur association rule similar manner patient medical record thin medical rule rule outcome pair correspond adjusted identify formally hypothesis method refinement find database prescribe two summary work base exposure occurrence ignore adjust remove patient period drug determine outcome gender refinement patient day patient patient date record date date drug ab measure record day measure quick numerous require present ccccc code ab ratio ab death unknown secondary rule thin contain version read code consequence read gender record date occur patient association record tell chi lift identify insight occurrence patient compose maximum lift maximum chi item patient record indicate value cause consider cause rule lift great lift occur patient population lift calculate base number instance lift maximum chi square calculated rule
svm compute hence binary vote vote opposite vote majority total vote weight win weight give vote stochastic call moment draw margin al tight account light vote inspire name whose state multiclass weighted majority design complex output classification generalize multiclass label section recall output give pac risk majority vote margin define margin let accord et function chebyshev inequality z counterpart justified thank elegant pac simple generalize important pac stand multiclass space recall look majority vote multiclass q vote realize version multiclass margin notion multiclass let margin vote regard strength multiclass draw vote classification lastly call example output margin majority sign come vote binary majority make mistake margin multiclass leave side hand verify vote necessarily correct weight mention margin differ multiclass margin definition decision multiclass strength true class combination versus base multiclass multiclass multiclass q binary multiclass vote equation multiclass margin relation vote term margin minimize much every inequality sum drawback minimize finally multiclass result able al multiclass margin label majority number output among label otherwise label majority label low square euclidean cumulative margin margin bind label vote classifier developing depend derivation minimize generalize margin multi distribution margin coordinate second calculation hyperplane
mistake row mistake efficient mistake dimension leave question mistake precision bound dimension mistake motivate instead constraint sample polytope objective unknown constraint change mistake ellipsoid coefficient change objective implement separate hyperplane mistake reveal preference relate multi problem observe show continuous finite give polynomial pac reveal preference connection learn prediction efficient compression efficient complexity meaningful reveal mistake price maximize also learn strongly leverage et adversarial objective know change optimizer constraint change similar distinct preference optimization vary think say problem trying predict lp partially choose coefficient learner goal predict learn mistake never use mistake program polytope take adaptively output first study give polytope learner polytope change change give finite mistake learner problem study polytope change refer study receive lp partial known polytope know observe define mistake partial example total sequence learn bound mistake bind mb put definition learner coefficient denote coefficient region change way polytope know loss generality polytope boundedness name uniqueness probably remove name write finite precision polytope tell encoding encode typical constraint program define need finite differently converse precision polytope without bound vertex polytope write precision necessary uniform mistake next mild polytope rank degeneracy organize present learn bind assumption necessary precision specify adversary force learn mistake like round rather finite precision part still move avoid complexity arbitrary observe polytope know day avoid inspire et ellipsoid mistake result ellipsoid assume represent day ellipsoid terminate coefficient objective denote denote make feasible write imply define polytope coordinate lie region write infinitely polytope informally arbitrary solution rate precision specify solution feasible polytope fact program vertex write vertex polytope satisfy adversary constrain run copy ellipsoid feasibility constraint define maintain ellipsoid candidate mistake constraint solve nonempty true lie ellipsoid polynomial mistake ellipsoid volume ellipsoid use n whenever mistake hyperplane ellipsoid current separate hyperplane ellipsoid tw formal formalize leave solve simplify lp prediction vertex remark lp solver exact vertex polytope mistake make ellipsoid find bind ellipsoid intersection constraint bit access solution precision separate ellipsoid empty number ready mistake lemma instead observe mean region choose objective value solution contradict hence rule round e ti predict outside along feasible prove initially mistake round adversary pick bold point matter learner predict return different guess learner adversary pick process learn formalize high procedure adversary matter adversary ensure interaction adversary output polytope actions adversary algorithm precision polytope day nr tr tr ta mistake mid middle top small procedure take learner mistake produce choose adversary infeasible optimal learner adversary pick point computed constraint uniquely feasible region round r r qr polytope polytope subroutine make every learner mistake round polytope present round return adversary lp subject feasible remain show new always bind check intersection polytope newly hyperplane high second equation accord hyperplane polytope intersect adversary polytope intersect hyperplane intersect unless happen never hyperplane proof modify furthermore modify newly add effect hence fall hull round polytope linear constraints eq denote hull randomize unknown problem mistake dimension constraint mistake polytope unknown algorithm randomize algorithm completeness open problem mistake achieve hypothesis form multiple generality denote consistent day polytope day consistent update consistent optimization wrong mention I change optimize algorithm maintain instance see polytope set round solve name select real round round repeat number randomization might adversary mistake round mistake round eliminate round product round mistake mistake rearrange first describe new input two line return otherwise empty j j j j subroutine identify vertex underlie polytope infeasible coordinate tv e exactly contain vertex exist separate disjoint interval note therefore subroutine eliminate polytope contain several claim interior interior take line must intersect denote place gap interior combination interior ready theorem know interior otherwise claim segment polytope three generality write belong hyperplane
pp valid replace theorem furthermore constant orthonormal satisfy satisfied sup proof theorem pp complete assume space notation preliminary dimension localize eq exist let localize constant straightforward spirit exist take basis sup n ks ks p ks l q dd dimensional explicitly k notice deduce find sup respect vector norm hence admit eq axiom theorem claim example theorem exercise proof investigate selection regard regression endow haar optimality calibration procedure call penalization slope recent penalization perform existence behavior slope heuristic thus method successfully wide applicability slope justification indeed study framework theoretically optimality calibration show validity validate heuristic framework selection histogram extend slope density slope heuristic histogram density previous optimality heuristic general model endow orthonormal basis sup element number intersection element assumption analytical bound risk treat context haar expansion noise model ideal penalty resample candidate mild penalization describe framework slope heuristic validate hold penalization proof upon conditionally assume variance independent sample follow dimension paper introduce detail p x pf pf square eq regression consider possibly empirical image estimator excess excess least give loss excess quantity depend penalization dependent aim analytic model selection provide exist constant note localize basis orthonormal intersection support orthonormal orthonormal localize state let deduce satisfie give moreover straightforwardly satisfied finite interval large notice assumption localize proof totally trivial argument theory orthogonal polynomial basis notation convenient existence strongly localize exist partition orthonormal p dimension constant I noise assumption give relation specify quantity strongly localize latter ensure uniformity along constant define localize collection model polynomially complexity want choose estimate concentration deviation uniformly collection put extra inside depend reasonably depend suffice low upper ensure assumption property term assumption especially inequality supremum empirical used inequality matter include extension unbounde state derive context need concern heuristic piecewise lead slope heuristic penalization def decrease power hold n penalty twice estimator satisfy oracle select dimension remove remainder ensure assumption regular partition theorem opt pp identify empirical excess generalizing endow localize check unknown level slope heuristic issue slope shape prove calibration linear optimal procedure situation remain ideal thank slope heuristic devote penalization ideal index propose ks
limitation especially perform multiple interact allow challenge mostly focus memory doubly link dimensional memory head head move nearby current head move move list fix position b recurrent sgd propagation pattern controller learn hard sgd task supplementary material seem robust well operator introduce partially rnn stack stack round sequence generate goal learn rule understand scope pattern model stack list rnns backpropagation prevent learn use baseline rnn unit hyper baseline validation short long sequence unsupervise sequence pattern evaluate rule produce sequence training epoch predict sequence correctly bold recurrent mechanism stack list c c action stack stack b b clarity stack stack first stack empty interact deterministic bold pattern count table report sequence rnn stack rnn either list operation use table rnn unable able generalize sequence count unit parameter round require obtain stack show discretization rnn stack element input second stack start empty stack keep track stack unit rnn use random repeat multiple stack rnn rnn lstm seem generalize unstable stack frequently explain choose versus stack discretize stack read sequence addition supervise token addition ask reverse order train choose equal less digit stack average rnns generalizing run bar previous example addition moderately rnn stack keep track sequence e read read write stack interestingly capture stack store stack stack result finally stack care carry state explicitly say cache lstm stack validation test stack corpus recurrent stack rnn lstm corpus capture similar bag stack well stack decay bag memory efficiently learn rnn rnn rnn algorithmic attempt problem motivate algorithmic involve discrete algorithmic stack rnn input output format flexible allow loop access pattern algorithm possess automatically hard recognition solve recurrent memory stack operate complex currently memory fix learn would like rest facebook team comment facebook ai research york deep approach limitation complexity simple recurrent model capacity show sequential recurrent memory perform various task major source recent world research explore neural successful task lead vision recognition commonly attribute hierarchical recurrent theoretical instead represent learn current art approach past well linear one layer method describe demonstrate layer guarantee deep architecture layer non currently deep pattern model interestingly deep capability recurrent net allow memory structure stack matrix net multiplicative mechanism learnable memory simple read stack among work aware neural research done study sequence generator nc nc short building predictive mostly discrete pattern memory precisely sequence denote symbol algorithmic algorithmic example simplicity focus unary binary represent design free short length controller clear external memory module recurrent stack stack prior problem supervision problem sequential token character design symbol stream recurrent network rnn rnn layer recurrent delay recurrent sequence token rnn encode token predict probability base token follow sigmoid activation coordinate token recurrent weight hide network token number token architecture learn pattern one capture gram rnn
pose variation part face part probe method enable promise probe pose variation face image subject outperform degree pose encounter practical access scenario subject section review alignment probe combine method face model mixture handle intensive propose illustration part constrain sub ii transform sub appearance iv pose expression probe learn locally wise neutral expression neutral optical mrf local patch pose pose give probe face image localization alignment pixel accuracy wise correspondence pose exist one across pose expression rely relevant pose variation model detection pose objective score detector star shape constrain base method pixel image rather appearance evidence similarity align constrain part tree constraint complex property beneficial also cope illumination subject method follow collection surface piece expression change furthermore prevent face canonical template template part vary arrange define template template database region canonical alignment align template alignment realize transformation transformation yu yu md affine group similarity satisfie little simultaneously compose assume subject could term corruption intra leverage extend base give encourage mmd part dictionary whose align image denote reconstruction unfortunately unstable often initialization flat face indeed face contain structure overcome incorporate structure individual motivated shape constrain transformation different term determine structured shape parameter tuple tree consideration structure alignment weight address shown propose dictionary face produce correspondence pose face intra subject variation match fit disk center white line link display vertex set edge compose transformation root use face canonical consider simplified order associate node edge parent child use distribution dp I thus gaussian assume independence tree structure network joint logarithm joint regularization structured term localization ignore associated pair equal constrain shape model also enable later probabilistic prevent degenerate jointly tree integrate strongly supervise joint block degree tree form structured shape globally directly shape constrain call keep tree shape main difficulty convexity constraint domain alternate fix update efficient apply note relatively learn effective issue moment gauss taylor e iy linearization lead follow optimize e repeatedly linearly converge show convex lagrange multipli alm augment lagrange multipli denote frobenius matrix alm search saddle I alm directly alternate manner turn form let q operator sequentially eq auxiliary convenience basis equation form td ig summation hence large part simultaneously describe expand converge x inexact alm update group alm alm inexact alm summary solve linearize solve alm optimize technique section alm linearize paper propose without perform aforementioned specifically iteration keep unchanged indicate denote fix propose jointly efficiently solve outer section initialization face detector bound location bound available structured shape initialization act template probe detector fortunately alignment bad time handle face subject image face align form part face similar face subject perform subject wise optimize variable alignment residual subject sort subject wise residual subject small residual dictionary together part align transform transform subject part recognition perform aggregate decision basic adapt aggregate scheme nonetheless basic choice illustrative module note detail inconsistent different combine substitute high part small line adjust average transform subject dictionary align I prune subject sort ns j recognition exist recognize iy label section associate present face compose part align form dictionary propose part dictionary relational different part shape dictionary part couple present algorithmic first align part constraint serve alternate dictionary face stack face part ni part base contain part correspond appearance region ideally due inter illumination word rank aforementioned model frame sequence face leverage align directly apply alignment alignment often converge meaningful solution show similar apply probe instead constrain part simplify notation part dictionary surrogate function reciprocal root row denote solve alternate strategy face illumination conduct illumination expression outer corner center inner corner r corner corner r remark conventional region window outer corner pixel dictionary design whose neutral consist part part availability parameter model evaluate face pose illumination alternative face face face face demonstrate robustness state face section initialize corner assume pose expression probe conduct automatic face pose face detector across illumination use multiple probe face label corner manual face recognition manual automatically probe alignment recognition term alternative similar recognition comparison consist subject third probe face use pruning base face strategy define accordance name pruning face pose fig face neutral large publicly contain subject span accord illumination illumination appear structured model subject subject specification pose neutral angle face image neutral viewpoint otherwise mention use section across pose different subject probe illumination neutral lr lr lr align c manual lr lr manual report recognition alternative degree compare manual improve automatic alignment consequently face pose even tell baseline probe viewpoint neutral give high knowledge recognition across mostly due appearance period illumination ideally vary nevertheless piece alignment face recognition extent overcome report view image expression alternative recognition pose outperform practical face illumination people near neutral like minor strength subject lr lr align cb cd ce manual consistent perform degree alignment weak strong final individual investigate discriminative exist representative face show efficacy pruning dataset pose illumination part align individually report align individual part recognition within fairly recognition individual efficacy part part alignment recognition align realize face ns discriminate lda local recognition pruning also stage align experimental previously ns shot face subject also shoot illumination summarize illumination section probe di lr lr align di di si report table tell ns propose part alignment superior face alignment confirm effectively individual pruning investigate algorithm simply prune scheme bottom table list recognition rate together difference original pose drop prune remove impact pruning non pose time recognition error recognition opposite precede standard correct occur opposite pruning correct introduce rr pruning pruning report partial probe image position varied experiment setting recognition fig portion face drop rate pose pose good require face subject equip different feature fairly lda method experiment thank run model advance probe face illumination largely except face probe subject subject multi pruning neutral pose automatic manual initialization know pose report case semi use pose pose perform degree similarity individual training face image shape necessary knowledge able cope face pose encountered practical control reliable pose detector coarse face pose pose probe face either advance al pose illumination normalize illumination table report recognition row automatic change probe work coarse fine sequentially alignment part alignment experimental coarse fine strategy work well reasonable pose drop consequently failure tell equally pose probe either automatically confirm propose face probe illumination cc ccccc initial pose manual auto auto auto detector follow alignment pose pose recognition alignment appearance part structured shape part probe constraint formulate regularize optimization experiment efficacy handle illumination pose change integrate illumination change research adapt computer thank produce report table equation array parent similarity seek minimum keeping update horizontal translation similarity u c f unconstraine equivalence find eq link holds link q solution solve step gauss update constraint nj k linearization problem repeatedly solve converge solve adapt alm lagrange q lagrange multiplier alm I ii directly instead single variable notation use inexact scheme alm update alm n specifically present appendix might manual fairly improve reduce alternate pdf wishart wishart conjugate distribution different give estimation wishart nh distribution additional weight algorithm practical face system vision face alignment key achieve face piece wise surface surface develop structure shape probe face appearance consideration tree shape integrate classifier part recognition face par robust face across illumination develop system past decade control variation cause illumination pose illumination illumination use multiple carefully choose image vary probe face subject illumination face illumination generate face image
predefine dictionary wavelet central adaptive dictionary directions mod dictionary representation two often entire central increase feasible practical database entire access compressive data compressive version certain inference problem within compressed study line spectral compressive compressive domain performance towards compressive compressive general random propose computation onto bernoulli apply scale compressive tracking setting improve share several attempt compressive roughly three algorithm compressive measurement inspire aim compressive minor take overall none aim compressive maximally large scale moreover none work give efficient extend compressive scheme efficient key wide general compressive dictionary projection review similar signal dictionary member minimize coefficient matrix pseudo nonzero unit intractable solution via measurement product column drop compressed attempt solve follow learn minimize code strict optimization distinct measurement variety omp find atom hold penalty compressive tc minimizer ki preserve performance svd attention special guarantee closeness generate increase close increase gaussian intuition random dense mp generate sparse see get fix increase increase theoretical analysis give insight number tradeoff also accuracy distinct accuracy random reduce matrix divide pl control nonzero average interested cost collect datum representation coefficient penalty term code except omp omp dictionary penalty compressive quadratic first block sum square coefficient relate atom block l l represent li k k synthetic plot successful vs svd compression observe computation examine algorithm propose method implementation svd entire dictionary atom draw normalize atom gaussian corrupted draw compressive measurement factor evaluate magnitude inner successful recovery fig trial practice may indistinguishable see reach tradeoff memory vs factor reach eventually level grow access dominant support science foundation science foundation award university university university center shannon electrical engineering university
immediately iteration converge embed reliable worse suggest large dataset use projection projection run time error drop monotonically transform monotonicity decrease important mention figure need randomize embed dimension must conditioning yield ccccc e ccc projection dimension method fix completeness run set trial ccc embed method rest method quantity relative objective quantity versus independent perform median report machine considerable electrical scientific research originally use randomization resource algorithm several remarkable computation ram importance sketch solution subproblem sketch construct review highlight modification problem scale parallel increasingly importantly though scalability come communications improvement chebyshev query advantage avoid profile nontrivial expensive projection strong perform scientific looking would helpful like acknowledge research office advanced project energy provide large hardware storage massive storage store currently datum later create daily implementation simple processing method algebra traditional environment load disk easily dominate greatly increase develop implement randomized scale environment randomize problem deal implementation projection algorithm randomization relate solve exactly pass empirical highlight importance various quality versus etc medium exist sized data lie heart many signal processing compute convenient structure arise broad range value object use adjacency graph represent region band similarly dna nucleotide microarray represent snp condition individual internet automate set record task environment typical machine scan hard disk make cost application low solution paper overview recent numerical environment apply resource regression presentation rectangular sized least robust rectangular hold formulate result ram parallel environment relatively straightforward manner aspect algorithm aspect parallel environment random projection approximate digits precision medium digit precision user return principle develop high interested medium precision traditional subproblem understand principle relate principle important high development number subject freedom sensor number sensor word gram document stock extend restrict attention rectangular arise learn apply list health million snps genome subject disease determination target rectangular one collect internet spatial discretization partial differential freedom grow exponentially increase reach discretization cubic time dependent stay depend discretization spatial especially number sensor wireless hour back nlp gram grow geometrically document frequency trading stock well example daily file size great gb restrict rectangular develop method traditionally extension approximation svd qr decomposition connection qr decomposition rectangular similarly programming special problem class development environment thing term scientific researcher versus database researcher difference achieve parallelism share memory pass alternatively massive datum describe framework computation require want evolve evolve evolve interested reader quick parallel scale go tends share core core core memory memory core parallel linear generally provide computation analyze highlight design algebra survey algebra focus develop algorithm iterative expect interesting development idea distribute traditional performing distribute basic idea special case regression ram general traditional rounding embed method review implement sized provide discussion general interested reader addition review overview couple overview central theory ram important principle extend scale environment describe l two precision randomization conclude approach give follow ls interest meta take matrix estimate exact corresponding element sample row subproblem subproblem return meta term draw zero indicate trial equal choose meaning algorithm construct solve sample problem quality uniform subsample simple implement easy perform tune leverage score easily probability error mass crucial meta ls generalize rectangular leverage two type understand randomization designing practice ram well environment l problem think low precision inverse span diagonal element row alternatively express qr thin row coherence measure well vector basis well fit compute leverage key need sampling na manner leverage lead informally euclidean matrix axis provide key must leverage think small nonzero singular value norm compute solve precision ill speed iterative quickly informally aspect quadratic iterate precision precision key quantity condition guarantee provide meta solution incorrect relate randomized provide type vary application preferable discuss notion score generalize long generalization important environment meta time former depend flexibility provide score exact approximate I qr thin obtain na ram original ls problem practical implementation meta run dependence roughly scale prohibitive moderately value three meta fast ram meta run hadamard perform basically random quickly compute statistical leverage algorithm run approximate distribution use construct extremely input sparsity input order term implementation ram run ram construct iterative hold coarse obtain precision practically iterative construct high approximation l subproblem iteration could draw extra iteration could still failure convenient application might answer undesirable monte scientific moderate solution algorithm might expect run bad meta represent qualitative bad ls go elimination meta margin bad asymptotic matrix several remarkable ram open large environment principle want want parallel environment meta algorithm principle importantly situation must algorithmic principle must section review paper section traditional solver list letter letter scalar g etc use frobenius norm norm letter span except g solution e give problem important special square l regression absolute error former solution optimal particularly make theory algorithm formulation homogeneous elsewhere unconstrained problem scale l rank minimizer length minimizer unique min length computing define define regression problem linear system number rank let large underlying notion characterize simplicity conditioning let minimum condition matrix always notion definition factor factor matter formulation matter condition establish nm application call instance solve algorithm iterative number problem take form system consistent unique ls min min right follow hold certainly respectively leave problem ls regression arbitrarily well condition qr decomposition optimal solution solve well reduce ask orthogonal qr exist consequence round matrix full column lemma provide subspace preserve sampling build square classic problem linear algebra detailed survey certainly beyond know direct reader min svd singular singular calculate aa unique factorization factorization complete orthogonal factorization usually compute qr make triangular qr determining solve least correctness replace svd factorization either normal expensive mention especially chapter van sparse square problem column storage method refer iterative minimizer solution cg min preferable cg arithmetic numerically chebyshev semi l problem rate affect condition state z l thus cg like remains estimate iterative lack regression program programming formulate solve convex solver come example easy due therefore solver gradient interior cutting solve regression discuss solver reader smooth specify use simplex solver subgradient feasible initial search speaking conditioning regression make step solve square problem choose smoothed divide zero theory certain assumption rate hard relate low distortion technical regression emphasis environment full lemma condition qr linear particular section practical matrix speed pass condition quality algorithm family round roughly speed ram one practical level consideration consideration round data matrix property geometric property point preserve building projection linear problem dependent time storage storage input matrix construction embed embed important typical implementation algorithm round round subspace vector preserve subproblem round complementary reason latter introduce overview lp low solver precision solver lp lp round round low round datum aware datum aware right align center align center node align center fast leverage round start round n dimensional respect find round convex graphic connect round round round ellipsoid volume ellipsoid contain lead hardness ellipsoid round round call see polynomial special convex hull algorithmic work round call oracle er er pass er focused ellipsoid rounding method slight fast call find slightly round separation rounding describe ellipsoid origin separation oracle subgradient initial immediately improve algorithm conditioning take computing immediately present give subspace aa optimal important observe embed scan portion guarantee observe take norm summarize well discuss distortion subspace qr distortion use see table detail run run depend embed method time ccc mn score sampling within algorithm approximate leverage score accuracy embed subspace embed different embed introduce result embed embed name run ct l transform exist trade distortion linear embedding provide distortion embed dimension obtain application deal first distortion distortion cauchy ct nc scale least construct sum half cauchy gaussian random variable large always well dense propose construction sample combination fail independently diagonal comprise ti assume power integer informally effect spread entry comprise cauchy small finally quality fast cauchy transform construction constant ct lead dense distortion dense somewhat bad distortion matrix matrice ms choose standard vector independently cauchy theoretical subspace transform describe summary embedding several aware preserve embed previous subsection algorithmic advantage embed even embed hard embedding probability random whether aware embedding could conditioning yes algorithm preserve sampling leverage matrix l result give desire several completeness regard subspace preserve obvious compute leverage involve form normally undesirable application score done perform pseudo score specific leverage embedding idea point estimation satisfie sampling accord constant say require gain theory suggest preserve give ms ms n ms n n use asymptotically quality application qr base summary subspace aware embedding aware embedding idea aware distortion subspace regression subspace embed depend reciprocal ms coordinate orthonormal nice basis subspace preserve dimensional nr n na eq least choice lead typically condition row compute exactly central multiply idea theorem preserve sampling mn ms eq subspace preserve ms constant exist compute condition norm norm speed sampling affect theoretical formulation complexity still worth norm explicitly obtain small trade implement preserve table round subspace describe subsection problem tool introduce subsection solve subproblem construct ellipsoid round able relative regression interested precision medium solution principle condition basis elaborate solve summary several representative ram section distribute environment example embed subproblem svd stable high qr pc pc normal equation iterative pc c reference subproblem low er order pc er accelerate descent subspace preserve sized obtain step construct preserve embed box fix mean matter distortion reasoning indeed solution include state distortion optimal sized approximate deal meta many require subproblem subspace therein simply aware subproblem use algorithm run projection alternatively hadamard projection solve subproblem reference therein aware hadamard combine input sparsity asymptotic practical ram parallel distribute particular still matter refined implementation preserve use original original somewhat detail approach invoke system run moderately embed low precision high solver solver completeness solver depend first apply solver system condition small constant iteratively nm author randomize use chebyshev detail solver use various approach run conditioning quality trade size embed trade computing solver work nesterov employ combination round method technique generally also solver medium several implementation environment regression appropriate environment among need result precision subsection implement environment comprehensive completeness describe implementation illustrate several implement computational subsection implementation solver design design high random provably chebyshev cs iterative step prefer formal system size qr svd length aspect transform cs couple nontrivial way start among choice conditioning depend certain system q parameter algorithm condition probability slow transform several large environment transform fast environment easy implement partition along big lastly properly random nontrivial dominant na projection generate number use cpu understand preferable communication cost account precisely fail slowly control expect cs inner require conjugate cs iteration multiplication need synchronization strong conditioning expensive advantage environment consider single beta frame beta alpha describe detail low precision precision solver scale environment implement standard massive regression step well importance base norm sample subproblem problem subproblem key thing note job extract basis three ellipsoid round er qr low qr qr er summary conditioning conditioning implement cauchy transform ct dimension implement manner consist denote associate collect qr complete subproblem several approximate solution original exploit pass say marginally expensive pass almost effort provide node cluster query take second took come almost free basic solution desire return approximation original ip collect row aside increase evaluate use several ct etc solve precision summarize although interior cut plane need pass dimension resource computation tradeoff medium precision pass subgradient large environment design distribute computation similar except high region ball solution describe first ellipsoid construct multiple precision solution pass query iteration query point per multiple use convexity query subgradient serve separation contain return perform iteration solve determine embed crucial part aware subproblem iterative low medium datum choose challenging matrix stress variant describe range uniformity condition four dataset uniform uniform bad leverage score condition score matrix score generate list matrix b random control leverage exactly bad good single determine ram first replicate na stack alternatively ng manner call stacking stacking summarize yield stack solution possibility
kronecker htb normalize magnitude background number show compare calibrate incoherent previous capability gain pass single rmse normalize ratio background outperform htb htb paper propose rejection multiple detect stationary clutter signal rank factor clutter result clutter detection gain corrupt training analysis experimentally confirm multi change detection gain achieve rank element via let right positive semidefinite hermitian positive semidefinite hermitian iterating complete hermitian iterate hermitian hermitian matrix project estimate clutter form filter hence improve temporal channel clutter covariance since h h temporal lr thus prove apply proposition target image stationary target well move due use processing spatio clutter covariance stationary clutter enhance target note clutter naturally low clutter kronecker provide corruption due move target theoretical property experiment challenge advantage exist track move scene activity cause synthetic particularly task surveillance area regardless work move object move move perform frequency frequency detect shift significant include low small imaging view detect limitation move lose resolution use multiple moving cause stationary especially grow integration detect clutter either scenario velocity filter clutter otherwise target search massive velocity acceleration often complexity compressive approach intensive design channel potential benefit potential move configuration include spatially separate multiple pass combination create collect fact pass long delay issue exist move target background track center detect target applicable scenario detect phase clutter threshold amplitude well dim clutter parametric channel generalize advanced develop adaptive use spatio training across channel clutter classical low large guarantee due dimensionality spatio rich freedom exceed severe overfitte coherent receive delay design angle across etc clutter paper due computational efficiency excellent clutter regularization problem none exploit spatio contribution exploit spatio structure significantly corrupt return th th bin adjacent freedom greatly exceed bin covariance particularly bin standard show introduce reduce note clutter portion clutter significantly training reduce significantly involve addition structural via subspace corruption addition none spatio temporal kronecker kronecker product two temporal rank pass covariance include recommendation rich covariance kronecker l method many prove significant kronecker spatial clutter calibration model hence kronecker l observe theoretical project spatial clutter effectively project thereby clutter filter approach improve robustness allow remain clutter covariance factor sum different rank product base filter algorithm temporal clutter highly result demonstrate complexity extension organize discuss extension kronecker pass move change detection give return decomposition noise return clutter scalar distribute return bin isotropic nature phase target filter clutter preserve specific motion target clutter ideal noiseless return return single consideration ideally locate space clutter linearly spatial clutter turn depend characteristic clutter interest exactly rank significant principal time return target bin j platform shift target lie outside clutter long time move stationary clutter correspondingly b f clutter filter make orthogonal project clutter exist orthogonality spatio clutter projection basis subspace algorithm kronecker additional available moving lie call spatial kronecker clutter datum note temporal relative entire move lie outside temporal clutter mse project filter small outside primary allow kronecker reduce kronecker enjoy benefit arise clutter covariance motivate thresholde problematic kronecker kronecker combine specifically kronecker svd b I separate hence necessarily structure covariance addition subspace training include create signal additive superior clutter rejection pca energy clutter product clutter share target surveillance interest determine change reference later appearance platform generally reference image change detect change scene move mask change scene advantageous target thus arise clutter subsequent change detection spatial form history involve calibration clutter subspace project away pass pass filter output suppose clutter lie spatial clutter lr component temporal assumption lr spatial asymptotic regime loss lr lr typical become kronecker naive spatial equivalent except kronecker trivial temporal reduce via spatial become stage project clutter achievable case clutter gain turn analog apply temporal first instead stage occur spatial specifically show loss give spatial fix target integration follow singular giving move target ideal sample regime clutter temporal small spatial still use release dataset circular move formation divide integration truth currently dataset complicated real resort roc gain experiment reference target target section plus bin kronecker covariance synthetic clutter learn spatio spatial lr mean training filter clutter slow mse go added testing clutter instead value potential potential range contain clutter move target filter response vector statistic
polytope percentage cover decision attribute capable discrete provide vector categorization output requirement formalize definition dimensional mmd attribute partially great space polytope identify coordinate class counting instance value constraint datum decision datum structure tree kind convert space besides classifier train range pass minimum range small find consistency sake range shape svm shape assign rectangle convert define axis family impose intuitively along rule figure space use attribute strictly pattern actual bind consider pattern specify attribute pattern attribute range denote find attribute denote follow exactly coordinate specify attribute mm xx xx xx rule convert element one rule conjunction pattern element space pattern low op include space op lower k low bind upper mind mind upper pattern bind exclude space space element fundamental merge span value refer space merge use principle merge principle subspace intersect prediction vector explain least algorithm merge notation way element intersect attribute cover space cover element word element cover property establish decision component intersection element denote first principle handle third resolve two create assign specify cover weighted average space intermediate introduction contribute mx space normalize indicate specialized base small metric pure meta learning use repeat overhead metric distribution attribute attribute formulae specific require differently ensure formulae merge merge algorithmic merge add contain xx xx xx new associate value value resolve create initially share add decision never handle else update previous handle solve space element return intersection cover space cover definition remainder remove remainder merge element merge principle principle partially avoid already hash map cache hash map update intersection intersection remainder add figure range range range age grey pattern intersect convert intersect contiguous age range degree range percentage percentage attribute show decision contain rectangular element rectangular rectangular remainder element use algebra element computational reason merge operator merge consistent another depend identity space merge neither theorem merge satisfie algebraic merge geometric empty intersection union intersection resolution algebraic therefore combine appropriately operator consider could keep result change show keep result add create consider unchanged relate intersection space involve element intersection intersection place intersection track remainder updating create remainder add respect empty space first element loop execute show identity second cover divide line different two strict particularly element discard lead incorrect add element vx mx vx mx mx xx yy z hz line unique space algebraic structure operator merge get remainder value result merge remainder merge depend cover intersection thus result operation operator result depend merged scheme sequence merge specify decision merge merged representation merging specify decision merge represent tree leaf decision application represent final two merge decision merge pairwise merge describe x x introduce decision space merge bias space merge merge third describe merge decision merging impact impact space reduce decision space large merging desirable merging show homogeneous partition among unit final merging scheme time distribution example stream time may trend situation soon receive merge everything stream want model receive specifie operation place develop prevent result number furthermore power decay decay generalize overcome limitation subsection decision space binary operator combination handle computation merging introduce within discard otherwise intersection assume corollary merge concentrate merge merging space cover proportional operator path leaf merging scheme node leaf subtree account root internal scheme account account fraction number contribute fraction value root merging final path show scheme value merging x n space account scheme mx explain early space merge exponentially storage provide impact merge merging value vx formula value bias merge one merge order use value vx mx vx mx mx mm mx x x vx mx vx mx mx prove merge vx mx vx mx mx vx mx mx vx mx mx vx vx mx vx mx jj mx vx vx mx mx mx mx vx mx mx vx mx vx mx decrease new create carry information technique time could consider space vary attribute evolve broad restriction introduce operator apply algebraic merging space intersect mm space intersect element line unchanged algebraic derive property every restrict element decision space element element matter take continuous range unbounded identity space space create line space matter merge compose variety merge create create operator restrict element significant two consensus weight restrict exactly operator definition reduce intersection merge intersect attribute element lie intersection measure identity sensor classifier instance classifier merge global merging classifier create classifier examine operation classifier merge operation classifier merge property merge operator show merge behaviour apply soon trend bias bias also homogeneous achieve bias storage bias operator use restriction combination decision rely element shape intersection large attribute space shape shape become hard direction shape complexity suggest complexity type choose many scenario distinguish type uncertainty concept element predict consider dimension measure since fuzzy become fuzzy value predict class yes detect element make contribution mechanism transform measure represent transformation element value vector combine solely belong confidence generate supplement raw classifier interpret reliability type classifier classifier decision support general performance large specific combine transformation uncertainty element consider type take beyond sample area essential system framework address idea algebra datum highlight decision tag result union intersection table merging element guarantee element value uncertain element combination answer advantage lie mathematically express helpful discussion research observation classification observe environment among integrate merging furthermore classifier merge setting ad hoc framework possibly classifier merge desirable discuss main mining setting decision stationary impact decay database partition learner ensure model storage develop also operators meta entity entity record database tuple change view classifier prediction often might global resource storage entity arise illustrate international classifier classifier way group key distinction p classifier specialize restrict instance classifier design typically classifier classify say instead operation several firstly classifier differ prediction heuristic g meta train undesirable cost transmission consider case meta potentially type meta describe previously validate researcher suggest question understand classifier combination mechanism formal algebraic investigate behaviour propagate local combination support kernel rely observation heuristic merge merge
reference signal controller illustration image display feature correspond experience previous value value ahead optimize planning penalty greedy conduct trial auto display relatively prediction advantageous compactly figure controller start current trial trajectory experiment stage display case gradually number trial case framework policy task use collect transition auto find good material detail policy computing nonparametric overhead day run overall policy exploit dynamical learn fairly datum rl learn action space deep dynamical controller need crucial learn feature mapping jointly term art rl learn quickly scale learn pixel acknowledgment foundation dynamical contract number college research draw thin em efficient remain develop fully instance challenge must loop learn ingredient use deep auto learn dimensional also datum crucial long lie predictive control strategy art action scale toward fully influence many decade devise process image summarize behavior environment uncertain adaptation rely aspect scene camera robot configuration agent environment want action space natural promise toward pixel reinforcement rl principled mathematical deal prohibitive working use efficiently keep experiment dynamical internal rl dimensional use pixel possess thousand dimensional dimensional network stack auto art parsimonious high successfully google amazon facebook rl deep game purely learn slot raw employ deep architecture find representation however neither either discretization low limit applicability exploit transition internal purpose employ low feed forward dynamical closed loop practically information however exception neural data direct access principle along suggest solve rl need scale approach propose directly unlike exploit classical horizon objective minimize cost tf tu control face additional challenge trial set practically agent robot video robot fully jointly deep auto learn forward control input property compactly dimensional predict paper measurement encode measurement neuron draw none execute begin node cm count neuron fill neuron try input count miss try neuron count neuron neuron try neuron try fill neuron try output count north node dim hide output reconstruct align count align tm decoder map high observation none scale cm miss try neuron try neuron try every try neuron try neuron try every neuron try neuron try neuron fill try align high dim align leave z node control iy tu h color represent dimensional image encoder decoder deep compute activation layer e image g k number encoder ty encoder negligible compact auto dynamical encoder allow step history input feed forward nonlinear model performance control section exploit ready piece prediction encoder predict image prediction image decoder auto encoder minimize reconstruction learn auto feature transition material gradient ahead multi crucial pca exploit auto auto compute high recursively auto restrict low exploit use policy optimal sequence minimize small sequence control signal dynamic determine control sequence trajectory cost associate trajectory control determine control observe entire loop exploit predict requirement reference frame encode reference function online work exploit predictive image good turn together learn states input inherently prediction quality dynamical learn controller fashion gradually without system collection close loop divide multiple sequential follow strategy collect use record simply apply controller close converge reference imply collect include suggest imply strategy latter choose greedy exploration feedback select initialize greedy happen collect propose methodology framework image input gray angle angular velocity deal
asymptotic member exponential family distribute random define diagonal invertible coefficient column th nearest euclidean summary value correlation undirecte represent magnitude correlation I vertex event equivalent hence relate poisson summarize theorem row reduce diagonal positive variate dispersion valid differentiable everywhere except continuous component member family unknown follow q note parameter detection sequel correlation density reduce subscript plot various consistent arise purely summary use detect sequence summary pre dispersion sparse summary depict theorem diagonal pre dispersion diagonal problem unknown post dispersion assertion stop away form member parameter change summary model due rule optimal fix supremum achieve ensure leibl density asymptotically use variation suggest dispersion row pre variance post wishart delay log alarm change value test alarm post alarm approximately kullback large delay value parameter divergence delay alarm fig quite accurate similar simulate size detect htb predict ij mining discovery parametric sense among detection statistic future experiment real random partially verification technology department energy nuclear na remark claim problem base change point consider row belong density parametric correlation stop asymptotically sequential detection sequence post change distribution belong finance failure change stochastic finance stock detect interaction dynamic sequence random time experiment case coefficient successive well separate change series reflect finance stock change day week consider problem dispersion sequence paper precise datum know detection neighborhood statistic big summary parameter treat screen mine specifically discovery decision time decision stop maker subject alarm change overview problem minimize suitable metric delay subject false alarm correspond user time false alarm pre maker sr family sr optimality formulation post change parametric strong asymptotic
organized smc simulation genomic integration discuss application reason result give supplement tool develop supplement block notation rest row index indice matlab integer represent alone entire stand nb iv diag decrease value b iv case frobenius onto rank gap tail provide accuracy detail relatively analysis perfectly recover show svd fail observe supplement block fails recover high addition include constrain relaxation k k smc row well small method would unfortunately rank assumption unstable failure recovery approximately rest sequel u k obviously obtain estimate subsequently recover use know locate principal h u nu u typically serious near mis specify overcome difficulty introduce know heat htbp first move front row clearly orthogonal unclear back row remove helpful r r rt recursively finally propose h v v equation singular break algorithm construct column nearly singular large non investigate theoretical property section lower together certain class approximately choice tune discuss corollary helpful explain intuitively dominant block necessary dominate theoretical theorem significant gap properly lead accurate recovery exist constant q thresholding besides involve u singular quantify difficulty hard class rank theorem yield principled choice generally depend setting rank least give randomly column decompose replacement respectively satisfy column thresholding break number row algorithm uniformly select replacement necessarily column break satisfie amplitude row amplitude row probability mean hard case generate orthonormal column sample haar uniform break parallel orthonormal haar measure column column break examine numerical matrix setting singular exist investigate setting smoothly long sensitive compare smc finally replication supplement generate diagonal accordingly different setting haar measure specifically I significant singular q major loss spectral loss perform small get large gap adjacent work singular fast htbp singular singular demonstrate continue well set threshold affect report thresholding column thresholde similar norm frobenius norm decrease high decay fast across optimal choice thus fix htbp cc versus column decay original generate randomly choice j penalize recover solve supplement set singular propose method substantially outperform frobenius next decay decay study regression simulation randomly fix table relative loss small substantially penalize htbp ccccc frobenius lr lr smc procedure integrate genomic cancer cancer relatively heterogeneous substantially year survival cancer majority iii iv disease year heterogeneity part underlie lack successful treatment strategy motivate genomic identify molecular signature help optimize example cancer genomic sample gene highly survival include cluster expression interesting survival compare alone limit validate construct denote facilitate alone measurement imputation purpose imputation previously gene signature cancer unique gene information imputation imputation gene expression remove platform batch indexing subject indexing thresholding suggest penalize imputation yield leave observed variable smc imputation recover projection predict survival select marker marginally survival nominal lead principal component marker survival imputation integrate study substantially assess survival fitting cox integrated cox observe individual hazard integrate magnitude base substantially se size reasonably study compare combine study meta improvement statistic suggest compare smc smc c summary suggest procedure accurately lead add imputation significantly improve confirm method genomic smc correlate pc outperform conventional method completion analysis adopt smc poorly number reasonable cancer signature highly pattern gene expression signature gene imputation promise signature expect imputation sequence particularly gap dataset analyze smoothly significant gap fast theoretical smc implement major decision thresholding row row thresholding thresholding close randomly close consequently row provide recovery implementation multiplication sd possible accelerate computation space work acknowledgment associate constructive comment supplement matrix completion genomic supplement additional theorem key main consider effect generate haar measure thresholde different loss similar row thresholding interested vary set see increase accurate keep decrease converge increase collect technical first singular perturb suppose n standard divided block submatrix suppose orthonormal submatrix haar exist know ba nr proof end theory q convenience extend p equality achieve minimizer p two symmetric replacement eq know matrix clearly need prove must measure beta unit construct net v imply n finish set corollary row row besides submatrix namely linear p perturbation yx u ax u combine n stand supplement finally lb work orthonormal basis denote exactly proof due assumption supplementary z nu besides denote span span space want try actually besides perturbation inequality v derive follow characterize supplement finally separately b proof fact break show break adopt hence orthonormal eq prove know prove construct differ fix number real small b svd specify use construct svd eq small q q column know hence theorem part besides take transpose similarly eq corollary haar find happen q minimization comparison base integer denote
gibbs sampler parameter denote discard validity check assess good ss require adopt estimate replacing ls square estimator employ parameter aic bic ss ml exploit practice square access ls estimator score compute impulse compute test noiseless ht identification evident despite cause quite scenario difference impulse response ht monte compare oracle ss ml l method ss standard non ss ml moreover introduce identification subject particular gibbs exploit conditional highlighted numerical experiment affect hyperparameter theorem assumption example theorem conjecture remark se introduce identification impulse response process kernel spline information start identification framework employ markov monte provide gibbs sampler converge substantial art identification system area communication bioinformatics identification constitute dynamic system cause standard identification square performance technique system propose paper tailor measurement fashion exploit performance specific handling quantization recently identify exploit identification similarly impulse system gaussian mean system identification purpose permit flexibility identification e hyperparameter maximize marginal integrate impulse mean bayes g system admit think solution end system obtain non hyperparameter unknown noise impulse response contribution show distribution criterion sample quickly target base popularity identification organize introduce dynamic system system propose performance conclusion end eq impulse characterize unknown fed time corrupt non whereas shall consider exposition paper measurable version particular guarantee system determine formulate system impulse response independent th row entry hold truncate know marginal think follow hyperparameter posterior improper prior support variance equal case assume improper situation mild measure sufficiently reliable obtain replace estimate recall unfortunately bayesian still hyperparameter deterministic discuss section bayesian model previous system identification drop impulse computing section use carlo function close solve special namely sampler e idea stationary state markov sample conditional depend density factor becomes gamma distribute carry redundant discard density expression mean covariance matrix density position propose ht input initialization initial draw sample clearly large guarantee number discard conditional draw iteration get
switch switch matrix stochastic focus analysis rule signal uninformative epoch time step agent exponentially fast inefficient costly demonstrate achieve rule communication proof concern behavior agent switch almost log assumption imply switch agent follow invoke graph existence real number connected node leave technical switching dramatically communication future focus would turn threshold strategy proof agent recalling since equivalent number signal assumption apply view lemma tv sure neighboring agent eventually since switch almost recall write preserve limit derive surely guarantee since identifiability take convergence per exponentially fast asymptotic vanish denominator routine routine nf attempt observe private condition true sense distinguish benefit side observation rely protocol propose efficient switching bayesian regime exchange informative regime efficiently communication preserve cost verify distribute attract decade application range sensor economic scenario spread adequate truth result instance agent observation achieve belief social benefit private learn unknown exist learning focus mostly individual particular seminal observational agent perform use linear belief opinion inspire rely time protocol rule et effective switching topology assumption agent belief change drastically private agent private bayes private use average refine opinion neighbor observe signal give private criterion switch bayesian vice versa regime agent use update non agent rule due evolve time mild able switching regime efficiently provide discus remainder organize describe characterization switching subsection follow illustrate concluding remark future direction proof denote number capital letter letter vector denote transpose interact doubly assign edge agent j agent decide refer prior agent agent triple assign consistently I identically epoch initially kl kullback divergence strict induce agent false agent detect false globally identifiable paper marginal true path every content signal guarantee likelihood make uniquely identifiable aggregate guarantee end node exist direct time instant mass represent agent convergence sure asymptotic sure formalize agent learn true surely realize agent initial opinion initial observe give calculate vary update detail evolve observe realize opinion alternatively bayes signal information elaborate update incorporate belief particular likelihood neighborhood refined opinion view repeat update contiguous interval neighbor interval communication successive private verify choose protocol recover case characterize bayesian protocol switching offer informative influence agent opinion private private b threshold case binary agent
consider even behavior hoc center lead meaningful asymptotic one less bootstrap usually performance trial independence sequel via cm trial draw count seem natural avoid diagonal I particular square behind independence drawing index h c c permutation permutation uniformly permutation index use avoid pick twice train sum however bootstrap q bootstrap resample thing randomness precisely satisfy unconditional c ts trial fire hz window line conditional approximate simulating times randomness trial approximate carlo obtain thank line pick twice trial full bootstrap even observation realization unconditional conditional visual unconditional unconditional bootstrap testing quantile conditional quantile value reasonable nh happen conditional surprisingly none three may eventually gr develop computation carlo exact line figure set approximated distribution basic paradigm nevertheless widely spread aim explanation one explanation center prove mathematical center particular explain mean sufficient replace empirical bootstrap denote bootstrap conditional fit see construct impossible quantity observable nan check u couple j nn correct similar computation show directly h black ts ts ts ts u first line simulate obtain quality either simulate accordance wasserstein prove accurate statistic explain computation trial correct finally intuition work exactly figure indeed account approximation finally extra simplification permutation may surprising rewrite action permutation quantile conditional conditional distribution close c work base work phenomenon permutation full investigate purely one critical approximated monte method pass acceptance conditional usually realization value despite keep test present statistic value quantile way may permutation equal indeed test slightly version note correction five trial u version couple e firing rate hz window length firing rate hz trial f exactly equivalent f small gray represent center window detect horizontal correspond vertical plain dependence dash negative vertical separate left dependence dependence cm accept c fdr fdr discovery discovery rate leave adapt test fdr run fdr homogeneous firing rate hz window trial carlo approximation windows correspond theoretical dependence ability publish experimental old delayed point front vertical panel sensitive light six place degree hold delay ms ps six green delay ms ms turn red pointing first signal ms pass occurrence change reaction rt release movement mt record seven hz filter hz window spike neuron isolate along behavioral event signal store pc trial rs consider expect confirm pair previous consider already presented detect permutation window detect behavioral may false pair neuron unitary count sign correspond behavioral black vertical bar ps vertical bar es bar response rs describe delay count focus recorded train count count naive distribution suffer sharp trial approximation turn bootstrap namely method independence test real simultaneously record spike train message method center approximation second phenomenon combine observation classical think center apply center statistic approach first line still figure correspond thank make run gr exact algorithm count elegant really possible long monte simulation use work bootstrap value trial bootstrap contrary adequate quantity precise behavior small two right decided delay count much apply independence several individual test use treat window name well discovery classical trial test take table datum exist likely thank conclude article permutation unitary event delay free suffer count despite prescribe control fdr trial compute sensitive reasonable definition delay count recently notion still question acknowledgment access centre de universit nice partly la bs dependence graph region cm event base delay count fr fr fr fr france nice france france keyword unitary testing investigate several principle unitary permutation delay test prescribe testing simulation single fdr negative area nearby electrical occurrence potential spike neuron one record spike train dependent detect among popular gr un apply decade vast amount therein main popularity precise period degree substantial develop rough level low hundred may induce idea keep level despite define shift ms want analyze value poisson bernoulli commonly validate accept spike train thorough surrogate datum particular trial assess trial available bootstrap paradigm assumption underlie method always whose parallel practice main intensity fire poisson practice replaced take work unitary event include delay suffer detection process propose delay multiple preprocesse permutation propose delay trial finally similar method share drawback sequel two point spike train neuron couple independent observation I copy notation couple x correspond expectation neuron distribution stand event stand denote delay neuron potential spike train due temporal spike informally spike neuron delay less order several count process delay eq informally count bin contain one spike count sequence count recent delay count introduce shift define discretize necessarily point delay count point informally spike one delay delay given sequence order length two point assign assign govern interval length precisely step point segment namely homogeneous poisson intensity require parameter hz delay linear advantage exploit train new corresponding trial statistic either practitioner would choice observable compute denote several paradigm reject reject quantity critical
tree illustrate element formula publish two empirical literature highly mining piece extension use social participant hard innovation participant numerous introduction refine variance design estimator propose effect eight large variance bootstrap perfect converge slow result network tree relate string variability chain sampling index primarily motivated markov chain indexing node text reference markov mix apply sampling apply visualization store retrieval various bias sampling mechanism mining seek edge conclusion extend entire graph highly rather theory mathematical participant friend piece literature refer transition simple walk friend usually piece tree child draw subsection give notation associate people adjacency graph element friend node matrix define pp less absolute spectral extensively one social bottleneck satisfy markov simple walk term estimate many friend population estimator thompson spectral calculation lemma chapter piece let reversible orthonormal lead step write play fundamental lead eigenvector process community east west walk east west partition sign look sign partition east west make concept spectral let rooted graph cycle vertex unless fix seed node parent step close denote property chain call state initialize stationary contain index participant sample example node person individual network represent height number denote characteristic e wish estimate subscript sample thompson normalization properly thus estimator estimator transformation estimator study independent negligible second eigenvector second eigenvector social graph west potential correlation contribute bottleneck irrelevant mean constant functional select node uniformly tree graph generating function function eigenvalue one absolute define eigenvector correspond closed variance number grow eigenvalue remain unchanged function change piece two transition q summing motivate grow tree imply term dominate satisfy exceed variance outcome interest bottleneck enough var obtain design effect necessary depend otherwise design give bound interpret subsection refer exactly future refer iid select distance seed node diameter define slow correct dependence design effect converge fast converge rate subsection political blockmodel study network indicate support feature node six feature negligible portion whether population white worker drug follow use large component political contain average display political colored lead emphasize eigenvector figure individual title give create bottleneck political bottleneck lead likely error display sample provide decay especially htbp display five highly display network eigenvalue great instead figure cumulative covariate composition population display panel composition panel leave center right panel average increase political standard covariate line jump indicate variable panel correspond make line covariate five translate correspondingly computable tree previous across size effect legend drug city tree study tree line strong tree present horizontal strength potential bottleneck legend tree line sensitive illustrate practice reason practitioner obtain ensure large tree much present effect bottleneck preferred sample achieve variance realize illustrate sensitive insensitive bottleneck critical identify drive network node theorem close form construct drive combine follow eigenfunction eigenfunction tree rate rate converge give critical ignore ignore match understand balanced require illustrate analytic simulate empirical political relate political quantity subsection design synthetic empirically tree sensitivity large distribution herein disadvantage fundamental obtaining error avoid eigenvector require define reversible node v x yx yx ease subscript yx yx xt constant yx yu xt yu p yu yu yu yu yx proof completeness proof inequality loss generality variable pairwise cauchy nb eq nb procedure repeat function towards zero sequence upper bind jensen next use notice fact fact upper bind di di k bit growth term idea apart two di k di j j di k c h di k growth rate hz di z contribute q fraction proof fourth balanced distribute denote moreover single generation I drop correspond iid correspond wish random borel exist variable balanced chebyshev constant eq borel theorem w conclude convert section proposition helpful thank liu helpful course web construct index tree correspond observation indexing chain sample unit popular estimator effect network critical social eigenvalue large bottleneck finite design effect grow long converge slow introduction drastically statistical take classical population individual sampling frame available cover response become typical survey difficulty require frame interest network reach people use reach go name sample united provide researcher reach population friend context population public drive technique population e people worker conventional technique international quantify population include center control health organization united serve result herein model walk person exactly model index individual indexing
shown model bm sample gradually outperform rbm adding rand cv trivial connection add complexitie rand estimating rand increase model sample size increase cv gradually rand cv principle preserve confident real could benefit way kl divergence sample real simultaneous rand complexity second third column rand cv divergence sufficient cv rand tend select visible rbm difference rand rand fourth worse small gradually outperform balance complexity could simultaneously useful reduction problem theoretical side principle maximally preserve confident confident theoretically general work orient interpretation bm building block deep boltzmann architecture sufficient abstraction describe flow transformation illustrate maximally preserve confident achieve tradeoff preserve layer architecture maximally preserve indicate fisher confident specific adapt series density plan incorporate modify confident could respectively partial derivation partition part cc verify coordinate imply fisher diagonal er unique asymptotically fisher involve parameter recall block ball surface kullback sample fisher fisher fisher exist diagonal matrix apply decomposition become rotation tailor index tailor mixed fundamental property ellipsoid monotonicity base hyper ellipsoid ball surface indeed surface ellipsoid eigenvalue surface main eigenvalue ellipsoid parameterize term axis eigenvalue ellipsoid spherical coordinate prove diag diag ellipsoid surface integral definite prove integral partition finite let sum limitation sum multiplication arrange order monotonicity ellipsoid coordinate extend ellipsoid hold monotonicity preserve dimensional standard ellipsoid monotonic maximize preserve top eigenvalue block element bound upper operation affect integral give maximum complete proof element bound x obviously diagonal diagonal great complement bottom equal hence complete vector l h I realize prove mixed projection mixed coordinate tailor stationary learn thus denote datum since define solution gives hence preserve complete proof complete expectation respectively equality expect divergence vanish completes prove uniqueness projection bm thus unique divergence fast treat bm learn gradient choice coordinate exactly ml bm distribution manifold bm treat unit visible projection qx theorem corollary em plus em minus height width depth orient focus feature situation limitation method scale datum feature aim consider oriented dimensionality space propose call confident preserve confident less confident parameter assess fisher manifold neighbourhood boltzmann bm perspective visible bm general formalize essential bm aim bm discard theoretical sample sample study series boltzmann belief stack auto encoder deep attention application vision language despite principle search difficult introduce region parameter could capture data distribution thus aims learn meaningful representation parameter describe since block deep architecture formal essential part density bm parameter underlie respect small occur trend moreover overfitte adjust complexity selection could criterion I confident universal probabilistic model system phenomena parameter reduce parameter adopt various criterion rao formalize theoretical general dimensionality manifold smoothed free restrict major difficulty choice keep geometric preserve project distribution perturb true surface assumption without well belong e define problem maximally preserve rao distance unique close fisher distance information assign free parameter appropriate free neutral zero use section advance principle close turn maximally estimator estimation normal covariance asymptotically er rao ml exponent opposite square respectively maximally preserve projection maximally e maximally effectively class class beneficial sample among reduce noisy maximally preserve capture dominant discrimination class principle fisher decompose two orthogonal parameter contribution former distinguish true confident minor reliable preserve confident confident optimal equation parametric reduction fundamentally reduction extraction focus offer deal scale contribution intrinsic datum rise maximally preserve confident confident one binary analytically parametric boltzmann bm visible bm bm unit certain propose scheme experiment develop manifold simplex foundation ig family differentiable manifold parametric coordinate multivariate coordinate coordinate exclusive index regard nan index indicate zero coordinate respect denote ij solve subscript order parameter index convention coordinate equation meet identity coordinate define great row regularity partial derivative measure carry inverse fisher tight consider coordinate call vanish influence uncorrelated rewrite j g otherwise generally develop proposition generalization information q probability three calculate g g p j give share n manifold could target reduction reduction construct confident distinguished confident one confident confident neutral confidence parameter assess contribution distance coordinate usage infeasible coordinate since orthogonality hold show mixed coordinate bm neural visible hide stochastically depend visible interaction interaction visible interaction visible self hide connection express boltzmann joint normalization factor boltzmann realize bm actually see role coordinate bm general bm unit rbm rbm sample ml use ascent bm maximize equation likelihood phase sample positive sample stationary second call negative estimation adjust gibbs phase avoid difficulty gradient follow markov run denote cd expectation end number dimensionality endow design direction coordinate define maximally preserve fisher rao learn stationary exactly stationary ml uniquely hide preserve unit bm distribution bm bm visible leave bm activation bm due bm manifold projection framework rule learn bm theoretically algorithm reach fix iterative property iterative guarantee hold projection proposition minimum mixed coordinate unique one proof ip investigate confident neutral equation part learn bm hide exactly confident empirically density boltzmann machine adaptively bm bm modify confident connection give hide restrict rbm connection visible unit emphasis confident among analysis use confident expect denote meet incorporate adaptively follow edge confident graphical comprise assess could follow infeasible tackle connection orthogonal decompose independent coordinate respectively fisher fisher note equality hypothesis nan alternative chi investigate rand cv perform connection fold validation topology adaptive artificial jeffreys dimensional learning cross kl fit various size generate distribution focus divergence give offer trivial result qualitatively variable number reported average could n relatively sample effect gradually cv rand cv performance connection column complexity rand term theoretical insight confident cv gradually rand cv explain preserve confident sample increase benefit rand column
show category image non property play role superior sparse solve reflect sparsity loss max numerical school sciences technology china china wang mathematical technology china china technology china china ny usa mail wang com recently successful capability incorporate sparsity representation code pyramid matching transform descriptor non character experiment show improve use sparse iterative part research vision machines pyramid matching model extraction treat document keyword text match document frequency keyword apply method image processing turn image histogram capturing shape object discard pyramid match successful model correspondence discriminative codebook generative partition segment histogram promising image illustrate sift extraction firstly descriptor sift descriptor extract image codebook descriptor code layer pool average sub image task chi nonlinear complexity svm impractical pyramid achieve art image categorization year like locality code image representation code widely pool mainly representation image however max sign coefficient condition consider representation sum coding section describe propose present two basic image quantization trade balance fidelity term trivial solution unit typically normally overcomplete e consist code phase restrictive constraint sc achieve much code image patch help salient model popular due sparse work image difficulty behavior hard structural incorporate follow satisfy component max pooling bring moreover propose convex non code iterative truncate nonnegative component correspond remove plain practice able value coefficient regularization remove truncate kind practice correspondingly short reliable solution correspondingly alternate repeatedly take convex iteration truncate detection name describe give extract codebook initialize stop universal l magnitude estimate pooling pooling define th row report comparison convex implementation especially window matlab gb descriptor widely use
u h remains show inner need g g follow h g h j nh g unitary nr f nd r fx ix n dx ix ix fx nr product uniquely consequence algebra department engineering university ann mi statistical problem measure mixture model parametric instead measure uniquely provide moreover latent identifiability hilbert base number measure realization random drawing mixture primary question concern mixture model identifiable explain identifiability consider iid impose group component call group sample mixture random element identifiable simple paper show identifiable per improve regardless mathematically measure probability arise naturally see concerned extraction sort structure popular assume question latent represent determine statistical utilize interested different perhaps regressor another relate dataset directly measure anomalous similar paragraph probability sort consistency require make unnecessary consider penalty fixing clearly theoretical couple important implication firstly practice twitter keep analysis lose result seem suggest technique significantly identify past couple decade application identifiable sample identifiable cdf mixture closely measure domain group show different rely theoretic basically collection measure technique tensor tensor proof totally algebraic previous tensor treat measurable dirac measure fold contain denote probability dirac unique ambient mixture probability follow probability want law mathematically bit construct integral principle must exist representation index measure mixture permutation henceforth summation summation minimal mixture component measure dirac define minimal integral algebra sign v minimal law derive v definition central object interest mixture measure measure modelling collection identifiable mixture complex practically random confusion describe literature illustrative map tensor separable demonstrate give dy dy g beyond tensor product hilbert unitary u h u hilbert decide associate span product introduce rest modify purpose connect product tensor product unitary u follow unitary unitary n u nh space hilbert proceed induction clearly finish without hilbert schmidt hilbert schmidt hilbert space unitary operator linearly inductive previous define time continue measure finally need technical derivative product nonnegative n proceed exist measure l l assume simply side normalize pair mixture share let lemma v derivative derivative equal multiple example p l therefore everywhere generalize first suppose evaluate yield p contradiction pair satisfy cf ff nonnegative without p ii since unitary lemma unitary r dimension h h I linearly conversely remove follow exist generality thus p I mr right combination let k know exist orthogonal z p apply p distinct measure whole
evaluate biology preliminary material conference paper preliminary analysis derivation comparative genomic biological make serve benchmark show outperform particular refine task multiple structure view cover formulation formulation box kernel another tailor kernel toolbox empirical artificial well dataset task comprise us insight special model regularization supervise functional term measure regularizer control trade easily task interested past discrepancy st st similarity develop multi allow similarity learn equip weighting instead priori automatically part comprise line include line also jointly make loss besides connection importantly novel special label independently measurable training point mt encoding view multi learn mx mt tw mt vector concentrate encode loss task r ms tensor space direct sum hilbert us hilbert regularizer insight paper representation primal problem base theory space review remove dependency need adjoint I definition adjoint identity prop example way retrieve index task alternatively index use adjoint map conjugate write prop mr mr furthermore q supremum optimum w duality dual problem partially primal maximization present formulation completely exploit first note affine exchange c c dual completely solve optimality assumption differentiable requirement analog kkt stationarity lagrangian note rewrite definition previous rewrite optimal introduce kernel generalize multi loss function several multi dual hinge increase start single task standard towards novel many multi conjugate hinge loss verify calculus q c hinge briefly may single onto thing first believe facilitate reader familiar task formulation yield non sparse multiple equation greatly simplify svm give mkl obtain special case case mkl corollary restricting task first case section definition similarity appeal fix assign pair example task publication domain two task fix idea base similarity task read express idea center following group within respective term regularizer cluster assignment regularization center assign least primal formulation may regularize constitute date regularizer framework adjacency encode task view regularize task laplacian invertible regularizer eigenvalue relation capture relationship task task dual involve weighting bioinformatics form furthermore consider sum kernel corner kernel dual interesting within consider also constitutes actually choice discussion kernel relevance work ahead novel importantly mkl engine mt mkl tree similarity mt mkl exist similarity combination graph adjacency laplacian extension readily task multiple kernel access suited laplacian weighting give prediction accuracy couple equation coupling advance formulation introduce task weighting decompose arrive term tree graph assume relation computational biology different task expect evolutionary history beneficial share terminal terminal node task discuss regard mt mkl require give square length scale use mt mkl weighting length scale hierarchy trade task transform scale algorithms matrix mkl implementation tailor large allow datum demonstrate set employ efficiently computable certain string toolbox convenient way mkl completely completely mkl along without step mx constraint eq whenever update need objective decrease epoch need keep date change result algorithm computation feature map cf iterate line stop define line kernel primal last objective alternate h task matrix precision initialize inverse satisfied accord store primal primal decrease hypothesis mt task kernel implement toolbox implementation classification furthermore optimization mkl solver use analytic cut plane novel computable conventional svm integrate perform require modification currently module lastly scheme describe module implementation truly large mt string interface module mild exist strictly contrast concern direct rd rr descent q set continuously minimizer cluster minimize data domain similarity iterate cluster corollary unconstrained put indicator shorthand problem sequence thus order recall initialize f q cn cn cn eq function finally trivially fulfil employ paper string fulfil infinite kernel exist representation map framework range control toy experiment diverse review experimental work closely one case investigate genomic computational biological early h task independently pool evolutionary power biology illustrate case multiple successful application computational biology joint multiple problem two experiment describe sequel beyond investigate framework generality evaluate hierarchical artificial generate vector inspire evolution hierarchical accord leave root node subsequent hierarchy tree leaf carry dot product pair figure clearly valuable leave couple mt mkl create node mt mkl mt union combine treat task report roc accord detail comparison mt roc mkl good perform margin suggest beneficial next simple considerably improve performance observe improve non mt mkl mt mkl accurately identify genomic sequence genomic whereby rna copy genome sequence genomic resource bring annotation nine take annotate select jointly learn treat initial similarity extract similarity genomic hamming rna genomic change evolution different class similarity refine mt mkl create task similarity mt mkl exponential task different collect include positive label consist mt mkl scheme split split validation ten set nine svm pool task improvement union individual indicate similar discussion improve least marginally individual seven nine individual attribute matrix speak propose mkl eight bad improve mkl achieve task similarity able learn mt mkl beneficial several believe biology potentially application multiple computer security present regularization refine kernel hermitian numerous primal formulation hinge integration toolbox framework could norm mkl primal efficiently solver special software machine toolbox term predictive intersection analyze outperform baseline theoretical good computational great instance international drug breast early helpful foundation well european support research foundation grant kl support duality real present reading refer introduction duality machine presentation conjugate hilbert g g supremum affine duality indicate conjugate semi appendix definition adjoint real hilbert euclidean appendix known duality hilbert theory helpful computation proper hilbert g hilbert space conjugate loss conjugate note unbounded supremum translate show rgb remark corollary center york york ny usa computer university cancer york usa computational center york ny multi task similarity refine multiple mkl general
via tune r training convnet layer select inference module integrate principled efficient extensive demonstrate efficacy convolutional novel pooling yield demonstrate capability statistical model ultimately map plane map multinomial block next set impose activation pool bottom initially sequentially bottom layer refinement learn jointly readily variational image hadamard product indicate assume layer top layer stage view plane pool map partition contiguous pooling pixel pixel location block pixel stochastically pixel equal amplitude associate block max learning part fig constitute excellent initialization refinement top generative process constitute w
analogy especially well suited sense lack make also token improvement al focus describe understand paper neural language extend traditional gram represent token indicator jointly train replace likelihood character model network gram incorrect replace normalization probability distribution consider extremely computationally log linear neural language training context window language learn neural embedding nlp dependency parse sentiment document name recognition parse among less word construct multiple representation extend document multiple dense embedding word sense cluster context expensive token learning mapping type embedding skip gram learn predict sentence gram model word vocabulary embed probability eq context sequence noisy consists sample noisy context w window maximum window noisy context noisy randomly sample tw skip sense induce cluster embedding token context word vector type maintain sense token predict predict token embedding perform jointly predict use current tw tw w close associate context vector let v global avoid complexity predict formally mean context belong similarity context sense observe embedding noisy skip predict predict word global sample noisy update randomly tw tc ts tw tc tv tc tc k learn number np vary relate non type proportional near vector initially vector cluster create new vector create online word r vector order skip gram skip microsoft pc l gram abc tv skip gram net try offset loop pre ball tv roll np run run run run operate operate np walk run run walk operate go limited run present neighbor associate various embedding near neighbor word compare similarity embedding parametric skip p skip ms sg seed sg seed comprise drug storage power physical agent np two related contextual similarity evaluate vector rate include contextual make overcome issue stanford word pair context noun noun noun noun evaluate embedding corpus since per contexts sense measure embedding k compute embedding address metric pair fit metric ignore select word independently probability cosine center report similarity human dataset r tf gram skip gram tf al np outperform measure bold face skip np np np tf representation face dimensional show well network dimensional perform slightly skip task give achieve art achieve measure metric since well improve word analogy introduce np compare state skip gram np show np model embedding fair mainly frequent vocabulary multiple top even embedding frequent art show present extension gram embedding word perform word sense type state word task token embedding nlp acknowledgment support center reproduce finding recommendation material author necessarily reflect computer science edu interest space embedding nlp single ignore thus usefulness skip learn embedding recent jointly word discrimination embed art machine corpus token represent dense value commonly help curse improve generalization semantic syntactic dramatically processing benefit arise volume considerable gram log high wikipedia day much common input name extraction parsing substantially continuous name extraction continuous skip supervision similarly dependency parse skip embedding recently apply notable prior string embed approximately contextual relate biology moderately space close triangle word example without triangle discover embedding multiple
order conduct synthetic remark preliminary experimental nevertheless nearly histogram sort add overhead need sort match quantity conduct use intel core mb cache ram mac os operating report error illustrative sorting point std sort take distribution gaussian gamma support size consider differ style width axis title north inner sep grid xlabel number ylabel shift ylabel label font style width list blue mark mark legend style anchor west format legend style anchor west style format anchor north east title gaussian density gmm txt cycle title beta table plot beta txt title histogram pdf give sort algorithm histogram constant piece record time exclude achieve result scale three nearly size constant note three sort sample run essentially desirable show achievable piece histogram shape hard gamma mixture beneficial rich piecewise next algorithm decay regime dominate htb exponent minus plus data histogram gmm avg histogram txt avg minus error error histogram gamma txt table avg time time high avg error beta txt x avg time high histogram gamma format sep std gmm avg std beta x n avg std table avg std gmm txt avg std beta txt std dominates demonstrate attention interesting contrast linear somewhat simple polynomial encode instead potential programming lp solver run far utilize small account repeat histogram bind give relatively root negativity root correspond find nevertheless certain regime leverage negativity proceed remark utilize arguably elementary build part ii unit satisfying polynomial root necessarily root root require sect still achieve remove divide degree act return formal notation formal representation fact give def root root run quantity simple compute scan eq right side expression root lie root root z pz lemma dominate runtime root let ps polynomial evaluate point return ok running claim correctness clearly always return ok choice since algorithm correctly negativity assume lemma since approximation definition either return satisfie theorem throughout focused metric naturally goal norm subsection algorithm discrete set minor piecewise distribution sample degree pi th hypothesis c algorithm distribution albeit slight modification set difference definition total formally well highlight modification move continuous notion vc discrete particular maximum disjoint interval property inequality guarantee projection computation efficient interval dimensional program appropriately generality polynomial use interpolation formula polynomial representation bound analogue feasible robust efficient separation feasible recall negativity fast polynomial non find root precision evaluate distance exist root kp ji notice integer argument section quantity return discrete separate current c mit li mit schmidt mit distribution piecewise polynomial piece degree draw yield structured family domain nearly consequence complexity meta experimentally demonstrate practice level piece piece fit interval iii finding efficient density claim density observe distribution fundamental history extensive estimating estimator estimator technique decade large body belong approximated number distribution precise define output density learn computationally whose time nearly linear additional agnostic misspecification family merely univariate family several decade yet understand surprisingly distribution mixture gaussian use wide variety three agnostic nearly polynomial employ context family polynomial existence approximations ingredient learn prior unfortunately polynomial exponent quite turn yield nearly wide range family domain nearly broad single stress number idea describe overview consider univariate function finite loss generality focus standard notion natural analogue boolean notion minimax access cf cc algorithmic result interval follow give tolerance h work slow approach principle efficient high run necessary level ideally show indeed good exponent substantially improve running remove information theoretic nearly nearly linear time natural family modular distribution order extend polynomial discrete design linear poisson nearly optimal basic structured family prove appropriate value target approximation minimize would like product theory family example family lead algorithm complexity concave piecewise linear piecewise polynomial show theoretic concave density nearly agnostic piecewise theorem nearly linear mixture concave matching theoretic show approximate piecewise agnostic time natural mixture family note several previously study cover unimodal monotone hazard concave yield nearly linear family unify way range structure modal hazard family provide aim cover rather power method crucial component know agnostic univariate distribution equivalent proper hypothesis search find efficiently roughly speak find mixture provide overview technique supremum gx probabilistic tool inequality let pdf theorem follow piecewise hypothesis quantity time involve main interval learn sub remark appropriately dynamic programming roughly formulate interval discover interval theoretically polynomial slow application particular hence fast lp implement run overall follow idea iteratively merge interval become subtle vc speak consecutive ensure run improvement roughly speak exploit inherent problem solve convex separation optimization ellipsoid efficient long research family structure sort pdf reader book early dimensional past decade start monotonicity concavity focus restriction monotonicity reader refer book mixture structure attention theoretical common address shape mle variant mle quite mixture piecewise polynomial spline extensively inference density moreover spline work mathematical non linear wavelet smoothness remark work unknown recently give seem run require sort logarithmic run iterative merging current show efficient constant emphasize easy distance significantly paper algorithmic subroutine find simple subroutine indeed exponentially many constraint section application univariate finite interval denote family set disjoint generally define metric norm measurable norm supremum take different vc bound pdf empirical pdf elementary fact finite set sign change k f ii piecewise top merging close approximation merge broad class hypothesis satisfy intersection hypothesis efficiently learn present constant capture many proceed merge merge algorithmic challenge projection coefficient polynomial approximately empirical give oracle allow solve feasibility checking whether k convert feasibility variant polynomial feasible semidefinite program simplify suffice set interval replace supremum finite set distance suggest solve black box application sdp solver run constraint inequality increase contrast achieve dependence sdp additional structure projection importantly separate desire polynomial equivalently interestingly significantly lp cut plane distance coefficient polynomial root large necessarily separation oracle outline algorithm proceed point assume reduce jointly distance nearly current variant biology maximum scoring segment exploit give variant nearly number point convert separate polynomial polynomial guarantee start generalization piece arbitrary integer histogram denote density g start provide follow single point sample merge notion crucial j th partition merge form interval iterate quantity eq error interval track error perform arrive respect formal trade piece output histogram n j te characterize establish run exponentially th merge imply construction loop substituting show generality proportional sample algorithm proceed learn return terminate rest partition boundary interval jump event throughout final jump j create jump jump create interval eq use ta suffice prove set jump eq indeed hx interval constant use analyze interval interval definition interval contain may jump singleton include jump interval contain assign sequence finally merge iteration jump triangle jump change bind distance prove lemma complete b use complete proving merge except error merge large suppose merge candidate merge iteration li te lt ti summing condition give plug since create merging interval recall complete ready general merge version algorithm introduce histogram histogram hypothesis intersection hypothesis find efficiently note class sample algorithmic generalize piecewise hypothesis variant I intersection ii good fit function distance efficiently distribution histogram merging hypothesis piece definition aforementione mild family subset whose domain hypothesis ti h ii main intersect formalize interval contain example histogram piecewise framework set interval piece note histogram degree polynomial throughout sometimes denote piece piecewise negative two intersect easy ready fix integer sign restrict pdf want constant iterative take arbitrary output hypothesis agnostic assume call computation oracle sign pi pr respectively abuse support take ready projection n n remainder explanation merging analysis formally proceed interval interval histogram oracle well eq keep merge interval partition formal explicitly merge defer oracle fix interval run time sample interval algorithm conjunction projection oracle turn main subroutine general merge unknown hypothesis minimize merging depend underlie present non polynomial degree polynomial interval formulate construct approximate combine exist achieve k convert feasibility empirical convert scaling pass subroutine transform empirical distribution feasibility also empirical set feasible polynomial polynomial consider original want find slightly relaxed version find negativity additive truly negative small ei collection write polynomial negativity constraint space lead could establish intersection comment intersection negativity let negativity encoding fix expand constraint cx ci ei ei ii ci ic worth note feasible negativity restrict polynomial simplify location replace supremum show suggest black sdp encoding constraint polynomially super run black box lp solver importantly separate allow interestingly lp k efficiently separation see utilize structure separation show notion separation return yes separate yy cc perform basic hence resort accept approximate separation hyperplane yes cx return hyperplane hyperplane note definition separate hyperplane membership hyperplane employ several separation oracle exist still use approximate oracle contain moreover radius return yes return ellipsoid cut plane oracle technical order suffice separation separation oracle e cx bind separation initialize need ball bound polynomial let polynomial coefficient px px lemma upper radius radius ac kp define length separation reduce volume feasible region reach volume feasible separation find conclude achieve infeasible radius feasible also easy polynomial change stay hypercube next constraint distance relate feasibility optimization binary carefully order canonical separation separation polynomial l small empirical achievable return coefficient cx maintain clearly begin trivially cx loop approximately preserve loop ball must identify loop consider inequality feasible thus oracle empty radius return c concrete cut plane method run lt multiplication separation complexity running time dominate iteration operation oracle ball combine run oracle run ss projection separation oracle ball define oracle along part c whenever polynomial separate hyperplane hyperplane hyperplane run none happen return theorem run claim formally negativity approximate negativity satisfy polynomial ok return polynomial approximate negativity eq prove negativity simply hence correctness correctness run runtime claim oracle describe v describe subroutine ki I yes v q left hyperplane argue guarantee hyperplane inequality indeed hyperplane entire claim define number number interval maximize q collection interval suitably length support consider maximize let interval maximum rhs q let otherwise exclude ei claim direction denote disjoint achieve maximum support interval follow set consecutive pi put way interval negative smallest contain associate sign support small reasoning pi I namely pi j transformation claim transformation pass solving consider compute present analyze description alternate otherwise consecutive number ki merge q compute efficiently store weight collection interval amongst formal definition algorithm let weight nontrivial return solve attain boundary collection interval every collection atomic maximal say maximal either respect suffice long atomic maximal stop iterate either atomic maximal let contain piece atomic contain subset atomic maximal since maximize attain subset every contain interval interval subset ever end sign modify property maintain resp left sign
fortunately recursion normally call max convolution compute recursive operation pattern total pattern normally still moderately discrete max analogous allow nevertheless convolution operation though first recall convolution piecewise max convolution belong moreover f slope fig symbol write iy ci x thus retrieve ic parameter c ic ic max concave let concave interpolation imply domain piecewise discrete sketch max piecewise concave sort piece slope clearly pass cavity convolution omit simply remove convolution step trivial affine mapping easily generalize trivially concave ensure concavity computation binary simplify tw w tw w tw eqs correspondingly eqs simplify order result drop different presence impose arbitrary single special configuration ti flip respect configuration partition sort index order correspond subtract order index turn index optimal sign f index define variable clearly get picture except shift shift turn shift quantity expression computing provide report update cavity sum e bp behaviour reasonably suggest vertical reinforcement value solution perceptron log bar fit show estimate critical perceptron layer concave iw zero e qualitatively similar around close connected layer capacity single layer still capacity unit test store thus demonstrate great increase clear due permutation unit replica effect tend intermediate state mix different solution make still help achieve constitute improvement bp limit extremely approximated ms approximation naive normally slow purpose show extremely max valid advantage break thank hoc additionally max contrast equation extensive weight algorithm achieve theoretical cavity full detail note text cavity analogous cavity change turn point change express global choose convention obtain consider cavity concave maximum fact expression cavity field consider simplify I I I I plug expression cavity algebraic I I I turn region jt jt jt f last kronecker jt jt jt I go cavity eq efficient np complete call algorithm independent update put par scheme bp interest perform perceptron inherent naturally break ms feed neural etc problem obtain assignment device test give example discretize sigmoid scalar vector input weight unit operate unit reach popular successful even usual drawback gradient presence minima slow circumstance problem even simple version become hard complete storage capacity device bad robustness application theoretical long storage rather network hard potential practical biological light upon origin hardness physics isolate energy landscape minima tend poor cavity instance show belief propagation correctly output association single simulation simplify version simple work complexity measure performance thank approach deal tree structure fully use least straightforwardly arise address ms reinforcement analogous reinforcement use temperature limit bp approach zero temperature ms addition error add field go small field layer ms bp use shall binary computation ms bp storage capacity time reach polynomially well rest organize present solve thank convolution complete detail implementation binary layer throughout unit binary transfer I evenly spaced unit convention single omit layer unit layer receive device kind also consensus machine like input would share overlap tree architecture generally capabilitie storage fully connect situation also possible fully machine symmetry since second throughout context desire association correspond desired extract random input pattern still extract usually teacher device rule low architecture student device student teacher permutation pattern
classifier algorithm applicable form use produce write discuss approximate equation simple average observe instance previously chapter prohibitive storage evaluation another optimal misclassification v surrogate regret bind surrogate usual work function linear throughout shorthand minimum classifying equivalent classifying include completeness straight cauchy schwarz quantity self similarity ex verify distribution pac include label add label flip sigma noise robust negative feature mmd see restrict divergence q commonly yy ex negative equally mmd solve take cauchy objective theorem classifier regularize optimize margin suggest high feature normalization idea kernel kernel density instance use optimal rule ensure feature kernel function class hull calculation feature linearity loss word pick usual multiple general pick generalization collection draw dependence correct definition application frank z begin point average choose run obtain term line available view minimize approximate concentrated originally produce super standard ex ex time use rate rate fast rate search cd approximation appear statistic closely frank wolfe sparse approximation material split disjoint use approximate sub separately tolerance margin therefore assess compression motivated mean mean compression classification n produce classifier let compression probability eq way tolerance maximum sample mean theorem suggest stop optimize justified theorem tight contribution highlight set keep classify mnist comprise kernel bandwidth classifier parallel split range step blue entire baseline test training obtain rapidly obtain roughly perform mean entire margin assess validity loss place show maximum discrepancy regularize machine surrogate speed evaluation margin relate degradation incur instance ie q x pac prior draw couple term refer furthermore linear identify posterior calculation kl divergence assume prior posterior identity restriction see weight vector begin theorem posterior standard moment normal quantity decrease bind minimize map define union posterior normal ip furthermore fix previous posterior normal hand final moment line cauchy schwarz eq recover take distribution minimize hand bayes theorem present assess generalization useful hoeffding draw feature least hoeffding yield distribution collection draw use hoeffding union mean result let say ball define grain assess decay hilbert exist diameter know equation
bethe bound begin experiment use mle analyze various matrix diagonal partition gradient regularize run entropy upper run display upper high regime estimator rw reweighte interestingly objective regime produce value moreover inequality rw regularize bethe quantity use fw compute already learn red incorrectly probability low blue nontrivial red force pick pick l c c computer vision sequence follow setup datum consist frames house separate frame toy angle label point frame divide validation split learn loss parameter house gap little difference synthetic tune indicate make probability match albeit permit bp approach unstable bp fail converge step mle go year student return student year return student preference obtain room major year train student live remain student node gender entire datum student create many feature indicator feature matching mle table coordinate ordinal ordinal value see agreement appendix perform multiclass hamming model classifier plot roc curve demonstrate false perceptron structured svm decoder obtain test bad pt hour take long image segmentation formulate keep image result approximately try naive subgradient descent slow partition loop run bfgs another objective optimize optimize parameterization mle fastest mle evaluate error describe curve average substantially smooth raw curve quickly attain low test error curve run inner loop finite run algorithm confirm convergence fw early low attain iteratively move value initially inaccurate value dimensional result prediction nevertheless move compute expense minute portion objective hamming estimate local classify dark region classify already run make essentially correct texture algorithm hamming get internal criterion exponential free energy free energy concave add enable maximization leave efficiently frank wolfe scale dataset coordinate wolfe rapidly achieve map practitioner either employ double svms competitive margin error fast compete mle simple work combinatorial regularization technique employ part setup part fw bethe energy derive full dual search fw appendix explain far low likelihood two sided evaluate evaluating display likelihood compute fw procedure likelihood small dedicated hyper ghz intel gb physical ram run run matlab combinatorial author matlab extension year period assignment addition student survey ask preference feature level student question create several student matching learn describe unit assume ordinal qualitative relatively indicate perhaps second predictor successful least structured employ publicly available code author try lambda configuration achieved ham significantly mle profile go usual hour study audio entropy local polytope show iterate fw decay subproblem wolfe curvature quantify linearization twice upper heavily influence piece depend unbounde fw curvature require part entropy depend look function long inside fw always guarantee produce inside box typically issue appropriate propose bp add combinatorial matching pairwise binary mrfs become gibbs assignment ever well define iterate boundary reasonable reweighted rough high bethe energy necessarily graph bethe proven argument argue polytope follow concavity bipartite perfect free polytope graph bethe written entropy concave entropy concave formulate describe fw match column ki k contain separate graph conditional arbitrary maximum matching denote absence item x coefficient single replace add regularizer bethe write bethe likelihood program linear constraint begin eq derivation mn sequel pairwise matrix additionally stack thus rewrite justify product later minimization objective maximization compact domain unconstraine function attain stationary simplifying iterate fw plug product fw bipartite feature able occur perfect link perfect discover matching neighbor polytope infinite match feasible overcomplete parameterization matrix treat learn fu e exchangeable replace parameterize likelihood bethe bethe convex mrfs span frobenius penalty reweighte approximate entropy n nh singleton entropy marginalization use identity stationary objective frobenius rearrange outer eliminate similarly reveal gram stack stack entry n mu v flip simplify sign later write h nm nr gradient w products bethe coordinate frank wolfe computing denote step contain row row row n md e w md add value wolfe fw perfect use code marginal marginal force call match use bipartite call affect aggregated bethe approximation provide fw substantially fw converge accuracy differently h fw bipartite perfect speed trade complete edge lda run sampler average algorithm termination accuracy bethe specify present ratio fw fw slow within affect advantageous include rand rand lda david many formulate predict structured output framework support vector structure discriminative apply posteriori decode likelihood probabilistic structured partition paper bethe approximation mle remarkably connection bethe frank wolfe fw partition single efficient double approximate maximum estimation outperform exist segmentation vision learn markov mrf conditional crf parameter learn regularize mle maximum include regularization principle ascent repeatedly gradient log partition surrogate likelihood perceptron map solver quite approximate map black box user abstraction mle goal practitioner superior offer interpretability time marginal approximate access solver bethe energy frank fw method naive fw mle perform marginal call experiment gradient solver accurate answer achieve avoid costly double first generic reweighte technique bethe style surrogate model dual approximate problem minimize subproblem formulate separate accelerate fw use fw test interact allow variety map max pairwise ise match apply pairwise binary bipartite mle problem mle method student sample statistic hypergraph ease well linear regularizer central compute work learn bethe recent mle convex free energy fw convex iterate define search fix step minimize independent parallel depend application combinatorial reweighte lp solver project onto despite parallel prohibitive large problem space perform fw convergence know fw include fw algorithm input uniformly computing useful
record distinct extract news follow experimental dataset yahoo criteria bandit algorithm dynamically ucb single single ucb algorithms ucb ucb algorithm assign context user actually quantitie subsample suggest apply user proper tuning maintain fair dataset run maximize suitable range payoff dataset make plot version test average variance observe small summarize yahoo retain far dataset record discard ratio cumulative payoff aim three consideration web yet point dataset fairly way consequence dataset provide huge imagine population yahoo moreover collaborative strong article yahoo item high chance user preference collaborative effect fact clearly win comparison suit start accord unknown c logarithmic denote w big big clear easy j achieve partition upper become relevant distinct partition influence yet worth repeat role whole kind partition ahead simply contextual bandit grow factor r shall exploit reveal replace term n become bad scenario result extreme single many operating scenario group similarity universe item group similarity provide operate encouraging operates simplify content universe conduct reliable annotation yet potentially research lack infer factorization technique subsequently clearly combine co far see adaptively item computationally amenable advantage stage get stage somewhat similar meta investigate content recommendation exploitation user item possibly clustering take advantage preference collaborative filter world show scalability increase bandit regret within web collect preference service enable interaction content recommendation recommendation web service core business web universe popularity service adaptation preference algorithmic interaction group similar e static specific type user community content clustering user dependent music cluster music change item could group tend preferred notion side group user base similarity item side user see suggest recommendation scenario movie recommendation computationally double simplified version technique double recommendation contextual associate exploitation work user tend item recommend cluster need different universe user clustering induce compare perform context real scalable exhibit bandit algorithm also hold stochastically prominent bandit content main information fact embed preference relationship click exploit filter technique typically collaborative user often impossible adequate aim exploit collaborative effect co technique batch one recommender g whereby lack suboptimal recommendation approach behavior clustering specific consideration partition dm user belong cluster like lie cluster significantly behavior u common thing context assumption threshold cluster set e unknown upon user value conditionally bound variance observe setting break sequence step receive user recommendation pick ti ta goal learner comparative theoretical interested bound cumulative regret learner extent good exceed aim section kind contrast content universe universe p partition cluster user induce induce I e possess common resort content make user method whereby preference relation rating collaborative group item item pair e grouping recommendation social beyond bandit specific piece bandit paper try assume combine seem large scale analyze bandit cluster completely like result technique lead rely user side emphasis recommendation see spirit author author none author specific effort dependent present rely tradeoff exploration exploitation call associated base feedback operate linear algorithm available subject initialize identity bind counterpart need regret repeat holding high practice order define compound turn compound exploitation item put emphasis compute receive algorithm perform user neighborhood compound di ni brevity aggregate ta tm drawback clustering base item fact dependent item make store clustering maintain similarity behavior universe affect aggregate user belong round lack reason despite drawback reasonably stream ii approximation generally exhibit good priori maintain clustering bandit description contain maintain clustering item clustering represent connect undirected index graph cluster cluster clustering exploration user item dm di ng u ni determine current clustering w brevity quantity ta update tn te tn ti te item single correspondingly cluster make unique item depict therein candidate eliminate depict elimination algorithm compute neighborhood accord compare split item new cluster user clustering overall main point square clustering unique side point item item side change item item edge user item get imply naive would allocation maintain prohibitive moderately usage approach start complete create randomly graph la
noiseless size test setting equitability across sample intend achieve equitability budget size setting independence insensitive level analyze lead size fast yield achievable compute translate runtime minute variable analysis snapshot currently improvement way algorithmic allow computation equitability power several dependence new exposition equitability independence community rigorous side exist state finding equitability noise achieve superior equitability equitability variable poor maximal examine statistic share independence perform independence albeit differ equitability independence testing much examine substantially high correlation power independence equitability estimator equitability characterization equitability give lead high expense weak relationship equitability expense runtime trivially runtime fast large result suggest rank strength sound imagine keep equitability examine enjoy alternative relationship type broad simultaneously appear preferable choice measure dependence demand equitability goal low possibility power evaluated set finally understand comparative analysis exhaustive date dimension equitability relationship statistic analysis hope enable precise trade one setting equitability functional explicitly possibility besides intuitive equitability interest method perform much bad equitability attempt scope relationship equitability add result theoretically equitability understand strength method direction measure variety superior different appropriate setting understand insight inherent trade allow landscape effectively ultimately understand acknowledge k constructive table use equitability analysis cosine cubic cubic shape x xx xx x u exclude due poor across portion drastically graph h quadratic xx statistical equitability power perform require ability result case expression wherein solve use population equitability worst worst interpretable shorter interpretable colored red interval white equitability robust analogous present supplementary setting size value present computationally expensive equitability interpretable length short interval color interval white length equitability factor distribution setting include material expensive analyze equitability curve relationship take quantify poor equitability model square low mid parametrize whose parameter equitability size use maximize equitability across noise test equitability material correspond equitability vs mi infinite equitability mi noise compute newly interpretable interval equitability mutual square correlation large setting composite curve noise analyze relationship strength quantify mutual mid powerful parametrize equitability equitability test sample table equitability material power curve relationship aggregate power power comprise area curve statistic parameter area vertical average relationship turn poor testing equitability suited equitability dependence aggregate relationship power relationship test determine relationship type relationship dot average case list line represent default set use equitability independence parameter equitability see equitability noise runtime series correspond indicate equitability equitability equitability present value test maximize equitability examine value test include constraint supplement test equitability default rbf kernel sample test pairwise computationally test median pair pair median analysis examine area range test statistic c pair c runtime method setting default information parameter independently test median present parameter equitability fast equitability interpolation point set example relationship low computationally exploratory analysis association accomplish compute possible examine assign type equitability formalize addition equitability assess independence runtime lead dependence include statistic newly introduce equitability primary power regard find relationship mutual estimation prove setting test trade runtime fast compute trivial achieve equitability relationship appropriate tool guide statistic equitability hundred thousand association within analyze pairwise association search common compute low scoring list depend statistic statistic zero statistic contain many relationship relationship fact though trivial systematically score relationship relationship crowd relationship list manually relatively small relationship strong detect power trivial relationship weak allow relationship exploration set goal many association task utilize equitability dependence equally formalize power hypothesis relationship strength nan hypothesis zero equitability intuitive functional relationship reflect determination respect possible equitability difficult measure equitability mind relationship efficiently computable relate coefficient essentially compute translate benefit extensive equitability power runtime correlation mutual hilbert schmidt framework rigorously equitability yield main conclusion regard estimation setting four art art also achieve outperform outperform alternative examine mean poorly detect power independence competitive albeit parameter equitability characterize equitability free consideration final conclusion concern find fast method run cluster near equitability take compute together conjunction achieve mix filter result equitability explore together first equitability coefficient introduce dependence measure wide setting power independence focus primarily performance comparison provide hope paper expand review equitability equitability analyze characterize tradeoff independence equitability analyze extensively definition relate informed reader coefficient maximal score quantify strength see goal coefficient statistical grid plane point analogously discrete variable denote empty finite g maximal achieve type population define quantity object characteristic population supremum jointly matrix shape type different relationship maximal maximal corresponding jointly variable population alternate characteristic estimator population original maximal estimate characteristic order maximum size grid resolution order let define prove consistent dynamic introduce prove consistent estimator contrast although whose characteristic matrix turn compute distribute define set grid analogously pair define programming include control remain consistent statistic maximal coefficient aim relationship absence independence maximal behind coefficient characteristic tend grow imagine bias equal power well bias independence goal noise maximal useful signal information coefficient avoid total pair consistent tailed control test ht p quantifying value equitability dependence formalize exploration equitability review equitability way equitability roughly equally noisy relationship viewpoint dependence equitability test distinguish relationship different amount reject nan strength test nan relationship highly power relationship formalize rigorously equitability specify strength relationship diverse relationship add determination respect return build intuition define equitability broad equitability interpretability equitability specifically reflect notion concept even statistic let standard relationship satisfy distinguish equitability question call equitability discuss property statistic equitability independence equitability statistical statistic independence extreme extreme say identify relationship weak would equitability differ analyse independence statistical hypothesis equitability equitability contain distinct relationship analyze b tailed test base hypothesis show alternative power instead heat instead consider set hypothesis plot result color correspond size right distinguishing surface attain row define equitability take intuition assign equally type invoke estimate exposition term equitability bad equitability reliability statistic reliable close interval reliable diameter v pdf pdf c relationship amount since interval interval analogous plot relationship range blue red interval interpretable interpretable reliable interval shorter interpretable interpretable bad case interpretable interval solid line interpretable interval red representative relationship reliable interval hull interval reliable reliable noisy functional relationship consequently reliable central b type distribution three sampling small union central acceptance define interpretable interval small close interval interpretable diameter show interval noisy different type pearson interpretability reliability statistic one time way basic one measure resp interpretable resp resp dependence reliable resp interpretable reliability resp worst oppose prove interpretability gain intuition interpretability interpretability interval case bad interpretability interpretable interpretability perfectly interpretable arise let interpretability functional depicted figure interpretable correlation worst shorter interpretable turn equitability assumption state statistic respect interpretable confidence interpret way strength relationship measure vice versa relationship strength statistical independence equitability news news one equitability relationship equitability require large equitability far conceptual equitability relationship mean write variable equitability amount relationship dependence bad resp abuse terminology equitability equitability respect set relationship detail review analyze equitability interpretable interval pearson coefficient trivial analyze standard relationship noisy equitability th reliable reliable enable interpretable interval equitability reciprocal length interpretable interpretable many large value sample relationship type different different amount interpretable thus poor contrast notion interval course equitability question relationship trivially perfectly correlation normal cc pdf v respect pearson central distribution interpretable interval relationship different interpretable indicate red case possible illustration monotonically use proxy relationship receive equal review equitability equitability dependence begin quantify equitability interpretable interval conventional connection set analyze representative coefficient total view exploring grid draw assign aggregate normalize mutual aggregation supremum explore grid except summation method use pearson statistic explore two grid differ test stable variation model good equitability equitability model outside noise affect outperform mutual problem inspire equitability mutual suffer superior equitability model substantial estimation equitability setting equitability test occur population effect equitability information insight demonstrate minimal superior bad average equitability noise add mutual perfectly examine surprising broad test examine equitability specifically mutual analysis show sample contrary picture equitability estimator mutual technical comment publish author theoretical see figure table demonstrate correlation poor equitability equitability equitability high equitability equitability correlation measure dependence examine demonstrate examine equitability perform interpretable interval present material composite relationship quantify heat map tail distinguish hypothesis composite come well method average set hypothesis list maximum achievable information parametrize statistic affect present across test assess equitability statistical confirm conclusion quantification equitability interval distinguish examine small equitability estimator task size test even variable see maximal test degree equitability noisy relationship property method traditionally detect deviation yield high nan statistical independence due fact hypothesis method highly simultaneously detect deviation display may detect relationship bad relationship course nan allow hard nan composite correspondingly suffer equitability lead vary type setting sample mutual outperform equitability poor equitability come parameter setting good power equitability demonstrate inherent statistic equitability establish case interestingly estimator maximal alternate expectation correlation hand lack equitability return result bind computable achievable extent relationship world relationship large claim understand crucial determine extent important development grow hope insight together enable investigation equitability ranking rank measure dependence assess dependence examine notably upon hypothesis alternative allow aggregate across gain view power perform determine last analyze depth use achievable performance large analyze examine equitability analysis perform relationship choose analysis similar material manner equitability analysis add eight noise level evenly range noise noise substantial sample level understand affect power dependence automatically need way eight type power compute power relationship integrating amount add area amount result power statistic noise choose uniformly power noise drop set threshold supplement power curve score type plot list choose parameter red substantial attain threshold material contain quantitative ranking dependence value method parameter several across relationship quantification quantification large outli accordingly supplement circle analysis supplement threshold use generally threshold material aspect ranking statistic closely coefficient aggregation summation mutual grid score grid via characteristic fundamentally promise dependence task demonstrate translate note power discrepancy due analysis intend equitability show use equitability trade final perhaps result method small method gene show observation true statistic relate detect rank relationship pearson correlation coefficient rank result believe set relationship independence power examine strength detect statistic question threshold systematically score equitability provably low threshold converse true test require achieve relationship minimal examine threshold threshold besides supplementary material grain analysis statistic strength independence type grain picture relationship ability power curve directly axis independence examine figure result choice base previously report substantially set test indeed strength dependence test grid base base summation choose result method find provide relationship examine maximal entry across relationship one promise exploratory discard burden power relationship use whereas time since power appear important eliminate power independence base yield test close art differ suggest trade additionally find test vary considerably across different whereas sensitivity substantially finally bivariate many appear last observation magnitude case answer wish high datum exploration measure already reliably thousand able relationship additional relationship strength scientific uncertain exploration maximize relationship think statistic inspire framework pose numerous challenge establish test parameter optimal equitability power
l est une pour es de mahalanobis des es les pour par pour la q eq iv e est e analyse en de eeg dans est en une se la dans la les les et es la est en n stimulus une li du stimulus les une et est ce un des dans la es par exp e par pour correspond une portion du si dans les observation figure des des et les par les es et une la projection dans des class es des des le rare se la de analyse une de des des svd une analyse des les la en dans la de pour analyse spatio des et es dans eeg signal optimize et journal statistic et mapping k variate lda spatio feature eeg eeg localize analysis sim r international cm eeg universit universit universit fr l analyse lin dans de lin des des plus par composition la des les des et des la de mahalanobis des dans des une la la cl es analyse lin es eeg abstract focus discriminant variability average row dimensional discriminant multi relevance separable eeg analyse lin es es est dans en es eeg une de les la les tr et est le est sup et la tr de de il une se la des en des es par un mod le de la covariance et de des les es se dans dans des eeg de ce est de une lin les en de structure des es se de la en la des les des des de mahalanobis les les dans en des de des es dans des une des dans les eeg dans un par dans pour eeg dans un interface machine la est notations pr se en de et extraction pour des de eeg des est de la est pour matrix l op trace la b bc aa de kronecker l mod dans le de dans des les pour r es r une de de les de covariance des q p
subsequence hold subsequence ne ne subsequence subsequence becomes restrict converge large v rewrite subsequence boundedness converge subsequence sufficiently necessarily nf large inequality countable going establish main boundedness q n conclude simply show hold virtue argument consider straightforwardly several need q know boundedness show acquire outline divide show triangle expectation conditionally n define lead second inequality rank one perturbation old obtain eq moment matrix write ensure n positive solution definition prove unique conclude hold define uniform derivation use take deterministic provide generic result purpose et entry zero I nc ne nz transform eigenvalue next describe successive moment generalize valid successive kk step theorem corollary large covariance comprise sample arbitrary upon advance robust estimator scatter sample infinity introduction behave asymptotically different enhance outlier mostly outlier thus robust estimator bring benefit estimator sample within class huber favorable risk estimation momentum big advance large dimension complexity fix e detection started adequate instance toeplitz structure particular scatter come regime huber classical unlike classical nature know robust successively majority gaussian arbitrary give aforementioned scatter take form implicit application limitation regime large regime several estimator scatter behave explicit fully nonetheless independent zero elliptical salient work elliptical henceforth normalize apparent versus fundamentally estimator outli comprise amount outlier focus scatter aforementioned robust behave easily find scatter impact outlier normalize demonstrate robust scatter outli control appropriate huber substantially remainder rigorous statement proof defer attention analytically case outlier either I conclude remark provide stand hermitian transpose transpose dirac unit denote stand part support denote order hermitian x diag stand matrix compose almost sure stand weak ni ni deterministic hermitian moment deterministic vector consider assume last column outlier merely request moment technical n refer merely consist discard outlier lack know immediate alternative estimate norm robust normalize diag arbitrarily bias detect significantly differ majority robust estimator scatter huber study precisely outli identification scatter vector shall scatter continuous increase equivalent accounting multivariate call show shall however estimator later implicitly assumption hold relation take theorem entail allow property implicit matrix matrix interesting outer product scale along outer expect set emphasis weight merely ensure outli impact especially small immediate eigenvalue follow empirical define via n nz equation importantly imply support determine respective k df n df deterministic implicit transform successive formula precisely formula albeit behavior quite weight relate implicit deterministic get insight property successively specific scenario simplify assume maintain assumption perturbation find remark eq shall denote arbitrarily individual involve specific one read simplify side left hand increasing depend come isolated calculus asymptotically choice scatter tend strong mostly outlier norm essentially thus gain dimensional coordinate font style yshift near white anchor north east font xlabel ylabel near bar width mark plot n ij xt come subscript let independent th n surely real function density obtain monte carlo average soon numerically figure suggest somewhat confirm observe approximation decay factor versus font style densely yshift near west anchor north east font xlabel ylabel major xlabel r j xt interesting arise majority become difficult general n impact enhance sample version n behave asymptotically neither contrary capable reduce outlier n outlier previously outlier depict n f optimally discard thus highly confirm show close tail contrary main matching yshift anchor near anchor west fill white anchor north east font xlabel near ylabel width grid major xlabel plot f n f f nu various estimator affected outlier interesting investigate moment case initial expect induce bias fair scale normalize moment successive moment relative scenario moment hold rather important moment ccc p ij estimator large lead conclusion address scatter elliptical reveal behave similar normalize conclusion suggest conclude behavior versus unlike normalized scatter compare value even might probability induce lead tune close rejection property measure asymptotically oracle estimator interest come isolated finitely suitably isolated bring important performance gain finance heavily isolate relative outlier one moment close oracle improve account observe successively regime estimator study frobenius nothing suggest alone quite sensitive outlier study essentially say noticed proportion versus towards let truly provide well behavior scatter finding aspect relevance rejection risk inherent nonetheless implication plug detection isolate global information
nice odd popularity logit link come simple maximum equation fast theory vast reference fraction extensive characteristic generalize answer ever logistic probit link second function probit binary bayesian reference point probit mining deal logit attribute experience regression link strongly support tendency give place rest word appropriate practically suppose represent complete chi thesis population development height weight logistic law even theoretically subject laboratory normal play distinguish role theoretical support observational material matter obvious logit surprising apparent due fair recognize definition probit logit yield two essentially determined solely mathematical convenience theoretically equivalent univariate characterization tend context throughout information view ability good work perform dimension life reveal link univariate sharp begin dimension increase organize general namely equivalence structural real section clearly describe demonstrate claim moderate goodness probit logit provide probit logit reveal input section spam conclusion along insight compare goodness goodness perspective ask prediction probit differ logit often logit function yield difference prediction mining force focus predictive equivalence binary classifier define cdf binary majority rule take component logit probit classifier give q one misclassification hx practice close form empirical classifier randomly tr te te te te te te te te split test error yield classifier dimensional shall say disagreement classifier say perfectly classifier space shall e version cdf almost perfectly standard cdf probit logit perfectly write thank straightforward link dimensional equivalent nonzero pm sufficient logit probit thanks deep logit relate parameter conjecture linearly intercept replication copy compute probit logit probit tendency replicate logit probit relevant sample generate intercept ir determination equal indicate logit entirely help determine probit coefficient probit logit slope neighborhood probit replication determination replication diabetes logit probit logit patient diabetes obtain use probit logit probit logit ratio probit around appear pattern variable theoretical univariate intercept benchmark logit probit logit give characteristic display cl probit ratio probit logit still around important justification simplify univariate intercept complete multivariate relationship neighborhood regardless consideration support confirm conjecture relationship probit know know confirm already noticed express pp proof mention equivalence logit without loss denote variable probability logit probit function equivalence show probit logit coefficient slope task approximate confirm consider probit probit logit logit link taylor approximate log logit ignore high expansion probit function derivation ignore order get c c straightforward inexact confirm derivation reveal probit link equivalence concern similar logit relationship logit verification replication link function consider function data error classifier corresponding probit logit replication realization four calculation replication skewness etc assess similarity difference similar replication bic verification cauchy namely eq replication test error suggest link almost indistinguishable equally htbp logit sd skewness min verification equivalence diabetes diabetes diabetes arguably use statistic diabetes link equivalent logit mean sd skewness min four bic aic slight link function goodness yet evidence claim logit simulate link percentage
penalty derive several variable put moment condition heavy general theorem residual enable imply nan x lemma provide indicate nan statistic also technical suppose cumulative achieve x f part indicate region theoretical imply exist achieve three fp positive entry nonzero false nonzero penalize estimator replication vary cccc lasso adaptive scad scad fp penalize fp increase fp size get k eq trace n complete term q third entry indice group step sum combine lead j pa thus complete limit follow remark rgb measuring dependence statistic distance adjust correspond nan latter error load root asymptotic distribution test domain strict asymptotic significance efficiently generic building result superiority word dependency various signal bioinformatic whether conditionally mean ij precision rich normality always real tail skewed propose gaussian network flexible still transform restrictive find graphical would propose natural way conditional give remain node hypothesis decide present economic nonparametric test function calculate test lead value dimensional hellinger conditional characteristic likelihood motivate respectively n I nk f independent I rest node proposal dependence rely heavily pearson satisfy normality non measure robust deviation nan hypothesis statistical measure relate include method consideration benefit distance independence true second correlation dependence test exceed distance measure propose test covariate estimate mild error main contribution conditional independence dimension covariate relaxation gaussian organize independence construct conditional theoretical type study demonstrate set short section technical introduce notation norm th matrix frobenius function characteristic tool covariance review section gamma nonnegative joint covariance show nan hypothesis corollary property surely corollary constant dependent test seem covariance true observe result step ordinary least ol statistic level nan one replace ol penalize conditional justification dependency
result performance positive lipschitz ac transformation minimizer adjoint let initialization terminate generate terminate stop show c lemma singular inconsistent subset minimize unconstrained initialization iterate current soft arise inconsistent soft multi constrain cut appear noise experiment turn satisfied illustrate increase satisfied cut error observe satisfied enforce always constraint choose option integrate set partitioning problem generate bi among constrained always link pose difficulty multi link procedure satisfy cyclic constraints recursive bi early split issue binary split specify derive class assume add constraint one vertex link constraint although cut derivation different sign encoding introduce towards classify vote point measure supervise give table spam uci mnist illustrate unbalanced problem digit versus generate show plot enforce cut value initialization even cut suit normalize cluster consistently case unbalanced spam vs rest significantly outperform cut moreover hard encoding multi leave versus middle normalize cut violate constraint ccc versus middle cut constraint right violate dataset breast heart mnist ccc middle constraint versus middle fraction violate definition technique relaxation cut constrain always moreover soft trade consistently cluster grouping similarity alone give item domain give al encode available ml short cl incorporate constraint performance constrain research base originally normalize spectral relaxation laplacian relaxation quite loose recently rewrite combinatorial problem nonlinear graph laplacian cut far balanced cut tight continuous approach integrate spectral idea modify order enforce another idea embed graph laplacian close original start encode link constraint inconsistent normalized relaxation continuous spectral fulfil present inconsistent constraint thus extend handle optimize trade cut violate scale problem non convex satisfie stop omit find supplementary notation correspond paper normalize cut vertex formally graph vertex vertex volume partition ratio weight graph element specify suggest allow follow theoretical statement inconsistent relax framework function q link show constrain normalize correspond relation violate quantify lemma partition connected assume minimize lead contradiction constructive lemma consistent constraint equal constrain normalize cut problem combinatorial problem combinatorial optimization elsewhere minimum non c diagonal equivalent particular indicator partition functional thresholding yield small second denote cut convex positively homogeneous convex homogeneous symmetric convex last change limit integration indicator shift follow choice normalize right statement show theorem hence immediately solve normalize constraint integrate correspond vertex derive cut link must constraint merge edge note many must integrate link merging preserve cut constraint prove merge reduce merging vertex vc partition reduce must cut
use different easy check everything induction repeat previous q accounting use coefficient conclude lemma writing couple jensen uniform choice I q since absolutely return complete contain recover straightforward calculation binomial difference replacement improve sample replacement set element hold course straightforward calculation probability eq theorem proof difference union inequality appendix improve discrepancy eq invariant improve de observe label final giving introduce study inequality prove tight process also relation popular complexity rademacher complexity argue finally combine concentration provide risk rademacher complexity widely empirical measure inductive thank attempt apply notion observe label goal correct many mining recommender object generate training realize cardinality source gain probably due fact imply risk bind essence second deviation risk compute follow emphasize learn standard inductive trick translate inductive sample inductive complexity make difference replacement rademacher application author depend contraction know loose dependence concentration without base measure include interesting continue empirical index function replacement nature limitation nonetheless illustrative expect empirical replacement constant remarkably new hold additive additive order low upper order suggest measure learn measure provide achievable low bound relate rademacher achievable also improve comparison show apply obtain computable theorem discuss advantage symbol finite set input function loss defer arguably statistical rademacher conditional fix subset quantity sign take simply rademacher important role learn complexity mainly replacement rademacher fix un remarkable additive I low boundedness necessary I conditional cardinality eq conceptually rademacher sign permutation contain minus idea sect multiplicative much also significantly improve relate class rademacher result even class map absolutely sect b multiplicative bind improve fix un sign bernoulli complexity notation mf sect risk set fix receive consist element remain learner predictor fixed measure nx hx h mh risk sect least satisfy sect conclude concentration expect appear meanwhile computable contraction dependence loose complexity label write note identically low slack significantly small latter cause show appear two time comparison also tight rademacher extra argument one simplify expect appear theorem eq marginal appear equivalently uniformly sample without replacement replacement jensen take distribution rademacher sign random plus minus
review music specific follow review section centrality describe experiment perform multiclass result conclude remark work music however algorithm summary automatic clarity coherence generic segment represent part song segmentation frame detect repeat along song self segment contain rank segment cluster output song structure boundary cluster middle produce strategy segment similar segment song summary extract measure modification ensure extract piece start calculate compute similarity aggregate similarity song pick summary another method filter filter segment lag frame filter lag classify pure feature frame selection duration human since summary naturally people generic song since human differ create meaningful segment specific allow variability aim human consumption instead specific research effort centrality social focus improve input weight corresponding sentence encode incorporate outer stationary accord convert sentence iteratively select accord number rank far item arrange list follow correspond item fundamental give start centrality rely cosine represent summary central sentence centrality rank graph build edge create sentence pairwise threshold use weight unweighted edge eq convergence total vertex high sentence reach sentence sentence sentence high sentence score text reduce dimensionality original building term sentence element number sentence translate composed sentence apply singular singular matrix relevant sentence select index singular sentence include sentence sentence value never never singular fall value sentence selection diversity low redundancy speech take centroid sentence different sentence previously sentence base diversity sentence represent centrality text centrality sentence sentence certain q set select sentence set recommend eq previously unweighted centrality degree sentence recommend allow sentence oppose sentence recommend corresponding sentence set centrality relate sentence threshold explored set specifically heuristic cluster distance first cluster initialize sentence document heuristic set several metric distance test multiclass music consist classify base scheme song address task hold comprise compare standardized step song summarize signal select process step classification feature feature per song first texture song centroid skewness ms frame solely compose task feature candidate extract consist music several music although differ similar deep solely usually usually influence post country music characterize repeat phrase builds represent several wide vs vs validation classification experiment segment begin middle end song baseline classification summary multiclass dataset yield dataset c library extraction operation summary algorithm music algorithm operate concept extraction sentence song vocabulary frame use assess obtain cluster centroid centroid frame vocabulary frame represent discrete nature song size since sentence discrete represent occurrence frequency exact representation sentence cosine size final sentence cover value size second size word instead weight effort music signal spectral try choose impact classify vs orient nature classify result htb second send thing vs task job distinguish drop full beginning section drop lose full heuristic beginning second htb c c send accuracy binary binary binary set task poor job describe distinguish perform worse vs vice versa segment classification equally reach improve performance baseline classification classification task calculate set analyze music go beyond classification look confusion obtain carefully case understand confusion identically sort ideal diagonal confusion diagonal show individual result classify confusion group sense present achieve accuracy share confusion explain music sharing characteristic virtual produce perform classify accuracy track strong important much information track remove instance second incomplete song blind summarize music extract interpret heuristic extract segment music second whole song time ccccc h classifying begin full classification using drop second low energy contain relatively part whole take begin ccccc htb ccccc classify middle drop accuracy get track way middle segment section though low track part vs nature human would probably unable distinguish classification drop mean segment ccccc f ccccc classifying second end section compare misclassifie mainly mostly share confusion second song good htb ccccc htb middle beginning still average feature feature perform well signal second may happen song sufficient diversity segment accurately represent whole song distinguish structural example accurately distinguish need part song summary classify detect relevance diversity inform important fit summary table claim classify summary vocabulary sentence weight accuracy middle lose section individual since middle perform distinguish summarie accuracy mostly increase summarie diversity several structural remarkably job classify original full data ccccc ccccc confusion section respectively second vocabulary weighting compare section namely select diverse part include able interesting individually increase htb ccccc ccccc confusion correspond difference middle combination second vocabulary word frequency weighting apply performance sentence tend get summary document even appear undesirable bad job describe song aspect frame cluster presence frequency representation section increase explain summary improvement ccccc c confusion show middle word word sentence weight algorithm use overall improvement section improvement explain produce htb ccccc f ccccc confusion classify summary specific vocabulary sentence weight create cosine similarity section lose summary confirm include middle remarkably improvement namely ccccc b htb c ccccc run sign rank confusion scenario second middle full drop full term accuracy e statistically speak accuracy previously summarie diverse amount second audio allow piece automatic consumption could
independence suffer type systematically assign relationship avoid power threshold across relationship equitability define threshold equitability prove equitability converse criterion equitability ask standard relationship low detection tail sample equitability low straightforward interpretable confidence assume say equitability make need imply equitability therefore minimal criterion equitability equitability equitability achieve sort robustness power independence relationship miss intuitive heart equitability low pre fine grain relationship precisely preliminary question argue perform equitability equitability commonly correlation mutual property size eq detection equitability prove serve primary purpose provide equitability context central power language formulate achieve connection current future exploratory independence measure dependence currently relationship whether one relationship however relationship equitability dependence independence understand equitability achievable besides relationship characterization set equitability behavior question equitability certainly develop varied new interesting case particular assessment formulate bivariate equitability change exploration ask author acknowledge r constructive useful legend type color equitability legend equitability correlation information use distance correlation require overall mutual parameter equitability specific david measure increase exploration trivial relationship kind tool nan variable attempt trivial relationship matter relationship meaningful impossible need set characterize equitability measure aim challenge statistic assign g idea call relationship interpretable draw testing moderate distinguish kind relationship regardless strength relationship kind strength equitability thought power interesting equitability evaluate like interest minimal dependence candidate pair variable evaluate follow size grow dimensionality meaningful gene analyze reliably thousand significant relationship percent pair manual usually characterize impractical challenge test examine small poor depend thereby list toward systematically assign relationship pair cause set crowd relationship would examine rank guarantee datum pose identify kind identify number kind equitability previous equitability informally one measure assign equally relationship regardless type theory object interpretable interval type strength interpretable act good belong estimator narrow narrow interpretable explain measure connection equitability interval hypothesis test typical dependence analyze distinguish trivial association independence assumption statistic strength regardless question equitability independence ask detect deviation also distinguish regardless connection equitability detection fix sample minimal relationship minimal relationship kind threshold strictly high equitability equitability converse equitability ask relationship strength reasonable goal example formalism relate equitability indeed analyse analysis equitability popular introduce aim good equitability functional power statistical equitability power dependence understand equitability discussion around equitability equitability accommodate variant allow theoretical equitability allow explain limitation equitability additionally connection independence question concern maximal information defer paper use equitability analysis hope paper equitability method achieve equitability goal setting equitability informally formalism equitability rigorously statistic dependence relationship idea way ideally quantify inference interval invert certain interval statistic narrow constructing use equitability use notion equitability appear discuss equitability practice equitability generally generic accommodate exist potential motivating often functional relationship coefficient determination correspond functional form statistic previously standard relationship interpretable ask statistic reliable sample reliable diameter illustration reliable hypothesis central reliable relevant interval question interval small reliable acceptance level interval interval interpretable interpretability statistic value interpretable interval correspondence coverage interpretable value sample counterpart sample interpretability small interval denote small interval region relationship value black reliable interpretable replace interval reliable interpretable imply respect summarize reliability interpretability interpretable resp say resp interpretability reliability average resp reliable interpretable imagine grain summarize interpretability accord equitability equitability case interpretability equitability interpretability equitability measure often use interpretability mean bad interpretability equitability interpretability reliability sample interval respect uniquely value reliability interpretability perfectly reliable interpretable build give example perfectly interpretable limit perfectly normal deterministic correlation perfectly interpretable reliable set bivariate normal perfect interpretability bivariate normal framework ideal contain guarantee interpretability perfect applie equitability requirement exchange equitability view tradeoff tell give vocabulary concrete equitability variable trivial variable denote functional equitability respect equitability functional functional observe function encounter enable empirical make equitability realistic distribution leave relationship past examine gaussian depend lack description noise easily besides allow might importance modification formalism design equitability functional relationship impossible introduce equitability represent variable conditionally coordinate otherwise interpretable serious first limitation point technical comment extremely arbitrarily noiseless prove model second limitation result address perfect equitability approximate primarily dependence part say notion equitability also mathematical equitability latter define perfect equitability informally equitability approximate notion equitability mutual scheme empirically conclude perfectly another perfectly perfectly impossible equitability property question remain measure analogy science np want approximate solution search appear practice formalism equitability relationship expect broad fact dependence population value trivial widely score equitability equitability evaluate equitability generate uniformly spaced maximal percentile minimal percentile reliable interval reliable interpretable equitability reciprocal large expect contain function assign type pair figure depict relationship depict way analysis interpretable equitability pearson relationship give interval indicate illustrate show relationship large interpretable line achieve limit relationship monotonically proxy relationship correspond equal see notion reliability interpretable intervals interpretable interval estimate define bad equitability size interval speak equitability natural interval arise equitability rich relationship monotonicity sufficient equitability obstacle equitability bivariate gaussians many asymptotically differ motivate exploration scenario relationship different relationship value relationship equitability specify population due effect depict population identical instance might correspond different operate interpretable interpretable sample interpretability picture though fundamentally data measure robustness make large functional parametric asymptotically perfectly undesirable exploration lack e circular leave next thing whose approximate equitability section though largely noisy application application functional relationship decide mutual deterministic possible impossible make equitability property equitability term interpretable via inversion natural ask equitability test alternative answer question equitability equivalently nan hypothese strength equitability power equitability analysis fix statistic reliable state equitability sided interval invert interpretable sense interpretability consider level reliable tail base nan interpretability direction maximal element provide reliable interpretable interval state describe equitability term power definition power uncertain interpretable versa statistic let interest right test hypothesis alternative hypothesis level composite complete parametrization independence give distinguish main result information tail nan view interpretable interpretability precise uncertain x set interpretable vice alternate equitability term two short lemma minimal interpretable give statistic know connection reliable interest level small test supremum ready prove characterization equitability statistical function statistic bad right tailed test power proposition fix equal closure closure illustration show pdf indicate denote interpretable power axis indicate interpretable uncertain equal statement prove claim show simply observe empty construction equal show already tell tail note option give equitability interpretable distinguish illustrated figure pair small relationship dash critical tailed nan curve power define instead heat nan nan critical heat result
period convention active hyperparameter perturbation bayesian optimization et al author knowledge depend reduce sampling hypercube literature avoid room optimize classical stationary poor slightly fewer clearly improve require counterpart find parameter many sample mcmc hybrid monte carlo monte carlo sampler slice recommend benchmark result remove statistically passive simplify surrogate among benchmark gradient low respect advantage perfectly figure apart regression bayesian fast less comparable fraction optimization intend per minute hour like bayesian local minima surprisingly computational cost case heterogeneity continuous categorical present section order reduce cost loop evaluation local fail converge fast real deep dataset row row second combine improve speed achieve art fraction idea surrogate efficiently gradient heterogeneity input optimization automatic reinforcement design popular rely form many rely stationarity able state computer economic function nice numerical point sometimes smooth g however many trial consume furthermore function multimodal although classic become popular method system adapt human machine automatic reinforcement kind box target available trial criterion decision condition improve optimum generality value associate function observation easily summarize generality remainder hyperparameter acquisition representation hyperparameter acquisition upper bind among n paper improve combination achieve reach evaluation improve also exploration drive gain simultaneous estimate variability comment bayesian deal space application regression base stationary many isotropic isotropic gps quite se represent smoothness variability function capture property behavior idea exploration achieve characterize mat ern instead everywhere might unnecessary may variability smoothness different scale nan reward actually attempt function process popular gp project stationary recently idea base input region combination gps gp even local gps region weight decrease second hyperparameter optimization hyperparameter hand bayesian root idea hyperparameter become problem learn kernel hyperparameter explicitly explore area gain hyperparameter surrogate reduce bayesian improvement confidence etc exploration predict principle expectation proposition hyperparameter add purely exploratory criterion need combine annealing focus several iteration gain pdf represent importance expect early result hyperparameter shape find highest relate certain numerically considerably accuracy however use require simultaneous perturbation classic space approximate perturbation although perfectly perturbation bernoulli I algorithm simultaneous perturbation also compute theoretically burden bayesian negligible respect update perturbation anneal many optimize discrete categorical etc model space acquisition optimize available implementation bayesian combine space reduce result
network filter much asynchronous compare weight initialization solid line deviation see line use demonstrate streaming evolve track subspace suboptimal contribution past track evolve forget old discount eq tracking derive follow keep output get eq outer drop last arrive discount descent neural asynchronous update rewrite discount activity local except discount suitably stay sufficiently two discount cost presentation numerical asynchronous modification section eigenvector row eigenvectors subspace initially subspace reach low rate allow fine filter jump rapidly neural filter project principal fall jump interesting lead slow decay extend filter error reach high small fine filter increase adjust neural filter error fall change lead slow extended memory past track asynchronous statistic describe initialization db definition text subspace paper make mathematically dimensionality satisfy possible single network show find neural input subspace show learn algorithm principal principal minimize projection component feedforward yet form anti update initialize stay mention exist filter orthonormal yet maximize input norm component however multiply mutual functional form feedforward connectivity architecture interesting converge filter filter feedforward numerical simulation subspace advantage principled activity inversion inversion local implement inversion iterate similar scheme guarantee guarantee criterion still suffer plausible network yet available stability start dynamic reach perturbation around stationary ever fast available generalization approximation recently bind dependence regret convergence speed streaming limitation neural use formulate exact pair stream reveal dissimilarity problem euclidean distance general dissimilarity distance derivation streaming version dissimilarity current past datum reveal approach datum implementation biological plausibility necessity much impose biology could conjugate method truly see generalization tracking finally similarity representation computation acknowledgment grateful discussion iteratively covariance hand strictly upper triangular triangular substituting iterate convergence symmetric gauss explore value recommend matrix multiply upper move component implement represent feedforward connection neuron plausible require algorithm rewrite result main text calculate evolve e introduce notation weight denote perturb network perturb w express definition relation eq side note stationary f next calculate go relation finally proceed far simplify get final skew matrix row row eigenvalue row remain e future linear treat separately start stability ease express convenience change eigenvalue could calculate eigenvalue stability multiply orthogonality filter would f contradiction eigenvector eigenvector eigenvalue share eigenvalue prove derive plug perturbation neural orthonormal decay perturbation make orthonormal perturbation perturbation restrict calculate perturbation therefore extract subspace neural cm title anti neural streaming medical york college tx keyword multidimensional typically dimensionality streaming adjust principled time rather principled bridge derive plausible subspace stream principled cost multidimensional plausible rule project principal principal subspace algorithmic theory process high input information million cell subspace project subspace simplify plausible dimensionality offer plausible reduction seminal figure point response furthermore transpose neuron eigenvector covariance importantly mean update neuron plausible derive square view compute numerous network example algorithm coupling previous able derive principled single rely feedforward neuron activity network update neuron anti derive perhaps come separately include contribution neuron numerous reduction biological plausibility requirement perhaps originally streaming collection reference minimize wide notation row span matrix rank point learning rule use neuron appeal anti rule b principal streaming datum traditionally dimensionality cost scaling connection neural implementation streaming organize minimize online single asynchronous analytically neuron numerically generate generalization subspace neuron compute subspace feedforward weight follow weight follow anti output similarity center vector space capture product datum product gram gram matrix center identity find minimize discover low cost pca projection rather eigenvector rotation stay evident cost streaming minimize stochastic compute batch output cost keep equality keep large simplifie far grow linearly dominate drop arrive term cost sufficiently admit close inversion plausible algorithm map onto asynchronous minimized coordinate optimal component converge mild assumption produce output meet well paper asynchronous nature subspace input set take principal analytical stochastic algorithm input project reduction rewrite eq matrix filter term perturbation stationary stability principal eigenvector filter cost optimal correspondence filter stability analyse pca network ode stability ode method time network stationary set zero input covariance perform weight update average average rest drop dynamical due neural activity assume algorithm sufficiently algorithm calculate stationary matrix collect relation weight summarize stationary state kronecker stationarity rule immediately stationarity rule equality hence ij e prove filter first equality ik f stationarity state filter partially question filter span eigenvector share unity rest value decomposition eigenvector span combination span eigenvector high eigenvalue need orthonormal row orthonormal row orthogonal skew h g arbitrary imply decompose row row skew rotation keep
recurrent network language explore phenomenon find translation new recurrent implement continuously traditional superior recurrent rnn input language nlp view translation rnn ability encode string elegant rnn task require store string imply short string inspire synthetic task rnn long far neural stack double stack suited processing structure language neural middle ground rnn implement powerful access read write unbounded constant result propose particular able range deep rnn lstm cell learn test string sequential reproduce perfectly input encounter training nlp translation symbolic base free rnns attractive symbolic algorithm expressive beyond capacity model neural network idea operation recurrent stack queue fully control obtain rather parallel explore powerful random operation whereas efficient believe task closely relate work continuous stack unlike operation mix across stack limit symbolic memory stack queue queue describe stack modify queue act stack traditionally operation letting intuitively interpret controller onto stack top stack formally stack form upon controller recurrence receive state stack pair dynamic represent represent maintain change strength first remove object lowest next remain quantity stop next strength current equation dynamic read quantity copy strength quantity preserve read strength scalar output read row traversal stack illustrate third step remove stack ignore stack cast differentiable arbitrarily case supplementary material neural queue stack exception operation read strength high read front queue top operation operate end call end write bottom follow read dynamic stack queue direction equation equation decompose update notational clarity neural except eventually affect read versa three module recurrent state input fully differentiable training controller exchange offer logical size controller controller enhance neural queue give neural stack wish replicate interface layer dot line take recurrent transform setup tuple state stack exception randomly overall read pass controller controller state scalar pass stack well projection projection stack pass stack next read stack tuple controller serve recurrent adapt queue stack module configuration enhance rnn controller controller linguistic non terminal begin root tune generative relatively training terminal balance terminal terminal generate terminal generate vocabulary word class reproduce level syntactic divergence english phenomena challenge unbounded recursive space include purpose subject rp gender free article gender translation english sentence noun infinite conjunction noun neutral article sample sample change observe measure well capability beyond length train sequence separately unlikely ever set read symbol begin next symbol take maximally likely softmax symbol produce correspond corresponding sequence generate formally end batch correctly predict th task benchmark evaluate stack queue enhance run size stack embed hide architecture stack queue enhance two deep come extra memory module regardless logical htb c stack lstm queue lstm minibatch across used gradient run seed number generator training calculate batch accuracy batch overfitte train coarse fine grain accuracy architecture deep benchmark well automatically lstm benchmark similarly across random try stack enhanced lstm yield sophisticated c lstm queue lstm pt layer lstm stack lstm pt lstm stack lstm stack queue outperform benchmark stack queue enhance lstm partially enhance whereas drop experiment gender task enhance order enhance htb sequence serve controller learn nonetheless regular benchmark expressive power ability perfectly notable finally queue solve stack solve simple controller operate solve flip sequence half reach token success deep lstm benchmark exploit short local dominate dependency overall rapid manner sequence enhance enhance unbounded memory capable act stack queue task task benchmark accuracy enhance accuracy require considerably
path explain insight imply bn add dag allow factorization effectively avoid tree observation help first bn alg htb bn terminal bn build build bn create create variable terminal alg creates hide build observable appear line create alg point observable variable root htb add add retrieve cache root terminal root create th find cache root alg build add observable variable variable hide alg build obtain find add build observable basically recursive scope terminal node child scope create variable associate child respectively current product node recursively child equivalently alg contract obtain alg bn add fig x alg indicate show add alg verify parent alg iff appear root terminal consistent parent contain correspond def bn construct encode induction height height bn alg consider root child root say iv ir iv ir ir contradict construct bn forest root height hypothesis separation rule note component follow child scope present variable take edge except add fact appear refer eq hypothesis complete thm bn observable scope include otherwise node increase construct hence consider terminal node node node variable root one edge variable node edge add graph size add alg alg construct bn consider child visit line create hide connect root observable alg alg alg alg thm transform know bn table bn tree tree size exponential bn representation generate power compactly alternatively bn add convert ac root leaf add pre ordering extent restrict add add normally refer detail add alg sub contraction node node pre order add add symbolic x x r r r r r x x htb symbolic symbolic appear weight replace common intermediate process allow symbolic operation internal add add view symbolic add symbolic add alg return symbolic simplify topological elimination use help symbolic add elimination detailed symbolic add alg recursively simultaneously root root node line root label variable line variable recursively simply create symbolic share rooted long share root alg occur contradiction share variable add rooted hence scope occur hand appear scope apply share root node appear multiply child line add rooted child correspond sum branch appear node two scope root link node symbolic create process alg line simplify symbolic merging encode suppose connected node symbolic add remove connect encode unchanged sum alg replace symbolic label elimination alg alg alg respectively htb add ordering add x hide variable ordering multiply alg alg keep symbolic add give bn return multiplication symbolic effectively redundant hence alg represent distribution illustrate bn one htb alg sum apply alg alg apply alg otherwise global ordering sum add topological alg implement preserve alg also view apply result bn add alg alg build bn add alg multiplication importantly branch product share scope view product alg end hide leave may add begin multiplication hide operation alg bound thm powerful informally exist polynomial represent key recognize proposition add record bn actually store sum product online quickly field bipartite relate depth low width bn consider height node sum path accordingly bn hence bn clique size minus tree reach bn author exist family much I substantially internal give convert exponential exist convert bn width high high inference precisely enable add compactly represent distribution width machine undirected network exploit practitioner resort require bipartite diagnostic quick medical causal independence insufficient present establish precise provide constructive consistency analyze tree also learn add correlation directly correlate causal relationship explore since bn convert example establish connection product network network key insight algebraic diagram add compactly bn independence generate bn direct elimination bn history inference process help paper depth recently tractable deep inference distinguish network task need approximated inference broadly clear introduction seminal understand expressive joint clear convert occur small bn encode belief lie context belief ignore due correct direction representation bn done translate bn compact prove adopt algebraic diagram add bn space generate bn structure bn generate bn add time constructive add give understand semantic bn recover relationship third normal bridge direction produce convert reasoning model counting count I evaluate boolean weight sum truth assignment important stream exact weight relate style exhaustive decision diagram decomposable form circuit broad field divide answer knowledge language target key shift common phase offline phase study recently convert mutually exponential closely relate decomposable formula dag enable formula count reader discussion class probabilistic discriminative propose apply classification later structure directly field promise activity modeling modeling investigate quite provide examine introduce capital random bold capital letter bold vector omit subscript variable letter denote subgraph arc refer need material reader familiar skip subsection whose characterize support characterize dag edge bn bn variable parent bn encode among topological bn bn variable bn conditionally independent probability admit reason comprehensive definition diagram domain algebraic diagram algebraic graphical function root dag terminal whose internal edge allow boolean discrete internal label take represent set henceforth add local constructive representation representation local branch internal node alternate node node bn generate decomposable boolean distribution allow add allow consistent transform network derive proof note refer complete let terminal complete decomposable sum every terminal distribution scope scope terminal nod complete decomposable f complete consistent decomposable let topological terminal node topological order pair iv iv network root expand polynomial law sum example iv apply iv jx x mf expansion otherwise generality expansion I eq root principle intersection scope due removal ensure local respect affect polynomial transformation alg complete decomposable v iv v root contract transform consistent informally iv union appear kind inside line induce root use nonempty intersection child build multiply link keep unchanged create node decomposable remove unchanged network affect line example transformation htb construct decomposable sub highlight highlight red outside dash
set connect balance even without membership constraint balance function hold partition function sf simplex equivalently continuous solution c cf note round trivially yield balanced relaxation exact e partition construct f k obviously simplex constraint problem membership imply exactly element I sf c lc indicator partition constraint enforce belong avoid column proof long indicator round constraint round solution continuous balancing round yield theorem control idea suggest image variation piecewise follow cut sufficiently propagate neighbor k sf c approach usage constraint symmetric balance asymmetric balancing employ prove result desire trivial balancing cut optimal construct non negative asymmetric feasible achieve low partitioning relaxation f l f feasible homogeneity indeed continuous relaxation obtain argument asymmetric set matter balance zero show bad asymmetric graph c c ff k homogeneity theorem introduce relaxation balance asymmetric balancing also relaxation indicate dominate illustrate toy desire relaxation enforce converge continuous solution round converge degenerate dominate generate yield partition solution round converge color point apart relaxation key paper problem reduce like guarantee moreover work negative balance monotonic eliminate introduce sf k sf hence proceed iterate approximation iterate result feasible monotonic ratio submodular convex element iterate lc j l definition subgradient sf sf amount change ratio decompose sf l rewrite break constraint relax I f I cluster unlike vertex automatically discuss use minimizing solve follow q feasible l sf l sf terminate theorem state predefine monotonic sf kf tt terminate inner strict terminate inner eliminate smoothness introduce additional lp lp material edge new equality ij l optimal constraint decrease finally rewrite yy order propose class saddle vector huge appear saddle introduce lagrange optimal value indicator negative elsewhere b saddle point dual q primal matrix introduce practical diagonal row completeness explicit dual iterate l k lagrange multipliers simplex ji primal iterate otherwise diagonal matrix adjacent vertex dual iterate give q per reformulate lp integrate label overall solve constraint sequentially via round see repeat double compute l b large belong natural fix top vertex membership minimal increase another lie center cluster membership depend well constraint degenerate stop see htb initialization sf li c kf sf kf c rank j membership f f f diverse method algorithm superior publicly code default initialization cluster similar initialization addition dataset variety uci repository nn weight news experiment term balanced balancing table report fraction construct per cut cluster font balance outperform across result show recursive bi affect cut directly asymmetric ratio cut significantly qualitative degenerate solution pt ccccc well strictly strictly good good cut balance method cut case integrate truth help cluster performance one fix percentage label truth cut initialization strategy work cut unlabeled method fail completely class news minimize cut continuous relaxation specific balancing splitting approach method enable integration link monotonic difficult ratio contribution acknowledge grant lemma sketch base cluster exist computation balanced heuristic weak original relaxation cut recently relaxation loose relaxation new minimization achieve outperform approach ratio cut pairwise formulate balanced cut cut
I state exist transition count occur state merely count without previous state take avoid characterized parameter prior tendency tendency markov chain depict explore short long coincide correspond stem perspective derivation section dirichlet reason gibb elaborate pt recall interested posterior gibbs turn moreover full sampler turn variable emphasize dynamic conditioning merely count leave characteristic chain sample concrete transition restriction unchanged probability suppose force take value precede regime join regime regime exist concern series alternatively think assign nothing regime suppose take table take innovation count different htbp py py currently sample normalizing mention sampler simultaneously transition prior draw take place conditional section study shift state change inherently estimate burn pass markov hence theoretically begin convergent initialize rather allow grow suppose reasonably change state chain normal shift change variance gamma derivation full conditional specify specifically occur realization see overlap range htbp change inverse gamma hyperparameter variance conduct burn sample period reduce dependence figure show intersection line demonstrate break true number specification replication detect change correctly point number case point estimating model gamma subsequent approach maximum posteriori estimate obtain newton together parameter result slightly burn great indicate explore besides model metropolis hasting walk value previous standard normal incorporate show table conclusion explore correctly change simulation bayesian parameter modal therefore empirical model first apply model mining poisson plot assume case prior prior perform sample metropolis hasting sampler burn sample reduce sampler draw sampler estimate probability indicator point intersection line location interestingly produce exactly one identify occur posterior deviation use deviation standard match literature robustness prior robustness replication collect replication detect one point find without number point number structural let index change frequentist autoregressive structural subject prior gamma hyperparameter sampler sample draw regime exist year change second mean deviation consistent estimate replicate whole detect change number suggest replication detect point model sense misspecification need change hyperparameter three discrete poisson result empirical dirichlet true change location appendix give conditional sampler q conditional gibbs obtain unknown discuss department finance business economics university economic economics finance chinese international economics university china edu quantitative management hide markov specification number specification model need sample state around normal mining united provided detect change growth model assume probability chain derive depend propose change change recent include change introduce random indicate regime specifically unknown period remain vector assume parametric parameter indicator take model transition regime point hide markov specify alternative vs multiple introduce hide right dynamic impose restriction appeal specify determine method state utilize dirichlet mixture provide introduction process study section discuss provide dirichlet technique derive dirichlet
lastly treat inversion still process work relatively obviously rely requirement set actually rest reasonably range accurate observe use hyperparameter recommend ht component component posterior component component last posterior ht component role propose complete spaced treat play another role univariate quick evaluation fitness evaluation would auxiliary although seem trivial truncation dimension sampler avoid approximate truncation stick break event break transform multinomial event model probit binary outcome certain difference finding full fortunately make easy simply l child functional analysis nonparametric spatial functional analysis provide nice interpretation regard sophisticate simple put novel assume latent utilize spectral enable miss transformation datum surface air spatial additive separable functional rapid interpretable spectral non nonparametric dimensional easily probably commonly spatial equivalent add base kernel therefore smooth light wide computer inversion fit solve many classified category reduction learn nystr om approximate top another approach spatial function thereby simplify refer utilize location reduce bottleneck multiplication one concerned whose determination resolution cost interpretation become commonly example correlation decay strategy issue focus spectral discover nice eigenvalue density time lattice school advantage firstly operation inversion completely avoid multiplication carry fourier complexity secondly exponential ern lattice rely property finite non effort limitation bayesian procedure treat lattice one operation stationarity local stationary process integration address call reduce approach issue solve bayesian example demonstrate capability dataset section briefly review study regard likelihood dimensional study lastly utilize consist north american dimensional gaussian z fourier transform covariance complex number angular frequency symmetric conversely represent fourier transform approximate review obvious lattice regular distance location increment location diagonal follow verify c conjugate straightforward eigenvalue inversion determinant express omit weakly stationary regardless assume always secondly frequency real backward dft dft either evaluate need dft operation multiplication operation involve operation dft lastly commonly increment location appear g one undesirable fortunately easily augmentation nuisance section substantial computational advantage see application two restriction sufficiently high lattice initially datum partially lattice lattice location beneficial noisy realization miss update complete smooth process degenerate form spectral scale range vector interest miss realization around use eigenvector transform imply note likelihood notion low covariance generalize value lack within element regularization definition covariance help one traditionally parameterization affect formulate widely function form ern close fourier covariance improper originally tr xx spectral domain xx quite efficiently range interpretation consist covariance utilize miss independent copy equivalent I distribution inversion sampling efficiently dft dd involve conditional variance index way would need create bottleneck avoid simply normal parameter update inverse gamma parameter metropolis hasting one focus however reality phenomenon edge appear corner observation instead rather dft dft represent distance cx kn knn cn remain proceed reverse fortunately exhibit edge know small value since manner point nuisance edge obvious double prevent edge appear practice repeat close zero one adaptively number negligible effect height issue might rely accuracy requirement solve lattice rest reasonably range quite even find recommend functional spectral approach counterpart bayesian inversion approach maximum estimation implement software cost lie subsequent failure cause correlation process either drive neither aforementione package square exponential ratio correlate reasonable possibly stationarity convolution location location small parameter assumption mixture process collection process define stick confusion frequency unless state context flexible stationarity capable address gaussian distribution worth correlate vector dependent consideration mean vector smoothness take vector stick break realization address flexible formation functional probit apply element range functional efficient functional process condition set update inside different besides functional gaussian essential probit correlate univariate multivariate distribution independent secondly assign component th multinomial sampling illustrate multimodal height ex n dots datum frequency frequency difference appear cluster stationary within subtle correlated curve switch middle process pattern become indistinguishable correctly estimate weight carry suggest interpretation transition suggest smooth suggest gaussian distribution process north american assessment surface temperature collect weather forecasting exclude year induce year intercept second second assume gaussian kx sx smoothing call filter choice spatial keep spatial additive last forecast correlation conditional space affect filter performance modification spatial multiply spatial accommodate benefit smoothing represent therefore drawback still assume limited additional example quite flexible describe model reduce separable ideal filtering create result impossible computer however easily solve ns functional mean process weight distinct exhibit change fouri three datum evaluate forecasting posterior mean location forecasting run take minute sophisticated hour converge converge smoothness parameter change subtle range autocorrelation weight allocate domain help nd rest change mean thank evolve change fix l rmse rmse ns fit l rmse rmse list rmse forecast complex forecast suitable filtering purpose interaction separable slight illustration plot temperature height height ns ns decay location independent contour covariance whereas latter parameterization application surprisingly sometimes unless close spatial association exclude alone ratio term density criterion cause curvature time change framework spectral new back despite ease estimate signal processing often locate characteristic prevent fouri dft study framework assume observational partial noisy realization surface miss underlie requirement integration benefit dramatically reduce allow feasible rather recently gradient compare major underlie enable sampling provide cause comparison study
g qx n minimize estimation slice challenge difficult challenging aim set framework estimation recall sir linearity reduction follow condition easy necessary sufficient version linearity let obviously identity hand side e therefore sir interestingly description linearity allow unable equivalence describe model careful consideration focus generalize concept classical new find flexibility dimension properly extend data theorem equation inverse regression reduction pt li chen tu institute statistical email tw south email abstract inverse base rigorous possible reduction formulation unify mahalanobis sir discriminant naturally reduction space measure separable phrase dimension functional linearity traditionally space column satisfy em variable problem inverse sir lead eigenvector sir construct linearity condition require x method angle normalize covariate new dimension reduction enable simplification permit reference therein model univariate response vector product covariate think functional straightforward may g inner perform case form dimension correspondence covariance consider vector operator space correspond careful rigorous restrictive message relaxed investigation formulate reduction problem subsequent mathematical kernel summarize outline either understanding motivate wherein dimension function fall require model model together sir standardized restrict equip inner bivariate linear operator say operator v ds dt ds induce positive semi definite definite equality definite strictly ball define map say ball bivariate induce xt xt easy verify continuous function integrable hence continuity chapter definition semi take expand convergent series eigenvalue eigenfunction sometimes write strictly orthonormal strictly still complete regression study semi integral integrable independent random obviously well solve problem domain hand require indeed sufficient classify group categorical feature reveal response categorical phenomenon categorical normalize within covariance form conclusion domain purpose classify response xt example regardless slice present main link reduction relaxed reproduce hilbert induce define range role equip interestingly product mahalanobis covariance study relaxed eq boundedness induce norm f bound il new compose three well rigorously extend stochastic xt st proposition exchange order double integral jensen apply eq large flexibility dimension define surely e reveal belong analysis dimensional finite component functional ensure integrate arbitrary variation variation degenerate sufficiently finite much subspace subspace ensure inner obviously I equivalent linearity condition quantity verify identity sir main problem lead requirement subsequent etc functional operate restriction lead relaxation flexible useful st define proposition therefore
decoder constant rip q combine column observation kk dt c singular good nonzero entry small quantity satisfy v union low denote exist inverse constant yy thus unique minimizer analysis c hold triangle unique minimizer sufficient exact minimization minimization relative hold minimization rewrite reformulate eq thus polytope positive vector hull x minimizer minimization c q c kt express vector q easy identity write equation rip additionally right side side I inequality solution second triangle set condition number q following hold q depend generality c integer lemma lie polytope vector q right eq want
previous state wavelet put probability ball satisfie consider wavelet continuity conclusion coherent investigate nuisance sophisticated regression model handle metric prior investigate slightly result handle adapt impossible mainly come function wavelet measure dr value retain nonparametric consider case design ahead covariate fix neighborhood design n n n sequel z ahead compact uniformly spread sense partition lebesgue covariate identically assume bound lebesgue consistency theorem kernel coherent wavelet whereas theorem soon satisfy may quite explain statement correspond coherent coherent state dp c dx true know let wavelet dp dx pd statistical assuming give let denote joint coherent necessarily complex window put positive function consider real wavelet wavelet situation get value kernel fix mixture begin situation make nature kernel function concern mixture coherent wavelet independently identically distribution denote variance stand product positivity neighborhood n existence exist function f c model dx computation let finish dense wavelet coherent converge lebesgue measurable proved exist converge previous follow write inequality nk existence test miss recall exist p f f f also increase eq f enough metric entropy desire concern ii finish existence test need correct complete metric equip locally bound closure series representation series intermediate compactly ensure continuity computation without generality moreover check bind eq choose small entropy desire remain prior probability decay notice nu nf replace consistency gamma consistency theorem verify weak satisfie condition gibbs mixture kernel duality dirichlet dirichlet apply I dp reversible thresholded approach rely measure chain set allocate dp inspire p tf almost surely sampler sequel particle unique iteration successively pn l stand allocate evaluation require knowledge hasting I metropolis scale parameter wavelet cn rmse wave spike corner cn rmse rmse wave angle spike corner run simulation find dispersion criterion run htb wavelet seem offer take carefully change dramatically estimator already crucial estimator opinion mixing prior fact wave coherent state wavelet theory choice carefully physical consideration geometry guide discuss wavelet candidate perform situation computation technique credible band credible good sample band efficiency done develop new necessarily kernel generalization done believe gaussian result measure weak consistency slightly different arise use outside measure properly prior whole care propose easy consistency allow translate fact dirichlet appealing decay jump dominate lead sparse reasonable bit conclude result desirable occurrence coherent quantum physics front square integrable group consideration could eventually guide right example namely coherent possibility entirely totally minor modification acknowledgment grateful helpful throughout article list laboratory france regression problem underlie probabilistic analogous density induce mix contrast function topological fact class way jump treatment locally separable variable definition poisson surely mean characteristic convenient interpret characteristic disjoint surely atomic let ia ia ia variable mean hence proposition mean purely almost surely number almost surely atom atom recall value theorem mean linear surely completely functional product jump jump either context random assign borel l consider functional spirit strong condition random say l consequence almost purely surely atom surely jump coincide atomic analogy case moreover measure homogeneity first z dx reveal context throughout gamma homogeneous letting turn onto characteristic aa two gamma gamma characteristic appear simply distribution dirichlet study bayesian literature complex computation derive dr assume support define borel gamma completely measure classical worth note satisfie strong convenient construction formalize next proposition variable v pi vx surely notice derive happen gamma normalization value scale variable integrate carry paper situation could obviously shall kernel overcomplete hilbert banach desirable function basis expansion coherent wavelet kernel promise leave haar space preserve representation irreducible strongly square integrable eq continuous subspace unitary irreducible say invertible carry simply identity normalize integrable mod carry bound onto closed sequel slight abuse notation stand inner complex product linear df real group endow canonical euclidean compact hausdorff haar unitary irreducible strongly however borel localization rewrite define stand mapping connect affine know endow topology topological hausdorff group leave haar irreducible well space subspace fortunately unitary irreducible strongly integrable unitary irreducible continuous allow irreducible call wavelet wavelet follow condition basis see previous wavelet basis wavelet dictionary b deal supplementary restriction kernel coherent topological homogeneous topological locally hausdorff interesting let locally q follow devote let u automatically verify dx u recall u bound rhs rhs g satisfied wavelet admissible b remark compactly take arbitrary recalling lipschitz rhs last rhs term exact thing let follow merge abstract measurable prior follow investigate property go
nature experiment estimate collective perform unsupervise move supervise formulation demonstrate base yield std mr lr see regularizer method oracle note though mr successful baseline lr svm show loo difficult illustrate actual accuracy compute nest loo loo whereas select value hyperparameter enforce result many bank loo colour influential weight introduction avoid also character letter shorthand else tweet truncation weight consistent sign suggest avg whether output assign inside train ard good spectral relate medium news reporting broadly relate central high mr benefit additional supervision mr drop bank explain effect case bank tweet positive positive negative classifier learn non stationarity collective unsupervise supervise domain find loo achieve return inspection learn reveal automatically data access annotate learn difficult allow minor difference stationarity community yield result baseline inspection correlation quite enable justify apart future explore feature contribute exhibit social ac media community determine belief collective ground formulate collective annotate reflect characteristic share annotate several thousand tweet seven successfully increase interpret act upon social medium especially circumstance spread twitter active turn attack child social medium text reject really trust setting access exhibit characteristic linguistic little annotated outperform multi collective great topic reaction broad tweet deal thousand annotate training build sophisticated novel classify collective adaptation show successful multi spread media flow group source tweet flow rank proxy information flow manually explore twitter along user former website match facebook false receive comment website none automatically automatic medium detect political twitter task define piece cascade classify cascade credible difference entire necessarily carry whereby classify classified support train identity pool tweet transfer classify unseen hard classify tweet set tweet temporal community r bank false twitter dataset annotate manually study medium medium support include tweet corpus tweet class refer corpora various corpora label modelling automatic without supervision five finding true rare primarily interested community nevertheless strong ratio ratio support vs tweet prove true third ratio come prove break many nevertheless overlap confirm event tweet first loo training transfer set leave formulation half training leave set may tweet classifier logistic word addition rbf first use naive nb component log treat mr setting jointly towards training tweet feature tweet gold bias regularizer learn present training use vector exclude underlie signal useful gaussian well nlp art widely use process explicitly handle imbalance process generalization mr central latent function specify degree function radial rbf lk latent map probit interpret q j integral intractable technique various calculate posterior propagation refine kullback true conduct parameter evidence ep gaussian experimental handle nlp compare rbf rectangular specify relate control way fact simple empirically frequentist lr use set unsupervised adaptation loo hyperparameter selection via nested loo hyperparameter
sequence get dd consistency let I write follow fail convergent basis focus multivariate transform consider obtain may row project orthonormal basis calculation obtain eigen structure generality independent ta ta rewrite obtain coincide easily obtain note basis estimate immediately functional discuss property concern prove cauchy sequence previous variance subspace add follow ty ny tt acknowledgement cover thm department universit di mathematics di via year concern explanatory curve curve instance chemical variable predict environmental weather pattern temperature variation year like spectra classification classical technique functional estimation contact range deconvolution reference base see reference term eigenvalue eigenfunction basis something properly justify proper give procedure automatically neither functional real link compact call identifiable space estimation pose related moreover reason come discuss influence orthogonal firstly parameter inferential large dimensional size introduce auxiliary analysis carry consider functional model deterministic collect correspondence treat among notation inner norm assume realization neither distribution quantity main focus describe functional paper small sub call coincide random support small space coincide function orthogonal odd present center straightforwardly verify analysis well straightforwardly generalized context entirely functional estimator least square finite classical particular mild pose compute reconstruction provide minimum situation avoid choose formally lie sub reconstruction consider uniqueness restrict reconstruct identifiable search sub thing perfectly projection component datum orthogonal vanishe searching suggest guarantee determine finite sub typically reconstruct imagine actually answer discuss central rewrite slightly operator I orthonormal explore relation describe orthonormal basis analytic since identifiable dimensional projection element square minimizing obtain behavior mention wide behavior sake define replace follow q estimate dotted belong eliminate priori choice observe irrelevant related quantity pointwise differ component hence sum influence explanation phenomena lie divide simulation simulation appendix mainly consider sub space situation simply fact influence analysis contribution take reconstruct among figure among pointwise mention due fact correlate put related close well contribution observable compose eigenfunction basis construct solid dotted coincide subsection bias concern introduce trade tp operator relation obtain eigen denote eigenvalue respectively eigenvalue one hence projection direction I total distinguish bias choice sub identifiable minimize suggest estimator situation estimate variance estimate discrete naturally chance minimize variance choose space ease notation variance coincide component priori different space principal pc minimize bias describe panel panel pc pointwise discrete prefer chance minimize choose discuss arbitrarily compute infinite closure countable I construct investigate asymptotic dd arbitrarily make consideration mention identifiable curve greater implicitly arbitrarily present consider analogous space define
estimate examine prediction n prediction error match theorem slope thereby scale theorem triple block zero normalization prove appear suffice sake shorthand scalar loss generality bi column make hold swap swap side moreover every minimizer lower complete convenient event condition may two zero independently piece show conceptually similar detail proceed derive contradiction finitely convergent unbounded claim let sequence beginning write definite unbounded guarantee must claim imply boundedness proceed monotonically guarantee minimum point ball eq contradict specification prove claim always minimum iterate shorthand center ball disjoint volume within air force office research fa office convenient priori let sub design minimum hence bound bind u minimizer interior say interior remain prove low ii since ta notice combine yield non global definition hold define triangle inequality find yield proof convenient omit shorthand representation construction matrix update statement suffice event ib claim final fact thus iteration assume integer integer suppose minimize inside ball function turn eq fu fu minimum contradict f prove argument induction well claim definition prove argument minimum belong derivative namely prove contradiction mean calculate fact assumption establish claim second equivalent hold chi square inequality ccccc berkeley edu department engineering california berkeley high estimator absence restrictive condition slow intrinsic broad estimator estimator square function together local optimum associate bind apply optima popular nonconvex regularizer scad penalty mcp addition regularizer optima broad local minimization typically bad minimax possible problem order feasible implement polynomial study computationally estimator question coincide differ fundamental classical counterpart explore gap classical risk high regression minimax broad coordinate separable regularizer include regularizer linear throughout entry matrix variant integer sparse parameterize triple ambient dimension use denote measurable take quality response obtain estimator denote constant deviation compute manner subset motivated heuristic basis pursuit selector replace constrained covering reference focus design satisfie certain restrict property know design matrix computationally efficient matrix conservative prediction error lasso question establishes result complexity condition polynomial achieve hardness possibility dense polynomial show slow namely base square wise decomposable consider choice least regularizer popular pursuit square nonconvex regularizer mcp nonconvex local argue bad optimum statistical concern focus descent isotropic result way paper gap dimensional closed application broad conjunction add fundamental gap remainder definition estimator illustrative prediction achieve statement provide lemma defer discussion previously link linear predictor prediction analysis regularization separable regularizer example analysis provide bound estimator risk adaptively ordinary instance counter slow achieve rate nonetheless design adaptively error square root minimizing criterion different root relative advantage require purpose lagrangian lasso minimizer square varied root apply square root due form nonconvex penalty due fan li mcp family scad penalty similarly mcp mcp regularizer illustration scad regularizer turn precise good matrix lasso achieve prediction information theoretic avoid include restrict good easy recover intrinsic et strongly column prediction absence constraint say simplify notation follow consider normalization integrate tail follow range could main theorem discussion consequence local minima local open define object triplet typical descent method converge applicable local sparsity separable coordinate eq statement see low estimator result adaptively choose nature allow choose minimum execution convex provide minima match bind separable regularizer establish existence bad adversary power minima concern typical optimization function local general g iteration also algorithm unify begin observe iteratively give stepsize approximate minimizer ball radius neighborhood thus iterative current algorithm choose close tie randomization terminate minimizer belong interior loss define global nonconvex ball powerful observation impose regularizers origin jx iii
operates score binary word minimize error train principle layer activation batch word target lexical translation lexical word take multiply sentence translation restrict book avoid difficulty sentence calculation word calculate sentence naive pre probability slow calculation source sentence comprise pre calculate target phrase word whole corpus phrase translation configuration add component corpus log utilize house phrase decoder nc corpora train correspond corpora plus news package big corpora domain describe conduct reduce sparsity describe house phrase decoder search hypothesis error weight optimize consist contain k consist investigate impact describe model corpora mainly corpus quite speed process testing except validation table cc avg send avg include linearity consume boost unknown value keep hidden gain strong baseline source context whereas use source show impact extract source appear l fr sc contexts include help case case good improvement cc system fr sc sc source vocabulary size indicate context architecture architecture consist train hour hour whole translation time decrease use architecture deep stick architecture big use big corpus quality domain similar model broad domain report corpus conduct experiment english order baseline would effect language long build similar integrate translation table cc system sc translation increment baseline come frequent word context baseline frequent source context notably sc phrase enable abstract representation dependency decoder help baseline linguistic resource preprocesse process linguistic resource translation language future try integrate linguistic feature might fp edu contexts discriminative deep neural network leverage dependency linearity model reduce approach art pair translation task attempt approach draw community huge address depend context standard phrase translation translation phrase pair extract corpus phrase segment context phrase leverage wide context phrase discriminative exploit predict wide feature discriminative hence translation dependency source perform abstraction input linearity lexical translation rarely share could model well address issue discriminative lexical train neural share word source linearity exploit semantic dependency among organize lexical translation translation model neural network provide experimental result lexical lexical decision present phrase base employ translation lexical advance propose either enhance model context approach directly target sentence gram phrase mt thereby boundarie phrase gram basically build principle joint linear relationship le gram translation phrase long context joint word source aforementione work essentially translation inherent exploit sequence word global context motivate et two belong another lexical approach name predict word model employ classifier perform lexical word phrase extract rich source sentence input opt network architecture employ global original describe finish decode determine individual train train word give sentence sentence represent sentence bag indicator represent vocabulary example target neural next sentence
analysis proceed summarize pca performance area diverse expression implementation package comparison robust pursuit discriminant pp projection consider datum remove along find pursuit pp projection dimensional reveal detail interesting surface etc extract analyze widely blind source pp projection extract seminal like pp variety pp outlier pp high space combine definition dimensional four seven later though technique fail intrinsic dimensionality existence intrinsic ensemble often data optimal typically unstable tend get contamination contain cause aggregate bagging offer one variance aggregation aggregation combination form equally weight th base regression bag outlier class come bb bag vote frequent replication namely bag learner bag b paper bagging help achieve robust classification space bag datum contaminate estimator subspace attribute bag proceed bag add crucial consist select building randomly draw replacement index build ingredient estimation bag tree make ensemble base replacement sample drop dimensional build base learner base b rf max max pair training size generate replacement denote turn base deduce outlier replication base aggregation bootstrapping mechanism learner performance life thing apparent training true error replication define throughout randomly assign consider none tune replication six microarray gene clarity completeness present performance explore analyze diabetes deal obviously dataset reveal difference performance method small available package ii please dataset qualitatively explore microarray gene expression iii data microarray subset dataset classify recurrent recurrent tumor observation cancer cancer dataset vi contain subject classify fail pp huber pp rf diabetes microarray stability strong aim robustness notice unstable producing despite initial finding general thorough study determination well generate density contamination capture contamination scatter furthermore study pattern parameterize identity dimensional one classification throughout paper use consider namely contamination mild contamination contamination performance e simulation look reveal pp appear pp pp huber pp categorical pp performance discuss much pp typically pp least low pp notice carry rate contamination clearly show contamination among rf whereas pp fail pp fail ideal namely binary large correlation consider technique contaminate regime number class investigate space dimension technique fact favorable regardless contamination htbp fail notice rf well strongly regime look combination class effect htbp htbp see figure contamination categorical contamination well take rf predictive rf realistic contamination thorough predictive performance robust interesting large rarely yield technique one remark pursuit seem matter fact pursuit fail regardless aspect two predictive suited lie indeed overall random forest yield explain early mechanism forest estimator subset crucially mechanism leave proportion certainly indicate early forest inherent forest robustness subsampling attribute fact selection inherently address dimensionality elimination outlier subsample sense connection however loose yield determinant bag forest relationship forest currently way acknowledgement express infinite ever especially flow powerful proposition conjecture corresponding institute university mathematical sciences institute ny usa pattern recognition subject research year explore performance special concentration variety forest ensemble inherently specifically robustness focus classification dataset precisely large class may consider building throughout shall th estimator build portion split recognition create solve precisely amongst discriminant classification tree forest tree relevance name datum instance state namely call small since common day especially microarray gene disease cancer matter consider six contain various namely brain traditional discriminant near neighbor mainly lead method even solution severe curse loose ill several achieve discriminant analysis microarray expression regularize regularize logistic dedicated typical contamination get ever outlier relatively extremely high ill outlier literature discriminant scatter apart discrimination fact reveal covariance estimation context approach explore stage technique get life datum reveal test context simulate contamination scatter matrix correlation value contamination location rest paper two present brief emphasis use limitation method life five dimensionality contamination correlation various choice prediction replication section six brief introduction work dedicate arguably group estimator discriminant object whose discriminant give gaussian presence discriminant lda share sample group membership discriminant give ik observe membership finally pool turn make robustness performance outlier explanatory cause discriminant bias need method scatter much deal version regularize application extensively area need subsequent contaminated dimensionality combination robust pursuit presence author aim know minimum determinant robust discriminant recently explore extension compare extension explore also performance one extend mention package immediate scenario contamination technique
ensure symmetry upon inversion back spatial subsection propagate gradient fourier transform layer apply inverse dft input dft remainder dft propagation gradient dft introduce symmetry distinct choice technique function project onto idea pooling stem observation input discuss technical detail advantage h map output pool cm n implement output map dc shift maintain central submatrix denote approximation take dft list conjugate symmetry special case subsection break truncation value treat individually effect various procedure intuitive supplementary material address dft dft layer apart gradient appropriate note convolutional employ pooling implement additional dft regardless spectral spectral significantly retain pooling representation information resolution precisely linear low pass uniformity respect mass frequency elimination reconstruction suffer dimensionality pool specifically freedom input specify arbitrary gradually pass small frequency outside central maintain output dft resolution stochastically truncation truncation axis outside nest e uniform high resolution filter cnn directly frequency domain advantage empirically layer seek frequency domain attain inverse dft cnn input mini batch back dft identical outline discuss change cnn explore epoch filter frequency transform domain representation pattern considerably considerably update cnn filter across domain weight capture degree freedom hand provide appealing basis characteristic often localize spectral representation observation filter tend narrow qualitatively speedup descent linearity exactly regardless whether frequency transformation space parametrization meaningful relevant modern rescale able leverage axis small optimizer element exist number promise future elaborate discussion lrr pooling maxout imagenet fraction parameter keep measure error achievable pooling b augmentation optimal pooling network effectiveness run optimize hyperparameter evaluation validation imagenet result pooling observe subsection spectral pooling permit number pooling control severe quantization bind preserve horizontal permit produce smooth choice pool filter size spectral layer output layer filter convolution relu nonlinearity height r think parametrization dropout layer decay weight decay epoch dropout augmentation hyperparameter want success perhaps hyperparameter assign map randomize constant setting attain cifar cifar competitive employ architecture deep generic sp remainder past match mark light achieve speedup architecture negligible parametrization cnn optimization architecture different pooling layer size architecture attain competitive classification deep equation increase difficulty optimization reflect considerable horizontal dropout spectral filter spatial spectral fourier transform attain variant optimizer find figure convergence speedup table surprisingly negligible speedup expect much room exploit spatial work rich spectrum spectral pooling allow pooling dimensionality significantly addition parametrization fast frequency employ fourier transform convolutional ed multiplication ed desirable transformation forward sensible nonlinearity switch significant dft difficulty impulse involve sum hence locality fouri locality spatial locality employ wavelet provide wavelet throughout machine great learn cnns center research support mathematic office science advance compute research department contract ac resources national thank school university school engineering science discrete transform speedup computation deep efficient computation cnn employ cnn perform dimensionality considerably per parameter flexibility pool output representation modification competitive dropout effectiveness complex convolutional filter observe configuration cnns result across science video cnns expense train convolutional key ingredient cnn deep natural demonstrate convolution filter computational arise convolution wise domain offer provide cnn study spectral representation cnns fourier filter representation mapping correspond unitary transformation however argue spectral representation show filter tend spectral thereby reduce optimizer align representation dc domain domain conjugate gray symmetry fourier transform dft decompose signal introduction dft input
random scheme develop minimax promise currently code scheme greedy sparse apply acknowledgement support nsf thank valuable comment theorems minimax clear non finitely prior integrate optimal residual r ci show schwarz show achieve suppose theorem asymptotic minimax risk normal ellipsoid argue problem reformulate last support provide parameterize optimal bind suffice attain jj j feasible show l solve feasible plug calculation particular base correspond favorable sufficient base follow choice lead complete write expectation show argument lemma algebra note eq equivalently universal apply schwarz need lemma inequality denote brevity inequality easy minimum small non increase simplicity go observe since hence achieve must switch thus justify fact complete suppose degree centrality central chi poisson weight follow recurrence derive prove replace prove thm thm remark carry storage placing limit bit encode estimator excess risk quantization level pareto tradeoff quantization space technique analysis minimax storage size place datum construct understand error let resource fall within minimax risk region pareto versus vary case storage procedure formulation problem naturally motivate instance analyze estimate send communication cost become lose risk quantization scenario cloud environment process store limit storage dominate scenario quantization tradeoff processor limit precision arithmetic computation precision motivation storage nonparametric estimation lie classical distortion theory minimax theory idea refine fundamental connection code wiener deviation periodic constrain minimax q bit total identify regime bit r pt bit quantization classical minimax regime however certain pt number bit insufficient preserve minimax quantization dominate minimax insufficient regime show regime error insufficient achievable quantization threshold surprising classical optimal risk achieve keep estimate smoothing level minimax compute least favorable source base mutual combine quantization establish achieve quantization classical distortion generation codebook determine source allocation bit use bit adaptively smoothness quantization estimation adaptive coding storage operating correspond regime establish minimax quantization regime outline proof procedure stein quantization detailed supplementary material obtain estimate lie subscript mind nonparametric subscript maintain minimax infimum estimator respect abuse minimax risk bayesian align whose support minimize integrated favorable prior lead provable scenario constraint bit formulate follow codebook estimate need store index simply encoder decoder denote storage constraint decompose expectation equality form decomposition due express abuse infimum definition distortion mutual denote minimax since low q goal least favorable turn simple mean hypercube show low risk estimation quantization large scale low mle close take constant ball scale tight bind show mean euclidean nb show asymptotic euclidean quantization construct bind budget euclidean ball case relevant estimation sequence stein suffice new code strategy clear sequel allocation determination detail bad series variational asymptotic minimax tight demonstrate quantization asymptotically space white diffusion equation wiener observe diffusion goal choose arbitrarily recall white familiar carry encoder section composition encoder decoder call risk asymptotic model specifically coefficient ellipsoid hold actually long expand orthonormal convert reformulate collection follow establish explicit distinct quantization regime play interpretation apparent q achievable asymptotic budget turn threshold value surprising classical first keep basis regime bit great convergence rate bind directly proof possible regime bit suffer regime optimal quantity term sequel constant constant dd regime bound choice parameter large favorable prior insufficient regime point third regime communication insufficient optimal long quantization dominate risk decay matter fast analogue simple informally term quantization regime outline defer gaussian term section concentrate prior q mean optimization infimum define classical distortion give analyze term quantization follow moreover close insufficient regime regime begin solution decrease decrease correspondingly optimal reverse water scheme third exist integer series three reformulate writing bandwidth function bandwidth choice omit argument solution express summarize achievable code scheme quantization together block system cardinality infinite index block ready scheme distortion codebook codebook adaptive block budget separately code magnitude detailed generate codebook kn encoder decoder optimization dependent codebook codebook keep normalizing row illustration codebook encode thin color gray dotted rectangle node color dot thin fill thin dotted thin gray gray kt store store index codebook get base reconstruct codebook theorem establish asymptotic constant respect remark estimation bit asymptotically budget namely size codebook suppose low radius codebook parameter function bit fact adaptive grow formally modification numerator expectation denominator achievable grow need code bound accord expect codebook similarly codebook inside imply achieve ellipsoid generate codebook universal risk ellipsoid code detail outline term
adjoint operator expand numerically take discretization skip minimize live functional fix initialize convergence result classical gauss example smooth popular proximal thresholding fista employ fista great backward see proximal contrary forward backward limitation fista convergence prove far know property fista rely step soft operation practice evaluate iteration procedure backtrack choose l z reduce closed simulation request motion smooth brownian motion bandwidth denote medical challenge know language framework instantaneous record varie fast study signal vary instantaneous patient instantaneous heart evaluate r peak peak non instantaneous heart cubic spline visually instantaneous sampling analysis fourier transform window please see extract instantaneous fail addition factor process signal take window window deviation clear dynamical hand instantaneous frequency blue add noise perform model noise snr noise signal frequency curve curve mode function harmonic function vary instantaneous find representation refer encourage several thing fista still usage take minute finish analyze length find sensitive choice choose behavior moment robust numerical theoretically study interesting finding compose intrinsic varying depend weakly domain window profile window signal show window ideal nevertheless nothing window support wide window standard signal note start indicate achieve window make effort determine kind energy window extract topic wu research support eq c part determine positivity amplitude condition eq q change globally change integer hence q sign inside calculus know n n loss exist last part thus know thus sign inside sign inside finish take without calculus hold claim come taylor cosine q contradict claim amplitude generality calculus claim contradict sign bind immediately control qualitative dt nf lf lt lt zero taylor second expansion component taylor recall short transform window need universal fix argument immediately q first claim particular v ft g define enough distribution hand result g lt v kt smooth bound right hand simplify clearly conclude finish eq g claim lt lt equality obtain proceed finish define also lt claim enough generality point choose condition argument lead assumption claim lastly phase function na lt kt dt na lt lt contradiction theorem definition thm motivate limitation signal compose vary instantaneous frequency harmonic signal property representation fista shrinkage thresholde numerical coin confirm analysis fista instantaneous frequency proper feature step toward signal inside traditionally commonly however compose harmonic perform behavior capture lot decade time frequency attract example short time transform ensemble decomposition solution analysis however mathematical foundation several limitation name attract issue introduce coefficient phase technique class share precise preserve technique trick wave mechanic finance energy physics signal slowly instantaneous want study signal instantaneous song wave kind signal frequency instantaneous requirement auxiliary function quantify fast instantaneous minimizer guarantee apply shrinkage thresholding fista way adaptive harmonic model provide fista vary frequency mode consist adaptive harmonic fix constant consider functional function kt c satisfy c c instantaneous control proceeding component reconstruct th component integrate eq realize spectrum ideal function restrict compact connected numerically function ft reconstruction property visualization achieve take equality hold due one h ft kt frequency kt kt k lt argument hold q ft observation variational discuss follow well carry come term might te special case know thus follow cf capture consider function plane restrict representation
substitute exclude stable solution correct among stable say negative number satisfy strict give set much prove stable cx cx cx cx I prove condition already stable otherwise process say right consistent merge otherwise consistent merge otherwise consistent otherwise continue process construct note stop kind stop prove equality thus main conclusion reduce problem sequence maximized solution follow v ty b prove ct v construction start leave consistent right combine get go show contradict correct mention main consistent obviously among solution prove solution solution cb cb cb cb cb thus note equivalent wrong say always text sub suitable consistent solution pt cm pt supplementary material dynamic specific degree accept document text mention text claim prove loss move around actually finally stop component exclude sort
repeat arrange half I define simple vector repeat span precede unique transform update possibly repeat also uniqueness express cholesky practice square scalar iterative practice precision svd machine p extend application image applie tr u tr sparsity iteration matrix l yy compute see minimize p tend study behavior code equivalently establish transform update corresponding solution orthonormal result coincide solution employ alternate minimization specifically eq update denote svd optimal solution unique zero proposition replace constraint additional proposition provide appendix orthonormal synthesis dictionary denote dictionary immediately alternate update orthonormal alternate transform computational cost matrix begin onto ball employ sort hard equation update computation multiply update exclude pre scale step cost coding scale overcomplete synthesis synthesis scheme also computational analysis converge iteration svd typically translate e constraint use barrier violate otherwise w unconstrained objective exactly equivalent objective minimizer whenever two objective unconstrained minimum constrain formulation unconstraine formulation error control avoid interested know alternate minimizer result alternate minimization g accumulation denote magnitude wise ij k x k iterate sequence decrease say accumulation point minimizer define arbitrary start large choosing magnitude local accumulation local optimum accumulation particular equivalent equally minima objective minimizer irrespective empirical transform insensitive initialization conjecture potentially provide illustration corollary convergent refer globally convergent set minimizer learn requirement distinction convex extra g tight isometry convergence control arbitrarily perturbation half accumulation half perturbation perturbation outside local maintain sparse could choice perturbation small directly adaptive blind denoise compressed sense formulation highly see solve transform denoise guarantee converge trivially p obtain similar objective monotone value moreover iterate sequence accumulation iterate minimizer accumulation proof result corollary corollary demonstrate transform usefulness alternate learn initialization sensitive initialization study provide understanding compare patch representation usefulness implementation code version intel cpu memory bit window operating system image representation patch mean stack patch display removal adopt compression denoise work experiment metric learn code lose fitting domain learn transform learn transform ratio transform patch serve simple surrogate information overlap learn transform transform scenario initialization initialize algorithm iterate code initialization dct kronecker dct second initialization invert learn instead row matrix zero show initializations b converge error decrease require importantly nearly initialization indicate algorithm reasonably insensitive initialization initialization dct frobenius norm initialization condition dct learn display call transform atom texture initialization equivalent appear somewhat permutation transform learn initialization provide differ experiment learn overlap patch propose closed compare version value determinant p fix dct analytical extensively compression near integer simplicity execute update figure plot learn transform patch dct fig c close via dct normalize close recovery analytical dct patch size transform size cf reasoning learn experiment number gap patch learn transform conditioning restrictive normalize error algorithm identical algorithm involve fast actual general another analysis blind source separation similar work ica representation learn ica signal correspond ica independence matlab code learn recovery replace sparse ica seen propose transform recovery approach superiority compression ica fig slow ica finally transform synthesis heavy burden scheme compression classical involve dct wavelet learn adapt specific transform adapt variety transform compression goal recover represent vector corrupt whose variance present denoise use transform patch patch representation unknown propose estimate iterate propose algorithm step average respective brief employ image couple fig different image compare denoise overcomplete synthesis denoise scheme matlab available website use setting maximally patch result transform denoise usually transform increase degradation overcomplete setting various denoise number denoise work pt algorithm denoise iteration step patch transform brain couple denoising obtained obtain well image average svd db transform algorithm denoise involve close fast c four average transform denoise denoise compute svd denoise svd method speedup transform mainly cost base multiplication see sparsity method synthesis sparsity since sparsity svd threshold threshold svd speedup transform higher point actual speedup information synthesis typically fraction speedup denoise svd noise increase increase patch image explore transform denoise involve close overcomplete adaptive transform denoising become well choice parameter use experiment simplicity bm bm adapt transform formulation square efficient become transform orthonormal synthesis provide establish convergent minimizer guarantee rely learn transform obtained provide dct provide well k svd fast denoise learn involve discuss extension overcomplete elsewhere easy eq subscript element optimal whenever consider corresponding term objective either hard fix discuss remain depend square root specific us svd index converge singular hand aforementioned limit behavior formula degenerate accumulation hand scale constant clear coincide solution alternate transform coding aforementioned simplify problem orthonormal update problem value latter provide proposition denote every accumulation r q tm subsequence obviously limit inner product sequence orthonormal diagonal matrix entry maintain decrease diagonal svd precede indicate accumulation similar difference barrier replace penalty brevity sketch use operation optimal ball minimizer follow vector prove property existence accumulation point accumulation iterate every accumulation every sense monotone converge transform global minimizer objective decrease thus explicit therefore bound sequence value generate least accumulation point convergent subsequence bound accumulation boundedness simplicity inequality denote k n unbounded c inequality follow triangle since section boundedness previously lemma accumulation accumulation iterate satisfie subsequence converge accumulation objective monotone converge accumulation accumulation sense subsequence converge accumulation define subsequence accumulation point barrier continuity converge accumulation iterate accumulation algorithm converge accumulation linearity r accumulation matrix subsequence limit equality fix sparse accumulation accumulation subsequence iterate sequence converge accumulation imply deal uniqueness satisfy similarly ii imply feed stay accumulation finally fix iterate accumulation equally minimizer g ng produce stationary hessian definite trivial sufficiently furthermore obvious code thus q sparse code give fix point perturbation suffice otherwise barrier trivially rest preserve preserving expand norm term trace inner x simplify q I aforementioned positivity assume neighborhood become lemma define simple x sparsity great column column thus follow zero outside thus proof discuss bound converge singleton sufficiently combine sequence singleton tie support coincide support one converge point case belong theorem unconstrained barrier function perturbation I ix repeat theorem signal certain transform dictionary synthesis denoise medical instead square transform alternate code transform step provide globally convergent minimizer transform practice insensitive present promising transform sparse convex dictionary widely year study notably specifically transform sense unlike synthesis measure residual transform capability model scalable synthesis briefly sparse reference difference find representation transform subject w contrast synthesis noisy np synthesis analysis code especially involve various synthesis computationally adaptation sparse adaptation synthesis dictionary signal
deep acoustic deep net neural net really really train improved discriminate highly yet acknowledgment model imagenet helpful discussion google google google machine train prediction unfortunately expensive especially net possible develop use compression surprising improve acoustic heavily single new full distinguish fine grain model expert train energy completely optimize requirement large machine typically training stage requirement like highly operate real user requirement easy extract model regularizer dropout train transfer already rich conceptual may investigation knowledge make keep abstract view learn learn average incorrect answer lot model tend generalize mistake probable accept reflect true closely optimize generalize generalize normally large however generalize train transfer class produce soft target transfer transfer simple geometric individual predictive soft target entropy much mnist correct high much ratio soft target may way information define rich say look little stage softmax softmax produce call temperature softmax suitably target temperature match soft target show model transfer consist entirely unlabeled original training find especially encourage target match target provide typically match soft target turn neural typically produce use softmax logit compute temperature normally higher produce class knowledge target case produce temperature softmax temperature train significantly modify soft target soft softmax generate target exactly softmax temperature magnitude gradient target important multiply soft target ensure remain roughly unchanged change contribute cross logit soft target gradient approximate transfer simplify high provide case attention advantageous almost unconstrained cost training could noisy acquire dominate strongly suggest ignore unit case net weight share pixel net small net add task match temperature soft target deal generalize translate transfer two give unit work try perspective digit error cause much increase optimize right get test training model error approach try relative hard indicate extract hard single improvement frame similar improvement preliminary mnist word frame recently relate acoustic train temperature unlabeled gap hard advantage usual ensemble ensemble though easy give total computation fine grained overfitte target dataset million label deep convolutional large two parallelism net core mini batch replica mini gradient server back parameter gradient server last send replica replica core put neuron parallelism two lot several train option fast number make ensemble contain many highly type low weight slightly coming sample remainder training correct bias logit class derive decide full even though confusion cluster party bridge cluster line algorithm obtain produce result investigate happen ensemble deal step according call use step take special empty intersection note empty class minimize class plus class arithmetic optimize carry c c accuracy baseline extremely completely show absolute combine model test overall report accuracy belong class restrict r delta accuracy disjoint cover class table show improvement break trend claim soft target hard target lot target could hard demonstrate fit speech early baseline hard target lead severe reach train target information remarkable early stop converge soft target effective train another training train frame test baseline target collapse
two correction copula study fix keep unchanged gradually get tendency simulated investigate work iid chose show histogram ten burn reduction uninformative towards correction seem binomial particularly among difficult cox datum one everywhere else section study correction future making set less extreme simulate appear copula correction general always circumstance intuition underlie strong case problematic correction limit applicability maintain normal cdf jacobian jacobian ix qx n I collect term substitute eq correction account thank provide r code relate study grateful hold datum thank comment correction mix integrate laplace low count somewhat particularly problematic new implement part package evaluation simulate indicate bayesian copula generalize nest part accurate extremely software rich generalize prior somewhat inaccurate replication new ten million red curve correction scene minimal let iid prior prior intercept precision histogram red show run quite experience difficult fix reasonably problem usual ensure asymptotic validity see effect parameter realistic amount spline smoothing lack parameter little add panel observation predictor bottom derivative near frequentist attention recent computationally laplace propagation type costly use speed requirement correction proceed present correction section show method work brief conclude remark hyperparameter observe want laplace approximation find curvature approximation skew approximation detail notice marginal approximation second improve approximation structure fix solution additionally correction soft define increase correction determine since zero close moderate large increasingly find let job investigate retain skew cdf complicate derive find simple case preferable skew simple version try correction significant show approximation inaccurate case correction iid per give sample correspond page difficult error see empirically laplace use prior correction chain million burn million posterior close reasonably mcmc parameter see show simulation difference mcmc deviation effect lower improve major effect value correct c correct correct discuss
infinitely density activation converge surely activate infinitely often activation bandit structure bandit activate well surely without strong law finitely infinite optimal activate time write trivially eq define space relation yield logarithm similarly large rearrange positive eq note relationship take unbounded eq sufficiently relationship follow eq take proof bandit population random slowly increase regularity almost surely additionally remainder construction herein effectively control exploration tradeoff contribution derivation upper regret policy establish ii term assumption law iterative logarithm action arm sequential sequentially population population receive time sample restrict though ix purpose establish important distributional population alone discuss bandit let indicate event population I interested outcome controller complete would bandit measure regret notational form explore pseudo satisfying give inherent controller reasonably gain lose activation reasonably value sum assumption law number modify sequence force play winner able case bernoulli bandit collection consist density respect scalar set vx leibler mild therein require policy slow population subsequently collection type population poisson bandit due set show regularity therein population satisfy far I sample population dependent asymptotically I simplification requirement policy together alternative way policy derive asymptotically reference optimal strict construct regret therein big paper assumption law type policy mild index almost asymptotically bandit large particularly sure growth motivation know utilize minimal considerably expectation might reasonably interested currently explore minimize path outcome offer sense intuitive policy essentially set explore experiment available high explore slowly slow grow explore enough bandit past implementation policy guarantee asymptotic behavior bandit guarantee bandit broad additionally index policy individually capture many policy behavior perhaps emphasize asymptotic basis essentially asymptotic policy good definition good satisfy mild condition bound order growth remainder attempt change dynamic remark roughly equally single bandit rarely growth pseudo way function bandit essentially good trivially e concave differentiable linear example policy unbounde positive concave let define way blue arc width bandit sample mean broken exploration bandit view variant sample good sequence convenient define sub activate state policy policy good statement theorem arc follow every almost surely exist considerably non fluctuation true go almost bandit activate result e linear occur brief whenever integer point raise activation equality fact discrepancy occur increasingly rarely hypothesis specify scheme break explicitly sufficiently leave tie statement make appendix ucb define blue width briefly mean policy large mean increase exploration traditional upper refer policy ucb necessary exploration simple simplify give proposition proof q optimal bandit application give regret hold limit leave produce prop verification minimal bandit activate hold bandit activate must activate activation hard indicate phase bandit roughly policy single bandit optimal bandit rarely offer potential circumstance optimal activate coefficient activation govern comparison fluctuation term assumption activate linearly assumption grant law iterate seem index optimal bandit greater dominant dominate reduce optimal roughly often bandit index policy essentially play winner fix bandit period phase dominate property problem side phase variance iterate bandit bandit activate infinitely iterate great alone theorem arc width short index eq restrictive since traditionally logarithmic trivially proof sub surely leave extend pseudo surely policy one thing improve pick sense certainly optimal always fix however bandit constant order present control theorem bounding fluctuation around indeed bound however sub optimal index sense bandit equally regardless quality sub rarely optimum boost bandit contribute bandit fairly matter discovery unknown though use measurement play proof distributional property surely law iterate utilize bound remainder independent never necessary bandit regard arbitrary multidimensional process remove way proof somewhat pseudo reasonable finite horizon regret complete policy bandit horizon bandit suffice result bandit implement hereafter two alternate phase mode play winner activation activate bandit least activation winner activation great activate phase govern winner mode activate bandit bandit least activation period winner bandit minimum activation activate follow hence repeat base sub reach bandit activate policy period winner loose sub bandit period winner bandit point relation time bandit activation activation additional activation activation activate
hull projection algorithm constant rank outer matrix unfortunately parameter trial find latter operation operation iteration case I force bad base cumulative current key perturbation regret distribute lead perturbation parameter achieve find perturbation instead random gaussian consist recent consider connection calculation computational dominate component perturb bad case optimum respectively algorithm independent bernoulli flip add closely grow distribute linearly analysis eigenvalue independently pose pca version operate deterministic top eigenvector online wolfe base perturbation except inferior method dense benefit pca finding goal good reconstruction guarantee study suboptimal due fact adapt perturbation clear use perturbation independently zero vector trial tune perturbation matrix skip instance matrix trial predict perturb coin half trial bernoulli quickly extension perturbation unfortunately force suboptimal regret counter dense counter regret force conjecture achieve regret w gain good gain share instance vector expert gain follow algorithm dropout provable share essentially bound perturbation rotation pca nk reveal receive call positive eigenvalue gain oppose protocol choose respect randomization algorithm matrix specify generate gaussian entry grows tune single choose accord rule sum sum picture trial variable perturb relate give sparse sum separately start eq eigenvalue generate orthogonal sum bound bind p nn proportional similarly symmetric furthermore function change integration convergence replace integration w last instance dense opposite old instance summation rule q sum second dense instance instance compare minimax regret instance case instance present suboptimal factor informally bound general soon since large noise eigenvalue par eigenvalue trial regret determined condition claim length suffer suffer little suboptimal bound minimax sense action follow expert small perturb cumulative analogy algorithm unfortunately perturbation would basis act differently perturbation would need eigenvalue try pca dropout perturbation coin flip record predict record natural noise conjecture achieve asymptotically conjecture standard set expert dropout loss gain often provable case word seem online perturb perturbation avoid top eigenvector worst regret close online generalization version support st summing trial since jensen eq plugging put pl technology university california machine vector parameter inefficient address learn obtain parameter trial per trial ideally decomposition per time line predictor minus key analysis achieve optimum algorithm per small paradigm
discriminative tune discriminative regard generative datum goal study coupling scale model empirical tradeoff neural mnist weakly mnist approach svm likewise rbm weakly perform classification label mnist deep currently good performing variant architecture protocol usually involve unsupervised supervised phase combine algorithm classifier highlight tradeoff system include usually additional validation set come set consequence net amount increase argue favor label account comparison net naturally approach deep learn architecture total label model label operate perform reduce number system likely number require fully hand outperform compete back finally unlabele direct integrate application involve inference observe regard evidence potential unsupervise direct favor integrate unsupervised supervise favor far appendix execution black graphic structure matrix multiplication ideally parallelization graphic cpu concept mini learn update gpu mini gpu usage cause negligible mini batch parallelization parameter choose optimize else gpu rate size test likelihood converge ccc mini std log std hide unit turn complete validation reason number available prominent account find start validation indistinguishable normalization softmax winner closely able broad learning set express proportional hidden unit iteration label value keep learning might stay learn hide yield step optima overall trade encounter gradually paper keep choose unit dataset validation decrease recurrent feed forward possible overfitte point hide normalization experiment rate early optima general long validation training choose experiment deep n em directly maximize free denote entropy depend parameter em optimize iterate posterior posterior expectation maximize locally maximize detail behavior sum respectively dynamic apply non result assume weight learn equation learn add repeatedly learn step form expression give product denominator regard growth give product approximation inspection approximation hierarchical step small approximated cd use supplement detail sum approximate sum supplement last furthermore apply find analogously convergence recover converge individually experiment verify come balance preprocesse applicable small explicitly free refer motivate unlabele parameter tuning construction require sparsity anchor near dimensionality reduction write mnist label test probably dimensional manifold classify support vector result number manifold investigate manifold dimensionality nearest take part testing state test label test low dimensional manifold combine result use measure tuned model combine model deep decay hyper use deep neural architecture paper validation publish contain deduce protocol combine represent system component boltzmann typical consist layer decode hide unit bottom discriminative comparison mnist validation sup test unlabeled point include summary protocol tune diverse report also protocol difficult exhaustive believe eqs sec eqs eqs eqs eqs eqs eqs eqs eqs eqs com de mm institute advanced university von study supervise often complex heterogeneous parameter work integration highlight define learn mixture classification network scale learn quantitative benchmark tighter competitive amongst label demonstrate applicability competitive operate favor strong focus simplify capability deep rapidly progress modeling belief convolutional network machine autoencoder hybrid unsupervised make benchmark difficulty mnist prominent example complicated architecture unit data initialization parameter define number sample initialization systems vi anneal parameter parameter drop early integrate require theoretically network architecture hyper g importantly hierarchical unlabeled seek optimize principle combine derive network thereby probabilistic minimal number allow classification model hand class within write generative model top abstract comprise layer occur digit final observe poisson distribution I observe note normalize weight dimensionality large order maximum seek come summation maximizing expectation em iteratively low log setting update update aforementioned transform happen scale derive neural circuit proper derive show plausible formulate make neural mode neural analog neural neural obtain unnormalized datum activity normalize weight assume assume satisfy th activity become limit rate derivation taylor expansion technical normalization activation long identify convergence sep sep black color black inner fill gray cm cm cm cm cm pos cm width dash edge yshift pt amplitude mirror xshift yshift black yshift pt amplitude mirror xshift yshift pt black yshift mirror xshift pt yshift cm cm cm pt yshift width center corner minimum em draw thick font mid label mid distance blind label em label unlabele em unlabeled block validation label em text width generalization height general right em keep half tuning training protocol label per set risk optima layer compute denote uniform range overcomplete far label separation problem use information second dataset belong topic comprise raw occur tf stop investigate total per fully set mean error value total initialization procedure describe good knowledge report task break binary merge topic work experimental setup discriminative rbms perform weakly optimize supervised validation compare l cr experiment r achieve error ff test decrease ff label early optima reach fully tune base set additional training free could potentially setting performance handwritten digit mnist mnist test gray handwritten mass minor font legend font xlabel style ylabel font scale false xlabel xlabel near pos ylabel ylabel near pos axis thick mark none black txt axis axis cs access height axis ylabel ylabel near pos line right thick mark black txt overfitte layer feed forward influence layer therefore recurrent unit substantially likelihood criterion early monitoring soon overfitte first influence datum start stop overfitte compare
spectrum exposition practice rarely strictly stationary usually stationary slowly change spectra range suitable determine integer extra average kind value fairly reasonably terminology localize sequence range integer please hand follow average facilitate enable driven approach ideal number index fact multiple relevant say wide frequency variation know window scale multiscale result low frequency channel recursively low frequency every simplifie range nonnegative via processing convolution later stage deconvolution subsequent convolution stage stable later window wider precede recursively two one nonzero transform recursive consider sequence doubly provide smoothed alternatively multiscale yield fourier wavelet sift et constitute choice whole recursive high channel serve desirable member gradient calculus calculate spectra classification regression integer integer locally infinite integer implementation computer obviously finite sequence furthermore locally stochastic filter regularity easily filter offset integer construction enable stochastic process regularity distributional versus poisson parametric convnet relevant nonlinearity stationarity local translation convolution correspond subsampling correspond convolutional convolution follow entry subsampling entry correspond absolute entry convolutional essentially convolutional filter precede subsampling acknowledgement remark convnet implement follow operation convolution complex absolute result average real multiscale spectra multiscale spectra drive configuration calculate multiscale convnet filter complex exact correspondence wavelet whereas analogy drive multiscale certain modeling series natural include patterns multiscale spectra nonlinear connect concept information original value theory leave far exposition nothing basic treat limit consideration doubly nonnegative range integer range latter sequence range terminology refer fact
face phase enough dm optimal arbitrary accurately dms dm good th adjustment approximate wish find achieve dm q algorithm provable stochastic discount weakly acyclic well weak strict dm deterministic e dm call differ exactly dm dm strict discount weakly acyclic strict strict path start joint equilibrium policy good path acyclic games acyclic straightforward adjustment introduce dm strict well would clearly converge acyclic next enable dms adjust strict path adjustment dm algorithm initialize iterate mm cl u ik ix ix ix ix ix u ix ix k ix ix cl game acyclic strict game hand define acyclic dm policy assumption q exist finite eq dm learn policy baseline experimental dm switch strict well dm switch dm algorithm cycle switch strict flexibility come baseline policy flexibility lead convergent behavior drawback phase inconsistent spirit synchronization hold th dm denote acyclic value learn exploration consistently equilibrium predict even set path policy converge equilibrium tx assign initial condition respectively non convergent infinitely w contradiction show initial w hence know dm dm random assign eq exploration phase dms finitely joint policy follow appendix combination policy either use uniform ia j finite number uniform dms deterministic v ix optimality introduce upper bind due acyclic strict minimum length exist q recursive part pick satisfy existence integer eq assume due borel borel lemma contraction lipschitz th dm random assign exploration finite dms finitely hence desire convex form dm use uniform ia joint policy exist uniform deterministic policy ix tolerance rate upper event since acyclic well exist satisfy rest recursive reasoning proof iii proof sample process show learn obtain q lead recursive part completely analogous proof part assumption mathematic university algorithm dynamic game stationary maker minimal also cost decentralize dynamic game weakly acyclic algorithm surely stochastic game desirable game specifically acyclic game oppose static game future applicable broad application literature stochastic dynamic game learn gain popularity interest generalize reinforcement dynamic lead set primarily reference tend real agent find play equilibrium objective serious agent maintain action compute use value action objective player learn enable learn optimally environment application information agent burden agent make analytical regard property game also attempt extend algorithm present computationally prohibitive quickly agent state action claim convergent equilibrium single game fp stationary past fp convergent even simple state agent move zero game strictly adversarial resemble centralized learning relate include actor equilibria stochastic game higher store give state specific rigorous viewpoint agent short localize reasonable system agent organize devoted paradigm standard first introduce present generalization follow paper remark proof technical system illustrate generate controller generally state coupling lead dynamic discount dynamic game follow dm dm states decision dm dm determine decision game induce denote initial dm make joint decision together well distribution select dm appropriate dm policy dm decision solely policy dm interpretation dm policy randomly set policy dm primarily policy dm find dms possibly cost may dms adopt represent ease policy dm joint perfect discount stochastic possess equilibrium define since may many game possess particular game arise application dms benefit possess primarily denote deterministic class game dms identical team although team useful model system game arise system dms necessarily team deterministic joint policy dm least joint low dms generalizing notion game weakly acyclic set well dm characterize factor satisfying denote call respect policy position say dm strict discount weakly acyclic strict dms constant learn factor visit often proper rate environment exploration randomly small however decision expect persistent team weakly acyclic games repeat singleton
apply obtain place paired basis pair next derive multivariate bound condition section exploit result derive definition independence bss derive bss satisfying invariance discrimination minima derive section derive ip define reference ip derive expression report empirical verification derive independence section article end conclusion f say hull say finite support volume surface constraint condition differentiable gd mathematical integrate side equation integrate respect bring let without let accordingly none derivative cx derivative get get prof combine case mathematical differentiable gd g induction fx integrating integrate bring integrate bring sake hold generalization f gd none zero loss f function equation integrate equation dx dx gx x x x equation base inductive step induction differentiable gd g hold prove loss integrate case side bring integration real number integrate respect coefficient ij ij tc zero number none bring coefficient bring prove equation odd base sake let obvious without f gd none loss generality pf integrate get dx integrate equation integrate equation dx g dx dx gx f gx g n x x integrating prove inductive step induction gd n bound converse differentiable gd principle implication value prove generalize derivative available differ constant derivative would decide prove bring equality strength prior conversely equal derivative line derivative everywhere lebesgue unity bound seem restrict say restrict natural think extend independence look similarity score develop matching dimensional product marginal pdfs independence variable independence find define difference pdfs independence joint gradient product difference vector analogy gd satisfy definition add pdfs order support result obtain hessian product hessian pdfs hessian hessian hessian component prove pdf pdfs bound independence bring interpretation independence vector goal quantity nonnegative quantify derive independence quantify integrable pdfs norm add invariance detail appendix pdfs invariant respect translation variance affect reason pdfs shape desire invariance invariance apply pdfs pdfs vector integrable joint marginal pdfs deviation independence vector differentiable marginal pdfs correspond standard deviation differentiable condition apply prove necessarily support pdfs order differentiable pdf prove independence blind bss generation instantaneous variable mt component problem bss mix matrix unique bss accordingly bss mixing obtain separation source contrast bss criteria correspond formal bss let transformation contain identity variable mapping trivial leave unchanged follow property source maxima discrimination spurious achieve maxima need scale whole permutation scale operation state widely divergence invariance accommodate without bss define invariance quantify property q transformation satisfied justify invariance predefine act per argument measure satisfying scale equivalence solution bss precisely bss solution mathematically whole contrast easily convert demonstrate whether contrast bss invariance scale filter diagonal prove normalization scale invariant norm normalize density invariance property definition verify bss contrast linear bss verify invariance trivial simplify start prove neither invariant relative invariant though apply density invariance permutation discrimination property normalization bss p bss source verify invariance contrast diagonal scaling filter simplify start scale normalization bss independence already mean invariance way prove discrimination advantage minima minima still respect also contain optima correspond optima analysis bound differential pdfs optimal thought imply spurious maxima eq indicate c independence zero indicate maxima spurious optima prove prove information prove gradient differential mutual mutual minima desire investigate obtained try small pdfs convert simply integrate prove serve integrated difference similar proved reason effort prove actually result proof spurious optima maxima maxima focus measure avoid estimation sample marginal pdf estimation pdf entropies bss conventional pdf marginal gradient pdfs derivative achieve base histogram estimation fast pdfs pdf estimation pdfs indirect estimation suppose though theoretically say pdfs derivative require derive light square select place sample require basis select place basis pair un pair location computation estimation note order gauss similar direct estimation difference potential reference extend concept estimator realization unknown pdf bandwidth usually definite single square q thus way avoid continuous theory information ip independence information appendix interpretation describe done bring field field zero potential reference analyze absolute potential reference local reference field potential neutral system express quantity reference potential example electrical circuit reference assume ip concept far derive field ip concept first question define rip quantity initially set place select basis rip analogous approach bring quantity place act potential linear select potential information rip rip integral pdf gaussian definition rip bring cross potential pdf reference newly concept location basis location sample space vice analysis scalar intermediate information interaction interaction due potential force interaction size similarly reference potential gram potential interaction potential already scalar x nb target close expression concept stage approach least direct estimation square mutual worth contrast euclidean quadratic square generalized estimate potential actual require correct assume select number basis parameter depend rip identity obtain r rip use rx ij dy bx rx rx ij symbol operation rip multiplication term replace hadamard addition entry exponent obtain multiplication number require available complexity measure term multiplication multiplication time reduce correspond directly number specifically place estimation multiplicative cost gx kx accordingly matrix nx l n nz way dimension explain previously number multiplication estimate equation q require equation complexity multiplicative pair expect accurate dimension mention article elsewhere calculation way intuitively convolution article correct think delta pdf justify popular square rx kernel multivariate multivariate major equation estimation basis replace rx let square least square contrast combination correct obtain kernel basis place term purpose rip rip estimation bandwidth multiplicative pair pair derive basis computation kx xy product estimation discuss write require solve derive verification ability dependent testing bss kde cross demand choice balance select uniformly distribute sample let generate signal dependent entry trial ht definition opposite uncorrelated component mean imply orthogonality rotation transform component bss give contrast dependence theoretic bss local design existence spurious landscape derive contrast bss keep test selection bandwidth selection selection bandwidth bandwidth bandwidth validation parameter distribution first type r suggest type type skewness skew skewness skew add experiment report justified bandwidth bss previous application set new every consideration demanding assume expect property vary ideally bandwidth parameter gaussian assume bring bring need rule consideration distribution solution derive assumption near gaussian estimation bss show ideally multiply comparative minima spurious minima unimodal landscape minima everywhere hessian everywhere accordingly measure ica contrast discrimination spurious multiplicative basis place pair computation verification bss quality vary vary require restrict future function negativity symmetry triangle inequality four condition contain concept distance geometry generalization space define norm
scale data factorization utilize randomize user already indicate category represent multinomial category corresponding category popularity transaction recent restrict transaction day recommendation pick product interest category brevity lift term acc different observe lift bias matter lift partly mf poor bad factorization binary factor construction incorporate c ccc mf email add recommendation early adopt base recommendation percentage customer split one item motivation temporal score attempt temporal g highly likely recommender keep recommendation hand time item recent feedback capture trend algorithm rank base acc map ndcg guarantee step reasonable offline boost base recommendation capture learning independently encourage practitioner add standard module recommender couple remain open though computational bias nature difficult power distribute need improve recommendation transaction recommendation encourage tail understand balance relevance popularity top recommendation shift dependent recommendation engine engine capture trend shift observe resolve criterion user online always boost recommendation encourage practitioner recommender temporal dynamic technology database recommender extensively domain movie music news recommendation etc recommendation content neighborhood base model recommender body work evaluate item user collect paper take real world website recommend observe product week week day user demand external demonstrate rank sale product week line figure indicate transaction moment among product website huge figure rd rank product temporal challenge recommendation ignore bad experience instance recommend na I deal discrepancy restrict model transaction occur day transaction severe majority customer across transaction wise choice fewer couple shorter small may appear window plot model window baseline train transaction imply collect training possible exploit transaction sparsity capture trend recommendation capture discard item bias capture recommendation recommendation model optimize bias carefully examine recommendation dynamic interaction proceed use generality throughout total user index user position recommender item typically email item recommend email template practice yield score denote user yield item default rank top scope find th item evaluation item relevance item rating collaborative filtering application item relevance particular user irrelevant presentation rank acc ndcg since top individual user many care item enter item rank essentially ap average position denominator relevant item rank map discount ranking ndcg relevant rank individual user eq item presentation convenience mention suffer transaction dynamic attribute kind huge discount external likely capture external factor temporal challenge collect day determine bias evaluation maximize state information user item score user find bias accord performance maximize bias recent transaction capture change analytically shall iteratively guarantee terminate step optimal total item recommend user index position score prediction relevance top bias aim rewrite difficult resolve calculate adopt score bias optimize next describe find item contribution one solely item simple assume user recommendation accordingly discard item four consider relevance eq item share irrelevant accord accuracy associate particular discard item item discard top recommendation item recommendation bias add enter change determine maximal utility convenience utility recommendation serve utility kb k f inf via subroutine present respect bias utility compute bias prefer utility possible enter recommendation infinity constant include recommendation lead negative exclude recommendation subroutine bias bias lead utility sort time single previous subsection update detailed b cb top necessary loop cycle item outer loop stop utility line item straightforward save record recommendation top recommendation user immediately decide whether update item recommendation bias need care similar recommendation user speed computation recommendation cycle utility terminate couple iteration state consideration recommendation utility change top recommendation always increase utility item top recommendation essentially remove consideration recommendation suggest remove positive transaction remove propose parallel size item couple heuristic might consider adopt save medium score memory keep top recommendation recommendation multipli instance recommendation remain item observe matter recommendation complexity type item bias frequently raw score tend past popular item appear frequently one negative item candidate without dramatically reach local optimal stop cycle yet criteria percentage threshold percentage bias cycle reduce time yet find optimize bias optimize recommendation ndcg item top play item rank rank utility change bias th nevertheless position example average recommendation gradually score rank position compute utility item currently item unchanged change item relevance utility item term utility move utility change original e top replace ndcg hence rather item find optimal bias item suppose potential change reach single item fine reasonably criterion candidate set u compute difference b subroutine setup include benchmark recommendation transaction construct benchmark split date user activity date week evaluate corresponding measure acc lift improvement lift behavior different quantity efficacy setting recommendation model chain current action another estimate consideration item instance however consider pattern action pick high transaction validate experiment recommendation well learn day transaction fine bias without consider change utilize long history long window user baseline transaction method train month item compute item bias metric item month specific bias probabilistic recommendation engine popular another incorporate temporal decay date decay work recommendation factorization music rating root stochastic application response acceptable conduct series experiment construct sensitivity base metric various table lift bold winner lift observable thank number look mind year thousand improve netflix competition improvement suggest recent trend propose weighted decay help bias individual preference improve recommendation especially huge acc map ndcg map ndcg cc popular bias macro tend shift item towards ground item reverse order test prediction respectively check overlap trend add bias recently
statistically mis specification indicator superiority initialize since initial allow reach likelihood vertical concatenation lag empirical block next asymptotic unconditional purely q corpus output filter whitening whitening fit perform scale extremely matrix vocabulary describe novel leverage scalable indicator corpus structure shifted use generative fit structure assess usefulness variety high lag approximate covariance corpus denote count corpus minus matrix operate second matrix strictly speak size co sublinear ignore rare unfortunately every word vector one orthogonal degenerate direction structure instead perform projection need fortunately lie embedding train level embedding train maximize might capture syntactic semantic release code algebra library consist modification public svd randomize approximate repeat multiplication low handle individually reasonable employ spherical poorly property diagonal modeling anti correlation would capture pass multinomial function prevent kalman filter conjugacy practitioner upper multinomial hard estimator poorly exploit minus low infeasible multiplication formula efficiently evaluate training formula usage use technique particular indicator multivariate maintain minus rank minus low em linearly transform post hand affect whitening minimize coordinate whitening obtain high whiten similar canonical correlation context successfully word embedding identify language appendix provide whiten ensure return psd manual correction unnecessary psd smoothing equation vector smoothing identical inference first alternative appendix inversion avoid token distance variable coordinate sphere highlight similarity commonly kalman rnn explore sec softmax sigmoid initialize steady kalman online kalman parameter nonlinear rnn sigmoid regime behave identity capture coordinate anti evolution rnn pass rnn separate word mapping dynamic evolution explore dynamic state study namely word likely reflect interpretable strict transition invariant salient community share express recommend get unsupervise pos response pos token predict pos embed include token embed set token kalman smoothing use universal pos tag original embedding york token maintain text type terminate broad maximize local pos network outperform word tune ignore instead classified tag hour train since heavily fast sublinear corpus tag v v table compare em initialize feature token embedding embedding local initialization find maximize force token embedding statistically significance level use vs initialization column universal tag contribute statistically significant tag expect context independent capture consider token baseline expect outperform pattern match long v highlight rnn application permit language lm context unit could restrict leave softmax architecture work rnn whether improve final optimization obtain hide tuned rnn comparison train substantially corpus popular rnn schedule hold decrease tuned value crucial jump start plot number minute rnn core far em iteration prevent rnn epochs initialization allow cpu table set find train specifically initial rate ignore require small set train embedding unclear baseline assign token syntactic single tuning rate rnn architecture leverage posterior work complete microsoft helpful comment empirical also various detail section scaling simplify coordinate orthogonal rather consistent procedure semidefinite psd singular span psd critical psd kalman structure datum fix unit vector ignore find factorize next expression kalman solve property use directly still solve form mc effectively ignore invert employ term inverse plus formula therefore intermediate filter equation worse possible indicator subscript denote along ignore zero therefore posterior term invoke come formula consider posterior smoothing depend model eq inverse inverse low useful algebra large example place trivial recursively invertible identity quantity storage compute inverse inversion compute contrast progress randomly nearly identical fact post hoc looking course em iteration quickly begin take find pos nlp ignore natural success word word posterior dynamical efficient kalman filtering learn extremely scalable operating employ task see recurrent neural recurrent neural reduce training nlp datum text nlp unsupervise independent context oppose context per token bank financial would dimensional obtain token embedding variable specifically moment perform maximum amount operate aggregate co discrete encoding token mis generative draw vector multivariate system desirable scalability filter simple learning moment svd co simple form multinomial second problem input employ token embedding pos name recognition pos apply local embedding token pos gain baseline embedding salient name kalman filter update rnn rnn however pass token left parallelism aggregate matrix distribute fully characterize consider given provide every invert substantially filter update key update actual
connectivity population set cover spread brain structure recover display obtain weight cutoff dark dot start involve different level structure notice generate even randomly cause optimization early tend dag constraint rich medical list produce edge connectivity happen temporal play role connectivity ad related increase corner figure temporal study mention connectivity increase ad cognitive patient significant change important one income connectivity communication increase communication dominate please influence factor limit datum disease imaging study worth exploration paper whole nc whole edge initial learn class axis quite weight leave fig adjustment edge increase mm kl mm paper variable discriminative achieve optimize classification demonstrate utilize bridge classifier margin problem analyze advantage discuss propose guarantee validation prediction improvement discriminative associate direct node exist constraint equivalent topological page eqn topological dag meet prove condition node similarly contradict impossible directly link path node compose another path node compose eqn eqn eqn far explain eqn hold topological come link dag topological order node come combine eqn dag iteratively alternate dag dag parameter column alternate dag x pa eqn I mean monotonically f second prove therefore optimization problem alternate contradiction ji ji ji ji order constraint regardless ji pa pa contradict sufficiently dag proof alternate eqn dag sufficiently edu liu university north hill usa semantic bayesian bn discover exploratory brain variable bn necessarily subtle critical investigation population propose power bn bring framework gaussian employ bridge discriminative svms convert fisher framework upon explicitly advantage framework paper ensure learn max probabilistic wide diagnosis bioinformatic effective infer dependency bn acyclic dag bn absence bn focus bn bn score use independence testing consistent include recent score approach define scoring candidate optimize strategy length vary heuristic dag consequently restrict exist tc tc stage identification candidate parent stage prune certain computational drawback never stage stage prefer gaussian lasso stage approximately demonstrate improve bn traditional bn task diagnostic purpose straightforward usage train bn high kind train bn optimize discrimination reflect difference bn stage parent bn face exploratory unclear explore discrimination interpretation interpretable finding mathematical term requirement come demand understand domain sufficient identify purpose accuracy also generative bn amenable generative necessarily discriminative share critical group unfortunately happen clinical consequently inferior discriminative exploratory aim gain discriminative bayesian discriminative variable brain moreover bn bn individual discriminative base gaussian kl employ svms learn learn maximize svms contribution framework include induce fisher discriminative svms bridge learn optimize iii advantage kernel induced simplify discrimination svms second learning term optimize upon motivation optimize svms framework jointly class discriminative learning framework contribution dag ensure validity simplify optimization consequently parameter block coordinate dag parent discriminative propose application early diagnosis disease newly image brain network attempt study interaction region image brain semantic bn connectivity connection pathways region affects find many disease novel disease diagnosis framework test demonstrate capacity conference publish extension aspect used constraint theoretically verify benchmark set structure learn method framework constraint fourth difference discriminative framework comprehensive framework vary organize review brain framework propose topological ordering notation occur summarize ht random represent represent pa realization bn bn background brain please generalize beyond example define bn direct closed path property pa pa gaussian parent I bn graph parent bn traditional consist stage parent initially criteria drawback true recover stage sparse implicitly regression optimization sufficient dag require ji ij ji eqn difficult solve method whole iteration update search node simultaneously obtain conventional network recovery modality sensitive traditional cognitive assessment diagnosis subtle brain suggest necessity brain complex mathematically align affine transformation interest voxel brain region brain connectivity brain technique dynamic causal bn causality bn lag network fmri bn note region employ handle relatively bn construction identify parent bn bn underlie context represent ad train separately likelihood may ignore subtle critical distinguish argue learn jointly essential discrimination integrate discriminative important divide incorporate train generative discriminative train influence paper propose kind discriminative framework induce discriminative kl performance base discriminative optimize subject capacity framework fisher discriminative kl model one map represent vector feed error adjust improve convert kernel feature learn way sample intuition object similar compute describe change direction fisher weight identity categorization induce feature gmm visual vocabulary despite success application mainly publish discriminative confirm separability sample fisher develop criterion log factorize pa learn kernel require possess property firstly separate secondly learn maintain reasonable capacity strategy separate hyper plane svms fisher leave minimized good svms secondly control model minimize parent enforce dag ensure validity develop margin optimize svms meet discrimination exploratory network different conventional classification bn learn reflect hence drawback regard discriminative framework discriminative framework classifier sample likelihood accord train specific couple well discriminative svm root relationship classification simultaneously explicitly optimizes implicitly optimize implicit svm direct outperform kl advantage bn optimization problematic increase mm fit two svm solve package mm optimization increase medium could hard solve kernel kl edge introduce vector word convert study body option encounter node handle keep whole compute option fisher pearson high correlation simple avoid dag remarkable selection discrimination kl mm selection step candidate tractable would could switch target gmm indeed simplify favorable property gmm propose constraint simplify discriminative recall h utilize implicit eqn optimization search could high optimization solve change induce obtain interpretability dag change experimentally solve dag solution satisfy optimization propose dag topological variable separable dag use direct node predefine bayesian direct dag constraint eqn topological order link remove compute ji ij ij whole avoid order worth note provide number could reduce long eqn satisfy topological dag however continuous noting therefore drawback point dag could single column absolute way alternate process strategy eqn dag warm strategy increase eqn tb datum initialize eqn fix optimize solution let enforce bias improve pa equal bias dag dag yet challenge noted eqn problem conceptually link framework analyze require deep investigation significant work aspect learn process connectivity brain network set follow h constraint ability recovery iii discriminative iv learn connectivity conduct publicly disease two imaging modality include weight mr cognitive patient control removal gray matter gm region normalization gm image node include disease gray mr patients nc belong partition partition image image disease use typical brain spread partition test class learn optimize column dag whole whether whole column implement code website whole attempt objective implement square provide warm applicable whole dag dag compare solution time arrange initial ordering average whole almost identical feature order feature order contrast variation change fig give quantitative averaged present solution affect permutation solve dag constraint regard order quantitative permutation average correlation whole ability whole since truth available due unknown conduct nine set come repository literature nine arc arc arcs arc chain arcs arc arc arc water arc whole eight bn mb gs tc tc repeat parent eight predefine nine bring behave mb show total number mis
term short term load balancing learn present balance instantaneous load context select short consider instantaneous instantaneous exceed transmission bs propose accord consider aware scheduling bs criterion velocity introduce rank candidate velocity many fair avoid enable fair resource cell base scheduling incorporate interface macro instant average mm historical associate past define capability balance reinforcement mm perform load balancing represent player I player optimize perform scheduling short step utility update select bs allocate base velocity mab maximize overall analogy machine maximize collect iterative select trade mab mm approach macro bss db due partially simulation basis scheduling utility mab part time maximize whereby mean action select iteration drop radius bs drop bss number large time due simulation velocity direction movement macro macro simulation baseline perform average filter entry cell baseline proportional scheduling exchange macro refer throughput mm mab improvement respectively hence mm throughput center throughput throughput base approach term update achieve approach hand aim maximize cell throughput gain also reflect throughput mab improvement classical approach compare fig versus bss bss yield per ms ms cases classical propose approach classical htb besides gain propose depict fig modify simulation setting classical higher obtain depict pp decrease high velocity half pp mm show behind load balance cell tries extend extend coverage history scheduling approach system maximization second aim cell compare method gain throughput base approach heterogeneous network essential provide user throughput cell however due macro base bss introduce additional requirement challenge dense base aware management mm cell use macro jointly long traffic schedule term throughput throughput probability approach expansion management reinforcement scheduling evolution heterogeneous approach meet ever capacity entail management across management essential ensure connectivity mobile user maintain service project macro dedicate key interference plus ratio failure probability unnecessary pp unbalanced load cell entail resource efficiency experience dynamically optimize traffic essential bias margin management investigate literature main cell speed simulation interference resource velocity co interference approach consider broad moreover adaptation reinforcement robustness selection formulate proportional problem load cell management jointly balance scheduling scheduling reinforcement base individually optimize schedule limited macro optimize traffic long term process history velocity scheduling bandit mab base improve overall reduce pp difference basic classical approach exchange among traffic cell individually optimize among mab aim satisfy capacity macro optimize result load balance short carry scheduling velocity average rate effort propose long solution traffic balance scenario load balance mab base short aware consider throughput enable fair enhance association paper organize system conclude transmission model bss drop
stand inequality element wise stand wise spectral hyperspectral ni slide adjacent column q abundance column abundance zero noise physical abundance coefficient namely relax say abundance efficacy lie characteristic I low justify abundance consideration correlate compose material although suggest abundance recently powerful g mainly nuclear norm impose another present spatial area mark safe abundance use robust already successfully contrary hypothesis low impose report sparsity low property knowledge sparse optimization q rank term fidelity term parameterize become flexible enough impose result accordingly estimate advantage low rank later surrogate attempt robustness consistency enhance weighting utilize knowledge simultaneously solve differentiable form regularizer optimization tackle alternate direction multiplier admm smooth incremental proximal exploit splitting function indicator function w differentiable require minimize incremental proximal consider recall proximal function soft thresholding perform manner ij ij diagonal matrix belong mind singular q thresholding correspond next projection nonnegative number utilize operator nuclear singular value computation projection proximity operator differentiable fidelity follow gradient approximately operator cyclic convergence select abundance norm constant analogue summarize table concern complexity complex svd take dictionary ill condition usual hyperspectral high signature convergence framework slow convergence robustness admm type develop alternate direction solve abundance consider lagrangian matrix lagrange case lagrange multiplier augment sequentially remain variable value lagrange multiplier alternate minimization cycle elaborate far admm remain auxiliary involve norm indicator regard minimization experiment highlight follow simultaneous incorporation abundance initially counterpart algorithm ignore modify low accounting solely say drop prior next simultaneously abundance entry illustrate fig eq db fig specifically exploitation sparsity counterpart see visual inspection recover fig rmse c early sparsity respectively abundance herein optimal rmse abundance abundance matrix snr db show rmse minimized region grid form subject rank abundance proper mean concern abundance aim end way depend range db realization rmse calculate white rmse obtain compete compare admm snr additionally slightly db method outperform characterize abundance colored actually hyperspectral structured assess method realistic linearly illustrate algorithm therein compete whole snr robustness c rmse rmse experiment highlights estimate abundance mixing mention simulated hyperspectral image consist block abundance solely row produce sparse abundance pixel abundance level rank description mixed pixel corrupt snr pointing row ccccc cm ccccc cm illustrate algorithm apply hyperspectral scene capture usa pixel spectral band nominal water low band examine algorithm term minimum dataset assume case pixel characteristic prove compare pixel reflect lie lie build dictionary signature display abundance map worth point evaluation careful inspection map algorithm produce similar pattern nevertheless algorithm reasonable meaningful abundance material hyperspectral hyperspectral sparsity spatial comprise proximity component norm abundance treat sparse rank two solve incremental admm call simulation synthetic art derivation computationally efficient scheme svd investigation relevant research abundance impose could topic interest work deal compressive processing interest know structured form suitably impose sparsity enhance paper spectral hyperspectral specifically two novel lie homogeneous region hyperspectral nuclear abundance determine slide window conventional impose sparsity abundance cost incremental proximal alternate method admm illustrate conduct hyperspectral alternate multiplier abundance attract recent year process identify spectral signature material whose generate mixed deriving correspond formation constitute call abundance vector two rise put literature address extraction research spectral signature available abundance fall make fundamental mix signature pixel latter adopt therein pixel signature predefine dictionary abundance henceforth treat due consideration impose various abundance regression attempt well abundance result idea incorporation knowledge justify dictionary pixel signature accept dictionary one abundance zero practically norm bayesian scheme abundance constraint incorporate region region among signature pixel hence correlation abundance lead development whereby abundance estimation spirit term dictionary large assume translate abundance sharing present abundance pixel abundance structure impose abundance regularize direction multiplier fashion propose image abundance central take signature
technique know originally stability bottleneck overcome develop condition recursion control iterate inclusion measure explain motivation stem property dependence vary reinforcement iterate stable pose reinforcement paper word markov finally application temporal weak literature guarantee convergence use involve show iterate outline specifically set reinforcement notation al et inclusion di guarantee absolutely reader km lx dx say chain compact great sequence dx dx x call dy control eq jointly compact lipschitz without integrable martingale contribute related assumption call value markov process state let iterate detail limit x limit set sufficient constitute lyapunov explain fixed choose dy sc h assume n dy paragraph hx h us c n explain early definition convex compact hx hx functional say hx sake convenience ax proceed ax convergent nk nk nk nk nk nk nk nk nk nk compact convergent subsequence notational contradict f nk nk fu nk tm tm n tt k control martingale noise surely follow rescale recursion q unfold expectation get mn claim surely initial almost follow stability iterate lemma limit c rl rl tm tm generality far know h kl tm tm tm tm n tm prove kl xt xt compact hold tm tm lemma sense claim consider manner p n hold x l mn nn ml hence implication see tendency fall unit jump inside unit outside jump outside trajectory fall since force jump jump jump use get contradiction present impose recursion couple stable measure list control process take compact space borel respect measure ergodic prescribe show proof reader dy tn tn bs almost remain hx iterate stable converge invoke converge idea several vector recall recall value control detailed exposition van impose condition convergence say van regular usual analysis range l component trivially n nx nx c auxiliary
number approximately serve compare early rounding mode layer format observe point near begin epoch appear upon procedure stability bit fractional precision baseline proceed sgd stop reduce precision tend despite round small gradient suffer result achievable error reversible increase bit bit precision format rapid improvement reveal promise network train stochastic round epoch high precision computation employ explore arithmetic fact processor throughput double precision concept conjunction stochastic extend share orthogonal use conventional point arithmetic neural network hardware mini batch descent dominate feed error propagation calculation computational throughput translate training processor high memory hardware hardware enable hardware cost secondly modern early potentially gain choice prototype multiply mb block ram gb typical dimension storing store memory perform throughput measure second multiplier parallel sa multiplier great subsection block ram cache fraction matrix store read request income l cache back compute external sa column array controller ip interface operation x integer capacity constraint continue multiply employ allow show logical array implement every cycle ram either column cycle element early accumulate accumulation partial datum round implement store external memory throughput operate example neighboring delay improve operate frequency cycle multiplication accumulate product accumulate multiplication throughput array feed incoming cycle multiplication output array length typically bit input typically less element round truncation bit operation unit feedback shift bit add bit capability determine excess bit detect max complement rounding result vs number follow examine result remove enable compact external memory logic unit overhead hardware array implement synthesis frequency consumption throughput efficiency w compare table resource throughput benefit series operate frequency flip network influence low unit specifically substitution arithmetic circuit come gain energy throughput precision fix computation conventional round fail adopt bit computation implement throughput efficient multiplication round little overhead software scale inexact deterministic across stack right hardware large deep precision context network train use round degradation energy hardware implement point arithmetic round deep hardware ability perform capability rapid architecture thorough search space hyperparameter come recent large computing effort include distribute compute thousand core graphic processor time natural apart computation context unnecessary moreover improve neural exception asynchronous reduce purpose hardware design need traditional often unnecessary overhead computational resource possible tolerance relax hardware software energy possibly computation hardware error ingredient develop need introduce preserve application towards cross co low precision arithmetic rounding fix point motivation arithmetic computation firstly point resource circuit area secondly reduce enable fit give memory capacity dramatically improve parallelism key finding exploration neural train fix arithmetic validity approach neural network mnist cifar deep network train round achieve throughput use arithmetic unit round module determine critical design analog digital surprisingly literature aim performance majority study focus implement train high study processor employ hardware throughput implement propagation present bit arithmetic understand precision network back propagation learn probabilistic rounding update precision requirement technique neural variant perceptron contain unit result neural million trivial precision deep dataset procedure recent hardware neural employ find achieve number round computation knowledge round neural low arithmetic implementation network correspond integer fractional number plus bit paper notation representation correspond length integer fractional small format limit precision set element result number mode round number fractional length weight layer format present investigation dnn convolutional network dataset comparison reduction error set bit arithmetic constrain intermediate propagation layer output update bias format start parameter keep unchanged one word length bit allocate fractional fairly restrictive choice implication perspective parameter update store loss information update significantly fix point format consequence progress worse near scheme oppose stochastic rounding maintain zero probability secondly offer limited relu hardware bit length carry simplify connect network relu activation train recognize handwritten digit comprise pixel contain digit lie augmentation perform weight minibatch minibatch size cross baseline achieve bit show two rounding mode bit leave bit integer portion seem network degradation either bit begin impact primarily reduce fractional precision zero rounding preserve learn precision round unable prevent ht rounding
noisy ica consider unit sphere turn slightly notion iteration complex hessian expand uk real derivative derivative kk iteration perform candidate inner ta orthogonality inner invertible largely issue noise free aa form aa inner define zero tc contain proper particular aa pseudo ia eq basis equation issue column tb choose vector couple remark modulus constant guarantee modulus convergence cubic limitation omit proof near make progress predefine k return result orthogonal modulus minimizer g e derivative let g minima maxima let recovery demand simultaneous row recover complement recover column achieve use aa kk kk kk kk c tb b access allow use ica mix orthogonality column td kk latent know find properly cite gaussian paper transpose space lot ica mixed sign joint sp paper trick paper pca rely sir highlight paper direction arrival problem thm thm thm ica blind ica assume signal linearly ica inverse datum recover latent however optimal measure interference plus even reconstruction propose model source signal improve ica provable practical original quasi step lead typical ica dimensional kk latent entry may throughout achieve center ica eeg removal signal recover column direction application processing recovery recover recovery impossible uniquely generally natural question inherent ambiguity part noise preserve optimality interference optimal b good signal remarkably different optimal noisy orthogonal whitening noisy even noise datum provably summarize contribution demonstrate optimal setting noise upon ica ica arbitrary complicated preprocessing necessarily experimentally ica algorithm noisy moreover noisy matrix infinite ica wise conjugate transpose transpose proceeding important somewhat arise ica nonzero contribution equivalently scale convention modulus factor addition ambiguity permutation indistinguishable say recover choice unit modulus permutation source define invert obtain align random indistinguishable particular ill define length mean even obtain source ill despite possible additional assumption portion source simple form inherent probably blind number generate vast reader book broad overview allow many recover ica mix noisy ica largely split higher another manner somewhat noise ica directional line advantage e latent signal sign algorithms tensor redundant also high complexity deal overhead directional derivative characteristic rely heavily upon continue start ica address practical issue ica originally sign ica source preprocesse unnecessary modify experimentally generalize source work attempt ica note noisy ica ica outperform first recovery assume recover precisely permutation zero constant ica recovery give kb noise define divided interference contribution access b estimate ica column white maximizes generalize spherical non proceed supplementary use even fourth nice algebraic ica random homogeneity vanish section unit sphere f property derivative ica necessarily unit f generalize iteration direction hide orthogonal multiplied treating update iterative converge rapidly one transform make orthogonal fourth positive may fourth sign preprocesse variant get issue distinguish ica one ica ica step notion inner candidate orthogonality issue inner invertible ignore issue aa noisy set let entry tc particular ia unseen obtain column unknown scaling choose unit fc fc fc worth couple remark notion exist guarantee choose exist converge cubic due omit proof ica test significant progress k convergence check meet recover however column demand idea simultaneous instead column using simultaneously find orthogonal complement within product k ta kk kk kk k fc fc fc c k zero let k us orthogonal gradient idea recovery algorithm step orthogonality column show td kk k signal tb tb inner product ica ica ica fourth ica design implementation free set gaussian opt reverse singular decomposition random singular singular choose datum ica aa noise variance variance ica sample power reporting error bar interval distribution ica degree freedom uniform unit create compare optimal ica ica use ica result formula apparent sufficient additive keep algorithm compare ica
type analogy detailed analogy question give pair implicit apply wind pressure answer temperature connect pressure analogy also model identify list form example two one analogy pair chapter book act answer book several act difficult analogy question combination candidate answer classification question also study classification ability majority odd one iii relaxed sound mean pick close word identify correct typical word close iii iv lose question close question require pick word list opposite question test ability identify select word typical ii answer type solver divide good framework basis question three component framework classifier identify different type short use build representation svm portion label include analogy component representation word answering requirement multiple word focus rare relation multi relation second sense challenge capability aim text employ label relational embedding combine sense relational word embed many text mining limitation single multiple study representation capture word pre cluster discrimination knowledge dictionary discrimination skip slide window input text stream slide window try word stream skip q word stream probability input output e vector window occurrence use skip pre learn embed weight regard context window calculate frequency window denote window calculate average embed context spherical representation cluster number cluster sense window include cluster denote vector online description word represent sense word representation vector close I match pair strategy recursively match pair find word sense use embedding different occurrence correspond sense pair word sense pair integrate relational skip entity operating entity function relational knowledge extract dictionary etc form triplet knowledge word sense word make relationship sense away word unified knowledge denote distance embed simplicity corrupt triplet construct replace pair select trivially norm word representation constraint norm commonly trick enforce representation adopt relation relation within combine derive relational sense pair calculation combine question online final provide section manually test type analogy analogy ii classifier set correct book effectiveness framework test set corresponding statistic list record standardize united student advance framework public analogy age analogy analogy overall education analogy analogy school master candidate experiment new baseline guess straightforward guess run design intelligence quite human answer question human amazon internet allow intelligence five question people collect take several high restriction worker american demonstrate accuracy intelligence worker iii age education continue believe sense latent distributional word allocation word gram sg word train skip gram skip window embed count multi ms word respectively publish multi word embed author frequent adopt propose method learn embed e triple adopt public relation build solver www computer question solution question wrong answer solver empty cause e analogy classification solver answer ccccc analogy school degree master candidate degree sg ms ms human amazon human question statistic participant participant age table find participant report distribution education background test participant hold school education education normal intelligence demonstrate mention accuracy empirically superior gram sg sg sg aspect performance ms ms sg multi sense word bring much rare contextual training linked train rare empirically superior sense demonstrate effectiveness adopt building quite indicate knowledge level intelligence answer various compare sg question incorporate word improve table accuracy type answer question human tend achieve accuracy consistent reach competitive involve amazon worker certain intelligence accuracy answer question different education people education degree tend common reach involve amazon master question potential human intelligence sg people school well overall assume school master candidate overall sg sg although concern due deep relation framework type ii representation relation design dedicate solver relation address experimental framework exist even exceed amazon involve experiment work early solve ai leverage intelligence plan enhance sense embedding bin liu intelligence standardize question human intelligence question measure understanding test solve automatically artificial intelligence apply simply apply word mainly relation among tackle challenge analogy word leverage novel word consider nature word dictionary solver outperform exist question exceed amazon might far intelligence deep embed intelligence deal experience test intelligence design individual adapt intelligence year measure test education successful life people still tag intelligence human like intelligence software activate attract ai design agent maximize success artificial intelligence interesting like robot answer far know limited develop solve test human develop agent question near question recent progress natural language nlp word advanced ability ai mean relation leverage however attempt application could satisfactory performance occurrence multiple capability embed aforementioned component first recognize type question usually sub type analogy question kind therefore effective question leverage novel consider knowledge word word retrieve dictionary tag word sense addition embed relation incorporate relational word effectively dictionary third specific solver representation relation representation find answer word question word answer question calculate offset answer conduct use experimental outperform question human amazon interestingly propose appropriate intelligence intelligence intelligence usually contain complete limited score calculate answer several age test behavior perform normalize less contain category picture question several detail algebra geometry logic reason identify element sequence effort develop method www computer program solve test score average number solver target level picture question recently approach
predict thereby reduce burden foundation practical multi domain demonstrate margin several far bn structure learn concern future parametric suited control false graph gene expression pc track dependency previously redundant calculation pc code maintain cache store research extension modification search procedure promise optimize multi several thousand label multi underlie joint composition need acknowledgment thank grant joint integrate lp j lp lp fact contraction lp label definition lp lp weak union completes consider satisfie composition lp contradiction lp lp composition contradict minimal parent form lem mlp suppose exist condition intermediate undirected minimal recursion I parent mb boundary lem boundary union property induction second markov define lp lp lp lp lp lp lp lp lp lp lp contraction statement lp thm lem gray gray network structure pc skeleton bayesian greedy hill edge series experimental hill bayesian structure use eight benchmark various assess return algorithm outperform goodness structure second solve characterize identify minimal irreducible factor condition multi classification multi encode term ten cover conclusion structural pc form motivate empirically effective suited source code boundary bayesian bn probabilistic structure bn acyclic distribution associate dag encode attract dag useful observational ideally coincide dependence identify structure terminology classical model basically cb systematically independence orient bn search ss evaluate hybrid attempt skeleton cb constrain consider ss present novel call pc skeleton scoring hill search constraint target variable think extra arise expense positive false among modification al later modify pe na fdr edge scale thousand series comparison pc hill powerful algorithm bn bn benchmark assess new dependence pc investigate ability joint challenge video annotation web page categorization large number category recent research focus exploitation dependency combination bn offer elegant powerful establish markov markov boundary theorem offer characterize decomposition ii predict thereby input burden solve datum assess comparative ability pc encode dependency structure mean indirect indicator make rest review bn evaluate show several involve artificial world learn empirical methodology theoretical raise issue future section upper letter set letter bn tuple direct acyclic node represent satisfy subset parent denote easy factored allow parsimonious enable determine huge determine relatively bn structure entail denote path parent parent path bn converse necessarily remain variable conditionally none parent drop separate adjacent iff find iff encode conditional via criterion dag show establishe undirected arc undirected edge class uniquely represent partially dag dag link dag orient dag equivalence structure learning sound dag database ideal test decide let denote four disjoint subset probability four z property z composition z x super probable bn bad need resort able cb ss cb relatively quick stop rely significance test unstable cause many error graph ss incorporate user database ss method deal prevent find optimal bn currently available like computational variable burden impose parent ability restrict cb method construct local around target bn scalability balancing accuracy hybrid min hill conduct show fast outperform reconstruction dependency equivalence greedy hill parameter heuristic powerful state art bn capable enhance algorithm bn e constraint super dag ss optimal vertex structural feasibility bn super structure scale hamming long prohibitive interest hybrid capable accuracy cb ss contain skeleton super small control discovery false attract false introducing behind cb induction bn handle cb method identification neighborhood scalability cb systematically check relationship target either datum discrete acceptance discrete global distribution multinomial latter independence test discrete table frequency c shorthand classic independence mutual mutual log ratio test differ asymptotic degree detail limit problematic small contingency implicitly conditional heuristic literature heuristic perform assume power count user heuristic reduce contingency freedom adjustment heuristic hybrid hybrid skeleton perform bayesian hill search cb subroutine call parent discuss view learner attempt produce learner pc false discovery fdr fdr node hybrid benefit incremental start extract severe size order increase reliability pc obtain pc think negative extra true particularly dimensional domain criterion say less severe original weak learner fdr aim proportion among fdr extension pe na modification incremental association control false positive estimate simply remove condition five indicator density equivalent think support suggest bic goodness new new measure generalize hamming distance learn learn dependence match undirected remove orientation sample learn pc compute indicator pair set generalize reconstruction two posteriori probability encounter practice score sensitive equivalent typically score rely gold instead employ network also skeleton phase value average depicted indicator express false distance clarity mention interpret regard advantage consistently negative expect come little false precision recall benchmark independence cb test worth capable reduce sample result much cc c alarm cm link dag obtain size bic training clearly dominate generalize network dependence datum heat find h rapidly plot sample average involved somewhat linearly h time slow mainly expense incur obtain pc challenge assign label throughout vector find function amount way suppose capture relationship straightforward meta binary method opposite sense independently lp consider label question capture label exactly multi attract purpose review fundamental wish involve feature subset r multi recently encounter minimal proper subset label behind partition j wise generalize simply mode seek minimal factor facilitate also characterization boundary depict dag dag minimal either ii parent address question markov answer markov boundary dag boundary problem decompose multi combination whole label summarize label respective boundary minimal lp boundary aggregate probable assess separately label markov boundary scenario without denote classifier exploit serve comparison selection label boundary evaluate effectiveness feature label dag denote minimal label dag forest classifier achieve good handle continuous rf rf default summarize come biology music repository come dim example label dl label continue bn learn datum biology image medical multi label assess focus decomposable score global term accuracy exact match label predict set hamming inherently label conditional fold table report report node dag figure illustration purpose conclusion inspection label several dags densely dag display figure conditional sep use many inspection dag bottom dag advantage information label beyond scope deep h pc significantly degree label degree resp nice pc consistently reduce negative discovery reveal mlp approach extract mlp confirm h interesting looking size mlp part lp see label decomposition ignore dependency expect mlp dominate dag pc compare mlp mb pc h sign statistically improvement pc approach without mlp mb pc identical hence column reach conclusion mlp average actual difference accuracy significant sign pair surprisingly feature increase due restricted feature though interestingly densely connect selection reduce input effectiveness balance increase burden drastically relevance loss boundary
roc lower u contingency table least curve pr contingency generate roc pr curve space dominate pr mapping pr true pr roc roc dominate pr curve skew balance unlabeled large pr loose demonstrate effect via roc table property confidence cdf statistic via strong law number empirical cdf pointwise cdf area curve amount positive ranking depend ci upper bootstrap number positive practice fraction unlabele violate within understand effect positive correspond ranking cutoff well correspond rank cutoff bad baseline none black corner none text corner use interval cdf standard aspects pr show estimate cdf cdf practice outside interval true roc pr curve close roc along curve strict plot treat approach pdf curve wide roc space assume yield poor estimate note estimation pr curve rough available pdf contingency upper base use efficacy approach curve positive cdf rank positive estimate sensitivity space recommend use roc unless good estimate roc curve thorough assessment set address project circuit logic grant research fellowship medical e van cancer cancer plan bm grant research grant assess performance metric label unfortunately many furthermore unlabeled example available able unlabeled bound empirically able roc curve quality critical rigorously evaluation often report summary metric area operator characteristic roc curve visually operate roc curve construct contingency table confusion contingency relate ground depict contingency predict predict positive false positive negative false negative compute contingency label label costly impossible special research cope labeling oppose partially empirical evaluation superiority mean assess partially label alternatively unlabeled certain circumstance describe show compute example contingency rank compute contingency table overall theoretically give low upper bind false empirically efficacy pr curve review partial classify learn logistic naive value belong typically belong order list instance sort value confident characterize value rank top treat complement prediction predict compute fraction label incorrectly contingency positive negative instance within instance cdf r eqs rank overall interpret label top rank x receiver curve rate axis vary axis cutoff point curve give positive draw straight line consecutive rank extensively assessment operate commonly range random perfect precision curve axis function recall positive positive skew set know positive negative unlabele instance unlabele negative focus available include negative contingency curve rank positive directly tackle theoretically positive knowledge disease either roc pr however positive available rough pr treat unlabele inherently cutoff positive treat small curve positive relationship include unlabeled latent positive positive implication disjoint subset overall r numerator minus r positive rank positive distribution rt three positive within equation equal rt lemma relationship rank contingency overall rank table require account unlabeled proportion positive example label greatest bind rank l r surrogate positive define u l greatest great via due contingency greatest bind contingency discuss contingency cutoff rank contingency create positive construct distribution contingency table rank describe construct cutoff decompose separately partial contingency table via contingency surrogate positive great via contingency positive translate require ensure may assign surrogate contingency contingency bind replace equation rr available rank upper vice versa obtain positive similar eqs deviation independent l know positive l equation violate biased occur application via cdf since perfect estimate cdf interval ci cdf ci cdf
directly fit reference therein copula posterior density type transform datum capture mixture model variational hierarchical vector variational posterior representation applicable require conjugacy within family assume jx prior vc mean field vb special independence differential c hold px parameter diag univariate concern simplicity copula margin include analytical copula univariate histogram shorthand cdf copula copula construct copula marginal pf offer variational framework representation j c j copula allow margin I c achievable kl optimize free margin c copula function accurate term posterior although form margin copula function approximation univariate margin j p copula still discrepancy margin compare upon update parameter lead calculus convenience incorporate exploit gaussian copula non decrease directly optimize one diagonal apply variational bayes j diag translate jacobian cholesky triangular detail deterministic stochastic introduce desire posterior normal auxiliary form illustrative show capture form normal gamma consider single observation conditional p sampling available fix margin jx nx cc deterministic special algorithm baseline gibbs appendix mixture automatically desirable polynomial uniform nonparametric simplicity add construct cdf parameter cdf exponential help chain rule derivative q term analytic derivation due expectation analytically tractable non nature locally solution contrast likelihood respective derivative derivation accord respect diag jj alternative way express gradient expectation already j derivative define polynomial low b limit prefer degree issue variational bernstein much simple proposal assume integration hold j gradient gradient turn enable stochastic constraint simplex projection introduce prior optimization summarize improved adapting seem kernel variational continuous margin involve cdf need entropy mixture pt j demonstrate able preserve margin constrain certain parametric form relax nonparametric construction refinement polynomial k gamma accord margin hierarchical avoid specify margin nonparametric base flexible univariate margin copula limited interpretability via pairwise dependence discrete margin parametric copulas rich dependency future copula automate posterior efficiency metropolis sampler mcmc proposal derivation model hierarchical variational combine copula log margin part width conjecture copula constitute unified construct posterior divergence posterior propose copula bernstein polynomial characterize posterior inference distribution represent intractable integral kullback bind tractable bayes field readily gaussian incorporation correlation continuous unconstrained skewed heavy tail
sample statistics nominal ess code summary ess st rd significant rate well size coefficient quadratic nearly improve mix lead much low strategy partition space beta mh diag dimensionality state partitioning burn interval sampling sd ess ess min max sizes trace confirm select part stochastic newton twice differentiable pdfs taylor series hasting dimension empirical assign dimension preliminary sample subset well call mean within subset potential go dimensionality tradeoff complex evaluation algorithm generate mcmc less correlated univariate metropolis proposal center cost generate hessian show next technique e software per sampler extend utilize computational cost gradient evaluation seem implement newton sampler algorithm multivariate local expansion newton advantage geometry capture hessian sampler independent resemble gaussian require hessian negative valid property easily glm initial take augment fast problem lower apply improve convergence visualization capability predictive distribution world pdf multidimensional univariate sampler slice rejection pdfs univariate sampler however less correlated development efficient black box importance machine recent development sampler carlo hmc view implement stochastic newton metropolis hasting proposal locally multivariate current equivalently everywhere satisfied much coefficient appear variation square glm metropolis close glm hessian hasting identical dimensional provide adjust langevin rate software write package extension mix argument diagnostic method review overview hasting package offer provide summary future research begin metropolis density mh property ensure transition ergodicity mh density locally density hessian respectively globally negative fit think counterpart nr nr mh reject proposal identical case acceptance describe begin overview responsible implement follow log function hessian multivariate prop prop hessian old old prop old old r prop new prop prop new old via call argument control choice newton nr control partitioning argument generate via fed via unnecessary proposal collect diagnostic probability series relative series diagnostic use illustrate method diagnostic visualization capability mode multivariate pdf lead small overlap nr mode accept next pdf mode argument nr guarantee nr maximum fit gaussian different derivative observation bad strategy partitioning subset subset gibbs argument list space belong convenience respectively rich applicable chain mh specialized adopt create specialized calculate summary include quantile effective plot proposal mh test log pdf iteration nr exhibit mix valid ensure end full arbitrary across expect value proper important mathematically expect apply incorrect interval correct require glm nr x primary ml rather switch remain beta diag examine trace plot pattern non peak pdf chain area plot select offer plot besides diagnostic sampler space iteration nr burn deviation sample sd ess min st rd max half burn argument counterpart function package observe rate contrary pdf secondly relative non pdf still lead good next want predict poisson explanatory distinguish predict predictive illustrate
pt mm mit mit edu microsoft microsoft unconstrained nesterov accelerate geometric ellipsoid method nesterov one verify descent view combination descent optimum search optimal rate convergence acceleration practice behind nesterov accelerate descent new interpretation reason possibility acceleration problem acceleration strongly convex gradient ball contain optimum maintain ball intersection radius shrink accelerate iteration contain intersection section convexity imply q recall smoothness allow factor obtain style pos right pos node pos node node pos pos diagram intersection one geometric accelerate optimum combining obtain already center ball ball formally easy calculation eq q fy fx fx fy eq radius see acceleration shrink hold ball observe ensure geometric well insight center square radius ball formula r induction case display fx fx fx fx square fx fx assume point intersection two consideration intersection segment reveal satisfie ball half ball ball remain g straightforward recall instance correctness iteration guarantee property believe integration new suited radius radius h initial fx kx fx fx bc bx bx descent accelerate accelerated method bfgs updating bfgs ii
input measure eq absence concept capture notion sample visual combination q cluster q cluster label visual number representative cluster max cluster list refine cluster base label share concept least compute label exceed label sample split break differ neighbor static neighbor seed sample create centroid centroid unlabeled compute diversity angular distance choose centroid gain speed diversity unlabele unlabele label diversity approach differ notably disagreement cluster entropy refine batch label decide batch newly beyond batch sample entropy cluster sample mean conduct divide training test video vocabulary divide part initially label test concept algorithm batch reveal label algorithm annotation score test start corner skeleton annotate set arm exhaustive annotate level frame computation angle maximum whole annotation dataset available research testing rest select datum positive size sample select seed method propose validate second baseline annotation round weight work assign top compare discuss since htbp baseline well annotation capture inter train concept base jointly pattern case al select informative first train early monotonic decrease ap annotation ap sample get correct concept dataset round baseline average run round indicate help concept model well believe one like propose combine novel uncertainty sample approach baseline retrieval reveal active learn recommendation author reflect author like member ari suggestion feedback operation base finding recommendation reflect additionally thank member dr suggestion feedback conventional normalize label good determine annotated learning combine measure novel diversity integrate refined level agreement corpus annotation retrieval video disk daily effective tool annotation tool treat automatic annotation whereby explore correlation rank sample relevance user query exhaustive normalize like annotate effort cost annotation raise way annotation require address issue active unlabele query provide system output label annotation typical engine annotation retrieval engine responsible labeling unlabeled system video retrieval combine measure refinement density art relevance generative annotation technique define label concept word word et al suggest annotation pick high task query pick video high estimate video fully annotate integrate selection annotation combine uncertain engine label use svm
constrain collect mini calculate monitoring statistic specify environment decision specify make svm precision exist present rate unlike approach detect presence label outperform detection drift arrive batch concept stream limitation performance extremely magnitude hence limitation propose identify fact imply define tn fp denote underlie percentage tp negative classifier characteristic value predict stable joint concept worth step pair refer influence note concept suggest even classification current stream use easily adapt efficacy detect drift concept detection estimator concept stable reject level statistic user level type point stable false every fix accordingly cost simultaneous inference correction run alternative na I na I four drift I four use rate r essentially previous decay weight instance class imbalance detector test characteristic test detailed significance level detect significance level concept c tr r c tr cd decay significance level decay current framework practice detection rate application control move fair learn geometrically sum variable historical take weight stationarity overcome reliable geometrically bernoulli random simulation decay carlo surrogate generate upper significance serve bound surrogate detection corresponding obtain bound four framework instance classifier class bound likely reach detection drift store since store new one sufficient cross correspond fail reach new monitoring cycle underlie decaying factor significance significance level bernoulli lb investigate bernoulli variable stable concept I change equation stable concept set distribute realize time imply thereby statistic compete empirically dominate achieve lag power exponential short lag limit alternative variance htbp km kk maximal lag rate regard detect drift kp lag occurrence drift detection observer adjust cost incorrect prediction immediately concerned occurrence drift stream provide statistic convergence long guide r synthetic public consider imbalance public dataset across various baseline poorly utilize bootstrappe generate algorithm classifier public create stream stream fed visualize detection point concept avoid redundancy present histogram remain one compare identify drift false numerous cover concept drift mechanism algorithm identify discuss typical three classifier drop cp cc cp cp count top bar bin mechanism consider concept imbalance balance imbalance cp cc class occurrence imbalance classifier unable alarm drift trivial false increment respectively imbalance ratio unchanged cp cp decrease decrease select cp cp public generality choose machine misclassification majority concept svm classifier adapt concept example hyperplane hyper dataset drift detection create dataset stream collection difference magnitude drift user topic table imbalance c balance balance balance imbalance imbalance imbalance balance balance imbalance imbalance imbalance concept early false delay rotation dataset drift drift experiment dominate drift superiority detect decay minimal early poor missing delay count drift count false detection simulate dataset table dataset concept span concept concept histogram predictive drift period bin span summarize particularly dominate performance small benchmark regard detect produce false delay drift concept belong allow work stream use user parameter detection term low require however change relationship observe concept new concept concept drift apply stream response underlie specify compare benchmark public span type benchmark term detection concept mining stream stream always income concept stream business intelligence data stream paper focus detect binary classification drift say predictor intuitively concept drift underlie detect concept widely use detection streaming employ hence detect drift false
useful ps call whenever agent correspond exist value categorization common place layer action layer particular operate introduction act traffic drive traffic among continue drive stop car category four combination color randomly choose action setup describe would compose four shown try associate rich connect associate action elaborate auto circle minimum sep font edge node edge enhance ps illustration green create step green encounter place newly create edge line dash line similarity third create place second left cause share red establish modify auto inner pt thick font node cm style inner sep dash bend right bend node auto thick font dash bend edge dash edge dash bend edge bend edge bend edge bend left cm sep thick font bend bend bend edge dash dash bend edge bend right edge bend dash node node node dash bend node dash bend dash bend bend node dash dash dash edge bend right node edge bend node mechanism categorization categorization ability recognize achieve come think red relate stimulus fulfil connect zero categorization realize flexible fulfil environmental adapt scenario begin drive green light irrespective opposite stop green driving phase drive stop leave color reward choose irrespective signal phase ignore action clarity explain later demonstrate connect correct correct build update present flexibility configuration network adapt fast fig display practice strong edge address configuration rate thick style minimum edge pt circle sep thick font node node node bend bend cm node inner font bend edge bend right cm size sep thick font bend bend bend x south north west efficiency ps agent function agent choose adapt phase impose action asymptotic action extent chance learn stimulus trying find similarity consider analyze environment different background different agent direction whenever follow irrespective call color auto thick inner thick right right dashed node dash node dash dash dash edge dash node dash dash edge dash edge dash dash dash dash edge edge dash node dash edge dash edge scenario ps two time correct never later step symbol twice thus infinitely many ps ps behavior color twice thick style circle sep thick font right edge bend dash dash bend edge dash bend bend dash dash dash dash dash bend bend edge dash edge node dash bend bend dash bend dash bend dash bend bend node bend bend dash dash bend edge node dash edge bend node edge edge bend bend edge edge edge bend dash dash bend edge dash edge bend bend bend bend dash dash bend left difference enhance ps successful enhanced look strong correct tend imply probability unity colored efficiency ps prefer reach enhanced ps precise auto cm thick sep edge dash dash dash dash edge bend bend bend cm bend cm edge leave dash dash dash cm dash dash edge leave dash right dash enhance solid curve efficiency completely reach asymptotic eq figure hundred analytical efficiency ease analytical curve qualitative difference x east dash blue dotted qualitative advantage quantitative fig indicate nonetheless network action majority voting occur agent whenever gap random ps follow simplify approximation actual agent efficiency look take argument lead show configuration conceptually action eventually edge correct present finally fact attempt approximation time plot simulated curve red cause whereas reduce nonetheless certain ps start analytic lead high asymptotic efficiency agent certain reach express cumulative thought express series reflect thereby converge configuration action give namely irrespective agent unnecessary would affect h auto distance style sep thick font style regular side thick font style size inner sep thick font dash dash bend bend dashed edge dash bend edge dash bend leave dash edge bend right bend node cm edge bend cm cm bend bend bend left edge bend bend node dash node dash bend dash dash bend dash dash edge edge dash node node cm edge dash dash dash node bend dash dash bend bend left node bend answer look efficiency lead action hand correct accordingly lead asymptotic e contain reduce irrelevant agent categorie irrelevant generalization make generalization efficiency also enhance ps tends show fig h c image east image west category expect action irrespective asymptotic efficiency eventually connect action enhance environment stimulus chance unless common recognition categorization classify new meaningful dynamical enable ps generalization generalization property mechanism ps arbitrary ps learn extreme introduce ps practical handle also conceptual enhance ps rather enhance dynamic walk give sophisticated complicated overall preserve inherent enable model rely existence induce generalization successful potentially create exponential intermediate agent efficiency agent performance color relevant present recognition particular predefine circumstance associate blue together even similarly generalization single category color single possible require far dark divide systematic take order addition structure whose predefine ideally even completely priori generalization thereby speed subject acknowledgment discussion project universit universit universit er universit cope simply several enable projective projective novel artificial intelligence recently perform standard simple reinforcement problem complicate canonical world car projective principle allow performance generalize ability extreme environment otherwise impossible ability act upon new experience extensively life need recognize light correctly traffic may color neither shape reaction aspect common generalization learn red signal signal action relevant whenever mechanism illustrate generalization handle traffic
berkeley edu microsoft com microsoft com grow make crowdsource important expert knowledge option address issue introduce voting utilize worker couple strictly proper mechanism guarantee together conduct empirical study amazon validate big ever deep demand grow collection crowdsourcing web service like amazon crowdsource worker perform exchange obtain crowdsource worker lack interface express several improve via interface mechanism increase labeling crowdsource label task question option category option worker require option formally involve single crowdsourcing extensively empirically vote worker wherein allow figure example voting system worker experience test confident alternative remain one individual vote interface worker mathematically problem belief worker set crowdsource multiple allow worker valuable question identification source item vote worker second worker worker worker answer worker second flexibility partial voting crowdsource partial knowledge crowdsource worker option problem couple compatible payment worker receive crowdsource payment strictly proper scoring want possible select option result prove compatible section coarse rely people belief payment requirement mechanism coarse select option belief enough also conclude report preliminary basic discussion voting candidate preference among interface crowdsource candidate option question crowdsource voting focus fundamentally irrespective rule design payment payment event payment strictly event strictly maximize belief however mechanism specify mechanism present score support belief question crowdsource worker response propose mechanism suitably worker free mechanism interface turn satisfy axiom adapt set presence answer design operate gold typically additional predict worker mechanism design therein weak absence option option assume contain question mechanism know answer worker individual worker option belief respective option belief distribution worker across shorthand goal worker worker formally worker option value worker select precisely option worker response gold question question question select correct answer select gold question none correct worker question option response payment worker note restrict payment crowdsource solely perfect worker answer amount mechanism quantity expectation choice question among belief question formalize belief select option sum option consequently option worker payment summation gold question worker choice assume extend general compatibility mechanism expect payment worker option correct theorem main result prove mechanism requirement mechanism algorithm compatible compatible mechanism amount worker attempt optimality uniqueness claim mechanism contradiction argument specific belief worker act belief coarse belief option know sure option incorrect among option thing payment make remainder devote jump without continuity first suppose generality worker belief option maximize worker payment select support option payment worker option payment mechanism inequality payment different support strictly payment upper strictly case worker equal maximize maximized component correspond set early expect payment maximize act outer respect question expectation take worker belief condition gold argument prove inner maximized payment worker act subsequent proof equality mechanism compatible lower thereby complete coarse belief worker belief compatible support belief worker worker belief mechanism belief turn break something desirable option worker belief contribute verify early theorem suppose question q eq follow result coarse select derivation mechanism belief axiom say worker question question select wrong option select option question gold wrong payment axiom notion qualitative goal interestingly notion value belief option present evaluation mechanism amazon crowdsource platform goal experimental perform basic check practice evaluate voting whether worker interface reason design quite respect worker understand unlike exist benchmark position primarily theoretical detailed publication conduct separate worker language display text identify display worker mechanism execute payment gold question payment mechanism start interface start fraction question zero incorrect voting interface payment interface payment entire author evaluation response response comprise selection option figure depict turn figure depict response select plot reveal voting approach successful fraction incorrect answer depict per worker increase
relevant arise becoming choose sample wu zhang discriminant ratio determine relevant one dominant range image ratio lie image give relevant lie dominant weight give choice distance successful literature combine instance retrieval distance consist query useful euclidean move space cluster relevant thus respectively set define apart set account relevant image inspire relevant database image define euclidean define score interval initially consist score mean high membership alone able reflect completely may relevant far centre relevant outli modify relevance score cluster evaluating define calculate database average database average scope provide retrieve lead increase relevance scenario image early retrieve follow change time relevance feedback one adapt precision evaluate total image retrieve scope equal scope involve return image relevant retrieval therefore aim retrieve new set contain retrieve sense retrieve number relevant consideration retrieve expect expect type increase value scope basic increase clearly establish really behaviour become counter hence evaluation remain irrespective rf search database image retrieval segmentation visually object indicator contiguous segmentation contiguous adjacent merge recursively contiguous homogeneous hand sub one contiguous segmentation terminate isolated image mean cluster view preprocesse computed rise mean segment equal cluster discard illustrate show pixel group detect correspond segment row image base segment motivate image segment database rank th th obtain segment segment denote segment segment th amount segment close denote rank similarity propose segmentation define image retrieve st relevance refer retrieval reflected perform henceforth shorthand respectively conventional involve propose scheme section implement relevance combination cluster density rw propose assign distance update every scheme refer brevity experiment whose table propose rw feedback difference lie initial retrieve rf rf additional apply rf retrieve image retrieve specify retrieve feedback improved propose seven rf iteration carry retrieval apply six retrieve rf five time retrieval retrieve set six rf cardinality retrieve image image integer six five rf rf accounting image feature perform image take five new assess retrieval rf iteration converge iteration rf discovery relevant retrieve retrieve retrieval possible crucial aspect standard describe feature texture colour occurrence colour value colour proportion pixel colour co occur colour position diagonal give colour image shape colour pixel diagonal single represent column indice researcher quantization specify co co pixel horizontal direction occurrence need demonstrate approach briefly name db db database subset db illustrative image database image per category database db db single idea diversity improve thereby justification mm database retrieval db db rf compare rf iteration c propose rw rw rw db db db db database evident table mark change rw scheme table table shorthand name identify initialize improvement evident case higher encourage discovery achieve high relative clearly lose c database db db iteration database present illustrate recall precision database db db content retrieval conventional retrieve query image segment match conventional approach segmentation instance employ scheme improve acknowledgment author put record ray school technology university discussion former student crucial insight ph thesis contribution dr university north university division institute road north hill mail ac ac com retrieval essentially extract database similar content image bridge texture human system typically use rf mechanism iteratively incorporate user give relevance retrieve approach vary work alone implicit incorporation image scheme relevance time suggest limitation approach keyword retrieval segmentation relevance direct consequence rapid imaging technology million generate source surveillance device security purpose scientific imaging home availability digital device internet maintain enable retrieve require indexing keyword file name index book result indexing picture carry represent school lead second content retrieval objective colour shape texture role attribute associate image scheme information segmentation propose retrieve rf efficacy approach establish six representation color describe couple feedback overview
right point take irrespective initialization challenge stand implement idea exist dimension finite vs ahead section describe trust address f f e carry section n invoke trick plane capture composition verify contain open instead avoid trivial minimizer locate study three eq region figure typical style describe look orthogonal plus perturbation argue perturbation affect ip behave u sound write magnitude simple aspect know ahead time knowledge appear spurious minima order information saddle riemannian trust region sphere unconstraine successively order approximation iterate order geometry movement trust induce subproblem trust region make progress typically radius dynamically accord text generalize manifold natural tangent iterate k riemannian riemannian usual riemannian subproblem ball trust trust region take basis equivalent objective trust ball admit numerical minimizer solve issue tangent additive resort tangent see movement iterate sphere geometric w small show decrease curvature trust sequence eventually move stay strongly converge minimizer quadratic sequence far include development dl application book survey summarie development recognition technique relevant current dl start algorithmic correctly extract generating require exponentially many subsequent study polynomially many ensure local however efficient algorithm dr show recover solving per provably overcomplete incoherent initialization refinement include recovery run super provably recover complete dictionary aside theoretical dl result cast show sublinear vector improve recovery nonconvex sequentially core blind separation nonconvex optimization nonconvex structure include recovery tensor pursuit signal refinement exploit refinement dl also guarantee optima complete geometry initialization algorithmic design may prove valuable nonconvex trust sphere build effort generalize algorithm riemannian reader survey development conjugate spherical trust riemannian manifold adopt either global critical force piece together entry similarly b u function dl pca ica multi nets architecture provide provable subject go recent deep ica matrix study assume row perfectly fourth overlap considerably help improve stability component connection help ask play role dr help landscape section even independence familiar figure broken sparsity level become believe generalized correlate case hand landscape grow minimizer adequate knowledge unclear objective distribution rotation exclude course bold capital scalar several specific work sphere ball integer confusion entirely vector th operator norm max norm distribute identically n bernoulli compactly frequently state prove unless otherwise note notation organize geometry riemannian trust sphere corresponding discuss main present main discuss direction major algorithmic deferred cover result proof reproduce figure find characterize landscape n n landscape orthogonal minimizer claim hold np n constant failure probability obtain sphere riemannian part stick capture x np n map minimizer sign local ic theorem local f x w q argument sign minimizer symmetric section cover sphere calculation sign local contain uniqueness isolate minimizer equally sense produce close row characterize high landscape hold simultaneously probability w w claim fail hold np p numerical np particular minimizer sign ic constant identical e quantity detail nontrivial landscape successively nonzero directional negative curvature depict constant every r c r h w page every hold gr page w qualitatively concentrate nice extend next every page w see next provide desire w w l fix x lipschitz l lipschitz integrate thing dictionary landscape look qualitatively dictionary condition define landscape u enough canonical comparison constant np see hessian cause section detail small large result local recover row one saddle motivate riemannian descent trust sequence minimizer asymptotically produce algorithm target one minimizer exposition try technical possible seek initialization iterate repeatedly minimize approximation iterate sphere instead approximate tangent space sphere obviously expect eq q next solution choose movement iterate read understand interpret particular f riemannian notion subproblem orthonormal recover classic trust region subproblem solve method numerically suffer approach e trust practical follow recommendation chapter backtracking base whole algorithmic procedure algorithm n r trust subproblem k general chapter produce probabilistic assumption guarantee iteration problem algorithm efficiently produce close summarize dictionary orthogonal dictionary dictionary orthogonal p riemannian trust minimizer set constant positive complete dictionary suppose condition positive p c riemannian trust region size solution near minimizer constant convergence target polynomially estimate relatively adaptive size produce relatively goal state provide analysis riemannian find nonconvex entirely nonconvex ultimately minimizer show start take consecutive unconstrained unconstrained iterate stay ensure stay gradient point minimizer continuity magnitude show iterate stay fact unconstraine mapping lemma magnitude gradient see section c r compare magnitude unconstraine crucial bind magnitude iterate trust region f p trust subproblem iterate make upper combine condition give also unconstraine detailed trust cc claim carry unconstrained k provide next forward calculation subproblem interior inactive q q constant unconstraine note claim simplification complete iterate actually nearby minimizer serve good proxy distance f relate magnitude point lemma component lemma make characterization locate sign obvious symmetric unconstrained iterate integer lemma local k line apply condition thus combine estimate moreover region thus proof ready piece together proposition enough lemma proposition satisfy minimizer contain interior connect respective sign component closure r minimizer proposition objective fix call unconstrained minimizer iteration consecutive step consecutive phase never drop continuity whole whereby minimizer either case drop many within unconstraine three drop stay connect component iterate finitely take step unconstrained decrease future must unconstraine unconstrained certain point minimizer iteration claim away target riemannian produce solution away minimizer simple linear programming round procedure optimizer sequentially recover row w sparse dictionary summarize relevant dictionary though recover dictionary resp start appropriately find target recover accuracy q show resp recover riemannian conservative produce row recover round input repeat procedure recover dictionary cn failure numerical constant towards correctness round desire provide already orthogonal round small induction procedure mind solution first vector correspondingly dictionary objective matrix learn keep setting reduce everything original effective lemma recover h small overall simple simplification tail inverse work might scale affect recover dictionary dictionary number algorithmic setting recover dictionary failure need round round round page argument compare orthogonal case part necessarily u v small perturbation perturbation matrix orthogonal bound see come instance dictionary prove solely rely c u theorem provably perturb q next identity suggest w obtain combine choice word return small constant lp round upon sketch plus round union tail inverse simulated recover row coefficient rotation l np entry bernoulli reasonably produce step size vary primarily vary pair repeat simulation sign e return reconstruction trial near likely recover lp round implement obvious dependency complete dictionary lp program approach recently improve matching obviously complexity intrinsic work suggest necessity efficient complexity proxy adopt amenable affect algorithm work directly sphere treat complete bound treat transform change transform current geometric structure plug play deal orthogonal nontrivial relatively forward extend recovery tight frame though elimination marginalization overcomplete marginalization behave coefficient likely physical ica characterization modify theoretical insight heuristic plus refinement mostly whereby initialization analytic highly coherent problem framework experiment believe nice spurious minima either surprising point predict modern reasonable generic initialization maxima saddle iterate saddle continuous counterpart theorem complete result state section deal convenient random subset moreover last line composition similar argument establish twice claim composite eq particular result integral form taylor w q result let q approximation f optimality substituting establish lemma certain work basically continuity need operator operator respective tangent place translation play lemma translation moreover e complete establish property riemannian fact lemma substituting claim f direct state work section canonical w riemannian differential justify enough scalar part apply tail monotonically decrease similarly tail summarize n ready result together radius define net np q e hc obtain take eq complete given consider get complete lemma local property manifold version k translation isometry theorem lipschitz complete taylor theorem q parallel define isometry isometry combine bound obtain need lemma dictionary cn least homogeneity consider random bound moreover integer simplify fix n numerical z n q cn overall lower suppose riemannian return nearest round take form vector enough return relaxation obvious objective thus solution necessary r therefore imply orthogonal subspace solution argument exactly basis complete define invertible lp eq imply v complete r e bind see iii upper large principal angle eq column rank projection onto span nontrivial pt conjecture width electrical engineering york recover dictionary modern machine give efficient provably recover contrast guarantee algorithmic certain spherical manifold provide riemannian arbitrary initialization presence saddle light structure spherical trust region landscape second geometry approximation c thank private foundation partially support grant thank mu mp possible model seek input signal signal role compression useful acquisition traditionally heavily explicit analytic constructive successfully numerous advance ever range classic modern multidimensional basis wavelet practitioner adapt describe signal challenge couple rare put effect codebook naturally occur datum challenge question ever empirical mathematical science discovered encode atom early process discovery application dl past decade process recognition deep architecture review development particularly learn great variety text genome series derive optimal manner hand dictionary increasingly promise armed sufficiently allow successful contrast theoretical application dictionary free desirable guarantee correctly important algorithm broad understand arise consider formulation attempt dictionary fidelity coefficient quality impose desire admissible orthogonal set optima unchanged example hull effective symmetry break tend play together relaxation come bilinear conceptual diagonal diagonal denote transpose isolated amenable relaxation unclear relaxation g problem sense relaxation provably hope dl seem always produce towards basis
without backtrack next time select multiply drastically reduce evaluation variety crf character recognition dataset name entity speech task journal pos letters image sentence syntactic sentence token tag correspond etc pos tag entity classify quantity gram pos case character token pos task assign syntactic tag token sentence data follow division development gram shape task competitive five variant average basic averaged py meta initialize dynamically classic power work comparison deterministic bfgs include online heuristic author sg non test sag give initialize sag size figure pass time gradient small different extra forward line evaluation backward computing use across strongly unique running indicate accurate plot exclude interpretable poorly include test plot plot dotted gradient size outperform move away early meaning early outperform view recursion outperform hybrid early relative rank change choose perform less effective initialization obtain competitive runtime outperform substantial good error far always reach speed pass reach e x x positivity lyapunov term lemma followed expect lyapunov add round q third ignore obtain get part proposition section strongly sequence probability except use choice uniformly maintain useful summing minimize side next x nf l nf l nf k expand l similarly b define lyapunov straightforward expand apply k simplify term k combine term row attempt constant require relations eq zero look place n eq expectation constant get error plot plot appear despite value exceed sag optimally tune gradient lack tuning could method salient utilize crucial performance perform poorly well obtain part label figure remove permutation perform bad perform nearly sag previously perform although performance set value lead runtime plot body pass independent except implementation tie result hardware thus little runtime hardware set runtime general runtime number several perform slightly runtime seem extra root implement runtime fast high relative bfgs bad runtime proposition rgb average use crf gradient improve uniform reveal training objective well well optimally tune stochastic conditional language label extraction parse name processing vision mrf dependency advantage discriminative building mrf disadvantage slow train crf due cost crf single training example advantageous deterministic single example deterministic method fast deterministic method require might optimization community consider vector objective objective like regularize gradient contrast deterministic low cost sag combine training example iteration reach accuracy fast convergence rate sag traditionally implementation sag step show sag binary sag tracking marginal crf drastically reduce sag uniform adaptively frequently datum sag particular fast compete task part speech parse optical character indicate sag outperform term require perform optimally error pair comprise standard minimize likelihood summation chain backward compute gradient solve grow thus like pass prove inferior strategy newton evaluate training traditional attempt improve deterministic example process reach dual online sag achieve fast classic sg cost quasi hybrid deterministic method slowly decrease method state fast algorithm bfgs accelerate stochastic gradient sag objective constant bound gradient primal sag stochastic iteration write slope example sag use instead keep iteration sag randomized gradient aspect classic fast major requirement nice use sag select regularizer crf likelihood gradient start track see normalize update sag nd py py I f would gradient fortunately dense scalar constant gradient small quantity change gradient unfortunately crf typically approximate standard backtracking since use value step monotonically slowly decrease g g py since stop see full zero difficult decide continue often memory crf marginals respect feature example feature write term take algorithm difference old parameter thing similarly pair store pairwise marginal
speech search infer probability stop variable stop factor affect model random capture parametric overfitte sensitive two fourth previous query query query hard parameter increase need result nice nlp use armed multi article improve click weather user want place search repeat treat online generalize thompson build hash term term associate query lda treat topic get collect predict adopt generalize thompson treat model original ten behind vote weight average vote rank feedback whether logarithmic ir ir wrong adjust engine date base user generalize thompson quickly identify cluster current sampling payoff rank search think high put pdf step update behind click observe rate position instead ndcg feature describe feature huge statistic one user term htbp short relevance level short relevance term time relevance history history level term relevance history time long term show history user user history cross user user number number time cross user history number time algorithm compare exploit default search engine ndcg default ucb article high yahoo thompson term thompson propose cluster thompson corresponding ranking htbp achieve default high thompson htbp generalize thompson train model model payoff explore suffer insufficient htbp see high htbp propose improve term short improve long term arm short use thompson thompson linear experiment efficiently improve rank popular armed bandit engineering extract lot style train rank novel way long term behavior use algorithm new default rank arm project start begin web topic I semi short user period turn query several keep query show armed learn quickly multi ad recommendation none armed web decide web web hard challenge user totally project project focus armed news user web search propose behavior site engine index engine treat query search really want movie read review infer interest previous query user want search engine return ignore interest display user basically technique context interest user past short user type contextual different user relate request search query refinement refine refine refine generalization refine mutual home home refine user profile build history current try interest example click technology article engine company try model behavior user generate query try interact user cluster current infer term behavior build interest profile adjust interact remainder solving describe short long term web search work short user good variation query benefit issue thing query express need return benefit popular people adopt click measure variation experimental show web significant click query click bad effectiveness argue query handle user history profile user profile represent history interest capture feature noun phrase tf tf bm treat inner profile user query train probabilistic look behavior construct history consist search model weight history query em estimate lot interest category category training interest distance user profile cover return web metric word semantic classify category learn history adopt sensitive behavior aware query query map tree effectiveness include click post category base occur prior query rank cosine search train behavior latent former n represent vector perform identify web page rank include decompose temporal segment discover temporal long short term search feature bandit news web news set article one click baseline try expectation click model article traditional regression predict estimate always bind thompson old heuristic lot posterior normally function choose however true function implementation simple
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paper propose deep auto encoder acoustic feature speech stochastic conventional analysis synthesis speech speech include auto simple encoder synthesis wu school computer science national institute propose speech synthesis feature spectral linear conventional synthesis text speech confirm speech dnn auto encoder current synthesis use probability apply language offer advantage know however speech model still artificial speech partly remove speech room improve spectral speech reconstruct fine lose synthetic experiment dnn indicate synthesis suffer due average due feature bring representation coefficient intermediate representation acoustic result speech answer propose deep technique linear synthesis extraction discrete cosine dnn denoise auto encoder condition finding use speech information dnn apply acoustic modelling synthesis dnn restrict boltzmann modelling hmm recurrent neural network modelling modelling work use auto encoder speech encoder base bottleneck group deep auto encoder technique paper binary deep et try discriminant discriminant encoder continuous calculated spectrum synthetic speech auto encoder encoder decoder encoder vector eq dimensionality frequently sigmoid relu map decoder mapping perform linear alone employ bias mapping encoding layer stack fine tune reconstruct maximize typical error mse e denoise encoder auto encoder auto extract encoder auto first corrupt pre auto encoder encoding locally layer layer desire decode layer deep corrupt train denoise auto encoder auto architecture back mse encoder training derivative represent neuron I il function l gradient parameter fine tune report encoder independently spectra evaluated encoder method synthesis condition synthetic use art synthesis dnn consist long extract spectra straight dim spectrum autoencoder sample extract encoder acoustic f energie reconstruction error auto spectrum hide layer decrease encoder bottleneck encoder dim acoustic input dim global randomly value produce good experiment dimension l dim encoder c auto report synthesis build auto criterion show spectra encoder denoise auto encoder reconstruct frequency part spectral distortion original spectra reconstruct spectra test observe auto encoder distortion compare distortion preference number ask auto da ask auto encoder encoder speech
map inner inner component tangent fx depend perform develop algorithm together crucial line condition ensure algorithm k equal inner product gradient algorithm strong condition search euclidean line find point wolfe find interval point condition euclidean detail reader behind satisfying initial previous gradient follow turn find minimize along geodesic point combine length numerous experiment effectiveness method good extract report cg notably fast except cg without manifold cg cg cg without iteration lrr rr riemannian counterpart model enable potential match accordingly yield quite suggest optimization may potential open algorithmic work effort extension rich class wishart prior leave simple actually fit broad enable incorporation penalty avoid easily easy moreover manifold optimization topic explore table pt thank author title title author corollary definition mit laboratory institute technology new mixture em box optimization fail slow intuition convexity consequence manifold match em highly encourage record outperform variability strength optimization tune method prove exist tool hope encourage wide widely variety process quick search maximization numerical conjugate gradient newton inferior programming covariance euclidean space especially boundary iterative affect cholesky also point program nonconvex stationary formulate convex resort sophisticated slower statistical high viewpoint nonlinear optimization em believe em remarkable substantially motivated observation positive numerical quasi difficulty implicitly simplicity em may justify manifold turn inferior discard refined outline idea intuitively whereas make manifold turn remarkable consequence ultimately enable contribution pt show key development key procedure help beyond outperform usual cg independent real show comparison manifold perform across em run encourage new mixture ensure service release implementation publish huge summary impossible line examine several counter claim think inferior purpose nonlinear programming method value amenable order convergence refined paper convergence gradient positive constraint suggest decomposition covariance single nonconvex add spurious report dimensional spherical near spherical matrix gmm branch nonlinear classic reference toolbox substantial study sometimes limited spherical gaussian focus highly algorithm background material serve establish notation quantity gaussian sample estimate q time seek compute manifold em canonical programming em especially stem cost incur enforce covariance avoid motivate take view manifold manifold applicable box optimization dramatically geometric intuition see manifold resemble convenient riemannian smooth equip inner product tangent manifold possess usual nonlinear rely locally join along short generalize euclidean convexity say geodesic within convex symmetric definite tangent entire riemannian metric nonconvex remain globally geodesic convexity coverage recent emphasize geodesic play convergence expect method much intuition geodesic close ultimately em subtle handling apply single geometrically suited invoke far reach impact transform prove cg manifold omit reason theorem maximize decompose eq must eliminate gaussian leave impact conjugate cg riemannian gmm likelihood minimum log theorem
forward backward capacity lead redundancy independence high strong may ignore many capacity highly pairwise dependent also contribute largely discrimination subset salient since dependence redundancy develop precise correlation effectively try redundancy introduce adjust redundancy organize review theoretic metric include experimental conclude study decade kind general aim class redundancy selection know correspond discriminate class label class distance may since weight feature remove extension incomplete still unable feature redundant accuracy speed method feature base inter obtain relevance separately measure class relevance feature score select fast correlation feature yu another method separately utilize would redundant mention redundancy identify index redundant ignore complementary discuss remain feature tackle former wang information mutual intensity relevance redundancy implicitly complementary complementary criterion identify redundancy complementary criterion mutual complementary salient identify salient although mention recognize measure among complementary select correlate select approximation turn essential describe fundamental unit assignment convenience hereafter logarithm quantify mi q consider amount two note mi extension mi q third mi entropy mi solve mi imply potentially cite justification mi write simply take top number decide stop assumption feature make mutually far widely recognize salient redundant redundancy variant generally relevance redundancy redundancy discrimination enough effectively redundancy word redundant may weak individual particularly microarray complementary modification identify complementary mi explain great significantly weak give word redundant conversely relevance complementary simultaneously redundancy complementary magnitude criterion sake q measure redundancy complementary relationship feature correlation among strategy well still suboptimal although pairwise handle relevance mutual independence word mutually identify able inter feature thus redundant I hereafter term select salient hereafter specifically give candidate feature candidate fp interference give status influential recall candidate interference fig green proportion pairwise distance complementary correlation ht scenario candidate show distant redundant complementary redundant feature candidate likely complementary value complementary correlation candidate reliability candidate distant candidate candidate complementary likely salient distant reliability make interference reveal interference dispersion likely complementary negative would redundant vice versa apply dispersion interference standard instability eq less interference salient less redundancy less dispersion interference end adjust e value negative account candidate piecewise use rather analysis relevance redundancy dispersion search candidate ht code th dispersion f algorithm repeat loop loop predefine one end additional newly add feature could record summation summation take fast ss f ff algorithm algorithm method representative review five selection describe algorithm relevance correlation criterion take fs fs redundancy mi select greedy manner measure relevance f note selection order redundant redundant feature eliminate method suggest mutual maximization base mutual information concern also select field feature ranking search instance throughout platform use dataset select implement conduct ghz cpu ram computer window validate method ten frequently six kp dna datasets tumor microarray mixed supervise feature r lr name class kp tumor I bayesian nn adopt nn kernel show check independently thus classification feature check feature result fold validation fold conduct classifier classifier collect classifier significant dataset range report classifier feature ten approximate fig fold rate different type classifier knn effectiveness consecutive select average fig superiority seven kp dna tumor breast cancer begin several tumor fig probably consider measure pairwise redundancy ignore redundancy feature higher select well never dispersion redundancy influential evaluation feature e respectively find inferior exist dispersion ht ht ht record feature indicate mm mm mm dna tumor cancer breast cancer avg rate c ten fold cross conduct evaluate sample correspond record fold cross significance notation test correspond feature high bold good average ten last see value rate show outperform dna error rate ten give diagram apply visualize box box represent red worse indicate l pt degradation level improvement level use ten feature less feature row among significant bold value classification selection na I correspond performance
estimate gradient ascent update optimisation intuitively mode behaviour beneficial probability alternative hessian result newton optimisation however hessian difficult abc prohibitive evaluate proposal problem construct local hessian newton make use limited bfgs set currently propose markov user crucial memory correct requirement fulfil restrict accept hessian psd correct standard remove hessian approximation denote eigenvalue smc discuss fix smc rely nonlinear consistent new variable model perturbation manner require intractable instead need distribution approximation determine formulation practical balancing requirement return study impact smc abc perturb particle tolerance level est carry I iw u case z I compose particle trajectory particle generate step particle deterministic smc unbiased carry variance high difference detail proposal particle together quantity close discuss evaluated gradient particle quickly lag reject step algorithm markov chain detail evaluate first evaluate serve comparison intractable appendix mix markov autocorrelation lag burn discard indicate uncorrelated imply index investigate smooth gradient dot indicate obtained see error log gradient minimize grow estimate result suffer parameter optimal alg acc median abc abc median computed monte abc pilot require abc probably match exactly model log use prominent financial presence jump return denote symmetric stable previously simulate appendix similar posterior indicate estimate volatility obtain smooth abc upper left obtain pooling dot grey density posterior prior quasi proposal enjoy require size provide hessian length experience simple derivative density prohibitive proposal hamiltonian algorithm likelihood resource provide national lag smc map accordingly use discard burn iteration discard hessian pre parameter gaussian denote distribution transformation simulate v transformation mail united mail division control mail hasting parameter proposal pilot inspire quasi newton hessian proposal inference likelihood application benefit new modelling return bayesian parameter inference intractable latent eq q intractable lie exactly far evaluate wise prohibitive evaluate effort intractable correct together inference intractable smc random walk evaluate view intractable also
rr rr c test e letter e e optical description rr rr c letter optical l rr rr rr test optical label unlabeled addition calculate lda hoc unlabele close repeat use follow criterion average likelihood denote test likelihood unlabele percentage time semi strictly log read supervise calculate training improvement supervise optimal l calculate determine average ad hoc supervise column comparison supervise pair likelihood show retain hypothesis average though reject equality optical statistically significant difference test likelihood reference number respective result strict sense carry readily train set turn indicate probably supervision estimate might substantially condition ad hoc concerned provide well bad look ad optical reason unclear seven nine set lda classifier immediate improvement supervision discriminant kind lda many classifier family make equivalent bernoulli study context class consist mixture concave outside supervise seem still appropriate need carry seem though try combine gradient ascent finally try rely interpretation consider generalized concept much broad merely make supervised principle generally certain concavity worked illustration lda acknowledgment de tu david tu I eventually simplification I thank give great decade like anonymous critical input laboratory image technology mail http improvement currently supervise estimate never bad argue upon example prove supervise well counterpart concept estimation principle refer objective estimate supervise explicitly improvement latter experiment improvement contrast discriminant parameter estimation widely far field diverse image quantum communication tool ml develop modern field satisfactory supervise however develop general supervise additional typically unlabele come improvement supervision contrary study theory improvement rather deal improve supervise essentially close estimate obtain supervise close estimate label supervision instance former strict relation classifier readily datum semi resort generally order applicable principle supervise objective account explicit refer treat behave bad supervise benefit conservative possible unlabeled encounter principle main theory core principle supervise section work theory semi lda really regular one employ tackle optimization problem section compare regular semi supervised put somewhat broad raise conclude begin put estimation lda principal early log take contain sample label pair class model consideration ml semi focus attention supervise lda broad review essence ml already consider rao exploit eq labeling estimation supervise parameter unobserve label unlabele maximize fairly sample procedure refer though classifier available label train give newly classifier unlabele iterate convergence initially unlabele remains probably suggest computationally tractable alternative procedure couple decade instance propose learn classifier subsequent self year possibly know semi likelihood treat label nuisance parameter come model likelihood maximization rely classical hard rather posterior thought assignment consider already formulation apply modern overview find seem different way soft assignment datum assign self make major aforementione suffer increase unlabeled behavior cause misspecification e class properly note supervise capable typically display supervised treat label datum might change go idea marginal subsequently density exploit weight author asymptotically semi procedure counterpart asymptotically behavior supervise learner depend set may problem previous choose marginal could performance improvement reflect year different take conceptual put relationship label exploit class prior supervise set benefit arrival adjust dependent well lda improve involve notably total refer semi supervise significant improvement classifier aforementione impose hoc constrained likelihood give maximize original reference reason applicability currently broad suggest find allow solution include label version unlabele instance well suggest solve ingredient augment unlabeled label second ingredient proper original parameter classifier need unlabele rao intractable overcome introduce possibility fractional result well behave classifier put appropriate supervise task priori constraint correct learner benefit may lead motivated method motivate fix soon make work concern technique take write function note ml lda supervise rao treat univariate semi set lda contribution lda finally remark contribution employ decision early widely self inferior contribute scheme currently generally applicable come make devise strict consider similar contain true unlabele upon information result obtain mean seem helpful supervise trivially fulfil take proceed strict improvement argue q likelihood label supervision lda prove construct learner difference incur first introduce give refer precisely vector simplex provide posterior likelihood dependence explicitly indicate side soft hard equation semi supervise express supervise datum enable extent possible deal semi supervised go assume soft labeling product ready general ml estimate maximize maximize lead nature objective take minimizer consequence never look even worst hard labeling expect semi estimate obtain supervision different soft adversarial bad case consider quantitative often happen imagine expect instance ill label nothing gain extreme exact copy semi outperform regular proposal semi explicit specifically briefly subsection demonstrate canonical eq fix fix concave compact invoke minimax allow maximization saddle unique definite follow definite pose parametrization canonical triangular square well invertible cf come back cl leave fix essential merely offset maximizer maximizer lda every average datum weight well supervise solution unique simultaneously e distribution unless empty equal upon prove semi lda vector continuously eq inequality label unlabeled probability pose eq expectation mean always subsection provide know look saddle optimizer know semi maximize equation calculate guarantee linear opposite experiment decrease maximum number addition reach every care quantity determinant covariance high fairly result obtain determine take semi start log raise firstly unseen compare concern remark
introduce abc simulate model intractable abc abc abc let abc model parameter generate decided accept py design possibly multivariate statistic empirical moment approximation operate rather convolution impose rejection use delta form j posterior use introduce simulate summary measure actual rely likelihood sl abc perform mcmc accept acceptance distribution quadratic true vector fit summary likelihood mle fit observe zero search close observe positive conditional embed operator statistic kernel need firstly I canonical map q implicitly transform non thereby nevertheless eliminate mmd abc element rkhs associate definite whenever bound moment w unbounde discrepancy mmd simply distance capture particularly mmd example use kernel rbf laplacian expectation mmd x n embedding measure f learn distribution insight essential measure nonparametric distance mmd distance yy almost summary biased estimate population simulate histogram mean mean shape notably differ true range term euclidean comparable abc rejection abc refer abc b j coarse estimate measure euclidean estimate good second algorithm match insufficient give inaccurate row sufficient histogram sample posterior abc sl posterior black bar posterior obtain close sl difference drastically mean red example inference system population model equation observation denote determine put broad draw vary drastically challenge b sl abc adopt mean peak abc gaussian three ran sa simulate use matlab sa split test set run abc value coarse error fig euclidean vector summary simulate estimate sl operate summary statistic abc attempt summary affect mmd discrepancy measure abc embedding take sufficient simulate world posterior statistic mmd abc widely rkhs capture domain datum adopt ard include adaptively gps save sample worth total abc approximate mmd university college ac uk contribution situation observe intractable simulate base choose summary rule summary incorrect partial paradigm manually mmd observe datum reproduce hilbert statistic scenario effectiveness abc bayesian likelihood abc applications bioinformatics abc posterior distribution interpretable natural phenomenon intractable evaluate integrate likelihood marginal computationally issue readily mcmc evaluate observation actual abc actual observation summary statistic abc partial rather poor difficult quantify summary summary transformation statistic minimum regression least square boost method summary focus may suffice still heavily inspire indirect auxiliary thorough review advantage control aic complexity principle way one exponential family cf sufficient dataset light
underlie opt latter mahalanobis simulate statistic compare projection ks test word multidimensional taking difference function cdf mahalanobis combine mahalanobis constrain root galaxy correlation sigma sigma root component explore keep permutation particle reduce percentile calculate good magnitude radius light two component target early linearly wise distribution calculation intel bridge ghz hour line consistent approximate behave standard deviation posterior fourth refine towards value behavior threshold iteration fast decrease r target sigma sigma sigma model blue line denote parameter parameter configuration approximate forward likelihood intractable combination perform pool particle solution iteratively improve gradually threshold automatic impact particle toy software prediction agreement threshold calibration application distance measure mahalanobis goodness infer configuration find abc reliable input map correlation simulation implementation promise modeling problem acknowledgement thank statistical thank discussion software grant national package website package package development commonly parameter observe various high parameter space hasting hamiltonian affine ensemble rely situation function direct make modeling include observe body semi galaxy weak simulation measurement example highly measurement process year bayesian abc gain attention need systematically explore space distance metric metric threshold data calculation efficiently monte advance sequential monte carlo abc problems instance carlo galaxies ia variant constrain disk formation simultaneously software base abc principle implementation calibration wide simulation software fast make mahalanobis represent refinement carlo control loop calibration method weak difference simulation reference property test error numerous couple purpose simulation crucial discuss principle abc consideration compare gaussian toy simulation conclusion release implementation abc algorithm find appendix set bayes probability derive rely quantify difference simulate sample accept retain approximated specify small abc q expensive available useful summary statistic amount refer sis explore discard inefficient rejection particle gain attention advanced population sequential monte smc sampling construct converge intermediate distribution besides abc likelihood framework therein adopt small abc represent position refer pool typically candidate pool assign iteration threshold pool update approximate pool choice threshold apply perturbation appropriate balance slow expect hand slowly fast often select ratio poor choice preferable sort particle distance typically ratio final approximated poor kullback distance desire proposal maximize improve parameter space particle current constant correct discrepancy population current explanation able increase good consist iid drawn distribution deviation seek mean abc py normalise data simulate standard flat analytic cumulative cdf express expand distribution variance increase abc acceptance precision parameter cost improper draw normal distribution standard define mean deviation percentile pool particle describe eq estimation abc panel depict display sample posterior analytical small beyond change approximated variance line image generator field generate use simulation consistent dark loop rather pixel analyse package produce identify
classification latter allow small somewhat feature condition p fan ab closely study mean normal index generate reduce vector generate replication furthermore mean j nm new sample feature selection threshold classify vector unique choose suggest variance ratio may generate somewhat analyze lead feature proportion feature combination mention single essentially reduce pure guess mention impact weak small combination misclassification see error tend large misclassification failure feature significant see comment guess increase significant improve precision exhibit observe increase grow contribute successful complex misclassification always become enough offset ht indicate class conclusion observe weak feature contribute class independence ignore preferable high acknowledgment foundation partially science foundation nsf grant dms dms start lemma q k l calculus since generally chi calculus logarithm generate calculus show obtain complete x truly obviously independent recall let consider central chi rhs corollary department department mathematic class significant distance class successful interesting counter intuitive classification accurate selection coarse nevertheless strong illustrate misclassification multi context interest classify imagenet http www represent discussion challenge handle dimensional datum name exceed solve require rigorous fan fan case normal selection pure although literature obtain classification therein comprehensive survey high fan ab generalization design statistical mention adjust pairwise hill adjustment support technique class lee lin expand well class affect classification number class hand distinguish thousand human reason class dimensional easier grow even feature sufficiently coarse classification small nevertheless impact phenomenon first attempt investigate rigorously impact accuracy selection class truly really separation carry observe close euclidean select condition require successful finite size finding indicate define without become misclassification error assume euclidean center satisfy follow classification misclassification class bind classification confirm evaluate assumption th infimum rule consider realistic irrelevant zero obvious chi centrality threshold th q theorem minimal feature correctly identify unknown feature j truly per increase weak become contribute effect coarse fine feature one truncate vector rhs result assume condition eq theorems minimal different order asymptotic consider classification fan contrast class define grow exposition separation distance
factorization discuss solve low solution challenge fail problem find negativity etc issue several explore th matrix variable norm appear name include projective atomic norm generalize regularization negativity norm norm many norm choice generally factorize still work several explore factorize q still factorize show idea factorize subject minima significantly unfortunately aside norm result force replace closely problem fact equal search converse minima formulation typically useful optimality local minima regularization application tensor form neural context network neural network single hide induce train globally optimal unit hide produce analyze idea additionally framework sufficient globally purely recall simplify use capital dimension dimension letter dd n r nr give element tensor slice along I slice along matching concatenation dot space image multiple portion gradient number integer general use tensor equal function direction differentiable return motivate family framework sum specifically tensor define slice factor requirement impose positively positively homogeneous k place restriction mapping must positively capture form several mapping q positively homogeneous column slightly cp outer multilinear apply wise training td relu linearity connection utilize complicate consider broad neural architecture hide x x neural relu architecture layer define fed layer positively map neural architecture compatible homogeneity note max positively homogeneous fall imagenet series convolutional pooling layer normalization layer layer take define transformation removal positively note however rely potentially change applicability architecture cast wide framework mapping positively degree requirement non follow positively homogeneous degree essential match homogeneity idea factorization function norm allow regularization place input slice factorize tensor requirement positively semidefinite formally semidefinite positively homogeneous framework variety semidefinite positively commonly compose multiplication power homogeneity combine positively homogeneous pseudo norm function positively popular interest semidefinite constraint also typically positively homogeneous transformation positively equal homogeneity formulation positively homogeneous degree arbitrary mapping additionally element compatible specifically give positively homogeneous x k xu uv scale convex infimum finitely sized dd size infimum u suffer issue early namely due complicated allow purely tool tractable factorize formulation build analysis typically eq factorize example intercept model possibly minimum note impractical optimize even solution would need desire merely next minima slice factorize initialization global purely local begin lemma relevant first verify positively positively degree recall concatenation tensor positively degree x infimum norm fact semidefinite infimum degenerate property infimum complete note homogeneity x x xx trivially factorization yy result property property gx hull equivalent w r dx I combine homogeneity pz pz factorization characterization subgradient dual give conversely associate concept norm solve still limit derivation subgradient characterize w regularization function factorization r gx x gx iw trivially gx I I also gx w x gx gx produce contradiction present local homogeneity local take p rearrange eq take qp rx z combine rearrange result preliminary ready corollary optimization problem begin factorization equality infimum satisfy minimum function subgradient condition condition gradient minimum satisfied optimality leave turn minimum f rx rx homogeneity rearrange take note side definition directional direction get complete result regardless give global local purely function global clearly minimum reach local theorem increase path minimizer must exist r zero generality scale homogeneity construction r recall x k gx theorem exist iteratively reach complete also meta outline global factorization bad case grow bind choice maximum require tb x I minimum define factorize terminate begin challenge alternate guarantee critical minima emphasize minimizer descent strategy guarantee entire space balance homogeneity mapping map conjecture result likely mapping save positive homogeneity match show factorization guarantee regard phenomenon positively assume homogeneous provide counter demonstrate minimum pg gx additionally exist neighborhood sufficiently happen always minimum origin always take arguably situation oppose positively homogeneous mapping depend decrease grow factorization uv decrease scaling factorization degree dependent choice column decrease neural note show neural positively mapping outline partial explanation replace traditional positively homogeneous output allow purely local conclusion sufficiently assumption initialization objective traditional regularization training form tend practice regard critical balancing degree homogeneity regularizer prediction ensure homogeneity significant deep noting limitation current framework state art must previously well however apply optimality reduce function entire limitation possibility future implement architecture advantageous experimental imagenet parallelization operate largely gpu focus mapping believe mapping principle future present wide variety problem analyze tool particular guarantee global minimum factorize tensor factorization range field common vast majority disadvantage associate typically multilinear idea
constructive algorithm analog theorem use line theorem begin analog complete low follow bring eq prove low case upper constructive smoothness role constructive greedy chapter q lemma get allow case argument case notation lemma eq prove choose note correspond low univariate kernel er nonnegative frequency enough imply need adjusted case old eq continue smoothness let case factor low case prove theorem relation class obvious opposite wide well theorem effect small characteristic smoothness prove kolmogorov class differently approximation detail study abuse ball lemma make constructive constructive proof bound typically dyadic depend let coincide mean wider pointed providing constructive small achieve traditionally research go paper approximation progress constructive still progress smoothness result present arbitrarily order right approximation detailed discussion banach constructive provide order respect system paper recent main mixed derivative smoothness constructive interesting history brief history system multivariate class smoothness definition periodic advantage univariate polynomial improve relation uniform constructive method result term respect interesting phenomenon discover establish decay fast bound constructive theorem smoothness time interesting late technique version use reference therein obtain order constructive powerful probabilistic suggest constructive approximation greedy banach chebyshev constructive inequality constructive chebyshev greedy give polynomial integer denote nonnegative exist constructive term polynomial formulate term typical constructive let upper constructive class point denote tn tn r q define proceed define embed control control logarithmic logarithmic scale smoothness recent book eq complete bound begin upper q establish take eq eq complete low proof give proof analog lemma weight one monotonicity prove case corollary combine bound prove follow univariate introduction convenience upper constructive wide build term approximation remainder function later q equivalent q index cardinality large include nonzero sum q continue q error proceed choose frequencie yield prove
form triangular window window adopt decrease incorporate understanding come loss minimize loss arise saddle rather minima gradient allow saddle reason optimal often htb implementation modification else triangular start policy cycle stepsize stepsize lr cycle describe start learn boundary exist example implementation difference number minimum policy cifar create curve rate cycle return start iteration accuracy peak decrease iteration well though policy policy cycle cycle rate drop quickly learn vary linearly minimum maximum vary boundary cosine investigated report briefly triangular follow drop fix learn policy reduce less limited number run investigation run policy need file either schedule factor half iteration number solver reader ask regard subsection question epoch epoch divide file cifar experiment cifar well simplify drop stop cycle train drop resource run addition iteration work stepsize convergence rule reasonable boundary epoch boundary epoch vary linearly reasonably experience idea approximate one repeat exercise twice set epoch maximum plot rate start fall choice first use two show cifar dataset start converge right reasonable rate get rough eventually begin reasonable exercise offer set policy wants slowly show reasonable reduce range iteration epoch maximize slowly use epoch applicable effectiveness method subsection policy cifar imagenet subsection policy policy valid use cifar run gpu gb memory architecture k memory htb htb website assume fairly architecture setting file website website fairly baseline recommendation optimally train policy run max start full file learn example cifar train lr max momentum decay snapshot snapshot triangular solver show policy four cycle rate cycle obtain classification policy setting might benefit policy derive reduce learn accuracy implement value linearly reduce iteration show table benefit compare file policy go substantially obtain method compare use rate dramatically tb cifar triangular cifar cifar exp cifar cifar fixed triangular exp exp triangular exp htb many website architecture parameter file architecture file website baseline avoid difference initialization number imagenet architecture file next minimum boundary figure converge set reasonable fair baseline necessary else apparent accuracy small rate rough drop learn reasonable policy accuracy file net test exp lr start display snapshot snapshot mode versus policy policy list table peak accuracy rate quite show table accuracy stepsize compare run architecture since accuracy around indeed accuracy policy around accuracy policy go finally comparison policy accuracy exponentially counterpart win entry imagenet file fortunately website architecture file use hyper file situation one cnn rather optimal architecture use hyper c max start max next run epoch increase result run cause short divergence accuracy file net display lr base max lr stepsize start weight snapshot snapshot solver gpu running limitation stage fully case peak cycle produce accuracy policy policy good guess versus architecture produce policy present benefit cyclic epoch rate vary near easy additional expense report present several cyclic test give cyclic drop cycle factor reduce learn tool train convolutional explore full plan learn rate perform furthermore recurrent analysis improve understanding acknowledgment author david david suggestion helpful laboratory name rate need find schedule instead rate report near accuracy tune often many describe reasonable epoch addition demonstrate training imagenet train network cnn give face speech car technology train global update loss book recommendation deep architecture say optimize optimize hyper one use worth tune know rate converge slowly rate tb monotonically demonstrate surprising phenomenon fix eliminate need tune near additional benefit see figure reach follow order magnitude drop adaptive
network recurrent expensive well rnn strength allow suffer less vanish hide wise w hyperparameter subsample network module correspond marker module return word notational convenience future module refer subsample embedding vector module basis module question use fact consist weight module input question reason answer module answer importantly fact focus attention fact pass episode summarize memory module fact important later allow pass retrieve fact ask reason iteration retrieve facebook place weight sentence intuitive module episode mechanism fact module episode mechanism question gate initialize practice vector help scalar episode sequence fact endow episode finally episode memory state attention highlight modularity mark fact important facebook end pass fact choose gate supervision pass module end straightforward sequence modeling wish one representation episode final modify replace pass final module go answer module send answer module module initial output softmax end token training cast error sequence gate supervision cross module differentiable deep network backpropagation application several task nlp parse sentiment analysis answer logical lack memory module sentence another chain network kind recurrent successfully speech recognition sentence relevant translation al extremely deep sentence sentence memory map sequence directly memory learn work recent network add natural language answering module cognitive human whose existence store module memory human spatial might argue form relationship spatial responsible module specific relationship module inference module human behavior answer speech sentiment preliminary train use development hyper early backpropagation employ word dropout facebook synthetic ability retrieve test th support fact conjunction support fact compound basic list path agent pass task pass begin training switch subsample module end token gate supervision modify episode gate result pick sentence list table bad long sequence recurrent model input suffer view module significantly require iteratively retrieve fact store slowly incorporate information sequence use sequence position part traditionally every classify speech journal iii split word produce th ccccc al acc state art reach accuracy model gets achieve stanford sentiment classification level fine grain label train result grain negative neutral neutral sentence classify positive train grain label neutral phrase label cnn mc ct lstm grain key et le cnn mc ct al sentiment gate function task grain list well incorporate experiment result lstm sequence special case input small english news use seq seq lstm seq seq lstm list nlp end albeit complex idea model multimodal input acknowledgement discussion style circle fill color color fill draw pos pos draw thick circle color color minimum minimum height natural cast answer language introduce memory input form semantic relevant answer iteration recurrent answer end facebook modeling speech classification sentiment stanford sentiment rely representation require manually answer text ability fact task cast answer like translation task name entity recognition problem sentiment like memory network fashion answer triplet question answer generally answer task require reasoning answer text semantic search reason retrieve fact answer pos tag take jj r est overview full detail generate answer module process raw input video signal nlp input long news article
select relevant show without conversely pls specificity false low avoid positive approximately whereas number relevant sensitivity pls selection process standard compression impact approach situation datum breast cancer level breast cancer work breast gene occur contain one restrict gene differentially condition expression express express center effect pls log observation split resample tune fold validation linearly space perform even prediction method simulation variance assessment h different important regularize usual convergence log severe converge tuning parameter cross validation pls hyper repetition c percentage coordinate component observation score axis compression technique would separate pls tune first method appear component produce discriminate procedure previously correspond pls bit easily combine differently log compression component sufficient separate indicate discriminate properly lead efficient evaluate use component depend method principle estimating size probability coefficient estimate method one expect positive expectation number positive fp relevant determine false positive penalization fp maximum false stable probability subset grid empirical positive stability gene log discover positive positive relevant gene approach compression positive support previously compression suitable dimensional small perform compression ridge iterative least pls logistic particularly throughput sequence consideration logistic properly glm framework appropriate ridge ensure confirm sparse pls pseudo pls moreover combine prediction turn technique pls stability validation eventually provide website f france france curse context number far observation thus partial pls perform combined logistic particular pls improve simulation ensure concern tune classification pls expression thousand concern patient breast implement dimension dimensionality challenge genomic record like gene classical classification method spurious unique call development statistical compression technique project information square pls correlate construct variable base mean contribute penalty constraint variable sparse pls combine introduce step component combination pls reveal advantage high lasso among one pls predictor correlate occur elastic net pls response adaptation sparse pls preliminary discriminant analysis classical pls solution logistic regression classification method achieve via reweighted least especially high difficulty rely step logistic intuitive handle propose pls step idea within generate make classical pls coefficient develop sparse pls compression inspired selection art especially increase hyper pls propose update soon adaptive discuss finish comparative eventually prediction breast year tp use predictor rely newton explicit observation observation interpret pseudo successive weighted issue give completely identifiability concern exist infinite ridge optimization regression replace ridge unique still exist produce ridge predictor suitable pls metric covariance center order intercept pls compression suitable particularly square covariance new continuous denote respective exclude inherent variable pls framework construct sparse weight response weight zero lasso difficult overcome sum concave penalty easily separate instead one stay metric case matrix product argument separate function covariance adjust constraint could lead compression penalization lasso yet classical constraint adapt penalty account predictor high weight successive square pls reduce base issue remain explanation iteration problem contrary optimize another achieve classification issue propose pls constructing choose generally high pls treat one add pls totally sparse summarize follow n pls weighting product replace identity precede pls discriminant pls classifier pls pls denote nevertheless concern pls separation compare art perform eventually reference perform solve maximization norm know elastic computation come pls pls da da package purpose evaluate compression aim crucial performance inspire redundancy within degree relevance predictor redundancy predictor block noise block block choose block response purpose selection tune validation ridge linearly space sparse linearly range especially crucial pls point analysis essential ensure proceed criterion follow low ridge systematically ensure sparse pls redundancy example contrary resp iteration cyclic confirm sparse contrary confirm seem procedure c consider suppose return value become uncertain hyper parameter return validation repetition variability adaptive cross validation ridge parameter fig contrary validation method return one consideration instability cross hand select influence validation precision deviation repetition another cross ridge variable determine
fashion protein similar protein diffusion vote reciprocal protein vote method diffusion capture long range topological local prediction figure advance diffusion capture association protein accurately neighbor majority vote level strength topological integration protein explain capture network effect specific vector fine topological visible single approach canonical vector majority vote function top string functional relation explain diffusion demonstrate plug protein multi problem toolbox representation radial nest five within parameter solely topology score observe improve supplementary diffusion novel biological exploit extend heterogeneous perform jointly optimize feature demonstrate exploit topology molecular network predict molecular demonstrate substantial diffusion accurately encode local topological predict gene vector informative describe protein term topology readily exist future plan improvement example simply straightforward ideal overfitte specie numerous annotate believe hierarchy challenge prediction evolutionary include identify functional module analysis discover term molecular network computer artificial intelligence mit usa mathematics mit computer il execute rna develop throughput htp two hybrid molecular interaction genetic interaction study often incomplete interpret thus functional annotation perfect interact likely type diffusion extensively context biological effectively propagate indirect gene relation gene researcher select distribution method mainly ability topological neighborhood still partially incomplete nature throughput high false principle pca effectively high dimensional reduce linearly project variance predictive machine application effective overfitte little spirit improve improve dimensionality design capture nature diffusion novel framework reduction topological facilitate protein idea topological run molecular node distribution multinomial parametrize node minimizing leibl divergence relative parameterized pca reveal internal explain variance compute extend heterogeneous perform apply predict substantial next capacity heterogeneous genomic resource string train machine node metric gene prediction svms test string annotation remarkably feature machine useful certain node characterize similar association interact rise molecular phenotype find node either select diffusion setting topological lie beyond neighborhood advance diffusion method provide extract information encode compact representation walk network vector describe topological minimize diffusion logistic walk analyze network probability take consideration identify adjacency molecular interaction protein entry eq control influence global topological place great emphasis entry visit current correspond iteration start probability state position suggest protein capture topological association instead simply achieve approach part quality dimensionality original spurious network fact biological incomplete greatly reduction logistic latent node dimensional assign context close direction inner frequently random walk fine topological retrieve function vector allow extend next model optimization take diffusion input find dimensional representation good approximate kl divergence probability guide entropy express objective low dimensional diffusion optimize respect objective use standard newton bfgs almost solution framework novel interaction variety network integration identify gene take weight manner integrate analyze topological confidence genomic bayes confidence network network specific integrate network mix extend integrate perform diffusion node distribution logistic assign encode intrinsic newton bfgs assess representation obtained consider protein network protein interact protein annotation characterize protein predict function topological proximity protein capture similarity protein close protein rank make topological protein similarity two protein representation protein cosine follow distance ten protein assign unlabeled protein unless majority vote cosine protein readily feature machine various exist formulate task machine svm functional annotation train assign protein annotation method annotation protein available via dimensionality integration network support machine able five validation string network annotation remarkably art diffusion string database variety throughput database exclude text prevent protein edge
syntactic implication far provide whereas quite complex perspective briefly full fail confidence occurrence consider provable observation jump nontrivial follow statement wrong explicit proper besides generalize implication partial implication proof construction value attempt generalize rapidly reach difficulty lack property identify turn find connection state implication confidence use connection partial enough use case implication boolean algebraic set partial implication explain partial condition linear programming situation surprising lp merely characterization decision seem follow program big alone attribute discuss section receive semantic simply terminology close analysis call sometimes dataset transaction thus attribute transaction simply subset attribute think attribute transaction subset attribute transaction cover datum set transaction transaction alternative write transaction cover implication set union fully partial implication else specify conditional partial note much symbol logic expression entail implication proper subset without properly number confidence lp follow real program feasible unbounded feasible arbitrarily objective call primal duality exactly infeasible unbounded infeasible feasible optimal describe partial implication start comparison notion implication consider consider start develop discuss otherwise everything strictly implication implication case confidence threshold nothing else say algebraic set implication differ equally well entail confidence occurrence everything course partial solve true interval intuition combine implication discuss incorrect appropriately cover implication implication iy seven simultaneously tight suggest zero state vector involve although elementary build hoc seven inclusion fail value generalization case somewhat subtle point seven proper clause begin move comment seven inclusion far intuitive discover right generalization turn getting discuss section discuss duality interestingly statement generalize merely useful characterize set characterization variant standard tailor apply consequence make dual play want intuition terminology usage implication e transaction cover cover intuitively extent weight give transaction implication mind read follow whenever weight non negative non negative combination say classical necessary sufficient confidence formally component q useful lemma weight implication transaction number time appear complete transaction w x yx yx union mean z z parallel resort duality lead natural lemma check correspond leave min w z ix programming feasibility really characterization prove equivalent prove certainly feasible solution w continuous rational component preserve feasibility objective natural transaction copy feasibility ii direct feasible reference transaction let number alone transaction dual feasibility positivity read non w follow call whenever characterization deal relationship partial implication x denote every smoothly prove v z u z would removed validity obviously contradiction read case fact know x iy I item j wrong low follow state partial implication k iy argue implication I l l I l l characterization theorem look say li easy solve theorem close get say implication homogeneity hold enforce homogeneity either cover implication homogeneity empty implication homogeneity nice homogeneity requirement show exist implication x statement cover cover read iw I put together get every characterize counterpart satisfie classical implication implication homogeneity homogeneity homogeneity else implication homogeneity quite subset fails note decide formula k ready implication equivalent l x k homogeneity iy clear l say index statement unless empty properly inequality fact proper inclusion inclusion lemma straightforward implication follow nothing since trivial assume empty suffice accord inclusion exist fail cover homogeneity rest non assumption prove case course consider violate lx lx iy ensure z also hold hold enforce homogeneity turn key result quite implication trivially nice partial implication nice bit exactly implication homogeneity happen symmetry conversely implication homogeneity recurrent concern implication lemma every implication homogeneity direct application rest hoc implication define confidence implication v z convention take occur negative make numerator point convention function inside comment ensure turn obvious well la un si partial set side reference notation first sort notation main theorem confidence x homogeneity iy iy theorem negative expression follow argue therefore also cover case empty max I enough point direct would state early max definition expression write turn right side hold additionally particular hand inequality hold assume definition max I w side theorem go one critical partial certainly theorem sound case find ab contradiction side big claim expression solution case check theorem say among implication however question fully
noise general lie union video extend via low popular approximate gmm recently develop density information able technique support award grant song h school consider setup measurement form suppose copy signal sketch sketch noisy result gaussian matrix equal copy average introduce view q far expand unbiased bx measurement may let eigenvalue mx ax measurement absolute tr power measurement definite apply let otherwise thus signal condition relate follow therefore nonzero positive semi chebyshev subsequently update sequentially q follow hadamard inequality conditional recall direction correspond eigenvalue eliminate eigenvalue lemma perturbation eigenvalue suppose stop step ideal q note exactly therefore ns else om lm proposition sense greedy sensing model may estimate setup matrix gap recovery sequential compressed sensing acquisition image large sequential nature problem either due fact streaming process compressed sense develop classify graphical exploit distributional signal seminal compressive gmm work general sensing refer consider greedy optimality greedy aim design subsequent mutual condition measurement mutual capture result orthonormal decrease eigenvalue consequence almost always estimate quantify performance sense proxy estimate theoretical include relate entropy measurement characterize covariance establish additional present initialization numerical example good greedy spectral matrix vector quantile chi degree semi definite sequential unknown sequentially goal assume choose goal measurement precision sense eigenvector eigenvalue measurement covariance illustrate dominate performance greedy sense note calculate iteration necessarily reach power k constant trace surrogate calculation matrix easy eigenvector power update take form decomposition update trace become bind signal amount hand hence characterize amount reduction number measurement versus roughly occur upper simplified suppose eigenvalue characterize versus covariance allocation prescribe reach precision establish extra reach precision give recovery level require expression require measurement one full covariance sample
scatter mention voxel lasso voxel probably voxel early region select voxel fold brain although apply diagnosis apply point nonnegative fista thresholding provide natural foundation china grant corollary analysis involve thousand million number regard lasso select diagnosis usually feature stability perturbation explore fuse lasso feature diagnosis incorporate spatial voxel optimize novel variation network nonnegative explore compare analysis sparse gain great statistic absence major performance apply diagnosis disease brain image interpretable instability mean perturbation include bootstrap unstable lasso dimensional result undesirable diagnosis issue selection less study induce structural imaging image brain voxel induce disease correlation prior brain exist label ad positively cognitive disease gray cognitive accordingly enforce name model non fuse fuse lasso enable select measure feature demonstrate model stable worth g although solve diagnosis ad make fused solver feasible regard propose solve generalize fista constrain prove use element post tv solve duality formulation include apply fast precision high sparsity people leverage underlie introduce strong voxel grouping voxel coincide various topology consequently overlap ideal problem adapt fact select necessary fuse successful induce nonnegative select stable positive partially support neither provide minimum tune loss variable assume variable suppose correspond many brain directional feature penalty tend sparse I spatially coherent also nonnegative select unconstrained systematically greatly encourage disease relate apply thousand mainly framework table propose modification scalable explore lagrange deal iterative follow constant kf convex file solve fuse show utilize separability term define denote element solution nonnegative tucker kkt necessary sufficient condition sub objective derivative lagrange complementary condition q variation perspective natural novel flow easily parametric flow inequality duality I proper cone file since rewrite generalize primal equal tv convex define lagrange multipliers derivative dual write k please supplementary file dual omit change illustration flow highlight illustrate flow take minimize cost quadratic flow minimum via include limit minimum via flow recall ij exist decomposition moreover possible devise flow tv efficiently prop prop transform duality flow compare solver node sample cpu ghz efficient diagnosis issue ad normal nc mild cognitive voxel nc classification use q voxel spatial adjacency
outlier significantly different non involve already outlier frequently version crucial recover pca contaminate pursuit construction universal pca assumption moreover convexity np hard pca extent relaxation paper aim reconstruction adopt classical outli observation row outlier introduce square regression ix I small error low default see estimator datum corrupt free minimization lead formulation separate method lead quite large error component state direct minimization desirable annealing approach regularization trivial section minimization convex manifold material identity nm concave employ technique linearization fix r u I pointwise minimum sum linearization concave tm u nm set eq I u ss u I coordinate concave matrix orthonormal symmetric semidefinite factor using svd singular uv uv tolerance output robust center pcs nu km I point objective reconstruction finally monotonic k u u terminate compute immediately small tie break minimizer k I one u I objective smooth neither construct data use handwritten digits digits mix proportion handwritten digits equal mix fig exclude value ground default simultaneously robust influence additional one supplementary material cc cc ccccc person visible end reconstruction person robust robust interesting pca sequence slight change whereas foreground model outlier pca background frame foreground performance water surface move background dataset feasible background algorithm optimize hand bottom crucial reconstruction cf foreground background mistake supplementary background water set present fast initialization sufficient runtime parameter r present reconstruction efficient default setting perform similar solve h partially partially extraction system cccc material background water surface object person frame person outlier cccc cccc water surface set fig change foreground rescale maximum note cccc frame water surface similar red frame person leave scene high reduce dataset figure please outlier compare scene foreground background foreground segment separately center main experiment method cccc cccc cccc reconstruction perfectly oppose large frame know principal affect put directly avoid often pursuit fast probably tool exploratory analysis reduction g see datum fact strongly influence outlier indeed one pc drastically robust pc receive lot attention recently pca base one pc perform however positive affine informally arbitrarily still upper inverse projection deviation lead non smooth search disadvantage compute technique lead poor pc vision
precise error condition motivate minimize u resolve label fraction coarse close need analyze lemma suppose regularity third minimize pose sdp clear aim minimize I u v constraints argument prove lemma equivalent theorem begin illustrative example negative matrix depend result tell label minimize illustrate matrix question assumption final happen high whether sophisticated acknowledgement order modification omit generalize suppose iid smoothness smooth follow sense point probability regularity interior invertible namely relate fisher regularity first l satisfy q tell wise combination write satisfied lemma hold follow u l side eq pt exercise remark example well rigorously characterize pac agnostic pac shift provide widely popular linear class conditional upper match sufficient estimation sample pac agnostic goal learn belong shift attention estimation mle active satisfied widely model multiclass field active mle statistic class asymptotic statistic label machine learn special involve consistency full generality goal minimize log estimate sample query unlike towards except low perhaps round likelihood either high implie observe classifier sufficient optimal classifier class pac generalization search style inconsistent case disagreement confidence active passive agnostic gain requirement agnostic active set classification kind random previous active regression algorithm fit increasingly refine partition refine complexity result exponential apply general provide selective decide whether label mostly estimation variate suggest select sample regression variate notion fisher information trace directly optimize consistency regression base work provide promise consistency algorithm apply guarantee moreover unlike single interaction single sufficient order begin pool example draw belong label example give py generate also abuse notation define goal label time matrix negative learning number brief generalized use hessian label solve sdp refer behind well respect essentially I minimize finer condition respect formally step case unnecessary skip directly step regularity quantify standard study regularity interior ix exist neighborhood lx extra essentially vast model lemma mild regularity estimate rate right main proof support l estimate follow
estimate grid mc correctly identify mc trial music considerably low snr db music fail figure note rate music localization source whereas guess rd set difference low music definition generalize huber sparse sparse unknown signal careful characterization huber devise simultaneous conventional refer heavy yet negligible usefulness localization application sensor array single measurement I unobserve row unknown signal q reduce row non non ij lead computational reconstruction accuracy recovery matrix level eeg arrival source process cs algorithm guarantee suitable noise condition meet heavy generalize huber huber value regression generalize huber estimate scale robust require huber matrix error simultaneously particularly obtain estimate challenge ill elementary huber devise problem offer outline background robust recovery section huber huber localization cardinality resp index column hermitian transpose row f function argument equip usual hermitian trace entry denote nonzero row equivalent statement operator possess set refer version index unchanged row suppose circular distribution density scale reasonable square l scale factor minimization l small residual imply even influence least use huber require difficult involve compute special start proper loss symmetric circular fix jointly huber elegant devise huber generalize huber scaling preserve property bound real derivative calculation minimizer minimum fisher huber behind sensible choice residual simulation aim pursuit due objective huber loss greedy recall offer estimate onto huber criterion iteration update signal matrix stepsize refer pseudo huber function divide build stepsize compute stepsize stepsize tune simply criterion n stepsize need adaptively control minimizer ascent simple minimizer solution fp huber huber case fp huber loss minimizer find fp fp iterate previous word consist receive wave point time array weight linear kt mt distinct represent known localization parametrize source cast recovery overcomplete source location measurement
block belief partly efficient train g cd present step incorporate deep propagation bp message two drawback cs continuous distribution necessity iterate message amp amp cs especially forward observation unknown corrupt zero subscript notation refer matrix order amp moment mmse contrast utilize inverse give estimate factorize posterior read calculate p ix refer seen know interested regularizer solution inverse within probabilistic amp convexity induce utilize gauss bernoulli gb ip bernoulli draw accord expression gb gb split normal control informative measurement infer course truly account signal truly sparse merely support instead specific site support identically refer reflect partition sub term support write mean variance gb nice mode coefficient amp amp support e coefficient existence dependency natural model support binary rbm rbm rbm train boltzmann restrict boltzmann bipartite physics joint visible layer rbm rbm coefficient hide sequel connection simplification rbm field first second order give energy hamiltonian free minimized simple I field within energy rbm equally visible hide factorization rbm visible site give fix variable eq equation line assume nmf find site activation literature use free tool nmf approach correlation situation make popular one way refer sake statistical recognize expansion constant show bethe densely proceed visible repeat eqs mean eqs action tool use rbm approximated solution equilibrium eqs eqs iterate possible arrive field balance enter reduce assume magnitude scope amp perform inference accord factor depict utilize rbm coefficient classical form amount bias visible variable amp rbm energy effect rbm amp influence hide visible thus influence visible respect within sigmoid nmf eq right represent observational text direct amp influence construct rbm prior term dependencie rbm unit successively via specify attempt persistent throughout amp force undesirable minima alg rbm factorization post value rbm factorization current smooth rbm sufficient post play role enter state minima observation side prior side rbm carry act show efficacy prior amp handwritten digits digit value binary support handwritten value rbm mnist set divergence cd epoch additionally impose decay draw distribution linear projection subsequently utilize digit comparison mnist reconstruction digits percentage successfully measurement visual reconstruction digits row correspond reconstruction amp gb propose nmf rbm factorization approach factorization column represent approach last digits top bottom amp amp empirically pixel expect least gb properly correspond rbm zero amp rbm amp rbm approach version nmf post test strict requirement amp desire number rbm epoch epoch use unit fig percentage recover successful reconstruction easily prior version amp gb amp gb upon rbm support support rbm gb demonstrate correlation amp cs factorization provide factorization matrix iteration rbm achievable oracle percentage amp rbm rbm exactly rbm order rbm attain necessary increase show rbms stack improve upon scalability inclusion rbm factorization burden reconstruction proportion require computationally amp rbm rbm support properly provide cs reconstruction superior empirical assumption
unfortunately guarantee rank critical point vector theorem exist since follow truncate expansion second assume eq rank nu u entry indeed feasible critical critical rest without increase assumption proposition correspond kkt partial cover strong critical critical similarly order bring point resp point critical orthogonal span dimension iff fx xx xx interior order critical second face restrict put latter result linear equality characterize result either globally rank dimension suboptimal strongly concave fx order critical point kkt convex unique optimizer exclude comment optima face global optima optima optima proper mild set refine leverage hessian assume order relative interior rank rank eigenvalue theorem critical mu np tangent semidefinite hand orthonormal space span vector span w follow denote likewise cauchy determine new add admit version nonzero raise q combine meaningful linear intuition critical optimal produce latter summarize fix global optimizer uniformly face almost answer face theorem motivation upper face attain inequality cover impose denote select independent pick iff linearly always indice select row slice index row see define py e ij ki subset constraint linear least constraint count use matrix expand term slice c c te furthermore expand namely k q attain make many combination confirm act conclude argue tight indeed repeat slice contribute yy pp pd combine ensure point kkt critical kkt point critical point apply conclude cut sdp second critical certainly solve practice empirically sufficient bad theorem provide require increase rank proceed move operator xx definition eigenvector bring cost compute simple kkt critical warm start iterate way yield kkt grow call list hope take integer I iy assume availability procedure inside zero saddle able make eigenvector saddle eigenvalue nonnegative return kkt kkt allow take discussion unlikely event terminate critical point optimize proposition compute critical kkt otherwise decrease iterate need cost decrease rank never exceed terminate admit kkt procedure toolbox descent toward iterate convergence unlikely impossible happen unclear ideally modify global order polynomial number step knowledge riemannian set light recent modification trust method polynomial critical region seem reasonable expect interesting drop return exactly point numerically exceed say future kkt I bound eigenvalue project algorithm thin retain slice return optimizer hope help bad remark remarkably low sdp follow sdp admissible strong condition solve sdp illustrative compete sdp orthogonal synchronization matrix relative transformation benchmark random rotation level independent solution equivalent admit phenomenon partly take solver run guess return force version top reveal empirically weak node node node problem solution return numerically close grow merely dominant synchronization gaussian naturally measurement square alternative minimize error orthogonal minimizer ij similar spirit relaxation subspace round program regime non noiseless tool nonsmooth prove typically higher formalize restrictive x ij ij kkt otherwise ij contradiction kkt block positive semidefinite show w ij contradiction apply particular thus convex small case even though two differ constant difference speak cost nonsmooth concave minimize smoothed note coincide affine function large explicitly appear consider norm minimizing appear kkt guarantee point strong concavity kkt reveal kkt empirically excellent quality kkt increase warm start rather convex pay lack purpose global orthogonal synchronization fact permutation modify synchronization notably arise computer vision permutation size perfect achieve perfect recovery phenomenon exactly assumption even cost node huber permutation vary horizontal axis square vertical close perfect get remarkably huber cost accommodate outlier still fast unfortunately grind global concave hand come guarantee solve start warm previous start identify appear optimizer optimize cost rapidly robust synchronization solver cost reweighted least successful preliminary work call compute relaxation orthonormal reduce riemannian effectively increase involved investigation riemannian portion usefulness bound smooth manifold cover address sufficiently kkt offer nonconvex kkt kkt give critical could compute second possibly spurious admit see sufficient perhaps investigate via regard cost admit second point acknowledgment author thank conduct research support research paris et synchronization cycle measurement convention condition exhibit formula attention rotation orthogonal reveal explicit formula check carry let dm semidefinite perfectly set inconsistent guess criterion spread evenly th root build part recurrence q ij j semidefinite unitary diagonal unitary operate use unitary proof general measurement none condition surely almost face minimal part er face select yy statistically vector vector surely recurrence almost statistically x ignore hold proper yy px q nontrivial combination scale density mass zero rgb line def grid step def pt anchor east proposition corollary example remark propose solve optimization form constraint block involve phase rotation orthonormal combinatorial cut exploit fact admit optimization convex characterize reveal kkt semidefinite magnitude fast well code consider identity far focus available application product indicate sign allow product correspond orthogonal relative rotation stack matrix correspond product define stack concern twice continuously symmetric negative datum encode induce invariance action group orthogonal cover max semidefinite block rank factor optimize cut linear constraint motivation relaxation linear dual may project initial discussion projection pay dimensional solution sdp name iterate search form matrix quickly operation seem admit intersection semidefinite cone affine subspace geometry solver phenomenon apply lagrangian sdp powerful bring great insight want matter guarantee nonlinear penalization solution like nonlinear build upon observe certain elegant put algorithmic true smooth replace nonlinear algorithm riemannian address invariance optimize equivalence full rank advantage geometry become difficult well justified increase practice often target quite theory nontrivial lift describe riemannian geometry frame toolbox critical riemannian gradient vanish critical critical unstable orthogonal determinant relax nice furthermore riemannian iterate iterate numerical tucker kkt condition convex sufficient kkt dimensional second order reveal point warm computation allow reveal terminate formalize rest improve avoid geometry tie reference cover grow view theoretical investigation call inspection particular face effect describe always tight essentially face generalize description concave critical reveal kkt sufficient mild critical point result convex paper efficiency solve synchronization rotation permutation note simplify exposition easily accommodate size development go complex relaxation numerous application consist estimating group measurement ratio model seminal class maximize proportional laplacian application cut interaction effect final matrix structure sphere appear notably fundamental solve determinant exclude pairwise rotation measurement model come camera sensor localization rotation separately determinant pick connected relaxation cost orthogonal matrix notably nonsmooth propose robust geometry equality count tangent sense affect tangent restriction riemannian embed thus problem euclidean tangent riemannian hessian directional use expression riemannian away next extra require achieve slice thin slice close consequently cost define compact matrix hope recover block proposition show consideration equivalent hull rank handle optimizing solve probably hull submatrix block semidefinite singular orthogonal conversely multiply notice orthogonal onto span remain span orthogonal may decompose dimension face face line relative face relative form interior call exist linear exist unique maximizer compact hull extreme point notably arise concave attain minimum construction proof admit optimizer value soon np hard figure admissible matrix lying segment admissible extreme construction full extreme support many meaning rank extreme singular proposition consider rank schwarz equality attain max admit extreme latter let nonnegative x though prescribe norm critical kkt critical saddle point kkt explicit critical point kkt bring ingredient proof kkt
denote visible unit respectively energy rbm specify visible bias statistical visible rbm visible boltzmann physics rbm rbm interpret free visible unit refer important rbm compute visible unit rbm bipartite however calculating computationally scale unit train rbm ascent carlo rbm computational nevertheless draw derivative sampling cd present physics inspire rbm free refer reader review apply rbm start possess binary boltzmann usually base multiply boltzmann inverse temperature apply write energy newly introduce external field recover transform maximize auxiliary variable inverse dd average configuration boltzmann free transform minimize allow temperature expansion carefully temperature arbitrary accounting obtain correspond entropy interact term I field order order systematic correction return rbm remainder theoretical denote hidden recover lastly visible sigmoid well stationarity condition utilize term consistency relation couple define satisfying field recall obtain weak coupling expansion couple system equation spirit propagation iterative rigorously demonstrate spin remain relation index careful minimizing provide running eqs converge present iteration second term estimation log rbm note utilize cd gradient ascent derivatives eqs obtain procedure visible hide bias merely unit respectively weak expansion poorly rbm recover calculate take greatly third easy include fact rbm admit triangle sum pair triplet exclude bipartite rbm couple fourth require utilize rbm number separate cd dataset handwritten digit comprise mnist rescale pixel construct consist foreground scene background represent image rbm visible study rbms unit adopt mini batch learn point mnist test present implementation mf great complexity investigate consistency converge instead iterate similar cd persistent maintain persistent epoch epoch therefore converge persistent iteration self mf persistent algorithm use comparison lastly evaluate rbm rescale raw design training stack rbm unit rbm operating term comparison train rbm cd rather training free neither momentum implementation however decay necessity rely compare free training average independent standard likelihood divergence log demonstrate advantage computational auto em mf generate rbm handwritten perhaps preferable certainly preferable mf logistic epoch comparison perform black raw refer training code pseudo log epochs implementation cd procedure find show panel fig fig ascent pseudo log epoch interestingly persistent epoch contradict common approximation gradient represent explain persistent epoch resolve informative examine rbm fig choose epoch p display rbm digit yet generate particle particle identifiable digits qualitative improvement digit visually particle possible rbm log confirm cd persistent iteration preferable fast moreover demonstrate surprising consist rbm training perspective rbm deterministic binary visible unit hide unit learn usefulness task unit map toolbox order place emphasis quality rbm training rbms training cd training yield cd persistent although decrease likelihood train datum raw marginally test imply real successful consequently treat interestingly persistent observe deterministic cd present rbms field I design rbms show I bring practical deterministic rbm cd algorithms file implementation online rbm monitoring throughout real show gauss rbms rbms field stack rbms jointly separately boltzmann deep generalize boltzmann hide difficulty expansion start
vector match sequentially accept word historical rnn reach last word natural information rnn rnn long rnn perform click manually label insufficient feedback limited click provide similarity exploit objective lstm query respectively learn sentence lstm rnn activation different input process learn lstm rnn effectively keyword lstm indeed keyword lstm read activation word encode entire reason application web query document cosine web task method word sentence sentiment performance design capture fine grain sentence recurrent develop retrieval also treat explicitly gram window pooling nlp capture dependency belong develop speech treat word encodes differ leave encode semantic embed vector reach view feature representation english convert lstm rnn lstm rnn sentence maximize probability predict sentence word summation bi pair plus linearity lstm letter gram keep encoder decoder jointly align translate rnns concept attention decoder discuss study train model sentence embed description ahead another rnn robustness example web ranking come equally salient limited memory ii word query robustness lstm rnn lstm serious use size lead retrieval advantage keyword require capture contextual recurrent embed rnn lstm rnn networks temporal rnn dense sequentially word sentence representation figure th code hashing letter gram recurrent word semantic bag representation whole use slide capture layer feature rnn neither sized pooling recurrence sentence express recurrent word architecture traditional convert letter supervision sentence detail although sentence principled manner dependency vanish neuron originally lstm forget gate connection fig gate forget gate gate vector connection connection connection vector consider forward pass lstm rnn model hadamard element good representation input challenge practice since collect manually semantic nevertheless widely web massive candidate usually record binary semantic query supervision signal embed achieve click engine complete please section adopt cosine similarity vector sentence length click sentence document subscript sentence lstm take corresponding want document rnn include query corpus document denote sample expression logistic accuracy measure cosine large help train rate momentum scheduling equation please refer nesterov momentum step rnn necessary derivation accelerate mini batch train incremental update back method accelerate convergence nesterov find rnn yet rnn model update update train lstm present scheduling gradient maximum length model document set number lstm rnn minibatch compute compute l l understand lstm perform visualization tool analyze question dependency critical embed semantic iii lstm rnn question lstm web search engine description query document pair supervision lstm relevance follow query sample try around query query collect web query popular retrieval finally pair query text retain comprehensive train lstm rnn visualize behavior output lstm reveal lstm keyword detect simplicity query keyword extract list interestingly cell keyword specific cell cell mainly keyword c cell cell cell cell cell cell cell al much make community infection infection infection health pressure cell cell cell cell cell hash replacement replacement infection infection health high pressure high pressure embed retrieval engine specifically rnn query vector similarity query document show standard mean discount ndcg performance rnn lstm rate train baseline evaluate performance fair lstm train include retrieval ir sake bm state document base match baseline ir latent semantic side give report base dirichlet smoothing lstm rnn significantly exceed good baseline ndcg statistically point sec come vector hide rnn lstm parameter number c ndcg ndcg ndcg bm rnn rnn comparison train rnn fig conclude rnn optimize please number epoch address short information sentence semantic evolve input detect due limitation human label signal user click engine performing show allocate finding example concrete sentence important web show outperform significantly work early deep model effectiveness extend include develop directional version propose embed processing sentence answer information structure cost present derivation gradient supplementary material recurrent subscript th subscript model please eq rnn cost nesterov rnn architecture omit simplicity parameter cell lstm rnn subsection subsequent lstm rnn connection subscript input eq gate eq q connection eq forget gate forget gate bias value connection bias q back material gate visible gate word word document value color title document good match reveal rnn sentence embed rnn train dataset document pair relevance generate relevance bad excellent rate assign assign lstm please score mean score assign generate neuron cell rnn rnn active neuron result query interesting sometimes assign assign neuron rnn c lstm rnn number neuron example rate human assign rnn lstm generate lstm neurons rnn rnn lstm rnn assign neuron neuron neuron table assign lstm rnn assign rnn c c c cell lstm derivation rnn lstm rnn subscript divide rule rule step backpropagation back use procedure follow rnn architecture eq diag entry entry substitute derivation gate gate solution therefore resolve bias equation forget gate derivation gate forget gate connection update forget gate substitute q gradient equation connection example lstm dataset document activation gate gate cell cell document fig lstm rnn three semantic cell state evolve gate cell word document lstm blue color try similarity query gate state valuable context information store
local minima dot act signature add represent pair local minima lastly adjust width edge strength cell describe paragraph evaluation minima discretized denote near neighbor ascent method fit multidimensional scaling mode minima denote plot connect proportional approximation consistency stability function nonparametric density like panel useful density estimate density work surrogate rather first estimator estimate square third difference define satisfie omit signature converge apply derivative rate also derive signature interested estimator summarize signature density region may unbounded kde majority cell pre drive case minima distribution minimum except move boundary minima unknown moreover statistical establish essentially minimize look good piecewise cell within cell good linear predictor cell except covariate finite moment assume use sufficiently critical counterpart regression difference follow show consistent estimate smoothed smoothing estimate cell assume hold eq regression via focus region covariate occur frequently put weight term version aim look good seek many similarity pilot find mode original lot effort come clear population quantity converge pilot construct pilot estimate theorem quantity test difference signature within cell density want simultaneously interested estimate strong reject quantile deviation control type control signature visualize approach provide use knowledge cell visualize algorithm compute region cell ratio every pair cell multidimensional center chart j adjust bootstrappe quantile visualization affect preserve visualize chart provide significantly cell connect introduce datum splitting describe cell visualize visualization slightly half certain direction cell alternative conduct energy iid useful goodness fit name recommend two sample version energy numerically use value quickly introduce testing consist split e kde cells energy cell energy reject correction provide use along twice test cell since density randomly split find second mse consider flow start whenever away close flow able bind flow move infinitely value eigenvalue gradient minimal absolute eigenvalue evaluate change whenever must pick sufficiently constraint minimal follow near remainder use bound flow lemma solid mode box minimum empty dot line gradient flow region equal flow within flow line close stop local complete eq eq region around fact critical boundary possible region note lemma link mf derivative constant condition basically extend bx bx theorem condition replace constant lemma x project gradient notation theorem derivative let near dx turn proof rate hausdorff critical distance constant depend actually small whenever point vice versa j small intersect obvious intersect bm f distance attain hausdorff project use connect segment normal space slightly distance hausdorff contradiction x sufficiently hold bind rand index mode specify small thus mode estimator x ab mode cluster boundary pair definition rand rand index volume probability vc vc process theory vc theory mode local generality sufficiently thus whenever equation note finite estimate local mode field far nx set note dx source different second level theorem part theorem assumption absolute eigenvalue prove second assertion assertion theorem version connectivity connectivity mode recall local eq convenience within region thus ji ij ji show convergence recall point define q within least majority set uniformly consistent transformation q q compactly within around denominator comes put equation occur take eq prove consistently extend prove define consider gain consistency far analytic proved split part agree second remain put matrix nonparametric regression solution one mode cell conclude prove convergent part region estimation part region estimation translate complete assertion regression first assertion prove assertion equation lastly k corollary give nonparametric high complex problem smooth define complex consist call intersection maxima minima generalization function roughly speak piecewise monotonic shift certain show visualize representation wish visualization circle cell large density latter dataset density visualize difference green circle neighborhood denote red chart ratio region complex estimate theory develop three two smooth boundary maxima regularity hausdorff satisfy mode let kernel boundaries rand kde use sufficiently fx visualization show function see test two information test application include vision topological previous stability pointwise sufficient prove regression visualization code use paper definition start simple maximum maxima plot intersection call definition derivative critical sign critical partition distinct minima call call degenerate critical function k c c ascent flow move along individual correspond point ascent converge flow element theory manifold manifold flow point descent precisely gradient start flow share flow limit function share dimension consist manifold thick curve thin thick thick curve blue manifold intersect see assume g collection subset function call manifold consist cell cell give manifold manifold use manifold fit individually cell empty intersection manifold measure boundary define boundary derivative eq measure difference function set hausdorff denote manifold show sufficiently derive collection manifold embed every span space simplicity column depend abuse notation eq space manifold gradient flow eigenvalue matrix scalar vx require flow move mode flow move derivative imply along behave behave flow move flow boundary thus minima manifold let f bound boundary manifold sufficiently close two st make manifold ascent manifold define analogous quantity consider vx result manifold imply nonparametric g manifold manifold density estimate regularity manifold plug regression regression condition consistency mode mode cluster mean shift mode use manifold point region describe denote boundary boundary boundary symmetric kernel exist l kde widely essentially norm function kde boundaries mode sufficiently small simply combine kde rate decompose nh estimating sense mode density way quantify rand partitions kde namely rand index versus rand rand adjust rand rand mode basically incorrectly mode nh asymptotically application plug cluster kde version level set level distance infinite cluster level distance design connectivity cluster mode clustering plug estimate kde estimate kde equation
simplify computed approach powerful kullback leibler measure demonstrate modeling conceptually link addition reduce unbiased information information aic widely applicable unfortunately contain nearly fisher greatly complexity failure analysis frequentist criterion account depend analogy aic applicable change frequentist test already canonical point interesting relation newly derive find approach fundamentally understand level bic suitable segmentation datum model write role partition place along axis red line represent minimum datum dot red respective change bottom point dash signal observation temporal shall q true cross entropy entropy expectation understand distribution change index mode fundamentally shall regular typically consider model zero consequence couple n define determination index change index transition binary study e g initialize sequentially greedy step say nest successive exist previous always binary process show panel statistically state mle differ binary explicitly selection entropy eqn estimator estimator bias phenomenon add reduce reduced parameter estimator cross q regular criterion aic generally approach complexity criterion frequentist use distribution frequentist parameterization small approximate direct eqn tractable complexity compute observe denote consider new parameter new essentially harmonic procedure aic model compute evaluate difference series complexity remain question first complexity compute state break information information per information constant kronecker delta analogy taylor expand parameterization fisher refer th parameterization eqn expansion variable information rescale coordinate interpret unbiased mle rhs forward intuitive mean dimension complexity complexity complexity binary eqn convention point segmentation ar overlap partition mle nest partition eqn since convenient brownian bridge walk brownian algebra bridge brownian step place relation disagreement illustrate bic bic justify observation bic always due clear fig constitute much large produce complexity complexity problem result twice complexity since circumstance intuitive pick change segment must reflect add must observe background discuss significance state occur significance complexity eqn identifiable change predict change aic predict complexity include slope exactly predict significantly aic fails terminate bic initially match simulate generate four family nest model fit plot corner eight true change minimize information red entropy plot split entropy green addition increase cross entropy green red continue compute panel complexity observation aic bic estimate aic nan hypothesis partition parameter test partition divide test statistic statistic expand analogy eqn approximation derivation frequentist statistic accept alternative equal interpret frequentist statistic determine also critical statistic base optimize large approach extensively apply four cell molecular paper information change point use criterion parameterization advantage frequentist analysis observation unnecessary develop hoc approach confidence level automatically information prior like bridge analysis widely author design perform global segmentation min information implicitly segmentation min plus iii terminate implicitly segmentation correspond positive acceptance plot false interval dimension analogous false rate false entire partition eqn relative bic complexity use change slowly normally iterate see
datum set noise drive analytic sketch mse depend moreover approximate diffusion step verify eq euler ask complexity euler eq constitutes compare euler reduce algorithm expect approximate reasonable q ensure turn satisfy regression model assume model fixed put subsequently rwm run iteration effective inferior drop mathematical expansion power size identify unbiased construct match euler asymptotic bias decay rate complete extensive toy analytic moment quantification error result batch ergodic function concentrate posterior mse express derive recurrence equation recurrent plug result repeat sum explicitly geometric sum q equation recurrence derive express term order elementary newton identity term compute e x newton identity similarly nn ix n nn jx lx knn nn p nn nn sketch derivation mse take analytic expression illustrate calculate e e n n nn deriving require mse es toy similarly explicit expression file cm proposition remark section large inform proposal usually whole propose gradient langevin subset accept reject sequence decrease mathematical central decrease zero bias size stochastic modify remove obtain obtain toy study euler datum markov langevin dynamics big fix many molecular expectation differential equation sde run scheme approximate dynamic one average finite use proposal algorithm machine modelling informed proposal require sde standard brownian appropriate dynamic ergodic arise approach would evaluation langevin dynamic generating subset q subset algorithm appear front unbiased limit behave investigate satisfie decay asymptotically achieve asymptotic size gradient hamiltonian monte decrease point analyse particular dynamic ergodic numerical measure apart invariant remain question arise lie respect tackle paper particular generalization power explicitly contribution modification asymptotically euler modification bias bound finite rhs extend main relate establish existence nice poisson decrease rate finding confirm numerical study toy bias explicit connection confirm equation match analytic expression toy importantly allow significant study mean square average second comparison euler quantify mse quantify mse useful proceed study regression observe behaviour weak apply finite long time behave euler discuss toy model section term acknowledge thank mi section preliminary backward section global expansion ergodic average invariant condition dirac adjoint generator moreover backward kolmogorov equation taylor rigorous taylor integer follow assume bound derivative fact assumption enough derivative growth regularity bound numerical deterministic assumption satisfy reasonable derivative order assume deterministic initial taylor series form smoothly depend assume coincide weak immediately integer remainder study invariant error weak time time numerical sequel ergodic assumption imply ergodic compact question though condition ergodic answer case relate drift behaviour euler solution ergodic theorem derive euler infer particularly equation generate step equation follow satisfy one ergodic average smooth ergodic deterministic similar difference euler explicit expression invariant key long say average discrepancy true time respect measure choose reveal formula hand case process study light cost calculate euler method likelihood imply lead corresponding presentation illustrate main calculation dimensional calculation q notation expand expectation respect variable obtain see expression leading contain error give theorem euler precisely make term see section introduce extra appear relate average like imply error front time way achieve time amount datum available since amount introduce modification calculation imply ergodic euler calculation appendix form simple equation contribution analytic expression subsampling variance extension average methodology adapt poisson pde transpose poisson equation sde average ds rhs control derive ergodic elaborate reader therein euler taylor expansion express summing divide remainder rise state derive presentation compact argument sde read unbiased diffusion result sufficient diffusion theorem believe scope notation instead drift write taylor yield sx yield h derivative drift suppose fix advantageous step mse agree decrease equation confirm toy poisson term assumption infinity twice satisfie v lyapunov sde e exponent subsample however even strong toy model simple posterior eq equation numerical read generate replacement toy illustrate clear theory obtain analytic moreover confirm toy mse derive section correct computational effort toy analytic match amenable expectation limit capture investigate limit law combine eq fact subset limit compare error coincide expression particular compare euler replace fact toy model simple integration section calculation discussion modify obtain euler point n rh superior outperform step euler evaluate observation indicate dependence bias asymptotic take step term subsampling form choice stay contrast consideration subsampling also regime small recursion letting subset see rhs restriction desirable need increase sde section bias becomes euler additional regime efficient computational efficiency accuracy far investigate variance study calculate
beneficial bootstrapping scenario report measure top reliably use overall predict miss kb recover entity type extend kb completion kb completion treat unobserve kb negative unobserved kb negative example positive largely method unobserve object systematically negative sampling snapshot kb kb kb type construct two set entity set precisely number negative entity type pair scoring type fact margin control experiment sampling entity fix global relationship exist map entity represent entity feature entity embed propose algorithm use scoring se te optimize eq modify handle vector refer ensure optimum certain adopt give update vector te te te te te initialize randomly ei te se entities projection se l column describe detail embed expressive model objective relation extraction entity type cast extraction entity ne objective entity model motivate make neither foundation previously method type min positive perform manual evaluation method fact data type statistic add recent snapshot effect entity example datum challenging feature description type description w w ne global objective ne ne nt model g automatic evaluation aspect system empirically perform final boolean wikipedia text map try yield improvement ne objective use show achieve score classifier among ne ne perform type expensive result show expect type despite popularity embed nlp verify correctness evaluation perform manual show evaluation kb incomplete result indicate automatic manual manually miss effectiveness highly type performs compare frequent bootstrapping technique supervision method would find prediction use linguistic entity word music frequently l g entity type focus entity type level develop kb kb new york article method sentence use entity wikipedia article perform kb completion relation extraction majority infer within kb kb train different suit comprise metric evaluate miss verify automatic publicly experimental objective produce baseline method future plan information link human get conduct microsoft computer science edu microsoft microsoft usa microsoft com knowledge kb completion relation extraction entity kb fundamental task kb little automatic completion information kb external wikipedia individual train consider consistently prediction baseline manual evaluation base correctness automatic evaluation miss write basis community contain fact drawback incomplete fact usefulness answer lead complete completion subproblem completion infer entity kb completion entity nlp extraction entity parse question answer add entity relation importance little publicly dataset infer entity kb dataset use snapshot kb could potentially evaluate ability already two kb predict add snapshot enable realistic challenging prediction compute ideal treat type equally consider measure prediction measure within type example entity side perform well global metric entity global combines type side low dimensional produce ranking negative side reliably confident entity candidate base summarize develop comprise metric kb miss relation evaluate fact snapshot miss entry unknown potentially use evaluation evaluate ability predict fact kb drawback snapshot early treatment snapshot later newly snapshot fact miss kb snapshot fact test snapshot subtle fact contain predict newly add fact hard enable realistic evaluation kb snapshot relation automatically kb fact kb automatic characteristic snapshot advantageous snapshot snapshot wikipedia fact snapshot treat unobserved instance unobserve instance snapshot example training fact newly add snapshot
kp long analytically multiple method stochastic deterministic counterpart inside partition sep fig protocol different approximate kf kp computation almost copy like uncertainty cavity distribution kf compute cavity f f cavity q kf sep keep computation sep consumption sep propose hidden factor hide variable eq sep next approximate need maintain point cavity n n involve retain also globally share piece sep accordingly assume k memory consumption memory sep year sep test gb ram huge test sep sampling probit probit unit sigmoid make moment analytically subset sep almost ep indistinguishable reduce nm mb mb mb protein mb year extension vi relationship introduction power alpha alpha q ep alpha change alpha projection return alpha alpha difficult motivate practical alternative power ep converge however equivalence apply sense stochastic spirit keep current discuss connection variational natural natural n another approximate compute form evidence zero recover current n reader local update global parameter typical return well imply factor power alpha divergence recover change notice dataset q interpret impractical like inner move n use technique outer average answer optimisation inner local optimisation alpha divergence inconsistent vi condition nature point average ep q fix
value enumeration slow exact permutation fix margin uniformly compute value exceed margin although per interestingly permutation test rather equivalent section use accord occur occur binomial large perform enumeration compute test enumeration accumulate large compute perform enumeration high need value goal since typically collection except dataset approach sample collection hasting mcmc proportion exclusive set collection reason mutually exclusive suboptimal may gene summarize collection use denote undirected weighted vertex connect weight exclusive identify exclusive first section graph set choose topology pathway two cut observe collection score chance evaluate significance score sum column permutation collection satisfy collection important cancer pathway exclusive primarily mutually exclusive biological pathway advance one analyze unfortunately reduce identify address one new contain surprisingly exclusive row one note run exclusive set contain cancer run summarize collection mcmc frequency collection marginal collection exclusive permutation gene high use test use ease exposition define contingency reasoning behind coverage gene exclusive high coverage contingency table freedom one gene contingency table contingency margin could result interactive visualization see website module marginal graph user dynamically module module module view sort search collection module mutation dataset publish mutually exclusive gene section result simulate dataset function tie e gene pathway show bar comparison adjust rand collection compare cancer sequencing pathway exclusive highly whose exclusive rate highly gene appear cancer exclusive section pathway vary ran set average rank pathway alternate rate dataset pathway able reproduce possible average pathway figure coverage rank pathway rank even extremely approach comparison respectively increase across much fast superiority score pathway simultaneously collection exclusive overlap non overlapping set important collection pathway exclusive gene pathway third total also pathway default consensus ari well pathway ari measure agreement partition ari partition ari partition maximally table fraction pathway ari furthermore ari dataset value varied multi range demonstrate parameter choice multi even fairly statistic gene cancer census version dataset sample besides dataset recurrent identify collection score remove least significance cutoff run match run identify collection significant accord census collection identify pi pathway surprisingly exclusive range less surprising dataset identify exclusive overlap report association cancer include spurious exclusive set handling cancer l without filter indicate difference filter frequency argue less biased gene mutation mutation gene breast run mutation cancer array datum repository availability cancer remove highly find without table supplementary multi include background collection high collection largely demonstrate frequency dominate coverage come gene gene gene real cancer run cancer analyze nucleotide copy gene detail supplementary section detail gene genomic circle indicate edge collection value output four mutually exclusive module fusion publication mutual increase number mutually exclusive module six mutually exclusive gene six together module target protein involve set clear notably show gene set largely fact pattern collection module include member pathway role cancer function significance unclear module include cell cycle pathway pathway module four include gene moreover cancer suggest role cancer perform breast cancer merge run introduce breast traditionally classify analyze include classification number module first association report module module relationship associate pathway module contain gene highlight high consistent associated form contain mutation pi pathway gene annotate part pi pathway study circle module p report breast role cancer breast cancer factor involve breast report tumor breast cancer note multi report tumor explain occur suggest breast neighbor tumor breast cancer report occur suggest gene pathway domain possible role exclusive interaction pathway due allow refined interpretation mutually exclusive collection mutually exclusive knowledge exclusive frequency rare large contingency genome datum identify collection exclusive superior real large mutation cancer identify significantly exclusive overlap pathway illustrate subtle pathway protein hypothesis analysis variety site annotation cancer cancer project cancer analyze exclusive analysis finally novel tail enumeration broad examine dataset biological adapt type supplementary approach exclusive ta wu correspondence define whose collection collection ergodic ergodic chain converge want proportion use exclusive high hasting algorithm chain modify ergodic desire stationary distribution hasting collection state bipartite apply exclusive number occur check weight examine sample significant relatively exclusive distribution initialization supplementary gene assign gene collection uniformly else gene note unchanged p algorithm initialization converge similar fail converge create perform initialization initialization initialization multi initialization generate precisely gene union gene union chain gene variation total million iteration initialization examine calculate initialization process stop otherwise iteration final example mutation variation million figure obtain clique marginal corner precisely log choose start horizontal show dramatically find first form corner slope line negative move run million initialization run equal define million initialization supplementary large meaningful practice large long mcmc since space collection large alternative approach g clique clique suggest collection detail contain array patient genome use annotation together expert protein protein patient copy genome include include identify significantly single nucleotide variant file copy output gene sample nucleotide variant cancer include copy remove value four instability level number unstable call contain copy number patient cancer genome approach necessarily exclusive pathway fraction proportion sample gene gene gene probability introduce noise datum match mutation frequency empirically breast mutation remove occur result run million initial start h cancer l cancer cancer page non pathway overlap non overlap overlap overlap overlap overlap non overlap overlap consist pathway rand choice use run value sample exclusive color dot occur note value co occurrence fast tail enumeration green dataset tie score pathway pathway runtime bar gene first edge slope subgraph h dataset whether occur grey indicate colored exclusive colored blue rgb combination mutually exclusive ta wu correspondence cancer heterogeneous disease combination drive cancer pathway pathway incomplete identify recurrence f pathway mutual expect pathway several important distinguish analyze mutual include exact sensitive detect combination rare simultaneous mutually exclusive simultaneous ensemble collection mutually exclusive multiple pathway cancer outperform real application hundred cancer reveal mutually exclusive overlap pathway cancer cancer international cancer genome identify genetic project genome sequence thousand role cancer cancer directly sequence datum challenge heterogeneity collection interact perturb heterogeneity cancer motivate development examine pathway review pathway database network lack specificity accurately examine require biological combination mutually exclusive observation observation relatively tumor pathway gene mutually exclusive de recurrent mutually exclusive module identify gene mutually exclusive protein protein identify coverage mutation approximate mutation combine equal minus coverage overlap co occur find chain carlo set identify exclusive set program weight cover exactly mutation gene identify however show sized require contain mutually exclusive frequency bar exclusive weight like common genome gene tail use exclusive occur test highlight exclusive cell limitation combinatorial subsequent algorithm gene mutation high dominate towards identify majority gene include version study model ratio identify mutually exclusive sensitive gene limit applicability identify gene feature prove overlap overlap option cancer show multiple pathway mutual signal introduce limitation outline towards discovery combination low tail enumeration approximation simultaneously identify collection mutually exclusive collection summarize result marginal pair enable include overlap without knowledge simultaneously discovery mutually exclusive avoid identification mutually exclusive approach cancer datum breast cancer type pathway novel breast cancer simultaneously identify mutual pathway first transform value collection two collection whose exceed summarize significant indicate corresponding pair algorithm mutation nucleotide mutation variety change binary surprisingly exclusive mutual motivate score four exclusive sample weight later least mutation mutation give mutual single frequency three low like common cancer recurrent spurious therein score test contingency b entry equal occur margin sample occur statistical reduce score product score nan set collection grow exponentially typically compute use markov mcmc collection pair connect pair exclusive remove call collection specify tuple measure observe rate generally unknown mutation nucleotide vary uniformly
theoretical applicable temporal scheduling markov stay integer demand service census decade common center indicate van care stay ultimately cause cost al study induce census variability show effect job census excess resource material frequent instance expensive patient census estimation term typically patient census optimal resource allocation body address census integrate less developed work integrate significantly quality et van appropriate characteristic account rs assume type trajectory employ analytical technique patient patient stay express call outcome lack suited capture patient sufficient depth interaction issue develop patient historical patient classifying optimally form cluster impact scalability patient heterogeneity many traditional classify patient diagnosis service work large patient insufficient patient two historical factor gender determine trajectory patient trajectory patient entire location must patient different another patient visit skewed census forecast classifying requires perform significantly second patient classification solely identify patient cluster gender define shape patient associate location statistically validate finally cluster phenomenon capture patient force group assign type closely patient estimate manner hoc gender patient statistically classify patient identify classification seek develop method cluster patient type interaction heterogeneity begin cluster trajectory patient census serve rs module resource schedule traditional novel patient literature cluster patient trajectory group patient module output cm van optimal resource schedule system enable maintain internal traditional hold review new detail follow validate apply study historical finally exist focus rs integrate rs least aforementioned characteristic develop integration rs account heterogeneity interaction various rs green patient arrival optimize resource scheduling either develop flexible consider interaction pathway al al al rs isolate feed forward ignore interaction address issue propose scheduling patient pathway interaction portion consider properly patient heterogeneity moreover patient census estimate arrival rate arrival patient census forecasting review daily largely hence week ahead capable forecasting patient flow multinomial prediction feed flow implement present challenge even variety source burden scale patient al scalability exponential distribution reality patient characterize patient phase combine network include inter relate variable represent causality type begin phase enter fail propose complicated interaction build mix e account scalability interaction heterogeneity htb patient trajectory heterogeneity patient van partition diagnosis diagnosis et planning provide cart etc attribute diagnosis find service necessarily patient share age diagnosis trajectory figure diagnosis trajectory literature high level methodology flow admit record patient location cluster combine arrival stochastic capture census level network census product accurate census programming scheduling patient initially patient stay another serve patient service follow cluster conventional applicable problem patient trajectory mix patient develop semi mixture model patient trajectory predefine validate important generality scalability patient patient type trajectory patient patient cluster estimation patient patient type patient trajectory number semi equal mixture henceforth different markov estimate sample state patient indicate patient stay death initial patient enter u length stay subscript sequence capture behavior restriction visit component initial array hold mass spend patient les belong trajectory transition probability transition time patient amount spend function sample trajectory likelihood observe zero issue conjugate pdf denote q patient cluster give membership unnormalize maximize iteratively regularity satisfied guarantee ultimately optimum distribution iteration membership plug kp kp kp kp kp kp yx kp kp number sequence kp kp make sequence convergence increase redundant test control type chi develop compare transition compare hold merge cluster detect cluster patient visit number patient u one input scheduling transition recall patient compute period day consequently express sum eq semi estimate stay compute research objective purpose integrate process method arrival create stochastic census scheduling patient throughput volume purpose brief detail focus patient cluster integrate optimization approach herein two develop census set section cluster trajectory trajectory one arrival semi patient type section create census demand arrival process arrival combine trajectory poisson demand deterministic assumption approximation reality widely literature control theoretically possible close beneficial patient flow toward deterministic management deviation incorporate approximate particularly arrival arrival patient take account arrival homogeneous poisson vary day week combine poisson markov process arrival census fix distribution define demand rs integrate census objective focus metric minimize maintain throughput perspective objective allow increase consequently patient present van formulation similarly schedule optimality apply approximation metric formulation patient attribute week unit day reward reward patient trajectory type day demand day patient day decision day patient plan horizon system day week throughput patient planning horizon column flexibility allow type patient project patient day set subsequently distribution patient leave side level management add formulation model constraint approximate limit census census patient able proper mix patient type week ensure resource frequently conclude type trajectory estimation integration scheduling develop accuracy patient trajectory flow validate scheduling simulation state sequence patient four semi different cover transition gray high mixture additionally probability generate output would general htb c c model plot drop indicate cluster weight find pairwise hypothesis estimate cluster generate representation estimate true probability show figure chi square test equality estimate parameter table equality hypothesis mapping demonstrate patient flow scheduling model rs patient type effect scheduling schedule trajectory patient use obtain schedule trajectory compare resource optimal utilize naive patient age gender diagnosis patient cluster patient frequency perform attribute datum generation attribute patient formulation show percentage improvement setup naive metric patient result optimum naive patient significantly benefit patient close optimum attribute lead understand resource indicate accurately estimate yield near outperform apply propose resource schedule historical patient complex system year include stay age diagnosis patient leave death
set algorithm table table iw column name dp compute exact method posterior assumption modular dp list dp iw show iw clearly use short computation significance dp meanwhile mcmc stability similarly much short case exception several thing term method large phase obtain dp occur letter child mcmc dp mcmc unable reduce dp please iw run dp dp method dp dp step dp dp factor hidden difference iw mcmc iw run data iw corresponding time phase less run dp section effectiveness strategy range across show cumulative mass iw note formula p five tumor letter five iw small dag tend local large number iw include structure node inclusion improve show small iw great datum sound iw increment case letter correspondingly figure increase increase running figure clear achieve case second dag mean interval respect increase actually decrease benefit strategy iw vary letter child iw run iw sample direct edge totally mcmc iteration discard mcmc iteration burn dag sample run sample experimental setting case run run iw demonstrate soon letter run program memory expensive iw child outcome run get experimental result show performance method edge bar deviation iw dp iw run run smaller compare iw iw real sample iw significantly show figure term mark compare iw advantage iw combine figure dp iw shorter iw second phase dp second dp compare good iw shorter iw letter experimental examine material experimental modular method modular force enumeration dag approximate tool estimation posterior edge mcmc comparison one modular showing fundamental structural direct complicated perform experiment uci well heart modular five modular influence length length intermediate variable feature eventually influence eventually influence combine path eventually influence path influence x x z jj iw set iw significantly compete mcmc iw compete modular support b iw try run iw times standard direct modular feature run iteration discard burn dag run mean deviation run bar dp iw correspondingly three modular iw dp iw method modular dp edge significantly two modular iw well real iw significantly p one consistently modular iw good value kind mcmc edge investigate mean iw significantly corresponding variance result iw good sample iw iw compete five investigate modular iv letter hypothesis guarantee case letter serve performance requirement dag come performance inequality posterior easily event bernoulli independently repeat success figure see much mark bar edge even trial side hypothesis reject nan hold next demand requirement dag sample show come guarantee hoeffding logic repeat estimator even histogram even hypothesis reject hoeffding table figure edge even correspond among sided conclude less clearly even sided testing case increment hoeffding hoeffding decrease figure edge run algorithm max clearly maximum furthermore side reject nan conclude hoeffding hold efficiently bayesian network dag sample dag feature interest show empirically estimator considerably outperform art without modular capable estimate arbitrary modular modular prior iw structure modular modular modular time modular modular bottleneck dp application able implementation use cluster totally memory proposition along eq prove probability proposition pmf assume modular derivation dag relate sample occur order consistent step get total formula pg sample sample p pg prove pmf bernoulli pmf ii almost since law converge random bernoulli thus central theorem eq denote mapping theorem theorem hold equivalent conclusion imply straightforward q define equal choose every dag dag exclude dag proper p p p f f I pg p op g order modular dp notational convenience essentially set lagrange q neither eq limit eq constant done prove lead whole done converge g positive super exponential thus algebra define g g probability g ng g ng ng nn g whole surely imply proof proof quantity proof essentially proposition direct dag dag accord feature theoretically estimator outperform art learning sampling widely various task network bn bayesian network dag whose represent direct dependency encode node parent semantic compact joint answering query semantic insight effect decade network unknown learn motivation learn make use predict another closely semantic discover domain context dependence different often interest semantic node directly influence interpret eventually influence furthermore node node cause learn interesting gene turn examine path structure kind structural goal discovery define dag find equivalent dag dag use single posteriori lead conclusion average challenging super special develop probability structural space capable handle bayesian moderate due big posterior via node limit path path compute path assumption modular soon dp handle mainly modular problem dp algorithm far feature independence hold generally user upon feature typically need limitation compute posterior one solution modular approximate model develop draw dags monte hasting dag develop procedure operate show mcmc outperform develop proposal mcmc mcmc converge fast mcmc structure superior mcmc nearly mcmc method operate order appropriate common drawback finite show competitive hybrid several work approximate order mcmc special form computational convenience please begin modular assumption modular prior desirable prior bias dag topological assign modular inferior please develop modular strategy dag order dag accord considerably considerably mcmc moderate applicable moreover desirable unlike algorithm estimator control dag address limitation dp whose restrict modular posterior various arbitrarily exact dp develop iw bias modular theoretically prove base empirically superior modular iw address dp usage modular posterior additionally efficiently prediction application avoid modular prior briefly bayesian network dp iw demonstrate advantage finally appendix proposition theorem node dag represent convenience parent dag dag represent probability dag assume global local local likelihood score local close score efficiently eq modular modular marked distinction dag often structural indicator otherwise average issue describe dag infeasible problem approach posterior sample dp algorithm space rather dag linear order vector dag order denote subset linear notation nonnegative define return example modular setting accordingly dp step step compute degree truncate technique extend standard algorithm compute define follow forward contribution show start take compute whole dp algorithm modular include dp step introduction backward q recursively dp contribution structure limitation posterior modular use dp compute posterior idea sample invariant able compute result constant eq modular proposition state definition choice parent order order non modular draw draw sample dag draw parent dag prove subsection show compute efficiently conditional respectively follow q ni appendix u sample order element element order sampling order sample pmf posterior modular result guide view dp answer oracle aid counting generator reach situation count information compute dp perform parent describe algorithm term order accord dag one dag sample parent correctness dag iid posterior algorithm take overall time complexity efficiency fairly dag tend maximum biological assumption widely experiment dag dominate moderate reach several thousand develop dag step dag exact construct dag take need sampling modular edge take take note order dag memory structural algorithm ii converge q limit iv hoeffding state order least require property obtain guarantee iw weighted dag correction sample dags dag check dag time hash time iw iw joint store construct construct iw structural dag method experiment property iw converge convergence represent cumulative mass sound sound interval sound dag possible tractable desire please note express equivalent give correction strategy iw solve desirable dag strategy keep sample sum dag problem instance dag maximum possible negligible majority dags dag cost dag sample expect time requirement hard disk store dag take dags dags dags dag sample reach order cost per avoid meanwhile strategy therefore gets sample times dag implicitly serve strategy guarantee dag large spend huge dag sample dag dag sample dag keep dag none chance get guarantee dag state actually ensure good art applicable moderate mcmc first compete hybrid mcmc second eventually hybrid correct come dag posterior shown converge fast structure order performance mcmc mcmc moderate applicable mcmc iw infeasible large limitation iv hybrid mcmc compete apply collection dag score use dag constitute advantage algorithm specify iw complexity spend good search increase cost one iw ordinary computation severe usually iw interval wider set memory please language demonstrate capability set include ten real uci machine letter tumor synthetic gold synthetic contain letter child vary instance tool pc intel processor memory extra specification maximum partial order art order modular partial mcmc implement language estimate state readily dag consistent c tumor letter variable moderate able use language dp posterior every modular prior therefore absolute posterior essentially j j modular indicate discovery note mean since base value one datum tumor child fair set cause run iteration mcmc sample partial p sample information order consistent great method inclusion efficiently modular edge compute order performance performance approximation come dag avoid mainly decrease dag modular feature table list cost list correspondingly total run precisely speak total running method report run relatively computing include dp six run step due factor percentage performance
require block voxel matrix human matter acquisition normally diffusion direction brain contain gb format format five memory requirement memory system research dramatically storage requirement life life via tucker decomposition std introduce tucker tucker specialize propose approach multidimensional factorization array tucker rd approximate decomposition decomposition guarantee array array e tucker powerful compression store tucker explain compression obtain tucker tucker store instead require store compression mean classical tucker allow represent dense core tucker array dense tucker achieve core array tucker decomposition array tucker core array entry entry store require storing coefficient location hereafter refer tucker std compare tucker model std compression ratio low sparse array show life explain diffusion voxel dictionary signal extend decomposition matter voxel orientation white evaluate orientation hereafter calculation represent arbitrary dictionary prediction signal predefine resolution diffusion contribution q atom fig discretization atom spherical sample spherical construction diffusion spherical coordinate orientation diffusion dictionary different diffusion measure direction signal voxel discretize life signal organize approximated combine diffusion atom orientation zero entry indicate fig sparse atom voxel give voxel store individual contribution voxel store index atom avoid signal std life matter prediction life life comprise sparse v direction voxel array use tucker equation rewrite array n correspond white voxel eq life tucker n vs along main diagonal combine sparse tucker life also decomposition introduce optimize std life build white brain std orientation entry voxel atom version life sparse tucker std life evaluate brain variety negative least problem hereafter exploit std life life core life std version store describe use report implement w result matrix std write life q kronecker write follow stand stack vector avoid multiply allow maintain usage plot memory requirement std std measure diffusion std function compare usage life storage requirement std analytically nod per store proportional storage conversely store std non core std storage n without straightforwardly fig b memory direction node storage direction std fig memory number storage grows linearly grow much sum substantial reduction memory consumption std become direction std approximation life discretization fig std std predict original mean validate discretization std original std original range show discretization varied difference r whole matter volume comparison relative std life fit std discretization spherical relative w weight std respectively test tucker original storage show std major original validate predict diffusion deterministic std replicate finding demonstrate life implementation life software std matlab tensor toolbox file mac bit software test computer gb ram brain ghz intel gb scatter plot error predict probabilistic computed single probabilistic base std discretization grow time human grow require collaborative community focus modern big application imaging measure white brain major put improve white macro micro level digital availability develop compute computational trajectory white matter node first compare connection identify compare validation establish accurate core matter global global plausibility estimate alternative matter computationally intensive routine brain population recently linearize method separate similarly family linearize modern new linearize road investigation matter linearize impact decomposition road population human increasingly resolution modern clinical report pseudo decompose life option parallel decomposition component atom voxel acknowledgement chen comment version support computer support c code tuning project block mm ia la brain sciences program cognitive science usa edu publicly grow increase availability resolution require modern linearize decompose matter achieve comparable approximation brain big datum mapping brain network tensor major matter diffusion probabilistic deterministic transforming start share brain collect share potential group analytic tool attempt extend replicate scientific fundamental modern separate paradigm landscape imaging motivate storage sharing hereafter show development processing challenge resolution signal algebra object dimensional array array tensor vector number dimension primary compare algebra multidimensional turn address convenient substantial reduction briefly array diffusion combination track measure property white measure health disease estimate white matter routine b life brain candidate white generate candidate generate white matter volume error predict use comparison method model fig predict diffusion signal linear within voxel predict diffusion voxel model direction voxel red voxel combination predict green voxel measure matter voxel organize long full volume assign use optimization notation voxel life life panel measure double fundamental task evaluate individual one substantial white matter spatial directional resolution modern size model linear primary describe directional primarily matter signal directional relate combination brain measure signal signal strength denote strength voxel candidate matter voxel non apparent diffusion f definite signal example exponent tensor semi ellipsoid radial tensor diffusion array array array array multidimensional basic discuss array notation definition array letter call th order array generalization denote capital letter e slice slice use array along array slice let index vary array slice hold index unfolding array array address dimension mean
big appealing scale accordingly hence interesting concern finite subtle analysis fail connection cluster conclusion view open question mini batch policy minimal inf two proof distribute mdp correspondingly assumption satisfy show transition matrix correspondingly trajectory cluster least bind trajectory obtain h h unbounded maximal distance must reasonable trajectory order misclassification occur quantity leave corollary problem static strategy accumulate context interact website gender age device etc website customer interaction focus horizon know context provable context optimize naive implementation extension process mdps commonly dynamic health management trajectory observe trajectory answer transition standard estimation method affect temporal diabetes influence measurement variable incorporate create mdp reduce generalizing incorporate static form transition zero instead sized world website website activity suggest ad relevance ad age gender device determine website mechanism http mechanism insufficient website gender observe website word trajectory page user profile take type exist device want child parent suggest interaction fashion user elaborate want ad user time spend website interaction optimize line solution valuable case underlie contribution present general provable horizon contextual empirical discuss case infinitely context reinforcement rl reader mind preliminary present derive trade bar research setup markovian consider hmms hmms mdp essence context arm bandit mab present user similar setup reward selection mdps architecture task domain gate mix different area could model learn source fit representation deal state interact environment differently mdps perspective relate model user type conclude short comparison know affect complex generalize hidden parameter constant context reward compose suitable state rectangular singular determining formally setup problem analyze extension setup mdp setup tuple py rx py rx interaction episode accord learner action accord environment learner q distribution mdp finding maximize cumulative mdp rl definition establish action correspondingly map cx model action generalizing model observable arise number problem behavioral pattern customer side gender aggregate available independent greatly simplify writing finally adopt setup episode episode adversarial fashion state distribution generate mdp maximize reward increase optimize trade exploration exploitation unlike rl setup reward cumulative reward agent cumulative discount true therefore converge similarly know apply correct regret respect notice though new loss different define value obtain agent explore classify trajectory mini batch mini batch distinct new act possibly way issue next length recognize correct exploratory part addition closely problem mdps formulation mdps transition uncertainty rectangular related trajectory impossible consider unless subsequently pose confident trajectory embed improve notion separability trajectory separable policy fulfil policy logical maximize distinction optimal consequence one lead area reward action distinguish still problematic underlie exploration open solve mdp require sample need identification infinitely model substitute simplicity trajectory analysis modularity follow state transition set score trajectory scheme inefficient exhaustive cluster adjusted policy step mention decide confidence could combine strategy choose set wrong apply estimate sophisticated would rl whose scenario constant satisfactory necessary assumption result trajectory visit separable enough sample trajectory model policy realization algorithm assumption full available supplementary material assumption mini relate proper scale require enough third correspond trajectory extension previous consider complicated scenario evaluate consequently regret precise context context application agent trajectory step trajectory latent length trajectory naive employ rl algorithm approach share context trajectory scheme choose classify choose exploit length exploitation decrease approach trajectory short trajectory regret contexts trade exist cluster part draw length sampling action correct context thus worst use bar deviation examine follow long trajectory cluster experiment plot trajectory present trajectory part bottom trajectory length episode phase threshold follow adjustment period succeed certainly trajectory fail episode sufficiently cluster quality simulate episode portion trajectory dedicate identify exploitation exclude
symmetry finally v repeatedly pick vertex reference try require high vertex less likely good vertex give lowest well around spectral obtain directly grow community g logarithmic extend sized community degree overlap sized overlap motivation vertex vertex path multiple short use would community attempt get relate connect vertex integer cardinality intersection vertex community around assign subset edge even generate edge probability approximately eigenvalue operator hold ie terminology ie ie get estimate start leave r determinant follow e r ij q linearity one product contribute determinant product involve dominate follow find graph nonzero list approximation set r e se use solve repeat estimate approximation input distinct approximation create take return median eigenvalue product vertice integer already course flip get away away equality sufficiently follow vertex input decision proceed follow I iv iv j solely compare vertex fairly vertice subtract inequality strict representative community give vertex without classification vertex number suppose minimize community run fairly algorithm follow plan select vertex one anchor vertex attempt community actually community approximation bad accurate output follow randomly assign g v comparison community otherwise fail bad vertex time classification half classification little two contain none could bad average classification together agnostic start use pick run combine classification explain require sphere comparison integer list community small rational numerator q small hold classification small classification disagreement classification define disagreement classification minimum disagreement remain classification assume community great overlap return combine satisfied run comparison symmetric row equal reciprocal rational run community graph draw without beyond multiply corollary row agnostic comparison average degree row positive l agnostic corollaries connectivity matrix grow almost exact recovery agnostic proving require terminology distinct eigenvalue order also addition sum vertex well key behind short graph vertex short refer whether good q otherwise combination rv I follow vertex symmetric gp vertex compute count return count plan rigorously choose randomly edge fact never change unless double something furthermore whether whether proportional deviation complete symmetric eigenvector determinant lemma draw independently ij r g convention probability bad vector I u e mi mi j mi li bound multiple r u mi mn mr mi u h ji upper constant multiple choice bad expression link determinant rr p either bad determinant desire establishe prove order entry exist draw eigenvalue eigenvector large eigenvalue case eigenvector fix within furthermore standard mean distribution sum poisson c randomly select vertex j e give randomly member hold c om n algorithm classify r hold good within sufficiently select least vertex r h ij allow pick member community probability long initial vertex run I j h om member community e c r om vertex combine result reliable tn e agnostic comparison rational small classification half classification discard classification classification disagreement remain classification community claim correspond great overlap exist reciprocal rational probability tn run tn tn vertex improve assuming find satisfy large reciprocal graph success half classification give classification least classification classification less ensure classification community minimize vertex work correctly eigenvalue run om tn computing degree agreement classification find classification take whole run om tn proof desire positive stochastic block community matrix assignment call assignment recall exact community partition assign contain community recover q hellinger maximization far often vector instead e community recovery recovery solve precisely recall th p solve exact exact recovery failure rate need algorithm agnostic use step run agnostic sphere slowly corollary agnostic second making whether node another step solve problem profile sbm mis exponent agnostic splitting output community intend large subset subset agnostic sphere edge node likely belong profile computed classification size claim density degree profile compute preliminary belong notion repeat profile graph vertex community community relative independent ba enough regime particular le give rely side approximate draw sbm plant reveal profile vertex classify vertex resolve observation poisson hypothesis take value distribution mean goal minimize likely condition e posteriori resolve allow eliminate decoding give p time hence h p multivariate regime probability ix jx high profile divergence word find hellinger although exponent previous recover however infer recover narrow hypothesis composite poisson determine true disjoint subset hypothesis belong belong note disjoint wrong hypothesis realization test minima sum therefore control ji belong community belong group group error profile draw profile summary prove let subset subset degree correct q information j ii ax previous testing classify arise misclassifie let drawn community density corresponding size within expect vertex apparent change apparent follow let disjoint draw assigning subset profile wrong vertex graph less hold vertex eq let neighbor least chance community report community report degree profile profile conclusion follow possibility result converse know hence let recover partition behind technical step handle graph step agnostic vanish sbm parameter unknown hold time independent negligible impact classification error rate adjacent multiply rate plus plus pt plus recent block sbm linear three regime develop require community communities ii regime degree enhance agnostic overall regime agnostic parameter achieve limit quasi sbm study block slowly regime provide background detect community graph science apply variety network try community people social recommendation classify detect sub tumor decade understanding appear sbm sbm canonical community connect recently back center attention practical allow massive due phenomenon latter sbm communities community identical intra consistency regime community completely recover positively sharp recovery recovery achieve introduce sharp showing detection conjecture besides detection recovery property ask community vanish misclassifie two almost recovery generalize discuss conclude achieve threshold shannon capacity constraint development field g survey likewise identify threshold community guide reasonable regime fail particularly question whether sbm answer question work community sometimes exception provide sbm sbm sbm community recovery sbm concern regime connectivity symmetric separate solve exponential snr solve recovery symmetric equal sphere comparison community cp snr must exact recovery consequence previous sphere comparison almost entry sufficiently graph draw sbm sharp characterize divergence distance provide operational divergence analog channel solve show computational gap community characterize community definition exact recovery model partition ii partition recover exact recovery close sbm size community community logarithmic degree sdp agnostic estimate cycle count walk community namely aware private regime obtain sbm result allow rely major open without agnostic degree achieve node solve know community ensemble plant connected independently specify draw cluster label reveal observe focus community either grow relaxed motivated average degree fact phenomenon regime partial community agree agreement recovery recovery take element community community recover theoretically solves algorithm exact extract merge community merge vertex high connectivity therefore sbm entry nonzero sufficiently agnostic sphere graph community decrease refined word entry eigenvalue whose eigenvalue k recall information partition next show without know parameter recall partition ensure agnostic run theoretically proof rely fraction main procedure clean degree recovery already use technique difficulty know vertex short short path probably value require vertex attempt relate connect compute unfortunately whether cause differ get assign hence define vertex independently equality ie simplify terminology ie p ie dominate get term note expression estimate r e linearity determinant column determinant dominate pick several
slice method focus recover efficiently regularization follow tensor view computational explore involve rank tucker analyze heuristic tensor completion tensor extend version date sample resolve provide provably enjoy order contrast provable appear tensor symmetric factorization contrast employ programming base hierarchy even moderate sized numerically impractical semidefinite program grow rapidly guarantee recovery alternate relie underlie symmetric optimally dimension careful initialization method solve tensor nuclear regularizer norm guarantee optimally rank finally base tensor recently optimally aforementioned conceptually brief complexity third setting well key approach ccc tucker unfold tucker unfold rank decomposable tensor nuclear norm computationally intractable completion random projection separable rest introduce setup approach set also tensor upon projection tensor case validate section outline direction vector matrix case character e bold character two tensor define inner frobenius work third array mode refer keep third mode slice similarly mode tensor decomposition tensor r r generality tensor analogous result tensor tensor indeed challenge tensor r tensor mild assumption algorithm somewhat I r set vector precede measurement separable formally separable third operator q definition extend separability mode separability decompose action linear act mode application involve mechanism indeed sense study compress sense tensor argue separability desirable precisely recovery example outer slice slice typical separable form ensemble unit sphere suitably measurement case projection process play development compressive section separable share rank tensor completion entry reveal tensor separable depend nature mode slice n trivial extension slice slice separable completion tensor completion due contextual recommendation separable mechanism separable mechanism gain interest appearance retrieval rise similar spirit notion problem recover measurement interested recover spirit sketch slice recover natural tensor sense sense rank slice also think separable complexity exact aforementioned completion extend natural separable sense mechanism separable measurement measurement third recall weight vector require mode distinct separable weight vector introduce formal measurement refer vector potentially concatenation vector concatenation subsequent diverse suitably efficiently unknown tensor ingredient bridge inverse problem thereby allow contraction matrix slice tensor contraction primarily role recovery slice completion x v w verify enjoy decomposition contraction singular sense trivial particular form degeneracy e slice slice lose contraction situation extend terminology tensor tensor slice extend degeneracy trivially almost contraction suitably situation non degeneracy tensor choose suitable ensemble pick orthogonal vector concern w ix sample sphere case pairwise completion matrix r k I I abuse tensor slice see ensemble appear underlying degenerate generic suppose n matrix suppose degenerate eigenvector eigenvector determine ambiguity one arrange common eigenvalue entry decomposition next build algorithm inverse bound algorithm essentially turn tensor degenerate compute lemma turn finally obtain invert exact directly pose preserve tensor recover recover input degeneracy pairwise compute eigen denote eigenvector arrange solve w u last solve linear obtain whereby compute minor modal factor repeat perform eigenvector matrix modal properly align rearrange column sign observation drive separability separability word act measurement act principled tractable informally define nuclear succeed exactly recover recover along furthermore non pairwise generic exactly recover follow meta low measurement pairwise succeed exactly unknown complexity separable completion degeneracy naturally start describe main recovery assume separable advance eq efficient computational study problem matrix form input reconstruct pair normalize last linear input separable optimal eigen eigenvector sorted arrange respectively compute eigen let matrix sort common rank arrange simultaneously column q output tensor v move situation separable also provable complexity bound random I solve recovery measurement detailed thought measurement identical us slice establishing random matrix establish projection problem prove second identical manner lemma rank nuclear heuristic succeed recover high sub event event take bound event also constant recover complexity bind let describe succeed recover lemma tensor modal generic recover determine high simply take failure optimal tensor become unnecessary directly step solve factor nevertheless problem fast exist problem minimize tucker rank consider problem matrix far note sense seem compressed need store matrix store operator perhaps sense tensor require sense uniformly truly dependent recovery instance respect completion reveal slice slice precisely reveal slice require distinct slice say eq slice measurement mode precise key slice index slice without replacement context tensor completion slice contraction slice use input incoherence rp p canonical basis vector incoherence condition slice slice slice thin incoherence slice slice max slice mode slice slice condition slice condition literature instance point restrictive incoherence inequality away complexity multiplicative incoherence slice suitable ensemble bound wise decomposition direct contraction incoherent incoherent slice satisfy incoherence slice row space slice incoherence detail tensor give tensor slice definition slice k unique theorem correctly slice incoherence corollary sample complexity requirement recovery sub event occur change degenerate pairwise algorithm succeed recover decomposition probability allow slice I slice scale step nearly order comment projection necessarily slice result sample replacement uniform furthermore completion rely nuclear norm alternate minimization completion slice remove slice remove slice sample order straightforward way remain notation omit third order kn interested slice pick may slice tensor define mode deal contiguous slice collection mode element tensor refer notation contraction tensor matrix third case either coordinate analogue lx straightforward rearrange get eigenvalue along line notion pairwise degenerate ratio degeneracy tensor appropriate ensemble describe input tensor degenerate generic eigen decomposition column zero sort order common perform simultaneous obtain call recover r l tensor n k third assume separable separability norm via two mode algorithm tensor provide successfully recover nuclear non ht k eigen decomposition column sort let match result eigenvector determine linear recover k k k natural specific operator expression k randomly euclidean distribute identity b equality ii separable recover solve nuclear sub lemma solution suppose solution theorem concern outline succeed exactly recover rank omit sake complexity constitute complexity freedom e freedom whereas achieve sample e third tractable operation routine efficiently enable straightforward tensor order entrie along mode distinguished slice distinct slice slice reveal straightforward exercise argue obtained slice correspond separable recover slice optimization recover correspond slice incoherence degeneracy slice high slice uniformly sample unique solution recover compute factor recover exactly summarize km mode slice furthermore incoherent slice condition slice sample degeneracy pairwise succeed entire support conduct involve suitably tensor projection completion transition plot unfold transition plot propose sdp tensor even sized section run via tensor whose vary look recover tensor repeat less number tensor recovery convert varied tensor slice slice figure method recover trial far scalable unfold method compete along compare execution sized magnitude
step backward collection hyperplane b input precision subproblem essential correctness hand vector long horizon modulus occur lead precision tool algorithm convergence tolerance guide precision stop gap give solver condition feasibility feasibility could increase long hyperplane value bring error acceptable interior active solver often interior employ relative condition modulus hand solver terminate infeasible negligible appropriately brief tucker problem interior method barrier infeasible stop interior find condition system desire precision address concern allow difficulty tolerance convergence failure avoid manually adjust solver readily feasibility maintain throughout course interior result feasible computational resource size conduct optimize level storage device write storage drop price meaningful wide storage device conduct network storage device high focused water distinguish storage natural minute price price update storage device hour divide time quite algorithmic technology stochastic investigate effect regularization storage device determine post decision quality aggregated version grid transmission line produce transmission generator include generators generators nuclear generator wind wind discretization period run generator forecast wind wind make use storage however optimize generators device grid stochastic wind place device wind high demand energy efficiency storage device challenging factor device transmission daily variation demand take store generation avoid peak period balance store challenge environment source information wind wind wind historical wind month period consider wind weather regime information characterize stagewise former appropriate stagewise equally scenario assume stagewise period assume period weather percent nine regime visit percent anchor north simulation grid coordinate align date zero xlabel ylabel cells sim true col comma sim col comma cells data table cell sim col comma sim col comma sim col comma sim sep comma cell sim col sep comma cell sim implement solver quadratic optimization problem tolerance storage term examine device resource stagewise leave uncertainty show convergence bound especially consistently problem height title stagewise major xlabel ylabel value baseline south grid title xlabel iteration coordinate height stagewise xlabel ylabel coordinate coordinate major title uncertainty xlabel coordinate h anchor south height title stagewise xlabel iteration ylabel coordinate anchor height major title markov xlabel coordinate south title stagewise xlabel ylabel south width grid major title xlabel iteration table second storage device stagewise uncertainty iteration application offline example day wind cut time day policy c c l iteration c c large long numerous world energy finance demand often trade ever grow practitioner framework cut hyperplane quality development approach consider energy storage grid transmission operator unite fast counterpart great gain several would involve regularization possible path exploration involve time coherent additionally like obtain insight also interest pdf definition pt plus pt develop period stage hundred resource finite assumption storage transmission propose exhibit counterpart gain problem keyword nest scientific development finance create practitioner meet ever speed reliability problem issue significant gain important understand implement integrated work program satisfy horizon potentially hundred period stage realization stage dual receive considerable attention real framework widely energy account area popular approach medium planning scheduling al portfolio formulation method plan transmission market traditional planning technique show planning operation system wind introduction fast time optimize grid level storage subject weather planning demand stochastic dynamic asset end horizon account transaction model consistent employ popularity exhibit resource introduce markov surprisingly grow improve issue technique method however framework growth require propose reduce around distant cut hyperplane contribution label stochastic identifying exploit post decision regularization optimal sample regularization avoid growth scenario exist computationally tractable involve exhibit fast classical useful resource work relevant wide practical work use set device transmission realistic setting allow state day increment produce period linearly horizon paper review relevant problem formulation traditional programming dynamic overview dynamic main contain quadratic method describe algorithmic tuning issue behavior dimensionality resource state normally handle originally eventually approximation nest scenario stochastic dynamic practitioner upper guarantee many version despite curse essence cutting class exhibit slow convergence curse computational period cut plane n problem enhance improvement utilize chen point randomness dual identical realization multiplier small use approximate solution correspond realization explore cut evolve stochastically go beyond side warm tree cut prune cut et al hyperplane large furthermore dual programming adopt scenario strategy scenario selection successfully problem low proper stochastic develop version develop utilize tree entire history applicability period sigma algebra throughout convention measurable surprisingly standard solution programming realization nest partition enough computationally information history employ dynamic state sufficient decision cost transition represent pre resource please necessary formally evolution difficulty among equation place method solution result since approximation high address concern follow semi definite matrix resource substitute converge solution detail present study kt tr subproblem pass value original problem include suppose approximation let forward pass k basic solution pass every algorithm converge forward tr tr tr since dual basic similarly function post collect forward cardinality complete know subproblem backward pass therefore vector know iteration obtain without formally solution current solve suboptimal basic moreover hyperplane update pass r occurrence converse borel event draw realization pass one large number draw update perform finitely exist realization theorem every vanish coefficient however rely might properly relate way overcome impose assumption finite iteration assumption vanish therefore complete proof drawback stagewise generally problem often involve temporal approach adopt address case occur vector autoregressive stagewise autoregressive period additional accommodate realization model autoregressive advantage stagewise independence dimensionality history dependent convergence rate setup increase information rather resource discrete probability current realization stagewise autoregressive constitute realization historical weather indicate distinct explain around forecast properly weather dynamic different weather regime inherently multiple approximation optimization problem
break tie perform obvious edge ensure cut find step stop label label subroutine mid short short subroutine subroutine proceed mid point step compute repeat describe sub procedure label label min label propagation entire sake completeness run theoretical subroutine subroutine connect label return learn labeling conceptually think operating vertex find vertex opposite search phase pick short set apart end keep short among connect vertex cut pick vertex label shortest shortest connect thick short short path subsequently find edge short boundary finally query main reason firstly keeps presentation extra incorporate know criterion label stop achieve secondly recover boundary handle quantification challenge let label edge graph pose underlie making labeling partition collection component induce partition cut boundary sc illustration correspond colored cut vertex correspond interested clustered component cut might easier cluster cut argue complexity instance term boundary first natural induce label large conversely label query linearly label labeling vertex n see knowledge unlikely vertex might expect easy see key graph sp g path fy fy sp g cut imagine pair discover lie pair search path label edge quantity exhaustive drastically cut edge rise third near connect component correspond component cluster turning observe cluster path resp define cluster path dot prove binary induced cut remark g strictly well integer version quantity term phase component balanced situation ideal consideration assumption various line ignore error outli component allow readily generalize case labeling argue parametrization show graph query property near optimal discover direction show query optimal begin observation label vertex label path cut b phase observe know entire search cut imply set lemma graph subset random least proof combinatorial phase attention algorithm search recall remove edge shortest short connect short label phase short even step phase length short current short length drop thing get nearly step span step end step boundary might query phase end connect label vertex end new greedy cover connect discover sense discovered cut cut component per run cut component found never discover bind put characterize rate active characterize boundary assume curve analyze cover difficult apply flexible result practice gap adapt boundary minimax less good knowledge class binary box counting boundary generalize piecewise smoothness connect realistic detail classification predict cell intersect boundary box counting set follow bayes connected component hausdorff label generalize away zero away check distribution satisfy bf regular lattice center indicate ensure lattice also label active lattice partition request noisy correspond near optimality excess active learn minimax logarithmic significant nearly achieve classification problem achieve enough require everywhere use regularity complexity arrive excess allow estimate excess function h experiment red grid preliminary digit originally smoothing gray pixel separate task vs might simpler randomly choose digits nearest vice versa thus node undirecte unweighted nonetheless edge drastically vs vote voting record uci repository create graph retain boundary grid positive core square algorithms b need query completely guarantee query sensible experimental analogue query figure experiment clearly quite surprising perform mind believe try national foundation health ai grant grant ai prop access noiseless oracle request label vote bound union fact query voting statement label cut long small union integer effect l hand drop conclude recall feature cell oracle access intersect boundary however fail correctly vertex cell detail affect oracle query take majority noisy oracle sequel assume free label vertex lattice connect randomly vertex discover component observe connect cut know add query component number cut cut cut query cell fix cut hypercube contain furthermore vertex cut easy order edge cut component since complete repeat mistake denote excess observe everywhere intersect bayes probability number eq observe large depend c w derive main near trivially true cut fine restriction expense clarity induce cut component component notice low complexity cut cluster value result tell sake evenly odd evenly compare manuscript disjoint vertex copy clique show copy obtain subgraphs replacement way chosen pick edge path way subgraph total leave label determined cut follow hold cut number give result particularly relevant active grid bottom vertex count connect labeling notice query set size way component top right cut box contain cut contain cut furthermore cut cut locate contiguous box box contain cut box bind walk observe walk must box box symmetry walk end box valid walk go restrict walk right notice walk towards end observe walk terminate walk make entirely move count attention walk contain move move walk walk start box end suppose walk block walk move move therefore reason conclude walk one leave bottom walk start end correspond cut two edge valid cut walk r observe cut sum size conclude discover like walk observe tell query argument interesting black learning application nonparametric classification label label select structure label short pair need demonstrate data performance theoretical guarantee demonstrate implication theory show minimax excess important problem keyword find classification suppose vertex unknown
reflect transformation naturally unique polynomial surrogate account us department office office advanced award de address field uncertainty reduction traditionally use hyper parameter despite coordinate infer infer seek parameter use inference end dependence hyper expansion acceleration bayesian coordinate enable avoid expand explicitly solution uncertain hyper feasibility propose diffusion spatially log datum infer profile profile markov monte inverse arise many information computational view challenge inverse ill multiple continuously affect error infer challenge motivated ability posterior carlo hundred thousand method prohibitive large acceleration surrogate much forward mcmc step instead pc pc extensively setting flow model inverse involve number unknown computational challenge arise suffer curse dimensionality hard achieve expansion endow process surrogate evaluate parameter quantification incomplete bayesian inference calibration hyper propose effect scale hyper accounting variance attempt prior uncertain hyper work methodology expansion term parameter expand pc surrogate hyper hence estimate extension explore base eigen eigen eigen form hyper utilize basis kl hyper distinction pc uncertain expansion hyper apply transformation expansion expansion hyper avoid complex pc even development start provide account uncertain hyper change describe role pc acceleration section toy conclude statistical include model briefly formula prior relate model rule express additive assume become issue discretization need uncorrelated problem reformulate coordinate lead q p kl expansion generally improve hyper generalize bayes covariance hyper equation model kl sample multidimensional suitable carlo rely metropolis hasting flow chart evaluation sample dominant realization value fed solver model demand computation prediction particularly differential motivate substitution polynomial surrogate compare resolution complete surrogate offline subsequently likelihood approximate use construction detailed section introduce polynomial together domain equip inner product belong covariance function semi eigen equation kind value countable eigen continuous constitute orthonormal eigen value decrease retain stochastic uncorrelated meaning expansion mean square kl converge employ inverse infer directly infer structure motivate introduction parametrize family section gaussian completely covariance function aspect covariance stationarity easily confirm large parametrize symmetric definite know uncertain covariance interpret hyp great importance trying understand traditionally infer end set noisy location hyper kl expansion uncertainty expansion model address uncertainty develop enable infer hyper kl formulation generality center uncertain hyper become eigen scale eigen ensure function orientation eigen covariance representative deterministic random order eigen orthonormal continuity coefficient computational purpose generality allow translate expansion eigen variable specifically observe correlate cast shall invertible every determinant brief illustration error center hyper parameter correlation affect shape eigen assume hyper important note kl mode small small mode case decomposition numerically mode space leave plot right plot respective eigen use decay expect average uncertainty note l average right truncation approximate subsequent reference estimate realization equation corresponding realization observe average basis comment error spectra report decay precision mode suitable numerical eigen eigen accuracy indicate remain achievable double precision continue average yet range decay reference low eigen represent vary htb indicate error hyper plot reference error increase truncation involve behavior report increase precision cl far highlight robustness coordinate even prevent determination eigen yield error small exhibit denote subspace length speak eigen represent process range provide less select accelerate inference process pc numerical section exploit surrogate efficiently handle uncertain previously provide brief error input let formally express uncertain parametrize probability restrict second functional dependent complete expansion form eq expansion mode functional lead polynomial practical truncate truncated total term pc exponentially expansion number expansion denote series series converge determination pc distinguish realization pseudo method reduce computational equation level project pc procedure result couple efficiently couple operator reference return problem want pc prediction coordinate hyper notation previous expand dependence advantage discuss change mode map accurate varie reference diag construct approximate mapping field variable reference propose relate numerical avoid trivial note non trivial reference pc consider complexity simplicity change order gaussian independent gaussian case using continue uncertain condition simpler high consist equation deterministic boundary log uncertain function away ensure set addition setting classical element piecewise time investigate approximate r error incorporate approximation approximation truncation pc expansion discretization inherent counterpart provide section mesh discretization parameter ensure dominate effect process mesh deterministic diffusion project subspace kl translate approximation period counterpart depict covariance indicate pc truncation dominant low use reference scale minimum competition effect increase cause translate low marginal standard depict global kl mode increase pc show covariance previous see dominant reference appear superior choice monotonic l function present plot local combine approximate report plot pc truncation depend scale close achieve error ensure remain hyper local monotonically contrary local error strongly surprising expect pc order account local error minimize base independent standard pc aim hyper value induce transformation quality depend well insight measure plot eigen high thresholding see reference exponentially denote maximal reach impact clearly much moderate pc direction low insensitive decay contribution short scale filter coordinate robustness minimal yield maximum yield pick minimal quickly decay confirm covariance ccc indicate average solution stress first readily alternative consider technique polynomial instead proceed pc basis truncation involve certain focused uncertainty mode contrast uncertainty variance additional dimension propose amount uncertain numerical demonstrate variance average covariance remainder pc expansion expansion problem solution construct expansion prediction addition measurement direct discrepancy could advantageous illustrate interest field parametrize inference serve accelerate comparison purpose value consider inference covariance advantage field use diffusion correspond test field base profile cl inference perform measurement profile time uniformly solution measurement noise avoid diffusion equation solution observation significantly fine log profile unless depict location observation x solution observation times gp parameter choose inverse enough scale contrast diffusion problematic orthogonal gamma moment uninformative specify determination give instead chart require new finding reference average dominant mode solve obtain problem unless discretization involve approximate extract component constitute sampler pc determine surrogate transformation remain enough present multiply distribution acceleration distinguish offline adaptive posterior bayesian assign infer properly explore chain kl show kernel density kde coordinate posterior gaussian distribution quantify divergence quantifie indicate quantifie find kl plot posteriori map finding inference brevity ccc hyper covariance divergence kl plot quality infer median quantile posterior quantile large attribute pre hyper everywhere infer inference use gaussian function median posterior profile repeat previous hyper mcmc necessary pre chain hyper illustrate
laplace operator generator sampling section grant spectral operator regular basis orthogonal satisfy respect additionally need follow generator reflect brownian motion closure basis drift volatility satisfy strong regularity obtain weak namely bernstein fulfil applie fix correspondence denote ergodicity ergodic eq define projection introduce wavelet scalar since diffusion verify unbiased action basis gram v g ergodicity high argument prove j scalar schwarz inequality arithmetic vx vx easy eigenvalue function j invertible estimate laplace measure distance I structure show uniformly strictly invertible view formula plug diffusion need generalizing eq via continuous unstable boundary choose n view density ergodicity due estimating laplace neighborhood latter ingredient find control achieve magnitude generalize use posteriori bound g bound tend markov difficulty theorem eigenvalue conclude volatility ill pose drift estimator rate achieve convergence infimum estimator bound sample strategy observation alternative admit measure operator accomplish use schmidt explicit generator present numerical volatility estimation throughout mean drift volatility two use euler scheme compare four sampling shape observation symmetric beta rescale interval mean together distance depict scheme tp sample process construct oracle minimize present root square volatility interval obtain carlo surprisingly n exception decrease across allow small beta distribution error surprising beta yield uniform integrate estimation carlo volatility estimator problem see beta volatility trajectory four randomness observation ignore estimator design time throughout generator normalize increase generator uniformly proof gap equal due tv dt e tv tv tc ts similarly tc mc establish general integral follow variance nk kf l ks analogously cauchy inequality kf l ks easily geometric exceed upper equal cauchy cauchy series first assumption furthermore choose event inequality I I enough constant bernstein order solution eigenvalue q take suffice inequality adjoint definite u u consequently obtain claim since j l operator restriction operator hence generalize invertible generalize eigenfunction expect posteriori error control norm operator analyze norm rather refer finally prove argument invertible j v hence operator schmidt consequently chebyshev norm r eigenvalue nontrivial set hold j subset j restrict finish bind j event choose eigenvalue start continuous first estimation yield determine taylor intermediate denominator event cn supremum process relate yx envelope inequality exist proposition proof derivative exist let proposition especially high event v xu jx xu xu jx xu xx xu bm b triangle volatility since xu separate zero cauchy schwarz grant xu uniformly hence furthermore consequently drift obtain bind drift bernstein proposition obtain j error b yield easily since laplace transform parametric suffice use distribution lebesgue alternative compactly wavelet vanish jk k generators converge uniformly alternative therefore yield eq leibler uniformly leibl account times tx bx dy drift know volatility assumption ensure define obtain generalization assume nx nx lebesgue ensure dominate n lemma kullback leibler divergence r r l schmidt generator contrast generator laplace functional calculus map furthermore suppose decompose term take maximum estimate generator prove compact adjoint order eigenvalue eigenvalue eq project x variational eigenvalue v I consequently project operator gap projection eigenvalue cauchy z operator see z remain eigenvector x v want sketch posteriori technique extensively chapter error v problem symmetric generalize matrix q furthermore eigenvalue normalize nx I generalize order eigenvalue furthermore orthogonal definite purpose posteriori matrix cholesky normalize eq condition disadvantage relate method suppose provide exact intend compare theorem hermitian definite reader convenience n exact b theorem thm thm thm thm assumption theorem volatility coefficient scalar diffusion random construct ann estimation minimax sense adaptive sampling operator performance illustrate numerical secondary diffusion convergence continuous instance price scheme discrete distinguish remain argue realistic parametric naturally arise bad observation study nonparametric drift diffusion generalize reflect scalar one hand technical difficulty investigate economic reflect within reflect option individual affect act force f reflect brownian
use choice slice receiver slice summarize actor step observe infer likely year aggregate tensor cover aggregate tensor interaction omit receiver country actor activity receiver actor tensor usa china actors usa tensor million six million oppose exhibit show insensitive report error mae error mae nz hamming zero z correspond portion reconstruction incorrectly across display dispersion leave portion l consistently low mae low mae nz score unobserved portion l score simply many zero opposite observed complement predict portion low mae mae nz score magnitude suggest non portion consistent unstable section compare explain sparsity issue focus interpretability factor index date range tensor infer span date year infer component infer correspond decade anomalous tensor summarize relation take interpret cut tie correspondence five year difference date range illustrate correspondence also sparsity receiver depict ten receiver actor central possess activity factor component component receiver exhibit top actor dominate relation international exploratory tool explore localize anomalous interaction reflect activity span two year date range ever become zero alternate poisson away estimate latent factor practice solve instability mle efficiency empirically performance verify equation specifically expression arithmetic ik except point factor replace clear inference lee implicit correction suffer bayesian contribution geometric expectation define geometric construct variational arithmetic inference implicitly reconstruction expectation difference geometric arithmetic illustrative arithmetic mode parameter mode probability monotonically depict arithmetic low upper arithmetic grow yield much obtain arithmetic geometric approximately figure geometric probable arithmetic expectation close factor geometric use expectation r mae mae top top e political dyadic event inherently phenomena green et analysis dyadic biased independence continue dyadic instead event opposite viewpoint dyadic conduct analyse phenomena researcher begin dependency thereby effect analysis line research viewpoint latent factor exploratory datum specifically identify characterize international dyadic event exploratory reveal interpretable capture persistent anomaly analytically variational tensor instability issue negative tensor factorization count provide empirical demonstrating arithmetic recommend subsequent inference tensor david helpful discussion microsoft york city part part nsf finding conclusion recommendation reflect david tensor structure dynamic interaction decade political record country country know international develop bayesian poisson interpretable salient predictive performance tensor variational counterpart use bayesian exploratory tool political infer international behavioral dyadic international social process pairwise actor actor researcher facebook however explicitly time dynamic infer process task international decade collect record country country e traditionally help international group study less scale create several set automatically extract encode dyadic event news modern differ previously behavior document micro behavior day although potentially picture international effectively new latent relation event component span china aim international concern north top top beginning component infer span lead attack month occur attack paper poisson factorization infer dyadic scalable variational exploratory international relation party localize activity attack dyadic aggregate element count country toward country tensor salient latent tensor derive dyadic rarely interact another non interact traditional unstable count tensor tensor gamma avoid validate outperform highly algorithm traditional relationship explain construct latent researcher geometric use geometric increase infer factor use involve bayesian matrix bayesian tensor factorization exploratory tool political demonstrate infer interpretable international year researcher dyadic extract news database daily parallel started collect dyadic event forecast political instability publicly integrate comparison code dyadic piece type target actor code country information hierarchy action sentiment action study international code datum origin actor country actor recognize dyadic aggregated country aggregate event element country toward country date entire set step tensor tensor million element zero must dispersion decompose factor representation salient pattern tensor common tucker canonical cp generalization singular decomposition tucker decomposition way count tensor cp treat count know tensor aggregate ik kn four example vector receiver factor length infer perspective reconstruction poisson e perform via maximum count often yield estimate matrix draw assume poisson rather impose prior full bayesian inference pmf image music topic recommendation bayesian pmf bayesian pmf impose gamma thus mass zero maintain tail induce prior define impose four prior latent factor gamma mean throughout encourage sparsity interpretability factor invert posterior hyperparameter approximate typically facilitate product gamma ik pmf variational parameter fit exact kl eq factorize achieve coordinate ascent iteratively update hold parameter auxiliary pmf omit update arithmetic expectation since expectation g q sufficient efficiency efficient even optimize via empirical iteratively variational inference intend support arbitrary tensor addition tensor describe use complement portion slice percentage non element dispersion portion report type absolute error zero mae nz hamming loss achieve slice four portion kl r mae mae nz mae mae nz nz z top top top e validate
pr py source termination message form backward combine reader familiar bayes refer probability source solely forward global focus maximization translate example apply kkt initialize n f usually numerically discuss block forward message distribution architecture follow building latent show finite alphabet architecture code label alphabet come heterogeneous powerful whole increase bottom belong box upper connection bottom stochastic pointing assume source child architecture categorical alphabet represent independent visualize draw index nm mx px propagate collect usual flow simultaneously follow rule incoming combine produce imagine act combine connect branch handle architecture backward message come towards branch message latent feed forward build generally collection spatially distribute architecture red blue architecture represent layer bottom patch patch latent message among subset build across scale system inference generative latent message forward cone reveal associate cluster image layer code role propagation delta root reveal specific system impulse response encode bottom variable backward delta diameter hide contribution message soft code pattern completion message patch patch patch example patch termination source iterative outline deep deep network large mode check check generalization specifically follow architecture pixel randomly select patch layer build connect variable another pixel bottom layer backward learn b block connect block pixel backward message learn block cover take car filter diffusion filter filter obtain car alphabet filter patch extract show fig previous use matrix learn mode generative distribution delta gray pixel symbol complete respectively forward quite forward represent orientation early visual distribution large scale pattern patch extract bottom considerable amount patch delta backward resolve quite uncertainty experiment pattern pattern never obviously succeed quite complete learn generalization easily determine space result paradigm successfully deep architecture retain contain internal layer layer choose extract paradigm salient structure object character different constitute proposal deep literature flexibility modularity network introduce type scale add let take care adaptation mix supervise unsupervised architecture complexity issue paradigm embed
version inequality variable could matrix choose distribution computing surely handle n rp z follow complete product subgaussian tail lemma go bernstein inequality prove bernstein concentration prove jx xx z ok therefore complete use ok ok ok claim bernstein state technical lemma control variance follow z ax si repeatedly incoherent recall subgaussian fix complete next ok sx ia ta si term separately column column si ok ok expectation ok ok ok ok ta ok ok ok complete bernstein ok ok nz claim truncate log bernstein spectral norm x xy ty xy bind calculation bind term eq matrix bernstein complete order bernstein form claim subgaussian hold tu entry subgaussian yy v yy apply bound notice yy yy ok variance apply bernstein number field implement field figure simple layer neuron output decode respect middle layer weight middle moreover top layer equip layer bottom outside remark update attention accomplish spike update weight execution new bottom layer neighbor decode layer repeat implement appropriately sketch implement neuron network perform product neuron even receive sequentially generalization principle top singular also inner onto required life accomplish stochastic fs include basis enable optimization minimization work result provable somewhat outperform heuristic give understanding minimization provable architecture motivation introduce algorithm work limit incoherent dictionary previous parameter improve upon application setting function think minimize allow leverage variant common methodology field enable datum convex heuristic alternate provable somewhat surprisingly outperform alternate minimization heuristic alternate design provable seem architecture field give code almost limit finally believe framework code patch number many wide turn appropriately representation improve efficiency brain sparsity constraint fitting tool feature important segmentation retrieval super building involve notion formalize dataset vector matrix whose encourage subsequent choose large overcomplete adapt give coding call recovery gave try give evidence matrix image portion author place familiar setting whereby datum assume appropriate lead surprisingly seem unnecessary practice energy hard constraint work mod svd fact method rapid progress design polynomial code provable guarantee paper discuss generative viewpoint success largely far new simplex ellipsoid behavior successful new time code simple intuitive important beyond efficiency architecture roughly weight entry potential also general analyze algorithm relative field see rigorous bound sparse case dramatically recent add literature heuristic solve approach representation alternate solve problem update appropriately rich heuristic neuron remarkable e plausible basis optimization problem many heuristic offer computational explanation possible rigorous sparse broadly computation instead rely solely code give solution give column work overcomplete give overcomplete incoherent next former work give depend alternate minimization close give square sparsity run time however exponential give error decrease geometrically work sparsity remark empirically alternate minimization appear much generative similar rigorous paper polynomial assume generate dictionary normalize pairwise subgaussian support noise motivation representation requirement expense restrict sparsity iid little like singular vector long stay incoherent include wavelet whose column incoherent relax permutation sign distance allow high sparsity framework instead try update rule provide good gradient heuristic plausible converge geometric complexity step implement algorithm sparse provable general technique use alternate rule whose initialize near error polynomial modification carefully project component along complement analyze variant however analysis initialize algorithm near high setting help return sample complexity algorithm admit implementation appendix currently logarithmic incoherent algorithm exponential implement operation framework sparse code simple context suggest new way analyze algorithm update opposite appropriately challenge identify progress lyapunov improve probability progress difference subsection identify inspire try convex strictly analysis turn propose though movement decode procedure functional specifically true gradient really movement flexibility different need quite either compare bit next bit please movement setup move expectation contrast expectation true gradient bias bias algorithm approximate error connect relate sa abstraction general look desire essentially sake relate correlate fairly flexibility decode turn variant take useful et em require gradient close plug allow decode step step check movement conceptually solution step guess natural sufficient correlate convex correspond strongly geometrically systematic optimization solve recurrence z second part theorem constrain iteration euclidean z far lie update systematic derive functional various rule move approximately direction certain course know update direction unknown nevertheless whereby converge update rule get optimum enable analyze sign rule approximate project descent error project along column currently go get early rule analyze denote decode main give useful expression generative order correlate lemma across take simplified essential suppose iteration geometrically defer show ng I correlate k ok mn proof omit notation assumption pointing direction extra triangle g I corollary apply I I ok mn lemma last notational I note close distance term right get simple I complete near previous crucial rearrange g q write convenient matrix side first straightforward use term introduce auxiliary matrix I claim iv moreover frobenius spectral norm ok ok last inequality put piece complete induction induction hypothesis inductive say inductive invoke prove large norm induction theory term initialize usual analyse give novel succeed work support vector correspond recent throughout execution suppose moreover incoherent k define theorem main invoke time order support share element show sample let q norm expectation v v sx ta ta j independent expectation whose rearrange next useful set recall incoherent subgaussian diagonal invoke eq q term q complete inequality invoke ok one symmetry collect frobenius om claim main conclusion go beyond idea minimization break alternate minimization random heuristic analyse minimization method correspond incoherent dictionary incoherent I ci ic concentration intuitively strong randomness independent subgaussian disk therefore subgaussian union complete even long correct randomness strong randomness elementary wise I I j choice j ok nr kk however expect hold bound proof mx jj r high subgaussian variable ci complete really require expectation maintain estimate proof repeat maintain give analogue suppose remark lemma although condition indicator invoke event happen let event happen omit eq strategy next linear lead contribute term ia mn bound gm norm term complete proof ready except invoke notice project constraint projection subgradient method nesterov seminal book detail update rule preserve property summarize follow claim second part complete fact increase analyze particular converge globally project ensure correlate frobenius norm theorem suppose choose proof substitute recall write q recall contribution early rewrite lemma imply near third project onto rule maintain avoid intensive difference repeat calculation sketch denote also hence term term verify remain note
gaussian place universal place fix say admissible moreover notice fix mixture admissible remark somewhat arbitrary choosing obtain fraction minimal fix recall statement obvious half also place half interval intersect split contribution first term interval since let p b bounded follow xx interval rhs intuitively close unbounded far since gmm concern interact need suitably parametrization linearly transform domain gets map fix write rescaled variance denote give say rescale property rescale follow variable know furthermore set follow suffice return algorithm inequality find back polynomial line know negative component valid k rescale round weight simplex check p lemma argument complete proof show univariate parametrize set fact zero algebraic plausible candidate satisfie attempt believe natural interesting proper agnostic mixture algorithm linear brief give agnostic mixture laplace addition mixture gaussian run class mention arguably difficult mixture condition parametrization laplace trivially care demonstrate parameter however issue introduce polynomial program degree taylor expansion laplace mixture thus complexity nearly constant schmidt gmm fundamental theory learn gaussian norm give univariate k ok logarithmic achieve good k replace moderately dominate run achieve recently state researcher progress approach offer agnostic gmm gaussian agnostic closely agnostic recently question proper fit solve entirely deterministic carefully polynomial encoding inequality apply learn class besides popular practice sample gmm encounter normal study seminal include biology gmm sample rigorous notion gmm community year relate algorithmic decrease hardness gmm mean variance mix weight variance proper equivalently total variation hypothesis unknown small htb ccccc learn cm ok estimation low bound arguably guarantee recover instance physical wish cost recent parameter show necessary mixture univariate tight give scale exponentially univariate quickly prohibitive mixture reasonable improper small work possible gmm tight logarithmic factor produce gmm disadvantage interpretability attractive gmm unknown somewhat proper many gmm require produce accuracy estimate increase gmm allow expression quantity learn expression density return produce gmm output peak smooth make interpret identifiability gmm ideally proper learn interpretability density recent factor complexity understand estimation nearly run become scale much density resemble exponential bind show avoid nearly run time nearly worth proper moreover arbitrary explain properly parameter offer generate assume gmm produce give sample truly gmm gmm rarely think phenomenon distribution far gaussians set pdfs achieve incur onto gmm universal pdf agnostic agnostic approximation hence agnostic produce hand agnostic gaussian agnostic open question see progress increase agnostic outline contribution paper restrict attention many understand main notation pdf let run optimize front instead learn gaussian closely make question imply time special simplifie bind moreover complexity density factor compare algorithm run moment algorithm offer agnostic guarantee hope worth agnostic significantly agnostic hence understand tractable agnostic guarantee gaussian mild offer particular applie appear polynomially polynomially convert purely learn algorithm mixture distribution learn available differ essentially proper gaussian core mixture invoke complexity piece sample obtain density fit gaussian entirely deterministic fitting reduction gmm solve carefully design polynomial solve system reduction polynomial inequality main technical directly fit challenging pdf gaussian gaussians e piecewise consist piece restrict polynomial must pdfs inequality easy convert restrict back proper polynomial good polynomial encode guarantee proper approximation require intersection piecewise polynomial integral difference instead minimize minimize vc theory exactly intersection norm norm norm polynomial inequality norm would exponential polynomial polynomial inequality give gmm gmm doubly work real ram different specify sufficient rescaling parametrization inequality length interval piecewise fit near serve simple identifying combine idea outline mixture gaussian carefully design crucial inequality lead polynomial inequality impossible summarize body work attention provable corresponding notion outline subsection picture current reference therein seminal work start long community g reader give tight parameter mixture univariate gaussians necessary price give complexity offer weak many attractive become tractable dimension smoothed knowledge take mixture unfortunately polynomial dependence underlie note necessary complexity component paper properly candidate construct hypothesis none conceptually gaussians component gmm component return moreover improve worse hypothesis polynomial advantage section utilize assumption introduce shape distance proxy adaptively restrict polynomial properly inequality approximation denote fx x measurable one paper pdf precision pdf iw iw k component k kk result work algorithm density estimation subroutine gmm estimate carefully system polynomial formally inequality either boolean formula predicate relation predicate inequality satisfie result let find solution case run gaussian pdfs estimate directly polynomial piecewise order piece flat tail piece taylor around expansion always exist shape polynomial inequality restrict q triangle encode estimate distance density denote family interval kf q property gaussian zero proposition gaussian pdfs distinct fact corollary variance distinct imply claim gaussian behave use polynomial live scale solve polynomial inequality work density obtain suffice solve follow use restrict proxy part polynomial norm approach per component weight precision quantification quantify boolean polynomial finally solve behave know polynomially good approximation parameter yield first run subroutine occur system precision could extremely approximate take arbitrary shift scale shift since solve apply rescale estimate interval note mixture large accurately clarity presentation ignore control appropriately point show overcome limitation formally behave estimate rescale support behave gmm behave parameter learn detail behave case illustrate difficulty proxy shape require zero polynomial zero auxiliary simply introduce run since polynomial lead become goal avoid dependence system inequality mixture gaussian contribute significantly q estimation succeed gaussian triangle connection norm attention computational perspective polynomial regardless encode convert solve feasibility encode norm section general particular wish optimize polynomially adapt algorithm well section setup let polynomial piecewise algebraic moreover assume membership formula predicate degree moreover satisfying polynomial maximum polynomial polynomial indeed suitable piecewise inequality hence suffice set constraint combine constraint piece encode know become polynomial encode integral piece set ii appear iv boolean constraint encode set permutation represent piecewise piece integrate inequality encode integrate expression trivially integrate integral individual piece valid depend fix moreover check tool encode constraint interval satisfie follow simply count addition optimize boolean predicate require encode quantification variable variable precision system system
yield grained experimentally show explanation augmentation extension architecture yield competitive mnist fraction unit deep turn input response linearity dropout multiply mask probability half unit input drop magnitude active hide formalize work help set learn function input need datum mean training region learn mapping cover portion help boost coverage make noise regularization increase hide view hide less clear globally transformation possible activity transform project noise linear input dimensional noisy bernoulli adversarial square equation generalize define generalize unlikely find minimize detailed subsection fortunately hide layer representation th layer sophisticated augmentation respect raise question answer trivial inside network without noise make hypothesis dropout adaptation projection differ always version layer large reasonable dropout project noise back view argument experimental hide section dropout layer representation roughly corruption present manifold span entire manifold along linear manifold non let represent dimensional map representation representation could overlap small case representation overlap insufficient representation least overlap representation overlap view data arc belong arcs arcs class separate arcs class create arc length arc arcs class total train single contain randomly run representation presentation number training mainly arc single layer mlp arc dropout result corruption view create space sample work corruption report noise anneal help supervise assume imply noise layer rich augmentation schedule propose alternative avoid schedule rather slowly change corruption sample training sample uniform show unit actual test non linearity function half run relu unit train noise give mnist cifar experiment dropout give mnist cifar relu latter seem mnist cifar inference turn binary result mnist cifar rely magnitude consistent hypothesis sparse sub region sub region run experiment mlp architecture mnist cifar study effect scheme mlp relu three mlp architecture hide mnist split epoch training mean percentage run permutation consist pca whitening preprocesse reporting evaluate fix noise noise increment experiment hide increment dropout noise scheme layer level increment experiment cifar dropout dropout input dropout train cifar mnist layer zero noise corruption plot layer scheme sparsity representation sparsity limited experiment corruption unit zero deviation varied level effect back input second input dropout back generate approximation solely epoch epoch generate noisy input hide bernoulli mask cifar dropout input suggest project neural generate drop cifar idea dropout augmentation suggest network plot layer dropout corruption cifar representation term class lot new work suggest augmentation give evidence view require understand augmentation evolve architecture augmentation space rich augmentation test work support education project research award grateful towards find property composition rewrite exist expand compose apply modification p ip percentage probability hyper layer sparsity start fall cifar probably cifar dropout
kl heavily auc semantic smoothing wasserstein favor auc achievable deal value combine loss two test illustrate relevant despite overlap irrelevant error example music car band water water water wasserstein wasserstein costly describe regularization connection index wasserstein encourage respect output tag baseline interesting work may wasserstein encourage equation relax objective w diagonal eq risk minimizer bind bound wasserstein depend empirical rademacher value call random control risk stability constant empirical let lem hold probability jensen role conclusion finish est kk space wasserstein lipschitz lemma wasserstein define probability valid around function softmax softmax proposition wasserstein distance wasserstein follow continue gradient theorem immediately conclusion follow trivially zero uniform softmax lipschitz singleton identity sample calculate theorem thm lem vector specifically mean th column zero equal h feasible plan wasserstein prediction proposition case become diagonal assign arbitrarily long meet q scale wasserstein define wasserstein distribution overlap mass mass cost q conversely give without generality follow alphabet integer pixel metric plan euclidean ground feasible direct support match respectively denote q notice obviously assume lem derive relation upper inequality prop fall illustrate multiclass label correspond lattice semantic similarity flip neighboring category figure level repeat multiclass classifier standard wasserstein average performance function tag yahoo filter well dominant lexical noun location noun proper vocabulary remainder occur select tag frequently occur tag art nature family tree water architecture car live cat sign road build new cloud tag flat master drift wasserstein loss wasserstein loss use enforce mini momentum optimize function redundancy semantic tag tag tag tag image pick run country build road house family education weather people b center machines institute brain institute mit international e predict output challenge output improve wasserstein distance optimize wasserstein costly efficiently compute efficient regularization wasserstein unnormalize statistical connection index wasserstein loss encourage prediction output tag yahoo dataset achieve superior baseline eliminate comment problem use scenario multiclass class imagenet challenge acoustic speech segmentation consist model metric semantic adjacency capture euclidean distance pixel classification task organize hierarchical label space also call presence metric measure severe intuitively incorporate metric favor completely r multi learning incorporate measure cost plan move match apply include propagation represent use performance multiclass indistinguishable category human label make class plane switch neighboring loss wasserstein loss prediction euclidean incorporate wasserstein ground truth noise section describe experiment cm contribution learn wasserstein divergence loss wasserstein loss base justify erm wasserstein draw connection synthetic real world annotation demonstrate incorporate decomposable popular value lead algorithm nonlinear post distribution distribution wasserstein simplex graph minimize dirichlet wasserstein formulate compare retrieval contour problem discriminative estimator cost space measure set shift recommend choose tangent normalized segmentation example shape represent truth simplex generalize unnormalized propose unnormalized measure penalty divergence result z division optimize satisfie hx u represent element multiplication unconstraine gradient hx smoothed enough smoothed normalize output relaxed wasserstein nearly b unnormalized learn wasserstein two full let risk q composition function softmax lie simplex probability constant commonly decay train achievable important minimize risk wasserstein multiclass multiclass risk note semantic prediction point capture wasserstein probability wasserstein intersection region metric space wasserstein plane ground wasserstein grind w pd incorrectly far
unbiased unbiased selective state selective selective hypothesis away selective scenario root simple hold stage root detail turn main inferential also law selective likelihood lead root inferential tool multiscale derive lasso shorthand select kkt condition shorthand simply subgradient note square root square root usual rewrite kkt calculation kkt write strictly speak calculation define quantity event question computing yield take closely usual imply one test selection explicitly design deriving eq easily computable inactive event inequality recall question interested q provide simplification show independent though use formally every dropping law fix specifically convenient convex otherwise goal q without like test nan q relevant law freedom computable truncation law truncate precise choose convenient parameterization inequality inequality solve explicitly one intersection convenience happen event event restriction residual fit inequality define event model observe inside usual p law score selective orthogonal consider separately pseudo setting rather estimate plug quantity law statistic statistic estimate consider perhaps selective direct inspection equivalent truncate law truncate root truncate pl estimate vary limit likelihood ol obtain estimator root call pseudo likelihood degree freedom estimate think improper density proportional recover estimator event lasso compare literature validation generate absolute sign lasso root discover ratio selective variable screen close usual ols root include comparison pseudo variable coefficient result ht coverage consider example various anti point ht mutation naive selective v p e drug marker predict quantitative square tc square generally impossible multiscale problem piecewise prior get noise level formalize problem part assume discretization recover inference balance goodness impose penalization motivation solve scan statistic get active knowledge level analogous subgradient selective l repeat q connect multiscale work author multiscale column easily highly screen kkt kkt repeat residual get solve residual copy genetic variation human genome dna link development disease provide useful simulation gm ratio ratio well selective splitting produce example demonstrate splitting dominate procedure second datum even yield fact gaussian formal justification consider square root description interesting partition group lasso sign identity select two affine affine stack partition set course setup law result reduce inference parameter projection subject affine conditional notation jx jx j drop set distinguish two either function law linear functions eq law active drop inactive event interested row law ti es ti es constraint possible insufficient otherwise estimate order must natural projection sphere ti e affine truncate scheme gaussian stage rather pool unbiased ht interval short interval use inference stage mutation datum l p p p hold tc table selective scheme tc name proposition inference level selection square lasso selective selective test inclusion select root level perform well root hold datum inference observe drug selective consider model natural canonical continue fair recent work assume though crucial selection procedure value focus suggest angle could carry unknown approach distinction tuning post equivalent square root modification eq lasso know lasso hypothesis equal parameter roughly view would enter lasso square root convex differently literature analogous use convex objective view procedure model square sign notation investigate kkt tucker basic selective describe broad post notably canonical begin specify formalize determine ask selective attempt selective critical vs selective error determine take distribution
connection trivial integer real usual lemma concept dct consider average impose dct concept previously describe formulae dct describe must generality signal nan assumption affect dct spectrum inverse admit derivation arithmetic dct average dct tailor average next specific derivation mathematical average exposition order average th act average algebraic one thus perform suitable recognize always consider impose act issue precisely find inverse say separate useful notice lead unitary sequence situation address inverse analytically dirichlet eq digital multiplying number bit possess regardless expression direct inversion tell unitary alternate unitary constitute simple dirichlet however formulation nevertheless indeed definition establishes connection formulae follow probabilistic integer zero assume signal value convert nan subtract consequently separately address problem fractional sample interpolation way let integer dct formulae construction take direct formula invert act weight sample fractional combination uniformly fact stem orthogonality transformation kernel investigate suggested function inherently act usual derivation summation expand follow run obtain return double block act dct approximation argument illustrate act example short dct widely adopt code subject extensive act identification interpolation fractional depict diagram act interpolation procedure fractional employ act average block act averages implement act combine function cf approximate dct spectra randomly signal distribute uniform result mse due figure comparable approximation dct cosine transform construction matrix v matrix accord nk matrix need construction matrix thus singular determinant unity inverse inversion element stems additionally express spectrum act spectrum dct due combination ii function term dct dct spectrum decompose part dct extend interpolation act thus invert inner weighting depend independent separate define weighting average eq row weight proposition indicate across application part q effectively relate function nan mean always dct signal establish dct consider algorithm nan signal contribution arithmetic transform ii arithmetic interpolation issue introduce arithmetic cosine devote dct computation therefore application dct differently formula framework arithmetic adequate interpolation exact dct length particularly exist error tend consider evaluation issue could adequate solution act order collect fashion act need sort dct concentrate computational perspective transform filter motivation arithmetic cosine paradigm existence act fundamental building arithmetic transform direct could attractive exact interpolation signal open topic author provide scientific partially dirichlet inverse examine dirichlet series result q find put product dirichlet accordingly final already format simply return write denote convolution conclude odd q admit form integer multiplicative multiplicative sequence maintain q dirichlet convolution possibly conjecture example cr cr tag de work electrical ab mail act act design efficient implementation new mathematical tool demonstrate interpolation provide application transform arithmetic fourier tool spectrum main nature part additional improvement expand dft fully arithmetic lack impose sample sensitive incoming sample necessary essentially two signal hand drawback sampling discard option sample method obtain arithmetic way order interpolation consider although interpolation attain acceptable block length consider length imply enough totally meaningful way spectral component domain testing build hardware consideration direct enhance dct describe dct obtain mean dft spectrum map dct spectrum goal dct cosine act method examine arithmetic average exist arithmetic fourier transform average tailor dct act exact
summation know growth know pick decentralize bipartite illustrate similar bipartite n exploitation il il player event il il event hoeffde bind combine unknown similar omit scenario chain arm irreducible represent I stationary markov since ergodic arm x max xx x max I ip irreducible reversible let gap obtain jt denote contain j j ji mutually player reward player successful play without player frame time player pick max xx jx j x x ts prove reward lemma occur tx jt jt jt jt n bipartite representation decentralize markovian ii de choose slowly increase arbitrarily decrease regret exploitation clearly similar equation ct il il theorem problem bandit bandit pick arm distribution design policy maximize reward player separate communication costly yield expect grow low centralized case algorithm work fundamentally address question incur decentralize solve relevance domain decentralize system cognitive system model understand choose bias repeatedly instant instance coin reward know bias coin help discover coin bias exploration versus exploitation must well option current widely planning sharing etc say formally policy could employ show grow slow subsequently generalize play pick multiple bound reward seminal simple policy logarithmic policy see interest well bind deterministic sequencing phase also appear approach like probability exploration policy bandit general finitely arm policy single player pac bandit grow bandit motivate network wireless spectrum try channel channel look user statistic wherein maximize channel dedicated channel expense imagine setting wherein user arm rise question decentralize inherent second index regret bandit decentralize regret logarithmic achievable reduce ranking solve decentralize appear setting make become decentralized quick propose mechanism player answer two question decentralize insight exact decentralized policy regret however partially policy stand spaced exploitation policy regret grow near assumption factor exploration exploitation pre long exploration phase policy answer fundamental question inherent cost cost policy introduce policy e chain reader extension markovian set simulation conduct evaluate compare decentralize previously know formulation prior work new study performance player instance much appear present armed instant identically process support distribution unknown choose history reward action arm objective choose reward mean solve play reward notion compare policy cumulative obtain formally player minimize policy eq arm take great bandit sensor useful computation unit index computed refinement compute arm armed bandit refer channel dedicated player however play arm signal signal regret hence communication choose player pick regard player player unknown bound playing time player want horizon know bipartite unique player computational cost communication cost distribute match exchange incur occurs expect minimize q define feature exist single single player bandit capture class policy al seminal index regret work play large mean arm play exploration exploitation incur study context quite arm gets play become unlike omit dependent numerically empirically outperform overcome frequently difficulty extend regret regret computation q worth sub actually logarithmic growth major encounter bandit policy channel player natural expect q frame length difference difference reward precision matching slight unknown policy player bandit generalization player exploration exploitation epoch try player choose maximum index policy exploitation player epoch initialization play arm compute lt l phase play arm play arm perform else largely arm exploitation value concentration reader hoeffding inequality range well bad e make arbitrarily extend bandit effort section generalization single exploitation communication exploration phase explore arm round phase represent turn end exploration phase player protocol aside bit contribute regret bipartite match line yield k k player process early bipartite stick distribute phase phase successive initialization exploration player play arm match player initialization set exploration arm number play reward trial success else obtain play player price prefer player player price spend consider term end add cost precision index bipartite matching precision run round precision two index close happen player bit specify cost communication communication constant phase phase player preference bid index player receive arrive channel optimal exploitation denote context bipartite index know choose de de ts note choose sequence thus slowly increase arbitrarily slowly sequence make close near policy cubic extensive performance propose policy respective scenario simulated mean ranking algorithm player policy compare known consequently logarithmic performance slightly account scenario see next computation retain logarithmic grow linearly present policy generate independently horizon million tolerance performance average unit distribute bipartite
relationship function copula hx continuous uniquely conversely copula joint margin continuous margin function copula generate transformation extend representation joint induction generator approximate joint start f x x conditional x px result formulate joint let convergence x unit hypercube theorem tensor n equivalent give belong copula joint minimal euclidean respect copula density function cx x j relation instantaneous process represent htbp approximate present result sde secondly unitary lastly martingale continuous dynamic close corresponding purpose n fx infinity converge function pp detail norm approximated would imply approximated assumption sequence chain generator function statement process converge generator let follow uniformly chapter path use generator follow generator convergence discretization unit locally write taylor obtain ax x x ax ax ax ax ax ax uniform ax x I ax ax ax precise see process infinity n fs n x generality theorem distribution consider integral sufficiently small second integral h like limit get eq way order approximate marginal term calculation n cn equivalent cn cn cn h h mix derivative h exactly term result diffusion belong martingale problem assumption smoothness sde extension martingale central martingale existence sde refer property sde per db continuous f martingale pose sample matrix definite ns ns ns r r local martingale martingale particular valid x relatively stop convergent subsequence x kt stop process bx lemma generator well uniqueness stop hence imply dependence attribute describe unique multidimensional interpret functional specification involve generalized process develop demonstrate obtain multidimensional start multidimensional decomposition define general generator multidimensional generalized demonstrate copula provide equation convergence multidimensional sense microsoft property max gray light gray department university college martingale decomposition generator multidimensional diffusion sde symmetric nonnegative matrix value sde denote c smooth compact aim address propose weak generator emphasis derivative decomposition approximation scheme generator conditional generator copula propose develop characterization furthermore support constitute little look multidimensional investigation focus aspect decomposition martingale continuous chain approximate dependence structure throughout volatility form joint coupling structure drive framework specification motivated deal representation process markov room far chain fact always treat algebra model multi whose construct generator characterize construct among apply discrete generalized diffusion approach associate multidimensional markov process approach theory diffusion decomposition literature manuscript multi process construct instant drift process interpret terminal extend functional take functional equation markovian framework process conditional diffusion coefficient project manuscript generators multidimensional generalize define copula specification paper organize introduce correlate characterize generator convergence approximation place zero measurable countable probability sx ty denote operator axiom relate law start assume contraction generator continuous property entirely fact solution compactly belong generator equation sde bs cs dynamic law drift operator specify solution rigorous formulation martingale straightforward sde martingale martingale operator representation uniqueness result define mention develop sde adapt condition growth theoretical construct specification process sde approximation involve characterization cross tensor generator algebra make correlated generator emphasis mix decomposition generator first illustrate sde approximation block approximate construct element n discretization boundary consist possibly complement construct building equation ax ax ax ax ax ax bx uniform alternative discretization generator markov process particular approximated generator process furthermore process discrete impose condition state law previous instantaneous moment coincide x z instantaneous local moment introduce approximated generator notation process generator multidimensional process matrix instantaneous moment describe generator markov jt jt two jt jt x rewrite independent markov operator dependence way namely dimension operator action state space approximate orthogonal introduction x k process approximate generator markov act local x cn cn mn z pn pn pn n furthermore differently conditional n x z correlation finite partial x j f note apply decompose j x observe operator act joint product discretize discretized act along difference pn intensity x l l mn pn pn pn generator bivariate correlate act hilbert generality cn z mn z pn z let generator generalize assume locally jt generator operator dimensional consideration therefore let generator whose matching x ax x ax ax ax ax jx generator instantaneous intensity report instantaneous calculate states probability rewrite py j py x px I nx jx entry identical impose large two straightforward independence multidimensional
rademacher complexity borel lemma university ann microsoft research microsoft research bb v prove minimization unique unlike first sequence predictor risk source hold minimization technical free space np even approximate computationally attractive utilize amongst predictor limit error effectiveness specifically address predictor minimize sequence converge concrete object minimum rather minimize high dimensional infinity straightforward predictor minimize draw say unclear highlight thus boundedness question minimization main theorem function set linear consist vector function countable formally instance arbitrary collect differentiable loss defer study loss exponential population size lastly define excess risk probability logistic sigmoid early limit probability hypothesis sequence existence existence sequence compact metric construct duality duality carry consequence utilize approach handle secondly conditional probability convergence property property resolution eq satisfie natural apply introduce exhibit make essential sequence encounter strictly generalization particular coordinate logistic put rectangular red region define separate negative convex attain norm grow lead sequence regression give unique resolution point analog uniform apply margin dependence achieve sequence behavior concrete g gx rr might relationship exclude desire occur true along metric choice equality proposition I risk classification risk example generalization classification error problematic behave provide consistency function see dense complementary small sided close collect symbol precede subsection continue view whereby drop integration subset symbol ambiguity sometimes risk excess risk paper meaning satisfie class denote continuously differentiable restrictive classification loss existence conditional challenge hypothesis space begin study hypothesis finite uniform problem exponential recall infimum exist strictly separate example achieve risk study minimizer convex minimization linear entropy loss different kind unnormalize dual unnormalized unnormalized entropy uncorrelated make reweighte note unnormalized entropy slope zero always unlike primal minimum attain differentiable optimality rewrite define q absence label q example value infinite space infinite construct finite minimize dual unnormalized return formally give rise negativity objective question intuition construct qx qx qx large absolutely include slightly objective finite one candidate space banach measurable allow function work space call detail construction begin convexity serve role th define ball ball q outside measurable choice ready introduce taylor space conjugate contain finite iii constraint via adjoint transpose adjoint map constraint require write fact constraint apart result dual well optimum look technical primal optimum exist use qx qx yx qx nan obtain follow differentiable optimum optimal appear prove slope loss informally dual avoid unless force fundamentally see call difficult build addition alternate positive negative predictor return predictor equal predictor margin receive always still weight prediction wrong origin risk along case orthogonality slightly perturb risk along pattern minimizer exist example minimize perfect achieve call separate non nan measure receive margin difficult hypothesis optimum call risk prove highlight implication difficult reason hypothesis optimum arbitrary measurable difficult optimal dual optimum attain maxima moreover dual optimum difficult main restrictive class twice classification loss lipschitz every finite recall easy take becoming give loss measurable let r r r control constraint difficult positive margin incorrect prediction increase minimizer reasoning risk must fortunately hold e difference split difficult four either turn control require range low derivative mass control technical include bound risk key need separately optimum scalar apply piece handle proof treat easy risk predictor actually view half finite generalization remainder focus eventually span optimizer enable application complexity generalization span take hypothesis correspond complement kernel subspace span interesting perspective risk orthogonal complement risk convexity negativity rearrange rademacher generalization consequence class difficult loss optimization force large finite hypothesis finite correspond canonical difficult set base imply control piece scalar subgradient canonical give db heavily influence point risk moreover instrumental care behavior lastly notable prove multiclass classification find extensively boost analyze convex without adjustment space allow treatment non point topology adapt hand appendix cover banach measurable finite p analog inner banach banach space bilinear describe versa banach endow topology imply topology topology topology pair topology bilinear construction compatible begin rest pair banach endow compatible topology give banach everywhere graph close banach close theorem conjugate fu statement equivalent optimality measure assume give banach bilinear measurable function establish conjugacy adapt say banach decomposable proposition page decomposable close decomposable pointwise optimality equivalent complete proof contradiction I segment connect pointwise differ banach map finish state duality adapt strong space close proper function operator maximizer discuss appropriate banach appendix banach space space begin around zero norm ball clear space definition summarize part function identically hold also banach space norm measurable must rademacher rademacher random variable rv rademacher absolute summation essential thank task control deviation approximate collection scalar rv z alternate iii iv consequently l census eeg letter horizontal quantity bfgs scale quantity relevant appear rademacher function please wide little observation primarily motivation depict conduct uci repository split logistic yield set logistic bfgs point split bfgs code relaxed termination order provide early avoid please roughly capture norm predictor error whereby behave whenever satisfie imply grant since first grant positivity precede property grant concave hold note increase statement continuity large eq concavity manually check loss let e consequently jensen normalize measure grant maximize positive expansion eq convenience start entail entail secondly entail entail entail everything bound already taylor lemma establish inequality tight derivation consequently mind agree along combine see go suffice measure note optimum evaluating turn moreover odd lastly statement follow h follow side provide equal particular imply norm dual taylor obtain banach space topology weak finite function mean definition yield decrease also therefore eq e law via duality banach space topological argue invoke denote yield remain continuous map finite decomposable imply show continuity finite optima since may condition start qx qx q adjustment q feasibility construction far attain consequently consider adjustment furthermore primal r inequality actually differentiable imply strict qx qx ir ix rx rx yield differentiable strictly whereby ii optimizer technical useful proof optimum satisfie everywhere strict nothing entail derivative coincide suffice scalar piece piece along every univariate neighborhood lipschitz continuous lipschitz obtain eq definition subgradient grant finish otherwise piece appendix part course attain attain imply consequently eq imply also primal coincide prove establish proof inequality finite hypothesis set dividing give prove apply control increase structural optima use whenever extra purpose suffice predictor lead hypothesis give difficult also optimum let already suppose define adjusted meaning must remainder continuity u u u u sign set r rv rv way q presence incorrect prediction precise control briefly logistic loss derivation tie pair handle thank old cf controlling behave advantage region rearrange suppose scalar construction imply interior image optimality taylor inequality final integrable side make rearrange taylor expansion q convert iv grant q jensen bind immediately imply convergence depend two function definition mean split subsequently control simplified apply bound definition equal apply construction depend desire show suffice contain expand case grow go text develop degeneracy problem scenario learn since equivalently mind relate boundedness hypothesis definition whereby hypothesis every whereby end result follow continuous infimum compact give
eq wise batch nonlinearity replace normalization need dropout style add target give denoise traditional encoder decoder calculate connection vertical connection connection unit vertical project wise connection drop sigmoid nonlinearity parametrization use layer low denoise multimodal multimodal ratio distribution decoder connection analyze path training combination determine much parameter encoder encoder ten impact auxiliary training evaluating model seed supervise epoch minibatch weight adjust accord schedule learn linearly last starting choose hyperparameter tune report validation multipli auxiliary task beneficial test hyperparameter choose worst misclassifie significantly lower report comparison boltzmann train include target target input connection connection input go layer initialization activation inference feedforward classifier train maxout autoencoder connection compatible supervise denoise training achieve margin conjecture due supervise unsupervised supervised find propose feedforward back quick implement function connection without many currently study impact cost work extend dataset autoencoder connection unsupervise simultaneously unsupervised cost layer wise improve significantly permutation combine auxiliary help hide generalize auxiliary autoencoder perform noise denoise autoencoder unsupervise show connection denoise change way permutation classification perceptron bold fully
bit h use instead bit hadamard normalization bipartite access pass immediately let integer give time ns position b bt kx desire suffice accord notation explain theorem algorithm form stack approximate accord entry index ta bb scale generality entry output sure integral ingredient prove kx ensure exponential progress corollary suffice deduce formal run vector position value precision entirely except along since polynomial time see computed assume represent trivially length operation take loop take time part light fact computation procedure number time loop present aim output satisfy goal arbitrarily round near integer integer range round entry near integer satisfy nothing round integer therefore note averaging position different round cause position coordinate add cause proposition focus achieve attain place achieve ingredient graph edge edge unbalanced intuitively pick magnitude tie break pick set unbalanced denote q equality regular argument note union regular denote goal iteration precise lemma suppose defer satisfy particular q triangle add subtract inside rip use use increase imply every constant pick sparse deduce plug triangle add subtract definition enough exponential guarantee attain end finally discuss proposition magnitude coefficient represent note slight abuse notation index position integer one rest pick hash output induce thus respect hash fall recover formally set coefficient later intuitively dominate correct implie must would produce bit bit note recall whenever equivalently correctly equality agree outside outside pick adjacent namely collect decompose hand claim suffice recover guarantee bin note intersect recall ready since h ti h ti ti ti ti si zero last lemma plug optimal randomized perform finally recall compute follow combine theorem randomize adaptive query access coin algorithm perform arithmetic one bit use ok proof modification algorithm absolute coordinate round vector suppose h hx leave randomness bipartite right contain contain right union vertex determine choose random seed shall observe result underlying respect assume conclusion deduce case note since depend coin determining hold analogue least stage condition conclusion lemma coin bad happen proposition independently choice particular rip particular support long expansion applying support take union thus follow tool analogue proof instead randomize random coin stage however conclusion coin union bind happen throughout error condition bad happen union final proposition least os write sketch sketch oracle queries sketch throughout execution optimize sparse take arithmetic operation naive sort nonzero entry add indexing update entry logarithmic would indexing observe run since corresponding copy additional per arithmetic procedure loop instead iteration plug value run proof alphabet q seed length irreducible polynomial interpret give shorthand follow construction provide choice theorem fix arbitrary explicit rd r construction apply mostly et require integer regard expand vector upper seed q desire upper task obviously nu nu n efficient algorithm integer polynomial integer imply claim hash normally technique construct seed universal family turn proof random basic universal hash family contain extension field map mapping universal element random nonzero uniformly imply universal ready hash tx variable independent min argument support hx z vector two express assign zero write rewrite probability universal family schwarz domain bind distribution set contain support california berkeley mit thm thm rough thm construction htb compute hadamard transform dimensional satisfying use algorithm start important technical tool use construction optimal deterministic general algorithm improve explicit construction would improve lead reconstruction exponent allow run improve discrete hadamard hadamard coordinate index entry variation dft hypercube notation hadamard well time fourier dft scenario hope signal improve rely process design efficient dft sparse last decade development efficient recent mostly discrete fourier transform dft cyclic focused hadamard term approximation dft development survey aforementione randomize desirable deterministic although subject transform one sparse dft signal run time run recognize considerable interest run reader think regime dimension like exponent query access deterministic give parameter compute constant approximation value exact goal formulate sparse recovery think resp objective comparison follow capture constant let bit output long constant fix appear solely state unbalanced use technique unbalanced immediately potentially dependence hope running theorem absolute simplify optimize exponent regard say exponent however randomness run improve use exist family graph construction rather large exponent asymptotic article hadamard transform adapt substantially fast randomness hash randomize access bit long internal coin absolute constant bad ok arithmetic sparse mapping coefficient bin locate position key need typically introduce parallel aggregated identify coefficient eliminate proceed fourier cyclic run h show guarantee aforementione implement e bin bin furthermore number different result reduce select still deterministic however relax mapping formal expansion mapping lead near need simulate black induce mapping implement paper observation explicit number mapping find construction yield extra query identify procedure query analysis considerably thank call isometry rip norm immediately hadamard linearity discuss reduce compressed notion unbalanced focus amount compressed sense reduction section main sublinear algorithm main deterministic improvement comes add let index coordinate agree elsewhere equivalent hadamard approximate k role hadamard equivalent hadamard computing equation compress sparse hadamard formulate thus goal present adaptive linear requirement isometry combinatorial need rip order eq rip approximation satisfying use normalization equivalent unbalanced formally unbalanced mention characterization matrix unbalanced rip absolute unbalanced recall xx regard hx distribution min entropy computable achieve equivalence unbalanced nb bt tx hx vertex choice bipartite associate unbalanced focus algorithmic hadamard hadamard sense focus discuss h rr bipartite query suffice namely real rip order constant give compute combine arrive absolute constant follow deterministic running adaptively query kx adjacency bipartite unbalanced satisfie rip constant assume sufficiently constant suffice product efficiently well explicit use technique list construction make result prove p parameter deterministic run ok absolute constant set derive proposition linear simply hadamard algebraic q v
covariate result efficiency elaborate efficiency depend bias thus strategy future focus recommendation cm de la paris france universit paris de france universit paris centre de paris france recommendation integrate online user influence interact system consequence recommendation via application offline reduce filtering frequently aim suppose interest system internet online social adapt obviously evaluate monitoring achieved click historical offline several real offline etc profile item rest profile various influence historical specific product etc limit strategy recommendation protocol principle weighting propose practical relevance constant recommendation recommendation user collection phase build public business organize describe weight scheme demonstrate relevance denote item historical recommendation build instant associated possibility item recommendation instant profile user scheme user business favor evolve classic offline calculate instant joint moment item select item certain lead selection process similar evaluation protocol moreover bias indeed recommendation discount major constraint business thus classical static value lead weighted leibl asymmetric useful reference reduce influence time offline production tend evaluation unbiased production offline evaluation seem class apply collaborative filtering consider recommendation suggest user quality recommendation estimate simulate repeatedly item select user collaborative user present collaborative filter cosine proportion associate item algorithm recommendation high method different collaborative one present h value day day date feature important notice score
volatility abc smooth estimate filter cumulative denote compute procedure return exception volatility vary return heavy confirm filter accord approximately copula residual drawback directly suffer accuracy tail cdf generalise nonparametric q denote location right low tail model marginal tail asset type kernel tail filter residual parameter make likelihood filter residual quasi newton present low tail cdf residual dynamical contract simulation perform resource provide national like thank discussion lee make calculate gp covariance log acquisition hyperparameter gp th algorithm sample obtain hypercube lee optimisation solve use direct implementation abc space use follow make fix lag smooth lag smc discard hybrid particle real world price contract obtain use particle additional setting adjust hessian smc abc tails newton solver department electrical computer university technology university nonlinear intractable likelihood sequential monte approximate smc costly novel smc laplace intractable abc volatility conclude enjoy comparable significant reduction optimisation portfolio use copula margin optimisation dynamical modelling datum extensively volatility stock financial volatility important risk management copula risk var portfolio bayesian nonlinear limit model aim bayesian nonlinear say wise analytical recursively computationally prohibitive denote denote abuse py evaluate intractable possible problem likelihood computation result apply standard estimate perturb perturb quantify challenge problem abc smc abc estimate suffer problem require mind optimisation extract posterior iterative operate constructing computationally evaluate approximation contribution introduce result pass computationally area alternative work combine smc maximum biology parameter usefulness inference volatility model useful investigate impact likelihood standard consider financial portfolio smc abc inference margin copula result considerable speed compare method use related alternative log estimate ii robustness approximate alternative bayesian optimisation latter simultaneous stochastic drawback problem practical parameter gradient ascent smoother wise continue motivation overview abc continue discuss smc abc intractable log construct exploration exploitation numerical highlight aim method construct complete square second log express hessian observation see around bernstein von concentrate laplace require obtain see laplace ii result newton gradient many suffer slow log posterior use optimisation gradient even costly obtain suffer variance estimate analogue log furthermore problematic main laplace approximation contribution difficulty create laplace surrogate surface aim create smc abc evaluate resemble around make optimisation operates sequentially surrogate use sample predictive process briefly account globally evaluate noisy log posterior model abc already hence predictive could determine heuristic phase method optimisation return gp acquisition therefore could abc alternative laplace proposal challenge write intractable except instead propose smc particle sequentially predictive quantity obtain dirac denote normalise dirac particle particle evaluate consequently filter model require algorithm abc variable express assume augment denote density non refer bandwidth density fundamental abc select particle first perturb transformation later assume correspond easily use transformation random perturbation simulation denote particle require wise use perturb reformulate tolerance est operation carry resample propagate extend trajectory particle particle iterative particle particle three iii carry randomly multiply discard small part implementation usually recommend practical application propagate simulate system third weight present denote particle implementation know alternative useful perturb estimator biased particle variance variance estimate use simulation repeat likelihood estimator unbiased view bias decrease likelihood von motivate laplace quantify abc construct surrogate posterior reference gps see infinite variable distribute gp see value see surrogate priori log accord gp infinite value specifie correlation encode gaussian estimate later posterior theorem calculate notation available parameter hence construct function posterior use obtain however application calculation handle computer demand filter gp decrease cost memory grow properly explore discuss issue vary space flat arise five ard function posterior calculate necessary experience major paper recommend plot mean mat ern ard function type equivalent mat ern posterior smoothness laplace weak smoothness property assume replace mat ern covariance square mat ern function choice regard design mean gp hyperparameter bayesian approach use e marginal get optima next parameter aim heuristic next available point log user optimisation evaluate test research field paper make recommend derive ei rule denote exploitation exploration maximum assume large peak expectation predictive posterior gaussian often add exploration area increase parameter covariance matrix similar exploration optimisation therefore two local optimisation gradient section put together abc outline combine log surrogate function suitable hyperparameter prior may fail converge properly hyperparameter log ii quasi iii hypercube sampling execute update hyperparameter pre interval hyperparameter costly recommend algorithm parameter covariance parameter est hyperparameter k direct compute direct finite execute suitable ei predictive extract map carry optimisation newton wise laplace approximation require log hessian hessian function choice covariance note solve hessian adaptive obtain accuracy smc smoother costly negative previously discuss optimisation noisy prior ei achieve convergence mat ern section usefulness accuracy rate impact tolerance second illustration return illustration management illustration abc algorithm competitive practical collect consider stochastic q parameter assume random increment synthetic use evaluate smc abc tolerance parameter posterior gold standard laplace smc estimate observe
scenario due improvement proposal achieve remarkable conjecture whenever know cause negligible organize include complex signal exploit study kernel kernel apply calculus value derivative illustrate conclusion transpose hermitian trace determinant sample real throughout gaussian distribution regression regressor relation follow model nonlinear linear square wiener gps solve mmse filter input input value herein immediate dealing output wish dependent resort multiple relate complementary joint construct additive follow circular complex training function factorize multidimensional proper complementary covariance jointly number positive definite marginal later equality opt choose tune well obtain previous output remarkable good nonlinear channel mse increase steady second try hyperparameter capability first train fig procedure indicate step criterion algorithm prove valuable learn similarity well underlie tb cm output widely study resort process hand end endow gps output availability optimization derive reproduce conclude complex cross output nan must part need prove exhibit end fully remarkable independent process produce output white iv integral value use value many fundamental often value proper uncorrelated complex simplify value paper develop value reproduce complex value input convolutional covariance pay preserve input maximize use derivative besides scenario solve challenge scenario deal signal literature benchmark remarkable proper kernel vast engineering complex availability processing value processing interest processing widely case complex uncorrelated conjugate useful simplify solution admit complex version improper nonlinear complex address network recently reproduce rkh regression mainly component regard author complex discuss involve besides neither may suffer improve previous isotropic value fed real physics drawback solution approach develop kernel although bring study know technique successfully gps interpret advantage provide hyperparameter likelihood benefit novel provide output reproduce output proper signal provide covariance complex kernel output part design part addition skew definite covariance pay
implement continuous specie one belong share describe semantic behavior allow previously unseen computational reduce time cluster collective repository solve first collect large software piece experience program hardware believe process practically infinite space collaborative one currently many optimization software cm repository find meaningful architecture mainly benchmark htb cluster software across several architecture experimental result update public cm repository mind pool distinct cover software piece set benchmark benchmark achieve high speedup optimization number benchmark distinct benchmark benchmark across piece help substitute manually practically improve approximation decide associate unseen unique optimization base software decide purpose generate repository software specie semantic hardware collect monitor software piece share resource os pass share one past classifier full htb demonstrate high software support similar I much share dramatically drop close behavior collective mind model simple cluster relatively program pool cluster remove prediction leave semantic ft count basic ft exist relevant hardware understand improve since feature counter correlation simple try manually code highlight add feature effect perform transformation speedup show problem machine realize large though highlight repository usually community practice share publish improve machine specie public benchmark various researcher switch share possible continuous share balance reduce complexity usage neural good help hardware problem htb methodology improve lack record behavior automatically customer software surveillance require distinct execution intel core image set cluster classify detect explain software specie gradually specie add statement additional branch rarely additional cycle dependency solve notice effectively convert customer self minimize energy tuning effort market plan software therefore identification extraction frequently consume software piece real program plan use extend interactive interface connect framework simplify integration care automatically dependency add repository methodology physics community benefit powerful source gradually furthermore immediately validate novel optimization across realistic system big repository ai physics characterize interaction share software piece fine specie repository possible like algorithmic specie continuously optimize hope hardware boost innovation implement idea regression include run particularly mobile device avoid subset representative interestingly eventually large enable repository support software engineering present support foundation fp thank ed feedback collective mind technology feedback considerable improvement version technology share graph wide user flexibility repository force possibly possibility crowdsource exploration dimensional apply service user mb web service plain line slow slow indexing option decide original open repository format fp project collective fourth technology publicly core collective fast orient analysis service separate share service conceptually collective collective mind kb mainly furthermore package automatically dependencie believe considerably technology collaborative sharing experimental interactive report term effort share interactive graph gradually past format focus solve program run time collective predictive available repository gpu scheduling video processing mp gpu exploration local frame processor apply active build refine share could cpu improve frame second believe vision towards collaborative systematic engineering framework brain quick idea knowledge innovation get consume issue experimental wikipedia extend fix error improve miss many entry unique collective module dedicated page allow researcher evaluation lead along useful tool arm international uk develop know hardware eventually run mobile cloud service unfortunately optimize keep ever ever change system code collaborative publicly solution help software gradually available connect collective repository characteristic exist hardware configuration environment collective mind consume mobile mobile science continuously track win solution hardware skeleton parameter minimize execution spend failure specie pareto mind furthermore continuously classify redundant one various software input set hardware similar technique create public realistic evolve benchmark knowledge gradually improve depend usage scenario continuously grow collective become hardware self management collaborative collaborative knowledge crowdsource active learning hardware validation pareto code interactive ever diverse computer power consumption reliability hardware availability specialized hardware working memory storage often run heterogeneous multiple mobile device cloud force software rely exist hope fast efficient scalable hardware complexity ever change hardware hundred fail produce code energy include decade evolve possible software empirically search combination implementation scheduling among choice execute program cost thousand per considerably performance life mobile device promise advance report discussion show miss software fundamental already grow exhaustive find good recent acknowledge continue efficient hardware development usage htb pdf hardware fraction whole hoc heuristic vast hardware furthermore specific run time mechanism costly practically framework execution vast practical engineering tune behavior conceptually hardware propose background ai help start gradually real software range whole line piece depend hardware possible requirement recent collective mind repository share open piece together meta gradually extend easily format describe build piece dependency hardware software include operate run library piece randomly execute window device device share resource mobile cloud service gradually cover configuration community continuously software nature biological treat piece continuously track behavior versus configuration environment record win solution minimize execution consumption size failure memory storage time software piece pareto repository project optimize specie continuously reduce cost hardware aware software hardware conceptually help create first knowledge public large diverse evolve continuously optimize benchmark knowledge gradually software apply physics community learn optimize large share piece whole library function consume loop versus global coarse gradually move fine include versus internal decision interactive already methodology adaptation include begin usually focus execution consumption deep combine ad hoc architecture large benchmark allow methodology sciences biology ai big datum predictive cm continuously win specie distinct community unified background extraction continuously add specie thus practically software automatically environment cm continue species execution predictive find feature complex intensive technique neural decide manual option exist predictive manually simple fast decision explain predict specie web link share software hardware wikipedia engine success depend active try use weight share resource specie software manual gradually work together describe partially share specie together gb storage moment share I benchmark validate help tune production customer consumption specie combination cloud mobile derive distinct class cover share specie manually semantic dynamic feature predictive analyze predictive end correlation isolate share counter code wrong classification substitute ad architecture verification derive optimization dramatically eventually involve software engineering development improvement hardware continuously grow repository unify service practical hardware design decrease development cost importantly side help international engineering present life motivate engineering example encounter briefly introduce collective mind repository collaborative associate across provide demonstrate continuously big datum specie predict realistic representative section demonstrate improve demonstrate miss specie improve optimization tune computational conclude development direction various brain inspire brain function layer popular choice modeling pattern image include implement fairly regular neuron receive weight input output process activation sigmoid many implementation filter activation switch neuron neural determine processing capacity correct prediction failure heavily total neuron speed resource include specialized co involve careful balancing versus associate cost consumption development price usage improve evolve modeling vary minimize execution hardware center cloud service minimize cost include consumption network surveillance mobile internet thing strict place hardware memory system year software engineering neural relatively straightforward simply select hardware tune achieve nearly peak figure configuration arrival hardware hardware would double software consumption dramatically htb mind contrast software execute market frequency core cache hardware gpu neural power consumption access parallel home popularity cloud services amazon google microsoft others experiment mainly service time improvement operate together numerous free software development tool may expect advance practically generate efficient software piece hardware nevertheless eventually decide validate sake start collect c system multidimensional choice whenever real experiment execution include code usage decide see room improvement fast many projection multidimensional characteristic cost track win solution minimize physics pareto filter quickly necessarily fast efficient require move improvement execution degradation achieve improvement execution old furthermore internal parallelization scale old linear scaling parallelization htb mobile architecture execution dramatically power consumption drop try specialized hardware execution hundred considerable development cost performance encounter problem cache scaling core static fundamental lack run try move language achieve similar software considerable ad hoc good program cost summarize software often aware improve usage cost cost balance execution accuracy device execution balance gain care therefore believe current performance blind engineering change improve innovation science technology start search could software relevant job account gradually software try connect keep track within production several severe difficulty evolve difficulty reproduce machine web service collective mind biology repository optimize tool whole considerably usage cost briefly help decompose software piece currently support major language community gradually optimization feature dependency software hardware characteristic cost unify format allowed formalize almost exist finding function piece run computer hardware software system software
rely result half give two classifier improvement change pac resp marginal next give hypothesis share source support upper inequality chi could unsupervised way appropriately two distribution hope control transfer definition theorem st france universit st france provide base pac bayesian theory improvement previous et way bind tighter easy algorithmic adaptation generate source generate analysis adaptation belong domain generalization express average pac bayesian new pac improve moreover appear design able term paper introduce obtain pac offer majority vote dimension priori quality identically accord pac weight low precisely vote related consider da al difference da target information source label sp majority vote domain recall usual pac generalization gibbs al risk notion r disagreement marginals eq h g reflect well favorable situation achieve source derive promising minimize domain
obtain let great lead real I value case derive eq therefore hand equivalent noise probability lemma imply desire test multiplier see phase transition sharp comparison plot phase generally cs achieve illustrate empirically grid reconstruction signal separation guarantee edu xu explore superposition projection arise application biology imaging rank subject sample show long incoherence separation interest superposition measurement require experiment practical model superposition choose acceleration medical digital imaging nuclear biology signal superposition superposition consider linear superposition satisfying word superposition superposition less ambient reconstruct assume order enhance matrix notation shown directly reconstruct recovery tend matrix nuclear I norm q theoretical via theory measurement nuclear minimization contribution exceed scale reconstruction toeplitz superposition complex signal signal accelerate superposition analog digital possible compressed uniform grid fall domain usually exactly fall basis discretization conventional compressed recover grid complex atomic minimization prove reconstruction separation frequency enhance matrix chen et play similar complex apply aforementione exist frequency achieve comparable accelerate frequently molecular monitoring chemical application etc et low signal result explain give apply incoherence uncertain diverse chemical sample require organize extend main rank toeplitz numerical base observation whose consist lead rank enhanced constraint row specifically let column propose eq rank correspondingly problem hard solve possible recover nuclear norm minimization likely guarantee require robust much degree freedom special interest extend complex guarantee incoherence available realistic diverse chemical biological limit applicability recovery theoretical ensure scale result incoherence arbitrary number small obviously get accord order choice error easy form orthonormal standard linear adjoint identity onto simplify introduce diagonal letting q case gaussian dominant recovery minimum nonzero gain minimum respectively let minimum condition concept atomic norm give set unit gaussian map corollary part gaussian unit sphere convert real matrix complex letter value therefore gaussian mean get desire follow eq gaussian instead give cone accord cone hull respect whose part I variance need r calculation satisfy singular decomposition define linear q adjoint singular decomposition subdifferential estimation width check choose get convert vector second q definition orthogonal check satisfie imply line similarly together prove give tight gaussian mean part constant proof relatively introduce idea lemma even integer kk k jensen inequality easy see ki utilize index specifically node I edge equivalent class class index accord traversal play bound reader concept two equivalent
forward recursion complexity array support increase exponentially recursion notably array problematic deal deal describe two name row share composite maximize em longitudinal satisfactory name composite row composite likelihood analogous construct column dependency due latent row composite array r request finite simulation simulation cover array small give chance quantify due composite likelihood another relevant aspect tackle particular support variable cross extend finite implement independence devise cross cell array index version estimate training illustrate utilize leverage inter intra comparison type throughput genomic rate mb genome process publicly genomic window thing aspect dna insight feasibility utilize methodology array genomic contiguous mb window along article assumption section section outline likelihood genomic offer conclude row basic conditionally give identically distribute column chain initial probability coincide parsimonious chain parametrize complete pair natural depend requirement mixture application section transform normal mixture suitable parsimonious incorporate impose mean observable comprise covariate fix obvious replace normal family array variable parametrization covariate full feasible array switch methodology mixture initial distribution implementation account covariate denote denote configuration latent eq denote trivial involve relatively strategy joint column compute becomes indicate use introduce indicator reference decomposition comprise latent comprise column matrix comprise step compute indicator sum extend v v step value update mass probability constraint also number way array time increase however latent variable prohibitive infeasible application propose composite row great potentially efficiency datum underlie simplified version variable row composite base importantly readily computed treat useful log satisfying assumption chain need target otherwise express composite cb z ij e definition term approximate model ij one e maximization finally mean composite column datum py uv composite row composite log regard latent meaning section former separately indicator express w cb compute update update perform simulation study assess likelihood another estimation datum array design typical application full likelihood fix benchmark suitably scenario scenario underlie row fix apart apart respectively parameter bias root likelihood likelihood likelihood table median median deviation likelihood c bias rmse rmse rmse rmse rmse c c rmse rmse bias rmse c c rmse rmse rmse rmse bias bias rmse rmse rmse row mass probability estimate approximation comparable available either composite approximation comparable approximation fact even sophisticated rely independent approximation former estimation composite approximation fast benchmark design importantly pass order run second hour size simulate still effect see average remain instead general row approximation e appear fast approximation iteration even consume large array composite row literature suggest information bayesian see select free maximum computing penalization complicate hessian rely regard deal independent composite cross straightforwardly extra cross splitting treat either half repeat base maximize cell remove result quantity average log consider quantity pair either maximize close maximum course derive consideration illustration study university human base four comparison contiguous mb overlap window try relate landscape dna molecular along genome author publicly window produce segmentation mutation simultaneously characterize utilize segmentation array comprise measure contiguous mb overlap cover table capture dna composition dna e g nuclear associate I site site bind dna level code standardize normal prior ht line nlp h rna ii average es dna n l l lines ac structure form site code ht critical number latent cluster genomic number distinct strategy composite compute correspond denote report log c result high composite likelihood one well obtain alternative compare similar quantification compute high model c towards base form genomic distinct table report latent convention modality order decrease modality table stationary ht figure color code way feature segmentation e latent posteriori predict horizontal represent contiguous bar top reporting code vertical dimension genomic reporting code black red range report green green characterize genomic vertical bar mark horizontal bar concern note comprising estimate small mass detail cluster proxy proxy include proxy region activity concern segmentation cover approximately estimate estimate e least characterize cluster feature cluster state feature cluster whose profile former feature cluster feature see represent alternate approximately window towards figure cluster strongly level strongly cover window strongly cover article array contiguous segment composite approximation optimization specialized show methodology methodology demand clear composite row
covariate hold eigenfunction uniformly bound constant depend cover square minimax follow eigenfunction probability least square also contraction rademacher find book van processes immediate result connect literature assume ball estimating assume underlying regularity zhang similar mixed norm yu highlight rkhs ensure hence question strategy estimator ball surprisingly answer challenge occur smooth minimax optimal rate rate choose optimal rate attain unit ball multiplicative obviously suboptimal attain minimax main theorem brevity shall assume argument natural whose specify nonzero jk generate clear set step ns ns brevity jk kx jk jk jk q jk dx jk jk jk follow specifically infimum measurable ny nx fix g cm l kullback leibler conditional ns eq yield constant q constant complete immediately q write lower together derive inequality separately value n h l inequality fact hand l exist constant probability event hold l inequality write light l contraction concentration h g gx gx n n g bind uniformly argument jx u conditional constant conditional hold combine c e exist constant event constant first step proposition corollary establish reveal component smooth sufficiently rate identical dimensional smooth rate curse dimensionality transition reproduce advance technology finance devote understanding challenge dimensionality development methodology counter fan li selector progress understand extent regression reliably van reference therein model restrictive alternative attract much attention past several lin zhang van yu couple amount certain kernel fix idea follow support product compact subset component rkhs clear model identify h obviously view trivial take collection another canonical unit interval g note additive representation define quasi h minimize rkhs number function interest optimal rate th interval imply side dominate always eq pay rate nonparametric closely learn aggregation machine combine kernel single achieve study david expect understand organized concept reproduce present basic use reader rkhs symmetric semi integrable hilbert completion product shall rkhs assume paper cauchy recall shall repeatedly later spectral theorem admit eigenfunction marginal delta
participant slightly ensure camera overhead video capture angle place face fig rectangular person meta datum player starting video videos pt videos rgb videos depth encode rgb videos distortion correct frame camera pose participant face participant whenever available object bounding box position track focus annotate divide absence segment video develop annotation schema draw concept social literature schema series question annotation schema social predicate focus object language similarly involve movement rate six student video introduction survey ask video video annotate ensure student accurately reliably detailed annotation h h conduct skeleton generation partial entire visible layer raw player average generation relatively case except similarity visible demonstrating effectiveness also across indicate relatively stable possibility show detect task skeleton actor novel essential predicate attention collection new audio visual dyadic research social discriminative conditional boltzmann generate datum purely decomposition powerful offer possibility substantially advance mid predicate layer beyond generation multi understanding semantic extend multimodal stream include audio full behavior make multimodal capture rule interaction automatic efficacy interaction interaction behavior virtual reality environment thus well robot interaction foundation establish systematically scientific acknowledgment nf view conclusion view imply david novel predicate sound social methodology collect game consist dyadic expect provide new research computational social restrict boltzmann combine combination accurate predicate exploit capability actual training mean frame decompose behavior purely computational determination human processing scene research bring problem leverage computer vision social interaction aid worker country course interact well worker success general would enable smoothly interact extremely useful identify detecting predicate facilitate irrespective interest aspect reduce trust predicate attention orientation social sense exist infer internal instead body social emphasis cognitive action interaction approach apart social action jointly insight involve reciprocal act joint behavior nest demand behavior participant interactive social interaction behavioral movement pattern social interaction establish essential predicate mind focus computational social multimodal deep model recognize discover past decade machine advance furthermore complex full body pose many activity multimodal multimodal dyadic social model maximize solely often unable incorporate hybrid address combine discriminative level discriminative model allow recognize propose answer question approach attempt detect social qualitative multimodal human variety must span everything lexical pt modeling essential predicate multimodal temporal model social interaction predicate multimodal annotate publicly interaction discriminative conditional introduce enable learn advantage result detect behavior sec sec specify explain sec quantitative sec conclude interaction implication science focus theory largely much show infer participant interaction sequence social dynamically participant behavior social require realistic human interaction detect overall state person external speech multimodal also model activity involve deal physical rich participant consist learn representation low input tend solely joint method single generative energy iterative consist discriminative train separately learn generative project rich unsupervised powerful boltzmann rbms building block train cd algorithm demonstrate ability deep representation rbms deeply boltzmann capture complex recently deep capable rich include rbms temporal rbms motion human pose phone recognition parse music generation describe similar prior define parameter rbm generative hybrid rbm define distribution visible ensure bias architecture h factorial case real binary layer code prove energy slightly rbms visible distribution hide function term history previous gibbs time model autoregressive visible equal history rbm vector phenomenon factor restrict boltzmann boltzmann however complicated involve factor layer layer similar equivalent visible visible label generation bottom type deal miss fig miss visible goal label cd obtain update combine respect expectation reconstruct reconstruction visible generate visible finally label activity recognition dataset annotation interactive behavior demonstrate contain relatively action involve person interaction collective activity dataset lack rich dyadic dataset dyadic interaction child dataset collect structured format child interact pre narrow social behavior child another focus study social analyze human format human different aforementioned activity limited coverage diversity activity class lack narrow behavior g issue game
enkf enkf describe enkf filter localization equivalently member enkf member assumption enkf enkf denominator gaussian read easy correlation successful enkf become calculation successful estimation enkf require enkf sampling proxy assimilation illustrate observation enkf successfully sample enkf condition success use importance success particle filter see appendix particle two regime success enkf observation moderate noise thus particle filter enkf perhaps surprising particle particle additional thus optimal sequential avoid observation past broad enkf filter uncorrelated illustrate show particle filter noise noise enkf sequential enkf enkf particle enkf think variance improve enkf improve surprising update enkf marginal I go limit go thus posterior enkf become inefficient enkf assimilation success filter fix observation hand filter also successful fix keep quality improve keep fix optimal behavior importance importance filter inefficient enkf posterior derive enkf gaussian localize enkf draw marginal posterior density localization draw posterior target since enkf moreover enkf assimilation enkf ensemble properly produce marginal various enkf attempt importance enkf carlo know marginal numerator evaluate write integral integral density know multiplicative constant exact enkf integral carlo become ensemble size infinity ensemble define approximated analysis ensemble however ensemble carlo enkf analyze weight compare report difficulty assume observation assume available difficult enkf finding impractical frequently enkf posterior brownian discretized order forward euler brownian motion step collect independent noise deviation enkf perturb localization ensemble necessary comparison filter coincide implicit consider enkf numerical synthetic perturb assimilation compute weight store equal collapse weight enkf member enkf maximum panel particle resample replace weight weight enkf assimilation plot panel particle filter however joint frequent enkf explain synthetic assimilation weight enkf member marginal enkf require connection enkf review feasible enkf may slower assimilation scale enkf require dimension problem may numerical confirm enkf localization system ensemble size small enkf adaptive multiplicative also implement enkf directly appropriate noise compute times deviation mse dimension section far plot numerical figure constant left panel dimension panel show find equation predict system moreover localize enkf direct sample confirm statistic enkf agree compute localize enkf also localization without localization localize enkf mse insensitive sensitive tune localization e satisfied vary change reduce state refine assume show enkf localize enkf confirm assumption confirm feasible scale linearly sample dimension also effect uncorrelated violate attempt posterior bias localization work enkf produce show typical tuning adjust adjust posterior mse size enkf enkf moderate relevant infeasible enkf dimension rarely behavior reason application assimilation case mid dominate far mesh region rich discover mesh refined decrease mesh refined assimilation mid process dominate would observe behavior enkf resolution mse observe practice mesh mid effect resolution assimilation summary mse connection feasibility theory find assimilation may sample operational assimilation summarize density assimilation current filter enkf expression enkf broad enkf suboptimal particle filter broadly enkf joint imply enkf sampling enkf suggest enkf explain usefulness enkf require ensemble size enkf suggest ensemble dimension bound connection feasibility assimilation enkf assimilation enkf mse rather insensitive material base science office advance apply mathematics program contract foundation grant dms office research pe would thank interesting comment feasibility discussion provide wish gaussian problem become note independent expression simplify component simplify eq denote covariance independent thus add assimilation also origin component denote one component find filter give substitute expression eq assume enough reach success filter constant rearrange tb tb tb tb berkeley laboratory mail berkeley california ensemble kalman enkf widely condition evolution condition marginal condition filter imply enkf marginal posterior localize enkf useful explain applicability enkf tune enkf mse sampling huge moderate model density careful distinction two model condition describe condition filter determine variance one develop enkf enkf joint enkf broad one assimilation cycle filter unless past observation enkf imply posterior insensitive error make assimilation derive weight enkf localize enkf importance question assimilation weather forecast forecasting assimilation cycle irrelevant enkf optimality enkf posterior enkf explain applicability enkf dimension connection enkf datum assimilation true ensemble investigate localization mse enkf marginal variance error mean localization mse insensitive huge successful moderate consider assimilation vector smooth dimensional discrete random identically iid assume covariance assimilation perfect deterministic explain careful elsewhere assimilation consider describe assume specify entire trajectory joint posterior history variable dimension posterior large small assimilation large record weather forecast frequently posterior hand kalman covariance require impractical case enkf use enkf make monte forecast requirement kalman filtering ensemble member member obtain sample perhaps localize perturb approximate marginal implementation enkf enkf particle use empirical e recursion approximate weighted function posterior need collect extend recursion account lead filter particle unnormalize else particle condition variance normalization obtain close mean contrast normalization small nearly posterior desire carefully recently logarithm must infinity summary condition success wish particle filter filter particle go infinity achieve cost bias apply assimilation assimilation question assimilation implement concern numerical explain variance assimilation datum particular forecast need intuitive feasibility assimilation deviation e extend high assimilation feasible covariance norm data assimilation uncertainty kalman kalman gain mild steady algebraic steady state kalman gain kalman assimilation rather tool derive asymptotic steady state assimilation steady small suitable frobenius define covariance connect frobenius correlation g correlate red large uncorrelated random frobenius norm imply project onto span reflect empirically fewer easy assimilation motivate effective assimilation noise actual large non dealing need frobenius norm precise requirement resource interested assimilation behavior dimension finite limit numerical weather assimilation problem pde connection limitation particle reason mesh discrete mesh mesh refine small imply feasible fine computationally tractable feasibility may hard assimilation large reasonable positive scalar varied system attract literature particle filter posterior scalar well steady covariance compute assimilation balance represented set sufficient qualitative feasible assimilation generally constant believe datum assimilation feasible dependent white hold grey feasibility assimilation else confirm assimilation infeasible counter intuitive scalar feasible theory label infeasible frobenius norm invariant rotation coordinate scalar sub structure
quantity generic tail adopt particular approximation posterior rather base theory incorrect situation valuable tool model marginal spirit interpret represent variate cdf variate mf jx account among component cdf copula assumption marginal copula paper distribution parametric copula limit particular approach empirical survey produce nonparametric likelihood particularly readily available completely replicate vector express functional sort profile moment result whereas obvious independent third towards quantity computation likelihood expensive repeatedly consider achieve posterior avoid crucial abc relatively easy new simple propose pseudo randomly distribution actual represent highly inefficient strategy available example concentrate abc expensive likelihood family importance sampling sir generate draw propose bc partially quantity set meaningful although statistical nuisance produce might robust inference parameter way important lack physical meaning estimate dramatically circumstance reasonable specify adopt frequentist simulate estimation copula paper spirit propose goal specify model assume distribution sample representation copula use copula skew parametric investigate mainly partially popular literature obvious imply robustness method essential aspect dependence disadvantage parametric demand posterior though density require huge iteration might algorithms avoid computational burden modify run among pseudo lie modification ht marginal distribution j ms draw row eq f store b datum know counterpart say nothing write know coincide actual rank original evaluate p general multipli produce quantity size comparative frequentist describe confidence interval construct frequentist three low limit nominal limit box posterior take computation procedure precise estimate median behave length length tail credible simulation short copula raw value histogram mass entirely close perspective follow describe estimate give many properly treat il il marginal represent correct variability work notice slight towards large incorrect report contain monte di bank log return available model student innovation may via return bank distribution package simulate parameter follow simulate simulation consider row return di posterior acknowledgement provide universit universit universit di simple make functional multivariate carlo algorithm functional particularly costly evaluate work
point reader field log relaxation space dominate relative propose sampling key collect estimate understanding likely future etc effort generate possible pt run adversarial news b rule marginal deep nlp e rule negative example conduct incremental system program last competition quality sometimes assess blind assess program development technique development speed news adversarial name refer domain figure six template four category focus system build base relation corpus million news page generation extraction supervision span spectrum collect sentence journal article precise text relationship text news write relationship e g belong pt learn incremental component write core ghz ram adversarial build program technique show speed development quality incremental sample collect collect combination similar system focus news competition six sequentially score cumulative execution take significantly time win indeed hour take minute fact extract end task fact issue differ incremental compare evaluating given understand total execution part extraction classical incremental news speedup free key speedup news across high update original acceptance execution extraction supervision speedup rule attribute contain contribute speed produce incremental low execution application interesting cause fact introduce large case need get factor therefore factor hour spend sample single conduct verify leave report evaluate impact sampling variational news sampling approach variational slow acceptance distribution change extraction rule sample use group supervision variational slow rate pt baseline switch group experience build quality challenge accelerate build incremental component statistical approximate improve face order keep aid acknowledgment advanced program fa program national office national national institute image research fellowship foundation american finding conclusion recommendation necessarily view find namely sampling provide proof describe rate summarize result markov chain represent difference assign event metric comparison sampler couple statement follow argument low lb voting voting variation unary exponential eq meanwhile flip similarly event could happen event state q event bound less least step heuristic behind variable specify iteration set rule dependency change want active call inactive next active inactive create new factor inactive conditionally appropriately group decompose presence inactive variable collection inactive inactive partitioning inactive line conditioning independent inactive separately variable pair impact avoid grouping inference grouping make simplification specify inference group concern hardness g cg allow reduce contain inactive heuristic accord active inactive variable I independently height pt connect remove variable set condition j j decompose strategy find compare actually fast less determine extraction rule news show change follow experiment hour common run execute phase variational finish many sample collect hour hour happen document analyst would efficient interesting distant supervision able create human intervention perform online method model last start experimental adapt standard outperform compare approach namely descent without new new training proxy stochastic separately pick grid hour pick fast loss epoch percentage within learning epoch loss achieve reach within sgd fast descent sgd converge drift stream keep resolve concept machine adapt change consider update model forget cause impact drift incremental learning impact solely focus target significantly drift require difference target amount change concept drift approach second learn target section work concept follow dataset email spam use testing train drift converge drift allow low loss term iteration use almost active change original concept due inference small current design component motivated aim improve design justify find able incremental decade incremental rely classic incremental technique operation individual like iterative segmentation database graph algorithm reference remark stanford stanford edu database extraction recent deal dark datum system combine learn idea help observe develop technique inference optimizer five showing speed task two order impact structured community profile effort processing learn community place emphasis extraction community common technique mix datum quality structure information database entity complex relationship assess complementary claim tuple tuple many actually massive far document count effort shoot algorithmic arguably question good use rapidly number language axiom rapid move quickly construction execution language perspective perspective logic language semantic execution go phase evaluate describe tuple gibbs output tuple google expect e subroutine loop computationally tb ram machine iterative use technology field drug recently compare system provide ten database precision entity win entry iterative arrive concept lead contribution inference incremental incremental feature extraction series make specify change systematically system due phase change datum compute problem work database new new change simultaneously use inspire probabilistic database technique apply incremental clear experimental diverse program find approach largely orthogonal axis change performance highlight neither choose optimizer experimental highlight improvement describe program development run incremental incremental snapshot approach fact throughout technique incremental outline rest paper depth development presentation system incremental exploration tradeoff description optimizer present study decade base system machine study improve quality build formalize ease accelerate hope feature google knowledge incremental pt language goal construction relational classic study maintain inference focus related work focus incremental specific structured low degree factor graph much examine modification graph aware single end build language first definition semantic heterogeneous collection system contain put schema may extraction integration illustrate system news article incomplete kb person linguistic pattern roughly indicate terminology four object system seek input entity person thing entity person another individual entity entity relationship mention span text refer entity mention entity phrase connect process entity detail end engine walk phase manually store database default sentence nlp pre speech linguistic parse loading type mapping entity relation candidate candidate mapping mention sentence mapping query must candidate chance extract feature markov user phrase phrase two whether people think say influence phrase indeed indicate return two receive explain detail arbitrary operate tuple allow example bag aware nlp feature dictionary specify ability specify rich entity rule helpful integration supervision markov logic particular schema relation mention label distant supervision illustrate system kb entity pair q incomplete world entity entity incorrect generate sentence people technique redundancy cope phrase generate largely distant supervision relation furthermore integrate unified logic semantic run phase inference obtain final confident repeat understand feature mistake facilitate find three aspect believe enable program program sense probabilistic provide algorithm allow extraction language familiar stack visualize datum user construct end system pay traditional time spend extraction evaluate pay go inform logic language logic implement weight rule across feature every rule single couple writing easy user user specify extraction allow bring optimization implication semantic way noise semantic default semantic give semantic evidence define weight semantic logical tuple user schema predicate variable supervision part specific class evidence assignment negative ease exposition domain boolean predicate variable rule substitution variable replace conjunction fact rule three q real add weight indicate world less likely motivated semantic boolean world allow framework compactly specify distribution database illustrate web could extract one vote think variable relation indicate resp vote size consider vote close semantic depend vote semantic give logical ignore voting level semantic raw count ratio raw semantic ratio semantic suitable want semantic theoretically even logical semantic less write resp expand way symbol create create weight allow model g indicate tie logistic formal weight return value explicitly construct learn triple node correspond identify possible tuple correspond datum outside database database rule ground semantic probability follow factor statistical value compute return system tuple efficient run incremental phase evaluate program delta modify factor modify relational operation incremental incremental inference change advantage decade input query schema output modify view technique schema additional tuple represent update delta relation tuple update delta delta execute generate modified variable overhead gain load present incremental produce incremental two phase access entire attempt information store call phase precede variable factor respect change study tradeoff infeasible use explicitly store fidelity store infeasible moderate sized world perform speed arise factor store update store hasting scheme store
adapt target design away prediction expensive another measure symmetry upper mse mae lift bound hypothesis cm la paris france universit paris paris france universit paris centre en de paris consequence percentage regression find weight mae universal mae study paper goal quality traditional application quality q risk choose accord opposed mae mse practical well order minimize mse rather determine theoretical see obtain adapt complexity independently copy set addition problem indeed fix weight therefore support weight use notice verify car mae mse simple stop car goal change result summarize expect loss relate practice challenge mse mae view uniform function introduce give notation supremum cover controlling supremum cover function unless assume mae cover number easy q get class classical mae bounding assumption case bind need indeed eq expression role hand might hypothesis covering role play mae interestingly replace mae vc dim indeed yx yx vc dim equation erm
algorithm converge fixed point increase avoid cycle expect normalize message follow minimum would stop reach threshold propagation propose uniquely dna resolve difficulty continue throughput short read sequence counting accurately repeat exceed example version population np copy contain protocol randomly assign one letter probability binary template copy template introduce per starting read generate determine template assign regime expect read template hamming read template bit introduce error rate per bit obtain computer template various determination count even high determining template bit generate read rate perform accuracy mis algorithm average simulation hamming distance incorrect red hamming distance correctly edge regime property propagation balance impose question dl grant work grant foundation award prior cluster real equation become section give x h ik h spin degree however pairwise spin hamiltonian transition critical critical vast spin aware alternatively configuration constraint run partition blue uniform partition recurrence principle use compute intuitively effect favor quantify average behavior blue fraction see phase sort balance consideration weight configuration function order estimate entropy cluster limit balance phase impose partition cluster cluster configuration recurrence configuration definite order prior knowledge recommend uniform choice simplification eliminate since message eliminate message give simplification dependent moreover shall difference write explicitly contribute contribution effectively eliminate arrive involve ahead ij become author cluster pairwise belong triple assign implement sequence random word noisy channel cluster cluster paper wherein govern describe describe assign zero configuration likelihood pair assignment different sufficient ensuring constraint acting triple determine affinity propagation graph message pass configuration complexity usage rapid force net force require calculate distribution illustrate trivial future constraint e pi pi edge assign edge hypothesis matrix cc belong blue edge count point triple triple figure choice consequence hypothesis calculate denominator solution maximize result drop effect result operation ij interpretation energy minimum term force pair minimize solution applicable datum separate decision edge condition energy section represent configuration maximize equation graph variable neighbor every represent square every variable graph depict
transition different wide series nest well drawback complex predictive proper selection procedure appropriate natural unfortunately fail regularity nan hypothesis aic bic introduce estimate ar general inaccurate criterion specifically gap identify describe add new improve goodness base non goodness fit maximized criterion process smoothly root whose polynomial evolve behind observe knowledge remainder organize follow gap state section specific model state assume markov emphasize parametric consider propose multi new effectively impact bad initialization maximization present approach point curve new distance turn ar fix approach ar outline selection elaborate subsection ar filter subsection distribution symbol bold face generate single filter instance x eq subsection selection criterion add goodness ar predictor filter ar stable within distance simplify curve criterion root determine filter intuitively root characteristic small filter h ar model algorithm compute root ar denote filter uniformly curve fig ar h element filter characterize center w w examine filter close update subsection explicit formula distance assume generate stable filter use integral conjugate assume mean curve reference general reason ar reference measure ar filter explicitly filter generate notation power reciprocal multiplication let give determinant associate identity b determinant mention statistic require calculate filter sample generation obtain I extent reveal equivalently summarize lemma procedure long generate recursion k z z z z behavior model hmm hmm px method ar transition series function auto assume brevity unobserve indicator nm denote multinomial word tractable maxima em step predefine criterion brevity unknown expectation side take value old omit brevity cause away optimum initialization choose time consume new initialization reliable technique series economic probability state close hence adopt initialization retrieve em ar mean update achieve normally around style elsewhere et al filter cluster curve curve maximize synthetic scenario scenario scenario ar generate transition mm algorithm ar aic represent gap filter root inside unit independent gap statistic gap filter root inside
strategy finite strategy allow maximally include win exist proposition sufficient existence win characterization extend necessary precisely maximally strategy property environment characterize precisely property check satisfy exist strategy general win sketch relevant briefly take n formula game win construction additionally offer check win condition property translate construct tree commonly property get state accept tree never visit state visit check checking solve answer win proposition maximally existence game special proof limited winning formula maximally solve number software extract strategy gr condition maximally extraction application greatly simplify enable performance strategy non thus game remove game counterpart induce vice versa win win induce counterpart run result win acquire correct respect specification move strategy respect priori instantaneous reinforcement reinforcement discount work form optimal win reinforcement discount reward case concern strategy implement word act game equivalently environment system strategy loss algorithm markov game choose well learn interact condition discuss ready maximally strategy win divide formula win system maximize maximally game ss reinforcement maximally strategie compute game include win win win preserve win strategy subsequent correctness requirement reinforcement algorithm win strategy summarize proposition maximally expect proper win many include win intuitively would bad demonstrate robot motion plan different win game first maximally compute strategy strategy robot square turn know cell robot go adjacent cell go adjacent stay current cell robot avoid environment always observable pos pos pos pos pos pos I leave stay ta change atomic proposition l j k requirement I maximally reward robot ahead numerical encourage robot reach environment possible available robot advance reveal instantaneous take cm spend extract action tuple ghz greedy simulation result adversarial environment system robot position environment strategy environment reach position step show converge coincide optimal strategy iteration example construct game win robot system robot visit corner infinitely often low cell instantaneous remain maximally move win ht example trade win system visit cell say infinitely gr game way add counter control move visit cell maximum satisfy extract maximally force visit strategy extract maximally maximally strategy increase game allow counter discount maximum discount system counter trade learn counter discount reward n study respect logic criterion unknown infer idea synthesis reinforcement provide specification need planning fact corollary synthesis priori interact specification subproblem maximally way satisfy specification quantify priori reward use establish correctness logic specification respect unknown technique specification specification correctness preserve sub overall demonstrate requirement motion plan adversarial logic specification criterion seem effective supplement description hand concern rule specification temporal quantitative criterion help encode subtle application system requirement e jump light criterion specification design human synthesis specification reinforcement unknown solve synthesis focus static know dynamic crucial nearby adversarial environment strategy criterion objective logic specification deterministic environment environment objective guarantee adversarial quantitative payoff crucially rely quantitative gain experience direct interaction coincide multiple reward use application process specification expect modify case paper optimize priori temporal specification decomposition subproblem part encode adversarial satisfy specification quantify apply reinforcement operating envelope synthesis guarantee satisfy specification win concept rest model care system external interaction control play critical correctness specification discuss tuple control action finite action win action correspondingly state control set action available game state exist assume logic evaluate take specify otherwise state infinite finite memory strategy win initial formula win qualitative win model reward maximize choose evaluate system nonnegative consider accumulation instantaneous run game add instantaneous acquire instantaneous reward reward acquire weight example payoff function instantaneous independent define strategy give used run environment strategy interaction experience environment another use game win condition describe formula player game win maximize give instantaneous necessarily exist win strategy maximize win specification w fig r cm auto white edge loop leave loop optimal winning
density estimator density memory requirement small requirement entire present mixture logistic gps lda competitive root bayes kullback sequence bayes differentiable appendix strongly divergence compactly sampling normalization integral meanwhile numerator perspective scale inference leverage advance optimization resort particular mirror descent gradient unbiased stochastic bregman mirror descent minimize draw stochastic mirror iterate prox density divergence prox prox resemble rule furthermore pass arrive appear prox stepsize imply mirror scan dataset iteration issue tractable may mirror prox provable convergent particle later map eq update reduce usual stochastic mirror regard mirror inexact prox mapping step give recurrence sub q sa scheme average prove experimental suggest behavior last iterate prox essentially involve intermediate introduce particle already guess deal case initialization interestingly two yet good guess cover sample intermediate particle q form prox mapping ignore unnormalized version working incur approximate integrable latent variable lda incorporate guess could summation several difficult way lead particle inaccurate develop estimator leverage mirror suppose approximate smoothing derive mapping location solution version location associate normalize weight function working show possess formal old kernel kernel bandwidth om kde achieve rate stand lipschitz far linearly kde dimension weight solve prox section mirror weight estimation appropriate particle location divergence ht pt input density px present mirror descent incorporate posterior iteration maintain exploit benefit prox either weight therefore connect carlo particles integral return stepsize algorithm monte smc importance smc gradient smc utilize visit also share algorithm approximate density density product efficient make scale assumption rate sublinear sequence sublinear posterior term convergence return algorithm give difference commonly monte langevin may proposal integrable stepsize particle mirror consist integration optimization particle overall convergence integral true density generate later kernel function old almost surely boundedness depend assumption especially automatically validate bound convergence inexact prox mirror descent main state stepsize kernel tc b om apply b solve recursion sake sample convergence ratio number grow divergence average decay decay first directly achieve conduct bayesian process dirichlet multiple mode deal conjugate sequential langevin dynamics sgd langevin variational variant inference lda detail please observation p tie make mode pt ccc pass langevin ccc smc bandwidth theorem batch burn langevin repeat times recover method fit fail multimodal density kind quantitative way visit understand behavior perform begin utilize stop noticed smc start particle langevin dynamic bad one fit mode sgd langevin contours langevin find mode simultaneously sgd optimize inaccurate totally dependent logistic regression conjugate handwritten digits classification mnist function identity first period langevin initialize prior distribution pass whole time repeat obviously gibbs need scale well search notice achieve comparable performance nonparametric datum carefully flexible bottleneck sgd optimize ccc sparse gp wikipedia year conduct gps smc approximation represent randomly select year map standardized induce hyperparameter sparse fix bandwidth median distance point particle demonstrate advantage initialize initialize report pass baseline synthetic demonstrate sparse conduct generate rbf times batch illustrate evolve datum see gp well mean sparse cccc iteration ccc lda wikipedia dataset document vocabulary estimate separate document since follow particle smc save set fix solely pass stepsize default burn provide search
contrast fine domain unseen domain table contrast pos dependency f bt l feature tb l pos google gram word syntactic syntactic pos embed syntactic embed syntactic pos cnn word dependency bilinear embed publish compare previously publish compositional ne tag tag help may make expressive introduce distinguish embedding fine report task traditional cnn less next feature template head surprisingly show binary template remove degradation remove entity need md attain nothing strong move engineering hand capture linguistic word strength relation back tag semantic back relation move lexical word embedding word dimensional space back designing easily incorporate incorporation dense value sentence certainly challenge mention extraction consider limited hand property contribute utilize compositional deriving sentence level word embedding compare compositional arbitrary type alone state art extraction obtain art approximation embedding open domain sentence contribute component parse application gold entity prior instance entity labeling pair instance gold train type ignore relation token report domain report additional task entity semantic entity entity report comparable super tag entity paper use tag table entity type greatly remove contain entity baseline drop gold entity unknown play role improvement become context head dominate make entity type unknown baseline tag resource encourage entity predict gold bc pm pm pm baseline pm c author md intelligence technology china compositional linguistic rich compositional extraction expressive domain idea learn embedding able tackle difficulty meet compositional embedding handle arbitrary sentence annotation global propose relation extraction compositional traditional rich relation tp sentence drive united people drive depend word generalize g operate lexical insufficient extraction lexical appear sentence entity significance fine information nlp feature word extraction lexical word linguistic help capture lexical embedding embed task parse semantic extraction capture lexical insufficient linguistic context compositional model linguistic embedding extraction contextual feature construction generalize composition linguistic begin construction compositional use feature capture compositional treat word embedding rise annotation utilize associate annotation stage annotate sentence annotation combine word sum compose softmax layer output dataset relation goal identify pair construct sequence tag name entity direct relation type towards embedding compositional nlp relation still e relation sentence extraction name entity boundary type available assume two entity relation standard within sentence entity complementary illustrative example develop representation capture word entity incorporate work use entity feature insensitive show word propose framework sentence annotation focus relation extraction benefit lexical annotated embedding embedding special subsection annotate sentence first vary sentence hand vector matrix herein annotation annotate sentence distinguish annotation task utilize extraction sentence embedding input case relation extraction annotate sentence many nlp highlight annotate embedding specific form polynomial specifically literature suggest powerful directly log model form annotate embedding previous annotate sentence produce entire formulate dot product matrix normalize matrix model fix bilinear feature drive indicate appear label embedding drive dependency label dependency path weight generalize across word share product smooth lexical nlp lexical lexical property lexical word outer recover word form therefore view lexical keep expressive cnns rnns zero sentence cnns rnns optimize word embedding gradient application equal gradient bilinear deep embedding training easy incorporate separate feature use feature vector path head refer entity name head two conjunction indicate entity whether entity feature entity result help embedding predict introduce lexical embedding embed linguistic annotation pos stanford gold entity entity tag embedding train portion corpus l set head h
code region optimality sequence convergence convergent minimizer well minimizer respect irrespective code outer large enable insensitive even convergence result theorem correspond except theorems corollary theorems blind compress sense cs involve patch level tune contrast involve set mr berkeley simulate include encode reconstruct step hard ball well p slightly transform learn well unitary transform application compare specific work formulation algorithm behavior reconstruction variation transform overcomplete recent use redundant geometric early method software implementation respective build wavelet image use image solve size suggest four fold overcomplete learn iteration overlap find empirically execute iteration per employ sparse code dictionary threshold patch vary linearly set patch patch dct initial fourier code matlab version implementation optimize execute computation perform intel core cpu ghz gb operating quantify mr reconstruction signal express db peak image root relative reference reconstruct fully k quantify symmetric whose deviation pixel filter reconstruct reference magnitude learn complex display reconstruct raw weighted brain raw fast spin sequence te fold peak reference execute converge quickly show e successive iterate converge theorem metric quickly initial measurement scenario db hand final reconstruction enhance db db problem identifiability noiseless scenario learn transform part frequency image experiment execute lead variable density normalize reference correspond zero reconstruction reconstruction various scheme well mark see various significant improvement db transform partially finally quality fold overcomplete rich show reconstruction difference magnitude reconstruct cccc reconstruction magnitude update since arbitrary establish lemma successive iterate converge subsequence index iterate converge accumulation convergent say argument respect convergent subsequence monotonic decrease lemma right together set patch always minimization unique minimizer combine work convergent subsequence finally subsequence accumulation coincide accumulation converge proof consider x convergent subsequence thereby inferior superior bound convergent subsequence subsequence convergent sequence convergent subsequence next accumulation set accumulation denote state subsequence iterate converge accumulation partial accumulation point simple sequence every accumulation svd accumulation iterate algorithm involve code every outer outer update use consider index accumulation full x due limit column l h l b subsequence aforementione square root svd subsequence get immediately establish accumulation iterate critical sub matrix accumulation algorithm subsequence iterate converge accumulation lemma singular next g finally eq easily thus accumulation establish accumulation point accumulation iterate small region subsequence index iterate sequence accumulation lemma accumulation perturbation perturbation preserve utilize equation side replace hermitian operation involve long whenever easy preserve support zero b therefore accumulation sequence region subsequence accumulation lemma consider perturbation column denote order perturbation b trivially energy preserving follow expand drop argument equation unique argument simplify scenario preserve perturbation b minor barrier set replace operator theorem differ local perturbation particular accumulation extend proof lemma finally theorem negative barrier unitary keeps otherwise national foundation nsf grant correction email signal know property heavily exploit medical imaging sense exploit domain synthesis reconstruct measurement blind compressed reconstruct well transform descent involve closed form importantly although blind sense formulation nonconvex converge objective define formulation usefulness image reconstruction highly measurement involve model provide medical compress sense convergence extremely popular application exploit sparsity patch reconstruct investigate subject aim scenario good image transform briefly review topic model blind compressed contribution base image various synthesis dictionary give disadvantage synthesis sparse deterministic hard various tend large transform approximately transform residual error rather image analytical cosine transform wavelet difference advantage synthesis compressed version fouri fourier recovery acquisition incoherent sense sense expensive linear compressed sense typically representation stacking measurement sense measurement typically choose orthonormal satisfying transform domain equation represent true rewrite substitution hermitian transpose synthesis np quasi replace convex problem reconstruct image cs fidelity depend physics recently image technique ct emission imaging demonstrate quality reconstruction compressive advantageous reduce scan time clinical throughput well reconstruct subset pixel compress sense compressed technique utilize wavelet reconstruct instead focus blind compressed compressed simultaneously enable datum drive dictionary transform advantageous application adaptation synthesis study transform prior successfully demonstrate compressed sensing image overlap patch typically learn compressive measurement much compressed method surprising method specific adaptation one primarily focus synthesis transform model involve transform simultaneously highly measurement propose transform importantly converge critical highly minimizer compressed compressed discussion work measurement sense denoise deconvolution imaging technique mechanism excellent visualization space acquire drawback affect clinical especially dynamic imaging application relatively slow imaging technique advance hardware mr limited mr physics constraint energy compress either aforementione enable reconstruction mr reconstruction transform reconstruction involve transform importantly synthesis reconstruct speedup consider mr amenable clinical describe transform blind compressed formulation efficient coordinate problem present novel scheme proposal compress reconstruction view particular constrained regularize reconstruction adaptive transform suffer high blind compressed directly object overlap patch assume patch along fidelity patch synthesis number sparse column learn additionally avoid scale ambiguity consider strong flexible patch code sparsity practice appropriate coding sense measurement standard level estimate synthesis dictionary reconstruction adaptive convex hard g synthesis code repeatedly convergence overcome aforementioned drawback synthesis transform transform effective learn constraint within arrive transform patch patch denote term number notice enable level patch p enforce range transform patch denote inverse trivially transform prevent degenerate repeat help ambiguity sparse representation penalty cause penalty together additionally control scaling learn regularizer condition I easy transform tend unitary learn via even transform previously well strictly orthonormal scenario representation denoise patch range well learn unitary transform regularizer follow formulation unitary transform problem p unitary sense jx jx simple consider error unitary w global p solve jx possible low notice satisfy triplet feasible case minimizer patch constraint unitary guarantee solve lack image minimizer propose admit code minimizer another minimizer permutation alternative problem replace problem weight p use absence space code objective finite constraint feasible potential unbounded iterate version constraint propose extend propose transform formulation scenario single extend image frame slice jointly reconstruct transform p objective video formulation extend compressed block formulation alternate code transform variable alternate step describe detail sparse involve follow transform patch notation frobenius set choice large choose unconstrained solution case index column page definition choose solution formulation variable keep p analytical term decomposition definite give eq solution invariant factor cholesky factorization eigen form nevertheless scalar assume convergence one standard accuracy aforementione give global minimizer non singular problem least alternatively solve exactly lagrange multiplier corresponding lagrangian formulation lagrange satisfie transpose real matrix unique algorithm patch direct sized gradient lagrange multiplier optimally j tw hx optimal multipli monotone guarantee rate solution practice loose solution minimizer obtain cg additional computation avoid way find repeatedly tune employ cg various matrix exploit enable show assumption efficient application structure toeplitz efficient case patch formulation overlap overlap opposite distance location clear patch around proposition establish block fourier encoding include column matrix except tw entry overlap image apply shift version correspondingly shift operator circular convolution sum standard state typically diagonal computed first factor case unitary assume matrix arrange first entry column patch operation correspond corner zero elsewhere extremely negligible aforementioned finding dft impulse efficiently fourier modality obtain obtain subsample grid multipli computed computation assume around image include matrix multiply diagonal q location lagrangian space location need newton obtain update fig would specific dct measurement sparsity iteration adapt transform code overlap patch repeat full svd hx l l set column generic p h z image ts w code transform iteration computation square root operation code therefore scale notably projection ball sort hard cost transform operation fraction zero application I etc cost patch pixel cost dominate hand various operation multipli latter take typically newton argument total
index column tolerance sample sample row random index block k ic general psd finish need prove guarantee form fully show compute linearly guarantee enable calculation computing discuss psd alternate proof prove linearly remark linearly hold terminate early early termination choose column nystr om I x x low right exactly rank vs random sampling b terminate trial exact guarantee recovery choose redundant accurate variety column contrast choose redundant rank gram nystr om precisely combinatorial nystr om indeed guarantee exactly recover column expressive although recovery step center psd rank column rank iterative compute column terminate machine theoretical guarantee guarantee selection redundant separate trial sampling sampling frequently select span step limiting equation enable must considerably computation result score computation dense nystr om mean form full result useful nystr om method gram finding slow uniform speed compute way regime complexity p slow regime make uniform competitive form many substantially expensive adaptively parallel appear column selection third adaptive advantageous run see low complexity accurate low runtime see nystr om approximation nystr om form impractical third problem class consider contain sum tune om random ii leverage nystr om iii repeat experiment uniform intractable store uniform sampling describe matrix fit memory sampling method tractable generate calculate consider dataset use matlab processor large competitive accurate convergence vs sample show vs second rate column second curve fair see consist point arrange kernel point characteristic age without dimension around cube vertex distribute cube impractical explicitly rather sample frobenius discrepancy score full representation compare follow matlab ghz processor gb show implicit mean mnist problem handwritten benchmark mnist datum contain image pixel similarity point hyperspectral band classify area assign class ground represent assign class light field dataset intensity plane patch stanford camera array angular point split uniform long frobenius discrepancy randomly entry eigen node core ghz processor core table size available increase among point sample two tolerance random capability subset consist size reference store binary color channel focus determine kernel maximum become approximation trial image kernel achieve uniform sampling leverage score full addition nystr om addition deterministic enable adaptive scheme know priori column leverage one must guess appropriate certain primary scheme accuracy example uniform random sampling well continue add figure second efficiency dimensionality reduction kernel nystr om clusters compute centroid compute kernel mean overcome observe flat leverage entire grow large primary runtime accurate give mean cluster gain nystr fast run multiple time nystr second run take furthermore sample selection invertible sampling calculate step gb infeasible column intensive straightforward appear higher random fast column take regardless selection compute computing data form less take residual sparse extremely matrix store novel om approximation requirement random exact demonstrate efficacy matrix processor size regime numerical competitive scheme able approximation cluster remark conjecture proposition g adaptive g gram approach reduction form intractable incoherence without guarantee recover numerical achieve adaptive cost complexity low psd machine machine framework require formation contain similarity nonlinear storing grow matrix increasingly store extend extremely nystr om rank keep capture majority ambient lie approximation information nystr compute broadly selection apply application range segmentation factorization success identically underlie sampling draw practice sampling far efficient uniform advantage entry adaptive computational burden reason entire form store require even store zero reason apply extremely collection possible kernel form candidate kernel matrix small sample un column principled predict informative matrix form operate select explicitly submatrix dimension make order runtime recover preserve sample column enable efficiency computation fill provide tractable adaptive apply accuracy comparable dramatically usefulness long entirely work processor efficiently column incur overhead node message pass interface om regime addition exactly recover greedy guarantee linearly independent inefficient organize introduce nystr survey important motivation behind column incoherence sis accelerate sis call theory determining exactly demonstrate efficacy approximate common machine nystr om describe write respectively denote product row sum describe matlab indexing index widely classification cluster lift linearly separable help measure frobenius accurate combinatorial complexity adaptive build nystr select residual large update accurate residual calculate sufficient obtain make point consist cluster centroid centroid describe zhang compute centroid exist mean np hard cholesky sampling parameter reveal low use example self expressive seed select point match pursuit seed properly dictionary use adaptive adaptive subset complexity address om approximation sequentially select nystr om lie column already add ideally significant impact span column incoherent already criterion find incoherent psd recall contain x ix select expand base upon denote second comprise index contain approximation entire apart sequential incoherent follow collect index element maximize value terminate
eps eps comment procedure kernel unbounded correlation difficult problem provide issue dimension eps eps wherein design inside section kernel suppose center optimize second figure contour length optimal move correlation center information length decrease become away reasoning show place area average eps eps eps besides another way length cost several experiment perform try another objective experiment lie exist point information length lie cost exist optimizer good phenomenon demonstrate minimizer near minimum eps eps eps numerical show word influence geometry evaluation adaptive closed loop procedure design experiment evaluate update hyperparameter g log function maximize gradient base sequential programming couple design describe adapt batch batch change rapidly batch leave batch size overall sense greedy design context maximum proceed batch greedy function hypercube endow weight exponential kx il l impose adaptive batch hyperparameter inherent error entropy adapt design batch hyperparameter outperform however marginally achieve error decay batch hyperparameter optimization experimental domain describe hyperparameter domain ill place new close size initially large subsequent design feedback large counter robust situation hyperparameter unbounded gp another understand approximation ever gp quadrature appropriate comprise advantage onto subspace basis sense grow exact quadrature seek inner q still external necessarily span rule subsequent express orthogonal version basis polynomial quadrature rule already decay interpret modeling impact regression space approximation rule gp kernel kernel function eigenfunction connection comprise difference gp identical immediately corollary case eigenfunction infinite consist eigenfunction quadrature rule clearly among generality quadrature posterior kx difference approximation source eigenfunction span contain gp case equivalent space gp simply due yield approximation example kernel construct fully polynomial eigenfunction quadrature rule bind depend condition sufficiently situation poorly condition difference approximation imagine invertible quadrature whereas proceed manner approximation experimental posterior eigenfunction quadrature splitting eigenfunction second term expansion contribution extra eigenfunction integrate comment special property eigenfunction thus integrate explicitly enter gp rewrite eigenfunction eigenfunction expect design eigenfunction error incur numerical eigenfunction eigenfunction order design ensure integrate capture precede underlie practical implicitly assume projection otherwise indeed vast regression properly expansion rkh accurate readily interact integrated variance ensure span eigenfunction correspondingly force measure uncertainty retain contribution subspace difference approximation extra tradeoff generate perform regression gauss quadrature polynomial degree kx ix normalize polynomial decay surrogate experiment closely surrogate approach evaluation approximate infinite eigenfunction emphasis project external figure error approximation gp improvement associate figure onto reference right panel show energy basis magnitude projection compute extremely high quadrature onto even projection basis begin decay error index function begin decay somewhat slowly basis lie red line error gp design spectrum error grey panel quadrature trend compare evenly design error spread broadly ht surrogate gauss quadrature gauss quadrature magnitude space similar experimental contain rank gp eigenfunction numerical like gp gp regression principle gp wide eigenvalue kernel endow standard eigenfunction polynomial version I eigenfunction present section adaptively greedy sum approximation regular infeasible grid dimension grid provide benchmark mit quantification framework domain endow gauss quadrature additive separability favorable use batch loop noise result trace relative f mx carlo h eps right hyperparameter trace design panel trace design adaptive evaluation begin error precision quadrature hand gradually additional add gp already trace fairly fairly indicate fast decay converge eigenfunction converge eigenvalue quadratic hand order order observe decay roughly dimension matter next note expect see well example product eps eps begin roughly evaluation error approach several gp reach relative reach attribute require interaction interested repeat hyperparameter adaptation length randomness choice start figure show square exponential typically hypercube domain approximation unbounded weight significantly challenging tail error gp using show well benefit include eigenfunction calculation batch obtain experiment future adapt iteration fall basis require gp family truncate kernel eigenfunction finite kernel adapt criterion surrogate design design criterion present integrated process continuous spaced avoid undesirable interpolation design substantial benefit strategy nonetheless greedy standard minimizing demonstrate adaptive update gaussian approximation simplicity gp use eigenfunction difference eigenfunction projection average quadrature quadrature node performance adaptive gp easily approximation couple domain current approach eigenfunction integral eigenfunction couple present work experimental way interpolation node radial basis compare quadrature great broadly rigorous reasonable ahead design adaptation posterior may effective author acknowledge research notation associate separate denote rest orthogonality rule mx ax define square orthogonality orthogonality property simplify expression take eq equality equality substitute cauchy matrix multiplicative j ax cauchy schwarz orthogonality begin trick replace come orthogonality equality arise arise come split mit edu experimental design procedure explore use gp minimize posterior integrated gp treat domain point good interpolation entropy mutual second perspective gp identify regression coincide polynomial orthogonality eigenfunction approximation set adaptive approximation function favorable experiment interpolation quantification computational essential design quantification system require large prohibitive expense surrogate relevant analysis computational simulation surrogate experimental view attempt input relationship obtain suitable choose contain parameter convert simulation include span radial experiment designs design mutual quadrature linear independent procedure preserve orthogonality relevant design surrogate space use approximation analyze minimize design process criterion criterion broad surrogate difference space finite call minimize involve add experiment candidate criteria mi alm criterion maximize alm criterion consider effect well design expensive mi sequentially maximize gain mi obtain domain complex direction computationally expensive deal combinatorial optimization explore ad hoc optimization sometimes design input help domain create barrier benefit complexity minimize undesirable cluster radial alm well approximation performance alm mi finally perform continuous good lebesgue raise quadrature use eigenfunction kernel eigenfunction assess design eigenfunction eigenfunction entirely integrate illustrate qualitatively quadrature select experiment outperform experimental test outperform approach paper organize part part review section section describe second background describe comparison gp process approximation posterior begin covariance semidefinite suppose simulation parameter evaluation result notation component covariance interpolation however ill condition result eigenfunction recall countable endow borel suppose eq operator countable eigenfunction form eigenfunction eigenvalue reproduce rkh kernel represent convergent let first countable locally compact strictly series converge absolutely compact subset regression later use kernel equivalently q polynomial viewpoint ill practically singular precision decay integrate point integrate become prior always integrate prior experimental integrated gaussian process choice motivate inferential consideration oppose procedure quadrature large add experiment location commonly call alm change h f mutual maximize greedy fashion candidate experiment greatest entropy mi perform greedy inversion contain set simulation set candidate invert effort aim reduce cost specialize kernel mi depend crucially candidate base differ approach avoid challenge follow optimization descent multiple experiment simultaneously design advantageous interaction account design take beyond early help design domain proximity location also address problem correlation kernel sufficiently expensive find design procedure one effort require experimental summarize computational experimental objective consider scenario carlo sufficient mi criterion candidate evaluation typically alm must location inversion evaluation design carlo pt mutual pt alm option minimization continuous candidate discrete design mi remain tractable possibility choose design play important role numerical objective write q become increasingly must similarly th data informative mode
word top near topic eight typically respectively understand difference like cancer topic return medical care share semantic fashion topic topic distinguish result lda qualitatively lda firstly explore structure inherent topic simultaneously acknowledgment cn mining document play language however relationship occurrence corpus good word essential representation lda representation topic space alternative show interesting model text document language nlp ir enable benefit similarity relevance past decade solution bag tf semantic probabilistic latent semantic know allocation relationship word document distribution word occurrence document lda high probability probability choose representative drug technology probabilistic language represent word document dimensional nlp word propose could syntactic relationship like paris state sentiment analysis paper answer semantic idea propose propose topic incorporate topic topic semantic relevance use cosine representation aspect list topic result achieve allocation lda latent vocabulary lda generate n word topic sample topic document infer latent meanwhile inspire skip word sequence maximize base skip gram skip window calculate representation w incorporate representation skip situation predict word skip predict word document infer maximize likelihood gram pz skip topic approximately maximize probability softmax without softmax stochastic sgd model dataset word english learn fundamental extract document choose document besides word word training contain million word
compatible adopt different problem investigate various user requirement wants meet practically reach statistical cluster respect generate first base technique result complexity investigate obtain standard setup supervise representation implicit cluster quantify define pac learn introduction notion capacity complexity combinatorial notion version mapping losse true embedding minimization erm successfully particular mapping paper organize section define investigate erm algorithm uniform sufficient present conclude provide subset denote cardinality respectively induce accordingly difference partition fx centers center minimize cost mapping mapping mapping formally mean let every pick mapping generating readily cluster output pac stand probably pac mapping representation size least regard formal pac problem good mapping class intuitively rich mapping cluster introduce appropriate address bind representation need erm learner sample implicitly access go minimize note studying formalize representative mapping respect cluster representative uniform convergence mapping therefore upper representative solution find mapping clustering learner interpretable mapping way include mapping solution concrete define notion introduce slight uniqueness say k mean cluster satisfie satisfy eq degenerate solution uniqueness require cluster arbitrarily small totally uniqueness mapping mapping uniqueness argue subset useful therefore rest paper next prove class representative algorithm feed make sure actually representative formalize mapping sample select paper devote provide mapping exist number small specify interested note mapping real prove complexity mapping beneficial capacity help easy analysis sample complexity class vc output function number pseudo generalize notion permutation binary union formally set able investigate mapping representative I value provide vc however provide dim sense class next introduce prove purpose uniform argue care class section uniqueness lemma previous let cover k basically cover mapping uniform result mapping constant cover hypothesis precisely e theorem number mapping notion mapping define let real define let l q value pseudo reason cover cover cover rewrite uniqueness show sufficient sample follow combine pac mapping logarithmic proof mahalanobis metric mapping result pseudo value dimensionality case value scale factor statistical representation mapping target cluster cluster finding mean pac notion erm technical uniform result complexity mapping notion mapping complexity uniqueness reasonable learner mapping unique otherwise interpretable uniqueness pac notion result define make challenging analyze open question acknowledgment proof uniqueness property note due eq term prove center ready first small cost follow lemma second david david school knowledge propose protocol relatively come datum align formal analyzing paradigm capacity spirit vc representation learn induced embedding task divide set coherent subset serve apply dataset likely dramatically answer critical
ergodic reversible spectral gap provide stationary random define version q spectral form spectral stationary eq length within multiplicative require average expectation average form markov chernoff bound result significantly bad requirement instead adapt tail iid directly article yu ergodicity state eigenvalue use frequency eigenvalue fact reversible difficulty invoke page whose exponential th root avoid return question fully notice directly suitable arbitrarily possibility width increase effect chain slow mix small confident empirical subject achievable sample path compute visit smoothed empirical bound sensitivity confidence stationary idea dependence interval include probability path markov frequency state visit lead slow rate strong inequality estimate form plug form mixing indeed estimate exploit confidence smoothed frequency call accuracy ergodic chain smooth sensitivity plug form spectral relate relate perturbation entirely observable g estimate computationally note reduce matrix implementation state concern input path ergodic reversible chain confidence spectral unique stationary q moreover obstacle encounter avoid establish observable martingale tail comparison validate bound remain part simultaneously hence interval empirical determine lower plug bound center confidence detail new interval u v furthermore surely apply generally I asymptotic random deviation help bernstein tail due divide contiguous block size resp let dd position inequality contribution block note ahead observe copy apply copy follow sequence let suffice surely block q norm sum cauchy schwarz gm simplify expectation third place last step bind result yu sx h product form marginal joint yu imply event denote recognize process chain integrate give time proof q conclude eq last step follow return combine obtain least combine probabilistic condition least probability finish replace observe recall trivial violate chain chain must markov swap construction indistinguishable eventually reach path length c chains eq ergodic reversible chain j di regardless must visit distinguish chain stochastically dominate generalized number random geometrically least visit smoothing positive ergodic unique stochastic martingale measurable union view constraint unobserve event hold avoid spirit quadratic conclude claim stationary unique eq matter start role central capture ergodic transition role j define analogously perturbation comparison p ji j establishes interval bound quantity q p q entry bound norm validity interval li improvement continuity proof term state sensitivity let compute bind fully yield derive complete claim proposition key true lemma definition bb c university article provide fully dependent prescribed stand previous either require additional interval zero length path sample require constant multiplicative low restriction place chain procedure achieve direction research work challenge fully markov irreducible finite stationary time assign trivial nx x scientific interest arise involve mix effective estimation quality knowledge develop construct non trivial empirical confidence reversible ergodic suffice main summarize guarantee multiplicative visit provide estimation estimator multiplicative notation feasibility interval unknown quantity chain task explain turning fail theory avoid interval valid vast literature markov instance converge surely central distribution deviation zero asymptotic nf result help limit behavior mean chain little behavior need evaluation numerous chernoff provide probabilistic corresponding bound identically due temporal dependence intuitively draw bound chain effort unknown context provide estimation asymptotic hence path another theory sufficiently yu mr providing case however estimate derive mix coefficient limit possible eliminate difficulty flexible sampling oracle generate independent device mix hand circuit transition probability exponentially large diagnostic integer expression exact elsewhere symbol
establish polynomial np clique show clique instance give input vx opt opt prove minus top write eigenvalue equation solve quadratic root expression decrease removal stronger exclude constant natural version clique subgraph subgraph maximum ds admit algorithm vertex every algorithm one determine polynomial determine clique distinguish clique graph construct let adjacency run solution clique contain clique opt clique radius graph edge distinguish clique every efficiently clique degree ds constant one clique subgraph vertex polynomial plant approximation plant clique definition axiom mm mm give clique establish exclude np weak exclude classic tool challenge interpretation component combination original significance application desirable obtain goal hardness clique traditional cardinality maximization sparse exist I radius eigenvalue resp zero resp exist clique fairly clique need order clique
code symbol symbol decoder distortion prescribe fidelity specifically ny alphabet fidelity letter operational indirect distortion block indirect shannon infimum mapping nd source code operational indirect rate direct observable indirect distortion fidelity criterion shannon source theorem imply q indirect indirect direct distortion computation alphabet perform transition probability letter introduce constraint lagrange dual give transition rhs non satisfy necessary achieve maximum number satisfy satisfied equality hold context auto source node noise plus channel hamming fidelity receiver correspond give source reconstruction view remainder infer fig decrease hamming distortion correspond case define distortion measure distortion intuitive interpretation make list step solution outline proposition equality imply maximize rhs derivative non decrease single conclude rh equality rhs special substitute q symmetric compare observe correspond domain decrease therefore increase reduce correspond vertical dash slope determine return unit system confirm increment bit describe less effective distortion intensity rate coding need provide tight convexity closure illustrate theorem summarize rh indirect hamming distortion give noisy channel indirect distortion investigate bit although conceptually model important distortion level error balance existence treat lead write derivative domain monotonically whenever since behavior root substituting remark department electrical stanford department electrical national height circle fill blue cm bernoulli compress manner consider noisy symmetric control classic distortion distortion function source rate distortion term indirect distortion close expression return increase bit rate distortion channel distortion allow average extension scenario source directly obtain noisy observation environmental source process statistically know motivation indirect source centralize restriction computational consumption local infeasible distortion fidelity criterion square describe description allow quadratic general possible paper binary hamming distortion introduce shannon provide source indirect indirect implicitly equivalence indirect problem fidelity fidelity identify compute indirect source
statistic reproduce hilbert mapping optimally learn pair scoring turn asymptotic see anomalous rank advantage false alarm nominal level preference sort x sort anomalous standard setup characterize high sec step rkhs kx cross learn adopt weight pairwise disagreement loss quantization complexity high training complexity closely ranking nn distance noisy datum raise fairly sec insufficient false alarm demonstrate next connection svm lie build connection anomaly detection natural ordering create apply detector unseen produce approximate unseen object justify linkage fall quantile toy example approximate consider quantization impact level set show appropriately quantization show alarm rate reject toy example demonstrate nominal plot appear reasonably algorithm preference pair vary surprisingly approximate notice generate level quantization preserve order discussion turn quantization alarm htb x approximate oracle density curve appear vary curve approximate peak step pair sort training stage wise algorithm stage evaluate binary svm bp point note input pair much worth distance adopt reduce stage analysis mention nn distance guarantee asymptotically reliable slightly purpose worth paper quantization conclusion assume none quantization tool consistency quantization arise score distribute probability nn cause confusion eq wish solution consistent fix rkhs rbf see claim concentration measure relate make except rank sample learn compute value test evaluate prescribed alarm region r nx integer give newly draw nominal lie denote contain fix permutation nk remark precisely draw score percentile bound asymptotically approach irrespective give emphasize context sort increase small low correspond extreme carry anomaly experiment sub isolate forest svm code configuration configuration fix bp use step calculate rank accord level whenever adapt routine version implementation statistic calculate rank vice rank average nn experiment nominal value performance quantization nn validation use disagreement vary ability bp comparison significantly bp surprising statistically appear marginally careful reasoning significance anomalous consequently extension marginally bp statistically resample see sec validation cv use anomalous argument smooth well account level unlike nn present discriminative ranking learn nominal asymptotically high region allow alarm rate approach state art grant st view interpret necessarily policy express ease divide point provide near among omit st near proof involve converge empirical concentrate hoeffding inequality step properly q last discarding follow check replace easily verify diameter inequality variable rewrite divide part relatively close show exponential converge x fu min fu min fu fu df df combine converge min rx bx mu lx rx l bound lx cd lx diameter upper relate notice loose c min r mr smoothness density bound bx r q min lx ax ax line combine input set variable main body small assume preference pair let ad svm rbf associate cover radius appropriately let appropriately kn us argument author minimize borel q derive consistency enough due concentration inequality inequality inclusion compact compact cover radius inequality covering finite cover attention disk radius also quantity enough schwarz tf tf n cover radius center covering rhs bound verify difference change ready result complete minimize asymptotically recover preference relationship give surrogate differentiable density j fx surrogate non differentiable correctly hinge svm preference sample proof line l pg pg gx gx I dr gx dr j gx gx requirement function increase six sum pg gx g pg g l I gx gx dr dr dr k gx dr however requirement complete theorem prove lemma set system measure event remain portion corollary thm parametric score nn accordingly train limit false alarm percentile result anomaly detector alarm decision superiority exist nn anomaly detection detection statistically deviation behavior area detection security surveillance parametric characterize class anomalous parameter provide likely view find least unknown test set estimation include estimate suffer high statistically unstable volume volume test avoid compute paper improve computational however stage runtime ambient test different db method reference therein possibly db issue specify work anomaly anomaly generalize auc anomaly scheme false alarm characterize leverage db identify outlier anomaly well estimate addition poor produce
immediate neighbor refer treatment interactive perspective reinforcement correlate assumption arguably way social norm take adaptive connect regret reinforcement algorithm isolate game past decision regret matching degenerate pass cyclic path visit among rather adjacent implement diffusion strategy make play human social interact environment sensor beyond physical sensor preference particular physical social sensor interact sensor reveal preference reveal micro economics parametric agent influence utility maximization preference single agent measure discrimination analyze relationship price internet service position page search google interact social test typically interact require utility detection player game extensively maximization agent utility resource social scheduling demand scheme introduce social refer tendency various individual similar relationship share motivate communication elaborate prominent comprise entity make sensor mobile device etc available decision action establish agent k k agent exclude bound reward cost outcome payoff function reflect privacy maintain link reflect consumption production content production sharing capacity restrict agent situation model economic literature formally form non action space e speak identity transform payoff market model cost far share communication among agent capture graph respectively agent except neighbor network aware agent outside social neighbor payoff decision exact however even know utility straightforward generalize form social present cluster network modification simplicity continue use paper part define correlate equilibrium show page pick equilibria several motivate adopt correlate simple nash equilibrium among equilibrium lead higher require realistic naturally agent future correlate equilibrium interactive decision private recommendation action agent recommendation draw action recommendation recommendation decide follow result neither wants provide recommendation agent device trust reinforcement attract much attention processing accord r ni average loss select action play action see bottom page strategy agent utility jump denote true update rely realize pt imply gain reinforcement assign positive fact name inspire enforce neighbor via regret regret agent belong utility decision uncertainty decentralize adaptive real diffusion game learn social share matrix connectivity agent immediate global identity denote represent one respectively agent combine realize rescale individual fusion neighboring belief approximate far use ordinary differential ode experience enable algorithm summarize protocol human individual ordinal place set update member group local summarize term part valid term action space force action play speak statistical large lead behavior correlate introduce evolution agent successively old vanish strategy ordinal action order exception equally see pick decision make play bad cycle equilibria assumption game conditionally agent chain feed sec derive agent individually time action profile space profile adaptation exponential profile enable evolution old decision repeatedly action agent straightforwardly evaluate global via recursion reveal global game matrix experience average characterize average behavior dynamical work work accordingly abuse notation vector rather represent process far represent euclidean characterize local global algorithm real exist agent follow distance usual distance global equilibria sense theorem play game exploration state collective equilibrium distribute fashion behavior equilibria polytope theoretic view rational agent sophisticated rational argument weak determined theorem differential inclusion appendix differential generalization path derive analytical illustrate c cm agent exhibit characteristic isolate agent arise social agent aim coordinate decision group place receive connectivity belief strategy reinforcement however replace algorithm view polytope quantify evident correlated equilibrium reinforcement stage share monotonically reveal preference equilibrium play setup address yes agent learn network fundamentally different compute minimum preference approach wish interaction reveal seek agent maximizer subject budget constraint micro economics google mx mi response external agent dot line aim nash equilibrium game reveal preference utility maximization mx select function non rule utility optimize amount social influence consumption agent external influence resource impact therefore resource impact provide maximizer maximizer exist concave scalar axiom reveal namely point remarkable feature theorem trivial concave monotonic utility way continuity concavity monotonicity demand comprise alternatively ordinal transformation function also ordinal geometrically utility finite hyperplane theorem social potential wireless network influence social agent p mx play potential fig external action formally agent utility denote action game individual satisfy nash satisfy nash equilibrium strong nash play ix nash maximization budget budget total resource nash concave game differentiable agent give statement consistent nash scalar follow potential axiom reveal single monotone several option produce preference potential provide necessary nash statement intuition connect statement provide concave potential response strategy nash equilibrium parametric nash involve determine feasible solve linear constraint polynomial short flow parameter fail nash result action noise statistical detect nash action consist feasibility clean nash denote clean nash hypothesis satisfy nash vs source eq give agent action nash equilibrium play game significance ii solution characterize appendix consider influence recall inequality allow feasibility straightforwardly guarantee less detection nash definition enhance adaptively optimize external reduce achieve dynamically external external influence density satisfy nash definition gradient simultaneous utilize estimate corrupt reach probe use indicator construct generate realization parameter accuracy probe estimate gradient compute per iteration exposition point detection apply nash real aggregate consumption energy market social network comprise agent agent section consumption price consumption system operator website economic rational self rational utility nash true associate construct management control consumption power consumption price set management respective price utilize aggregate consumption maximization nash detect demand power consumption model construct potential external agent price several behaviour weather define denote day action aggregate consumption respective budget unit aggregate consumption satisfy utility result consumption agent limit suggest aggregate follow variance term satisfy utility provide fig h consumption day start west east maximization stochastic pass consumption independently maximize grid however concave surprising result distribute demand management consistent nash test power nash construct agent prefer marginal construct potential suggest prefer agent give consumption agent player agent theorem agent preference improve program attempt site facebook twitter agent comprise depict tx detect social known detect tweet request tend behavior human twitter tend tweet far human tend connectivity network friend political consider depict fig agent design detect eliminate account network denote response agent account action agent agent static budget agent total total resource agent query friend capture agent decrease resource limited friend capture attempt respective agent preference fashion agent maximum concave probe distribution agent represent magnitude iterate fig via gradient decrease allow reject optimal external influence type error test satisfie optimize allow distinguished error statistical agent agent equilibrium game agent reinforcement equilibrium correlate wherein network topology rely follow agent attract set focused parse construct equilibrium probability example example detect property consider ordinal nature human behavior step proof omit adequate reference characterize system represent differential inclusion differential provide inclusion nonempty convex make form markov least characterize invariant take light protocol successive affect strategy regret use stochastic piecewise derive limit simplex joint profile kronecker product process define
x kx kernel scalable gradient idea inspire recently maxout random maxout feature locally boundary component interesting linear interpretable map linear bandwidth kernel involve product linear map feature main introduce analyze maxout advantage training utilize avoid take maxout scale classification unsupervise maxout follow pca data reduction visualization maxout locally maxout maxout relate approach set corpora maxout maxout maxout unit give precise description maxout maxout unit maxout study shall consider dot z independence expectation hx I maxout maxout random x polynomial proof material maxout unit linear estimation piecewise linear q locality value particular case coincide oppose understand locality non linear radius locality pool look effect gaussian vanish hash qualitatively locality qualitatively far apart radius closeness size pool radius hence locally linear solve reproduce kernel I derive linear radius locality maxout random feature map definition dot therefore I get sufficiently next incur translate convergence rkh locality hash cx jx w c qx j z binary hamming sensitive hashing show maxout space allow linear classification locally hilbert functions maxout approximate dense belong unit loss intrinsic ball cover radius dimension use intrinsic ball moreover suitably iid give gaussian dense approximate replace practice maxout feature q numerical choice training example loss bound relate quantity dimension maxout risk achieve nonlinear class locally precision maxout statistical function suggest nd space live g dx q f f ce e c e q projection assumption us study maxout first element value locally interest locally dimensionality reduction empirically efficacy
possible change combine sparsity encoder lemma discussion paragraph inside previous separate reason discussion interesting interesting show happen constructing h batch encoder achieve iterative feature though iterative flexible able iterative encoder separately bind sparsity parameter simplicity slowly additional encoder compute encoder vector encoder vector h loss encoder example message iterative encoder sparsity encoder reconstruction encoder encoder vector getting encoder encoder sparsity tradeoff iterative reconstruction trade algorithm non row increase batch encoder every row encoder zero non row first encoder batch batch encoder k define quantity encoder eq q claim summation handle modification use method apply n h auto matlab implementation implementation ghz intel processor gb ram measure expression psd matrix hence try explain information versus achieve sensitive goal preserve possible across come choose optimize figure highlight also give batch comparable despite bad comment run exist run ten notice considerably concern want error exist finally mention optimize run empirically fast version accurate version considerably fast call generic specialized future cardinality optimize reflect preserve learn dimension preserve much asymptotically relative yes encoder connection clique pca rgb axiom york ny enforce generalization improve interpretability auto give asymptotically tradeoff give algorithm feature pca auto encoder transform encode bottleneck close encoder preserve auto reduction encoder encoder important construct low auto auto auto encoder decoder encoder map perhaps linear auto encoder pca linear auto encoder enforce encoding map encoder formally auto linear encoder decoder encode feature reconstruct reconstruct minimum reconstruction formula perhaps encoder information approximation k chapter encoder opt opt kk k early visualization feature extraction simplify component application desirable dimension direct significance gene financial application seek tradeoff ability reconstruct introduce column zero encoding original interpretable know formally rr seek let usually clear singular view matrix k k v I ir frobenius ij encoder first encoder optimal set approximation pca result encoder encoder simultaneously selection loss pca low bind information pca construct simultaneously first must orthogonal batch extract provably guarantee theorem factor point provably iterative construct guarantee approximately iterative experimental performance benchmark standard predict produce auto optimality nonlinear auto prominent auto address auto encoder lot recognize factor encourage use scaling attempt rotation thresholding iteratively residual projection get principal pca straightforward max clique see problem maximize generalize know via reduction hard pca maximize historical symmetric var decomposition loss auto encoder explain maximize explain view capture symmetric lead historical approach capture explain sum information variance unconstraine minimize information encourage objective solution reason information general intrinsic unsupervised secondary encoder constraint symmetric translate suboptimal encoder place loss convert approximation inequality relative explain immediately give careful orthonormal diagonal factor constraint property typically one compute quality completely satisfactory sequentially interested produce aware exhaustive require perturbation exhaustive heuristic direct convex refined tractable simple greedy backward develop greedy branch run backward principal quite theoretical aware polynomial guarantee optimality additional negativity give construct sparse guarantee trivial rapidly decay guarantee apply pca clear extend apply guarantee top good approximation optimal pca encoder polynomial pca box encoding column selection construct provably auto finally modify main prove iterative f column vector range span sample e r rr non zero jj span whose kk reproduce kk k mention quickly projection main guarantee rank rank modify rr algorithm invertible modify factor actually strong column locate compute compute additional svd asymptotic running multiplication affect produce encoder e satisfie say rank sparse encoder decoder hence conclude find give approximation simplify way main black box give run r follow notation initial approximate svd ensure reconstruction part expense time reduce still reconstruction give expectation least factor run right via svd approximation also recent deterministic detail achieve apply constant approximation de randomize step appear result trivial increase entire time sparse auto auto encoder ok encoding identify decoder rough sketch doubly construct column ok ok ok ok combination getting doubly encode decoder reconstruction column doubly sparse encoder identify reconstruct entire black encoder obtain provable rank column sparse pca choice construct result deterministic expense guarantee kk dense show pca error require guarantee encoder optimal encoder optimal encoder approximation much challenging approximation sparse encoder combine single loading factor algorithm produce sparsity linear achieve compare require constructing
eq martingale construct respect sequence eq give respect apply suffice I obtain relation cp give q follow step relation pp op facilitate account mean ki h h n cauchy n h h h take account obtain c k calculation term take independent e h e k n e h absolute term hand cauchy term way notation obtain n lagrange behaviour nan hypothesis relation involve relation neighbourhood write similarly last term notation n I pn pn obtain relation follow give n norm n n n I nan hold op right hand side independence q q hand side q relation inequality eps fill stroke cm cm cm remark proposition corollary coefficient variable increase increase test nan behaviour ratio test easy practice test nan law fix find asymptotic confidence phase carlo statistic technology numerical refer explanatory infinity traditional cm type principle precisely type penalty parameter penalty penalize number explanatory consider lasso penalty concern reader review automatically dependent model accurate type devote high refer paper interest explanatory converge cm technique fairly problem approach traffic hypothesis brownian main maintain change presence change adaptive lasso estimator paper choose number criterion propose yet sample make behaviour empirical likelihood first change point vector explanatory vector coincide observation variable suppose explanatory depend depend take enough last use confidence nan assume change first present two notation define need theoretical study behaviour test statistic analyse accuracy confirm improve coverage result cm notation assumption likelihood ni identically distribute thus view introduction variable obviously empirical likelihood likelihood nk r nk n lagrange optimal probability I lagrange empirical take account respect j imply pp pp n jk tn two k I remark restrict study instead statistic particular empirical lagrange brief notation convenience matrix bold square throughout denote generic line even formula whose begin notation simplify notation phase follow matrix also nr particular kronecker explanatory assumption need keep property high dimensional change n np nc q k p op op op bound probability assume point expansion assume asymptotic statistic hypothesis build phase also model change obtain nonlinear degree different normal intermediate study asymptotic behaviour emphasize break without nh first concern convergence rate valid lemma lagrange multipli accordingly follow give lemma satisfied n proposition satisfy statistic approximation proof suppose hypothesis lemma establish asymptotic normality statistic explanatory appendix presence essential way proposition ii nk I n n immediate nan confidence asymptotic theorem build quantile normal simulation calculate firstly monte replication coverage cr divided explanatory ls quantile convergent ti iy relation calculate give less change hypothesis accept know iy absolute compare statistic point fix phase test phase system apart fact consider theoretical multiplier easy unknown estimate convergent conduct term statistic theorem approximate ie relation monte x monte model cr error distribution mean subsection throughout monte replication study behaviour power cr summarize give corollary trend accordance without power error cr cr precise false carlo replication quantile respectively nk exp calculate large hand approach table give power jj even remain unchanged cc cc asymptotic involve region parameter coverage nominal coverage level phase improve critical value generally coefficient decrease approach divide proof proposition theorem lemma proof one n probability relation equality use notation follow pn use pn pn pn q n p hand
define wavelet wavelet wavelet wavelet coefficient moment wavelet wavelet hilbert transform real wavelet hilbert belong hold condition coefficient stay energy coefficient vanish vanish moment nan frequency response generate wavelet additionally shift phase response generate addition operator wavelet wavelet toolbox signal illustrate wavelet wavelet translation scalar preliminary investigation analytic analysis analysis assess wavelet figure frequency change well low modulus analytic wavelet high view scale high wavelet wavelet low signal wavelet modulus fouri wavelet individually occur continuous anti versa pair wavelet fourier wavelet derive fourier write illustrate wavelet transform symmetry additionally fourier wavelet potential proposition corollary example cr tag cr tag mail de mail wavelet wavelet kind symmetry wavelet like wavelet wavelet transform like wavelet analytic wavelet wavelet wavelet idea introduce wavelet function symmetry odd present name analysis simultaneously odd wavelet represent account scale one derive scale compare play wavelet coefficient harmonic component fourier scale hence associate odd vice naturally quadrature xt look kernel analogy transform replace hilbert order allow fourier wavelet review result wavelets transform eq variable hilbert transform fouri transform operator define hilbert impose interesting hilbert function odd vice versa wavelet anti symmetric anti verify wavelet wavelet explore proposition view transform wavelet fouri hilbert coefficient generate wavelet belong straightforward conclude moreover energy coefficient vanish vanish moment vanish moment proposition wavelet number moment wavelet define look analyze asymmetric kernel write jt naive observation wavelet impose coefficient wavelet also energy wavelet transform energy ft moment vanish moment vanishing follow th eq moment fourier like nan multiply typical signal fourier wavelet design real framework wavelet
learn order albeit connect base marginal dependence fs label conditional mutual link co occurrence g generally graph intuitive benefit together parent vary near parent possible loop bring one ignore assumption make problem segmentation valid problem th highly correlate discover burden specification structure involve discover impose priori order label label fix pattern parent label label frequency information sensible try maximize parent label dependency dependence follow outlined matrix mutual placing leave corner vertex discover seed parent label direct parent case construct classifier output l w computational calculation easily thousand limit search label building pattern acyclic cycle consume employ classifier construct refer monte interpret provide undirected dependency classifier py compare comparable term connection refer argue undirected typically easy relation construct undirected slow stage notice ensemble improve build seed label majority voting follow concern exclude classifier py ct discard description classifier majority discover sec direct sec alg sec alg chain discover like achieve present improve scalable list represent dimension ensemble experimental summarize collection dataset vary familiar community th sort represent feature number instance train complexity time address problem metric return true logical label relevant audio image biology text text text localization localization confirm hill beneficial cross display increase relevant scene confirm hill performance sensitivity list classifier fit logistic probabilistic logistic e obtain well svms accuracy recommend tune classifier wish focus use problem within framework svm sgd implementations hill hc cv hc confirm run use hamming run superior run namely dependency exact measure orient whole achieve competitive full method consider possible initialization improve hill strategy initialization regard scalability note roughly instance run approximately conclusion large compare instance running show require run wise finish within hour gb scene local music scene medical local avg match scene medical avg test test rank table place bar span average rank critical bar method statistically considerably strong match particularly well hamming score propagation behind localization table present second finish gb memory dataset music medical local avg music medical avg rank base bar overlap slow I desirable application prediction segmentation sensor real arrange detect person segmentation synthetic top room light sensor arrange window thin target target come light light source target corner h localization sensor arrange around thick line horizontal axis divide th pixel active sensor inside detection color simplicity specific avoid super let position triangle corner triangle indicator pixel pixel sensor low much see sensor height create corner light room light sensor source initialize k j check ji consider also interested study directly received consider trial success probability prior map robustness algorithm address beyond achieve remarkable emphasize property exploit knowledge sensor result sensor precision correspond obtain ct table detailed section increase fine explain see measure ct label consistently complex contrary label model experiment particularly dominant ability respect significantly method surprisingly structure base versus difficult justify modelling dependence improve dataset scalability crucial ct place order computable information match surprisingly much begin overall statistically significant able excellent number time test prove experiment structure degree project c vs deep learning feature vector dimensional acyclic one parent uniformly probabilistic generate qx normal consequently equal dependency show generate term truth discover visually discover appear improvement confirm random difficulty range hard number leave medium medium accord novel describe monte carlo procedure scheme I order py py py bayesian fully similar sample independently py graphical exact chain monte technique target adequate undirected configuration repeat py py conditioning neighbor py ty certain burn task scheme chain converge produce mathematic signal communication de circuit multi become increasingly year popular multi label classifier particular constitute ct method scalable important competitive multi problem scale thousand even thousand keyword chain multi structured classification multi instance rather label correlate allow expense increase computational label classification label binary attract deal interest development author recent recent show vast learning relationship gene tag category news may relevance add month gender month simply irrelevant receive however treat multi paper family integer binary vice versa focus effectively large feasible many tend label present scalability show powerful suited deal cascade label possible chain complexity label main contribution novel highly ct impose ultimately chain ct capture essential among efficiently underlie label sequentially probabilistic measure ct namely thousand label ct run naive ct seed outperform ct organize formalize various strategy label dependence augment theory early carry two ct secondly competitive typical output prediction localization segmentation conclusion help complexity dependency carlo require notation traditional take instance example many possible finding np find around excellent discussion complex measure co label latter author measure incorporate edge network occurrence mutual hereafter moderately dataset final end rather involved graph inherently demand input course strongly particularly try label label facilitate train label build dependency learn following treat label find conditional dependency small sized suit pc
overall overall sum magnitude square system generalize group unit tight size capacity induce convexity apply regularization equivalent novel base regularizer network capacity per unit regularization capacity norm magnitude system depth far aware capacity feed forward unit reference begin scope significantly characterization perhaps natural use analyze feed forward often depth network light difficulty optimize compute direct acyclic graph income special output node add rely input internal output propagation wu graph refer network depth direct feedforward vertex partition layer layer maximal connect layer per layer also shorthand h parametrize mostly relu relu relu several convenient exploit share function pt hinge share property hard threshold activation without change network f g relu realize unit point simplify calculation layer weight gives realize norm weight go network group type parametrize q context group regularizer impose constrain incoming regularizer e magnitude decay layer multiplication summation activation homogeneity relu activation different change q two family class effect rademacher excess minimization complete treatment exact particular rademacher typically effective capacity class rademacher norm prove induction show neuron supremum attain output highest rademach complexity layer top rademacher magnitude capture node layer width magnitude happen whenever omit width countable allow complexity investigate width subset vertex hypercube first layer connect desire sign layer recursively copy unit add rd layer copy network h p h repeat process understand dependence width depth scale group avoid construction factor logarithmic indeed width magnitude control capacity theorem offset use indeed sufficient condition convex independent specific weight never weight instead alone result complexity may respect combination pf convex show place side side interaction come come complete proof homogeneous hypercube input weight value match vector connect pm qx hidden unit connect construct case weight except ph focus constrain incoming separately unit show two network relu consider ability suggest unit look input product path control motivate node weight going really regularizer look aggregated weight regularizer finite emphasize edge go unit magnitude edge regularization imbalance think regularizer refine consider dag path notion proof f combine allow bind non depth width independent bind limit network make problem kernel allow two network e infinite unit bound output unit beyond control width immediately width might hope might make easy however point require relu indeed apply hardness intersection two layer conclude efficiently pac even increase efficiently pac even margin moreover short learn even version corollary time though might input output feedforward prove homogeneity clear incoming vice versa homogeneity triangle function complexity input output vertex since observation per graph depth internal subgraph tree per unit think fully network mean unit low stream unit intuition power deep network feature encourage learning task multiple per impose namely share vertex pick edge copy vertex edge incoming vertex incoming nod vertex regularization overall weight system generalization guarantee input lead capacity class even increase depth condition unit confirm convexity leave interesting characterization show net convex neural net network consist unit class finite unit support think impose similar see relu equivalence overall constrain unit bottom geometric equality balance homogeneity hope might make intersection conclude learn polynomial subject assumption margin short discuss rely allow infinite still depth derivation depth avoid already depth unbounded increase depth arbitrarily graph toward equivalently arbitrarily sensible dag sensible depth generalization unfortunately width bind dependence avoid anti lipschitz per avoid anti lipschitz proof base inductive argument relu inductive argument rademacher hull class inductive complexity depth need complexity example negative cone control feed analyze control depend parameter control controlling size control guarantee perhaps regularization still necessarily depth although multiplication behave differently experience relationship novel regularization precise tight though require go beyond bound real class regard independent control another expressive going provide expressive monotonically increase continue related decade circuit might resolve correction rademacher rademacher class represent width activation bound induction neural rademacher upper establish lemma find cardinality rademacher complexity linear bound norm rademacher p member hypothesis inequality output technical lemma contraction lemma lipschitz class maximization rademacher independent defer obtain rademacher apply vector row equality prove less fx side reduce need show since need rewrite inequality hold inequality true proof neuron contraction lemma absolute absolute anti lipschitz anti contraction lemma rademacher proof df satisfy homogeneity homogeneous triangular df df iw activation moreover norm establish df homogeneity homogeneous
ice member ensemble ice help rigorously characterize uncertainty even systematic error also environmental projection rise ice linear model process principal component mass ice make substantial level rise ice pose substantial people ice sufficient ice west ice might future rise east ice might increase surface air ice may next rapid ice significant population people level rise future ice characterize ice importance project future challenge ice advance ice many representation important ice towards investigate confirm knowledge uncertainty contribution characterize key produce ice projection ice challenge expensive effort ice model significant advance ice calibration limitation sigma rule choose error set ensure robustness assumption run ice thousand setting applicable three ice ice height area cover approach even ice analysis base aggregated quantity spatially approach standardized approach probabilistic calibration approach unable utilize source ice ice output observation ice core last period whereas rigorous dataset aggregate ice profile make suitable aggregate uncertainty would properly ice propose calibration appropriate ice pattern ice binary absence extend calibration linear framework considerable pose logistic avoid latent variable computational burden ice ice extent ice presence absence ice west ice large rise ice include rapidly ice interior west ice terminate ice ice thin contact ice elsewhere line narrow separate ice ice know ice ice ice core smoothly ice equation describe simulate ice computational description remainder organize output use introduce explain computational spatial formulate approach describe pc application discuss implication west ice conclude boundary technique capture challenge enough basically evolution change flow slide surface ice manuscript nest span west km nest grid boundary store year appropriately present bp modern improve proportional record year ice prescribe ice modern reach day year basic linearly year temperature longitudinal correspond observe ice decade future ice west realistic model ice poorly effect uncertain ice model uncertain vary hypercube comparison stein reasonably range relatively r confirm design parameter generally recognize important ice configuration see tune previous design input cube infer ten day geometry modern map ice ice modern original cover entire grid binary outcome presence ice figure observational observational dimensional pattern challenge pose structure development computer calibration new calibration binary spatial computer calibration stage computer calibration infer observational calibration make easy identifiability model binary specify framework construct discrepancy possibility parametric uncertainty fast spatial datum example uncertainty identifiability issue considerable challenge feasible stage description framework calibration spatial detail observational computational inferential challenge provide approach challenge notation output spatial cover pn ni ice calibration outline calibration composite subsection reader reasonably fit computer represent trend matrix definite output setting estimate calibration observational model approximate run dimensional spatially zero parameter observation choose infer remainder model bernoulli write eq output conditionally model approximate natural parameter input spatial e function vector maximize ill pose respect fit predict calibration consider systematic discrepancy observational observation value parameter match covariance process standard logistic q discrepancy define posterior density carry face computational challenge spatial describe point also straightforward pose parameter flexibility result process natural pose computational numerically infeasible function cholesky decomposition covariance translate correspond hundred thousand compute ghz moreover store double calibration step challenge discrepancy integrate cholesky matrix scale ghz challenge previous build principal inferential dealing help approach remove integrate process cholesky decomposition covariance subsection logistic detail closely q rewrite q th th minimize iteration iii minimize find hand side plug ij imply mt I log function th result local maxima algorithm need ice study ice input make projection numerical real ice calibration lack identifiability design hence provide calibration ensemble cover entire lattice cover field supplement overall grey grey area percent principal rate less construct synthetic observational choose run truth mean considerably experiment realization discrepancy process supplement realistic synthetic observational approach work ice ice problem complicate input member potentially select truth reality operate actual ice ice ice pattern construct ice spatial follow realistic observation run observational patterns ice root rmse spatial location observational run select common ice pattern ice pattern synthetic observational ice ice step observational ice ice recover discrepancy well supplement panel component validation describe percent cross validation confirm precisely error translate prediction binary informative prior input range metropolis hasting integrate look standard size accounting plot density plot change system intel pairwise sub ice slide display reason density peak around recover value dispersion compare informative probable well limited modern ice load effect ice ice relax towards modern past position estimate likely able constrain mcmc projection build ice ice convert mcmc projection show ice projection comparison likely ice synthetic truth year cover true ice change projection match ice volume ice due term ice discussion ice dataset focus ice uncertain though behavior interior ice relatively ice accumulation slide ice advanced maxima ice ice warm ice affect ice rate main cause ice increase ice thin interior ice line advance regime effect ice extent ice variation little consensus highly ice et al decrease extensive interior slide coefficient slide ice pressure drive stress slide mainly hard slow slide soft fast slide relate stress velocity effect ice profile affect coefficient modern ice unconstraine modern advanced last area cover generation ice advance wide range slide slide modern ice evolution year response ice ice year lag deep ice represent elastic sophisticated model suggest short west know ice fill narrow nearby ice ice coarse size sufficiently lack plot mid close somewhat small find correspond nominal primarily affect ice little modern ice favor less nominal weak well slide quite valuable validation modern direct effect realistic ice affect significant influence ice profile recent relatively consistent ice modern relatively thin control ice projection leave volume seem counter impose day drastically early advance discuss produce modern line despite run time tail difference year value unconstraine well past long future level occur reasonably behavior consistent recent study ice calibration framework discrepancy identifiability issue calibration specify discrepancy pixel exhibit persistent see hoc detail discrepancy discrepancy section ice paper ice application discrepancy true create scientific subject run present projection base simplified representative pathway model modern day position expect lead well constrain mention future direction formulate calibration enable information geometry ignore ice ice ice incorporate information zero multinomial beyond calibration challenge comprehensive formulate calibration approach computer describe ice result calibration uncertainty ice principal analysis
university california california bayesian big year attempt efficient scalable monte monte hmc probabilistic construct collective geometric whole along unit optimize scalable lead substantially statistic principle powerful several decade underlie mechanism bayesian uncertainty reveal landscape global method computationally intensive inference require simulate intractable simple algorithm explore inefficient successive state autocorrelation movement effective tend quite low convergence slow hamiltonian hmc walk hamiltonian state distant nevertheless hmc explore metropolis fully space dynamic method hmc geometry space improve hmc automatically practical big analysis scalable technique bottleneck big evaluation involve provide balance accuracy cost common base contain retrieve criterion strategy effective subsampling surrogate substitute usefulness moderate approximation effective whole randomly unit criterion implicit subsampling result framework geometrically manifold follow overview explain detail present finally devote discussion energy always analytically statistical increase however simple method metropolis become especially dependency geometric target dynamic auxiliary momentum explore jointly quadratic correspond log multivariate hamiltonian system stepsize accept probability simulate hamiltonian hmc generate separate q l u distant proposal autocorrelation acceptance hamiltonian allow efficient exploration although explore fully model since flat use geometrically substantially call riemannian geometry target hmc automatically adapt identity commonly hmc fisher hmc hmc shorthand partial th element p dynamic mechanism time reversible dynamic use deterministic reversible however require intensive hmc contain energy included walk geometric quantity evaluation example potential hmc mass inverse extremely expensive involve remark piecewise use develop method difficult extend due grid commonly learn limited computation inverting space train neural surrogate function approximate accuracy network incorporate criterion subsample easily complexity layer layer number hide provide neural network feedforward network activation scalar define hide weight unit hide unit bias q hmc network standard algorithm base alternative key full subproblem run hmc collect state accept explore region train collect need hmc surrogate function hmc function construct parameter extend version pp p l partial pde hmc hmc ess mcmc ess monotone ess size step hmc stable effective acceptance discard proposal iteration unit logistic improve counterpart logistic regression observation design parameter sample x I independently energy everywhere hessian surrogate step hmc substantially almost compare ess marginal hmc present figure experiment speed lr bank hmc lr hmc pde hmc hmc next bank dataset feature make uci machine repository bank datum summarize hmc counterpart intensive pde inverse coefficient pde flow medium coefficient pressure forward govern pde coordinate exponential length eigenfunction operator define kernel endow target particular expansion add uniform grid hmc function local hessian directly diagonal two positive part surrogate posterior hmc hmc hmc improvement efficient hmc hmc substantially hmc metric hessian remark usual bottleneck another example complicated simple pde evaluation use neural surrogate huge advantage experiment high improvement amount datum increase scalable explore space neural surrogate explore completion problematic demand involve apply chain dynamically history drive force hamiltonian well function sampling well distribute dense well gradient h state journal chemical physics relaxation intelligence physics letter dynamic markov b langevin methods journal series pp mathematical via langevin dynamics international page online
histogram brownian learn capture stock price correctly clearly learn question hard edu collaborative kalman filter collaborative filtering relate collaborative filter evolution brownian whose parameterized dot product relevant brownian moment filter multiple interact dynamically evolve drift handle posterior via preserve calculation manner similar quantitative evaluation million netflix dataset qualitative stock return make historical learn interaction rating example netflix movie star rating amazon allow user video user filter address preference recommendation make user interest predict collaborative filtering include outcome dyadic want team b game wish predict stock price exchange effective factorization prediction dot prove powerful relate wherein sample recover otherwise treat handle static meaning latent learn batch assume movie team stationary good information dynamic collaborative evolve exchange tracking student competition collaborative location fix multidimensional brownian motion since motivate real problem ideally event update preserve calculation evolve probability location location filter call stock parameter volatility market geometric brownian motion approximation nevertheless basic factorization collaborative filtering evolve modeling collaborative kalman filter review collaborative filter dynamic basis generally speak factorization incomplete location latent location approach collaborative distribution depend product logistic probit could probit value univariate unseen interpret collaborative consider let rate eq rate location influence way collaborative filtering variation development drawback temporal goal easily brownian motion briefly review mention many filter technique temporal filter naturally arise develop kalman filter relevant kalman filter vector linear vector additive state evolve markov current previous write inference observation state control dynamically observe initial require allow analytical calculation extension require continuous kalman filter variance drift difference make extension brownian address drift present present highlight contribution smoothing store datum handle inference evolve brownian learning ability significant probit ordinal observation movie rating model gaussians discrete motivation behind collaborative kalman extend section kalman framework object brownian function dot focus rating etc kalman prediction probit problem real partition region denote star rating class fall relation rating rating binary unlike collaborative times rating may problem stock modeling arrive interested inference discuss multidimensional brownian duration event tu tw j kalman filter multivariate tw tt u tu iw dynamically index assume posterior continuous extension u tw posterior interpret design equation also kalman drift modeling delta drift move make information concentrated impact allow dynamically volatility stock notational specific straightforward motion define ta brownian state ta ta tc play brownian motion model purpose volatility geometric brownian motion modify integration imply integral treat posterior inference evolve model control vector point break presentation inference part deal overview inference present stochastic update brownian motion eq g subscript perspective multidimensional motion generative draw brownian drift unobserved become inference learn parameter time update note modify note skip directly change first posterior unlike kalman analytically tractable typically nonlinear kalman filter employ evolution state instead field tractable hide kalman calculations approximate factorize approximate variable significant advantage define divergence true posterior equivalently variational convex variational involve add entropy require hold ignore notation calculate pz z ij truncate ignore result ascent updating appear mean ty depend fall normal cdf evaluate u symmetry update j tu iw mean covariance therefore update derive infer brownian individual u process since depend posterior eigenvalue second expansion give respect remove expansion modification make necessary integral time interested tw j unbiased expectation fast approximation collaborative netflix contain movie rating movie rate roughly movie movie rating rating half star across movie user stock datum measure time total stock million require set dimension standard probit parameter geometric brownian motion netflix try share movie shift overall user movie landscape probit link equal width set netflix account star online vb case vb use batch map version probabilistic pmf probit estimate pmf probit mf mixed membership factorization drift sequential big variational algorithm iterate user single exploit expect limit movie instant drift figure histogram dynamic movie netflix vb map rmse algorithm hold value small omit discuss online modeling preference evident difference introduction space vector improvement batch vb pmf em variational variable probit rating find treat generate probit important well compare pmf pmf rmse static calculate rmse rate realistic interested rating calculate rmse clearly bad look user movie test batch netflix
item presence ib entry entry independent order item general comparison origin meaning show arbitrarily informally arise form shift shift one identifiability throughout denote score vector case special case form q concave see hence comparison purpose entry remain eq error pair induce edge determine time sequel central play laplacian ordinal related measurement dx ti jk nj k laplacian semidefinite eigenvalue eigenvector jk j jk emphasize comparison quality identifiable semi norm sequel norm natural metric estimation semi generalized model arise naturally discuss norm risk square semi pairwise statement etc numerical sample parameter subsequently b semi model consequently risk factor follow carefully construct difficulty construct semi norm laplacian appendix proof minimax euclidean norm present minimax norm ordinal instance ordinal strong concavity maximum estimator next topology topology complete depict simulation draw follow plot average reduce predict predict normalized curve conclude pair analogue pair eq conjecture risk ordinal topology identical pair suppose pairwise ask worker option assume theorem risk pairwise via understand exponent depend every item underlie quality score sample selection item large unit identity visualize choice item hyper contain observation represent shift observation invariance concavity also propose ordinal hope true ensure hyper connect every comparison hyper also assume illustrated model concern score make concavity little give cauchy yield scalar concavity goal capture scale minimax subset well understand human storage recommend restrict selection depend noise set incorporate define wise comparison comparison topology call comparison reduce laplacian early comparison square bound respect take dependence occur multiplicative pre one standard square dnn euclidean always evenly across nevertheless analysis require understand precise number pairwise comparison application one compare demonstrate play laplacian comparison design comparison application good topology pairwise comparison setup popular topology evenly distribute sample unweighte unweighted evenly concentration inequality see let laplacian q context square norm interest eigen scale laplacian small conversely scale matrix eigen minimax claim see condition claim strict canonical euclidean ordinal spectra regular graph various text graph edge complete spectrum scale risk condition regular scale equal respect give minimax optimality bipartite partition set comprise say node edge eigenvalue regular laplacian star scale minimax condition star bipartite scale discuss minimax risk associate edge pair every regular xx class strictly cycle spectrum regular exactly dd turn apply result minimax establish arrange product second laplacian angle risk low lattice hypercube pair hamming path scale laplacian bind know hypercube practical prefer conjecture minimax scale optimality establish topology path one uniformly replacement line worst predicts topology give topology uniformly random chosen determine pair comparison generate score estimator model employ average employ respective uniform draw set packing choose eigen generation construction low packing procedure graph packing identical use packing star procedure laplacian star shift bb various topology star consistently star graph phenomenon plot vary bad simulation multiplicative factor describe experiment conduct amazon henceforth crowdsource choice platform individual put exchange payment along worker complete worker answer worker allow question involve american require worker ordinal choice question worker circle worker area ask topology involve aggregation follow via validation employ pool worker experiment estimation topology consider relative consistent graph well bad misspecification case outcome effect misspecification consideration address approach comparison ordinal enter answer figure ordinal compare approach fine ordinal binary argue ordinal convert ordinal processing estimator ordinal set conduct seven investigate response obtain subtract conclude collect subtract amount ordinal task select several important knowledge conduct worker paragraph audio sound worker audio tag clarity relevance image worker relevance independent collect distance audio relevance std std ordinal std experiment worker worker ordinal ordinal answer access ground truth answer remain ground truth compute ordinal ordinal important measure interface see worker first website ordinal one directly ordinal response unlikely set table show sometimes significantly ordinal explain inherent human human first evaluation typically ordinal particular introduce assume discuss ordinal contain ordinal encountered vary bias give evaluation also recognize human allow evaluation cost clarity versus ordinal form reliable pair comparison noise comparative preferred measurement answer help determine response evenly accordance assume priori ordinal gaussian capture ordinal deviation retain set order bring specialized gb b place minimax setting risk risk coordinate measure number reduce model ordinal treatment ordinal reasonably error scale consequence result allow base ordinal well minimax exact sake completeness ordinal setting three normalize three execute iteration procedure select ordinal practical system pool worker ordinal pool convert task audio coefficient result rest error observe per close mistake reflect table estimator incur ordinal experiment mistake whereas outcome go remain need constant order address present broad preference demonstrate utility choice ordinal potential improve mechanism effort noisy characterize finally useful variety future semi pairwise acknowledgment office national foundation grant dms support microsoft fellowship ordinal low argument see minimax estimating index refer packing construct pack risk construct packing require auxiliary q result bind lemma dimensional vector symmetric semidefinite nonnegative denote collection jj eq fact choice construct packing prove claim give z calculation case prove kl b bb bf claim ordinal squared purpose subsequent prove minimize differentiable mle semi mean perturbation convenient semi convexity piece claim inequality lemma mle ordinal need verify strong hessian simply ft di parameter lemma concrete remain control component n square reduce fluctuation moreover mean straightforward v I gaussian imply algebra universal tail low respectively minimax euclidean norm ordinal part semi norm describe section substitute upper comparison semidefinite nonnegative scalar specify later vector boolean keep first z comprise coordinate remain choice trace turn employ z b k packing integer scalar subset boolean hypercube length j entry j apply pack large claim consider packing claim turn paired likelihood w quadratic convexity hessian lemma quadratic application lemma tail quadratic form see appendix since integrate yield semi bind state packing laplacian packing remainder identical except requirement packing square form q n claim respect suitably prior vector bayes lead distribute n apply iterate subsequently proof laplacian underlie bound lemma rescale take verify dual log follow consequently hold likelihood n eigenvalue one shift invariance imply write every pair evaluate I well recall define since chain k employ k z il b note r final scalar whose let packing appendix satisfy j yield pack b applying prove packing make k prove bind follow theorem minimax b show scalar whose later construct set j vector yield general element pack packing claim three construct packing calculation pair j k n state exist hyper suppose hyper edge correspond schwarz v span I desire eigenvector eigenvector similar argument matrix invariant collect useful form valid variable jensen complete laplacian graph semidefinite decomposition diagonal
binomial distribution l stack bit estimate window code show comparable ccc replace isotropic noise reverse trajectory probabilistic sampling rest image long region delta know train train technique model reference utilize available roll use radial basis generate roll successfully train simple occur use multi layer reverse trajectory figure perfect convolutional architecture illustrate direct previous mnist variety result mnist likelihood comparison window image consist circle capture complexity therefore probabilistic texture straightforward evaluate figure novel toy real test algorithm stay often extremely algorithm diffusion noise make distribution sample evaluate straightforward posterior extremely helpful sharing window office office foundation conditional f f u distribution occurring occur finally perturb trial b roll initialize normalize radial single hide x step share top pass sigmoid restrict roll successfully bin step binomial perceptron sigmoid reverse independently step share pass sigmoid restrict successfully bit nearly convolutional pixel sized convolution per dependent consist identical combine variance perturbation apply wish goal make use multi output vector every map convolution step mean multiple power perform convolution resolution pointwise nonlinear relu operation resemble multiscale multiscale dense act image act pixel follow nonlinearity experiment pass pass dense pixel generate image cifar dense pathway convolutional pathway dense pathway scale pathway learn family computationally essential equilibrium physics systematically iterative learn learn probability generative thousand probability model additionally reference probabilistic model suffer unable describe rich flexible normalization generally flexible expensive variety tradeoff expansion variational kl contraction score propagation parametric novel pt flexibility structure multiplication model log convert physics chain rather chain define probabilistic evaluate perturbation tractable explicitly full distribution analytically target utility roll sequence handwritten digit mnist cifar cccc identity slice reverse gradually term process generative decade develop develop directly paper early motivation advantage idea physics static bayesian method multiply learn training inference challenge inference objective inference restrict generative layer layer production time em summarize develop flexible trajectory reweighte improve generative train match equilibrium autoregressive recurrent deep extension tractable train attempt distinguish mapping marginally factorial density learn mixture mixture causal neighborhood additionally datum network generative compare experimentally adversarial idea equality learn markov slowly constant trajectory langevin realization diffusion use stochastic kolmogorov forward kolmogorov correspond kolmogorov knowing cccc cc b example b forward diffusion datum distribution learn finite process figure first generative probability derive distribution multiply denoise label gradually behave rate perform q binomial diffusion binomial diffusion kernel binomial distribution reverse binomial continuous step forward trajectory covariance bit flip binomial f reverse transition provide cost time would applicable include assign sample reverse trajectory evaluate trajectory reverse identical sample require substitution correspond quasi static physics amount provide jensen entropy divergence derivation trajectory correspond equation become equality reverse diffusion probability perform regression gaussian flip performance right schedule greatly improve estimate schedule move diffusion learn schedule overfitte dependence additional hold partial binomial gradient ascent forward diffusion schedule per diffusion image identical sample note display multiscale object produce especially high task signal multiplication produce x distribution difficult variational autoencoder distribution treat either perturbation multiply demonstrate diffusion multiply multiply
address initial answer question initialization b c estimate perform comparative among state study different compare real text identify method initialization candidate background sec discuss along propose sec finally conclusion sec mixture optimize objective mostly consist step maximization likelihood use multinomial use accuracy em solely focus significant value function initialization detail investigate em empirically discuss order automatically generate set select optimal need model consider focus model task novel exploit hierarchical cluster hierarchical agglomerative generate mixture difference employ require among inefficient dimensional simplification use approach propose statistical selection method list criterion select mm likelihood recently mm aim comparative generate investigate range datum complete cluster order assume multinomial distribution multinomial mixture k estimate maximization maximize number expectation q sample update certain criterion meet parameter examine value start short initialize short stage assign component probability step conditional candidate model generate cluster explicitly generate model employ em begin continue within em mml detail propose multinomial agglomerative permit three dissimilarity select merge merged cluster symmetric measure dissimilarity criterion besides use linkage criterion merged issue mixture base function employ minimize criterion augment model criterion complete likelihood add classified minimum message mml probability select minimum criterion graph consider idea point minimize root automatically generate method experiment simulate adjust rand ari stability ari discrete count type type verify generate sample dirichlet determine text collection cluster reader construction dataset initialization list consistently discuss sec selection strategy trial trial trial initial em dataset follow good competitive provide second term real emphasize accuracy experiment real generate discusse among implicitly method initialize initialization maximum threshold fig illustrate method stability bad real outperform perform performance fast prefer however prefer accuracy b compute sample
evaluate applicability see two possible dataset tractable approximate acceptance ratio illustrative simple dimensional normal distribution illustrate misspecification assign flat mh isotropic stepsize set reach acceptance applicable posterior coincide reference decrease bernstein von gaussian center true minus simple subsampling lot help pdf chain pdf basic pdf histogram fit deviation mh bernstein baselines mh run depict green green later tackle run combine equality importance batch note assumption propose artificial batch multiply justified posterior gaussian go infinity accurate property estimator batch multiply kernel estimator additional simplify chain assume independent target final bandwidth importantly batch grow propose disjoint poor approximation approximation propose transform kernel transform interpret extended copy conditionally first approximate speak propagation cavity cavity prior term batch iterate simulate cavity distribution fit cavity computationally experimentally posterior convergence another research avoid introduce wasserstein median propose wasserstein develop mean robustness median drawback circumstance valuable contain batch technique appear datum issue multiplicative number often rely unnormalized version useful several potential scale mcmc describe mh present access surely unnormalized pseudo mh realization replace mh considerable practical application particle mh paradigm worth investigate problem large mh qualitative mh preserve invariant mh might largely difficult pseudo involve mcmc estimate fix variance log likelihood ideal mh access generate quasi large autocorrelation ideal implemented recommend keeping ensure actually incur mh describe marginal mh unnormalized without replacement unbiased log denote unbiased interesting whether almost surely nonnegative use recently possible generalizing unfortunately shall result typically relative result poor define estimator replacement integer value whose decrease ease computation geometric correspond tail finally let mention logarithm difficult proxy appendix order impractical geometric truncate reason section expect mh rely none yield satisfactory exploit methodology datum methodology posterior expectation unbiased biased mcmc mcmc suggest algorithm expectation unclear whether alternative simplicity experiment also make heavy follow admit marginal pointwise evaluation must lower log replace target bernoulli lower exploit previously see e mh sampler equilibrium related evaluation explicitly specify pseudo elaborate extend unbiased unnormalize precisely pz ib accordingly obvious unnormalized give note choose integral integral similarly evaluation speak although pseudo propose disadvantage require integral tractable advantage first sampling require compute bottleneck avoid explicitly resample easy marginal hence particular ergodic ideal explain variance let notation tight variance big tight bound meet fix fraction outli log figure taylor taylor cases evaluation per initialize try small evaluation joint gaussian number evaluation replace algorithm expect behave chain towards slowly pdf resample propose rely pseudo langevin iterative iteration update sequence score langevin mala proposal mh mh compute decrease zero analyze target central slow carlo unclear compare subsample need reach practice stepsize recommendation choice subsample iteration mh subsampling draw away variance get big lead length pdf k pdf length p length figure subsampling monte carlo hmc hmc inspire stepsize explore approach suffer heuristic demonstrate rely violate ratio example likelihood result accept average proportion use acceptance often easy benefit inherently affect mh acceptance simple unbiased unbiased likelihood approach review attempt could try proportion newly target suitably assume new draw likelihood pe pe variable I r index contribute nontrivial rescale simplify denote roughly decrease binomial probability unlikely subsample roughly broadly contribution likelihood end likelihood much unlikely contribute subsampling also absence naive subsampling nontrivial additionally variance log keep mh entail gain evaluation term likelihood naive subsampling allow tackle mh controlling ratio author theorem justify likelihood equal mh correct target estimate inexact inexact chain approximate ratio tail tail decision result mh likelihood introduce correspond pseudo proportional pe ideally want variance log subsample point importance weight obviously purpose subsampling e spline total subsample likelihood control probit use full suffer unclear good proxy fit train obtain likelihood subsample take acceptance note level confidence aware anneal annealing mh application receive attention apply mh mh like incorporate rely average use mcmc approximation aforementioned initial slightly effect seem remarkable inaccurate choose subsample explain posterior fit gaussian include tail dataset subsample tight choose log ratio heavy tail go von run require illustrate use approximation size pdf pdf pdf plot overall mh little inaccurate also assume amount care recommend sure realistic note way obtain subsampling weak approach impractical illustrative potentially subsample illustration soon theoretical independently replacement inherent acceptance exploration accept move develop heuristic anneal presence formalize symmetric assume make third px I px apply px yield find probability accept move check whether acceptance target concluding distance control positive ergodicity walk temperature bound proportion even case control moment subsample inherent acceptance ratio one upper refined log likelihood ratio eventually sure inequality lipschitz know inequality yield bound take bandit subsample size acceptance take probability improvement ideal mh regression mh ergodicity mh speed ideal show uniform ergodicity extend result geometrically scenario sampler require equilibrium subsampling rely lead result sampler access estimate take subsample likelihood even basically pdf basic basic pdf basic inequality worst theoretical come locally gaussian bernstein von good gain example current sampler target empirical confidence sampler tool proxy likelihood act variate taylor center obtain stochastic evaluation expansion sampler proxy outperform represent log require less likelihood evaluation combine iteration mh figure big give detail basic basic taylor proxy likelihood act start confidence replacement generic empirical concentration one think deviation likelihood emphasize concentration concentration refer reader correctness confidence implementation ht px n keep track already replacement bt bt bc tt geometrically cb accept introduce performance confidence sampler w likelihood w control variate lower proxy ratio acceptance equivalent checking replace q confidence ergodicity underlie mh sampler mh confidence exist proof straightforward algorithm concentration confidence lead turn availability proxy assumption although basically identically possess third typically expansion section expand around obtain choice defer section bound lagrange proxy mass concentrate estimator taylor proxy section concentrate inaccurate insufficient subsampling gain mode close chain agree whole iteration I dataset reference nd taylor expansion example gradient need aim proportion ideal mh easily generalize number section evaluation subsample mh fundamentally first contribution budget flat stop rule loop confidence meet eq bind grow strictly slow often dominate lead bind logistic asymptotic proposal loop consider taylor correspond standard order time dominate asymptotic good set posterior drop matrix assume say approximately growth third case independent still factor avoid load proxy relate quantity build disk database http www define taylor expansion within gaussian consider start single report number fraction likelihood evaluation roughly evaluation break barrier reach http www edu tw dimension without require complex sampler mh cauchy solid chain vs dash run iteration drop every explain stop
plot show direction median learn infer round mp direction ty u gp candidate localize physical direction conversely model allow draw condition normal lead pc via pc condition eq pcs form candidate directional average sum conditional gp px mp choice zero absence localization average choosing absence inaccurate generalization fortunately infinite target substitute label reduce eq allow approximated train band tend negligible low sub band derive consider minimum round gp eq whose represent improvement minimum expect loss analytic weighted expect model denote cumulative low improvement explore property fast empirical human gp mean efficiently little query criterion direction gp covariance gp subject round necessary improve query selection close localize horizontal plane towards head near localize confusion angular initial localize median plane plane consist report alternatively play independently min mp trial proceed round direction trial conduct case large occur percentage gp error human report direction exhibit variance test may future measure human localization via variance ht sound nn accurate randomize full develop query offline model localization product improper formulation ability model parametric receive direction output summarize spectral response base achieve training sound localization subject test base find direction infer error interface localization localize close intend head regression sound localization possess remarkable sound localization ability subtle receive human arise acoustic wave head reach center head absence head transfer wave direction allow accurate audio function base sound localization acoustic event receiver sound solely receiver actual robot receiver transfer map place direction envelope near neighbor nn spherical coordinate learn self map horizontal median coordinate frequency posteriori transfer closely relate match leave right sound filter space frequency source computing measurement belong maximally pair sound feature measurement label filtering address number report ordinary ol regressor perform poorly inaccurate linearity predictor include arise parametric fitting datum linearity gp place weak joint observation observation represent predictor spatial smoothly model gp allow function realization select goodness belong hyperparameter learn without cross intrinsic high dimensional space feature uncertainty variance tractable thus linear inference observation training set vector make linear evaluation observation general gps method encode select prior observation selection inference large dataset active research fortunately quality overcome previous work dataset base interpolation gps model relate refer vector standard collection aforementione quantity sound gp specify train belong ht selection run cost randomize trade cost density linear accuracy cubic respective localization show gp small informative spherical randomized simple selection sequentially iteration subset add subset generalize small spatial audio resolve front back directional indirect measure query report new pool graphical interface move towards target develop query localization rank candidate along unknown propose give candidate determine would within recommender select query refer cosine unlikely round cost reasonable adapt predictor round base query predict trade require model prediction fortunately suit probabilistic query gp sound invariant transform representation e mx mx alternative common gps specify share gps physical topology thus inference three gps front back report posterior realization depend differentiable r mat ern covariance dimensional specify gp product mat identical product function time length bandwidth hyperparameter remain large length smooth hyperparameter maximize derivation realization likelihood sampling sampling large represent goodness mean covariance assumption different compare without domain order size dimensional sample conditioning operation compute store exact intractable practitioner cost expense analysis evaluate demonstrate localization despite gram dominant one minimum mapping contain space dimensionality kernel analysis gram derivation eigenvalue capture dimensional score feature apply allow reformulate problem contribution gp pd mp belong subject hyperparameter iteration domain use angular gp train infinitely model direction approximate sound pressure continuous well fitting sound source record spectra suggest relative intensity pd compute dataset total energy specify pd mp lead eigenvector suggest feature train angular separation predict direction method train parametric across mp low across ol log ratio predictor pd feature insufficient pd ols gp gp gp randomize select input risk efficient good approximate determine optimal exhaustive prohibitive evaluation greedy rank risk minimizer consideration future see gp full condition input point input risk empty r rr specify must gp covariance approximation remain evaluate require class point update define gp posterior eqs evaluation kx
theoretically hinge converge option dataset difference option converge machine precision option clear conclusion smooth pick check output let equality follow arbitrary q everything independent treat proof quadratic reasoning hold lemma claim theorem exercise question com ed ed ac introduce adaptive ascent sdca empirical minimization method adaptively probability dual variable throughout complexity distribution theoretical propose risk minimization supervise identify throughout differentiable lipschitz problem considerable attention recent year usage supervise erm analyze include sag gd gd consider coordinate sdca operate erm problem primal sdca select dual attract considerable attention year alpha advance mini variant review naturally randomize individual coordinate exist work assume descent subset coordinate allow optimize individual variable smooth primal explicitly typically quantitie development field literature probability aware piece resort heuristic theory method sometimes sometimes observe progress call propose selection construct base quantity summarize theoretical algorithm describe provide introduce element conclude technical numerical find highlight propose stochastic dual provide rate analysis enjoy rate sdca effort issue algorithm heuristic variant computational heuristic make sdca computational summarize first method variant numerical experiment appendix dual follow dual dual update coherence alternatively coherent primal ti mention solve exactly deferred appendix lipschitz let constraint obtaining hold defer appendix special case lemma hand proposition know q therefore plug bind primal dual letting optimal serial sdca convergence sdca next suggest sdca probability program theory sdca direct quadratic corollary sdca know propose variant avoid issue set importance v n tw divide epoch begin epoch one option update every intuition coordinate set p cost gradient probability epoch affect epoch result probability begin could sample even correspond work choice begin choose option close option serial one sdca differ sdca iteratively yield fast sdca uniform probability change take change epoch epoch calculate operation need
convergence property figure employ reach consensus state assume note mention show algorithm fix consensus reach weighted average node node address issue exploit devise distribute parameter identify weight estimate assume identity identity I close maximize w w close centralized extend centralized centralized weight problem pdf evaluate node attack figure roc curve learning see detection learn weighted detection scheme average effect performance conventional consensus attack show certain fraction exist consensus robust consensus technique operating fusion update effect attack interesting remain future vary topology analytical certainly attack topology question topology incur fast conjecture k paper fully exchange information absence fusion characterize steady detection detection address problem perspective robust attack node consensus devise datum statistical distribution node base distribute consensus datum attack literature traditional detection framework comprise phenomenon fusion fc decision many scenario centralize fc fc become bottleneck failure absence hardware wireless medium alternate decision one local exchange consensus algorithm consensus explore approach update summary fusion combination network global author consensus detection small fast consensus scheme however consensus author distribute weighted fusion show consensus detection performance gain consensus detection attack attack attack originally attack attack address centralized attempt make address security recent work attack exclude neighbor significantly different attack detection large mean step state node distribute spectrum sensing threshold adaptively eventually isolate exclude perspective early centralized way identify potential outperform design fusion decentralize attack arise node recognize identification operating parameter complexity update operating approach differ base exclude consensus contribution follow steady consensus specifically statistic probability investigate degradation detection conventional weighted expression weight attack likelihood operate enable adaptive fusion paper organize system attack study consensus scheme section mechanism effect attack link link define six consensus figure laplacian base sense summary summary state consensus steady state statistic phase decision phase energy detection node instant instant detection product square gaussian chi square freedom chi snr represent snr detector give algorithm step node continue information consensus nothing fusion update come neighbor node asymptotically reach information initial final consensus state I iy special average consensus whole network reach consensus node decision predefine threshold beyond reach consensus paper refer final test discuss attack scheme degradation average consensus base attack network node update suppose strategy prescribe attack w jk ik initialization allow completely appropriate adversary perform attack consensus consensus attack attack investigate past literature see theoretic mechanism capable consensus attack attack introduce initial devise influence attack try iy reach interpret assign node detection performance always statistic dominate statistic analyze detection performance analyze fusion weight value degradation cause attack attack detection characterize strong global h coefficient receiver operate general monotonically fraction need statistic denote free topology whole consensus process reached analyze consensus detection presence attack conventional consensus detection characterize steady alarm degradation loss generality index rest define n tw condition coefficient n local coefficient seek iy computed substitute condition appropriately attack make coefficient zero insight plot coefficient global statistic consider node give figure consider channel gain coefficient zero six attack less blind fusion scheme analytical detection alarm consensus come function notational denote clarity derive two later generalize arbitrary j j summation pdfs normally variance result eq due nature behave strategy mean node behave index notation arbitrary node ji performance scheme alarm network detection alarm iteration represent expression false alarm node combination cardinality represent compactly w tw wise w j ta alarm gain result figure node attack assume consensus detection alarm consensus consensus figure attack probability alarm consensus alarm consensus discussion detection approach sense weight assign issue datum assign high weight final dominate detect type choose conventional use e concern propose robust average issue propose consensus weight node want statistic make every majority otherwise treat consensus attack detect satisfy
mf predictive predictive tree forest mf implementation default use simplify propagation ep incoming message uncertain message numerically crucial dataset dimensional message message predictive uncertainty distribution show red unobserved region predictive rf prediction confident mf quite region mf region smoothly red predictive uncertainty mf blue expect predictive less smooth compare mf experiment compare forest variant process predict delay eight namely age year need arrival week month month dataset state art predictive train create use split train mf rf comparable rf significantly split stationarity burden gp well exhibit result gps phenomenon forest achieve rmse significantly compare useful decision forest achieve large believe rmse failure forest gp forest complex regressor forest split batch rmse rmse rmse rf novel scalable methodology regression sensible application planning scale delay demonstrate framework provide art rmse uncertainty forest acknowledgment share helpful bl foundation research fellowship college newton international ep fp grant agreement appendix hyper use hyper optimize integrate noise variance unknown fraction could actual belief propagation leave optimize increase tree tree depth assume fs fs fs post pt fs mid fs alg college department sciences forest efficient application quantification demand measure uncertainty measure poorly variant tool application area goal uncertainty model perform bayesian within tree light finite carry propagation design typical probabilistic demonstrate far quantification exist forest little cost performance turn probabilistic delay forest gps little gp due ability accurate also prediction application optimization balance exploration exploitation unfortunately gps cubic computationally non parametric scenario dimension progress decade big recent randomized popular achieve popular forest variant rf computation attractive regression task forest yield classification forest gps decision forest distribution move away smoothly uncertainty uncertainty smooth exhibit desirable combine gps probabilistic decision forest forest usual forest probabilistic specifically prior compute randomization property demonstrate forest well online set focus organize forest forest regression discuss forest uncertainty rf ii forest outperform gps term negative probability large regression uncertainty introduce forest task begin adapt completeness review decision describe n task label point uncertainty triple root strictly binary partition root represent parent denote denote location b rectangular put point dimension precisely restriction split forest I justification recall sigmoid encode child parent depth hierarchy align hierarchy marginalization intermediate change test branch mf hierarchy smoothing goal note hence message analytically hierarchy structure scale hence efficient compare message tree study chapter inference pass bottom pass parent message two child top pass first compute child message message root node parent fashion reach hyperparameter detail appendix optimize ideally optimize computed descent individual marginal close solution assume estimate reasoning increase depth tree assume tree stop split identical splitting label stop homogeneous strategy sample forest tree except gaussian rather denote branching reach branching away branch create instead prediction I prior bias predictive mean gaussians w think
class score pattern build appropriate great class develop quantum pattern class generalization introduce formalism purpose discrimination pattern specify quantum state quantization dimensionality also quantum form quantum extend present mixed pure state enable superposition quantum quantum interested representative classify order quantum pattern represent normalize superposition calculated product first list quantum require feature dimension quantization symbol onto quantum quantization crucial view correlation define quantization obtains classify unknown execute average quantum classify pattern accord great value class superposition quantum state quantum quantization pattern recognition perform quantum mapping calculate pattern vote however case utilize superposition base distance represent numerical start vector quantization subspace appropriate context transform represent interval order fix state space require encode quantization also require fix state encode element nearest sometimes function pure procedure implement system ns join ns join ns ns quantization representation represent unnormalized superposition flat score consider number normalize superposition represent eq representation amplitude note extension example encode pattern counterpart tuple quantum represent vector quantum representation obtain space introduce scheme dirac notation interpret quantization representation quantum dimension one main classical mechanic describe space describe quantum case system describe use kronecker product quantum classical expect tensor structure encoding correlation scheme separable quantum simple separable representation consist eq map feature dimension implement ns ns ns k plus norm v scheme execute real separable execute specify overlap quantum state represent th separable allow pattern produce describe class separable quantum assess positive classify class appropriate ht fig encoding symbol present introduce enable world quantization dimensionality quantum state require execute framework conduct possible separable representation mechanic work quantum beneficial security view party quantization quantization apply framework formalism quantization advantage approach possibility measure use classical information use measurement acknowledgment support project quantum theory imaging author would thank via suitable introduce computation resource quantum illustrate difference example discrimination machine merging processing engineering pattern recognition research quantum problem quantum discrimination connect development quantum mathematical formalism suitable recognition main method quantum investigate separable flow processing execute standard computer introduction dimensionality utilize notation initial consideration quantum purpose classification propose encode datum formalism quantum describe basic datum formalism
pattern form order sup sup sup b sup b sup sup sup sup sup c database change turn order construct worth develop length general sub pattern pt sup sup c sup sup sup sup sup sup b sup sup sup sup b sup sup c sup sup sup sup sup sup sup c sup sup sup sup c sup sup sup sup sup sup sup sup pattern contain binary sequential binary partition sequential sequential important support partition consideration length search pruning mechanism explore space exactly top interesting mining regard depth branch exploit anti space lower sub depth advantage node extensive maintain open store give explore locality variant explore disadvantage larger explore search evaluation exponentially burden explore adapt sequential ensure moreover sure space explore differ sequential specialized end note hereafter ensure late search queue item pattern queue queue repetition adapt expect detail sequential regard queue queue top sequential empty space database detail leverage extension directly bind exploit regardless use expect depend pruning dataset maintain set sequential explore prune high scoring first propose bootstrap quick limited bootstrap depend domain es ms add sequential high leverage make availability sequential application sequential sequence website visit represent transaction location series visit customer market action course patient dataset extremely rarely make scientific addition assess discovery discovery sequence discover scalability public select book many extract sequential database topic largely motivate mining aim demonstrate extraction use dataset token token flat generate accordingly shift make average occurrence average deviation embed sequentially associate embed select extraction extraction allow would return datum embed pattern use support leverage ever extract sequential frequent item explain introduction pattern compose frequently chance support difficulty extract highlight point single embed embed pattern rank pattern base approach large embed embed recall decrease three factor keep extract increase consider pattern interesting mean counterpart sufficient mining whereby overlap independently though actually within top dataset illustrate table match adopt embed pattern represent extract depict pt pt depict pattern actual next extract pattern base informative pattern actually chance pattern difference explain token relatively frequent exactly happen pattern token appear frequently see pattern show conversely leverage embed already appear observe thus effect accordingly hand introduction sequence accordingly depict pattern two although attempt extract without filter discovery sequential mining mining book domain build dataset book consider abstract sequence highlight regard vocabulary sequence characteristic avg max pt modification build book sentence processing reduce stem say make word linguistic meaning use sometimes extraction report critical one make extraction correspond repetition frequent word support example case seem extraction book similar claim contrast observe approach book retrieve give focus surprising know classification typical english frequent surprising ed vs numerous statement activity mostly decide frequent book past contrary rare observe early sentence person place english indicate compose say mr mr way almost sentence finish top pt pt sentence cc pattern pt cc cc pt sentence cc pt top pt top paper cc pt finish showing report surprisingly order extraction fast encounter top bootstrapping leverage relatively make extraction high little twice base extraction slow efficiency high interest turn pruning book book high execution long support extract top pt support pt pt pt introduce definition sequential specifically efficient exact exploration contribution introduce constitute high leverage validate consistency question issue heuristic integrate background knowledge common measure highlighted table assess pattern pattern joint differ investigate pattern joint framework china research research air office scientific research office contract fa rgb university edu framework exact pattern combine expected concept measure top carry confirm consistency approach introduce efficiently interesting measure core paper early pattern appear something happen body pattern frequent even event completely independent large database create frequent even problematic frequent pattern interaction within novel insight database issue even repeat extremely frequent relatively table five frequent pattern rest frequent sequential observe permutation frequency discovery high individual surprising contain occurrence enough believe probability occur length event demonstrate frequency sequential item determine pattern frequent capture argue pattern capture see frequency possible close question pattern database contradict sequential standard involve support formal motivation specifically tackle order high leverage note extract top expect research discovery sequential section conduct pattern sequence pattern occur build different sequential derive score strict episode difference independence consider overlap partition important reason expect activity sequentially parallel nan frequently paradigm paradigm relate
galaxy treat apply algorithm version splitting result panel bandwidth panel smooth panel panel right third coverage panel leave display smoothing smoothing see optimal smoothing method coverage integrate risk bootstrap estimator bandwidth selector work coverage limit instead learn risk risk select selection tangent sense projection eq hausdorff smooth curve parametrize end q property rr eq convert bound law contribution small define think du du small bind simple lemma property angle imply angle projection onto tangent bound projection since similar hausdorff du side combine conclude note derivation work constitute asymptotic nx bootstrap consistency easy nu nb proof replace lemma let uniform distribution q second assertion nx similarly need pick restriction department university density ridge coverage generalize two risk select tuning parameter estimate dataset density thm thm definition density dimensional characterize high vision imaging density universe detect propose shift algorithm modification usual algorithm adapt geometry unlike mesh mode move project gradient nearby kde act bandwidth despite coverage generalization integrate expect smoothed selection choose parameter propose dataset follow independently identically order whose dimensional curve dimensional manifold nh play role kde detect smoothing smoothing bottom introduce coverage risk geometric concept let hausdorff length area projection uniformly ridge random b dx bound distribution eq cover omit lemma link hausdorff call project apply set nice outlier risk hausdorff outlier risk kde datum manifold half du du rule minimize select principal curve different coverage count self monotonic bandwidth derivative high coverage risk pick estimate concept dimensional cdf contain coverage region cdf link diagram manifold coverage diagram use similarity coverage diagram serve nice cdf mesh consider curve coverage diagram green curve difference expectation take risk analyze prove particular risk since jensen bound define orientation collection associate specifically usually eigenvalue require r require direction common condition kernel kernel assume compact derive coverage eq density thank jensen convergence risk square theorem require decay come bound derivative converge give hausdorff density density appear agree hausdorff nonparametric converge fast
review weight adjacency length input shift operation adjacency filter polynomial q input signal depend filter polynomial describe method note similarity present section selection consider learn describe typical identically desire lead optimization sample useful dynamic evolution value power correspond undirected structure autoregressive coefficient conditionally accord mrf assume give matrix evolution process time ij ij use graph challenge single adjacency unweighted property process define single weighted model describe non graph new relate discretization partial differential signal framework series index sample represent graph sample graph discrete follow form collect affect influence limited provide continuous signal index describe first approximate discretization grouping arrive see causal naturally fit model big true graph product sum jointly basis jointly eigenvector term q adjacency cyclic shift represent direct dft temporal arrive estimate time series wish adjacency first follow represent represent ensure adjacency first nonconvex find locally near instead separate polynomial must mutually ip q still hold naturally coordinate sub formulate regularize incorporate especially new running except ic linear autoregressive drive white follow correspond maximum posteriori framework extended process nonlinear value loss convex prior polynomial estimation reduce complexity find term polynomial polynomial adjacency matrix polynomial function generalize outline adjacency filter behave initialize ti denote number maximum count estimate direct minima initialization summarize discuss basic estimating problem norm nonconvex like ensure coordinate global mild objective level norm block descent extend section converge compact produce appropriately choose optimum nonconvex example vary temperature sensor year location unite least square projection reconstruction mrf matrix proximal descent estimate create element draw make thresholding scale direct os enyi topology edge ensure stability arbitrarily result stable form unit additive simulated graph generate sample structure basic initialize graph direct see individual direct qualitatively sparse sparse close thus see extend square error extend across different carlo mse decrease propose plot produce mse compute decrease suggest total error interest temperature average temperature united pass filter cutoff mrf distance neighborhood city estimate consist index odd training testing different prediction since experiment see corresponding task compression prediction entire train leave proportion well mrf sparsity mrf perform training error mrf truly capture dynamic process temperature mrf estimate mention previously axis correspond produce mrf produce magnitude pick west east wind country multiple south chain knowledge experiment estimate describe believe model underlie temperature sensor process tractable vary
cubic interpolation framework global interpolation interpolation perspective naturally enable toeplitz structure gain scalability target grid ht conventional induce form perform w x perspective induce approach well popular rbf expressive induce interpolation nature go global cubic ability greatly induce function versus interpolation greatly scalability gps interpolation kronecker gp though write cubic evaluate model particularly approach superior similar efficiency recommend understand comparison recent storage le locate input consider ghz ram accurate gp predictive likelihood evaluate induce induce sort randomly use cubic figure indistinguishable absolute generally great ht cubic interpolation c average reconstruct entry cubic interpolation form weight black show vary point interpolation cubic grid blue input distance linear interpolation mean upon induce input allow precise interpolation less long coverage cubic interpolation regular weight global kernel interpolation red correspond strategy global kernel interpolation cubic interpolation reconstruction qualitatively cubic reach find combine cubic region ultimately runtime linear cubic large general go interpolation interpolation cubic interpolation great boost without runtime give importantly cubic alternative test perform covariance matrix moreover yet toeplitz accelerate gaussian dataset large provide discover rich statistical representation improve process show typically expressive suited great first place arise inducing require computational necessary suffer efficiency gps gp attempt product operate figure distribution even sample sophisticated intensive taking instance enable underlie likelihood gaussian process py kernel wish equip learn induce grid induce figure reconstruction reconstruction provide whereas unable reconstruct kronecker induce second hour induce point sample exploit method effectively exploit kronecker limit well exploit multidimensional scalability series exploit structure computational placing induce grid create toeplitz exploit ht automatic speech deep sound series consider context datum use large contiguous region grid locate therefore direct inducing point toeplitz scalability show induce mean log scale empirical value correspond runtime hundred runtime confirm loss cubic interpolation gain induce less runtime generally infer curvature function unable induce tend add scalable gp require computation overall induce induce form cubic scalable combine kronecker gains toeplitz arbitrarily locate input show ability induce expressive kernel learn improve improved order magnitude simplicity major strength explore interpolation create entirely new scalable strategy could remarkably perspective interpolation well induce point induce need combine model orthogonal benefit recent process kronecker toeplitz provide motivation toeplitz hope improve understand dark medium gps kernel framework quality induce interpolation covariance mechanism scalable kernel choose kernel scalable point alternative enable toeplitz substantial additional scalability require fast expressive storage gp sound process gps exactly flexible capable learn expressive requirement gps contain empirical thus induce method large computation storage inducing induce purpose require require number expressive learning exploit toeplitz advantage induce exist highly accurate scalable kernel dataset lattice product make kronecker locate input likewise costly similarly restrictive require grid induce kronecker toeplitz scalability induce critical covariance induce induce scalability fast combine exploit show interpret underlie help accuracy induce point interpolation interpolation strategie cubic interpolation create kronecker toeplitz induce point toeplitz gp computation kronecker gp require input view toeplitz locate input gp induce order magnitude popular extension toolbox simplicity generality make easy scalable gaussian section structure interpolation reconstruction sound conclude process vector yx gp collection fx kf gaussian nk jx kx fx smoothness kernel example rbf hyperparameter target additive yx covariance gp evaluate depend obtain condition separate calibrate fit complexity hyperparameter integrate iterative covariance form complete theorem prove eigenvalue asymptotically bound complete eigenvalue approximation pca determinant marginal approximation expressive input space general setting toeplitz complementary toeplitz stationary kx kx spaced toeplitz along kx toeplitz product fast g gradient storage grid gps computation flexibility scalability box require point suffer major predictive accuracy perform expressive valuable large structure gain requirement grid place induce method kronecker toeplitz algebra complexity computation computation product cross covariance induce wish bx u ix weight extremely interpolation per weight grid regular use distance weight expression approximate gp essentially induce make directly exploit toeplitz vector cost storage
health institute image big http www fellowship foundation google finding conclusion recommendation author necessarily reflect view nsf tradeoff space empirically cubic notation secondary one observation type resp resp resp large matrix involve size input relative two analytical differently dominate expect output channel type versa approach magnitude strategy channel demonstrate ratio output channel dimension fix channel vice versa optimally cnn narrow major validate heuristic scheduling device contribute follow protocol vary ratio show fraction scheduling essence gpu also use peak scheduling tried estimate device speedup optimal edu present compatible end examine characteristic purpose convolutional neural architecture employ cpu throughput improvement cnn directly cpu hybrid cnns networks cnns research application recognition cnns perspective database concern contrast cnn technology key choice cnn modern offer grid slow microsoft project cost effective generation intel cpu parallelism likely continue user center issue amazon ec neither google compute study architecture conduct open cnn call version output bottleneck convolutional execution technique focus tradeoff layer batch one standard multiplication multiplication compatible library intel art pick optimal depend ratio usually dominate optimizer pick space automatically network contribute execution system fast simple achieve convolutional cpu device create cpu gpu typically gpu reach almost use argue hybrid layer ec gpu instance core cpu throughput become effective homogeneous open question amazon ec end gpu describe definition convolution operation popular operation convolutional element index kernel operation problem highly multiplication convolution layer take channel transform tensor multiply phase multiply multiply create back strategy correspond sum let indexing array slice describe e submatrix dimension expensive expensive balance create multiply trade start index let eq multiply expensive balanced spectrum either phase expense call q two approach multiplication intermediate appendix conceptually experiment report number fusion discuss optimization discuss partitioning partition partition convolution partition indicate default process process single parallel split partition partition show indicate partition currently parallelism layer model share decision simple heuristic input device gpu conduct compare library cnn system neural architecture use cpu version gpu imagenet diverse ec machine illustrate per tolerance concentrate throughput find thus remain compare ec instance iteration image b ec cpu instance cpu physical speedup cpu use fast increase conv appendix probably comparison cpu gpu gpu core run gpu cpu expensive gpu fact gpu however far magnitude associate cpu suggest cloud services gpu microsoft google train deep cpu validate accelerate purely cpu gpu training running convolution operation gpu hybrid batch image run gpu run ec gpu bridge cpu core report figure group significant gpu batch cpu cpu gpu core available hybrid gpu cpu figure ec gpu execution per give speedup number parallelism fully layer briefly study focus improve although decade specifically multiplication due year library framework
entropy bind proceed lemma always eq combine lemma assumption study randomize reduction reduce projection result randomize reduction hinge large address randomized regularizer reduce dual mild dual study present randomized method communication scale datum continue application bioinformatic finance vision medical critical solve big great dimensionality reduce also computation efficient reduction lot e hashing latter refer randomize examine reduction generalization model strong rank assumption separate weight assumption regularize randomized leverage solution scale dimensional support compare example dual implication randomize hashing compare mild assumption subsequent interpretation exploiting classify people genomic important designing addition performance exist feature propose feature dual recovery reduce rely realistic ii analyze smooth iii method projection set analysis application distribute learn combine benefit address optimization receive problem especially dimensionality reduce employ develop ascent observe communication let solve parameter primal randomize randomized reduce reduce dual previous reduce random construct matrix recovery recovery error one dual reduction corruption original strong plan limitation different relax assumption contribution trick sparse solve regularization whose reveal later understand dual regularizer svm hinge non smooth hinge q change give reduce margin smaller reduce reduce square hinge loss primal eq regularizer emphasize dual error trivial assume set complement set remark dual proportional x bind assumption contrast require x dimensionality affect recovery result hadamard hash om recovery low connection signal theoretical smooth recovery restrict eigen restrict another restrict restrict eigen condition recovery function bind sn recovery nearly smooth quantify except top satisfie distribution dimensionality reduction lemma reveal subgaussian universal hadamard projection dd ii hadamard typically compute choice possible provide sample randomized hadamard sampling universal projection apply random mp dp dt md satisfie please speed hashing dimensionality analysis rigorous recently hash hashing algorithm denote rademacher random equal write type hashing remark compare remove extra discuss another one consist reduction random aforementioned randomized sampling implicit explicit scale method recovery transform introduce randomize smooth bind om idea relaxation show conv conv supplement entropy arrive remark result amount e sn section conduct contain split norm report hash randomized function hinge aim motivation affect vary among randomized reduction trial square hinge loss indicate magnitude variable support consistent dual decrease increase large certain making threshold recovery exhibit threshold consistent trend recovery much square loss sufficiently solve sometimes experiment randomize multiple distribute distribute associate among total communication reduce distribute original demonstrate effectiveness stochastic ascent reduce high use reduce problem record running step optimization ii recover communication method stop run improve
salient query guarantee discover left size example query fact three many query various fail query completely recover satisfie recover query good conversely good hence salient return recover video sign image tie video initially unlabele goal triple crowdsource platform amazon type crowdsource worker ask specify feature task worker ask feature assign non triple pick triple triple except figure run datum gender early choose face gender learn feature compare adaptive baseline adaptive pair worker show tag ask return feature complementary discover many distinct relevant two fraction hamming say differ redundant sign face product triple triple feature terminate done replicate seed triple discover dataset adaptive triple sign face triple obvious choose distinguished order learn feature poorly distinguished algorithm run query guarantee feature hierarchy triple efficiently feature partition feature discover induce belong agree average indistinguishable discovered perfectly distinguish benchmark function query scatter compare triple require indistinguishable triple triple adaptive pair rapid decrease discover triple discover introduce framework query inefficient feature experiment three set theoretical prediction unlike detect redundant human process avoid redundant feature place outperform feature salient face language product direction would investigate type challenge crowd direction work attempt salient imagine aggregate addition different feature salient diversity crowd triplet low common low also otherwise recall adaptive double feature rest one let triple low triple triple feature triple write child child triple set triple nice property triple query feature map combine therefore triple subtree make triple begin observe query step feature see query leaf leaf query moreover feature queue currently explore induce subtree root underlie subtree subtree triple query star subroutine stop query algorithm draw double query p yx follow similar lemma randomly triple triple query query discover feature discover maximize discover triple course minimize triple choose argue triple triple random pt theorem remark ex ex ex microsoft research ca microsoft approach discover crowdsource crowd member common two display provide binary discover triple adaptively label hierarchical simple similarity recover less finding discover statistic crowdsource discover merely label hand address crowd diverse salient name datum example face salient gender numerous binary feature think mapping feature refer string describe value crowd worker exploratory machine feature learn significantly compact exponentially grey crowdsource ask people multiple word phrase example fail ask crowd worker tag sign american gray tag tag equally none could discriminate sign inspire prior familiar belong name present crowd triple three example common feature meaningful datum choose triple salient triple distinguish sign size salient people often triple address feature crowd worker accord label necessary eventually require annotated discover query triple salient adaptive say gender assume thereby avoid equivalent face illustrate hierarchical orthogonal applie across large state query adaptive response e car sophisticated independent random find use expectation since incur moreover worker second feature batch label image feature learn ask query one seem could green similarity theory arbitrarily discuss hierarchical section triple adaptive triple query feature analyze expert approach crowdsource name automatic representation e summarize inspire worker right task times query vision order direction crowdsource show worker positive example feature reduce adaptation order grain terminology incorporate finally receive every triple distinguishing return already discover imply common triple definition common two child child child distinguish never observe triple particular internal node example leave parent leaf leaf distinguish triple figure triple would uniquely distinguished distinguish feature close point say specify advance triple answer could anonymous think purpose least proper non least query figure discover choose specific example child leave internal order discover triple must triple triple fail answer leave set difficult triple triple triple triple triple insufficient motivate triple moreover proposition query feature tree pair determine identical maintain queue explore discover initialize default feature initialize f jx query query label consecutive go feature triple efficiently internal child internal branching root tree rule feature terminate triple star triple use limitation defer appendix example face thus feature abuse much even triple among crowd triple always
analyse classifier kl moreover majority vote case classifier x point kernel express linear precise pac domain idea decompose risk expect disagreement disagreement label domain weight latter domain divergence limit worth note recover well domain relate enyi specific measure domain estimate sample hypothesis last old p bayesian adaptation divergence contrary measure domain oppose hyperparameter disagreement successfully need make adaptation negligible pac bayesian attractive consider different necessary sound domain marginal xy unsupervise interestingly correct capture pac bayesian justify generalization present precisely simplify h equation function eq shorthand jensen markov extend pac bound quantity appear especially interested possibility control help parameter disagreement source consider obtain similarly reason thank optimize domain real h disagreement respectively separately combine suggest minimize former disagreement choose ignore result hyperparameter tune hyperparameter trade optimize linear prior center e x figure adaptation linear achieve equation descent starting give trick weight augment rewrite fix rbf positive resp firstly decision toy label label domain clearly succeed adapt misclassified disagreement target secondly use amazon com benchmark review product book attribute originally rate simplified rate product appear ten domain process tf weight perform adaptation book train co train amazon com thank reverse search parameter logarithm report algorithm label unlabeled report evaluate separate imply method reasonable algorithm overall six increase confirm novel improve risk bold l derive context majority analysis domain joint source crucial divergence domain trade give major domain tackle name adaptation future aim extend divergence like covariate issue suppose two domain actually disagreement give weighting bound depend enyi second unlabele rgb universit universit france issue express majority upper trade source disagreement target easily rely pac generalization propose machine learn corpus e unknown wants study computer vision etc simple spam adapt another significantly different often importance divergence bound pac focus vote tackle domain fashion share seminal bound trade marginal ability adapt paper adaptation trade upper half disagreement bind relation source distribution weight target domain pac analyse closely gibbs predict draw return call expectation risk pac bayesian literature deterministic classifier study disagreement seminal analysis
obviously approximation allow sa intensity cardinality target target previous birth particle array acoustic sensor frequency sa measurement cell standard formula sa give factorial moment predict cardinality filter describe update distribution filter efficient tracking proposal equation design sensor sa filter accurate cardinality filter exploit approximate contain I distribution label particle label process label birth update time rewrite birth extract mean gaussian alternatively density kde follow cluster sake exposition survival target mass newly constrain probability constraint impose since track posterior subsection density proposal survival target birth survival birth proposal construct survival eq birth summary target exist newly target code multi use proposal grouping particle death newly generate target I k ip q k lead particle sa cardinality match cardinality prediction important designing proposal snr snr multi sufficiently estimate general become detection multi filter seek proposal match cardinality individual compute specify component cardinality account individual density straightforwardly choose preserve label mass cluster construct resample much thus implement time weight probability track elementary construction sampling cardinality newly propose particle sample ix code implementation k q evaluate transition bx k ix I transition due proposal technique computational load sum problem particle transition guarantee k target sample target proposal closely spaced target k k coordinate modulus amplitude velocity walk fluctuation complex q tracking measurement usually refer measurement consist power return cell mean white symmetric spread cell cell coordinate cell centroid I complex denote cell write statistically random reduce define I centrality distribution cell modify hence form template e consider incoming highlight grid notice target share cross consequently snapshot domain report initialization mainly propose incoming particle velocity towards position birth particle symbol acceleration initial state birth birth target assignment fig fig increase target enter scene thank convergence difficult target due fact retain target confirm behaviour target reduce phenomenon surveillance confirm challenge spaced multi filter enable filter result confirm spaced target use snr approach problem devise target target interact satisfactory load separate spaced target cardinality target mean measurement process number measurement process edu surveillance area call model finite estimate lead version bayes filter however straightforward smc feasible sampling space multi sa recently multi bernoulli applicability application closely target label filtering problem arrival array wireless amplitude tracking merge multi track spaced target contribution estimation sensor dynamic base typically collect measurement facilitate efficient tracking important target trajectory real describe target assume target generate belong target preprocesse raw set usually application space standard approach may adequate case make information necessary require advanced measurement time superposition multi approach smc method passive application target indistinguishable approximation target inspire propose target sensor track framework label enable track direct problem arise propose particular call sa design particle latter require spaced definition label sensor target particle label density numerical present section present description sensor report random euclidean practical tool notion integration theory standard system model specify notion integration multi contain measurement target denote target target track history track simulate technique tractable omit history multi recursion prior equation start target h convention density index satisfy interpret object term exponential family form give set existence probability track need filter principled bayes tracking capable tracking target describe general fortunately target greatly transition drastically subsection present particle complexity review ensure order integer target target kk k completeness multi recursion posterior integral computational requirement filtering technique approximate approximate integral converted density particular dominate measurement integral interest construct generalise context vary particle w k k value posterior z p description filter
high word ability direction work second accommodate arbitrary apply quadratic variational appeal structured penalty lasso group penalization systematically track promise exploit screen quantify response penalization rr rr rr em ar ex simulation design adaptive original ols intermediate calibrate control rr rr r rr r rr design em ols ex sparsity stage purpose possibly estimation coefficient estimate penalty procedure benefit transfer stage operate distinct crucial calibrate discovery setup gain range art discovery explanatory attract attention decade development penalty selector show relevant various recovery apply differ coefficient accuracy numerous stage predict stage relevant operate candidate accuracy assess perform ol lar relaxed modify reasonable marginal bridge statistical perform regression method rely relevance select region hypothesis testing bootstrapping differ rely stage summarize estimate support variable focus first coefficient improve genomic regression quantitative use penalty variational optimization hierarchical detailed benefit inference empirically ensure coefficient family false discovery criterion reliable input alternative numerous perform fairly nan hypothesis expect one usually error genomic false appealing context attempt follow path first consist permutation computationally theoretical approach credible define projection function semi propose whose build later extend aggregation response unknown dimensional vector rely lasso tackle define hyper fit throughout discuss regression numerical acceleration apply address approach view adaptive viewpoint mostly formalize formulation ridge coefficient return proof instrumental define dependent implicitly determine stage process primary retain estimated stage small large variational bayesian covariance matrix assume model stage two approach second state support recovery reveal regularize condition incoherence state truly variables retrieve provide irrelevant covariate correlate rate slow noise estimator procedure produce small restrictive experimentally design ol variant adaptive consistency since believe low interest estimator relevant predictor ols decay lasso ol respectively mean experimental validation thereby theoretically prediction performance actual penalization cross commonly penalty follow ols regression arbitrarily ridge adaptive choose apply serial overfitte sense result optimistic conclusion test dependency denominator randomization set exact exchangeability exact multiple take variable estimate test approximate block screening prescribed design lrr ex c permutation test ex satisfactory level either calibrate conservative prescribe false level explanatory problem propose among correct calibrate show establish clean procedure partly fact relevant establish describe remain validation mention independent rely summarize modify procedure devote stage gain overall illustrate set regression screening stage design cross importantly support limit support implement respectively criterion datum truly truth analyze present application genome association infection variable variance know numerous magnitude relevant varied predictor mean unit variance dependent mean belong position relevant randomly distribute block belong enable noise discuss design report three medium medium drastically compare variant mean conclusion measure coefficient display snr snr ex group lasso high medium ridge still slight step improvement benefit ols ridge adaptive par option jointly optimize stage respect penalization improvement compare study mainly focus setting beneficial addition design considerably penalty within cross post lasso screen ols ridge optimization penalization serial se though viewpoint also square discuss representative setting feasible respect control discovery significance true false negative control procedure univariate variable selection experiment noise level calibrate stage calibrate procedure ridge original ols ridge rr r simulation ar ex lasso numerous determined model poorly lead high statistic stage block group ridge line ridge dash ols dot rank dot line mark observe consider far design group approach high level extremely systematically dominate weight regularization ridge base plain improve ol ridge bring thus table procedure threshold level always clean design dramatically gain ridge variability regression follow ridge ridge bring stability c lasso ol r ol r ols dark red lasso clean significantly high sensitivity dark red box properly selection procedure statistic affect less origin design compare ol univariate testing screening perform lasso calibrate rr rr ex em ex ar ridge ols correction univariate design dramatically original gain effect beneficial transfer ridge difference ar original ol univariate calibrate rr rr rr rr simulation ex ridge ridge ar original testing calibrate rr rr rr rr ex ex ols ex variable wide infection study identify genomic influence rna level infection
enough sentence order vector need close additional run record fast computer error lattice learning problem great relevance well variant adversarial succeed time due run time grow learn question simultaneously number draw linearly consistent open whether make ia current elimination noiseless two confidence mistake strong pac function consequence statement mistake comparison relate comparable improve thm remain gaussian make explicitly keep track general observation seem explicitly make give learn use conceptual carry complexity run instance run immediately get confidence complexity consideration runtime bad exponent result thm pac possible enough ok example instead randomly mistake hide specifie lying take element cover factor every example vector learn think intersection gaussian increase collection sufficiently size sample order contribution reduction noiseless learn exhaustive search drawing devise well presence non amount noise rate open attribute hamming weight associate work mistake round round boolean must predict mistake learn mistake example unknown learn pac complexity example e consider pac fix half mistake pac learn concept mistake mistake run convert pac learn introduce correspond pac source rate every mistake bind mistake per round kn red pac statement pac mistake mistake mistake mistake well see slack divide roughly divide run round improvement improvement pac model prove thm basis large constant later arbitrary part let ensure km k apply conclusion rest reproduce span n kn tt nm kn tf happen efficiently elimination ready describe begin space ia ia constant individual enough thm fix lem maintain obviously terminate hidden mistake make notice begin whenever make mistake predict space reduce factor lem learner mistake learner store vector treat subspace elimination lem ease calculation round simplify ignore term eq gain learner mistake calculus long improvement mistake mistake noise rate half
examine theoretical cv formula fact predictive firstly cv second equivalent optimizer secondly lastly mathematical relation also minimize average statistic design regularization depend foundation enable call hyperparameter design hyperparameter optimize likelihood rational procedure minimize paper average generalization validation validation important property study applicable leave cross validation cv hamiltonian turn view criteria cv criteria estimator generalization value equal equal whether cv average prove regular cv make whereas maximization second loss sample take hyperparameter minimize random loss generalization problem criterion employ determine hyperparameter region cv criteria heavy enable candidate variance cv small new formula eight section section learning devote proof theorem result discuss conclude future definition bayesian statistical learning real euclidean probability train nonnegative call study improper normalizing predictive generalization even asymptotically validation leave calculation markov chain monte method define widely show paper posterior cv cv pn however asymptotically equivalent statistical regularity minimization method marginal minus marginal likelihood eq method integration improper hyperparameter minimize minimize several notation condition definition mathematical arbitrary candidate proper denote loss posteriori estimator depend equal log minimize define simple power adopt convention suffice mean automatic summation order singular set infinitely time parameter unique minimize convergence regularity matrix invertible almost inverse well k k finite let q equation regularity say definite satisfied hold singular machine expectation mathematical reason definite saddle order outside integration zero empirical mathematical relation define k w w note relation average relation manner mm calculate eq k k k j k empirical mathematical relation hand mathematical derive prior hyperparameter find relation asymptotically asymptotically unbounded phenomenon discuss asymptotically relation map replace average neither generalization prove regularity exist theorem true learn mathematical self average estimate hyperparameter widely regular directly point nontrivial example note improper prior prior therefore relation minimize exact cv nx criterion definition cv n nx z px numerical ten thousand collect prior firstly average average deviation give std std hyperparameter candidate hyper compare choose minimization interval proper average std generalization whose improper whereas energy hyperparameter h predictive design c std prior immediately derive hyperparameter cv calculate xx yx criterion nx n z free conduct dimensional identity candidate hyperparameter variance rigorously hyperparameter phenomenon contain std lemma arbitrary px training leave test log leave validation generalization proof px w I px px px n w w j half half px n px px px half obtain mm function map v need mm integrate region taylor expansion respectively give fw l u derivative define use notation remark matrix sum pair combination symmetry odd integrated follow sum put complete leave hence odd eq condition moreover n map proof equal parameter note map therefore apply therefore lemma eq term complete five mathematical average generalization prove sufficient prove define let odd even number q prove prove lemma eq q subsection minimize
computational limitation limitation model approximate evaluate summarize principle meta expansion krige model new krige formulation pc krige krige pc krige spc spc krige employ least determine orthonormal polynomial polynomial krige meta krige spc krige iteratively universal iteration case polynomial trend select loo model find krige krige spc krige krige error analytical function result pc good distinct experimental design krige preferable spc kriging reduce spc krige research realistic reliability idea design input zero level reliability instead everywhere initial interest add preliminary investigate carry france research fill blue width text center height text height text center width minimum height rectangle draw fill minimum height draw fill text blue rectangle text rectangle fill text text height width pt fill inner sep input vector two less researcher paper modeling polynomial krige regression set polynomial pc krige validate benchmark reference krige perform large limited asset demand computational model keyword model meta pc modern make order ever complexity structural new behavior acceptable basically input parameter understand eventually constraint similar common dedicated aim physical possible fidelity modern high fidelity typically mean run may day material apply exhibit simulate know refer probabilistic model variable prescribe joint assess uncertainty consequently onto system uncertainty call input million compute architecture surrogate surrogate capable predict input realization analysis among option construct meta focus non black input realization provide additional knowledge build expansion krige machine extensively investigate decade polynomial expansion krige expansion expansion polynomial variable traditionally expansion partial differential expansion obtain among method specific though treat especially code hand call projection review development spectral compressive polynomial paper reliability optimization find meta technique interest krige known gaussian krige meta interpret gaussian find field structural reliability krige technique toolbox r toolbox attempt krige bridge aim distinct powerful call krige sequel accurate flexible detailed organize krige approach combination distinct meta section benchmark analytical equip algebra variable denote capital letter realization low letter capital low letter system behavior introduce uncertainty pdf joint pdfs note variable model input value output cast expansion orthonormal vector index input dimension independent margin orthonormal construct candidate variable summarize orthonormal classical pdfs ht orthonormal polynomial orthonormal uniform beta bx multivariate polynomial compose multiply polynomial handle series truncation response accurately truncation consist bound total polynomial maximal denote polynomial polynomially truncation thus tractable highly input vector low interaction polynomial thus interaction degree tuning maximal decrease small interactive univariate retain vary part solid line represent index define candidate next consider repeatedly evaluate model realization decade pc namely square minimization minimize evaluate empirical denote error derive read function polynomial able output thus type regression expansion thus quantify l residual respect pdf error auxiliary validation rarely purpose model analytical eq normalizing variance output number polynomial sample experimental design tend phenomenon predictor vanish loo general build n loo theory computational loo would determination special analytically build proof build model technique krige assume response realization p value trend variance unit variance auto various mat ern autocorrelation generalization mat scale call shape euler modify kind simplify apart correlation part krige namely krige assume trend trend unknown unknown formulation trend sum pre krige krige discuss define krige auto hyper calibration parameter eq response development approach preferable autocorrelation assume family cv shall discuss multi minimization algorithm cast distinct algorithm quasi newton genetic differential evolution algorithm spc krige behind procedure polynomial trend universal krige procedure orthonormal trend universal kriging base error krige figure box blue box represent spc krige realization node auto distance cm lar autocorrelation block sequential pc krige distance cm prediction line lar lar prediction line line lar pc krige spc krige krige krige orthonormal algorithm spc yet krige consist iterative add auto calibrate pc krige meta model meta minimal leave error krige universal krige trend polynomial ranking box universal krige loo mark cm cm autocorrelation node block node krige auto auto loo loo line right cm auto loo loo loo right right distance distance loo loo line krige loo I cm prediction krige krige krige view krige meta model trend part valid loo pc krige meta krige comparison krige illustrate analytical evaluate verify new krige approach uniformly distribute input two gaussian original benchmark analytical independent input use method sensitivity analytically behave smoothly point space eq consider x function last function pc multivariate model hypercube meta model ordinary krige spc krige krige krige combination gradient order compare output value krige meta follow variable uncertainty carry th percentile boundary dimensional approximation krige krige meta minima maxima analytical component high global characteristic component whereas ordinary frequency lead input visual krige meta design choose yield second result ordinary show spc krige krige ht modeling meta sample ordinary krige perform box spc krige krige bad overfitte pc ordinary krige due polynomial krige krige spc krige ordinary krige krige perform value krige well krige accurate spc krige range ht relative generalization error purely model design sample krige relative generalization model need error behavior among krige krige approach sample krige significantly require properly surrogate krige resemble case capable analytical various function fig estimate highly input previously fig qualitative pc krige quantitative combine krige visible krige traditional approach follow krige low error whole generalization meta pc kriging resemble ordinary krige experimental design krige slightly spc krige krige experimental accuracy input krige approach like pc krige krige whereas pc kriging resemble whether krige provide accurate pc krige combination krige accuracy high computational krige spc krige intermediate
specify reconstruction move filter might significantly variation step modification expression noiseless next convenient perform majority side sparsity varie perfectly idea entire successful k independently probability reconstruction length albeit number close condition get insight q suppose signal tell large significant ignore remark inherent difficulty establish namely quality continuous reconstruct quality arbitrarily reconstruct far stochastic compressive video compressive signal introduce resolution instant collect decompose foreground typically background respectively take measurement tell foreground still measurement give compress mean suffice perfectly foreground frame motion image laplacian perfectly integrate nothing show compressive background translate diagram frame pursuit depict motion construct past rather foreground former texture yield prediction foreground operation model take reconstruct obtain output input optimization input foreground proceed current subtract module foreground frame z l dot dot dot op op op op op op op op op op op dot op dot op op highlighted technique generate decoder motion motion reconstruct consider require motion metric spatially smoothed vector coherence outlier motion far field linearly belong overlap block pixel predictor motion neighbor pixel pixel scan frame correspond pixel white white green white green black red black white solid red dash line dot line illustrative mostly background reconstruct reconstruct foreground k visualization oracle gray oracle figure gray reconstruction sequence sequence frames people office top panel fig show background frame several background foreground frame fig frame reconstruct foreground dark setup sequence true small value initialization adapt frame camera e isolate sequence motion block mention frame improve reconstruction solve remain frame benchmark cs oracle fig measurement estimate foreground frame quantitie fig fluctuation clearly advantage foreground standard oracle cs recall cs line oracle though fact small less resp frames frame reconstructed show quickly fig relative error around solver varied reconstruction error frame foreground ill condition frame closely figure figure noisy oracle figure gray gray cs oracle noisy noisy gray gray cs truncate measurement vertical e compute yet curve shape noiseless reconstruction order online reconstruct dynamical minimization perfectly explore background real image video sequence notice perfect reconstruction increase time perfect hence function independence recall definition assumption consist contribute probability contribute conclude event c event condition sparsity step prove inequality state third value expect note component contribute cf event independent equal simply value condition use corollary conjecture rgb rgb department electrical engineering college uk mail n ac electrical engineering university mail laboratory e mail reconstruct sequence signal support evolve nonlinear recursive compute compressive problem image background image foreground reduction respect background compressive reconstruct sparse signal signal evolve otherwise describe nonzero reconstruct measurement online reconstruct measurement measurement formalize online measurement acquire possibly sparse invertible transform f sequence signal track wireless application background detect video example surveillance traffic image g compressive expensive uv compressive video access frame conventional video frame notice foreground pixel frame sense minimization reconstruct foreground frame mention exception compressive compressive frame succeed cost unnecessary measurement address problem online use reconstruct foreground area foreground extra foreground frame give frame fail adaptive sparse reconstruct optimization measurement know reconstruct say generalize pursuit show draw measurement require reconstruct small pursuit furthermore choice irrespective free address reconstruct use estimate require compressive side contribution summarize contribution adaptive reconstruct satisfy compressive motion next acquire word know characterize compressive background incorporation number measurement another make fundamentally prior reconstruct signal sparsity slowly contrast operate theory slowly quality e side extent pattern sparsity nonzero use establish compressive background performance conclude proof reconstruct signal limited overview solution control terminology filter kalman filter e across filter knowledge state incorporate procedure kalman filter signal time instant nonzero signal compress sensing assume vary slowly number measurement assume along relate work include knowledge kalman measurement name briefly overview probably reconstruction scheme euclidean problem version replace measurement problem characterize computing assume measurement signal review
error interest solve problem regime present sequential scenario pass prohibitive minimal thought contradiction default sequential fashion assess enough evidence nan stream enough even far statistically easy conclude confidence look fraction dataset discard expensive test subsampling know hard subset suffice could resource address rest subsampling entirely stop would suffice devise formally mean q think parallel stream resort memory keep track single vs process main issue apparent multiple testing observe rejection conduct reject chance correction conservative produce walk move intuition law alternatively algorithm batch automatically stop difficulty near nature type follow imagine fair coin assigning tail keep coin basically remain envelope early walk behave play role new fair flip walk around envelope outside envelope hypothesis practically examine asymptotic version depend empirical tool independent non contribution control non uniformly control uniform desire stop propose intuition detail automatically concentration always absolute observe binary coin may bias detect test alternative size involve deviation hoeffding nan fail reject reject fail reject sequential arrive time size sequential define threshold rough argument statement introduction sketch formal corresponding result test threshold control type also type concern asymptotically iterate non sequential insight p n treat small powerful sample sequential early motivation use reject statistically distant alternative work coin full would q examine therefore low bound binomial test stop definition hoeffde first infinite geometric fact stop soon could almost good formalize precise non reasoning sequential seminal line testing implement upon result together heuristic e clinical trial perform loose version bind though scope current nan stop apply may additional cause practical implement rigorously test template refer batch importantly simplicity denote h v u nh h sum ensure large estimate base specifically whenever least consistent test test specify interpret unnormalized statistic assume moment subgaussian special suffice schwarz difference however analogously batch tight control computable priori concentrate around run simultaneously combine uniform inequality deviation analyze test bernstein walk exist theorem calculate time test generic control type basically constant result basically favorable high prove tend argument identical hold walk finally bound version walk capable extend outside scope stream I word discrepancy mmd mmd ball correspond population theorem mmd iff differ alternative sample kx kx limited testing sequential test note batch similarity testing get type error factor also early stop independence involve test population quantity remarkably characteristic joint conclude process b I calculate scalar assigning expectation batch n design previous section control hand present sequential nonparametric testing alternative analysis term desirable property empirical type compare importantly essentially early present simple extension setting theory next form upon v concentration inequality p second goal moment prove bernstein however version convenient follow take define
occur need recognize occurrence encoding speed stream meet number discrete transform dft apply spectra decision tree firstly fourier spectrum aggregate spectra spectrum memory reduce fourier redundancy advantage arise new combination recognize stability stream secondly devise thresholding compression obtain concept optimize dft remove potentially vast literature drift recurrent exist fall broad category store meta learning mechanism match drift method past belong category concept design unlabeled datum issue explicit recurrence overhead module whenever difference estimate concept store repository show outperform global dynamically classifier individual meet accuracy acceptance train ensemble function stream conceptual together build set version et approach consist classifier classifier concept concept drift state incoming classifier instance threshold validity also design delay similar accurately express base classifier newly receive confidence learn al recurrent concept concept accuracy observing design dft tree highly future dft improve classification maintain spectra parallel dominate fourier spectra match current fourier transform dft turn dft apply et distribute capture decision algebraic representation preserve represent dft fourier give jx coefficient fouri approximated computing storage overhead fourier consist thus mechanism capture classification fourier inverse dft expression value thus avoid tree fourier symbolic classified calculate maintain forest hoeffding tree drift divide tree fouri pool maintain encode hoeffding tree forest drift fouri spectrum spectrum fourier maintain pool ensemble ensemble pool ep carry reference structural similarity describe discuss special place h ep structural root datum set randomly select hoeffding forest pool read classifier forest pool classify embed detector window classifier correct else drift identify perform fouri dft threshold pool aggregation pool ensemble pool pool hoeffding attribute create good empty pool income forest pool drift signal drift instance good term thresholde help change tree high accuracy repository whenever concept spectra repository nature identify subsequent good pool classify subsequent point classifier prior dft reduce redundancy pool firstly check whether good spectrum pool threshold step succeed dft produce make pool pass integrate spectrum separate spectrum spectrum great structural similarity currently evaluate disagreement decision disagreement exist ensemble pool update ep single remove alternative aggregate spectra define fourier state call ep et show sensitive high energy great thresholding energy obtain inherent tree iterate spectrum energy order drawback proportion energy fortunately equal coefficient energy single spectral denote order compute exist extension illustrate character begin vector without validity cardinality straightforward optimization increase process speed calculation character absence present hoeffding computation generic domain schema optimize inner product character schema else exactly combination illustrate character occur beginning position validity attribute dimensionality value save multiplication case scan vector overhead multiplication coefficient calculation large derivation fouri aggregate spectra represent produce spectra produce different point spectrum spectra set aggregation stream advance still use express q spectrum produce drift accuracy comprise inefficient bottleneck spectrum major advantage overhead effort initial spectrum extend attribute transformation define split tree integrate integrate account attribute expand incorporate schema expand spectrum add attribute expansion remain index position unchanged position integrate spectra produce localize attribute essentially implement mention focus assess effectiveness ep consumption assess ep pool impact spectra significance generator recurrence stream know ep recognize generate span occur stream order challenge add noise inverting instance spam spam spam informative attribute datum price move price dataset simulator file four scenario record every instance velocity feature maintain stability take velocity directional moving average window instance spectrum reveal approach employ storing concept repository advantage reader refer comparative spectrum mind design practical dynamic stream accuracy take minus sized stream entire fig individual across strategy dataset contrast ep follow show clearly dft fouri spectra store environment memory limit large ep pool spectra ensemble examine usage time claim aggregation ep recurrence counterpart segment concept occurrence span concept represent recurrence ep concept aspect consumption assess consumption influence great store repository accuracy consumption use exclude memory relatively experiment pool ep spam present memory consumption pool forest repository distinguish focus repository exception ep hoeffde together spectra produce chosen candidate aggregate result provide memory high ep ep achieve consumption figure benefit apply aggregation fourier dft application stream variety maintain relatively spectra ep aggregation reduce dft computational term process speed r dataset hyperplane spam fast even though simple suffer counterpart ep current winner tree fourier pool ep aggregation strategy effort stability drift demonstrate expensive operation aggregation yield work environment ability minor dft application mention coefficient minor capture inherent provide dft therefore experiment aim non decrease level clear interesting tolerance ep counterpart similarly interval metric superior performance explain generalize
sample line median network network average prediction bayes nearby oppose confident cast bandit set label receive receives reward receive thus receive reward measure difference reward achievable receive take sum agent various bandit task agent scalar represent action output selection keep tuple buffer size bandit heuristic trading exploration exploitation greedy policy good figure compare bayes agent greedy purely greedy agent enough greedy explore agent nothing approximately explore begin ignore almost perfect learn good classify mnist digits performance bayes comparable demonstrate non problem allow reasonable unseen bandit bayes automatically exploitation readily scale optimisation asynchronous furthermore readily gpu david comment backpropagation compatible compression principled comparable dropout weight plain feedforward neural network reinforcement uncertainty train confident correct shall use call introduce rich representation contextual overfitte decay principled build upon backpropagation predictive little systematic exploration greedy task network rather learn computation exhibit learn thus instead learn train unbiased gradient inference network form neural integration upon gradient prior attain dropout relate deep modelling variational apply unit unit might thousand network magnitude make optimisation large hide allow complementary averaging problem thompson weight great uncertainty network naturally decision make deterministic understood neural learn network classification conclude brief view probabilistic give weight categorical softmax pass normalise gaussian square input map onto several layer transformation learn likelihood backpropagation placing upon weight find map eq laplace answer expectation possible configuration accord label test item expectation ensemble neural practical learning parameter sum shall trade satisfy readily interpretation length prohibitive various certain express density probability transform posterior work proposition monte backpropagation algorithm inference bayes unbiased gradient learn trick operate great apply unit fewer unlike complexity require close combination variational family cost posterior term weight draw technique common number use part cost posterior gradient find kl compute bad cost variational posterior gaussian weight obtain gaussian deviation posterior posterior optimisation pt pp calculate calculate parameter deviation backpropagation remarkably learn deviation prior posterior simple diagonal minibatch partition kl cost epoch partition scheme heavily influence largely useful influential contextual bandit persistent context choice different yield expect present context agent build reward action use pick importantly receive difficulty absence train upon exploratory action perform model neural weight observation high reward explore sometimes exploitation leave future investigation thompson popular pick exploitation pick pick suboptimal thompson bayesian treatment thompson sampling pick probable often fast thompson new pick sample thompson adapt neural pt sample variational posterior receive receive go mention decrease variance trade reduce monte pick posterior action pick begin converge selection focus discover estimate lead exploration mnist contextual bandit task classification ensemble dropout sgd sgd sgd mixture mnist digit training image label nine mnist generative etc shall improve ordinary feedforward softmax label exclude augmentation dropout attain around sized digits set used hyperparameter learn rate protocol descent mle size average bayes sgd unit initially overfitte dropout converge bayes expensive dropout slow eventually bayes dropout posterior
efficiency rwm hmc hmc riemannian langevin ess mcmc kk ess normalize overall measure computer illustration start bivariate gaussian box limit unit rectangle directly panel density panel rwm hmc hmc truncate seed method overall reasonably ht c truth repeat set covariance obtain discard overall hmc efficient rwm hmc rwm propose state reject constraint efficient slow high spherical hmc handle suit ccccc rwm hmc e rwm analysis tend group magnitude estimate optimization alternative method call penalty term replace represent full sampler spherical augmentation handle particular distribution hmc algorithm propose box constraint evaluate diabetes coefficient sampler spherical hmc ordinary ol choose correspond shrinkage vary spherical fix comparable compare impose tight low shrinkage spherical hmc substantially hmc discuss section fact family call residual sum square constraint magnitude allow force exactly model bridge flexible effect shrink limited bridge constrain domain unit ball apply estimate bridge diabetes spherical hmc tight fast shrinkage reconstruct take value translation form function gaussian type box sided spherical hmc form low upper unit form component side absolute discuss summarize rwm hmc hmc effective sample implement spherical hmc normalize ess interestingly hmc ht ccccc rwm e hmc hmc popular model draw mix proportion document assume draw assign probability semi collapse index factorize method stochastic riemannian langevin dynamics sg langevin use mini batch hasting langevin regard step refer sg sg modify sg follow metric logarithm volume adjustment sampling assignment conditional exclude decrease predictive method assign training test document calculate train document wikipedia vocabulary project text evaluate hold mini sg sg sg compare show list sg early stage number increase sg absence fisher sg sg sg introduce sampling sphere explore map mathematically framework augmentation original slack augment energy geometry volume adjustment total take advantage split efficiency split lagrangian velocity momentum avoid requirement embed could spherical geometry introduce directly start informative spherical geometry augment could might able add benefit future explore possibility spherical augmentation elliptical sampler slice sampler general infinite involve infinite f drop quickly increase geometry sphere r map introduce induced define metric metric dot euclidean call canonical lead fact foundation hmc invariant regardless right side ball view system jacobian way term canonical yield form canonical metric way c invariance analytic determinant determinant determinant canonical metric matrix lemma inverse change measure functional compactly coordinate chart cd euclidean geodesic q symbol cg ij cg rewrite augment multiplying obtain sd td x spherical coordinate element volume change measure jacobian matrix jacobian determinant need jacobian determinant weight follow note jacobian splitting hamiltonian usefulness hmc dynamic discuss split start hamiltonian eq lagrangian solve solve first dt dt g u tt discretization difference locally expand equation prove eq k ft iterating provide hmc error right determine monitor intersection coordinate constraint approach norm constraint leave state hmc boundary solve find element find consequently determine instead find intersection sort order intersection sign point k constraint constrain domain adjust velocity velocity u point v h v conjecture pdf figure distribution lasso probit domain challenge commonly novel augmentation constraint domain sphere sphere generate remain back spherical augmentation computationally sample state hamiltonian use example process lda modeling constrain geodesic lagrangian commonly statistical bayesian regression probit many copula intractable simulate estimation improve quite zhang due target deal mcmc typically proposal ensure boundary impose quite inefficient especially domain alternatively remove inefficient explore involve norm map augment explore way implicitly remain within focus hamiltonian carlo hmc discuss modify sampler boundary go create boundary approach henceforth inefficient domain follow propose hmc handle interesting type applicable present brief overview hmc variant underlie idea norm type spherical augmentation hmc constrain evaluate method section devote discussion direction hmc improve upon walk rwm propose distant current accept distant proposal numerically simulate hamiltonian denote denote common symmetric definite often convenience hamiltonian define density sum hamiltonian evolve equation available need time sake numerical usually solve system metropolis reject metropolis acceptance although hmc explore rwm geometric hmc riemannian use position explore sphere hamiltonian riemannian endow momentum hamiltonian unfortunately manifold hamiltonian become product consume g follow g reversible volume change jacobian determinant satisfy condition throughout term handling type restrict augmentation manifold ball hyper way target change sphere recognize chart collect sphere discard affect apply transformation adjust change corollary respectively transformation result implicitly handle impose original illustrate sampler move translate boundary original hyper constraint hyper constrain ball thus ball type rectangle side spherical augmentation use carlo particular hamiltonian however generic vector domain vector variable formulae present dd spherical like change transformation energy coordinate system hmc sphere coordinate could convert later besides handle hmc technique efficiency hmc endowed v sample define hamiltonian change chart minus derivative adjustment contribute extremely adjustment adjust integration v hamiltonian recognize standard hamiltonian augment due invariance appendix velocity c z hamiltonian g equivalently symbol preserve hamiltonian approach euclidean avoid assumption hamiltonian split lagrangian follow appendix tangent energy riemannian circle sphere analytical detail define simulated discretization size improve computational efficiency rotation proposal show henceforth h accept dt hmc metric sphere g start dt hamiltonian change potential energy round spherical adjustment minus leave volume adjust estimation integration x xt recognize hamiltonian explain invariance see
generate output input store across machine could represent structured consider communication alone work training support inspire constrain alternate multiplier admm primal theoretical show convergence solver method training model organize briefly introduce implementation discuss present follow input together minimize loss limitation focus well applicable still valid constraint view set write one respective solution satisfy fix method iteratively work well iteration work work current work distribute store call instance store index inference call optimize inference consider round single cut plane inference small work working dual inference multi core parallel challenge iteration use compute use communication form box problem apply distribute box constrain decide communication eq tune decide decompose sub locally solve exponentially machine coordinate machine inference two scalar feasibility directly base admm reformulate unconstrained mention split minimization maintain update require problem intuitively solve problem regularizer close equivalent penalty function minimization solution environment summation require add average consensus consensus outer update parallel weight equality augment lagrangian multiplier add objective duality saddle substitute next consensus sub problem convergence require convex converge consensus convex satisfy admm converge parallel many rely select sub problem work solver communication machine depend usually grow binary depend decide solver see feature index inconsistent across part speech pos machine observe order machine round incur huge communication overhead tackle issue hash strategy use unique function feature weight vector task hash use string integer environment note hash structured task distribute quadratic admm study extensively use primal rely several fortunately leverage framework modification non method dual outer affect costly many consume essential low structured perceptron method major optimizing perceptron simply guarantee converge single also effectively inference require modify minibatch update machine share unclear parallel part inference setting assume communication overhead whereas communication substantial environment core significant speedup expensive two pos parse dp pos label aim tag sentence tag tag speech pos viterbi word tag dp sentence structure syntactic parse formulation high liu evaluate parse score word correctly portion test machine result reasonable quickly machine perceptron train separate final eight conduct experiment implement set split eight partition fast method inference round communication improve realistic time parameter inference communication identical communication round admm ten speed tune affect fine set use solve
membership membership membership component update respective minimize log likelihood due accounting partial membership minimize maximum update seminal minimum message length mml message mml model mml scheme encode generic summarize encoding encoding message length encode cumulative fisher summation ji j encode message goodness negative mml minimize message membership update respective mml estimate converge model need estimation consequently quality mixture thus reliable tradeoff balancing aspect determination mixture evaluate mixture function balance quantify bic integrate complete mml parameter far criterion mml incomplete address tradeoff mixture criterion mixture vary component score convergence effort get optimum issue arise play role split merging component enable optima notable amongst mixture select two split leave component candidate depend simplify mml bic facilitate mixture component chance like scoring lack mixture fit explain address limitation propose conjunction comprehensive mml approximation infer establish outperform widely mml adapt modelling idea perturbation merge improve assume mixture child locally optimize mixture message update separately split merged operation merge great message length consider possible operation give component mixture good chance none perturbation explain split critical lead desirable splitting mean reasonably unchanged subsequently perturbation result improved component mean direction work directional three spherical describe moment estimation minor axis submatrix dispersion root equation hence maximum note minor axis parent align onto major minor axis plane split align op standard let sphere co second child angle mean parent start child component locally child serve parameter mixture component membership adjust remain component estimate em choice merging determine closeness identify close component kullback consideration membership component weight membership respectively em merge analytical form two kl ac respective normalization constant analytical b spherical find leverage state splitting merging attempt optimality mechanic suitable method refer reader example angular axis green mixture heat figure visualization spherical co transformation elliptical contour surface shape pattern begin split component child initialize mean explain child optimize two message bit ht merge figure optimize child subsequently optimize result component use algorithm illustrate splitting splitting mixture note split different show em case em state mixture improve amongst perturbation split splitting component integration black unchanged colour merge third carry figure depict split merging component splitting initial mean child separate optimize message length improve mixture merge appropriate close accordingly g perturb step ht explain message mixture component increase mixture overhead mixture weight mainly message correspond optimal associate examine variation beyond start mixture total message length reason decrease increase curve marginally increase thus encode parameter affect minimal gain log increase effectiveness ability mml sample procedure moment ml mml version form parameterization correspond parameterization version show inconsistent dependent mml expression likelihood estimate optimize require metric evaluate kullback divergence various compare use statistical mse decompose significance value figure ht considerable compare mml great map mml mml kl divergence compare moment compare bias mml mse mml tradeoff lead bias mml also proportion map mml traditional considerable especially size bias correction mml reduce moment mml base estimate bias far mml estimate competitive parameterization inconsistent therefore avoid mml regard parameterization applicability mixture directional chain atom position protein atom sphere radius atom co determined atom set form directional protein protein consider publicly comprise pair directional describe mixture distribution previously explore concentration use mml taylor mml truncate comparison work equation estimate modelling employ determine optimal employ mean current reasonably apart good chance form terminate iteration involve split merge model terminate iteration iteration mixture merge appropriate mixture terminate perturbation result search begin component split merge observe curve component mixture behaviour component increase method infer rise perturbation final characterized behaviour iteration search steady increase message length series characterized component message decrease search terminate increase dominate increase mixture mixture effective visualization component include contour sample heat concentrate characterize typical component observe component compare mixture model model mixture protein directional modelling reflect explanatory power mixture parameter complex show encode part bit message hence gain cost length lower serve well explain component mixture great difference gain bit low message length address mml htb c length bit million bit circular contour shape contour contour onto contour shape depend define explanatory enhanced compression descriptor directional protein previous descriptor base uniform sphere due mixture nan description provide model use offer oppose angle equation surface successive corresponding mixture encode accounting distance atom total obtain descriptor infer translate enhanced compression mixture bit compression follow application protein directional demonstrate mixture model nan mixture component million bit mixture mml computing mixture separately encode mixture mml use two introduce constant parameter negative aic bic modelling criterion follow heuristic mml perturbation mixture determine criterion bic involve estimate mixture em algorithm note mml em section mml aic em ml ml estimate approximation mml mixture mixture bic mixture moment resemble mml estimation mixture bic employ change evaluation c mixture maximum bit bit traditional aic bic minimum reach value behaviour initially decrease likelihood difficult appropriate trend distinguish moment log huge amount part message mixture encode length apparent moment ml mixture length ml difference encode mixture obtain unlike aic bic mml criterion also component mml htb htb aim project traditional aic evaluation mml conjunction search amount empirical vary range fixing change infer mixture base mml see mml trial agreement complete aic result great distribution unit sphere theoretic square traditionally moment maximum mml transformation estimate mml traditionally use conjunction demonstrate model protein spatial modelling mixture describe descriptor modelling task biology minimum message length von fisher modelling statistic grow science biology directional directional type surface compact von density comprise distribution unit sphere respect model directional generalize fisher form characterize value parameter scalar compare distribution factor additional directional statistic pose difficulty also lack order achieve balance suggest alternative relatively interpretation equation orthogonal vector axis spherical analogue serve surface argue unimodal gaussian spherical surface value scalar entity infinite importance angle mixture joint task bioinformatics complex mathematical estimate approximation considerable effect mml reliable parameter ml mml unlike ml mml map mml invariant state estimation use parameter mml take account parameter state determine mml framework inference part encoding encoding parameter select parameter base mml mml wherein demonstrate mml outperform one directional datum demonstrate reliable mml base traditional mml model mml well traditionally also invariance mml compare model protein distribution serve mixture model directional organize mml framework highlight explain associated construction likelihood parameterize mml implementation constant derivative mml base describe section experimental discuss respect selection paradigm criterion overview mml develop per probability per event code event require shannon comprise bit result give odd hypothesis rigorous compete message vary explain may state come message goodness fit generalized set reasonable derivative likelihood mml free lattice quantization comprise mml difference ml mml ml effect consider minimize map estimation self state precision incorporate determine volume region center multiplied probability use compute message encode precision comprise directional parameter parameterization relatively along axis rotation matrix transform orientation axis support co determine show bring plane operation transform axis subsequent rotation angle major minor axis plane orthogonality preserve orientation angle axis plane matrix scalar concentration contour spherical visualize relate allow correspondence elliptical understand vary spherical contour contour moderately major figure maximum ml require density widely estimate formulate estimate derive moment moment alternative adopt n normalizing moment step align co transform axis frame angle variance axis rotation define direction require low submatrix decomposition subsequently dispersion rotation orthogonal transformation transform axis standard coordinate axis transform axis moment estimate moment eigenvalue quantity simultaneous equation conjunction estimate limit approximation accurately optimization library obtain estimate minimize solution numerical routine root start previously moment typically explore compare distribute n non space drawback formulate give space density derive prior angular density uniquely define direction mean spherical surface angle determine orientation major minor angular definition range conditional joint density reason consider parameterization map invariant space characteristic inconsistent parameterization non linear prior j parameterization f likelihood across different estimate various dataset show random size parameter library conjunction derivative optimization maximize observed counterpart obtain ideally maximize
massive long term twitter explore via stream reasonable power adopt twitter share facebook friend activity find twitter stream without content devise reconstruct tweet study individual response suggest average exposure production tweet occur average exposure positive relationship response whose week experiment highlight occur highly differently different highly equally likely adopt interaction evidence yet avoid consequence experimental reliable forecast circumstance sentiment text tweet positive sentiment score provide text twitter linguistic rule correction medium tweet positive range neutral single sentiment express tweet define positive sentiment tweet range sentiment tweet tweet neutral bar neutral neutral tweet negative tweet author tweet baseline previously neutral tweet baseline proportion bar sentiment tweet collect consist least tweet provide week twitter collect user tweet produce week produce hour precede tweet tweet english contain medium video finally tweet target annotate justify english medium attribute sentiment tweet choice limit last limitation tweet week discount reconstruct user sample medium study tweet within description user finally separate tweet neutral focus intensity facilitate overall piece intensity hypothesis medium various pass via online typical interaction essential ingredient reconstruct tweet allow whether correlate response subsequently study purely observational control differently sentiment neutral tweet exposure tweet less one fig positive tweet tweet neutral positive amount exposure tweet model notably sentiment tweet neutral one perfectly neutral observed positive test significant neutral far illustrate narrow negative positive negative response response positive seem neutral particular another call tweet sentiment tweet formula fraction tweet large since tweet produce precede hour publication allow represent calculate tweet stimulus associate tweet value fig response neutral value stimulus illustrate show stimulus response stimulus stimulus response suggest strongly stimulus follow positive stimulus neutral neutral individual tweet cumulative tweet affect user previous measure tweet stimulus focus tweet user tweet produce prior tweet proportion neutral proportion sentiment tweet user tweet proportion determine euclidean distance sentiment determine nature stimulus exposure neutral less equally likely adopt adopt great tweet similarly distance tweet vice versa tweet fraction tweet fig tweet exhibit content presence select fraction tweet positively affect positive versa adopt high one perform twitter facebook control exposure design nan highlight user history response week number insight exposure negative exposure response suggest divide significantly adopt suggest observational possible separate entirely dominate observation user vice real mixture experiment need social medium channel million individual day produce daily medium micro communication therein facebook spread typical person massive content unknown consequence use content
obtain single source distance improve cv computation distance computation simple source computation source distance storage inherent query graph coefficient node sampling probability usually sample choose adaptively space distance size metric running dominate computation size base cv suffice node need computed randomization introduce precisely randomization ensure obtain probability relative exceed polynomially node identify initialize node dominate computation node graph short path uv query member compute apply computation query show include either cv cv relative polynomially section establish base regardless lemma close algorithm distance l since substitute eq z substituting contribute v z k consider situation uniform expect suppose uniform node close position sort uniform particular iff choose accord z uniformly z conclude per precise definition close distance accordingly space median within median close close node within mean distance probability proportional distance substantially large show universal contain node show partition node remain definition u z v nz corollary lemma grow concentration conclude apply chernoff node exactly contribution variable range chernoff expectation node equivalently high apply estimate relative polynomially query metric space would mention way step involve source ii alternatively polynomially small placing source computation contain uniformly select polynomially error mean polynomially probability polynomially establish identify polynomially distance provide important property relax high computation base total small time sample query relax definition claim property quantile distance distance close v well verify yield universal polynomially distance computation point sample quantile quantile candidate half nc polynomially error establish claim useful single computation apply coefficient apply return ensure cv correlate must amount computation polynomially polynomially single source computation base sampling obtain sample base probability single source computation treat apply space start overcome explicit calculation first nearly little satisfy p c replacement compute unbiased cv definition therefore direct hoeffding inequality size obtain probability relative exceed small efficiently like express probability v uv relaxed desire guarantee detail use pair condition within polynomially next subsection efficiently implementation fairly completeness independent replacement obtain independently replacement arbitrarily order associate interval randomly draw sort pass sort completeness describe sort set operation draw independent iii exponential transform sum hence precision identify scan point randomly point take jump sort scan array point size randomly bad per improved tree total contain clearly practical per cost search universal computed set claim computation upper say quantile get eq side side far thus wu c nu eq use rough accordingly definition apply weighted estimate query guarantee algorithm uniformly node factor size vertex decay lemma pair use ideal sort high bad adaptively another computation compute node sort exact consecutive mention completeness provide apply metric compute iteration fraction current exclude increase point linear arbitrary probabilistic create copy times max absolute weight g copy weight ab bc ac complete truly algorithm compute short show detection undirected triangle negative number triangle detection computing instance triangle construct problem union copy correspond complete negative claim path cycle see direction contain v v correspond regardless shortest path correspond shorter path get u triangle centrality distribution average point upper value low satisfy inequality z half node spread centrality consider node point whereas isolated network contain separate centrality comment cv point diameter restrict single still weight entry obtain respect nonnegative symmetric observe close row member therefore universal realize triangle absolute embed universal intuitively size distance inner product stem something whereas large like centrality reflect consider weight particular sampling extend node equal respect point perform bad median exact median separation fewer adaptively determine approach node maximum apply skewed hence easy sample well median z median working simplify necessary estimation increase way smoothly minimum tracking stop correctness note estimate summary weight weight even skewed cv factor single probability term space sample depend surprising linear section thm conjecture thm thm thm thm thm corollary thm com il cs ac il conference volume query fundamental classic centrality popular measure study social novel insight relation via fundamental centrality computation metric preprocessing estimate use computation error ensure error exceed polynomially structural centrality study measure classic closeness centrality term closeness centrality closeness centrality centrality reflect ability fundamental cluster distance centroid datum consequently relevance relate distance distribution cluster advantage parametric similarly near knn distance knn target outlier carry incorporate classifier demonstrate uci repository accurate knn notion centrality extensively aim provide facilitate metric correspond length path input specify graph round distance node input mention difference application relative imply distance centrality list sum give centrality metric seek computation node worst seek computation computation perform nearly unweighted pair suffice sum also distance node suffice relative scalable easy path tree contribution weight estimate statistical suffice ensure relative polynomially query probability least compute probability sample expect sample
every hx h part turn part depend set may combine guarantee error aggregation lead begin depend probability satisfied turn invoke fact every clearly nontrivial class g dictionary assume type equivalence useful ball member fortunately concerned inequality contraction principle bound establish side involve contraction high symmetric multipli concentrate exponential obvious contraction base totally definition theorem corollary theorem theorem mathematical institute national act fast happen always attain procedure procedure attain rate norm span square integrable space let predict effective cost measure squared predict behaviour minimize risk take distribution mind sake minimizer exist choice predicting present extend loss function sample independently hope produce random minimizer integer learn set potential class respect endow find feature study rate scenario bound alternatively rapidly decay tail subgaussian however explain fast function unfortunately mean often common fast scale rate imply reality straightforward construct simply matter size correct happen mid procedure rate suitable see precise statement thus reasonable fast rate must reflect highly ambiguity fast optimistic roughly location accurate intuitive optimistic ignore mid time procedure attain optimistic location select belong framework function survey broad restrict goal aggregation attain optimistic minimal aggregation require denote avoid confusion optimistic risk erm outline follow description analysis specific fx minimizer minimizer q one look note every optimistic straightforward important close closed product target independent cm satisfie rather low class small purpose norm class process index star f f x I rich end belong convex star shape around symmetric belong measure indexing statistical view indexing form detailed explanation role definition define reason consider happen coincide cm parameter multipli excess square multipli component satisfie term erm perform optimistic indeed base straightforward verify convexity interested satisfie exist least optimistic rate optimistic ball constant specify cm happen upon select extend erm optimistic assumption include structural assumption optimistic learn sense good let wish occur achieve achieve optimistic thus may occur stress attain optimistic procedure may depend let assume every notably side central median class member norm result find consist dependent way behave apply erm element contain observation follow always f lemma whose excess relative small type perform somewhat one construct know oracle independent case q convexity apply recall identify way class empirical estimating mi nx hold right nonempty matter ready aggregation eq w one follow right constant event cm devote event original dictionary unlike aggregation carry excess eq claim apply n f therefore every every thus q thus verify complete proof focus show provide properly constant main low f ball obtain rather every follow proof thus approximate truncation probability absolute next claim observation similar constant note set class abuse write minimal norm th minimal every minimal observe large n fx union equivalence absolute constant end fix observe contraction argument g q fix name function f f mf mt j u ni v star shape standard verify sl k note provided therefore specify eq rd lemma least every f star sl homogeneous star ball suffice ensure distance member behave tail certainly mean distance unless obstacle use median functional show little need small estimate independent copy application independent copy applying depend eq
attribute object multimodal merge build softmax layer multimodal predict word share rnn model sign sentence stage image start sign pick word softmax generate weight two layer word embed use compute wise one note calculate multiplication multimodal softmax softmax activation role encode back softmax operation equation dense dense word reduce parameter decompose multimodal activation intermediate strategy accordingly element wise scale tangent well view intermediate multimodal softmax sharing layer rnn model without increase concept connect softmax concept suppose meet sentence annotation consume unnecessary whole access fine model cause concept decrease originally concept concept learn learn specifically word cat associate cat fix sub tendency word think similar associate word new datum tend overfitte new concept enough intermediate change baseline activation layer original bias fix call strategy sentence word embed lstm layer part speech image example description play cat contain sentence playing model see word cat vision imagenet category useful classification vision combine vision effectively new annotation image novel learning release image annotation around description nc cat cat accord instance annotation check whether description cat validate validate concept tb nc cat nc three dataset learn concept ms construct contains create derive contain activity concept sentence label sentence describe annotation ms sentence annotation sentence firstly imagenet train secondly rich description compare cat concept nc annotation figure randomly separate table issue dataset base add test pick cat denote testing add image organization nc three publicly available publication encourage future description concept b evaluate overall generated previously conduct comprehensive evaluation concept cat cat dictionary follow sentence sentence represent testing nc set always indicate balanced measurement show concept tb share original layer model layer well performance scope validate task cat word cat term mean layer activation affect deep deep deep represent tb ccccc c nc b cat deep cat deep datum add sample base image concept stand set image image concept base set deep achieve improvement novel concept test reach demonstrate word deep successfully exist sufficient similar helpful easy whether shot randomly image cat randomness training limitation consistent metric full indicate blue red nc dash line line shoot scenario draw show performance train concept nontrivial metric b deep deep l deep deep ct base deep model nc firstly counterpart secondly rarely imagenet pre train vision deep cnn describe concept requirement annotated nc ms effect decrease cat table nc learn semantic show successfully new word sentence low concept concept ms cat word embed novel vector generate visual learning sentence task description image describe method allow image novel large concept particularly share validate university com task description linguistic new module improvement share scheme suitable task prevent overfitte novel concept dataset construct task experiment effectively learn visual without learn concept observe description parent slow accumulate enough concept children quick rough meaning sentence previous word describe vision field handle new sometimes novel learn category add novel however concentrate mapping novel computer task concept concept child seem call visual sentence novel address large amount allow dictionary extensive concept concept validate rnn base large object provide method multimodal e retrieval method recent multimodal rnn performance task sentence model three use cnn semantic language
inequality get lipschitz use introduce probability expectation really multiplication somewhat proof attribute significance coordinate property apply definition consequently since enough clearly belong write therefore assume know estimate q trivial fx overall b k second get assume say k get careful aggregation contribution version totally symmetric submodular f li corollary si jk polynomial several notion value boolean function influence total finally apply generalization influence degree degree corollary lemma far learner pac independent agnostic value eq learn access draw remark respect use range pac agnostic learn rely agnostic degree excess hold arbitrary uniform model pac submodular function two influential function fit example simply variable degree plug uniform example output time use find influential ensure variable therefore analogous degree coefficient substantially simple proceed exist set set define outside namely equivalent need bind number degree fouri coefficient submodular relevant submodular random example least satisfying run use exist close distance triangle suffice prove j index simply estimate q within lem time application chernoff obtain desire follow least run influential completeness corollary agnostic submodular submodular exist learn bound submodular self exist monotone stick variable fouri coefficient bind careful low concentration stick closely relate majority know see kx otherwise bind sec ks w j k dt use condition pac submodular use result submodular variable learn class bit particular random boolean small every boolean monotone submodular function time random hx ok construct middle whenever notice see example give hx fm tx tx convenient switch notation indicator set verify j fs j fs fs fs fs fs monotone function algorithm pac submodular boolean access translate overhead use approximate overhead choose statement function mapping formula give verify corollary low reverse sensitivity spectral noise sensitivity sensitivity noise satisfie follow result w exist obtain low every exist linearity coefficient differ uniform error theorem slightly weak learning argument imply prove pac function constant small pac reduce use achieve imply example low spectral concentration submodular function self function construction code briefly concatenation string bound self point therefore exist differ fact property hamming point q avoid calculation hamming code via bound algorithm learn query reduce variable bound uniformly otherwise function example could obtain use bounding acknowledgement thank useful discussion convergence generalization unknown value rademacher rademacher variable rademacher view complexity self bound rademacher show small class monotone self remove definition since affect membership naturally ia fact fractional equality proof also study geometry place rademacher use definition clause negative lipschitz monotone self f bounding claim submodular equivalently play role combinatorial algorithmic theory polynomial norm fourier concept approximate norm function exist degree improve well previous monotone submodular technique reveal nearly learn fouri hypercube notable object research devote boolean hypercube attract algorithmic game analytic apply arise rich real focus property fundamental value analog play combinatorial game game submodular find function return algorithmic characterization rademacher play role bounding class well function polynomial define standard polynomial degree degree concentration low well analysis number algorithm rely crucially low polynomial application privacy approximate norm degree analysis noise establish degree subsequently value decision also give bind submodular suffice addition show submodular self bound learning class function notably motivate polynomial approximate know approximation approximate low meaningful variable learn work investigate denote bound submodular bound via suggest self bound low picture substantially rich class complexity bound summarize submodular prove submodular k k f c show general class constant total namely bound match upper small approximated polynomial free degree constant bind approximate within improve lower prove even total fx upper spectral total influence discrete measure pair particularly set drop due presence submodular function constant careful degree self fx monotone random submodular behave individual threshold norm result everywhere show namely rely boost prove submodular partial warm submodular substantially simple totally symmetric function factor exponent quite actually concentration require totally additive applie error new translate use brevity describe model improvement query ask low learning algorithm exist output far use run use complement upper nearly tight value see statement algorithm proof bind learn testing pseudo boolean unknown function submodular pac value query use value define submodular combinatorial optimization lot algorithmic expressive self monotone submodular natural necessarily many especially approximation shift invariant submodular range way equivalent nonnegative monotone monotone appendix full rademacher vector well tool learn equivalent monotone bound broad broad generalized function bound bit lipschitz condition normalize value bound possibly monotone submodular function equal coordinate function monotone non decrease assume bound learn consider error relative negative submodular bounding scale additive scale within rely value combination expansion exactly df w iff df fx approximation function e particular lead seek degree define follow f k maximum define clause arbitrarily fix achieve maximum since monotone fx w w fx fx fx j although bind actually tight prove choose coordinate order fx fx fx ci ci ci ci ci
feasibility derive utility homogeneity example unclear explain explicitly inter variation demonstrate size allow though article believe regularize recent advance collect adequate sample ratio necessarily estimate zero often covariance still heavily biased size achieve unbiased trade therefore focus aggregate achieve allow stable toolbox sophisticated simulation experiment explore serious helpful much herein marginalization ease drop subscript eq recognize unnormalized inverse n reciprocal normalize simplify concave analysis concavity two proposition consider mixed concave sum since gamma well log multivariate log characterization log proposition lemma see irrelevant assume hold data generality continuity due monotonically increase positive definite partition possible infinity since combination go definite log hessian log conclusion directly reference page compute derivative q elsewhere derivative straight structure I hadamard vector space expression definite need hold point z stationary zero multiplication substitute ax inversion lemma whereby need ax yx px wishart compute straight maximized recognize model derivative zero yield imply I number estimate scatter rgb rgb lemma fp science technology innovation jensen university covariance class meta meta applicable intermediate basic compare homogeneity fundamental correct mle poorly become ill condition approach central utilize covariance standard analysis pca linear discriminant quadratic discriminant example gene fmri many expand list publicly sample become effectively ridge precision still dimensional scope total exceed exceed major cancer genomic laboratory accounting assess motivated interaction contain gene method limit genomic ordinary various inter treatment effect effect meta inter analysis study model observation multivariate hierarchical ip probability generic notation pdf respectively inverse freedom pdf generalization inverse wishart exist wishart control inter homogeneity correspond homogeneity versa around inter variation homogeneity preferable much especially near therefore parameterization remainder x pi study wishart wishart evaluate cf arrive eq expect state likelihood concave concavity proof defer fix definite lemma state therein moment pool observation know pool scheme maximization derive compute conjugacy inverse wishart expectation wishart likelihood scale precision appendix current update inverse repeat iteration maximum derivation defer keep iteration subsequently numerical define estimator pool estimate utilize log implementation yield disadvantage identify maxima saddle likelihood class heterogeneity construction large homogeneity estimate homogeneity homogeneity hypothesis equivalent wishart become test leave hypothesis however nan simply number acceptance region likewise addition denominator add approximately intra intra class know meta well determine ratio variation abuse let interested quantity variable proportion study law variation equality agree ij need fourth observation wishart cf imply continue term substitute expression naturally straight plug estimator gene though variance exist precision stability datum variability contribution homogeneity low suggest high covariance color identify module color key edge dendrogram color color weight plot outline simplicity employ analysis study identify correlation agglomerative hierarchical cluster linkage minus arbitrarily height produces name color heat hierarchical module show graph hierarchical top gene cluster gene yield repeatedly suggest homogeneity select sd fit next module relevance use module base go term significance module highly c cn cn cn cn chi col cd cr cd col col col il col check identify overall os r treat module cox os module module matrix module represent module gene interesting module arise result survival analysis mark identify patient outcome manual screen module l degradation suggest activate degradation possibly express activate activity system il also link pathway associate poor outcome degradation central study link poor disease enhanced cancer thought role interaction manual screening go module meaningful covariance application discriminant utilize discriminant suppose variable suppose lda e assumption yy intermediate forward implement derivation analogous determinant simplify generalize becomes consider cf procedure correlate design perform similarly bad belong multinomial class round observation
high one regard period period long due change number period equally long totally uninformative one big figure seem uninformative prefer focus divide depend multidimensional example volatility depend price associate period period six frequently forecast day use viterbi evolve visit evaluate reliability predict come mention method idea define day one assume look sometimes wrong period divide vertical dotted line day make consideration day general secondly stable look graph day day really general say due change lead moreover limit instability publish researcher similar past day average average day plot influence strongly quantity give formulae book forward variation consecutive wider predict continuity keep amplitude variation error small analyze type work noisy right behavior recognize temporal time three forecast daily unstable mathematically forecasting problem value inaccurate respect price viterbi try mixture appendix want construct show divide depend hide viterbi belong state histogram gaussian try histogram htbp model see associate seem collapse mean around htbp model leave right mixture histogram clearly leave totally uninformative base mixture portion htbp figure forecast use ergodic show one closure vertical line testing period htbp htbp figure forecast consecutive normalization use model notice state price price lowest associate consecutive complete htbp mm cm remark hmms forecast hmms financial depend subsequently analyse previous literature put primary b phrase market big market change easily access market online call trading rapidly spread trading consist software decision market lead interest intelligence finance neural network hide hmms focus paper hmms forecasting publish md study paper forecasting analyse critical daily create set value year build test behavior impossible predict organization description markov firstly choice issue relate lastly understand hmms explain thorough performance datum conclusion mention finally hmms present relate initial chain process matrix markov q describe visit emission density distribution hide observation continuous emission gmm constraint let denote hidden emission gaussian observe process price daily determine dataset structure way turn state ergodic look put financial context price probable adaptation standard us ergodic right implementation step divide divide consist group assign gaussian worth note use transition matrix choose uniform group zero transition transition upper triangular leave right prove wu maxima three researcher great scientific advance wu performance combine start idea hide actually state idea prototype abstraction build method gap optimality try
prove slightly weak uniform ergodicity base kind lyapunov perturbation error lyapunov take supremum take obtain setting lyapunov lyapunov provide transition lyapunov uniform ergodicity constant requirement constant ergodic measurable lyapunov q constant vx p suffice assertion state requirement perturbation play first ergodicity quantify second appear norm measurable finite restrictive classical perturbation chain see restrictive perturbation ergodicity imply ergodicity ergodicity relax requirement lyapunov p ergodicity imply ergodicity limitation separate role uniformly lyapunov uniformly ergodic measurable lyapunov eq vx know fix real consider lead q finally yield proof complete distribution lyapunov far comment kernel calculation interpret family perturbation state converge bound begin study consider quantitative perturbation prominent namely hasting langevin let autoregressive q variable I say moment easily transition exist metric e assume wasserstein distance l obtain imply give inequality distance emphasize estimate w obtain eq assume thus eq bind autoregressive also ergodic imply analyze value norm difference measure inequality perturbation metropolis hasting analyze either hasting wasserstein ergodicity ergodic kernel uniformly ergodic interested realization serve proposal metropolis define probability step form work sample define accept else reject proposal unable evaluate force behind hasting algorithm random variable algorithmic work independently else acceptance still hold acceptance wasserstein transition transition form acceptance random independently uniform arbitrary fix variable analogously dy dy within eq wasserstein perturbation bind acceptance satisfy lyapunov p essentially acceptance probability numerator separate part integral remain suffice make subsample moreover choose arbitrarily obtain combine eq norm geometrically sufficiently main geometrically measurable denote probability transition ergodic measurable lyapunov number e possess stationary set use easily corollary lyapunov function assertion corollary wasserstein last statement follow corollary instead many corollary ergodicity imply satisfie drift argument proof satisfy sufficiently ergodicity metropolis hasting algorithm langevin implementation overcome approximate langevin mainly base noisy langevin gibbs field define lebesgue langevin euler discretization sde langevin diffusion value sequence variable diffusion stationary stationary say depend random distribute normalize carlo substitute define markov langevin algorithmic draw call independent fact langevin lyapunov l stationary kernel argument irreducible lebesgue weak thus set stationary ergodicity q statement consequence right side assertion fact obtain perturbation langevin number independent determine ergodic number remark state result lemma thm theory address question markov reflect difference flexible two markov satisfy wasserstein ergodicity condition monte lyapunov estimate geometrically autoregressive bound show quantitative estimate approximate version prominent metropolis hasting stochastic langevin carlo mcmc computational respect unnormalized method application available demanding see small two expensive likelihood contribute evaluate approximation rely moderately random subsample value naturally cut metropolis hasting budget bias bias discuss understand behavior bias bound ergodic chain implicitly appear ergodic restrictive markov chain provide perturbation wasserstein lead mcmc wasserstein distance turn consequence wasserstein geometrically ergodic chain ergodicity extensively use exist generalizing finding noisy gibbs integer space mapping equip measurable another markov ideal like simulate perturb actually transition wasserstein distance marginals wasserstein transition kernel wasserstein ergodicity condition assumption curvature lyapunov perturb transition number eq constant lyapunov wasserstein parameter weight supremum lyapunov wasserstein always satisfy suitable q denote corollary autoregressive former turn perturbation chain kernel geometrically ergodic ergodic suitable drift ergodicity wasserstein ergodicity moreover pair measure wasserstein observation carry wasserstein perturbation bind ergodic hasting particular geometrically ergodic markov geometrically ergodic distance quantify replace weak easy control theorem perturbation lyapunov distance perturb langevin refer concentrated measure eq next equip property linear homogeneity obvious fact form nothing transition define notation measurable whenever exist interpret introduce curvature convenience reader finite interpret property argument geometrically ergodic transition finite reverse complete linear know stationary additional assumption eq estimate trivial metric indicator set dx apply triangle variation obtain consideration exist ergodicity impose metric contain ergodicity exist wasserstein ergodicity eq resemble impose wasserstein ergodicity wasserstein distance high transition stationary approximate use chain namely initial markov chain call whereas perturb bound difference suitably step wasserstein perturbation measurable lyapunov p vx induction w allow existence lyapunov weak uniform ergodicity
successive behave allocation like example recall budget must limit arm budget concerned identify arm recall merely arm however impossible final converge theorem successive allocation set successive recall uniform allocation factor stress merely fall back ensure bad uniform successive strategy practice observe experimental compose minimize x xy mp output algorithm train validate hyperparameter I mf I selection minimization necessarily generalize give output iteration compute eq assume necessarily issue loss sigmoid put hyperparameter namely arm develop section hyperparameter uniformly log scale within region valid range sufficient cover grid grid source package bandit arm evaluate leverage nature machine robust fashion review relate context show hyperparameter optimal without explicit function reject hyperparameter assumption select per song prediction sample dataset year zero variance small error fast term plot successive successive top exp validation require allocation successive svm rbf train hyperparameter uniformly trial hyperparameter allocate calculate use store magnitude fast successive respect successive iteration plot ht consider bi objective objective hence hyperparameter rank choose scale result arm trial observe two particular successive successive theoretical present direction analogous arm variance switch cost intermediate memory resource various balancing bayesian considerable notational ease infinitely limit read single element consider singleton successive algorithm identify arm consider subset one complete showing contradict prove state run loss last envelope involve arm almost integer effect favor simple rearrange interpretable uniform follow arbitrary bandit literature objective good identification stochastic framework know set leverage nature algorithm cast hyperparameter stochastic good identification resource promise hyperparameter setting accuracy method become widely simplify accurate hyperparameter learn search optimize since many nature working scale evaluate intermediate partially example show hyperparameter via stochastic ask hyperparameter vast black box fully make attempt intermediate work form simple build work multi armed arm hyperparameter intermediate loss widely applicable bandit solution remarkably exist bandit fail exist fail two main loss monotonicity non obtain costly case compute drastically bandit identify set confirm theory relative standard baseline applicable good identification behave source subset paper set survey work suit set baseline experimental bandit identify average versus try maximize latter analyze arm objective stochastic suit cumulative necessarily suit good arm h observe recommendation receive continue arm armed stochastic multi bandit choose loss get versus good arm future loss arm play loss adversary generality start game propose arm probability adversary stochastic loss something behave minimum return arm bound decrease novel I hoeffding stochastic increase tell nothing even decay consequence reject possibility arm arm good attain despite challenge measure idea stochastic activity decade major branch set budget arm total algorithm attribute amenable propose successive non successive fix input minimize many decide suboptimal long discard successive elimination elimination implicitly e ucb generally exhibit undesirable behavior observe c observe loss elimination ucb exp total arm cost partially validation market probe case assume horizon budget total observe arm include popular minimize cumulative practice successive particular attractive along observe loss budget successive originally propose novel arm predefine bad repeat arm initialize k k budget attempt progress effectively parameter notable worst ever place budget total sample eq next identify without assume wise addition order final relative reasoning heart theorem tb arm return merely arm
turn refined partition local modularity sophisticated heuristic modularity approach biased tackle correct take degree mix usually euclidean space edge pair draw maximum markov position euclidean network similar latent practitioner extend representation vertex mixture approach bayesian inference look block look sbm latent distribution parameter ik ik node extra sbm heterogeneous account generate scheme well posterior adjacency parameter dependency derive tackle variational alternatively gibbs sample even unfortunately tractable either criterion aic ic derive criterion case sampler development sbm deal take assume subgraph look cluster belong vertex cluster relation vertex involve multiple multinomial distribution sbm replace bernoulli allow belong last order dynamic evolve time temporal hmm linear usually focus choose social affect future unobserved structure like highlight approach homogeneous poisson continuous removal occur approach graph build large literature machine target case numerous graph point least instance undirected biology neighborhood use early appear tendency become decade single quite common main idea extract whole graph supervise two adapt graph distance couple technique numerous adapt specific graph neural neural process vertex leverage structure maintain already process neuron solution consist building dissimilarity graph base relational difficulty detect far expect class belong complete bad complexity determine subgraph np nevertheless numerous solve problem computation introduce substitution series individual cost least costly transform graph kernel reproduce generalize numerous walk choose relational machine mean surface vast know extract efficient topology lead supervise exploratory analysis label concern vertex vector via challenge consist vice description issue two source medium temporal graph ignore issue propagation lot task actor another generally massive spread decade object complex etc etc ignore numerous cm universit paris f paris france commonly object straightforward formalism many scientific computer give introduction rely include supervise algorithm usually cluster focus topology static graph edge evolve deal evolve infer numerical characteristic balance source challenge especially locally context object interest give full possibly complete task produce cluster graph rather associate model scientific pathway science tie actor web network powerful tool extract complete survey refer highlight share goal cluster finally technique take often indicate short generally unsupervised vertex share connection general vertex define topology top vertex community appear densely connected group effect discover maximize score
hundred set human activity segmentation comparison dissimilarity poisson process measurement vary detect occur task show contain allow performance several point theoretically block regard dpp block semi psd assume pi psd conditional proof trivially one hold definition proposition pt minus plus plus pt exist map need present great application successfully difficulty detection new name preliminary point candidate study candidate dpp dpp conduct estimate effectiveness demonstrate five process elegant model selection quality diversity dpp subset q write quality item angle diversity feature measure assign quality diverse problem attract note inference greedy submodular decoding minimize exist computation take time become become nevertheless almost block fig inference replace inference kernel size item similar different away manner mainly aim detect period refer segment candidate change candidate quality diverse state g preferred point dpp purpose meanwhile almost block g far dpp become decade broadly classify frequentist change location include run improvement e advanced monte posterior big world frequentist core testing general statistic past window move metric value threshold ratio kullback leibler divergence explore threshold determining study heuristic dominant peak discard peak close require threshold dpp metric create preliminary candidate much treat point dpp conduct dpp obtain final diversity contribution name rest organize brief give theoretical real dataset interested dpp ensemble almost diagonal zero bottom leave namely diagonal contain dpp sub matrix sparse sub index correspondingly square definite consider motivate element see dpp partition correspondingly c c tell sub largely dpp map far dpp inference mc invertible rewrite map represent area zero block recursion objective optimize depth essence ic jj I subset item perform inference c almost diagonal series inference sub kernel comment optimization sub conduct depend wise greedy achieve sub dpp dpp lemma third apply dpp unique whole block leave study partition partitioning dpp block every size bottom overlap adjacent diagonal area note partition way value obtain balance small achievable partition smaller small adjacent sub inference empirical illustration fig greedy inference realization randomly sub area next vector separately step entry kernel generate partition greedy map original use baseline run much drop within map plug play inference ask connection map dpp relation dpp successively diagonal kernel eq dpp correspond conditional belief conditional fed selection result form allow map entire kernel latter small information incorporate generate kernel conditional map average error bar input ii let dimensional interval segment explicitly denote interval new build dissimilarity arbitrary dpp popular decomposition kernel magnitude view angle diversity allow construct utilize metric candidate create move adjacent window likely value peak value select
second unconditional guarantee tolerance rely describe state suppose batch monotonicity complexity monotone success absolute combine distance estimator monotonicity apply modal sample failure monotonicity modal modal weakly monotonicity modal describe work complexity original distribution sample straightforward knowledge though sample preprocesse step sample able make many correct question without approximate whole answer completely require learn restriction strong query access original ability make query distribution outperform query per first monotone achieve latter constant query exist sampling sampling decomposition let monotone fact without piecewise following find j l index turn constant complexity already strong indistinguishable close unless take already strong guarantee namely access well detailed level idea decomposition approximate histogram certain weight get quantity carefully monotonicity gets correct ever actually occur state interval constant let I monotone time argue significantly total otherwise closeness monotonicity jump mixture start two consecutive observe loss generality one monotone close assume eq I sum satisfy distribution put monotonicity suggest immediately output sample normalize whose non increase non monotonicity monotone claim bind sample adapt consecutive c proof previous otherwise closeness consecutive interval parameter describe completely fashion failure nan sampling monotonicity idea since monotone monotone apply scheme monotone inequality derive triangle latter feature query sampling imply query dual follow fully access expectation possible cumulative usual draw monotonicity cumulative query approach group interval optimally group every coarse fine correct correct lie inside boundary inside overall monotone kind global average inside monotone boundary last cumulative access problem e contain allow keep correct implicit query coarse monotonicity optimally linear ensure weight issue go correction budget entirely used process remain effectively end match exactly correction essentially weight include weight extra allow correction whenever way first correction query every point inside inside boundary weight water employ context order algorithm use budget sure never remain portion whole distribution sampling correspond know access cumulative access close close reason hereafter entirely cdf query quantity average subsequent water budget correction back average remain increase spread uniquely explicitly cdf fourth core subroutine perform correction range average throughout element great move stay middle whose maximum px ie say spread water allocate amount would pour total amount weight move weight full I happen might yet reach list portion w distribution cdf two monotonicity sample ignore extra budget try partly implicit coarse inside monotone even process e remain regard monotonicity proceed correct monotonicity execution water procedure allocate budget pour use beginning ensure use first domain support exactly ensures pick initially weight modify either pick draw event observation process determine uniquely define query query stage identically distribute bernoulli take select begin fact monotone fact length weight also detail therefore increase moreover construction monotone within explicitly sample change water act stop prevent remain monotonicity violate consecutive guarantee monotone consecutive budget guarantee indeed budget pour uniform weight correct would pour ib distribution constant jj hard g process total variation close satisfy monotonicity separately jj negative minimize clearly moreover ps px px allow conclude process budget allocate put differently amount budget write j element execution monotone monotonicity disjoint boundary consecutive j monotone add correspond minimum weight bring total put finally main theorem taking arbitrary j distribution hereafter denote original monotone access call approach original try detect miss sample perform shall follow appear utilize subroutine miss error batch proof next describe approach influence remove weight miss model interval monotone occur interval length element monotone monotone get monotone monotone weight e I violate interval weight end domain move add distance p monotone claim draw detect exist hold monotone inspire equal take hereafter partitioning care big monotone must potentially big element monotonicity xx claim observe miss modal monotonicity far conditioning either fall close suppose accept reject soon interval observe rejection guarantee failure explain I n n obtain probability kolmogorov derive close monotone conclude finish proof apply distribution bound encounter part weight quantile estimate correct stage hereafter convenience phase pass return either monotone second observe thus support monotonicity case denote expression fact define eq claim close monotone monotone distance n interested uniformity domain allow amount task arbitrarily naturally slightly bad query hereafter construction correction fidelity closeness complexity construction term uniformity domain extend level idea first von closeness drawback lie operation argue sum exponentially uniform however get close guarantee distance precisely distance possible close get essentially enable distribution combine idea use generate coin whether bootstrappe describe arbitrarily uniform von uniformity failure cc bias coin applying retrieve truly repeat time precisely variation variable parameter bit failure take bit hereafter view group distribution close key definition exist uniformity complexity extend draw ks bind k hand side far might achieve even n exist uniformity n support p get query yet bootstrappe exist uniformity guarantee bootstrappe recursively recursive resp j applying get recurrence recurrence give upper conclude randomness improved fix determined call easy observe generalize result unknown unknown sample could conditioning rejection argue great generator let cyclic moreover non imply sake hereafter generality absolute overhead possible discussion uniformity dd generator k h correctness fact independent uniformly random kk x break find union event amongst h event h relatively ideal therefore adapt rejection uniformity uniformity uniformity cyclic query uniformity query complexity rejection identify try draw randomness provide source source random bit uniformly distribute distribution property goal similar sampling uniformity extra setting difference randomness bind min input variation sampling since weak additional sampling use extra bit unlike sample original distribution tight result particular randomness low apply extra bit conversely uniformity min also vary uniform even sample could even subroutine truly sample correct learn monotone access monotone element sample take start approximate cdf additive define fm element effectively purpose k k check quantity also access leverage output direction interest example property agnostic efficient consider use sample correct follow work set testing query either get domain query cumulative provide one sampling original black succeed precisely batch get test vote well test straightforwardly generalize fully finite denote property I unique element variation g furthermore tight inspire question one nn l nk range obtain thus notion distribution testing fix distribution access call output least relaxation item fix output concerned setting namely call dual access oracle behave cdf put algorithm precise distribution exact variant paper contain omit sake arbitrary recall obtain monotone part lem claim theorem prop conjecture claim acknowledgement acknowledgement rgb support nsf author microsoft nsf grant grant situation address act end connection utilize expand applicability property algorithm improve algorithm proper analogous cumulative obtain sampling monotonicity stronger monotone namely addition restrict miss significantly learn whether additional bit require correction process distribution bit sample methodology work gaussian natural define methodology correct much principled include question inherent studying challenge science basis draw modeling address fashion propose methodology property property within distribution hand correct defer describe state informally measure guarantee distance number need correction naive learning find accord inefficient agnostic sampling monotone get output guaranteed necessarily learn return distance reader truly need sample quite although simulate truly random tradeoff furthermore parsimonious extra bit correction factor reason track separately main complexity agnostic approach throughout arguably illustrative challenging insight sampling non probability body cover decade e detailed list reference monotonicity wide concavity hazard risk evidence monotone direct implication correction shape constrain begin implication existence imply class dependency learn exist sample agnostic family monotone poisson binomial sum next algorithm efficient agnostic agnostic hard third estimate imply latter decide rigorous general get bound low specific application achieve improve various turn monotone cumulative well monotone distribution approximate histogram small carefully decrease monotone also access cumulative cdf monotonicity complexity level combine approach correct coarse fine correction within monotone mostly concerned distribution provide sample concept cumulative summary comparison access query access justify sort query implement overhead unless otherwise formal mention although definition present total analogous definition give draw internal notion allow close desire convert access access sampling evaluation property oracle term simulate query improve oracle sampling query maintain ensure
present detection convolutional cnn cast object suitable weak pointing converge boundary network object object accurate ap architecture recent cnn object vision human limited application object thus visual recognition move towards rich image understand pixel level many still box existence far region score far level detection mapping bounding cnn think must room cnn regressor straightforward detection integrate object cnn bound aggregating combine name bound box pointing leave bottom corner recursively direction fed network bounding box object slide cope state single everything performance window single include direction mis box object verify strength model primarily detection contribution fold suggest box aggregating involve bound regression art performance class task decade handle part flexible compose object severe another demonstrate competitive numerous variation activate vote location recent development detection advance cnn imagenet detection region cnn represent activation cnn proceed probably object object proposal feed pre cnn proposal mid cnn activation convolutional svms object proposal merge fed regressor mis despite limitation quality fail procedure reason agnostic proposal improve quality reduce cnn individual component feature box r cnn another cnn detection train cnn map rectangular mask object approach directly estimate method proposal leave bounding method also unify verify component detection operate extension summarize fig cnn feed directional corner bottom input possible involve right go image f let prediction feed corner corner return f corner end instance corner end detect project bound box image detect box activation several benefit portion proposal window object proposal carefully maximally proposal guarantee object reach terminal obvious corner fig previous detection cnn weak direction short length strong bound convolution connect layer filter layer layer compare adopt please refer detection prefer max force bound direction force maximally bound enable return prediction final conv conv decisions size conv relu max make operate devise quite image form test decision possible pair evenly case process original multiple augment generate satisfy positive top fig area vary complex scenario scenario multiple always narrow instance among instance follow rule generating iterative stage way bound region ratio scale region feed fig train cnn select portion region portion remain average loss extend instance verify effectiveness proposal direct coupling cnn separate use feature meet discriminative study maximally activate proposal ht l proposal svm score toy human reasonable pre cnn svms evaluate svm cnn svms classification compare much weak correlation neuron body start target reach maximally activate discriminative face human b initialize merge follow intersection box feed finally merge detect single include proposal separate merge several bounding box procedure us region feed entire body combination instance instance logic sure instance include boost region slide paradigm feed slide window successfully require input layer layer compose regular fed spatial activation significantly slide window method cnn diverse ratio scale also slide window feed scale image obtain thousand slide window feed scale aspect box instance region proposal produce fed image box merge decrease box minimum average employ bound regression bound detection employ box refinement box b window initialize fed chance reject false fine localization bound merge final verify primarily detection human wide object beyond human center stage decade nonetheless human image still severe rigorous verification human primary class compose web diverse pose variation set ap rigorously evaluation ap value one relatively training weight optimize pre conv conv initialize conv conv whole regardless parameter follow length feed prevent divergence scale scale aspect ratio accord box set merging refinement ce method without refinement ce achieve refinement score extra extra ap refine
recall lemma need cyclic permutation proof abuse notation rather since satisfie thus observe respectively simply exercise claim novel stochastic via order suggest attain sgd guarantee assume deep experimentally form every loss categorization I later generalize regime stochastic sgd basic index random gradient perform mind much distance control stay keep simplify around q subgradient two use notation j instances specify distance frobenius regularizer easy sgd rewrite measure give conditioning reason become conditioning consideration converge equation possible would naturally sgd algorithm rely pick minimize assume lipschitz matrix equation optimality condition know norm eq optimality typical lead order correlate nc scenario relatively sgd get back issue update overhead sgd time column invertible sgd intuitively capture deal show enjoy speedup advantage become runtime runtime sgd magnitude follow survey discuss variant preliminary showing find technique choose twice differentiable method dynamically coordinate hessian utilize hence compute convex example preferable newton method case base direction batch approach operator applicable obvious see adaptation bfgs approach estimation hessian low aforementioned order sgd always hessian approximation tackle rely gauss newton discussion come guarantee approach name algorithm adapt algorithm along form several convergence bound discuss gap storage time apply update rely diagonal replace iteration previously start update form positive denote apply update given equation inner inner provide lemma family proceed inverse informally combination identity subsection rank choice denote eq straightforward leading eigenvalue diagonal recall previous require decomposition fast calculate behind depict blue vector random coincide multiply right probability formalize set outline feedforward neural composition layer predefine training network amount function fully connect perform transformation usually variant gradient backpropagation affine calculate calculate unlike layer iteration note process cause main gradient descent technique convolutional weight convolutional col besides convex conditioning particular batch nesterov describe initialize choose random accord uniform mnist view house input function output channel pool kernel convolutional size layer channel relu affine channel prediction architecture conv conv affine relu affine training test multiclass axis height log legend legend align draw black color blue sep crcr color mark option solid crcr b width height axis legend legend align align leave white black table sep red solid crcr width height legend legend align leave draw white color crcr mark mark option row crcr e e style legend align leave align table crcr solid sep crcr much terminology architecture conv x conv relu summarize width axis true align align white black row sep crcr color red option solid sep crcr height legend style legend cell align align white blue sep crcr color solid sep crcr width height style legend cell align align white black sep crcr color option style legend align align leave
penalty conjunction convexity fidelity estimator efficiently far improve estimate matrix equal estimate column perform sparsity lasso selector observation location entry coincide node nonzero column need diagonal coincide entry construct dependency course estimate estimator straightforwardly partial infer clear mean absolute b concern describe dimension without panel variance relax maximum ingredient graph feature present study term residual variance rv likelihood estimator column wise sparse linear briefly handle next section present figure accuracy estimate detail introduce present material unknown note equal integer integer complement singleton matrix every diag transpose square note element row resp whose give resp element th row column resp pseudo element wise norm sample unknown consider present estimator diagonal development always center variable two coefficient residual j j estimating consist vector solve sparse estimator use explore empirically different perhaps offer trade complexity lasso matrix fit matrix element small separately computable even appeal preferable column lasso establish investigate estimation prefer root confusion scale aim estimation therefore gain insight natural consider matrix lasso expression orthogonal subspace vector multiplication intercept reduce follow residual estimator regression residual coincide maximum likelihood variance use distribution therefore complete handle similarly result observe usual estimator quadratic independent maximize likelihood lead eq recall view decomposable maximum clear bit rv truncation sufficient satisfie j jj provide discussion entry combine proposition variance contradiction estimator explanation really certainly parameter ignore allow explain proposition denote b jj b kk independent view variance jj jj fourth truncate right hand probability inequality p suboptimal lack constrain proposition error loss generality consequently therefore relation entail entry estimated risk th apply write maximize vector check condition precision necessary semidefinite b positive maximum jj cost last jj jj aforementione differentiable point derivative vanishe provide jj b note trace jk j give differently quite entail description set component vertex cardinality denote indicate class equivalence connect distinct vertex matrix h h readily introduce entry path connect reproduce argument estimator belong connect compare proposition estimator outperform symmetry belong ideal ideal systematically outperform variance estimating need original th mention early quantity always provide convenient keep connect estimate correlation ij somewhat replace necessarily connect loop depend choose try span short combine algorithm short path span threshold short tree span root tree increment behind span tree favor contain partial correlation aim minimum spanning ht follow ols rv rv enforce symmetry raise issue relate penalty measure intermediate penalize responsible trade constraint enforce symmetry extreme coincide play role feasible equal assumption objective feasible set feasible lipschitz algorithm indeed descent upper hessian unfortunately value loose resort descent f experimental explanation symmetry penalize likelihood precision square lasso rv follow rv without rv comprehensive experimental diagonal precision order many situation possible six several precise matrix use matrix equal diag diag six entry ij diagonal entry row introduce q result experimental compare follow section rv residual correspond symmetry maximum estimator diagonal conduct experiment matrix column root penalization commonly universal lead fairly scenario square entry root aforementioned value scenario correspond know scenario include experimental empirical precision configuration configuration estimator rv replication expect r table along error conduct square lasso cone root rv square lasso follow c error rv root rv root follow ols rv ideal empirical reflect comparison preferable residual variance symmetry refinement vast majority slightly bad size quality use happen step quality estimation mostly estimator thank nearly variance rv variable graphic vector central ol root lasso convergence speed fix rv root rv c rv ht c root rv follow rv rv explain term reduce suggest fact entry include suggest look contain short path weight among span tree short give tree path among bad complexity construction tree graph operation connect node short short degree complexity try related weighted short node component overall choose root large variant well rgb rectangle rectangle circle circle circle circle rgb rgb cycle cycle cycle cycle rectangle rectangle rgb rectangle circle circle circle circle rgb cycle cycle rgb cycle rgb circle circle circle circle cycle cycle early descent scale one coordinate descent perform size opposite increase constant mathematically speak operation gradient thank guarantee start iterate limit tuning parameter cross validation choose geometric range result plot nearly choose introduce entry precision copy precision commonly residual significantly symmetry mle numerical entry noisy small realistic root ordinary conduct accuracy residual mention introduction novel observation partial
nn per jj nn market jj nn relation concept trading evaluate ground especially inherent relation cluster discover algorithm model per ranking vocabulary multi noun phrase relation cluster strongly strength association sample relation coherent relation reasonably summarize position though reality toward second restrict basically pick related organization entity topic resort city near france entity exclude seq notably tendency form entity rather entity issue relation clusters persistent relation look topic cause allocation topic absence special set share illustrate incorporation syntactic syntactic pos tag basis concept like core flow rather notably requirement word explain absence broader share explain relation relation behave much like topic word semantic content likewise although syntactic abstraction away word syntactic one syntactic overlap relation first relation group level resolve believe modification lead significantly inference minibatch sample correspond chain corpus iteration plot achieve level contrary explanation inherent good redundant corpus although discover drastically alternatively need simply variance yield computational drastically reduce likewise minibatch tuning base simply far redundancy appear case would gain gibbs sample regime hundred million document also advantageous statistically efficient corpus amount store minibatch structure promise extraction modeling assumption coherent relational moreover acknowledgement author like improvement model explain mathematic alternative setup fail however fortunately much approach analytical variational collapse goal hold fixed sample use document document mean batch document gibbs unbiased natural little bit work ensure iteration follow minibatch document without burn n overall update learning update note update track raw code actually global manually remove limitation extend hyperparameter begin variational objective depend give use fisher indeed identity fortunately know calculate analogously gradient thus obtain second centre author amazon amazon access web scale automatic base large corpus unsupervise machine text recently tool scalability obstacle rely scale sublinear inference qualitative extraction web corpora gradually automatic resource relation extraction inherently encounter unbounded encountered need success probabilistic unfortunately prohibitive time impossible incremental training model lda apply extraction process support stream plain variational able qualitative fraction reduce include unsupervised show major prior group cluster sentence document use separate entity word syntactic paper feature vocabulary size feature document notation set scalar discrete associate sentence draw assume access pos name entity name entity simply kullback posterior small impose factor entire minibatch document carry iteration parameter relation variational times carlo supplement minibatch origin supplement likewise natural scheme consist article new york time entity leave sentence nn pp vb seq pp vb seq exclude
maximize size respect minimize depend like several evaluate empirically dependence initial weak nearly problem another way aggregate partition experiment construct walk computation completely cost aim minimize channel good compression motivate procedure completely differ section contain benchmark useful general ground element see political political truth graph ground truth partially manual process introduce error ground see partition political particularly fit sbm ground also easily check ground ground problem sbm sbm partition degree community real parameter mix community edge go community lead disjoint separable graph boundary become community community normalize shannon entropy mutual partition coincide take otherwise overlap overlap community propose refer somewhat subsequently set community paper really communitie varied generate respectively correspond standard deviation bar experiment precisely material run algorithm identify reconstruct good performance evaluate result overlapping observe operate subset vertex recall random start note set start set community overlap community via introduce community algorithm overlap benchmark run value obtain benchmark alg inf less detection situation note generate detection community long overlap know clique average sbm define consider model assume follow partition iteration sbm step somewhat bias c p linearization whether versa amount third number two path p proof material note length essential argument never require adjacency consider path target algorithm result initialization initialization algorithm observe behaviour suppose truth cluster original part find precisely usually use construct co q regard cluster often initialize set choose repeat empty precision course also imply standard finite indeed rearrange j j jj non negativity equality iff recover iteration proceed plan state chernoff theorem binomial lambda use sbm initializations random sbm node graph next denote generality expectation count component degree degree obtain least union hold node union fluctuation sum necessarily independent consideration often omit degree total concern write ds ds lambda equal union upper probability assume intersection conclusion quantity expectation shorthand relevant assume partition statement individually probability claim union inequality two large thereby prove expression similarly c thus finally length path path concrete path let path path type expect hence consideration concentration bound neighbourhood set argument lemma obtain concentration one lambda order conclusion carry precisely full make shall either without note deviation guarantee deviation hold satisfying assumption randomness partition satisfied discuss claim symmetry reverse inequality obtain examine write denominator quantity summarize q proof plug use similarly proceed enable expectation fluctuation incorporate satisfied specify degree size case set overlap specify multiple specifie number community software overlap specify section strategy instance use threshold figure spectral package run final euclidean k improve different return somewhat different benchmark discuss sense heavy different sense cluster overlap overlap cluster post community non try processing comment regard structure discussion restrict setting community heavy remain belong community hand node belong community community community almost intersection property start chance return much measure chance common explain graph section thm claim thm proposition thm definition community walk therefore easily benchmark community benchmark performance previously prove stochastic community cluster problem subset connectivity subset connectivity rest happen community application may instead individual node application communication traffic design biological meaningful arise survey euclidean transform weighted survey community depend whether question weight direct another allow overlapping notion several adopt benchmark produce computation develop recovery graph variant diffusion entropy non space measure short walk certain detailed evaluate benchmark find alternative insight reconstruction evaluate random benchmark significantly similar performance performance spectral modularity clique benchmark introduce modification enable detect overlap community overlap benchmark perform evaluate stock sp correlation return right stock overlap community community node belong community fact community algorithm benchmark largely stochastic block reconstruction mean suffice first purely probabilistic motivation shall analytically analyse name spectrum behave probabilistic generally dense constant reconstruction dense fix size approach differ spectrum equally behaviour study graph sbm high concentration inequality conclude remark sbm mention view euclidean vertex diagonal walk community algorithm walk notion sampling vertex randomly start
sequence hx penalize erm expect penalty candidate overfitting reveal penalize summation solely nearly distribution inequality fulfil straightforwardly extend minimum prove bound excess reach erm summation attain rank minimizer summation solely value take measurable denote independent x x base pair sample degree rx rx yx q equip notation statistic key noise hoeffde true section goal cardinality rule candidate finite optimal rule simplify version nr fulfil constant minimizer whether occur excess fulfilled least soon reach ranking minimize cluster subsection sampling expectation realization replacement approximate computational scheme bernoulli explain result interest subset cardinality power general belong draw second inclusion probability inclusion sample scheme bernoulli given observe plan inclusion plan addition bs survey role wide survey scheme view refer account seminal conditional eq equal situation sample advance choose among inclusion probability else sample result obtain thompson estimate collection symmetric assumption incomplete statistic without replacement eq highlight perspective replacement advantageous replacement stochastic loop precede section ignore technique provide machine sgd investigate gradient estimate incomplete statistic draw space machine eq rate compute gradient unbiased estimate statistic construct draw symbol refer k ki alternative sampling statistic build function smooth one show small estimate get variance strategy risk variance average potentially form combination summarize convergence use form strategy report literature semidefinite function gaussians mean share overlap proportional respectively handwritten digit class consist image extensively benchmark reduce retain unit merely involve subsample erm gradient erm scheme risk statistic pick statistic empirical pick project random testing risk pick reach average accord modify reach compare strategy reach quality dataset though quickly scheme reduction erm compare analyze complete subsample gradient sgd incomplete paper incomplete statistic sgd complete experiment mnist project batch namely batch sgd risk mini sgd sgd complete batch size comment large strategy support though rate sgd incomplete similar strategy sgd incomplete lastly expect gap implementation small mini batch size hundred wide learning estimate risk seek optimize increase functional involve sum rapidly become implement counterpart pick randomly replacement refer novel deviation learn preserve certain situation occur extended scheme bernoulli show purpose base beyond experiment technical proof independent jensen use argument rademacher rademacher symmetric sign variable inequality shall eq convenience sequence k ki equip notation observe almost surely virtue vc major dimension q next independent I assertion formulate turn straightforwardly first proposition assertion direct assertion combine h observe triangular yield whose reader follow integrable assertion h start direct apply nx x decompose expect pick complexity incomplete statistic criterion successively notice prove follow assumption theorem fulfil probability straightforward assertion namely bind observe solve partly slight fulfil q assumption virtue eq union proposition focus sample degree argument easily express variance hoeffding see subsection equip orthogonal hoeffding decomposition eq center nh nh nn pointed subsection sampling replacement asymptotic rate big metric estimate average highly expensive moderate term procedure functional feasible empirical risk replace drastically base refer rate erm result describe approximate incomplete version sampling technique stochastic erm numerical display provide evidence largely empirical sample erm develop erm essentially rely study maximal deviation average expectation adequate assumption purpose inequality wide deal range recognition view pairwise empirical error rule statistic hoeffding ingredient studying allow establish maximal deviation erm minimization df statistic law pool moment establish mean representation functional completeness investigate average sum symmetric integer number observe eq representation refer illustration copy couple find pair close bound function hinge estimator one two minimizer framework algorithmic robustness without contrast naive subsample seminal contribution calculation summation index solely simple index incomplete replacement stress build replacement involve summation depict incomplete statistic sampling pair base population overcome issue depend summation
improve bf vs notice correct quite inaccurate since lead student interpretation accordance standard approximation likely indeed look bivariate marginal shape elliptical nevertheless laplace pressure write choose student error bf integrate whereas assume variate student take jeffreys improve laplace comparison comparison dimension integral reasonable amount correct period draw use consider draw df equal inverse modal c laplace improved laplace correct student bf vs approximation normalize bf offer model laplace approximation laplace due shape approximate fix generally write give integrate density consist three involved group r five group constraint experiment result indicate successful model successful pair across define type normal separate follow approximate mle approximate laplace modify approximation effect poor observation specie involve integral random compare laplace approximation approximate marginal improved laplace take gb ram iteration mle method laplace laplace approximation laplace compute derivative beyond second order joint computation three method available effect particular specie laplace start minute converge laplace mc gibbs sampling quasi laplace obtained improve mc quasi approximation approximation improve mc mc intervention laplace widely frequentist second inaccurate improve show superiority laplace three method well gold demand scalar integral parallel accurately integral burden enhance analytical strategy example analytical automatic evaluation david unimodal necessarily tail asymmetric heavy always regular unimodal laplace quadrature quadrature accurate indeed package effect perform quadrature large quadrature point seem context package plot consider reproduce plot use file reproduce skew reproduce file reproduce analysis code file reproduce datum notice package require package corollary multidimensional integral frequentist shape laplace inaccurate asymptotically standard formula also dimension frequentist superiority comparable keyword expansion integral likelihood numerical integration involve smooth depend quantity density approximated quadrature typically accurate curse feasible low especially sense reach value alternatively increasingly dominant expansion laplace formula computationally convenient expansion analytical tuning unlike monte carlo moderate may monte carlo focus laplace marginal frequentist widely bayesian approximate density bayes bayes framework effect likelihood markov nuisance interest draw posterior invert laplace lastly laplace context joint survival longitudinal improved laplace integral achieve order setting behind density normalise easily approximate point laplace spirit optimisation demand requirement method detail issue rest section background improve final remark ease twice differentiable minimum laplace sample line interval panel laplace achieve laplace order propose give multivariate identity skew marginal skewed degree df control ordinary student identity df normalize skew density multivariate dimension figure standard higher worse obvious laplace work skewness versa quality laplace contrary improved laplace answer scenario similar laplace however substantially less laplace approximation variate modify illustrate method application focus aim know maximum mle analogous compete package available material
numerical references langevin energy boltzmann detailed study langevin dynamics balance formulation actually demonstrate beneficial improvement acceleration relaxation reduction correlation steady state interact intractable etc direct monte master promise convert initially method relaxation efficacy accelerate report relax initially realization simulate multiplication step realization transition transition approach detailed balance bc reveal superior confirm mean solution present provide validation study langevin evolution alternative fast convergence langevin write equation equation instantaneous function steady divergence analogous bc context master steady correspond force bc confirm different reach present analogous bc steady review mcmc langevin equation find present several artificial force convergence analyze briefly review simulation fundamental evolution master equation degree define master cp satisfy transition numerically perform update master steady state define balance bc restrict aim generate ss x ff temperature nontrivial bc confirm satisfy bc cp nontrivial bc lead construct matrix follow system former propose diagonal configuration accept summation element diagonal summation cp element bc cp cp symmetry read find rejection symmetry probabilistic flow efficacy theoretical eliminate accelerate relaxation reference conclude efficacy come rejection decrease element sense efficacy method yet understand skewed replica transition master equation swap transition give manner balanced impose steady share satisfy bc simultaneously bc bc transition instantaneous master equation bc master symmetry rescale transition matrix present author prove rescale acceleration drive rescale emphasize rescaled transition dynamic master langevin force wiener freedom give equilibrium purpose hereafter force fast achieve langevin flow define steady vanishe therefore equality steady satisfy steady master hereafter thing divergence bc spirit divergence steady vanish divergence change ss steady particular equilibrium exist confirm formulation transition force solution indeed demand detailed free must fact denote element probabilistic element unity force careful instability term temperature instability force divergence analogous may permutation reach permutation scope nontrivial bc composite analogy denote flow define system divergence system steady state system ss nontrivial solution consider force steady confirm flow immediately exchange monte carlo extremely performance equilibrium naturally existence analogous mention study publish subsection nontrivial confirm interval obtain location transition omit argument right side derivative backward confirm probabilistic flow first heat regard excess heat signal coefficient degree case ratio probabilitie excess heat also confirm heat artificial implement artificial flow confirm artificial force numerical restrict force minima locate equilibrium n x tn minimum potential relatively equilibrium force accelerate system evaluate ordinary plane case trace exist additional force describe area steady state however steady without addition observe steady integrate steady lose avoid show confirm integrated reach limit increase decrease efficacy beneficial addition method spin temperature region method remove xy steady mathematical additional force steady red plot stand location tb origin acceleration steady force mathematical understanding acceleration relaxation steady define operator eq q one use transpose vector hermitian boltzmann nontrivial force effect additional force nontrivial express reach p anti hermitian right side anti hermitian steady operator hand quantity anti hermitian part free nontrivial additional rewrite x x relation trivial one k vanish reach force satisfy free anti hermitian system anti hermitian accelerate relaxation master equation transition matrix master characterize large operator identify gap force eigenvalue asymmetric operator restrict
user depend namely bit htb histogram protocol privacy parameter user else set compute private server server jj privacy privacy privacy basic utility unit contribute complete error suppose modified reflect round round simplify proof occur least recover item round algorithm whether counterpart item I unbiased triangle get eq claim ratio eq numerator hand depend pp condition decode implicitly assume perfectly tail conclude pp private histogram error protocol discuss protocol heavy lack interference user heavy heavy factor hold simulate choose repeat protocol times interference free channel hence eventually item list separate channel protocol protocol result frequency frequency random protocol description server access randomness integer run result string construction problem protocol section protocol namely construction modular protocol moreover aforementioned object theorem idea condition separately heavy hash item holding report user channel user choose protocol get interference construction channel eventually item high list know result oracle list frequencie suffice pairwise independent member construction hash length family hash server assume access input generate random server string user protocol histogram user input privacy initialize heavy ki v v pp modify item add user frequency hard construction cost sub algorithm frequency estimate protocol verify discuss expense public seed public need privacy protocol given differentially observe protocol channel separate hash channel user seed channel user item differ item channel privacy note privacy report ratio channel protocol privacy argument paragraph v list frequency mention frequency item implicitly change expression item v list implicitly item seed independent get hash report random execute component basic work string sequence kt step sign sign otherwise pp pp protocol protocol htb ii k frequency histogram pure pure meaningful trivial algorithm clearly private differential privacy upper histogram section construction show construction construction ms construction inefficient construct statistical first bad use channel uniform item whereas scenario bind second channel proceed derive mutual item output prove scale scenario channel mutual information mutual information together let denote simplex corner probability assume differentially report arbitrary randomness estimating estimation randomness randomness error hoeffding prove technical histogram I fashion respect define case example probability turn hand application hoeffding mechanism sake contradiction item follow hoeffding contradict notion channel randomize define uniform variable user item independent copy output replace let eq independently follow input consider scenario iy equality show complete follow scenario uniform item user apply fed output report output wrong minimax reach show n nd claim denote discrete dm bad bad information variable originally pure differential corollary therein next iv differentially private similarly finally imply complete channel denote differentially private put together iv complete histogram low advantage inspire multinomial pure make modular differentially private protocol bind state discuss scenario draw independently bad show frequency notion otherwise channel mapping every compare channel local whereas scenario user item low result would show channel scenario derive mutual item local namely composition algorithm imply channel mutual information inequality acknowledgment nsf university award grateful frank point transformation compression pt lem lem lem lem lem fact lem lem edu give protocol match differential privacy individual user differentially private server protocol heavy along user come universe protocol run time regardless one protocol either low adapt result server protocol transformation software web management want want collect datum provide store private datum call signal server figure provide public visible randomness local private remain protocol equivalently v protocol estimation local privacy local differentially private web task mention protocol coin server value universe enable analyst summary definition together item histogram implicitly estimate frequency frequency item implicitly measure frequency structure aim never item estimate frequency may list error ignore histogram oracle analyst oracle query oracle item retain universe home page summary histogram useful protocol storage user protocol satisfied protocol frequency protocol histogram heuristic none accuracy differentially histogram protocol regardless public coin also code construction inefficient take rather query oracle determine execute frequency construction sublinear protocol recover either server idea error encoding server receive decode protocol low sense g universe likely heavy unique privacy essentially copy protocol add private protocol protocol private computation frequency long efficient protocol give item appear random instance implication equally distribution protocol universe item differentially error universe assume proof simplify framework develop privacy modular low differentially private protocol possibly state mutual theoretic unless modification compression coin model common string dp protocol user bit server efficiently protocol literature protocol heavy public utility privacy transform server apply expand short send generator transformation rejection fix player whether kept ignore local privacy rejection procedure little quick far frequency paper protocol task add relevant every function exponential technique recently class giving protocol low protocol algorithm sense protocol hash heavy appear context approximation arguably root evidence see g add introduce construction use tool ensure generate hypercube symbol pick choose bit j construction encode serve purpose construction item basic string jx uniform bit choice depend randomness output represent hold matter long input randomness important utility outside send server public server receive use private provide inefficient private heavy oppose provide private I enjoy depend finally review histogram clear later binary mapping constraint decode element fraction mm encoding decode several construction example thesis describe basic construction ensure differentially private bit string special vector pick string later construction bite special symbol special construction item bit string uniformly eq uniform bit index privacy om output come privacy hold randomness help ensure come outside send server public situation server receive bit private projection line construction differ oppose estimate heavy local use copy add pure differential carry private efficient denote theorem parameter affect guarantee protocol randomness generate public shared server note construction generate much less independent construction protocol frequency protocol input privacy ni server compute length user note need bit part fix item give htb privacy guarantee frequency privacy oracle private set error randomness output input rely behavior inner formalize independent taking also hoeffde triangle inequality least probability hence term upper give efficient private protocol provide construction difference randomization user noise differential guarantee oppose difference carry oracle construction compose binary user report frequency give use oracle inner encoding aggregate protocol generate bit generate take construct differentially user bind theorem rely aggregate encode detail efficient histogram frequency subsection construction simple call heavy general heavy hold item special symbol represent item construction constant protocol differentially private input output protocol encode user code result server redundancy item require length constant say relative error asymptotic behavior several construction thesis example encoder part know convenience hypercube encode report code server round aggregate near argue combination round sufficiently close describe code note report report htb pp encodes else user server server compute jj construction differentially private directly differential privacy rounding code since unit contribute hamming complete common least frequency error reflect round alg simplify part heavy occur show suffice round would item unbiased part triangle apply distribution claim use bind fact assume depend pp pp correct perfectly remain tail property provide private histogram protocol sub protocol lack interference idea separately heavy cost channel item choose large protocol heavy assign interference construction channel item heavy item hash item separate parallel frequency protocol like frequency item pairwise hash whose input seed protocol description user server access randomness generate string see random string histogram protocol protocol namely private oracle private problem modular internal protocol object show lack interference user heavy create computational channel user hold channel item sufficiently large repeating time heavy free one channel eventually contain hash overcome separate protocol result frequency oracle frequency output item purpose suffice distinct uniformly member efficient construction seed instance family efficient family hash return server assume random server result hash family htb efficient protocol confidence heavy empty ki n pp pp pp pp modify set unique frequency hard protocol frequency sub protocol compute protocol overall user bit expense public construction rely public string seed protocol differentially private protocol differentially channel protocol channel seed hash report assign channel user seed hash user channel separate channel privacy ratio put paragraph frequency satisfie mention implicitly zero theorem mention estimate user item run channel every occur without interference inequality tv channel whose channel heavy run channel possibly channel event guarantee frequency item algorithm actual item great frequency complete give protocol private distribute report add bit original mention transformation technique private protocol server protocol iv public report string server server report randomness output give bit generic generate string ni ib server server obtain desire protocol cost transformation probability transformation computational privacy bit protocol bit item right side iv construction iy iv note take randomness two fact protocol thus protocol affect sampling formalize statement necessarily protocol point negative randomness bound least randomness respect transformation discuss protocol protocol efficiently local describe hash channel remainder user report uniformly random execute see public string kt dependent user public string compare bit step sign sign desire otherwise step per pp nm ii privacy th group I encode construction oracle bit protocol differentially private protocol bit protocol follow item least pick lead transform private histogram report bit expense add randomness bit protocol mention introduction transformation general compression let server I randomness output simplicity string statistic server server report report randomness output algorithm report htb generic bit protocol privacy independent string ni iv ip server server collect report transformation done preserve protocol bit output item right hand privacy public string user p iv iv e public string upon user server view report take randomness protocol sampling sampling transformation transformation efficient protocol protocol histogram bit error key statement histogram differentially protocol protocol argue compute efficiently parallel channel seed hash independent component user get hash remainder uniformly execute basic string kt kt corresponding step item sign step pp bad protocol protocol full seed construction oracle local low namely pure meaningful yet low upper histogram section construction yield construction construction ms construction histogram inspire expect pure model item user show low error estimating frequency obtain low channel noise input output output first scenario channel local generate scenario normal scenario user directly argue error lower next scenario derive bound channel item privacy namely first scenario mutual information item denote denote item differentially local generate report arbitrary fix randomness identically distribute observation error minimax estimator maximum distribution eq argue hoeffding asymptotic result minimax proof give first private sample directly section turning case application hoeffding sake contradiction item use contradict channel mapping user apply report proof input consider replace iy complete scenario user apply independent copy fed local let output wrong hypothesis incur minimax nd establish claim probability mass density simply continuous bad bad inequality iv pure local like iv inequality differentially similarly complete channel hence write fact differentially private claim v n complete lemma frequency advantage algorithm technique bound estimation modular show prove differentially protocol formally sample refer reader version draw independently bad right distribution maximum item bind notion item uniform channel scenario channel output second scenario suffice true channel lower proceed mutual item channel namely composition channel information conclude r university grateful helpful frank lem lem lem lem lem lem claim protocol bound local differential individual report datum histogram frequent item along implicitly user whose item universe protocol necessary regardless computational efficiency protocol run adapt al public need server know protocol preserve computational people software anonymous may share datum want collect raw subject collect provide store private also server summary public visible randomness extensively private protocol v describe bound local private protocol web basis empirical mention show protocol public coin set bit server value universe label wish enable analyst frequency look summary definition algorithm computing item histogram produce least list estimate oracle measure distance list never contain frequency may price ignore oracle retain frequency universe home page financial histogram useful protocol communication protocol protocol frequency bad protocol histogram bad protocol match bind provide polynomial differentially private protocol protocol coin server length time code construction inefficient state oracle query determine protocol execute construction previous error construction sublinear piece first protocol recover player server value server decode protocol input idea low compressive specifically use hash universe item run unique privacy essentially copy sense private oracle protocol protocol differentially instead show regardless communication protocol efficient protocol instances rise item frequency remain instance bound minimax bad protocol universe local item much contrast differentially private protocol achieve universe assume frequency simplify statistical privacy modular private protocol possibly state mutual information one theoretic bound unless compression et yield coin server player string dp bit server transformation probability protocol literature protocol heavy randomness public affect transform protocol particular expand seed send transformation rejection public keep server bit server quick reference recent work protocol specific task paragraph algorithms utility measure show statistical query show related technique communication give error guarantee basis basic protocol theoretic protocol idea large stream compressive protocol hash heavy fourier arguably root provide close g context subsection introduce construction describe basic construction user private one basic input either string vertex hypercube special represent pick bit string bit bit clear construction bit input unique special symbol serve purpose special situation construction user basic bit string uniformly e output pair bit every depend randomness om note output represent bit output bit come privacy matter independent
score main expert semantic form really indicator choose indicator function qx qx select indicator principle g take binary indicator generate indicator highly redundant imply consequence naive naive bayes assumption estimate use motivation classical property performance motivation successfully expert probability indicator basis allow discriminant box kind grey long performance mention previous realistic selection range backward method chapter redundant binary remain note recommend text book see give feature former fortunately backward strategy efficient subset provide already subtract time evaluate backward backward apply arbitrary move magnitude expensive g simple mi search cost cost classifier incremental base incremental generally addition take firstly lot performance mostly reliably classification however additive identical enough feature estimate uncertainty decision note conditional poor scheme indicator obtain section identical example example belong switch standard white trend anomaly trend trend amplitude step point central area score slide window explain variance successive window detail value train class good subset choose among feature report mutual mi rank classification search good step set backward vice improve one last backward summarize outperform redundancy indicator construction mi tend redundant indicator obtain mi error backward search forward backward always allow accurate ordering infer procedure alone move search select performance reduce filter avoid guide greedy comparable performance search good recommendation mi cm cm universit paris paris france france method datum study indicator reasonable context body available simple
figure fold plot scale change figure demonstrate explain protein fold strongly pair instance change fold across time show vs fraction fold fold change vary across constant three example analysis serve proxy variability protein combination figure pair protein pair result observe due focus distinguishing investigate gene protein fold gene may predict reliably fold cumulative error indicate protein change gene fold level specific post functional gene lack protein noise conversely ratio explore across ratio gene reflect biological specific define ratio divide median gene evaluate significance variability type gene pool across use quantify statistical result go protein small trend account fundamental type far set fact highly mode increase tf decrease indicate variability across reflect noise quantifying demonstrate level noise far take account rna seq mass large systematic protein ratio dna sequencing bias quantification systematic minimize intensity dna sequence quantification quantification variability start reliability reliability simply reliability proportional strength estimate replica rna seq protein derive overlap quantified estimate two figure b take account reliability measurement protein variability reliability reliability reliability variability protein level across type leave accounting explain level remain mostly post reliability lower large measure noise reliability explain post likely major determinant variability highly level role full distinct poorly role protein degradation protein level estimate protein level contribute post bias influential protein systematic bias variability within post error alone correlation protein figure reliability would level account estimate thus accurately quantify mechanism strong post noise figure indicate post substantially level significantly dynamical response variability increase level must across cell bind protein synthesis specialize degradation substantial big much post perturbation fold less due substantial contrast cell acknowledgment thank constructive grant institute grant health gm research fellowship contain rna seq level human similarly corresponding normalize protein multiplicative factor raw additive normalize choose baseline normalize measurement set conduct normalization specific scale level raw median represent protein protein protein noise level way error via correction measure datum decompose signal signal decomposition correction variance observation estimate reliability define fraction measure variance simply protein estimate protein level de protein explain c region identify restrict attention group go quantify relative protein exclude large protein likewise noise specific eliminate variability ratio conduct vector comprise index index kolmogorov ks difference systematic fdr fdr group due mit edu abstract post type specific contribution factor determine protein factor variability protein type variability find level protein contrast dominate protein fold highlight type specific introduction ease protein level protein proxy level conversely set protein independently case classical division understand trade principle assess contribution protein mostly protein level mostly post view quantify correlation absolute mix many protein variability protein across condition biological interpretation implication variability protein protein gene protein widely refer source protein protein order across principal protein orthogonal distinct different counter intuitive conclusion illustrate context gene scale measure gene trend counter intuitive correlation large conceptual datum demonstrate
expansion empirical eq uniform drop eigenfunction hold theorem expansion correspond drop also one theorem also obtain lemma let theorem expansion among iid moment exist emphasize validity er hand expansion expression contain expansion asymptotic small resp enough valid expansion actual approximation even zero hold addition possibly fix hilbert denote correspond indicate readily compute fix h equivalence long limit completely discuss optimality draw heavily suppose c speak rank infer computation order cauchy schwarz arbitrarily moment eigenvalue optimal expansion index still converge result eigenvalue eigenfunction weak remainder write measurable allow flexibility j physical dependence concept popular etc consider k p require sequel follow resp sharp reference relate result weak introduce well h structure ht follow hold mild decay assumption provide gaussian explicit condition structure impose assumption assume addition condition simultaneous important relevant stop rule develop require point corollary also covariance uncorrelated run serial perspective appropriate condition general exist weak take correlated sense approximation appropriate optimality already substantial reference therein datum basic plug lead choice result remainder section necessary bias optimal decay note bias negligible regularity quite decomposition eigenfunction natural term decompose degenerate setup result convenient include assumption present b discuss condition quite condition essentially common degeneracy encounter let analogy eq general transfer remain substitute correspond place version bound provide pointwise uniformly whether assumption everything express general optimal hand drawback interested mention mainly therefore desirable precisely uniform turn occur model imply valid everywhere apply section dependence precisely routine reveal analogue depend mainly convenient decomposition eq recall j nc n convexity require geometric condition process note mix surprising relevant general simply impose already lead simple polynomial boundary variety though boundary usual degeneracy already mention analogy expansion assume assumption eigenfunction proposition little explicitly assume may theorem normalize j operator mention test test serial stationarity many canonical augment increase since account view rigorous minimax theory estimate cf amount see mention necessity control field learn base highlight usefulness theorem reformulate framework convexity lead condition particular result range allow principal typically reflect degeneracy usually necessary discussion hilbert relate fundamental detailed discussion impossible theoretic perspective current cf mutually x assume independent involve truncation motivate eq sharp general necessity control actual deriving expansion goodness confidence reformulate change underlie hilbert condition validity discussion validity possess decomposition eigenfunction sequel let operator eigenfunction decomposition candidate operator eigenvalue eigenfunction distributional q certain see elementary schwarz validity relation hence complete shown schwarz invoke bound cf note assumption cauchy schwarz follow since rearrange schwarz similarly markov inequality combine inequality assume assumption recall cauchy schwarz claim follow sake ready readily treat note triangle inequality eq triangle cauchy schwarz lemma give hence first treat proceeding claim proceed iterate lemma claim follow result valid proof find manner function polynomial concept univariate polynomial invoke method partial moreover one denote suffice consider lemma fix eq since may slightly weak sequel note space want compare prevent necessity coordinate key ingredient variable q sequence grant alternatively may bernoulli theorem major tool subsequently reduce iid resort simplify sequence iid h l role finally n sequel suppose lemma grant p q constant u let q clearly monotone put j pm bm conclude set balancing arbitrarily grant theorem covariance claim iv follow elementary computation establish schwarz grant note obtain remark combine gaussian require denote gaussian follow adaptation result lemma slightly adapt mean covariance covariance proof employ obtain next verify deduce remark verify readily lemma gives hold next remark theorem first need assumption observe moreover schwarz eq verify proceed inequality far bound hence verify bn b routine calculation reveal claim turn briefly elaborate difficulty main objective course everything imply well sufficient guarantee validity order work truncation detailed long equality eigenvalue eigenfunction observation heavily reference eigenvalue theorem step verify deal introduce observe schmidt analogue put obviously hence remain proof convention absolute line write large version schmidt eigenvalue eigenfunction next tool summarize result sequel notion hold b j j n proof throughout proof analogue see claim argument computation routine calculation establish due representation cauchy apply establish proceeding get claim establish follow hand get ba schwarz application readily proceed grant construction desire lemma three lemma validity j note uniformly end separately schwarz times fu elementary calculations j sufficiently large manner obtain I virtue establishes show way difference assumption triangle inequality convexity q establish large b lemma conclude routine calculation q complete ready proceed grant sufficiently large j j proof cauchy schwarz triangle select virtue negligible precisely due get thus proceed lemma suffice replace bound uniformly frequent sequel consider establish preliminary yield relate via triangle give proof exclude complete omit analogue computation proof theorem bernoulli shift e inequality readily derive contradiction assume converse side arrive kolmogorov also get conjecture assumption mu expansion eigenfunction lag eigenvalue spectral gap underlie memory process study deviation among show extreme rise construction also asymptotic transfer covariance operator latter functional become comprehensive overview assume usage introduce convention eigenfunction functional principal empirical eigenfunction define lag j fundamental eigenfunction eigenfunction result become important well bound cf simplicity unfortunately perspective covariance operator expansion prove name correspond heavy structural spectral assumption iid also section presence serial generalization dependent general avoid previously mention derive expansion optimal dependence allow short memory weak memory strong condition optimal application eigenvalue precise mild high dimensional interest particularly cf section functional outline key assumption introduce notion weak discuss expansion context additional emphasis linear memory eigen section long prove devoted proof involve multiplicative give denote complement variable sequel convenient assume j consider variable part sequel depend
observation x tx probability natural likely paper correlate follow concentration finite obtain conclusion suppose condition lf lf measurement keep cascade number require cascade quantity interpretation node time simply infect cascade cascade necessary constant goal necessary recover cascade obtain sparse inverse cascade cascade output true follow sketch kronecker kronecker validate empirically assumption probability state art extra benchmark approximate evaluate algorithm real social network graph edge recently kronecker report cascade ic obtain commonly link compressed paper signal research direction solve adaptively cascade david grateful feedback suggestion proof section show lemma give q martingale assumption lf surely apply union proof contradiction positive get mostly rely show concentrate around cs h thus h mn cs guarantee exist w distribution set support follow draw quantity together graph cascade figure algorithm benchmark fix cascade fast algorithm cause time cascade increase linear expect slope large overhead fact seek recover cascade approach sparse cascade include graph recover parameter context validate empirically graph extensively diffusion take place presence infection graph precisely design cascade diffusion goal understand decompose focus single discuss identify among graph cascade influence cascade recover influence cascade non state step literature cascade sufficient contribution cascade sparse notably cascade cascade able efficiently robust recover prove guarantee tight survey cascade conclude edge active decade approximate cascade later network discrete cascade obtain algorithm one analysis decay limit cascade need suggest obtain recover weight cascade prove close wherein consider standard recovery strong exploit kkt cascade analyze model orthogonal describe node condition previous event state mutually graph eventually reach probability word cascade describe diffusion influence become node uniformly source verify cascade overlap cascade start infected treat cascade context cascade extend cascade direction consider transition probability cascade problem linear fact diffusion true special draw linear cascade indicator cascade state become time step condition cascade provide node sample interpret link cascade cascade either remain infection mutually succeed success stay cascade terminate rewrite cascade inverse cascade fortunately independently color cascade stops fix blue node time cascade cascade discretize independent temporal discretization whose infection interval infected indicator infect interval contrary cascade remain random property discretize induce cascade cascade link intuitively infect node get cascade inference become inverse cascade central present work cascade influence parameter note cascade mle prevent overfitte control cascade decomposable write equally measurement step reach horizon cascade node deterministic contrary cascade condition concave function iff cascade regularity lf hold cascade lf soon data regularity cascade cascade equal explicitly case constraint obtain verify need exist add regularity assumption program recover parameter cascade estimate sufficiently recover provide network support exactly relax decomposable henceforth focus omit notation analyze edge influence parent standard symmetric define q cascade binary vector lf let convergence rate different cascade
computation denote instance correctly classify check f h quickly always sign discuss situation incremental completely optimization incremental completely framework suboptimal solution compute gradient optimization gradient current gradient obtain gradient proceed gap stop incremental sensitivity analysis optimization sign low upper become sensitivity task describe summarize experiment repository small lr compare incremental part use approach conduct core ghz gb use rna million task compete problem regularization parameter nonlinear case rbf task speed also trick lower bind select well meaning error stop trick conduct operation increase observation mis classified compete table computational compete tight novel provide actually instance particularly relatively instance three plan propose stream prove function theorem complete square noting compactly obtain lagrange multipli multiplier rewrite constraint strictly active let optimal write obtain convexity inequality rewrite multipli technology technology introduce framework problem use remove quickly update classifier incremental although large completely incremental might expensive novel update without actually quantity cost property advantageous instance update demonstrate applicable sensitivity bound provide sufficiently tight incremental sensitivity leave training logistic support simple except acceptable thing care datum particularly design instance remove instance add solution linear efficiently framework original helpful reduce incremental computational incremental learning expensive except incremental mean complexity great intractable completely every nice actually unless want interest update suppose could change minor modification order propose quickly compute depend unknown classification low bound linear specifically denote linear obtain upper score compute property advantageous number update bound useful sensitivity test interest make positive mean label instance available study sensitivity closely relate design example literature check classify update exactly fit svms exist build idea sense bound obtained propose exist bind inspire safe screening coefficient use lagrange multiplier optimization actually contribution bring sensitivity develop framework cost depend rest organize describe task present low upper update addition discuss direction present first conventional incremental propose study train use conventional incremental update hereafter denote label remove instance index instance one want modify classifier predict class label give consider represent classifier represent empirical control differentiable example include logistic case number instance entire difference small use incremental algorithm incremental work incremental learn conventional novel make inference framework compute low upper eq computing depend update quite advantageous base propose framework classification problem sensitivity propose might element new j e toy bound bound beneficial making decision practical task old blue bar us sensitivity test follow tight sign update relatively entire size would demonstrate empirically many case toy blue bar indicate unknown rd sign instance update validation leave regard bound use correctly leave classified mis classified bind
hold case outcome occur frequently research recent finite maximizer although rigorously uniqueness maximizer stochastic unique maximizer satisfy imply diagonal maximizer maximizer entry maximizer nonnegative column proof correspond outcome maximizer proof unique maximizer element maximizer first maximizer follow form entry row third third row uniquely maximizer sum discuss construct maximizer let observe result marginal subsequently conclusion sample power dominate dominate fix increase marginal power decrease increase weak increase power methodology power specify instance guarantee summary randomization depend marginal difference outcome power marginal fix difference conclusion confirm easy sharp marginal easier reject sharp potential table contingency limited amount become negligible asymptotically paper sequence hypothesis systematic quantify hypothesis retrieve discussion distribution example university recognize several researcher construct large systematic ordinal outcome power randomization datum treatment extension contain contingency nan introduce randomization refer permutation test test randomization test randomization ability clinical trial describe status improvement ordinal limit assess power test ordinal assess randomization utilize assessment power randomization construct nan hypothesis experimental unit identify sharp hypothesis literature assess power randomization super finite population potential outcome something refer indeed ordinal outcome quantify nan develop alternative hypothesis close vary study power randomization proceed review focus ordinal outcome randomization sharp introduce sharp nan discuss test systematic hypothesis report demonstrate assess randomization completely randomized experiment unit ordinal category bad category value potential treatment science unit potential summarize view later reduce science play hypothesis p marginal outcome treatment take unit otherwise unit assign element science early observe miss treatment assign cccc summarize outcome manner similar term represent unit outcome count sum express sharp nan experimental unit choose suitable specific randomization randomization involve outcome hypothesis potential outcome assignment mechanism assignment simple realization randomization nan note significant ordinal question sharp issue examine power create sharp wish joint increase hypothesis infinite way make create intractable make tractable impose follow interpretation term distributional effect alternative sharp scope help hypothesis introduce quantify hellinger quantifie sharp nan hellinger intuitively distance sharp nan however need hellinger sharp complete picture commonly categorical hellinger solely rely joint subsequently hellinger converse hellinger interested power randomization test hellinger increase construct maximize constraint minimizer maximizer construct minimization somewhat associated potential outcome minimize minimum problem however expression maximizer stochastic triangular proof first satisfying note element define upper low p j p entry corresponding column
model discover concept human expert assign direct exercise exercise correctly recover pre knowledge dataset non simulate case hyper variation additionally simply get exercise virtual student virtual concept response generate two answer student knowledge exercise single difficulty student get exercise difficulty student model classic theory e guess student time affine exercise understand incorporate simply exercise exercise student concept sample student usage core complete across contain working govern agreement design privacy accordance particularly learn interact site student often self topic public student school latent direct set influence equation concept occur far node exercise nd ask dependency graph concept tag pair remainder product obvious student answer correct baseline generate relationship education expert require intervention coherent rnn education particularly annotation pattern input disadvantage simple hide require amount suited education small rnn future take explore dropout pose literature space model student input track especially student task develop program able include material efficacy propose acknowledgment many thank support cp appendix exercise exercise probability display less element capture get exercise google stanford edu stanford edu machine interact education effectively student knowledge task inherent challenge utility rnns student rnn advantage encode human domain capture substantial improvement range suggest promise knowledge rnns education open access grow cost building trace student future student hard delay already tune content show gain machine could inherently complexity human brain use education rely deep along allow representation student knowledge code main recurrent neural auc knowledge benchmark model annotation exercise generation formalize interaction take student next interaction tuple combine exercise exercise tag exercise whether show root problem correctly single intercept incorrect intercept prediction exercise next interaction visualization show prediction exercise type previous exercise tag assign exercise leverage expert annotation absence modelling predict inform diverse education cognitive social influence complex macro motivation challenge micro human complex process knowledge pose heavily aforementione nevertheless concept answer incorrectly formulation assume knowledge learner difficulty extension suffer mapping onto concept exercise several refine concept exercise mapping gold cognitive analysis domain learner process kind observable behavior although present require exponentially implementation restrict discrete hard code latent make overcome limitation analysis predictive combine combination adaboost forest feed ensemble limitation requirement work model kalman promise expensive recurrent neural neuron hide evolve input activation education rnns notable rnn ability point lstm instance translation amount training result suggest successful student formulate govern diverse property presentation individual rely principle attribute model rnns sigmoid student response upon recurrent neural rnns map illustration compute state successive sigmoid parameterize input recurrent state short term lstm variant rnns powerful forget gate thus retain make easy interaction complicated transformation equation rnn interaction necessary convert encode student interaction tuple combination exercise exercise space encode assign dimensional compressed recover constant sparse tuple exactly encode vector deal extend complex student interaction
structural illustrate triangle convolution represent vector feature evaluate subtree follow eq parameter fix layer design allow short propagation position tree feature effective tree base tree number vary feature fix rest explain detail subsection subsection deal third problem training illustrate leaf leaf g noun phrase stanford leaf node embedding rnn representation convolution process subtree window node child associate child indicate leaf child straightforwardly exponential node cnn add amount dependency dependency representation lead child node cause weight parameter window traditional c convolution g position believe much reflect relationship generic convolution parameter parent child frequently occur sentiment offer education bring year obvious slowly question place base topology size technique deal heuristic generic criterion pooling include pool slot neighboring slot pool approximately aggregate intuition global pool heuristic include slot preserve slot pooling b pool slot pool low position obvious aforementione slot pooling improvement address slot pooling dependency tree word order order slot position sentence extract pool efficacy method along pooling task penalty detail evaluate sentiment question conduct widely discriminative stanford review setting prediction fine grain strongly neutral negative coarse versus sentence neutral prediction simple class neutral class discard take stability place difficult list control comparison consistently outperform rnns extent flat cnn show important sentence tree information rnn integrate far evaluate model sentence plus split target entity location svm rule cnn cnn c convolutional propagate model various art traditional utilize code human engineering knowledge classification reduce extent architecture qualitatively light mechanism pool fair reasonable tune hyperparameter consume largely hyperparameter sensible hyperparameter report protocol different initialization summarize complicated sensitive pooling mainly serve necessity deal experiment slot pooling literature pool efficient rnn achieve sentiment typically epoch ccc slot sentence group smoothing sentence length comparison rnn achieve overall slightly bad think fair sensible contain confirm analysis sentence rnns rnn difficulty explore convolution propagate especially long sentence mechanism neural process pooling ultimately supervise back come layer pool pooling tend vice process global sensible slot windows sentiment convolution pooling know mostly sentiment result tree convolution window window root act child see window sentiment window neutral sentiment window sum window novel discriminative sentence parse build denote variant achieve high sentiment slightly outperform state art task convolution sentence effectively useful li xu cn com software institute china convolutional modeling model leverage either dependency sentence structural aggregated max underlie detector effective feature extraction state dedicated rule effort visualize convolution modeling aim capture sentence g various attract attention nlp community feature engineering dependency subtree dedicated one svm specify sentence advance neural bring language considerable propose unsupervised learn word real sentence neural automatic learning sentence cnn recursive rnns cnns neighboring capture inherent structure parse rnn composition parse long cnns rnns variant recurrent review combine advantage cnns rnns whether rnn propagation cnn propose neural call parse tree tree variant convolution subtree detector slide entire parse sentence extract dimension architecture feature propagation path learn sentiment outperform understand visualize code result website present architecture discriminative convolutional cnns processing language depict classic convolution detector sentence window word convolution position detector parameter concatenation convolution pool fix size convolution effectively interact though convolutional local
role rewrite less demanding finitely finitely section polynomial exercise chapter know exercise follow regular sequence finitely dimensional whether polynomial polynomial lemma large arbitrarily fact span example swap show suffice hold identity block nonzero polynomial involve proper subset polynomial contain univariate polynomial evaluated observe since common root elimination chapter conclude partition polynomial uniquely contradiction exist choice position prove satisfy finitely dimensional independent basis imply indeed case unique system equation finitely mention permutation row large say uniquely useful relation hold column r assume arbitrary hold matrix column follow satisfied per form denote independently term sum may rewrite last follow ready fails first fail characterization set finitely unique uniform lemma remark circle font deterministic pattern low completion university completion wide previous provide completion miss finitely observe matrix guarantee contribution set derive sampling unique high column observe attract attention range recommender filtering entail study know approach require coherence gap theory loose condition incomplete finitely agree infinitely additional guarantee depend entry characterize question main characterization pattern complete condition finally organization formally leave statement additional nonzero location binary entry dimensional span entry place may column subspace subspace consistent entry consistent entry different paper result sufficient guarantee constraint turn introduce express column block constraint determine redundant indicate statement hold almost zero statement one main necessary sufficient hold every subset satisfie follow uniquely follow satisfied form form column satisfie hold satisfie define little additional uniquely pattern prohibitive especially pattern satisfy probability scheme column independently least vector compatible I observe formalize coordinate degenerate minor determinant determinant measure almost every subspace subspace exist infinitely hence system become impose subspace characterize finitely many expand equivalent eq recall may subspace nontrivial infinitely infinitely associate subspace observe
behave presence trend structural var variable value causality topic phenomenon use short recall focus particularly autoregressive move detail reference therein indicate total observation lag period similarly denote j j ty variance simple draw identically variable average respect resp ty concern relate time past completeness process ar forecast forecast square ols regression choose line closeness make hence give slope represent error nature later iterate take process time condition eq rewrite lag act follow k te variance constant covariances ar generalize ar random zero operator rewrite framework four e regressor forecast become forecast biased inference perfect regressor perfect introduce predict quantity forecast forecast coefficient ol estimator forecast p forecast refer estimate actual forecast forecast mistake make forecast actually occur measure forecast arise estimate estimate useful forecasting regressor claim lag check causality coefficient optimally choose lag aic bic minimize choice bic residual contrast time aic main large choosing length estimator proof relevant due trend trend persistent movement variable trend trend time trend focus simple stochastic trend forecast u forecast drift time stationarity trend bring issue coefficient series ol autoregressive toward cause trend interval valid example namely statistic hypothesis mistake hypothesis reject true hypothesis statistical probability call sampling nan hypothesis actually significance reject hypothesis approximate standard limit see g moreover call e model trend detail side hypothesis u standard test one hypothesis hypothesis ol alternative deterministic must analysis forecast section model forecasting var future autoregressive var period appear process impact variable model regressor lag case say var consist form normally var ar var ols reduce vector u compactly pl pl pe stationary lag polynomial stationarity root say exploit follow l u bl bl z remain move accordingly vice versa lag length determine var rule past typically include lag full cycle lag carry lag six lag capture year residual component usually decide lag lag use limitation amount forecast lag let ol residual aic compute modify replace set lag minimize iterate forecast compute var forecast use main forecast forecast forecast forecast period period apply compute ahead forecast ahead previous stop forecast ahead var ol coefficient question multiple series variable predict another answer determination causality g concept causality alone coefficient past nonzero similarly cause formal causal nan causality x p tx ty ty degree freedom statistic critical significance reject effect cause drift root equal difference follow random accordingly random walk trend say integrated trend say two say integration autoregressive stationary test unit autoregressive stochastic trend reveal series process always trend integrated coefficient integrate say decide exploit expert qualitative graphical checking common perform statistical initially ol see augment test concept extend say need see line regressor number ol past ols two relationship regressor along lag multivariate influential focus introduction regression correlation check goodness devoted application able example augment statistic exploit causality test information approach constitute core second aforementioned acknowledgement author acknowledge excellent dr give constitute core autoregressive time series work project deal depth study effective make forecast concrete framework major mainly consider causality trend useful constitute core present whereas project present data concrete area causality bic criterion trend
dimensional space physics geometry physics neighboring matrix geometry play important determine neighborhood grid become mrf provide preprocesse involve random enable pdf statistical article briefly definition terminology computationally explicit interpolation interpolation four potential current connection present conclusion research n euclidean boundary hull n tt transpose interpolation observe field regular grid estimate spatial field denote probable give accord integrate statistical analysis distribution mathematical property group spatial prediction grid whereas bandwidth support notation bandwidth indicator triangular neighbor choose local bandwidth accord euclidean space determine compactly infinitely avoid zero bandwidth sampling depend purely compactly support imply kernel near neighbor fail bandwidth represent point value field normalize preserve unity configuration energy lead explicitly local maximize joint functional following h average fluctuation euclidean dimension two curvature influence motivate positive parameter contribution curvature control bandwidth functional partition condition metric intuitively justification average positive multiply increase multiply sign space express symmetric square identity otherwise kernel index curvature term h follow row vanish eq diagonal expression non I maximum likelihood leave calculation operation datum bottleneck optimization computationally efficient requirement store use use follow base reduce apply interpolation involve vector exclude negative respect validation matlab function initial parameter use assume lead optima value optimum optimum use global point unknown value functional give mode network concern interaction term neighborhood interaction sampling case bandwidth control illustrate distinguish difference weight compactly kernel imply pair contribution denominator point weight combination q energy functional precision ph bandwidth determine neighborhood illustration root side root diagram represent whereas represent term eq minimization mode modify vanish transfer element analogous krige predictor validation obtain compactly support neighbor distance ii k well dominant term computational time investigate approximate double denominator analytically evaluate double minimum globally estimate respective mae rmse measure mae mae reflect optimum slightly optimum quite covariance truncate dark online correspond area online value scatter respective daily automatic monitoring spatial exercise study thus comparison investigate measurement release corner simulate dispersion value magnitude rate measure hour h set gaussian minimum optimal illustrate isolate low sampling along boundary convex hull imply validation recent mean spline exclude cross measure range I mae map generate interpolation cause bilinear interpolation bilinear interpolation matlab b interpolation text measure predictor I error bias mae rmse I mae determine use leave point show whereas spike plot vertical range variation latter determine ensemble believe partially slow function ern slow variation parameter degeneracy may recently involve exponential quadratic last table result include optimal close counterpart high variance cross validation exhibit variation precision non imply sparsity matrix rely estimate thus reliably pearson correlation low extreme interval mae rmse neural look mae rmse close validation describe I rmse near neighbor knn employ knn determine estimate locally imply vary locally neighbor algorithm type improve use location input value output variable long plain version complexity fix neighbor alternatively locally efficiency estimate mean leave dependent version involve regression neural similarly surface case correlation abstract suitable imply curvature coefficient characteristic length correspondence establish rectangular available datum case parameter reasonable initial support neighborhood contain least arbitrary exploratory run help global approach use cross validation functional use involve measure square investigate search investigate include herein local optimization lead method performance improvement set bridge machine covariance implement neighboring algorithmic missing point cross validation efficiently rectangular grid calculate without function store large big spatial investigate extension model case web address laboratory manuscript space operational education european machine framework model scale require application present employ combine idea physics computational define involve mean sparse matrix expression interpolation number avoid big expect abundance similar scientific engineering field design
intersection contradict step therefore yet time customer change increase bandit mean despite foundation fundamentally dependent pattern account implement greedy exploitation period period understand correct present insight would want exploit excellent exploitation yahoo heavily series substantially contextual management regret classic pricing maximize price mix explore pricing exploitation yahoo front goal ad g classic static trend customer peak trend google query com would ad trend ignore classic armed bandit analysis arm pattern many form case effect beneficial customer certain day yahoo front article short much well explore article insight key exploration period likely make optimal suboptimal simple reward multiply know multipli high explore reward approach general micro I playing period round price dynamically result reward armed become explicitly care arm exactly suboptimal multipli classic bandit logarithmic inspire exploitation phase mining competition recommendation yahoo article article would short period available click article keep alone almost percent allow setup work differ work dependent multi bandit arm exhibit multipli accordance work know analysis discount ucb slide ucb change stay extreme exhibit brownian motion differently rather one work extremely micro occur length multiplier certain threshold stop information round propose consider available form dependent trend reward assume advance periodic google etc draw time arm mean let round let play jj otherwise play update provide regret reward last round initialization expect low playing make finally arm good arm high reward period usual rate harmonic want compare usual greedy suited set multipli discount divide greedy present play suboptimal arm reflect moreover usual less easily give jt jt jt positive mean reward arm lower greedy hard exploitation arm generally usual output jj mt uniformly play arm appendix logarithmic hold term regret bind give initialization play arm good probability decay jt grow remark appendix jt jt arm quantity remark appendix grow arm change w md w order interpret happen multiplier grow brevity abuse still little probability choose arm qualitative choose bad decay greedy direct consequence slow typical much well greedy introduce ucb multipli reward exploit good round reward call order let high total round high define short play regret achieve way sequence decrease amenable type ucb bound wrong arm choose proof leave calculate weak regret ucb regret high collapsed term round threshold round ucb soft ucb gradually contrast threshold reward exploitation enhance effect multipli exploitation let define follow q correct decision chernoff hoeffding second jj mt grow regret soft ucb bound identical reward modify multipli three type multipli wave want find far peak arm game figure wave normally arm choose make play essentially good multipli reward multiply estimation period website people peak multipli click ad show obtain comparison version reward discount round divide algorithm usual present multipli figure figure bernoulli success advantage reward ucb use h initialization mt output loop initialization jj mt play motivation scoring exploitation well recommender yahoo news recommendation yahoo unbiased evaluation challenging trend click arm news article access handle lead key insight although feature feature substantially turn insight gain time arm appear show arm drop global trend need modify stop exploration click rate stop explore old stay old
matrix ignore argument dynamic world normalize stationary toward equation stationary generate exploratory plain global system rule stop decay generate motion system lead signal avoid activity give plain describe sect novel propose body stay perturbation body external vary change initial act individual external influence quantitative learning sect control base position harmonic example xx u x factor average sum replace integral stationary dimension carry jacobian pair complex observe intrinsic generate sensor complexity physical deterministic inform exploration notion gain tool tool paper proportion weight human body degree arm robot use position perturbation position sensor force appropriate internal inverse sensor realization behavior reach exploration predefine work produce motion never self quality robust phenomenon circumstance understand broken symmetry individual symmetry break nature phenomenon root dynamic sequence situation author acknowledge discussion comment gm grant receive european union grant agreement institute system fundamental behavioral development provide answer unit require high level construct learn specify level intelligence surprising specific modification break brain plausible target investigation argue evolution brain small progress mechanism signal transmission enable understand organization however gap bind leave open question interact self organization gap circuit acquire make local together together early rule however like scenario learn drive generate perspective determined action self organization past contact relate behavioral feature together bootstrappe behavioral plausible naturally raise question whether aim encourage yet decade paradigm ai ai role environment close hypothesis argument present generate phenomenon self behavioral dynamic without construct motivation support material overview generic realistic link joint angle measure like implementation drive force simplicity complex external controller code neuron transform sensor feed forward activation type simplicity translate network particular intelligence plan brain require organize internal never mode dynamical body environment feed forward pattern rule plain simple robot controller law rate would proportional multiply concrete fix suggest time activity derivative focus demonstrate sect behavior main reason behavioral involve trivially lift entirely outside physical control let assume basic sensor approach realize approximately relate cause time lag r never exact reconstruct copy formulate write scale dynamic fig decay principle relate controller step together physical explore time idea neuron experiment threshold dynamic periodic inverse difficult different lead unfold motion pattern separate basic sensor capability development behavior matrix relate robot force constraint classical method alternatively order self organization experiment make whole exploratory behavior normalization latter feedback strength perturbation active maintain see sect edge argue life development finding see sect system strategy long spend behavioral reflect spatio pattern simple local rule eqs understand behavior generic call biased central obeys geometric also physical system transformation sign angle expansion situation learn symmetry breaking preference breaking picture break symmetry event rich scalable symmetry break newly propose specifically study due lead specific behavioral cause signal model simple calculate inverse back neuron neuron shift correspond classical pointing way little neuron latter term additional effect fire unchanged accomplish enhance intra series experiment demonstrate self organization interestingly behavior seem system turn task orientation always neuron start first study stage common understand cause abundance assessment robot neuron choice motion primitive duration realization storing alternatively normalization gain threshold another behavior formulate contrary dynamic nevertheless mention converge create large highly coherent depend combination body couple definite threshold include organized ground additionally behavior robot slightly fall arm first contact ground force create shaped environment interaction either different behavioral mode video meta stable perturbation mainly external observer look robot explore body change behavior generate inspire degree joint delay row step two record e interaction environment dash line indicate preference forward back specific body break forward backward backward break happen break desire way element additional facilitate circular delay sensor implement inverse organization connection delay forward backward posterior link direction anti two room behavioral first anti result see connection leave pattern transition front additionally fig second resemble observe subsequent delay sensor subsequent opposite side anti smoothly decrease delay sensor type small delay call loop cluster supplementary switch robot video sequence video notably motion pattern physical body robot interact dynamical massive robot massive force robot angular velocity action recent immediate learn process lead robot eventually end meta periodic crucially mass must large feedback external observer say channel search physical position robot force role couple axis force robot immediately rotation otherwise opposite video robot find influence variant experiment interestingly robot need much two upper body robot indirect sensor actually correlation occur force guide robot specific paragraph interact coupling let exchange extend drive force induce perturbation sensor lead eventually video outside level effect local plausible enable self show self determined without level orientation understand exchange organization artificial neither task specific system discrepancy artificial dynamical reality self multiplicative behavior evidence adequate behavior interaction argument edge allow break dimension human nature apparent goal search open nature dynamic
increase dt sn field variant name particle filtering interact theoretical extensively algorithm problem arise state framework occur recently class address space generalize smc support term smc sampler static regard smc particular refer target arbitrary distribution weight sequentially process mutation particle distribution aim weight n n n give filter interact particle sensible would construct asymptotically integrable denote example one follow modification smc develop final kernel define particle backward correct framework distribution smc stage whereby via mutation particle reweighte via incremental importance weight measure particle diversity commonly reduce detail weight particle particle use mutation kernel new w incremental importance weight sampler specification subsection basic sampler explicitly utilize expression need perfectly acceptable sequence normalization constant correct lack importance normalize obtain population event smc insensitive depend target follow cell let eq particle construct sequence population probability smc sampler posterior weight n n abc tolerance draw l reverse describe mutation particle abc incremental bias n sequential carlo joint mutation weight incremental induce identical smc marginal joint sampler sampler theoretically use suboptimal approximate approximate backward calculation denominator abc posterior smc develop abc specific target call anneal abc consider abc emphasis increase abc construction may consider smc procedure discrepancy stop online discrepancy tolerance fitness particle indicate take perform particle tolerance equation specify level cdf distribution quantile stage abc way tolerance schedule adapt local approximation efficient tolerance calculate strictly mutation smc sampler specialized mutation genetic search mutation particle mutation mutation smc formally mutation operator genetic distributional provide mutation follow select mutation say stock figure asset month spread ease financial asset chi operate book option book meet relate consist visible book process exchange matching engine range size limit sufficient construct much picture previous consider aggregate volume level volume per spaced construct precision fitting auxiliary describe volume subsample price volume interest auxiliary purpose compare indirect term make basic estimate term agent reasonably produce realistic intra daily reduction dimensional summary volatility dynamic inside spread instantaneous volume bid summary employ characterization capture specifically mid return suitable interval aspect parameterize volume bid side volume distance algorithm specify tolerance force iteration specifically procedure schedule employ force specify result test tolerance estimation configuration low value quantile tolerance show forced schedule run mutation compose mutation component degeneracy practice efficient simplify mutation mutation eliminate weight numerator denominator produce issue dimension due nature cross particle exclude possibility particle section abc trial axis distance intra day volatility price intra smc obtain estimate agent replicate feature relate price dynamic clearly value particle sampler standard optimisation evolutionary present discussion pareto intra high weighted figure intra repetition particle volume market ht result abc respectively indirect multi estimation abc easily introduce distance metric ii procedure consider multi ii vector framework parameter vector use mutation kernel solution abc return comparison highest return former non simulation financial market particle try fair comparison mutation operator highlight difference smc present firstly procedure suffer particle degeneracy mutation mutation secondly default iteration particle degeneracy additional front chapter stochastic demand asset exchange real perform via inference adopt result estimation perform adaptive compare indirect class quantitative finance attribute book primary intra every exchange place one fundamental dynamic important attribute asset recently asset explain real range truly pointwise trivial addition amount big day asset one calibration calibration indirect inference adopt algorithm widely search mutation sequential sampler smc sampler abc framework finish european secondary chi x recently structure financial market financial market provide drive market participant trade matching mechanism stock together pure york stock exchange stock exchange hybrid operate less trading activity chapter market participant allow place order price execute price order execute immediately order opposite book trading specify interest book display snapshot book particular stock chi market share share share remain share execute immediately enter limit book second share simulation model allow day trading process behaviour market participant financial limit market order depend model instantaneous mid high bid ask price modelling intra dynamic stock volume well reproduce address hand intelligence consist trading order possibly consideration impose action reproduce feature market tail later realistic agent financial asset necessarily price output reproduce one prominent power law intensity limit level unlikely modern presence strategy dependence clear present formulation representation model rich propose section abc purposes section book intra present stochastic representative reformulate computational day every bid ask side dynamically evolve respect equal price low price interval price interval bid book away ask book away expect bid remain price period course bid price order away subscript passive order order order understand reference price start level bid assume activity uncorrelated top unlikely impact price volume volume model unchanged interaction level inside band model dynamically evolve observe feature modern present detail include agent order model level random book side order ask order activity class model activity interval passive agent order vast majority typically execute particular level order generally consider small assume responsible market ask market participant execution take market execute bid ask side trade multiple market asymmetric information market size include order bid ask vector order furthermore order level conditionally stochastic order arrive bid property multivariate cox l intensity monotonic distribute accord skew kind skew copula importantly scale furthermore order independent si market maker activity limit order place evolve trading day place bid allow capture structure level bid analysis frequency dependence skew exchangeability intensity bid tail occur intra market produce bid depend book equally would occur skew copula process skew intensity bid therefore order specify stochastic model exception number order level order level c construct latent matrix k cox distribution order high order queue assume remove critical market ability adjust activity execute maker removal activity market evolve intra reflect nature trading limit order ask bid ask skew copula non exchangeability stochastic bid ask book feature mean market adjust volume order create new occur trading addition principle limit order volume book instant representative specification intra daily book market participant market participant type throughout day often market order limit chapter activity model dynamically evolve place market order compose component stochastic structure k order furthermore stochastic opposite order l bn accord cox k transform intensity strictly monotonic mapping characterize n k ki real dataset volume level ask stochastic agent model specification describe demand type intra synthetic state time denote obtain base transformation map activity incoming event limit process section trading activity simulate trading day algorithm model
run dataset gradient crf learn report average learn amount practice sampling computing require partition function structure prediction pair test significance legend anchor log xlabel ylabel bandit collect performance collect explore exponential thin tail hold detail performance affect experiment train increase improve trade probably prediction often produce hard legend xlabel height coordinate improve train varied quality crf hyper map keep condition policy affect hold multipli derive variant generate change deterministic predictor derive long still upon stochastic recover achieve ht legend style xlabel ylabel coordinate coordinate become within trend across remain unchanged include supervised capacity serve robust batch feedback risk bind hypothesis away dependent robust principle prediction optimize rich family learn massive classical generally learn non loss function extension relax assumption handle etc research rao anonymous constructive principle batch bandit feedback g ad query bandit feedback user click ad nature problem score generalization constructive minimization policy optimizer prediction decomposition objective enable efficient substantially datum record recommender ad little interaction contain record e feature describe g recommend news article feedback article read feedback provide partial feedback fundamentally correct prediction g news article provide feedback bandit feedback interactive control offline cross perform etc batch system control estimator develop interaction bandit algorithm evaluate perform sufficient unbiased policy reason pick conservative generalization error structural minimization family constructive nature principle bandit derive optimizer structured prediction decompose variance linearization effective classification problem verify support detailed risk principle parameter structure output optimization discussion approach fall give incomplete feedback estimate prediction supervise feedback generalize sophisticated cost offset allow perform output sized batch bandit candidate exhaustive approach gradient family equally expressive build develop optimal doubly additionally focus inverse doubly estimator tight concentrate historical stochastic like exploration bootstrapping allow strategy pick bind successful problem like supervised risk armed bandit regret contextual bandit beyond approach implication several application feedback warm armed bandit pre retrieval evaluation bandit ht supervise hx p xx mh could indicate assume fix unknown hypothesis note hypothesis notational convenience assign interactive system observe sample indicate risk maximize user wish interaction batch assume historical collect system typically supervise ideally need full explain set candidate test candidate collect comparison hypothesis principle supervise outline fundamentally feedback class equation finally constructive motivate learn bandit optimize well serve analogy risk minimization call pick true additive translation multiplicative risk crucially equation degenerate degeneracy arise objective conservative selecting via biased estimate similarly hypothesis avoid unbiased h un particular tail sampling really explore region favorable algorithm classic g structured machine use weight multi news label could simply concatenation bag assign efficient think multipli induce deterministic gradient classification interactive predict system document feedback correct loss outline analogous adapt yield equation prior sound batch bfgs box optima implement code need variance obstacle develop spirit multi svms converge local optimum training objective decompose differentiable optimize overall taylor concave approximate w tw du iw iterate proceed input popular problem structure svms simple crf essentially independently outline experiment collect repository range l supervise bandit policy collect principle arbitrary stochastic policy crf amenable loss supervise hamming incorrectly label false negative I I h feedback g learn supervise report loss rw hamming agnostic handle parametric
elimination wise appearance length representation per frame popular space align feed train spatio train cnn attention simply activation last combine mechanism frame cnn cnn element cccc cnn global video description incorporate lstm outline incorporate feature add mechanism incorporate otherwise whether performance video generation estimate maximize description description backpropagation word maximize probability validation stopped update sec cnn cnn confirm temporal convolutional report automatic first use four line comparison beneficial exploit benefit attention exploit temporal free across exploit temporal structure fourth consistently design reflect description intuitively table reflect description present video corresponding description model fourth video description correspond video dataset perform evident second mostly panel frame type cnn correctly oppose simply work video capture temporal structure frame wise fine grain motion consecutive frame order global propose learn frame decoder generator empirically validate approach indicate model furthermore result text use addition preliminary gain possible leverage generation direction thank acknowledge cifar li universit universit e universit universit universit recent recurrent neural rnns motivated video static video require dynamic description temporal video description incorporate spatial convolutional short temporal cnn tune human motion automatically select temporal current dataset large challenging pair video description open activity challenge vision hour automatic video help index online video conjunction synthesis video description visually description consider challenging automatic carry contain moreover video description sentence characterize typically frame interaction actor evolve amount vast collapse fused video description generator exploit underlie argue video structure grain motion information characterize action answer action localize time consecutive temporal video refer action video video summarize elaborate description focus salient salient video framework video appearance frame input train neural network frame collapse via simple entire ignore two collapse paper introduce exploit encode structure derive spatio base recently context translation generate attention small frame generator activity see start descriptor abstract action emphasize video generator effectiveness temporal domain call video description per video much track movie video use encoder decoder capture spatio promise static frame description method suggest temporal generate mechanism experiment static throughout video video exploit cnn describe approach purely description decoder generation decoder neural encoder decoder encoder encode representation sized architecture choice rnn length symbol input convolutional neural cnn generate encoder encoder decoder choose type rnn decoder step internal symbol rnn run recursively symbol detail choice decoder automatic video cnns successful recognition beyond recognition task cnns variety task localization vision representation cnn feature activation layer video temporal spatio temporal convolutional neural cnn recently capture cnn build representation preserve local motion descriptor frame sequence divide spatio histogram orient orient flow motion sure extract cnn convolutional activation relu max pooling convolution relu layer temporal video feature temporal feature spatial width height frame within finally combine concatenation image frame position train cnn train recognition activation relu pooling feed classifier compute aggregate dataset video video complete architecture cnn use convolutional ultimately temporal feature cnn duration action complete video averaging exploit frame action entire event allow video duration short duration attention xu exploit structure exploit temporal video averaging vector decoder weight reflect relevance temporal feature video input decoder summarize previously temporal return together normalize attention unnormalize score normalize attention mechanism allow subset frame temporal selective inclusion decoder exploit temporal empirically fig graphical illustration attention generation study work domain video embed video tend translation video generation low video representation intensity sec sense closely recently static approach apply et adaptation generation limit recent annotate observation description accurate description visual content well activity base cnns
hardware gmm apply l nearest mainly cross illustrate impact fig big lead accuracy table report art part cnn train layer imagenet deep table already outperform respectively stage annotation utilize conclude bag superior pooling bounding box demonstrate gain gain effectiveness view particular representation method fusion pre train benefit detection classification paper instance solve problem exist work level ability boost annotation framework extensively art experimental ability possibility scalability possibly improve establish criterion noisy proposal secondly suitable candidate pool directly select proposal pool without aid bounding box framework edu sg yu zhang advanced digital sciences yu com bin laboratory technology china edu wu laboratory novel china edu university edu sg great cnn enhance discriminative feature framework framework transform multi object proposal view representation proposal exploit label utilize ability boost category art method power combine deep art dataset availability cnn great task powerful problem generalize category application propose address label recognition level label indicate truth box label utilize tune train cnn representation diversity image car force hundred car multiple fall instance feature region densely fail poorly inspire utilize weak supervision label fortunately many strong supervision box solve understand ambiguity obtain box bound box annotation boost unseen category label two multi tackle task separately view utilize box form view spatial configuration weak label tune cnn view encode view global representation view train classifier balance abstraction local similarity thus enhance importantly utilize encode distribution proposal feature partial bounding box overview area vision machine cnn multi cnn adopt label recognition cnn cnn global dataset different image imagenet different global method help improvement weakly max pooling score employ multiple scale achieve multi view deal learn combine view supervise employ separate contain propose several study combination computer vision mainly vary utilize near vote linear analysis tailor neighborhood learn local satisfy train metric metric vary region space formulate label recognition object proposal detection become e target proposal background object classify multi complexity scale single need object cnn detection selective good hundred enough proposal cover category every instance particular employ selective object selective extra ground truth box proposal selective search find traditionally optimize alternate typical ability limit applicability efficient originally image assume parameter k mixture mean whose soft assignment probability map proposal abuse representation proposal proposal feature proposal subsequently utilize good accurate local intra class focus spatial view enhance effectively encode configuration candidate local determine candidate relevant particular ground label could pool ground object study local study metric mahalanobi satisfy eq certain symmetric definite decompose distance equivalent map transform cnn metric nonlinear minimize loss encode discriminative distance instance margin encode neighbor positive neighbor class distance employ near belong learn discriminative specifically replace neighbor utilize label abstraction build pool extract cnns convolutional conv three convolutional specifie apply pooling row full conv conv conv conv conv full full st st drop soft pool pool encode information near neighbor structural although linearly issue extraction inspire incorporate neighborhood encode extract proposal feature pool label label label annotate label utilize bounding box ability unseen category exploit visually close category box annotation cat cat train exact annotation visually object exist strong supervision boost validate feature view label representation view cnn convolutional fully connect fine full cnn dataset train network process subtract image also relevant currently involve stage cnn network logistic label vector iy fine seven last connect large use view execute tuning tuning process object fine logistic final margin neighbor fine accelerate seven layer connect view instance label evaluate challenge dataset dataset ap test cnn pt em train cnn cnn svm extract layer train representation imagenet visual limited imagenet first layer cnn layer replace adaptation train target extract object image cnn level tuning label tune final
auto encoder attempt contraction denoise chain perturbation decode deep direct generative neural carry likelihood stochastic descent secondary network variance gradient finally feed forward feed directly use transformation space learn cumulative density mapping sample exactly train minimize mmd minimax use adversarial specifically generative layer top indicator long enough draw pass map vector multiple layer architecture relu layer output cc jointly define sample prior pass neural net network model indistinguishable generative carefully involve mmd mmd kernels generative complicated auto arguably capture reliably encoder create visual exist manifold mmd reliable literature refer operate auto produce code auto encoder generate code proceed greedy encoder fine encoder code mmd final encoding add encoding layer term manifold code motivation denoise play crucial determine mmd optimally open heuristic advanced approximation use kernel span range mixture sufficient obtain weight kernel far equally well result drive difference behave write especially factor maintain gradient issue mmd usage kernel overcome mmd minibatch update draw minibatch mmd minibatch mmd minibatch pf minibatch generating pass train minibatch throughout straight protocol density model scale gaussians search auto dropout encoder unit tune likelihood mnist stack deep adversarial net stochastic net dataset competitive significantly outperform despite mmd effective decoder powerful produce visually appealing reflect window likely perturbation transformation along decoder noise learn merely visualize distance merely tb mnist highlight red box interpolation go interpolation highlight highlight aspect explore uniform space correspond datum highlight right bottom projection realistic attribute change gender simple framework call moment network approach maximum discrepancy indistinguishable mmd trick explicitly compute moment use minibatch descent training combine model fed original auto generative simple combine mmd readily mnist achieves superior demonstrate discover implicit manifold direction mmd alternative mmd criterion possibility time develop literature possibility expansion inner explore mmd long grow minibatch original advantage statistic entire another like treat encourage well generation latent explore posterior straightforward way predict recognition auto mmd fair representation attribute mmd could statistic change sensitive variable utilize encoder create complex possible convolutional create color acknowledgement david helpful regard provide deep generative feedforward recently generative adversarial adversarial technique hypothesis mmd simple statistic sample model backpropagation generative encoder mmd generate produce combination compare baseline mnist area deep neural memory task recognition translation supervise feature recognize promise inherent good offer generalization qualitatively assess generative matching begin easy draw network output quickly oppose mcmc necessary boltzmann recently adversarial unlike difficult backpropagation behind maximum discrepancy minimize discrepancy moment trick mmd minibatch contribution encoder behind encoder code encoder rich encoder model original decoder auto encoder yet effective produce generative mnist face dataset comparable baseline include available set ask answer question sample statistic similar distribution formally follow mmd match high moment write product
ising make early bayesian core neighborhood distribution base pseudo follow candidate conditional differ define equal joint equation conditional marginal support equal support marginal everywhere describe clique neighbor clique regular manual point square clique array array determine derivation normalize array define involve summation correspond summation neighbourhood j array array array produce exact array neighbor array exponential term summation require show compatible conditional distribution conditional conversely conditional involve q exercise essentially exercise array neighbourhood neighbor whole simulate sum pixel odd version wang exercise simple case improvement wang sampler obvious book therefore odd powerful property gibbs color exercise find exact involve sum though reduce overall normalize joint associate conditional resolution exercise use arbitrary joint compatible conditional obvious probability proposal proportional step propose multinomial neighbor purely especially value wang exercise simulation account sub grid sampler piecewise explicit directly chapter appropriate correction boundary loss therefore every give mean posterior mode differently error basically look first look risk equal pick sum allocate reach configuration produce necessarily minimum experimental way check different compare deduce slowly run symbol integral call neighbor pair full conditional virtue exercise ise modality ise distribution invert pixel ise eq modify model simulation color image perfect site modify cx col col col k x cn book increase manual histogram grid detection cutoff empirical simulation replication opposite decrease manual histogram return manual grid empirical ise posterior statistic exact could tolerance albeit high computational cost gray chapter marginal q correspond mode locate expectation n jeffreys give associated bayes identification standard proportional gamma distribution proportional binomial proportional distribution proportional power transform conjugate exponential observe observe conjugate proportional choose integrable jacobian unknown part make impossible separate exercise student since jacobian integrating lead proper inverse conditional get conjugate student normalize student q scale prior jeffreys generic location py pz z determinant constant jeffreys provide change parameter jacobian negative value density interpret conditional product eq cauchy integration discretization compute regular summing product density inverse fix dimension operate conjugate prior arbitrarily family unbounde limited hence appeal un normalise post sd post constant integrate proper level seq seq cover alpha low simulate z go vary probability take go ratio go varie ratio less analytically maximal denote sigma manual bt evolution factor denote example denominator integrate evolution xy exercise numerator numerator normal simulating follow bf seq le bf bf setup express normalizing exercise denominator happen support evaluate integral normal density degree freedom finite denote student exercise integral show discuss student density simulation experiment lead three infinite alternative symmetric integrable proposal integral since exercise integrable integrable student infinite integrable density behave importance weight cauchy tail importance face tail evaluate use nu df df col col gold code variability huge jump series manual bt importance solution therefore take text exercise integrable run r alpha alpha alpha show considerable improvement evaluation manual gamma bt evolution three importance evaluation gold cauchy sample harmonic simply harmonic mean inverse pi repeat true invertible transpose produce use word column invertible nx linearly solve solve equation check lm call intercept x q conjugate give gamma identity integrate q virtue q n n exercise sense matrix unobserve predictive q covariance matrix deduce mn gamma produce prior nc xx associate joint show g orthogonal obvious generate eigenvector subspace dimension determinant jeffreys predictive predictive jeffreys prior therefore limit gibbs irreducible sampler work rotation axis respective center radius connect manual mean thus similarly next sampler depend whether start disk bt gibbs sampler conditioning union jump probability disk distribution irrelevant role derivation symmetric consider correspond full full conditional compare gibbs conditional gibbs col sample true marginal namely col bt gibbs output posterior associate ga full sampler associate conditional full eq q code conditional gibbs sd remove grid le seq log post post col add sample fit bt gibbs superposition conditioning give corresponding family associate eq conjugate n canonical function case poisson poisson link generate pdf q property table margin margin cell inverse parameter write independence sampler gamma proposal metropolis al shape mcmc else col col col illustrated manual show fit target histogram proper behaviour example acceptance gamma histogram maxima histogram output mcmc distribution iid cauchy estimate normal cauchy check graph function random metropolis code x df df df checking length mcmc n mcmc mcmc rt valid mcmc comparison manual cauchy noise convergence noise number iteration curve observation induce acceptable comparison gold mean metropolis sampler al mcmc mcmc al rate mcmc sampler repeat call output three matrix take range something manual valid range l remove loop replication running simulation manual sampler metropolis hasting may converge problem approximation iid hastings walks compare code book sigma exp I version n quite improvement hasting target mixture top histogram target probit flat prior q inner volume infinity probit exercise additional power nonetheless traditional control limiting include intercept probit bank intercept need add code function lag autocorrelation produce bank intercept factor manual book intercept posterior perspective take magnitude last covariate book inclusion intercept bank probit coefficient intercept flat iteration right last latent variable probit introduce respectively mu mu application bank output manual unobserve py iy I symmetric obvious mu inverse call library conjunction complete irrelevant regression x introduction gibbs variance truncate respectively base call library mod x probit coefficient sigma inf mean sd inf sd sigma beta mu sigma function represent figure manual manual converge induce move transition compare bank code iteration respectively bank probit intercept histogram iteration auto bank probit k n approximation simulation suggest book adaptation x c mean log bf complete get individual tag loose capture obviously summation sign acceptable keep sum obtain model book constraint r c joint distribution conditional exercise q capture indicator derive necessary life history consider approach schwarz begin likelihood case accounting constraint accept reject replace marginal still know normalize eq uniform matter normalise examine associate reject even thus output fall within accept reject algorithm trial accepted parameter conclude trial probability uniform fall surface fall repeat accept bounding performance exercise constant log ta x nt pi prop prop sd degenerate rapidly moderately uniform accept reject section gibbs conditional conditional special gibbs slice exercise since distribution density high conditional sampler slice inverting take case exercise rate reject algorithm exercise normalise simulated algorithm since since rate lead eq reproduce exercise exercise deal simulation modify return n test consist monitor rate return test q I show beta distribution get easily exercise law number coverage vary coverage vary schwarz group location devise block joint distribution special partly hide individual constitute capture relevance conditioning time past location conditioning location unobserve block addition bring block extend respectively mixture bernoulli consider mixture define identifiable restriction solution part classical usual solution box trick box identical box allocation ball ball equivalently box count remove extreme value partition sample partition empty indeed partition mixture unknown eq q latent variable prior propose imply normal allocate value factor sum sum exercise em apply test influence representation imply get possible implementation tt sd sd sd em em em sd em em sd manual manual increase start start convergence independent sum clear sampler parameterization unchanged since x gibbs therefore loop allocation mu mu mu repeatedly highly illustrate manual seem start sampler iteration show exchangeable weight necessarily exchangeable imply apply proximity map annealing also simulation integer share get neighbourhood behind anneal first concentrated necessary simulate whole convergent slowly iteration consider application mean modification simulate since normal hyperparameter modify version gibbs z move guarantee walk behavior proposal ratio logit walk show simultaneous power normalize simultaneously importance difficulty normalize denominator run experiment influence observation share global get increase global define neighbourhood anneal concentrated mode large necessary whole convergent map increase slowly iteration much requirement book immediate simulate simulate p prior conjugate trick mean gibbs iteration I compute completion mean difficulty around sample close may many simulation mixture unbounded likelihood sample procedure image sum partition partition allocate single term unbounde grid mu seq length seq surface like manual exhibit illustration unbounde random depend bt tw bt bt h suppose show stationary moreover sufficient ar autoregressive integrate jacobian expand eq main close predictive distribution acceptance also reduce write hasting current indicate iid double impossible stationary ar point satisfied p eq associated ar define proportional posterior integrable integrable coefficient derive recurrence expand q j recurrence derive root test degree denote reciprocal polynomial
transform author investigate transform fundamental symbol coincide et transform know although investigate admit relu challenge transform admit relu approximate b transform space denote sphere respectively write class refer lebesgue dual polynomial smooth compactly function distribution compactly smooth decrease function consist vanish space space identify treat list range may associate fourier eq hilbert principal satisfie denote design fouri fundamental proof eq fourier leave right dimensional every respect old transform dual way say admissible provide belong class admissible q radius use symbol parametrize b absolutely immediately transform justified follow one integral integral iterate integral immediate right always exist dominate write remarkable product color depict absolutely convergent wavelet absolutely distribution coincide ordinary locally integrable contribution balance theorem converge pl get versa composite verify restrict use l f balanced assign table fact approach direct instance et admissible reproduce always instance laplacian accord convergent non stem thus dual transform fix classical uniquely mf uniqueness condition formula fourier domain domain key constructive derive constructive fourier slice transform theory say zero admissible exclude unique distribution product modification example problem origin equivalent change condition justify admissible pair construction admissible admissible k computation satisfy almost addition convergent fouri convergence theorem fourier transform another wavelet reconstruction formula q three p p j v identity j limit fundamental identity j pa formula proof control constant control compatibility fourier admissible admissible converse correspond side lead lead lebesgue relation relation identity k reconstruction topology uniquely universal recall checking table list activation belong subspace admissible approximation activation truncate z tangent radial dirac rbf derivative dirac proposition slowly function contain relu belong tangent obviously accord since bound proposition z derivative recurrence recurrence rbf derivative dirac derivative theorem admissible z k z calculate moment admissible even straightforward admissible odd obviously kk z e k perform experiment reconstruct image theoretical list theorem admissible necessary admissible necessary activation sigmoid sigmoid dirac relu activation derivative relu unit step dirac addition examine activation admissible fourier polynomial origin fourier domain numerically integrate radial grid potentially bias cccc fa truncate power function line draw reconstruction draw figure theoretical reconstruct rather completely fail necessarily work reconstruction dirac fail reflect difficulty dirac origin fail relu lack reconstruction relu work note step relu reconstruct sigmoid reconstruction fail consistent polynomial transform origin draw dotted cccc real dotted h cccc derivative gaussian dirac step relu h signal define reconstruction formula b list result fairly bottom reconstruct dim understand cause pass z unbounded activation universal property neural coincide construct respect wide activation traditional rbf truncate relu dirac precise admissible expression transform distribution existence distribution suggest convolution converge balanced coincide dual long admissible transform unbounded admissible polynomial exclude condition direct consequence sufficient investigate reconstruction slice approximation identity reduce inversion latter construct filter
frame similarity expect representation extract accomplish temporal temporal formally node divide induce diversity divide constant long weight ask happen initialize initialize matrix zero extract meaningful representation limit despite initialization might get reach image temporal frame experiment change result negative weight weight encourage matrix exclude sign constant absolute initialization accuracy express architecture detail convnet imagenet yield good imagenet choose study drop dropout fc transform convolutional pooling convolutional start imagenet network large evaluation split spatial convnet negative convnet fc lstm composite lstm convnet initialize ia spatio temporal convnet look whereas convnet look label edge detector spatio temporal convnet combine rgb optical flow spatio convnet single optical optical flow give slow fusion slow give softmax score optical slow fusion lstm fusion slow flow early fusion lstm give result slow additional average ex stream convolutional fc convnet multi stacking spatio convolutional net create spatio incorporate spatial convnet despite limited label video outperform scale label video dataset spatio convnet initialization performance label challenging incorporate spatial train spatio temporal initialization spatio convnet convolutional temporal video method video contain shot frame video represent continuous live temporal information video recognize getting get training spatio facebook consume require nearly spatio severe convnet imagenet tune video solve overfitte give model representation tackle propose several way convolutional temporal representation video convolutional dramatically overfitte video dataset learn improvement convolutional layer otherwise spatio convnet spatial video appropriately composite lstm example nearly match classification action mainly drive extended deal video traditional spatio temporal video orient wang propose trajectory feature fix fisher vector lastly classifier distinguish state dataset availability deep motivated scale label video deep spatio temporal originally propose et convnet fine tuned frame extract dataset show deep early spatio convnet dense flow extract video
band dynamical benchmark offer scenario multiclass classifier vc vc capacity generalize svms support paper describe dynamical system analogue circuit use ordinary ode solver uci repository improve accuracy reduction vc minimal system svms widely machine employ application cut edge utility demonstrate svm formulation quadratic programming admissible margin suitable constraint identify involve system algorithm learn vc value undesirable svms vc imply give equation recently show term minimal exact vc solution programming generalize benchmark dataset svms term use far support instance well variant dataset paper solution vc dynamical converge minimum vc classifier analogue system complexity modelling domain provide hardware implementation base dynamical attract significant attention last decade real realization circuit recurrent system interesting biological solve problem symbolic memory among integrate programming propose network surface breast diagnosis dynamical system barrier present neural programming extend originally recurrent programming bound wang wu neural include dynamical mathematical assignment liu demonstrate neural constraint utilize introduce minimal describe system discusse conclude remark sample dimension interest separates show vc imply close capacity minimize minimize attempt small make misclassification error fractional transformation lead transformation machine capacity formulation determine hyperplane map solve equation primal resp dual denote evolve couple moment formulation equilibrium element equation represent occur resp order differential determine coefficient matrix use make positive positive asymptotically imply trajectory equilibrium matrix redundant converge assume hence equation find vc classifier aim augment row matrix find notation represent multiplication represents matrix show equation also term matrix write equilibrium visualize equation system namely dimensional belonging draw normal variable horizontal axis fig
every edge locate within community graph spectral assignment assume dynamic divide label permutation group random bp vertical recover threshold phase increase occur choice agreement network deviation numerically overlap plane bp along notably large indicate structure weak algorithm transition zero transition agree past bp large especially away transition bp spectral threshold instance draw derive mathematically limit threshold assume structure previously develop community edge give latent structure bp dense consist edge handle step extend evolve continuous hard regime factorize describe fix regime theoretically possible believe group grow far include handle network independently annotation thank helpful financial research science grant fa u air force office advanced project foundation author pz us asymptotically consider detect well labeling latent sharp chance recover latent difference internal sbm phase membership I call take add temporal adjacent imply locally become temporal along copy community label respectively multiply distribution eigenvalue node temporal two branching edge give temporal adjacent version temporal give rise temporal time spatial describe expected child multiply population ref large correlation recover fix make arbitrarily dense imply community threshold fall two pass edge correspond noisy leave propagate rigorously static community conjecture theoretically community run exponential graph control bethe propagation bp cavity carry asymptotically backtrack perform way infer node time belief propagation tree locally nod marginal temporal neighbor pass along edge temporal fig illustrate scheme message node asymptotically partition linearize community verify average reflect permutation symmetry system unstable due perturbation parameter bp equation system bp factorize correlate latent spin simplify equation study message factorize solution linearize version static sbm linearization equivalent backtrack rewrite message deviation linearize neighbor derivative amount eigenvector jacobian derivative bp relatively backtrack convert temporal ta uv use eigenvector absolute compose node cluster vector community group complex circle give panel area region static point dynamic principle linearization eigenvalue non backtrack bp equation I real value unity
use construct flat test level prior encode object class predictive hierarchical prior prior flat prior hierarchical model degradation model characteristic hierarchical model improve decrease large visual appearance frequent large abstract appearance category formalize class share statistical strength sharing understand prefer hierarchical one avoid sharing perhaps prior suited place great category appearance maximize primarily lead hierarchical might political science census growth answer batch online batch relevant image whether batch prediction formulated training category sort reconstruct assumption learner one predict output suffer observe attractive circumstance bayesian think employ robustness speed allowing give guarantee future unseen generate particular simple bayesian follow explore extend certain glm logistic proof defer answer question important derive gaussian parameter lead great robustness next hierarchical gaussian complement learning theory prior model finally prior parameter small online must observe make glm py py choose modeling analyze world scientific application py density observe observe learner predict py py cumulative aim fix glm log rely loss bound glm py throughout remainder first understand likelihood require scale versa py py logistic respectively proposition learner distribution write spectral uncorrelated repeatedly choose attractive deriving regret directly posterior analytically logistic regression generalize originally glm specifically glm py matrix derivative hessian mean p multi label class belong combined likelihood make desirable bound develop pac relevant receive probability generality pac consider predict bind distribution pac risk regret fix attractive rely pac require calculate remain risk erm proof see l choice tight almost circumstance satisfy poor state erm bayesian predictor generalize small batch bayesian bound prior robustness share statistical strength selection relate choice hyperparameter hierarchical parameter place freedom one multivariate heavy tail decrease exponential cauchy recover place prior yield regret bind behave would heavy thought switch logarithmic behavior concave possibility priori might moderate value guarantee logarithmic large robustness specific choice bind bayes nr bayes cn c prior prior allow statistical strength provide answer condition strength hierarchical preferable sharing strength formalize framework number carry begin problem manner number example idea pac learn hierarchical bayesian pac task theoretic typically goal learn alternatively advantage use online receive task observation image source source accord learner observe place one level similar discuss qualitatively prior significantly nz hierarchical shot learn source make make large task cs cs hierarchical n shot provide new source case consider investigate bind two prior prior let appearance would constant furthermore vector explain poor performance class appearance parameter vector object class investigate space dimension example prior bind l regime g meaningful interest introduction dimensional desirable feature selection dimension achieve performance increase induce bayesian lasso convert prior place laplace seem dimension put lasso though unable match common induce place exactly density spike keep regret increase component ensure logarithmic maintaining generally n choose exactly zero choice purely reason set theoretic benefit hierarchical main three specific prior likelihood widely often substantial generate mechanism offer analyze variety employ represent hyperparameter group create complicated simple one result important hierarchical use result insight batch statistical risk applicability acknowledgment gr comment anonymous constructive presentation u fa air office scientific engineering fellowship proposition cumulative taylor f assume z combine yield theorem q q define occur reason nz py ct observing factorial odd
large document technical least intuitively say topic something reasonable assumption appear context large arbitrarily probability must q dirichlet namely draw probability reasonable range imply large well essentially enforce topic view notion separability sense topic document topic document appear mass word let non topic call document sense topic specify number initialize support topic matrix topic proportion multiplicative document correctly identify support large topic topic actually initialization within document effect multiplicative improve technique one c something large come existence local anchor anchor furthermore show decode quantity evolve system evolve iteration quantity quite topic support portion roughly devise document try common false positive never say might topic topic look initialization user document show work firstly anchor assume word topic proportion similar small document appear recent discriminative word large topic difficult roughly range topic proportion reasoning follow document kl proportion carry first long topic anchor progress anchor word show long dominating phase keep drop reach show dominate identify step finally improvement round begin positively topic large proportion try whenever quantity kl f kl minimization respect anchor namely use simple maintain estimate divergence estimate able anchor never analogous manner round argue reduce initialization section remark proof dominant topic weight dominate document anchor namely satisfy proportion sufficient proportion anchor word word explore relationship anchor bad initialization anchor word low ground truth even fairly stage variational challenging significantly new leave important acknowledgement helpful discussion use equivalence say want study specify initialize kl incomplete modify need sure topic maximal dominant multiplicative topic property multiplicative step logarithmic document effect cause multiplicative word incorporate correct support also fairly mean version among rest support put way imply value variable everything kl variant thresholde namely unique large constant number document kl problem kkt topic satisfie appear support topic next topic document support former easy look would one finite happen document estimate appropriately variable iterative correct side split notational proceed crucially topic support topic document topic appear word certainly document summation vanish however update document topic call analyze topic correlation assumption independent belong topic claim establish throughout together topic keep improve support without certainly statement lemma proceed prior document document denote ensure equivalent look ever upper step constant support look kkt word certainly j I enough corollary nt c conclude take consideration value correct iteration achieve portion technique keep quantity iteration evolution message pass evolution average track track tc c iteration quantitie cause update alternate update constant iterate well precisely tc kkt part non previously ic claim contradiction little translate c upper imply certainly monotonically decrease interval absolute case upper c monotonically hence tc tc proof proceed lemma tackle upper fc get want yet relationship get right function fc put suppose tc kl minimization respect iteration iteration satisfy tc lemma immediately proportion entry within multiplicative section remark iterative proof show iterative incomplete work whenever topic lemma happen kkt separate ic argument furthermore prove guarantee completeness fairly recall goal support I present document support topic speak devise try determine topic positive common ensure topic pair test say yes intersect support find roughly document word formally f min j min correspond neither contain inside remove remove list analyze proceed describe determine topic min topic contain let contain topic belong property topic denote belong word belong least say intersect say pair intersection show back round many dominating topic difficult track discuss formal state anchor document document dominate minimize quantity say something min j variable start anchor min kl kl respect variable f kl upper f dominate c r identify distribution p outline anchor identify word topic assume throughout topic prove claim outline virtue initialization enable iteration basically value lemma suppose j variable eq constant get kl minimization anchor document minimization optimum b kkt condition denote j rearrange need place intuitively view combination unless belong topic multiply something say reasonably large somewhat let anchor topic split update q partition three contain vanish word topic upper b ok dominating contain inclusion document dominate third reason actually preserve show initialization word j prove induction cover topic topic seed document claim low hence claim almost q certainly want min establish previous logarithmic number factor cause support discriminative word correctly topic crucially mass outline word discriminative word topic cause decay topic belong dominate discriminative whenever zero maintain multiplicative anchor word correctly whenever j claim I j c expression eq get claim maintain document I certainly focus furthermore bit upper inductive show equivalent together belong drop let topic belong hold constant ic b namely topic least multiply want bound term eq weak property di dominating belong must document since topic correlation claim want I complete claim correctly dominant maintain discriminative large point follow anchor discriminative kl simple since increase simple plugging exactly state combine anchor word discriminative c l l bc achieve dominant l bc l proportion want bc roughly argue round support discriminative word multiplicative correct support discriminative kl whenever kkt summation however contradict want correctly identify namely analogue basically back support thing deal quite anchor word iteration eq brief look nothing upon read prior happen specifically weak dirichlet dirichlet topic negligible prove elsewhere include concerned cc ic proportion document relate coordinate large prove small rest topic proportion correlation correspond ss claim individually since hand since inequality dx eq dominant topic namely topic whenever big show topic probability mathematically prove jx dx write bind ratio dx b dx way analyze divide integral portion much difficult last come pick part imply claim let rewrite little proceeding c course large c finally portion dominate portion portion x dx first expression evaluate numerator inequality portion expression bound separately low dx simple independent inclusion still within topic modify handle handle filter want variational handle previous section argue common argue common progress argument fail scenario study small certainly b mass dominant loose really proportion dominate require clear topic include break calculation generalize sketch outline variable prove multiply inequality fraction word document want correct discriminative whenever support furthermore multiplying get c indeed equivalent easily deal alternate lack anchor way document behave like contribution intuitively show correct current tc tc max claim contradiction translate side upper certainly monotonically interval difficult calculation lemma give fc lower get next lemma suppose ic c dominating sequence q upper simple eq bind side prove side hence bound analyze expression let bx indeed derivative fairly gm sufficient true proceeding expression bx bx bx bx bx bx j j ic c c c c finally estimate word dominate dominate document high topic word total union thm proposition section inference efficient infeasible despite popular current theoretical understanding effectiveness inference base optimum topic show inference provably learn satisfy topic expansion introduce anchor topic prior dirichlet prior introduce modeling role initialization fairly lda variational insight might nature force combination estimate eventually reach document word document span heuristic phenomena machine variational like practitioner alternate relaxation easily theoretical know guarantee quality direction algorithm relate convergence em algorithm setting another set appropriate initialization alternate minimization ground prove address provide assumption initialization strategy converge number difficulty somewhat closed second variational operate introduce stress identify rather understand behaviour method method variable generate q term achieve back step compute set scenario common relax e min pz e step none family approximation guarantee ensure one optimum explore optimum model model prior topic pick topic pick result document topic type commonly assume satisfied one popular originally vast theoretical relevant context paper work work word anchor topic topic partially topic work certain say support topic topic almost word one sequence long prior
triplet triplet multiple work sketch generative parameter mlp exponential focus diagonal depend triplet triplet multiple dependency potential integrate variable exception variational come crowd maintain flexibility cost goal maximize likelihood capture triplet involve latent class model equation resort employ doubly stochastic highly provably maximize evidence standard bayesian speed benefits monte inference define approximate resort mlp parametrize act predict latent input write evidence low act need infer inference variable resort trick variational unbiased expectation predict variational inference unbiased sample shape learn drawing triplet time becomes learn triplet carry carry combination oracle upon inspection component form variational autoencoder triplet triplet happen formulation force share information triplet implicit teacher turn generative model act fine grain run belief network determine performance absence basically belief network information hold datum hold triplet datum crowd inform act proxy loss distribution momentum optimizer crowd triplet run graphic processing hour logistic house natural comprise train counterpart digit perform density unsupervise unsupervised observation autoencoder oracle crowd identify digits triplet picking report visual inform model clear benefit desire triplet triplet task prediction dataset comprise condition image take source varied way depict change appearance apart variability identity depict person unsupervise use architecture series crowd ask simulation question upon present triplet enforce question term light similar typical answering accurately require understand variation concern physics tackle detail crowd triplet question match produce answer resort distance angle angle triplet influence question architecture unsupervise fair triplet inspection figure representation hold triplet triplet hold crowd drastically infer representation inform face identity inform pure inform flexibility predict unseen triplet unseen suggest inform crowd learn image content purely triplet slight space physic beneficial label knowledge automatically improve triplet triplet space strong well result unsupervise generative triplet contribution probabilistic triplet knowledge crowd rich result improved ability predict reasoning system medical theoretic distance unlike commonly approach triplet framework conjunction infinite partition spatially topological vision segmentation shape oracle work regard crowd multiple promise conjunction amazon finally wish mention bias study usa program institute york york usa systems label effectively parametric convolutional building representation compressive criterion inherent lose semantic always label crowdsource implicit feature take advantage come algorithm standard demonstrate image drastically triplet crowd supplement label shape vision development system use image video crowdsource practical representation crowd decision typically employ subtle automate reasoning system deal crowdsource frequently noisy expert alternatively similarity understand initially come apply object question oracle paper learn flexible observation learn human observation train interpretable exceed purely case representation robustness base instance crowdsource community crowd similarity constraint crowd embed probability rather fix density attempt learn assume similarity ask weak similarity probabilistic introduce adaptive crowd without mind triplet feature crowd flexible model employ learn work multiply usage crowd performance generic fine grain categorization density estimation proceed probabilistic crowd triplet principled combine crowd triplet graphical transfer triplet knowledge explicit remainder typically give object visual infer order group object observe crowd evaluate report close candidate repeat procedure triplet treat oracle internal structure function latent approximate oracle mapping conditional arbitrary uncertainty crowd triplet capture replace define triplet motivation heavily raw statistic provide constraint explicitly quantify
name cyclic low adapt impulse periodic incorporate simple state minimize call monotonic integrated algorithm minimization admit simple solution prove implement operation due double slow especially propose acceleration minimizing organize formulation present review section review convergence iv acceleration spectral vi present conclusion case letter matrix low scalar denote field denote phase complex consist element diagonal form diagonal stacked minimize periodic periodic metric express frequency rewrite expand square constant theorem e ignore simplify tackle objective nonconvex modulus follow step subsection simplify look good periodic zero exist length sequence impulse name low directly simple alternate eq summarize ease nk convergence author point two exactly minimize local even original minimization although show long periodic sequence view adaptation periodic show initialization although original metric acceptable problem metric general briefly scheme adapt periodic mm method refer difficult simple problem reference therein include suppose mm optimize function produce accord update generate say construct meet easy decrease eq second follow mm algorithm easy purpose first construct hermitian hermitian easy check problem define tr p eq n know maximum eigenvalue ignore constant sdp relaxation yield sdp rank scope complexity sdp solve amenable apply compactly k element absolute clearly choose max h p diag rewrite closed solution although minimization twice minimization minimizer integrate derivation carry periodic apply periodic initialize max k diag kx nk monotonic maximization minimization scheme sequence guarantee converge convergence algorithm point introduce minimize smooth tangent condition refer exploit property consideration real optimal obvious problem versa facilitate analysis minimization follow ready property algorithm limit respectively denote n fu accord assume converge subsequence also ignore easy I denote equivalence problem multiplication compose similarly product computationally sequence say describe principle speed noticed convergence may double scheme carry derivation acceleration accelerate adapt accelerate originally accelerate cauchy combine em update implement mm update wise accelerated summarize algorithm general nonlinear constraint project descent property backtracking adopt repeatedly e maintain original practice backtracking need loose original accelerate approximation iteration preserve require rather mind kl fu choose worth note take know guarantee backtracking choose satisfied choose result monotonic call backtracking require initialize repeat n max diag h ki k u k backtrack cognitive designing correlation satisfy like band low sequence challenge algorithm spectral constraint band denote hereafter express denote minimize constraint transform q follow derivation rewrite step list acceleration readily elaborate initialize repeat diag n spectral pc ghz cpu gb numerical propose design matlab code website book measure mf sequence algorithms criterion n uniformly fig different backtracking accelerate mf computational complexity among three length accelerate mf versus sequence curve versus sequence average sequence accelerate perform property accelerate initialize accelerate accelerated evolution shown initialize accelerate local minimum contrast increase probably also tell initialized accelerate thus frank probably htbp sequence impulse periodic exist periodic correlation h periodic sequence periodic correlation db
capture positive accuracy tweet score range neutral sentiment tweet one positive sentiment define range indicate extremely tweet vice tweet label negative sentiment tweet tweet neutral score sentiment calculate tweet neutral respectively adopt contain public tweet twitter sample social stream store tweet english contain medium month capture sentiment content processing content external etc dataset comprise tweet facebook produce distinct user tweet process sentiment score calculate sentiment neutral neutral account capture around neutral tweet overall distribution skewed observe tweet example tweet thousand opposite panel second pass tweet number tweet sample represent concerned sentiment diffusion sentiment dynamic popularity tweet function express fig b original tweet speed diffusion reflect second original fraction tweet never fig tweet report tweet comprise tweet tweet times tweet bias twitter reason average fig capture well know broad content popularity medium mean toward large spread broad neutral collect spread neutral tendency favor neutral far accordingly negative spread albeit neutral tweet negative generally twice cascade less popular consequence popularity sentiment employ discover axis represent proportion tweet occur color dot event discussion black number bic well star mixture investigate affect popularity exclusive occur twitter topic tweet roughly public tweet exclusive never appear study aim many type characterize three proportion tweet peak tweet proportion tweet produce peak popularity simply day exhibit tweet expectation maximization gaussian determine three gmm spherical fold validation bic quality vary bic different mixture bic process determine optimal four agreement optimal gmm four represent tweet occur axis peak popularity discussion blue dot exhibit activity quickly complementary discussion dot reach reaction discussion quickly decay discussion dot balance peak discussion black event attention accord exclusive observe obtain b example dynamical length day center peak day fig event capture game nature generate peak namely release exclusive immediately discussion rise band car cause death activity discussion perfectly reflect discussion day peak four day stay lastly event namely ed away four tweet tweet proportion four class symmetric discussion bar tweet evolution sentiment class twitter useful neutral discussion positive peak sentiment constant notably amount much average average content event intuitively carry stay dataset popular discussion dataset discussion toward complementary happen grow shift toward short length discussion exhibit high level around average ability scale tweet role medium diffusion find light sentiment affect spread neutral pass publication original post positive tweet might interpret reader chance neutral analyze tweet individual prefer tweet neutral amount neutral clear popular quick reaction negative neutral reason online environment aim diffusion ensure reach sentiment discussion signature discover sentiment yet event brief medium like twitter sound suggest relate etc vast characterize future political match release positively represent room etc characteristic exploit finding practical consequence relevant dynamic sentiment crucial want policy effectively policy management recent highlighted diffusion information yet spread short piece social strategy social medium
cm cm cm rmse versus data purpose set rmse multi sharing across task gap suffer lack interpretability compete lr rbf task approach electrical treat period model hour e g burden unfortunately discover consumption interpret framework electrical hour load forecasting challenge year temperature response moreover weather day year daily every day forecast response series set half lrr rmse lr additive specifically hour lr compare comparison signal rmse roughly interpretation fig temperature display b hour per day give index temperature covariate hour model similar shape tf lead load prediction intuitively air effect hour activate tf air occur tf hand tf active tf represent transfer function transfer tf tf temperature day learn additive fitting conduct recovery distinguish coherence wide range realistic correctly underlie compete term predictive demand multi competitive learn extract correspond customer improve scalability involve million task j j recover sufficient condition bc omp recover correct residual pp bind upper eq l l atom select therefore inequality term assumption conclude bc omp select correct correct recover orthogonal onto span relation dimensional flexibility key benefit transfer identify output loss interpretability cause interpretable forecasting key exploration corpus establish sparse new fitting transfer step pursuit whose result literature latter experiment world baseline method yield interpretable structure datum extend benefit additive task additive dictionary learn widely extensively theoretically successfully machine ingredient covariate additive manner flexible allow good additive understand face task additive independently several firstly domain expert visually transfer essence interpretability human view independently task model overfitte overcome introduce novel output task sum task interest variable obey law neighborhood use consumption pattern novel univariate transfer function covariate represent covariate specificity task scale use constraint recent field fitting update transfer scale function update transfer constrain matching pursuit bc omp extend pursuit coherence accurate extend theory transfer function learn analogy update experimental world demand accurately learn addition propose method outperform baseline candidate interestingly correspond small fraction independent benefit experiment maintain small decade independently task impose close weight live linear group context family enforce involve covariate significantly common across transfer covariate dependent expert candidate transfer transfer even thousand moreover covariate relevant hence input covariate small involve paper review formulate model explain sec sec real datum load forecast multi task performance comparable baseline interpretable customer vector refer element product n denote pseudo moreover moreover count operation pseudo inverse moreover column entry wise negativity constraint additive diagram briefly intercept represent covariate transfer transfer continuous covariate commonly model spline spline spline spline use estimate intercept consider centering constraint convert optimization center basis specifically ij equivalent unconstrained order commonly add simple easily quadratic regularization solution additive share denote zero new combination set function covariate constraint prevent transfer capture new candidate non function task offer wide keep non negativity interpretation transfer could represent opposite effect lead high demand task lead demand illustrate additive diagram covariate set task task model transfer set diagram arbitrary similarly done sec transfer spline denote spline basis rewrite j jx nj iid find p square residual avoid overfitte prevent note transfer regularization spline strength inherently find code minimize desire exposition analogy us scenario equal define lie former constraint code entry constrain efficiently approximate challenge linear solve successively spline mod sparse code dictionary coefficient global recovery bc provide recovery right superposition belong simplicity atom nonzero develop correct atom bc omp omp omp algorithms search select dictionary bc impose constraint available atom previously bc omp recovery bc omp omp conditions omp recover every superposition whenever q call similarity leave omp omp recover correct close j behavior sec inner j jj seen estimate population transfer coherence definition bc omp bc omp recover coefficient detailed condition coherence recovery particularly interesting application strong function sufficiently statistically bc still succeed recovery recovery satisfied bc omp valid omp sparse representation reader relate recovery author build incoherent coherent condition correct show word exact atom drop weak correct atom signal appear rather spread throughout omp differ whereas assume occur section implementation sec sec result section load problem background additive demand forecasting example transfer dot dash line represent lead transfer transfer correspond transfer regressor independently parameter task regressor implementation additive task package step centroid prediction centroid aspect simplicity likely give moreover world application parameter manually domain interpretability goal parameter employ bic know experiment evaluate affect problem empty transfer per large checking currently approximation eq candidate transfer simplicity task transfer randomly number iid distribution assess transfer synthetic set treat show estimation accurate coincide highly noisy prediction rmse significantly synthetic despite limited task improve rmse method experiment perform task fourth model complexity task basis simplicity theoretical numerical lr half customer come survey economic indicator number business time customer consumption total customer stochastic aggregate signal measurement hour day week covariate test transfer method outperform perform slight
backward convolution bank channel filter minibatch multi demonstrate ability fast convnet gpu spatial gpu domain speed shape execution time throughput measurement efficiency device peak gpu kernel evidence create great reach suggest create gray include create code deep minibatch channel filter channel convolution offset parameter assume scalar algorithmic multiply count wide shape implement convolution inner batch dimension calculation single coordinate multiply contiguous input column offset extra integer multiply operate contiguous block datum convnet employ programming load index arithmetic load convnet gpu half memory gpu extra share source create gray scheduling allocation include implementation project interesting also implementation advanced texture load increase texture indexing calculation memory share share iteration store output share bit point share load limit per demonstrate power parallelism project perform load combine convnet convolution multiply patch filter block block basically adjust indexing calculation indexing map patch channel offset offset add offset compute replace loop channel patch offset location map boundary graphics gm gpu convnet architecture bad efficiency ratio actual throughput peak throughput gm processors core device peak throughput count single execute processor independent speed efficiency unnecessary strictly multiple number mini size modify efficiency actually imagenet minibatch computational efficiency range bad patch loop compare section reach due intensity respect memory cache device additional minibatch device limited experiment determine efficiency significantly efficiency network efficient
account joint interaction next score drop include dropout give abuse notation removal remove sort also sequential marginal view measure contribution look removal measure look include challenge involve anomaly many benchmark real evaluate ground truth evaluation address issue work construct number benchmark second supervise learn construct analyst evaluate expand recent describe methodology systematically create anomaly give huge world benchmark huge anomaly benchmark benchmark create anomaly frequency anomalous main class represent rise anomaly union anomaly detection benchmark anomaly point benchmark experiment anomaly detection benchmark benchmark anomaly analyst distribution normal formally analyst specify describe obtain analyst uci curve use computed anomaly lead analyst anomaly metric number reveal detect anomaly evaluate analyst detection normality analyst analyst drop value threshold experiment uniform result consistent choice remain analyst anomaly benchmark construct set normal analyst learn include generative practice arbitrary require widely discriminative possible subset evaluating subset learn subset cache need learn anomaly al commonly detection experiment anomaly detector describe et al range benchmark ensemble learn replicate threshold varied ensemble mixture retain approach address poor model bad local optima number component gain advantage straightforward form density gmm model marginal easily dimensionality analysis obtain analyst regularize analyst primary reason accuracie competitive train must fold worth evaluation framework potentially choice analyst bias beyond scope replicate qualitatively type benchmark point multiclass concrete anomaly benchmark uci benchmark benchmark total anomaly benchmark serve train dimensionality number analyst publicly across optimal analyst subset minimize threshold constrain produce compare method represent simulated analyst benchmark anomaly rank compute six choice actual rank use analyst threshold value across anomaly derive confidence first note focus anomaly top indeed also percentage point anomaly observation qualitatively choice suboptimal detector outli anomaly constrain room worth note set correct reasonable expect version compute computation difference nearly performance exception gamma statistically advantage interaction critical anomaly explanation able recall evaluate make appear dropout normality score see overall dropout close weak signal early decision difference produce feature often marginal make robust early decision achieve score dropout prior investigate decision detector quality detector method detector analyst compute minimize sequentially constrain show result detector detector reflect sequential news motivation reduce analyst analyst rather arguably less desirable perspective often sometimes contrast anomaly observation indicate reasoning anomaly leave open question anomaly benchmark gap benchmark marginal generally decision method uci point http contain anomaly represent detector infeasible analyst thus evaluation overall approximately train domain quite rank rank evaluate domain figure clear marginal well particular indicate combination analyst weak decision number significantly outperform bad dropout detector little difference independent detector outperform suggest outperform recommend computing introduce require correctly detect quantitative evaluation explanation overall prefer introduce usa edu anomaly detection present anomalous analyst instance truly security unfortunately anomaly anomalous leave analyst evaluate anomaly detector analyst order analyst confident analyst effort investigation explanation number reveal attain one contribution novel quantitative evaluation analyst benchmark artificial simulated explanation anomaly benchmark several insight identify anomaly anomaly generate distinct point correspond meaningful anomaly security statistically activity reality true analyst decide anomaly analyst face analyze challenge especially interaction feature case even anomaly detector pass analyst recognize key due effect overall anomaly analyst improve reduce analyst anomaly intend side effect analyst reduce analyst detection effort anomalous analyst effort focus contribution intuitive score present analyst analyst acquire information anomaly security analyst roughly feature reveal minimize analyst identify anomaly contribution quantitative comparison analyst benchmark ground anomaly analyst evaluate reveal reach methodology anomaly contribution detector operation support finally fourth contribution empirical method anomaly real provide investigation recommend insight paper organize review anomaly anomaly concept section describe method compute introduce quantitative method framework unsupervised detection relate supervised aim classifier instance propose produce relevance score score classifier point anomaly anomaly detector anomaly detect analyst may consider chance detect anomaly analyst effort toward anomaly propose analyst attempt efficiently indice feature appear order consider notation feature onto analyst present analyst able make base feature add give analyst analyst continue analyst able process terminate early normal incremental analyst point anomaly reasonable expect analyst anomaly consider amount reduce amount analyst effort monotonically measure reveal analyst anomaly analyst detect anomaly quantitative require access analyst term section issue wide anomaly detector computing detector either compute particular detector avoid former attempt indicate
name survey exist try attribute string measure heuristic name etc high f around collection citation attribute along medical incorporate previous predefine similarity specifically obtain specific accuracy specific dnn feedforward many initialization deep rbm sparse normalize method automatic al big dnn recognition convolution neural train back propagation outperform method mnist complicated process yu feedforward recognition ability robust variant state feature normalization feature dnn author name explore system author combination many compute dnn learn exploit type neural layer parameter initialization scheme activation perceptron network stack upon connect correspond layer correspond dnn interpret direct approximate posterior dnn layer layer layer give denote express vector hide unit output represent output basic output become input next sophisticated compute multinomial dnn vanish traditional activation propagation algorithm issue adaptive dnn ability affect layer layer layer achieve however slow capable yield overfitte hyperparameter base experiment fold change hide hide unit create network size dnn learn must proper representative datum publication author attribute accord survey string match automatic author name author digital library present build acquire library name instance name author name van etc author name dataset create author understand label person research totally detail number tu le pair publication publication record original fold tune hyperparameter overfitte split percentage particular pick cross network layer unit hyperparameter layer hide many five size hide unit high validation predefine apply nn forest bayes respectively suitable predefine implement architecture implementation term separate test predefine set achieve use predefine feature relatively predefine set method predefine k forest c learn dnn evaluation benefit moreover automatic feature expert knowledge dnn learn feature successfully capability complex research overfitte hyperparameter tune early stop architecture dnn vanish train sigmoid sigmoid thank smooth activation dnn label train dnn improve hand automatic dnn ability complex recognition object raw rational create pixel dnn deep network feature automatically author name ambiguity additionally general system architecture author author dataset significantly predefine feature achieve prediction predefine extended solve open author benchmark unsupervised encode research university technology artificial intelligence digital deep deep neural frank com p edu name ambiguity decrease reliability retrieve digital library automatically ambiguity architecture name dataset contain author name significantly outperform use predefine method achieve relatively compare use predefine ambiguity publication author name appear name distinct reliability digital library author task digital library author
relationship case discovery structural proved distinguish cause depend functional attempt field salient speaking notion property one separate several purpose concept relevant context originally inference likelihood idea suppose conditional parameterized marginal alone say variation three strong weak subject restriction view imply posteriori exploit parameterization applicable argue parameterization causal pass fail compare distinguish cause without structural see algorithmic condition kolmogorov paper explain adapt weak statistical statistical nonetheless exploit direction two random draw density one set adjust setup say admissible take depend clause ii defining say operate cut computed concept generate contain conditional model concept also cut operate bayesian independent sufficient sufficient generating cut marginal model bayesian counterpart classical cut equivalent cut theory regard relative operate cut generate note requirement respectively otherwise ii trivially meet take trivial situation iii violate posteriori htp sep fill circle fill circle scale width sep circle node fill circle width circle draw node circle circle scale width sep node circle circle draw circle draw depend sufficient modeling unlike super regime relative observational causal common cause relative indicate determine separate argue direction turn direction admit causal infer sufficient accordingly indicate causal direction familiar identifiable cut cut give identifiable identifiable situation non gaussian follow mixture gaussian involve independent alternatively priori priori see direction direction identifiable illustration shape parametric constitute bayesian cut nonparametric bootstrap coupling examine cut htp bootstrap density estimate literature assess parameter e g clarity bootstrap pair bootstrap drawing replacement calculate estimate statistical test give point evaluate evenly length mapping mapping setting imagine effective case observable previous argument bayesian way assess determine causal input involve module input three possibility non direction replacement sec otherwise simplify estimate covariate shift exploit avoid flexible verify effective bootstrap estimation density method inference base find e generality otherwise instead nonparametric take optimizes produce posterior likelihood functional estimate conditional point pe test experiment similarly center size row nan hypothesis actually schmidt independence reasonable dependence width row evaluate bootstrap method ground truth data two examine causal geometric causal assume effect plus additive cause way reason use replication htp multiplicative super additive various inspire pass sign control range purely purely multiplicative effect control produce situation different datum see able causal replication although try replication speak improve replication include structure influence exponent change performance four bar correspond tend significant ht situation reduce pair direction seem scatter
derivative contribution literature langevin mala mala special mh discretization langevin drift diffusion proposal depend mala way sampler monte far current proposal numerically integrate often necessary hundred time propose achieve reasonable mala hmc rely user scale reasonable efficiency propose mala hmc manifold associate bayesian model admit take namely fisher associate negative choice demonstrate highly suit langevin hamiltonian rarely mcmc sampler derive simplify manifold negative hessian explore note paper degree software definite application overview full hessian cholesky produce hessian whereas present potential exploit generality handle cholesky region zero primary adaptive leave stationary hmc length mala adaptive step regime standard implementation methodology pilot illustrate methodology realistic model methodology mcmc provide admit log di take simplify mala mala hasting distribution symmetric matrix sensible explain modify cholesky adaptation aim cholesky decomposition low q diagonal negative definite particular cholesky positive definite see optimization ingredient newton optimization basis simplicity sparse factorization upon request hessian cholesky metric reduce q iteration metric chain rarely near tend chain approximately typically take small proposal mechanism local scaling region derivative observe transition occur region chain proposal variance phenomenon target degree see rarely region point tend cholesky computed modify effect generality derivative fisher undesirable effect either work purpose duality mala hmc mala integration hmc length single mass hmc absolute provide integration comparable across area pilot trial relative trial step evaluation note energy probability hmc mass target update individually mechanism updating depend manner position specific momentum start proposal equal l typically tune produce acceptance acceptance candidate equation reason approximate integrate computationally integral computed numerically obvious condition consist mh accept markov distribution gibbs block trivial former mh g make metric see problematic around forward simulation figure dot adaptive pd figure present see length selection step actual energy criterion discriminate behavior point reasonable backtracking application necessary backtrack iterative scheme inspire search length fulfil factor great length inform observe include slow may tune refined long dominate computing need considerably costly hmc tune translate hmc average burn however tend acceptable moderate selection typically contain scale act step substantially interesting overall rather q translate thus distribution walk consequence primarily primary section consider step modify mala highlight behavior mala posterior non part parameter space response regression allow follow estimating ess monotone ess second spend computation carry ghz intel core gb ess ess ess ess ess mala apply burn write time ess time comparable remain run mala value panel latter visual panel trace forward depict panel show density current first realistic illustration return take default prior specifically truncate p exchange return daily also include constitute log observation mala hmc hmc take attain rate around addition one th th mala cholesky potential problem prior table provide cpu ess per repeat sample iteration cpu burn package partially remain mala produce per matlab mala produce indicate little add modify cholesky depict various diagnostic mala highly ill informed illustrate different mala stationary regime mala shorter approximately show base enable mala progress keep setup indicate stationary regime visual indicate small zero minimum eigenvalue confirm iteration indicate useful take short little curvature direction acceptance mala minimum ess ess ess ess mala mala probit probit adaptive logit mala logit probit mala probit probit logit mala logit probit mala probit probit logit mala logit mala logit mala logit probit mala probit effective cpu times regression logit correspond link probit correspond fisher adaptive adaptive length selection inference response model include compare propose methodology specifically consider response inverse correspond probit close logit fisher whereas hessian still definite probit collection datum reference logit mala work mainly numerical occurring burn time simplify logit full mala minimum produce per logit mala interpret length version produce iteration approximately reference mala par hmc coincide slightly effective sample favor mala consumption cpu roughly logit find mala denote improved feature hessian rather relevant paper make cholesky produce hessian simplify mala metric near curvature length also develop implement intractable even admit factorization example propose perform mcmc dimensional hmc need soft enable
achieve performance augment mark extra table validate effectiveness learning cnns train much large k table method perform good category r training crf extra extra extra extra crf extra crf extra crf rnn extra crf rnn extra message crf reveal new gradient message potential cnn network conventional learning segmentation demonstrate effectiveness usefulness deep message readily acknowledgement arc future arc fellowship gpu appendix show segmentation propose cm van university centre output great segmentation scheme cnns message potential calculation significant perform crf cnn potential class contrast output cnn functions exponential cnn message large good method cnn message method deep attract deep convolutional cnns combines cnns feature complex relations cnns cnn objective cnn crf joint segmentation cnns sgd optimizing require calculate typically require repeat slow avoid repeat final prediction potential goal final prediction potential optimize learn conventional crf prediction pass cnns calculation efficient cast cnn potential potential pairwise etc scalable iteration message inference learn procedure message crf message message pass apply method semantic achieve intersection union report image cnns several resort joint crf first train apply dense crf post processing extend jointly rnn crf cnns dense implement cnns unary train unary potential capture jointly explore estimation explore field predict mention cnns crf incorporate pre train potential learn estimator rather potential machine propose relevant idea estimator learn potential train traditional regressor cnns deep cnns end style feature factor thus formulate goal crf sec learn review crf cnn approach mask function crf factor potential index function unary unary segmentation example predict image joint also calculate marginal exclude crf np hard type apply belief propagation reweighted message pass mean crf cnn joint cnn optimize crf construct cnns add minimize log crf segmentation mask stochastic sgd function easily compute apply bring direct calculation infeasible iteration repeat marginal extremely expensive cnn training usually huge sgd thousand approach infeasible conventional crf message potential calculate conventional crf approach follow message marginal discuss involve message estimator function calculation message depend message variable input learn estimator depend message estimator message estimator pass asynchronous message pass simple asynchronous simplify pass pass propose learn variable estimator interaction neighboring denote encode nod connect estimator become hence incorporate however general exposition describe inference dependent message clearly message message iteration inference become formulate formulate cnn output denote network indicator correspond implement cnns analogous design convolutional feature one crf likewise feature node connect exclude pairwise factor pf fc feature vector input alone final generate final connect please refer potential potential class dimension significantly increase computation tend training estimator need output write ideal variable marginal truth ideal marginal problem message network aim learn formulate neighboring node factor neighbor fix neighboring generate classification thus traditional regressor contrast train cnns learning procedure define sequence message aim quality contrast potential function message fast might preferable neighborhood effectiveness inference c c cat person train tv crf crf rnn message publicly background category set
outlier contrast return r outlier represent example character ten different template aim ten image centroid template average appear preserve reason font template information template may adjust provide centroid template fig fitting risk update covariance affect eigenvalue thereby bias another soft svm slack employ sample reliability hyperplane hyperplane outlier maintain separable kernel trick replace dot effect map inner pca projection eigen covariance procedure trick dimensional transform high dimensional possibly hilbert equip definite kernel trick typically dc laplacian flow show formulate instance general slack regularize far derive lagrangian dual applying trick dual simple trick normalization become maximizer coincide simple minimize variable globally efficiently necessary practically iterative either grow nearly minimum object construct outlier regularization prevent overfitte recently connection introduce slack object insensitive slack sensitivity parameter slack serve lead problem geometric optimization regularization labeling understand lagrange dual analyze n v n w lagrange multiplier three lagrangian infimum rewrite minimized z dual quadratic qp vector tc comparison elaborate characteristic template use dual therein four case lagrangian multiplier feasible constraint must iv become objective effect primal increase force deduce coincide lagrange lagrange dual dual except upper bind occur frequently simplex nonlinearity space handle space costly impossible product mapping immediately trick allow propose template applicability experimental original dot dual mnist handwritten image digit respectively carry sample take transform two centroid visual eventually centroid b test centroid region necessary aggregate confirm put template centroid close carry consist reflected reflect implement centroid comparison regression neural exist kernel test validation determine auc template nn template centroid template auc r produce svm regularization shape term follow distribution sake visualization discard membership depict color code result incorrect kernel correctly separate data membership method compare base original iterative carry one object vary fix window equip intel cpu ghz mb gb ram digit additional consequently regularize long show vary present microarray variation gene expression patient add runtime dual form demand take less time well change runtime design toolbox exist summary formulation form advantageous otherwise form desirable form complexity number primal visualize template vector imagine image stand step class word vector aggregate set correlation increase case maximum average slack mean effective outlier reveal result gain consider vector slack r mention fig representative picture extend fig plot correspond expectation fig fig fig analysis fig become close last fix experiment correlation template vector tendency know kernel certain approach originally context classification problem optical automate aggregate template application misclassification neighbor setup nonetheless limitation demand wide drawback instance present result successfully limitation original outperform classification adjust flexible addition benefit formulation regularization make appeal range analyze code become challenge example font automate neighbor bad procedure despite optimality drawback limit wide practice handle dataset suffer limitation improve regularize approach programming problem incorporate slack derive trick handle make propose accurate grow r run substantially solve primal neighbor qp trick neighbor parametric classify largely retrieval video indexing tracking location nn computationally intensive neighbor break dimensional space approach complexity adaptively neighbor template match pre representative use near template create template template classification aggregate template template minimize risk originally propose font optical character automate protein cluster despite theoretical limitation wider well propose limitation original centroid maximum member outlier add return angle represent appropriately r find robust represent centroid aggregate template ten image
line copy different note slight abuse translate condition denote chernoff note projection span result km singular soon condition second statement condition combine line k smc step find span share span extract u bm b rate find vector find extract split store whole easy span thank column span km km first first technical arrive belong span signal estimating span span desire compute process count zero entry zero law process orthogonal negligible km noise span thank split sampling simply span reason see multiplied lemma signal thing result km span use schmidt become q compute ready analyze smc first check contradict obtain high analyze required pseudo store sparse sampling bit finally thus circle q equality stem condition satisfy replacement nu matrix proposition sum sample replacement q dimension k w equation recover remain part satisfy eq lemma find w k upper entry independent last equality stem markov high v n eq randomly u km b k b k stem km bs k u u b high stem k n os km inequality inequality conclude k km mn projection linear span mp k mp u mp mp stem high independent markov x x ji ji j observe therefore thus k km k basis linear span orthonormal basis row ns chebyshev lemma indicate probability row coincide since markov inequality schwarz stem sample column streaming completion interest store column matrix sequentially miss memory propose streaming estimate original vanish linearly ambient store output number stream computational reconstruct noisy constitute collaborative attract recent motivated system amazon netflix google propose user rating naturally translate completion row resp user rating rank inherent similarity extremely become store rapidly matrix designing constraint system collect reveal request store particularly recommendation system actually understand machine technique approximation consider constraint detail exist work transpose singular svd unitary contribution let truth wish noisy observation assume ii large tend corrupted observe entry operator wish observe tend infinity completion streaming column among smc streaming completion complexity matrix construct assumption accurate square exist denote small satisfy iii main result paper smc asymptotically accurate pass soon rate smc estimate soon ambient approximated output smc store treat observe instead singular column find keep step right extract vector right singular matrix top easy schmidt process orthonormal span top vector realize follow subspace span row define operator number survey recent rank could build block section follow rank approximation efficient show absence rank propose program spectral reconstruct asymptotically depend adapted presence improve memory guarantee streaming approximation pass column matrix appropriate memory think accurately asymptotically mn f mn mn easily mn algorithm asymptotically recall stream reconstruct arrive identify pass right propose frequent direction sketch problem kk kn efficient use stream pca apply author randomly I low randomization way reduce algorithm survey devise rank mn sparse mn operation explain asymptotically completion initialization qr deal column information batch arrive column algorithm address singular value dominant vanish entry become consider interested provide estimate singular spectral simple observation constitute basic theory signal order qr issue avoid entry decomposition diagonal whereas diagonal remove dominant spectral present analysis decomposition suggest random theory argument loose ensure separation spectrum need span nearly analyze store store value store memory apply
place plant community whereas easy increasingly high measure plant partition mutual information partition decrease resemble plant er modularity clearly range mixing indicate performance inequality mix er modularity display benchmark four clearly much modularity modularity former benchmark graph community community grow threshold go structure modularity worse large consistent prevent er cm modularity community indeed community achieve complex develop accurate kullback text internal dominant observe exactly ignore use read ignore q divergence interpret move community density approximately straightforward logarithmic odd positive dense density significance expect significance alternatively leibl significance average internal arrive hence significance generally general entirely significance size dense community write dense partition keep mind low significance general repeatedly refer community group usually recently form optimize analytically clear discriminative modularity suit detect early method fail describe simple form compose node reduce establish protein interact connect scientific pattern heterogeneity node triangle diameter another real densely connect know share community field numerous detect structure ultimately regard partition either implicitly community modularity extensively belong wide spin quality give energy mechanic unable capture fact modularity community measure probability systematically modularity different framework describe divergence enable analytic compare modularity significance nan model behave addition formulation develop good community especially graph apart community quantify kl difference believe improve link overlap consist contain overview relevant provide bt number density edge community mm internal total edge internal draw edge way internal observe population derive difficulty implement mainly may assume consider elegant appendix measure metric case probability lie community whenever otherwise look dense generally original base binomial approach former link replacement fraction internal edge would eq development dominant binomial binomial become replacement negligible partition benchmark meta solution develop straightforward fashion initially move node long aggregate repeat contraction within keep edge keep initially upon total number community essential ensure aggregated graph move aggregate internal notice formulation aggregate improvement move internal possible edge total self loop internal complicated summarize graph weighted version graph internal unchanged open flexible suitable review compare still comprehensive modularity extremely method detection want partition original modularity assume nan internal total community modularity compare value sum difference model node refer modularity modularity propose nan assume every enyi er plug modularity version eq er modularity kullback leibler unlikely eq sometimes tight make likely modularity sense modularity consider circular neighbor exclude group consecutive node er modularity reach indeed modularity partition many whereas go er modularity go later er modularity simply unable partition show axis present approach describe internal look measure immediately significance contain community unlikely many community elsewhere random exact statement asymptotically significance eq graph leibler great benchmark scale resolution base divergence internal compare difference significance affect actual moving decrease significance intuition confirm convexity kl divergence kl appendix
architecture directly result retain neural restrictive consequence restriction plausible nonetheless influential development recently accuracy imagenet benchmark relate indeed act affine transformation abstract class neural another cluster activation abstraction composition sparse high really quantify similarity goal focus architecture suit large amount nuisance variation two derive bring nuisance enable build well representation task address limitation principled google face architecture art face benchmark use architecture crucially classification learn good basic behind objective light connect objective perspective triplet nearly global explanation dominate equivalently thus deterministic indeed implicitly criterion employ irrelevant nuisance transformation multiple abstraction interpret iterative recent understanding deep upon physics construct boltzmann rbm block physics configuration configuration variable long correlation range fluctuation show analogy goes create exact real indeed irrelevant multiple level strong literature summarize explicitly variation level abstraction factorize dual purpose exact product iii regularizer overfitte iv configuration justify low setting process deterministic vision invert powerful powerful currently employ limitation room refer follow broad back impossible top unlabele task moreover affine transformation capture phenomena figure clutter brain feed back dynamic unable unlabeled amount overcome design extend new rule summarize explore design realistic assumption cause include translation rotation perspective graphic geometry order computer graphic enforce initially affine transformation weight rotation nuisance transformation interest motion away camera template google directly scene representation example encode depth useful thereby benefit image rich representation inference equation geometry limitation correspond million order nuisance result learn contrast accelerate acceleration change implicitly probable global however potentially max message optimal wide temperature smoothly well pass variant approximate bayes em knowledge top feedback task low feature suboptimal higher implement top message properly outline convert top network implement pass inference max top scene understand segmentation clutter bottom feature principled define back propagation often algorithm inefficient implementation approximate em whose probabilistic contrary much exploit e bottom top fast computation sufficient sample covariance efficiency substantial fast tradeoff moreover although incorporate visual static video static input cause change little supervision would external supervision supplement enable focus nuisance factor difficulty purely discriminative technique thus armed enable benefit principled manner dramatically power encouraging factor hybrid generative relaxation perform naturally parametrize likelihood natural label achieve world classifier sensitive mis specification principled training span spectrum discriminative stochastic descent back em thank thank detail instrumental maxout consist template operation also claim proof eq limit hypothesis independent channel abstract feature assume matrix dependent template nuisance application matching template c maxout template reduce convolution operation activation unit us max pooling relu activation relaxation single convolutional consist convolution operation relu random nuisance switch independent evidence nuisance invoke noise free datum specialized structure derive mathematically nuisance line calculate log pa cg use write expression operator invoke v v modify w constant outside drop operation single traditional origin relu log likelihood measurement relevant important miss hypothesis relu activation relaxation maxout nuisance distribution write generalize exponential simplify greatly play familiar partition importantly counter maxout robust discriminative mixture maxout let nuisance exponential family maxout except generalize describe definition serve generalize quadratic depending amount label many world scheme still avoid data dropout consist neuron output activation neuron rely every piece evidence force prevent adaptation detector perspective show answer yes dropout generative completely miss noise free miss dropout em train strategy training utilize discriminative generative miss iff pixel intensity nuisance variable g since soft yield sum soft yield characteristic step generative however end derivation distributional generative flexible mathematically q pd partition allow pd label feature discriminative random subset sharing since intractable sum monte yield theorem definition mm edu pt computational outperform task visual speech numerous nuisance variable unknown speed recently layer large massive success system capability fundamental question analyze architecture probabilistic deep generative learn model lead system convolutional decision forest human expert wide array complicated object image despite orientation challenge nuisance nuisance simple inference car lie object challenge decade vast approach nuisance super capability learn linear computationally massive amount training architecture convolutional see recognition localization recognition part speech forest image system fundamental success focus raw amount coherent understand architecture framework insight learn principled design improvement latent nuisance combine class nuisance extend transformation abstraction enable pass via optimize lead nuisance additive directly insight answer suggest pathway improvement wide classification etc task nuisance focus paper follow map onto insight provide operation proceed derive suggest promise include generality generative model operation extend define layer represent inference feedforward propagation enable message probabilistic pixel multi intensity channel seek infer prior chinese crp classify map image likelihood bayes rule high amount formation nuisance transformation endow generative model image subject nuisance problem graphic engine c add practically poisson identically distribute iid function nuisance categorical probabilistic mix natural statistic gaussian image q matrix generalize I nuisance label gmm direct figure depict gmm fig image location pixel patch pixel patch nuisance variable pixel image nuisance omit quite maps nuisance template speech recognition might represent speed amplitude alternatively part represent nuisance approach one sp nuisance choose likely inner product definition compute approach conventional ms ms mode nuisance image global configuration target nuisance image justify setting approach commonly sum max amount inner product template nonlinearity nuisance isotropic diagonal simplicity treatment generalize manner non quadratic depend please ms impose type assumption template pixel configuration see accord operation generally pool see fig explore model inference implement sum message b several recent normalization know come later batch principled implication unclear probabilistic assumption old arise image filter template template value template fully object template layer object pixel position likely ground air connect equally present invariance global template relative occur filter overcomplete filter convolution convolution factor pass nuisance image nuisance must two parent might undesirable force active formalize via image patch whereas patch switch inactive strong correlation neighbor must prevent realistic measurement sparsity extensively spike coding variable think nuisance q bias soft unit relu modern last line assume drop world abstraction hierarchical accelerate abstraction concept category inductive bias exposure informally light natural giving summarize abstraction order concept image face detail level abstraction specify identity face specify fine fine location pose e close without fine shape continue fine scale pixel intensity channel image refer become increasingly abstract detail preserve high level essential overall back conversely move concrete formalize process deep start abstraction object overall pose follow level intermediate detailed fully nuisance level abstract nuisance transformation fine abstract concrete bias clarity factor location hierarchy incremental form intermediate path bias covariance kind face location whereas direction generate abstraction function composition product factorize exponential parameter number enable hierarchical deep difference model detail example classify image deep iterate fine coarse bottom abstraction infer high section explanation clutter top essential hypothesis fine pass class top make use classifier message fine coarse abstraction infer coarse fine abstraction infer intermediate mention fine since template eq image diagonal affine classify omit clarity fourth max index channel intermediate care infer coarse pass suffice since relevant integrate eqs eq simplify come feedforward th layer yet softmax regression layer network miss level fact label probable configuration interpretation become clear need input feature model label discriminative train know since discriminate classifier learn define procedure generative discriminative employ relaxation obtain discriminative mathematically likelihood extra introduce old generative equality differ generation pass family conventional ensure line relax learn optimize classifier step e arrive discriminative comprise probability relaxed brain world transformation graphical introduce discriminative generating label instead imagine via classifier graphical parameter represent relaxation discuss invariance fine abstraction switching variable layer relu activation eqs discriminative conjugate training day train mini instead dataset light development sgd generative classifiers discriminative inference configuration softmax layer end activation detail interpret back discriminative batch principled motivation pass generative help misspecification issue slow require datum light connection new insight work answer question importantly fail see explore insight graph interpretation convolutional pooling layer operation apply generally architecture type neither hoc entirely two neural table knowledge generate template via affine thus examine exercise mathematically notably study computational also strong representation visual suggest brain factorize appearance search image maximize cat train million result image store maximize mathematically seek score class ic g ii activity individual activity patch pose activity patch fig nuisance pose ig key factor image formulation classifier factor nuisance architecture learn prediction classify use systematically variation location much information latent nuisance possess variation nuisance require evidence traditional distinction supervise unsupervised ill bad formulation capture nuisance parameter fig show believe notice early probably nuisance max serve purpose nuisance forest machine seem arise model vast array explanation understand quite successful segmentation task prominent use pose track microsoft medical wherein distinguish quite expert section assumption regard nuisance switch additive mutation nuisance category evolution heart derivation possess cast sum message pass series determine branch tree node question repeat reach leaf leave class posterior decision classify input send average evolutionary category start root template randomly template template mutation parent mutation parent child evolutionary template assume add course early distribution exponential family mutation irrelevant nuisance evolution individual abstract template sequence local nuisance many additive incremental intermediate evolutionary path mutation add template evolutionary evolutionary path start end leaf leaf specie template q additive last sum iteration layer go fine deep decision difference evolutionary underlie mapping single decision label histogram miss sec treat understand interpret configuration wherein world infer leaf inference critical gaussian finish need function internal analogous thus discriminative counterpart relate forest
surface subject optimal respectively cross subject via evaluate examine hold actual response hold voxel false discovery small panel correlation panel know interest separate examine canonical bold cca canonical identify weight accounting delay second voxel surface canonical subject movie frame low voxel module canonical correlation module integrate complex hyperparameter easily datum fmri response movie movie stimulus component voxel surface blue four voxel voxel colored variance overview initialize hyperparameter range cca two hold finally variance component fit movie response surface voxel colored interest outline mapping response accurately predict histogram correct bar component subject response movie stimulus canonical plot voxel negative indicate blue indicate four negative positive variance component colored rgb rgb false false false frame title cca valuable covariance cca cca similarity across fmri subject resample template introduce module implement regularization gaussian cca functional response across movie demonstrate set discover cca subject response analyse similarity covariance analysis method relationship decade variety field cca conjunction ica apply find subject mapping activity network subject technique call functional instead voxel alignment space equal dimensionality relate correlation input quantify package implement cca matlab cca toolbox analysis package either minimal seem active package option kernel cca cross validation cca fmri activity multidimensional cca onto basis theoretic mutual transformation zero maximize q pair subsequent analogously uncorrelate precede canonical less equal constraint reduce generalize q canonical extend multiple additional cca trick cca relationship cca project x k transformation invertible accomplish norm analogously partial least become cca ill cca adjust lagrangian mathematically cca pls maximization pls reformulate cca thought cca machine package include variety mining develop many cca cross pls module orthogonal exceed kernel cca module develop seem development access use keep library minimal module experience module organize class file main common implement object implementation visual representation create dataset cca dataset break first cca dataset procedure overfitte explore example include np var random np signal component var var var var var cca component cca train cca cca object cca initialize component retain use regularization canonical retain validate employ cca return canonical quantify correlate complete mapping dataset well correlation prediction actual canonical disk load disk use library save object share module object object initialize regularization point default retain integer default kernel cca boolean type regularization cutoff compute canonical weight hold prediction number boolean object parent cca fit dimensionality argument held dataset dataset dataset feature format compute cca mapping method accept file object load newly array default array canonical retain test integer default analysis get cutoff canonical weight hold default proportion select boolean value object include object differ validation range cross validation fold split hyperparameter large base repeat hyperparameter aggregate fit rest proceed analogously demonstrate software run fmri natural movie find canonical similarity fmri response cca mapping subject hold bold subject movie fmri collect day
likelihood estimate use denominator jacobian dimensionality strength sx classifier job ratio loose point make particle community towards concern value face nuisance associate generate concrete training balanced datum quadratic py px monotonic various surrogate lead discriminative monotonic function ratio must parametrize parameter nan alternate could desirable convenient training target input specify via classifier roughly flow generate expected loss influence g histogram approximate point point run generative demand particular context impractical instance human intervention classifier pre parametrized compute value thus concrete realization probability language imagine estimating depend dimensionality cost generative leave optimization problem future presence event describe uncertainty physics response associate search test generalize use discovery pseudo generate correspond correspond signal alternate hypothesis approximate one alternate advantage help classifier capacity relevant region perturbation I continuously signal htbp feed event evaluate presence nuisance thought typical usage machine systematic parametrize nuisance propagate uncertainty statistical improve parametrize correspond ratio outline distinction classifier work stochastic try uncertainty offer explicitly event intensive approach element intensive integral response handle even detector minute single conceptual detector offer evaluate initial practice measurement physical describe measurement mass particle cascade decay involve parameter nuisance coefficient detector learning associate pearson work access goal pearson generalize perhaps formal general validate classification metric parameter classifier different use discriminative make purpose calibration importance estimate decision naive various approach calibrate individual event calibration non provide calibrate take machine algorithm scalar achieve dimensional approximate generative available dimensionality distinction come classifier real know correct go outline perform repeat work correctly identify classifier approach eliminate approach parametrize jacobian factor parametrize evaluation likelihood avoid occur ideal technique high multiple univariate score parametrize simulator offer approximate bayesian free frequentist formalism strength separate quality calibration involve estimate difficulty calibration perform calibration depend dimensionality complexity result run simulator thank challenge early work project grant grateful science national york university e particle physics statistical evaluate likelihood function parametric central summarize information experiment need key area science result test interval simulator generative describe process measurement often impractical prior construct ratio datum calibration search particle simulator detector processing confidence interval likelihood test discovery single event feature process label typically simulator interpolation parametrize improve hundred search utilize supervise high physics library multi decision tree progress classification lead test long true composite parameter suboptimal high event event demonstrate intensive optimization motivation extend usage classifier nuisance scope result expand offer require parameter frequentist separate target discuss map aid statistical value density interested alternate lemma state powerful evaluate able density increasingly act instance high physics forward generalize prove ratio test form discriminative discriminative classifier generate extend result generalized ratio parametrize original e make accept reformulate per event assume vast exist generative predict bayes contrast classifier model threshold model explicitly learn familiar lead confusion current confusion traditional simulator motivated domain source confusion classification term ratio include lastly frequentist generative produce data mix monotonic desire target ratio test well think map point
sequence apply gram gram embed datum gram skip try maximize training constraint assume training center number representation softmax softmax sampling softmax negative utilize train size thus biological size qualitatively gram embedding dimensional van volume property protein well small calculate gram g metric score gram strength task protein total distinct family represent summation gram sequence select form negative machine classifier protein prediction sequence distinguish structured protein bank short set length protein neighbor distinguish typical binary positive aforementioned protein protein gram average protein short sequence maintain distinguish characteristic use release order sequence protein protein offer implication visualize different criterion mass van gram qualitative analyse neighbor diagram color accord gram group protein prove useful classification reveal bundle module mcp terminal protein tm tm ab tm tm tm type tm tm tm tm protein protein protein bioinformatic ability character protein sequence structure protein fraction set structured protein protein bank order visualize reduce histogram overlap gram occur structured protein exhibit histogram binary protein sequence gram protein shorter trivial case maintain obtain classified protein specificity present order region analysis size comparable c see visualization sequence reveal column comparison experimentally additionally versus order region classify accuracy respectively dense biological call sequence diverse meaningful physical chemical dense representation family show protein visualization providing interpretation discriminate sequence protein furthermore http edu visualization matlab request advantage embedding train encode biological deep application bioinformatics representation phase future aid wide problem prediction extraction identification block department california berkeley usa division berkeley national berkeley ca berkeley edu berkeley biological sequences method bioinformatics visualization protein identification protein protein use evaluate classifying protein family average obtain method addition predict protein database database region rich classifier distinguish protein bank protein embed diverse information regard pre train deep bioinformatics berkeley visualization use certain language biological sequence dna protein language language information conceptual analogy natural process nlp deep discover encode biological protein bioinformatics visualization protein structure interaction biological embed dimensional skip explain embed work database subsequently qualitatively evaluate classification protein average obtain family mapping sequence database database classify bank protein region advantage embed train sequence base tool datum available language processing http berkeley edu visualization become machine year main approach store establish inspire item fashion store partial description item store close generalize attribute nlp semantic syntactic embed define characterize neighboring word context consider amount via way architecture gram vector using skip gram vector representation show degree seek unique biological sequence interpretation skip gram train sequence sequence chemical purpose protein sequence bioinformatics prediction domain illustrate tackle protein protein protein relate protein similar structure gap motivated identification sequence classify study classify protein classifier structure sequence perform van volume secondary protein perform protein frequent exhibit result balance study biased protein secondary abundance role cell biology protein protein study characterization category region experimentally order release protein present nuclear mostly computationally identify interpretation work protein confirm furthermore
lemma result suboptimal arm suboptimal suboptimal arm gap corollary main difficulty dependency random gap layer arm bind number arm depend empirical gap allow suboptimal prove small fraction near conclude bernstein empirical probability reader define event arm strictly easy mean problem since number chain rule get arm select arm know simplify enough add small conclude assumption follow distribution arm reservoir budget denote arm union know large write define notice assumption follow constant depend large v imply implie cm rgb many chance try certain number design cumulative learner simple infinitely bandit arm either multiplicative extension several near small case horizon sequential decision number action make classical action choose among call choose round respect cumulative simple bandit small number decision extension infinitely many action impossible try practically face extremely large first already past whose sample challenge infinitely armed bandit respect bandit good arm sample arm order reasonably difficulty ask link selection principle subsample arm want chance sample good least tradeoff infinitely armed bandit problem reservoir learner work give optimal constant reservoir characterize specifically reservoir reservoir achieve limit factor arm rate arm sample reservoir bernoulli spirit distribution arm refine even factor prove low even sub case reservoir come tradeoff likely output therefore arm selection cumulative regret optimize problem want optimal select minimize finitely attract develop aim find arm arm ultimately infinitely bandit number arm available relevant biology even infinitely achieve regret infinitely armed extension concern replace hoeffding bernstein extension treat precision implement algorithm implementation simulation arm optimize cumulative multi armed infinitely many bandit many infinite class example bandit setting consider assume infinitely arm classic assume reward arm combine reward restrictive contrary arm armed call arm set arm already always learner time output assess simple regret p right arm reservoir arm bound distribution domain distribution hold also relax different imply arm assume distribution classical standard first infinitely bandit present bound corollary crucially depend depend algorithm problem regime result one characterized rate characterize regime selection tradeoff correspond reservoir put mass close good reservoir come parametric regime output close arm reservoir many arm arm exist regime rate one interesting difference cumulative e valid value simple fact exploitation everything explore reach cumulative practical implication examine base arm reservoir arm eq maximize ucb lead order divide logarithmic classic almost confidence regret infinitely bandit target cumulative regret number arm sample regret whereas interesting ucb specific specific number arm select ucb indeed decrease gap infinite infinitely arm useful refine update seem hand constant present easy state main characterize regret constant enough short sketch result control control mean arm high approximately time eq suboptimal arm suboptimal arm high since suboptimal arm time factor except consider specific result conjecture relevant either practical reason particular horizon exist corollary concern highlight present include theoretic general state near small e imply particular limitation simple modify bernstein display note general modify variance sample armed term refined variance thereby term proof conditioning immediately corollary arm bernstein minimax problem minimax sense proof immediately theorem rate therefore bernoulli discover valid particular distribution bernstein decay arm rate limitation general discuss limit never yet regret enough good interesting vary therefore index extreme exist tail hill estimate slightly prove arm directly estimate arm q concentration assumption bc assumption ht reservoir run knowledge note enough imply situation minimax round ucb use learnable reservoir go loose another question quickly trick double size away ucb air modifications straightforward algorithm simple modify air regret regime performance regularity different
get reduce approximate sequential monte carlo techniques employ expect gain use summation available filter b use update update particle particle particle particle markovian achieve draw accept proposal usual hasting probability material smc describe rely heavily fast accurate posterior filter might achieve development kind simulation nature space integration need repeat calculation subsection way infer third leibler approximation dimensionality simulation show supplementary material distribution accuracy numerical calculation extend technique detailed calculation find supplementary material hence mass poisson distribution analytical unimodal feature study depth equation filter available type integration likelihood subsection alternative normal rewrite normal sn u un multivariate laplace transform employ proof derive sharp laplace normal unique stay laplace equation scheme limit kullback calculation particle well approximated interval unimodal decay quantile normal derive pair focus poisson tt unconditional distribution state tail tail eq standard expect upper quantile tt likewise moderate true far low tail equation simulation practically quantile tt help calculation example template assess template compare methodology frequentist inference community sequential status example discuss work ucb sequential totally chance integrated intensity b gs template primarily illustration affect clarity presentation perform large template template methodology easily adapt use template k l calculate ten filter available gs smc bias gs gs apply case gs case motivation use template filter scenario model gp scenario prior correctly template matter gp emphasize template gp approximation instead integrated intensity differ intensity away truth compare prior region marginally outperform posterior adaptation lead template truth prove potential practical gaussian process choose optimize alone review work bayesian sequential experimental fundamentally optimum criterion seek aggregate bayesian design global bayesian nonparametric seek integrated mean paper sequential know paper aim understand target study literature intersection differ lot scientific formulation inferential even though literature measurement gp focus active conjugacy analytical approach capable deal conjugate poisson generalize nonparametric even also sequential neither criterion study inference portion none demonstrate nonparametric ed conjugacy nonparametric observational monte overcome computation log cox process exploit log emphasize mainly study ed employ sequential carlo easily nonparametric work topic design application theorem third gain equation p b b b b b p sn sn te te n sn ts sn multivariate normal sn un calculate fine j b log ie iw xy I w w k k w I I k w j w suppose know update e k k already w w j k k k w gauss quadrature km c g c e illustrative simulation fundamental approximate histogram path job approximated particle bind observation filter weight markovian move ny l ci p ig break axiom criterion lemma notation remark proof em energy galaxy able object average perspective employ accommodate combination know study conjugacy monte efficient tool volume distribution inference process normal simulation usefulness literature inferential technique general spectral log cox fitting branch characterize rely strong relative ease fit exist template object classify source geometrically weighted template template however cost observation necessary observational strategy template specification filter equivalent specify filter aforementione scientific specification datum role preferred oppose advantage motivate firstly good secondly find np hard armed sequential smc efficient calculation sequential difficulty come combination make gaussian conjugacy several make knowledge ed accordingly induce uncertainty goodness posterior serve primary proxy filter play finally address iii convex template nonparametric motivate study design ed statistical well ed
control cf depend show nf minor detail hellinger formulation proof inequality notice follow corollary lastly inequality conclude let vector result brevity j j conclude proof first bind argument arbitrary low proposition identity matrix q similar algebraic direct consequence property toeplitz strictly triangular imply regard simple integration combination conclude one algebra n q ultimately grant additionally cs partially nsf grant edu definition example lemma section claim macro format cd fix set domain arise likelihood minimax salient exploit capture covariance plug asymptotically contrast standard theory applicable class plug dense covariance change point observe detect shift temporal audio eeg health sciences advance algorithm asymptotic theory context exist work shift temporal statistically less structure modelling detect change collect base change normalize series california despite consideration researcher seek exploit temporal detection single contribution lead detection possible algorithm simplify process point observe sample moreover nb infinity one fundamentally framework arise distinguished distance gaussian process study asymptotic domain asymptotic suitable bound observation three dimension also adopt speech finance cast involve smooth pls pls current pls asymptotic procedure account procedure aforementione obviously ultimately suitable treatment first aware establish domain set second vast increase domain focus carry detection ignore suboptimal fix domain serve spatial spaced asymptotic notation perhaps work author gaussian ratio bound probability hold test statistic work direction wish chapter test variance sum random test continue admit ignore dependence level regard phenomenon x covariance negligible accounting test domain contribution paper suboptimal fix domain moreover detection new statistic increase domain encounter fix considerably challenging one need way point process increase domain show account analyze base upon ratio draw know dependence asymptotically increase domain setting hold structure class term determine spectral density decay fix confirm confirm paragraph covariance know address scenario plug method covariance covariance consistency regardless estimation plug situation plug inconsistent case contribution integrate exploit property absolutely toeplitz matrix classical norm either several beneficial detection focus one gaussian hypothesis spectral adapt plug subsection focus shift mean devote plug estimate serve optimality discuss numerical experiment assess section contain direction proof present auxiliary appendix inverse toeplitz useful stand minimum maximum operator length one matrix ij usual inner pm ij mf p fu fu fourier denote absolutely toeplitz nf na nb nb nb stand lastly gamma present datum account underlie assumption kk subsection mean symmetric denote toeplitz integrable definite exist depend satisfie hold bound origin closely link origin section turn essential statistic fairly explicit spectral regardless density class function close assumption rational real unit lead note since favorable easily assumption set mean zero endow toeplitz set datum observe symmetric toeplitz view accordingly denote fact symmetric f real universal scalar regard infimum necessary infinite polynomial state condition fourier common analysis toeplitz proceed change composite domain satisfie either fix setting restrict spectral admit increase domain moreover hypothesis alternative notation occurrence time nb denote change state tt nt union composite generalize process essential threshold depend false alarm also substantially exposition propose form unknown present unlike function explicitly account specifically asymptotic estimate propose approximate matrix indicate later let plug strictly consider domain consider subsequent section detection false detection take choice become seek result admit polynomially gaussian polynomially decay admit exist depend constant several comment role various choose apply appear shift tail notion standard observation implicitly challenge sample rate one hand appear closely nn possibility small shift detection parameter variance size analogous difference process chapter small lead shift smooth jump satisfy assumption label parameter jump contrary exponential class previously since great algebraic effort easily process regard super actually detect small jump quantify assertion smooth turning describe satisfy admit scalar comment exactly fixed set structure play factor gaussian arise encode r priori take plug approximate method definition way assess focus dependence parameter space base mle detection affect namely component fix domain grow detail zhang show consistently behind existence mutually absolutely model realization generic refer induced measure word algorithm equivalence grow shall exhibit performance fully whenever dense dense contrast estimate approximate proceed state non sequence small sup must remain speak weak condition sup norm estimation result condition imply denote horizontal covariance scenario base upon compact wu likelihood determine strictly speak verify imply correspondence space weakly consistent validity assumption sequence regard satisfying scalar depend c n q brevity drop vanish namely enough previously clear appear detection finite situation equivalence detection aspect rate plug plug disadvantage handling difficulty introduce isotropic explicit formula class parametrize example f associate rich efficient inverting significantly cost compare accelerate generalize approximate flexibility rest matrix eq broad condition decay give show satisfy du et show label proposition ready plug satisfy universal give theorem appear theorem universal covariance asymptotically plug plug big plug almost surely apart usage test detection early highlight accounting fix nan alternative hypothesis fix control mild hold covariance paper integrable jump increase least guarantee although detection error vanish verify early vanish jump vanish qualitative hypothesis shift mean deviation careful look reveal small standard bn nu test distinguish shift remark gap critical drastically change favor universal exhibit optimality study use compare especially thus increase preferred jump theorem plug theorem nearly density demonstrate minimax optimality domain spectral consider restrictive k generally speak contain spectral density condition qualitative class satisfy stand rational spectral admit salient qp although spectral consider due add density kn kn n result near optimality density study although establish near optimality plug spectral satisfy conjecture broad turning domain size shift unlike distinction assumption mention term universal test detection control simulation goal assess fix fix regime sample area receiver operate characteristic roc refer auc standard assess alarm curve roc curve pure line origin auc realistic compute auc repeat repeat correspond accord covariance shift shift curve roc curve k assess literature dense due estimate apply figure detection smoother impractical slightly auc figure plug polynomially decay compare detection panel rapid remarkably jump establish section study detect recall figure apparent plug covariance show choice furthermore detection robust estimate recalling method analogous rate panel decay exponentially decay panel function polynomially decay r diagonal satisfied case exhibit slightly gap auc curve polynomially decay presence regime panel display jump three dash exhibit auc structure plug nu right panel estimate panel display three curve green plug nu introduction detection plausible subject thorough probabilistic extend extended rigorously minimax detecting recently specifically scalar intensity detect observation optimal dimensional domain study behaviour paper valuable intuition minimax aside set detection proof main establish toeplitz proof interest algebraic derivation save follow unknown triangular addition brevity nu n z alarm non centrality lemma demonstrate eq suffice universal lower bind identity nd alternative show indeed algebra get arbitrarily note inequality obviously hold large identity verify basic yield inequality conclude proof tight w upper imply cf g obtained get universal universal n kn proceed manner precede cholesky refer gaussian observe n tend get
trajectory crowd small label least crowd try give reasonable small make positive example trajectory expert expert bad match auto encoder layer occur layer layer try build auto encoder corrupt corruption max tune negative stochastic mini prevent datum percentage neural network find infinitely trajectory across sec pre make fix cloud cloud grids size represent remove normalize trajectory preserve status trajectory g rotation status propose trajectory dynamic length match cumulative match strength maintain contribute cumulative length match index di ic local later complete programming di contribute cumulative match precede match encoding order rotation object normalize contribute e path give form tb crowd platform platform crowd tb cloud trajectory via object vary quality likely task successfully leave dash unlikely successfully line collect crowd crowd web platform see virtual web without expert presence user unseen manual web user cloud manual object start demonstrate fig select object use work like video bar show gray user click bar trajectory full expert similar object experience show cloud move try modify orientation pr position extra gray click minus remove occur bar update status open user click broad build point example crowd platform trajectory expert amazon completes ask complete object manual follow take raw microsoft fusion cloud object dataset model robot never see trajectory execute pr able turn right two trajectory pr model dataset test show dynamic mt first column manual consist mt mt intuitive percentage mt value survey find mt trajectory robot reasonable test fig trajectory axis sec locate orient differently successful dc slow successful difference turn switch object last transfer affect method find coordinate sensor allow tb learn node last cloud picture visualization cloud trajectory show select point cloud high activate input execute object fraction robot never correct planning trajectory correctly execute trajectory visual exactly object expert collect purpose compare crowd see believe crowd well crowd amazon vision art handle cloud expert pre object cloud even still extremely multiclass svm accuracy trajectory large outperform art give give access test lead time difficult modality design axis language act turn node act deep modality give noisy label handle give show crowd learn modality pr robot front language cloud trajectory cloud language robot trajectory controller successful pr project website introduce never see formulate structured output completely modalitie cloud trajectory deal crowd platform non expert deep baseline dataset crowd share learn engine acknowledgment building prototype website useful discussion microsoft award office point part highlight pt pt object human program planning object formulate plan structure handle three modality collect large test language far show robot never person visually read manual possible human vast experience differently shape generalize water figure build key object share similarly among completely even robot never machine able previously similarly fashion carry name rather shape completely robot cloud find appropriate trajectory experience consist distinct object part trajectory object part cloud color use object label center operate unobserved successfully trajectory previously cloud noisy black robot machine could execute environment variety object object alone rather rely understand cloud plan key modality cloud language crowd deep impact language architecture modality collect expert expensive presence robot work web platform outperform train expert previous entire oppose sequential state validate via web platform present key planning via object incorporation planning modality crowd dataset activity image use part however object activity reliably object many need direct focus detect part pour predict motion sequence perfect pour instance interaction track vision daily significant demonstrate trajectory via transfer sub sensor sensor trajectory consist status translation rotation origin r pr rotation instead euler angle trajectory trajectory translation linearly spherical trajectory transfer conceptually necessarily inconsistent make compatible new modifying rely object object challenge depend variation object degree commonly angle configuration need trajectory small orientation error execute modification robot position orientation object modification via cloud many object base individual part translation limit object lastly even different shape cloud even angle intrinsic shape part trajectory compatible trajectory space configuration principal frame rather task position orientation
capture decrease solve recurrence epoch resemble recurrence gradient incremental study sgd scenario massive grow dataset increasingly impractical discussion empirical necessity version competitive practice benefit type algorithm understand instead focus effect use face consist example pc generate normal spectrum finally point spectrum theorem vary rate particular slope log convergence graph ref thm exactly occur dotted line guide higher explain target fix rate dependence due spectrum plot second considerably despite pc variance good argument initialization result exponential remain incremental scheme result analogue calculation q extra lemma martingale n nm n version instance pick therefore draw chi freedom draw independently square one freedom characterization specifically second function finish unable reference suppose concavity apply deviation q use establish inequality length value eq q repeatedly shrink shrink n lemma reasoning careful finish pick summing yield lemma lemma hand expression expectation claim epoch yield bind theorem eq write q b recursively recurrence q finish definite thm thm thm draw wish eigenvector fashion maintain new give finite sample principal dimensionality reduction project top prohibitive update estimate eigenvector study elegant closely proportional point covariance paper estimator eigenvector eigenvalue treat q achieved point identical term recently lot descent convex non end system lie analyze initialize unit time receive next perform gradient adopt show progress behave time forward initialization average sensible fail coordinate far p pp ne x likewise initialize orthogonal remain avoid problem intrinsic random update ignore merely interested progress potential random initial normalization update likewise state final knowledge eigenvalue sigma outcome include x pick surface sphere initial rate epoch drop argue times epoch use argument careful specification denote space moreover consist include build martingale argument conditional step size time nest subset also proof yield line perspective generative matrix batch computationally intensive incremental bad case iteration purpose recent attempt inherent present pca mild detail sigma nu improvement measurable follow appendix identical henceforth quantity additive term monotonically decrease arbitrarily close bound away recall advance skip unit little bit much establish surface sphere start recurrence generating nonnegative ty show define derive
bias arbitrarily number ensemble sequence bias process sequence denote infinite process produce coin single come particular trial vanish however statistically mean another almost fluctuation process bias future bias information mn random variable mechanism drive random relationship excess quantifie correlation second quantifie past due fix ergodic mutual closely parameter estimation theoretic identity continuous differential entropy choose finite come divergence divergence cover ref parameter realization consist noisy fire spike train transition generate probability entropy x x hmms internal entropy generate hmms divergence h trial one might suggest multimodal normality posterior carry essentially rely log behave bandit process normality calculate normalization normality posterior generalize statement arm normality attention limit capture error distribution normality correction infinity long decrease entropy know recall ergodic immediately recover likelihood excess spin lattice finite divergence ergodic period alternative divergence language text empirical sec analyze asymptotic aim asymptotic normality recover aforementioned power analysis utilize ref recover storage require amount accurate inherently observer note arbitrarily excess statistical infinite still excess reach predict necessarily accurate agree ref proxy introduction choose within least choose important process store memory highly vanishing generalize bandit ergodic even trivially transition variable thus agree criterion look spin challenge yet theoretic ergodic look forward structural biology semantic human operate separate transition lead signature thank member upon u office nf nf sm student fellowship berkeley fellowship intuitive suggest truly complex condition familiar truly complex purely interaction allow spin lattice critical power law spin autocorrelation asymptotically configuration surprisingly dynamic spin ise lattice evolve lattice configuration spin stochastically past future imply finite spin familiar concrete near otherwise take possible give bound directly standard ise lattice bit excess entropy continuous global spin utility order likely maximize temporal phenomenon spin purely truly first contradict ise lattice interaction coupling strength iteration concatenation bandit observe logarithmic fig node cm bend auto every style draw loop style leave style style loop loop style right loop leave loop loop loop algorithm color green infinite build process familiar known multi bandit theoretic understanding highlight distinct divergence ergodic draw consequence resource divergence structural hierarchy truly many biological phenomenon measurement neural language failure transmission grid apparent particularly challenging reflect resource store parameter memory require resource analog mechanic suggest divergence since resource divergence sensitive inherent organization uniquely indicator date tractable construction relationship construction class repeat vary stochastically trial trial stationary memory decay many trial past future bandit answer remarkably process memory mechanism select insight derivation structural universal property unique present estimation divergence past divergence divergence bandit process structural principle learn view truly simplified introduce ergodic process theoretic approach review alternative construct structural statistical approach highlight discussion nest organization relationship hand difficulty sample predict attempt express persistent consequence closely resource specifically minimal reconstruct series randomness description complementary resource logical length series irreducible randomness though model fortunately
bootstrap avoid analytically use highly favorable analytically example scalability complexity bootstrap bag little fast distribute computation advance digital technology lead phone health inferential crucial correctness hypothesis traditional inferential storage massive architecture massive methodology massive expensive addition assign computationally conventional assign uncertainty deviation commonly apply two obvious computationally impractical volume point process advanced computing massive even variant subsample problematic output bag make massive datum massive subsample module bag massive store moreover subsample module process compute construct assign weight massive bootstrap yet number bootstrap sample expensive commonly modern estimator demand numerically primitive ls original statistically bootstrappe massive introduce low robust bootstrap possess scalability significantly low subset robust avoid point bootstrap possess conventional scalable system consistently accurate preliminary present conference review bootstrap implementation consistency new section big idea process store estimate g confidence interval etc great confidence often informative plain estimate little scalable disjoint bb resample replacement assigning subsample computed population within subsample module module produce effort computation subsample nevertheless thousand impractical even complexity modern maximum likelihood estimator primitive statistically robust l face even sufficient outlier fp equation dependent value bootstrap conventional accurately reflect correction need n statistically compatible distribute system massive many pca combine desirable method smooth fp bag little scalability computational complexity burden estimate drastically replication bag let subsample randomly replacement equivalently random weight b distinct subsample bag compute distinct low allow fast computation confidence modern robust draw form module bb subsample generate form assign weight initial solve compute uncertainty disjoint datum set subsample module estimate uncertainty average subsample order statistically robust random robust continuously widely subscript different estimator iterative iteration obtain turn need present fp scalable step obtain modify follow let subsample b replication statistical bag observe outcome denote dirac n subsample bootstrap b side class pf fp appendix quantile word pr q robustness break bag b upper minimum proportion subsample sufficient proof n general estimate accord reliable sample quantile one bag former latter b explanatory quantile close quantile sample draw quantile efficiency significantly estimator important setting purpose e variate variance scheme efficiency original bootstrap simulation compare side setting right side draw subsample performing step e distribution well bad average along uncertainty see element n r performance assess estimator ls setup subsample maximum module start add subsample module bootstrap sample illustrate contaminate robustness setting accord theorem sufficient bind choose multiply resemble world lack accord setting upper estimate multiply still proportion severe lack robustness face contaminate make comparison deviation compute cumulative report relative error time remarkably
bound entail old regularity know entail regret optimal optimal nonparametric run average net note regret indeed regret entail well explain technique build perform simultaneous cm scale aggregation exponentially forecaster competitive instance extension competitive increment increment scale gradient core lie use already present argument scale unclear besides spirit square online contrary build use discretization g linearization aggregation consideration suboptimal exponentially forecaster address lipschitz slow multi crucial linearization type section design efficient old knowledge concrete proof endow sup small net subroutine extension algorithm extension minimize loss simplex simultaneous convexity function jointly k algorithm derivative scalar variable map sup vector value assume upper partial tune bounded cm sake prove bind consist main aggregation level scale cm proper net kk cm definition follow predictor tu define exactly vector output forecaster apply j weighted forecaster define initialization j n n n predict ty low k high new weight vector exponentially forecaster average type forecaster tune introduction pp exponentially forecaster tune I address associate corollary sparse high dimensional spirit see yield could slightly regret slightly spirit omit know advance forecaster adaptive prediction modification regret multiplicative constant know advance adaptation tune without advance even forecaster useful small regret also assume forecast round measure exp convex loss bound rr depend loss quantile regression aggregation replace proof level cm explain tuple forecaster assumption cm differentiable convex norm k ki value ix f ff inequality jx b inequality substituting entail b claim apply cm infimum dirac monotonicity substitute tx jx tt convex intermediate weighted forecaster tune since intermediate prediction square concave infimum g set f f eq obtain expand fx conclude exponentially forecaster exploit net complexity exponentially actually class sufficiently enable adapt technique quasi net easier exploit algorithmic viewpoint old class introduction forecaster net viewpoint quasi regret see fix role approximate piecewise constant follow quite ai construct net net dyadic discretization note every partition mf cm restrict cm see continuously fortunately combination n two dot line function maintain exponentially dyadic parallel round combination aggregation simultaneously u u cm nj aggregate tune e every make ax assume dyadic dyadic tractable fall round overall factor complexity tractable round polynomial tailor future process reader g g subgaussian I z lemma close technique maxima formally approximation k f provide adaptive horizon advance basically vary rate positive integer k k kt jointly convex loss multi jointly variable bound eq k correspond exponentially forecaster page expert therefore well e note lie need upper conclude proof appendix sequel exist norm continuous denote introduction forecaster modify net viewpoint lead section special function consist net discretization let play fact discretization fix constant final define set define ai constant cf net net net discretization partition note partition refine dyadic set center consecutive level denote replace definition play refine partition take look piecewise polynomial value replace let f tractable precisely algorithm parallel round fall multi aggregation jj simultaneously instance loss define convex competitive j exponentially average forecaster q prediction ax nest h logarithmic factor partition nest partition nest follow proof defer appendix assumption theorem nest h complexity omit paper classical linearization splitting gradient eq last vertex side sequence correspond exactly weight vector output weight page g eq last conclude net lemma x c c fx ai aa f fx argue fx x ai induction lipschitz figure illustration indeed ax ni na f ai ax derivative width setting choice conclude p q mx fx mx f cp fx nx I ni x b triangle choice n explain incur small cumulative regret inside previous fix time new fall multi perform low thus follow two aggregation start one apply forecaster theorem appendix instead norm gradient mx
make threshold analogous centralized condition thresholde nk n nk nk pn nk nk nk nk centralized nk nk shot procedure average study dual regularize form evaluate estimator cost central l server average back average form store form round remain machine solve scale row jx estimator l k ty l l j j nk ty show subgaussian converge usual subgaussian show match centralize subgaussian plug subtracting take similar together grow grow machine average estimator comparable exceed threshold grow machine exceed term sparse dimensional setting first estimator machine communication bound communication risk regression impose sparsity mean bit need bit among algorithm amount far establish communication average machine generalize coherence subgaussian thus bernstein ns obtain desire union component conclusion lemma express norm subgaussian subgaussian recognize nx variable constant simplify union subgaussian subgaussian subgaussian recognize j proposition simplify union lemma remark lee equally college devise one shoot key dataset machine modern dataset distribute arise work fit multiple machine computational bottleneck processor shoot highly round pose challenge design shot popular datum locally master estimator produce average multinomial non average centralized machine stochastic sgd subset dataset thing centralize recently study erm mse erm erm match centralize erm optimality setting number set average erm order centralize erm erm suboptimal centralize erm work generalize risk minimization regularize minimization rely beta min strength centralize rely aforementioned min ensure recover square restrict machine desire contrast average work correlate make study divide hypothesis devise average centralized idea estimator thresholding average nk nk nk total machine aforementione minimax optimal factor lasso shoot algorithm centralize average subgaussian subgaussian seek regression norm regularize signal processing develop say lasso nearly solution support bias proportion order gain nothing average lasso formal bias estimator lasso ordinary ol coefficient correct incur shrinkage term incur previously refer term depend play suggest form solution proportional variance refer generalize let q keep feasible subgaussian feasible subgaussian subgaussian occur see lasso decay suitable require condition positive direction relate call replace right side gaussian zero constant extended subgaussian design covariate subgaussian subgaussian q occur probability consistent suitable intuitively large dominate empirical part typically subgaussian event occur probability practice lasso estimator condition lasso give convergence rate bias small constant incoherence occur plug subtracting set old generalize occur piece decay fast decay comparison subgaussian variable variance nm
universe output approximate private release natural problem complexity minimum number differentially pac give privacy sample pac al suffice class possibility private private pure function universe vc properly privacy improper class pac differential sample complexity approximate differential privacy et show properly I threshold learner complexity proper pac privacy properly datum universe extend threshold threshold complexity concept vc totally properly privacy interesting characterize proper private learner learner differential leave possibility improper pac privacy sample question present improper point pure privacy infinite domain et point privacy grow give mechanism domain mechanism inherent countable improper pac learner function pure privacy release analyze sample totally universe every totally order universe four privacy problem release interior query release threshold kolmogorov distance proper pac thus bound release proper prove interior interior universe privacy h p database I simply output point hence universe size hard database reduction interior differentially mechanism hope construct reason failure always output distribution similar mechanism extremely generally feasible evidence computational hardness al class polynomial hard al proper universe concept need private continuous show class learner pure characterization private sample necessary sufficient subsequently equivalence representation way equivalence pure learner private al show boost query guarantee answer query showed transform private learner error learning minimization denote order domain differential early show release noise answer random variable say function laplace let sensitivity add preserve differential algorithm access differentially private mechanism overall privacy guarantee differentially private differentially private second argument close query solve complexity size every interior least monotonically construct differentially sample element set differentially private agree private private inductive depend fix define positive differentially private appendix combinatorial call interior code recently differential induction claim database differentially private mechanism claim construct follow ny sn ny bn sake contradiction differentially private interior solve dd sn dy iy agree private pair adjacent database eq argue great agree first digits interior point succeed interior agree digit except database fix randomness st everything else fix except union give desire contradiction introduce q induction get upper point bind mechanism low set proof guarantee solve totally order interior interior idea paper full solving provide goal pair element length recursion number several exclude least one pair agree agree random agree scene element agree element follow element agree two one good use technique recursive finite totally differentially private probability construct database small universe every element construction present tool primitive denote finite database universe define domain finite database maximize mechanism solve specifically exponential differentially sensitivity tn differentially private building differentially define domain database sensitivity growth without function choosing mechanism differentially private solution approximately instead set gs lemma utility mechanism slightly result bound function choose mechanism execute growth quality contain present privacy kn choose l start execution database recursive database observation recursive call input recursive call database motivate value agree element common pair stability randomly bad distance leave side place element approximation database recursion least guarantee w help identify induction call e mechanism output suffice recursive call database element choose pair close except continue database inductive pair agree least agree element thus mechanism exist mechanism satisfie case agree condition good output hence analyze hence big fail appropriate execute call differentially induction recursive call denote differentially perform call database call consist denote database similar database probability recursion private preserve argument formal first one database exist permutation composition desire whenever database execute differentially mechanism composition get close gap roughly interior recursive limited affect privacy preserve assume preserve privacy grow exponentially result low hand recursion pair new ensure change limited input affect element element pair pair twice database carefully dependency think pt pt common twice every database acceptable still begin input ensure close agree identify randomly element pair change every rapidly interior learn kolmogorov proper pac translate bound version write row query release private approximate answer count simultaneously query query universe answer output database interested release counting interested relate release universe pmf qx qx count totally cdf totally domain f distribution kolmogorov totally order cdf release collection counting closely release empirical distribution theory query sufficiently approximately agree answer privacy consideration privacy incur actually improve privacy computation large offset result mechanism differentially private database row let replacement row answer qx return accurate appear differentially differentially private algorithm operating database replacement row run result private adjacent database index sample index subset consist since observe privacy e n nn n concept take concept example unknown probability accord unknown output precisely respect target hx cx error pac pac class target distribution draw coin otherwise improper hypothesis statistical necessary concept suffice properly agree recall totally threshold sample differentially complexity differentially private require database database include correspond concept distribution threshold measure later analogously similar learner cx cx take without error every database case learner privacy every concept every differentially bound complexity next complexity size fix denote contain add concept exist marginal distribution consistent must hence consistent privacy show differential privacy multiplicative complexity properly error totally domain differentially solve interior point error private accurate pac learner pac learner differentially problem complexity solve interior every argument learn totally order domain differentially private solve interior differentially private properly threshold differentially private differentially private solve private interior threshold privacy hence apply differential reverse direction size change hence preserve proper totally yx suffice chernoff probability output consistent concept generic free uniform differentially pac learner concept differentially private proper result empirical database draw sample subsample replacement lemma complexity private learner extend general release private item bit dependence much even negligible separation sample complexity private private vc dimensional threshold vc obtain class vc totally concept differentially private learner hardness proper concept different concept datum universe consist universe element database example fail hypothesis differentially proper complexity note element concept evaluate justify necessity iff complexity point evaluate use sample identical show require complexity element class proof toward contradiction differentially proper use essentially output hypothesis consistent example contain embed axis axis axis element evaluate ni observe private limited entry differentially consider execution correctly generate observe bad random axis axis axis axis output contradict hardness find threshold universe algorithm fail algorithm query release private approximated differentially private answer prove query release predicate differentially private differentially private pac sample differentially private argument incorporate contradiction query complexity contradiction construct apply database answer every cd nc cd prove reduction database specifically totally order domain maximum every differentially mechanism relaxation interior require technique bound note ask work domain construct distribution every private mechanism solve generality totally domain contain infinite take mechanism increase bound unlikely note problem mechanism problem fix sake contradiction bound solve interval universe tt bind idea learn point evaluate packing privacy concept whether countable length resolve impossible even infinite privacy countable countable collection hypothesis differentially private pac point finite proper however consequence clarity loss suppose sake learner countable subset hypothesis establish sequence packing constraint infinitely disjoint construct wish hypothesis anonymous helpful suggestion guide point reference privacy mechanism neighboring database r ss two every fact imply hold fact output r must follow output gs k complete analysis choosing mechanism define fail solution event choose f gs get mechanism code address digital content piece digital content copy content user produce copy hide copy uniquely informally produce still provide certain traditionally assumption require show code roughly speak work nontrivial accuracy algorithm satisfie mean back differentially solve prove object traditional lie order interior follow order randomized codebook symbol coin adversary subset say security completeness completeness error probability take could code use interior produce copy existence interior lower solve completeness differentially algorithm solve database codebook replace differentially private eq construct interior idea allow interior every interior domain user domain completeness perfect suppose behavior user nx sn nx sn digit every codebook codebook maximum digits agree agree digits agree codebook security check completeness consider code produce output index agree prove perfect prove let lemma somewhat reduction show interior enable accurately release idea reduction differentially private quantile input release strategy tree generate leave node leaf sample path sort database block value quantile final differentially formally describe differentially mechanism interior succeed database include empty sort power n dd rd differential release let noise partition partition accord differ block cf moreover noise vector sample produce succeed suffice execution succeed give section theorem remark fellowship support technology information security fellowship computer differentially private threshold evaluate otherwise first differential privacy impossible require grow technique apply properly pac learning threshold differential bind separation concept differential properly extend direction small construction bound differential privacy pac threshold differential aim analysis privacy sensitive individual privacy differentially private introduction effect individual differentially private nevertheless rich many compatible privacy individual infinity still asymptotic vanish get
deep boltzmann loop train architecture stack let construct factorize distribution belong parametrized distribution evaluate form normalization dy fy dy maximize reach maximum implicitly find decompose divergence pz obtain low analogous seem important conceptual divergence give training quantify train bind remove normalization problem combinatorial bind follow direction see subject argue beneficial light formulate model let train close bind bp original prefer maximize minimized decomposition bp approach optimal intuitively qx complexity k wide range wide concentrate sigmoid bx ba sigmoid important equation evaluate eq estimator update reweighte derive gradient supplement p concern fully observe final determined computing normalize weighted individual basically optimize contain random proposal sample use proposal algorithm proposal candidate resample accord resample resampling evaluate relative end procedure resample l k l kp k draw sample distribution proposal chance cover weight mixture include equip distribution straightforward initialize p gibbs multiple chain converge bottom fix approximately reconstruction provide map give estimate conservative probably might likelihood experimental would normalization equation select experimental result various discuss competitive description initialize implementation available http uci repository mnist dataset translate robust distribution describe method gmm importance general repository summarize offset conservative generative evaluate c connect dna web auto ar gmm lower assume unknown accord train otherwise converge epoch importance report unknown reasonable digits digit sharp biased one obvious correctly variability assign probability seem one source variability gray short stroke gray pixel variability within one would propose detailed failure learn relatively ability highlight digits reconstruct partially layer proposal goal formulate stay weight whenever approximate inference weight occur quality roughly symmetric control plot p mechanism totally proposal visually indistinguishable show supplement show sample generate database pixel gray bernoulli proceed rapidly epoch mostly learn face epoch sample variable hide hyperparameter generative automatically inference derive likelihood deep generative multiple layer force model different approach solely train something future serious attempt normalization know would enable tight bound test report differ mnist attempt could certainly make apply involve nature generative always direct might make suitable task observe least wide range choice parametrize assume eventually suit training acknowledgment uci experiment hide letter web sgd research university unsupervise challenge problem dimensional auxiliary help fit start run
il operator backtrack implicit fix yielding implement use minimization matlab implementation bfgs mx dynamic consistent prediction figure bethe root bethe hessian construction cluster available direction replace pass type computation theoretically transition lead receive european research european fp agreement department physics sup paris france universit paris paris paris task application aspect problem reliably performance reconstruction propose completion bethe hessian negative eigenvalue bethe discrepancy estimate matrix reveal analyze random statistical mechanic neural efficiently matrix depict root square empirically compare exist infer entry motivate collaborative observe widely question complete assume address reveal question motivated generic detect reasonable expect existence impossible achievable root square estimating unknown provide rmse call rank eigenvalue bethe completion rmse exist contribution construction via spectral method use part spin phase transition fraction element call observe reveal reconstruct difficulty reveal entry per shall rank iid algorithm parametric analysis completion algorithmic associate programming low considerable completion entry rmse empirically achieve regime proceed observed reveal reveal entry resp singular decomposition keep ratio consecutive minimum discrepancy initial first improve replace different minimization detect community spectral traditional spurious singular show backtrack bethe spectral reliable inference performance completion analyze spin mechanic transition rank unable completion see optimization careful adjacency refer bethe hessian q parameter neighbor assume center numerically solve build bethe resp rank function alternatively negative bethe possible backtrack weighted spectrum bethe backtracking next motivate wise infer illustrate bethe justify compose remain positive belief size bar convenience eigenvalue bethe bethe free increase shift eventually merge increase plot uninformative region motivate graphical perspective generalize bipartite bethe problem minimize call bethe read degree model study decade shall well bethe energy energy initial expect correct critical mark appearance spurious minima bethe approach bethe detect retrieval look hessian bethe hessian involve vanish involve derivative remain bethe hessian picture motivate backtrack equation bethe bethe mathematically simple handle use statistical mechanic rigorous argument investigate phase transition method mechanic bethe hessian repeat computation cavity interested mechanic phase vector sensitivity random perturbation existence condition exist spurious condition meet bethe critical implicitly population remarkably population compute suggest matrix become simplify stress regime decay simple expression limit
variant hilbert embed markov hmm predictive exploit future reformulate stage instrumental hmm instrumental identify use dynamical reduce dynamical auto determine method similarity encode distribution next three learn approach distinguish work limit observable uniquely state handle choose window instrumental noise window estimate system rather multiple whereas regression establish consistency regression perfectly hand convergence section main theoretical instrumental true regardless regressor triplet input instrumental equally well successive convergence rate possibly estimate ensure invertible future mean future closely think main quantify regression independent regression satisfy q proposition hilbert schmidt hold test result proof go theorem generic main regression estimate operator estimate exact show use vanish middle replace eigenvalue completeness example bind estimation g rate regression generalize next address functional embed regularization error accommodate dynamical define assume dynamical stable get perform well onto x sense since prediction demonstrate learn specifically limited feature pick history window reduce consistency attempt student interactive computer question student learn student correctly answer question incorrectly transition summarize observation solid horizontal maintain represent answer student publicly call generate geometry student knowledge typical attempt correct iff student answer try student constitute discard sequence length begin handle history window regression important sample observation training regression incorrect restrict binary correct incorrect reasonable state observable predictive indicator denote statistic conditioning simple result hmm fact hmm incorporate knowledge intuition unlikely event aggregate observation indicator optimal must learn exponentially parameter predictor easy window length train logistic advantage paragraph need predictor near approach logistic close variant two split train split error split depict accuracy turn outperform expansion increase non work supervise propose stage stage history future estimate successful system history identify latent exponential increase scenario would like extend framework dynamical instrumental framework I estimate start restrictive invertible zero least equation learn specifie easily observable eq constant move realistic dynamic span hmm rp rp p replace regression sake detail rank decomposition rr operator use replace estimate b b tb bb possible history column represent possible give eq extend observation follow enough singular invertible match instrumental steady case gain specify marginalization instrumental regression regression action variant reproduce space operator possible implement non parametric smooth depend produce arbitrary conditional weight expectation action application kernel regression produce combination training operator shift future condition stable state regression estimation regularization bernstein state surely q eq q recall test error perturb version covariance operator effect regularization characterize effect text define basically capture addition sample regression important observable quantity depend respectively quantity effect covariance positive assume surely let union set suffice remain suitable similarly argument term case test regression define eq unit invertible operator triangular error assume ab v I rest triangular within x x cd geometric harmonic eq cox allow simply union instance projection proposition remark edu cs edu cs substantial interest state dynamical system algorithm tradeoff speed despite predictive sometimes practice contrast literature belief predictive restrict linear view instrumental dynamical learner simply effectiveness propose non linear regression outperform correctness substantial belief observable could invert often intractable sometimes invertible replace th seek moment discover tool expand consider remove difficult hmm expand observable brevity call offer computational statistical hard state prior structure dynamical remove directly derive analyze implement require difficult discover average track algorithm well fact interpret instrumental dynamical supervise additional ability arbitrary regression problem
hierarchical exactly model dispersion positive relation independent bound straightforward unweighted weighted semi scale size bind unweighted semi insensitive subscript inversion identity employ lemma method estimate quantity proposition group weight positive definite define invertible proceed statement contradict nonzero full hold positive semidefinite far one show suffice converse exist nonzero norm force moment force invertible imply additionally together follow force support size show markov imply final assumption q imply probability set choice weight sense equivalence efficiency effect assumption force imply exist neighbourhood asymptotic heuristic neighbourhood q write let identity series lemma three establish concentration bound near estimate asymptotic infinity certain weighted estimator consistent share estimate bounding constant bound assumption unweighted away infinity bounding constant asymptotic result define eq similarly asymptotic go typically condition go unweighted weight show thus necessary away infinity proposition establish consistent force force immediately proposition let show efficient choice bound assumption set sequence condition two result estimate multivariate random identity er device show quantity converge follow lyapunov ensure converge normal cauchy schwarz thus standard normal proposition force zero normal moment estimator counterpart study hierarchical regression describe logistic simulation similar behavior group replicate replicate draw wishart freedom splitting evenly replicate draw exponential point draw multinomial proportional equivalent drawing give rise group effect zero draw variate variety population empirical base estimation moment two programming language procedure laplace implement splitting estimation split combine estimate average implement quasi likelihood package iteratively fitting procedure maximize detail intensive loop outer serial tuning evaluate replicate indicate error visible moderate still appear panel method likelihood factor without include validation substantial improvement computational utility estimator recommender specifically user movie moment hierarchical minute serial hour section preference user rating recommender user population meaningful coefficient specific available rating datum rating movie star rating star movie rating relate rating use effect let specific plus indicate star set encodes movie rate movie reduce list covariate assign score score category list category zero predictor motivated intuition recommender popularity movie rating measure whether capture user overall predictor depend past regression rating order treat vector ordinary likelihood specific action child rate robust logit popularity rate movie movie recent review movie movie review rating rating assume moment rating compute approximate come normal marginal marginal approximately elliptical evenly spaced normally part bivariate look coefficient look association estimate follow affinity intercept tendency user action child movie user like movie action tend prefer movie movie prefer popular movie tend preference allow diversity encode regression also primary system rating compete ability strength obtain generalize vector linear likelihood model fit moment maximum randomly review set test fit test user aggregate average indicate error global flexibility model outperform estimator general hierarchical model unlike proposal predictor appeal large rely assumption distributional likelihood estimator procedure mild asymptotically fix effect vector linear hierarchical group size theoretical apply theoretical condition ask handle hierarchical hierarchy feasible implement derive guarantee care obvious proceed crucially conditionally coefficient likely impossible application datum item popularity perfect solution fall within simple predictor context normally datum volume continue advantageous computational primary concern demonstrate keep gain improvement speed author anonymous reference suggest heuristic minimax consideration show weight use eq make unbiased minimize minimize let denote square must lagrangian multipli eq like measure generality eq ideally minimize practice find instead risk attempt find gradient constraint solve situation computationally expensive hierarchical simplification weight motivated correspondence practical use semi consideration replace invertible define first case compact value identity force sum write matrix cauchy schwarz event least standard markov event triangle bind define replace imply identity force note force assumption lemma define follow identity along inequality give definite semidefinite exist case positive last q let eq markov follow define imply fixed pointwise similarly cauchy lemma right triangle put lemma markov normal write fix maximize depend gradient randomly th rate value appropriately recommendation effect effect simulation th well iteration perform simulation linear regression logistic gain reduce statistical method competitive perform four loss simulate sample choice replicate replicate generate effect covariance effect wishart degree freedom draw predictor I compare moment weight choose implement maximization programming language free likelihood package splitting split compute separate combine implement intensive c serial time random effect loss replicate base slightly method consistent method method range size panel computation simulation likelihood procedure procedure clearly fast follow exact consideration gain reduce term statistical efficiency competitive exact heterogeneous strength across likelihood hierarchical computational propose hierarchical consistent recommender application compare standard method hour minute multiple sub exhibit city period member ignore independent account observation specific social reference book describe detail explicitly hold accounting variability hierarchical second well prediction latter relate recommender system user item user preference specific recommender mixed item specific collaborative base preference similar user many recommender system iterative high letting denote effect likelihood maximization profile likelihood computation cost computational proportional substantial sparse exploit structure impose constraint estimate situation processor estimate cost cost processor reduce likelihood criterion descent series lower fit report propose extend population parameter exist alternative locality dominant procedure across computation factor demonstrate moment amount likelihood stochastic gradient descent propose first mix implement example cost dominant time validation time notably split running moment procedure improvement regime trade introduce detail exist procedure asymptotic normality simulation method additional support lemma generate individually jointly goal specifically associate predictor vector dimension let predictor dimension effect expectation population far within group give lastly identically distribute assume region random exploit root typically generalize non relation hierarchical linear response relation q variance dispersion get group get hierarchical inferential formally effect effect vector response computationally fitting restrict denote density maximize likelihood maximization newton iteration optimization use technique describe expectation quasi profile software maximize likelihood negligible additional effort construction unbiased introduce first value denote notational convention matrix multiplication invertible subspace symmetric relation concatenation matrix matrix due negligible practice handle cone semidefinite employ similar modification continuous continuous consistent exist allow rank estimator approach estimate consistency minimize weight discuss practical alternative option call unweighted correspond second call option call two semi set repeat prefer unweighted variance group unweighte big specific much conditional weighted light prefer show effect moment base estimation nonlinear moment exactly relation relative moment theoretical approximation
velocity buffer buffer buffer could also arithmetic buffer save store compare store bit integer point save fraction miss backward much net hyperparameter optimize rapidly train run allow hyperparameter advantage show several scheme would previously impractical net employ heuristic hyperparameter validation schedule choice intuition argument objective directly jointly rate neural separately sgd mean schedule initialization seed seed initialize mini batch enforce stop rather optimize schedule optimize deep neural network choose layer choose several optimization include sgd minibatch conjugate section momentum meta size cc elementary meta meta elementary demonstrate schedule average seed iteration optimize bias initialization scale bias initialization line total subsequent interestingly first say penalty neural role hyperparameter network improve individual neural simple see neural network layer mean might relatively hyperparameter scheme automatic determination view gradient transformation gradient objective augmentation procedure show example optimize label respectively label light class difficult remain sgd treat vector distinction rate exactly reduce elementary optimize come adapt domain recurrent neural net net think architecture weight hard architecture illustrate pixel character character alphabet character learn separate alphabet net generic like filter maintain specific alphabet weight absence quadratic weight weight infeasible implicitly build structure penalty net alphabet three diagonal matrix correspond diagonal level similarity character character distinguish character constitute learn correlation low partially equally share interestingly share input weight separate tie pdf pdf pdf pdf low force weight hyperparameter automatic ad software package development provide access internal automatic containing loop code back later engineering difficulty address practical explore issue learn dependency depend hyperparameter thing elementary depend network thus issue sometimes make elementary learning induce gradient uninformative term relate gradient illustrate phenomenon layer learn high gradient become uninformative maintain rate approach minimum problem learning rate relatively stop meta magnitude meta gradient optimize limitation overfitte objective I rough guide hyperparameter hyperparameter affect regularization discrete manner choose closely derive l bfgs update hyperparameter crf I available exactly contrast exact gradient hyperparameter learn converge svm loss weight tight optimize selection likelihood gradient hamiltonian several reversible memory dramatically approach could base optimization could incorporate computation require much elementary gradient chain mostly small update parameter dynamic memory elementary evaluation long trick paper derive procedure compute momentum approximate drastically reduce requirement hundred gradient validation something infeasible allow automatic tuning training tuning detailed regularization neural acknowledgment thank helpful discussion device advanced institute technology reverse individual derivative input output reverse mode differ work reverse opposite final nest scalar scenario reverse clear imagine intermediate mode scalar multiply vector jacobian general yes case multiplication sparse jacobian reverse intermediate maintain drastically reduce requirement reverse gradient exact validation gradient thousand include momentum initialization parameterize exactly descent momentum machine system penalty specify size initialization choose crucial hyperparameter selection gradient demonstrate automatic performance optimize gradient mode allow cost elementary hyperparameter gradient compute inner loop elementary I reverse describe technical gradient eliminate high elementary parameterization give flexibility explore hyperparameter back elementary hyperparameter momentum exactly gradient descent momentum continuous momentum reduce gradient allow thousand hyperparameter optimize fine initialization preprocessing insight optimize asset backpropagation allow compute backward evaluating loss obtain either mode force difference would make entirely infeasible hyperparameter however I approach sized maintain reverse division multiplication bit concern carry reverse require repeat multiplication learn procedure end usually unfortunately reversible set ideally initialization hope inversion another move analogy dynamic analogous generate
visual inspection designing recurrence analyze system build adjacency recurrence interpret network extend recurrence paradigm classification use compression distance another way build adjacency extract different unclear topological relate representation image angular transition apply convolutional neural classify previous inspire rescale auto da imputation mse test learn cnns da explain introduce framework encode angular coordinate actually cosine summation angle duality complex framework bin encode real rescale series value angular radius factor span angular span water encoding map monotonic time coordinate inverse second opposed coordinate preserve future rescale angular cosine angular discuss angular angular rescaled accurate transform angular interval angular angular follow transform time define inner type angular preserve dependency increase position top interval angular reconstruct level feature deep trend illustrate extend markov sequentially given identify quantile bin construct count quantile chain frequency quantile quantile dependency step demonstrate get loss overcome divide magnitude quantile bin bins temporal denote axis matrix position quantile quantile actually encode probability interval illustrate special capture probability quantile overlap aggregate subsequence encode nn fast raw control n trace cnns classify signal processing pre facilitate compare classification publish compete window bag space classifier recurrence pattern symbolic svm bag convolutional multiple map parameterize shared producing cnn learn overcomplete sake please details cnns illustrated bins window time bin enable construct small discretize quantile construct size kernel size quantile cnn soft cross image cnn penalty factor finally classify test use prefer help selection provide without cnns compound sake generally overfitte generally high note rescale time series map image mapping image mapping show later rescale come hand image variation signal dynamic recognition pixel encodes static depict channel image e channel static embed classification tune compound classifier competitive state time series previously mention function series uncertainty among come ambiguity precisely predict missing series break manually noise randomly transform data auto encoder note add break help last break train model apply back series broken help recover extract series imputation raw input da run change hidden type imputation shape remain da four totally imputation initialization descent repeat report mse mean complete sequence imputation unknown interestingly da perform sequence gap full imputation mse da raw well predict always mse imputation mse imputation raw stable performance raw raw trick augmentation dimensionality datum information image da utilize temporal spatial subsequence stable full mse mse contrast cnn neither interpretation edge angle cnn work illustrate reconstruction six map cnns eqn cnn patch essentially move nonlinear integration consider dependency benefit preserve temporal observe layer cnns dependencie convolution image preserve address feature cnn orthogonality advantage
bit put differently player bit capture payment player analyst select player uniformly report bayes nash positive maximize affine player bit player payment informally payment induce payment agree bit note would equal payment rule induce report regression unbiased nevertheless preferable desirable consider vector r follow condition close decrease significantly variance invertible depend large although expect ever follow notion provide small spectral theorem whose restrict attention ball simplicity useful differentially nr r jointly short lemma imply payment observable differentially jointly mechanism jointly differentially output player sensitivity player follow sensitivity arbitrary neighboring database say r r neighboring bound change estimator satisfies show differentially database differ denote compute output choice computed noise difference fix arbitrary plug p joint differential computed subset theorem differential use player uniformly theorem report partition differentially nash lemma player expect run different dataset player differ lemma noise add preserve privacy symmetric profile player fraction player player cost player player report I fix database player q player sequence database input player run database ridge differ database report exactly player maximize inequality add equilibrium group accord compute let report strategy compute ease estimator nash player receive individually player privacy unbounde maintain privacy player compute group report ridge follow algorithm player ease bound payment low bound payment player receive input input payment payment payment payment utility mechanism negative next require analyst budget payment bound receive payment thus q continue since infimum expression feasible region finally cd jointly differentially private bayes nash fraction report estimate individually rational fraction mechanism analyst cd fraction union bind private guarantee report approximate achieve among thing n due dominate nash fraction estimator dominate term third rational player bb bb final always entire simplify bind suffice individually rational fraction mechanism analyst n characterize draw must identity suffice calculation q minimum n contradiction eigenvalue case n contradiction mechanism far upper depend report privacy define expect expectation player report mechanism I privacy bound log particular composition setting differentially private report player sketch inequality specification privacy inequality utility ic g I mx I fc interpretation loss plus convexity eigenvalue function prove convex identity strong convexity reduce requirement quadratic strongly strongly notational ease denote loss denote th coordinate identity thus positive vector sum also psd corollary formal hold individual privacy mechanism guarantee participant individual loss immediately poses differentially model exist mechanism privacy sensitive well challenge computation fitting perhaps fundamental experimental many model learn hold analyst task must medical trial census survey behavioral currently massive hold interested enough wish influence outcome either benefit directly concern necessary design mechanism proper tradeoff budget participant concern privacy easily handle clarity privacy hold analyst linear analyst player computation minimize analyst cost establish player analyst differentially private pose differentially bias individual payment issue mechanism appropriate mild technical square receive positive individual assumption experience loss accuracy attain establish provide vast decrease effect technology agent cost cost would interact series paper acquisition agent concern vast operate agent lie private cost privacy explore notion bring technology prediction report presence concern report privacy player report report agent player simplest sophisticated accurate deal private ability regression different context analyst consensus come agent loss show minimization albeit establish agent datum without receive approach consider agent privacy body risk outcome perturbation instead even choose perturbation mechanism characterization preserve set technical preliminary review prediction privacy player eq analyst infer player player properly analyst specifically measure analyst physical extract medical record list player preference lie either payment analyst privacy design mechanism take perturb response negative informally mechanism accurate budget accurate mechanism part player guarantee privacy detail player rational formally throughout analysis independently ball discussion generalize uniform response conditionally support boundedness sensitivity finite natural response finite support imply response estimate eq minimizer unique estimator regression classic differentially database quantify privacy intuitively differential privacy output payment player insensitive like sense shared neither publicly player publicly payment mechanism publicly comprise exclude payment consider portion mechanism outside jointly private player observable q privacy also require payment differentially private player deviation payment roughly intuitively report emphasize learn privacy mechanism differential notion differential feature treat privacy certainly attribute medical medical response response paper privacy privacy modeling player characterize sensitivity analyst privacy describe cost incur differentially private payment utility assume arbitrary bound increase privacy player intuitive imply privacy player assumption quadratic hold cost formal reveal formally conditionally conditional illustrate idea report player concern simply player formally present spirit player payment depend agree produce report bayes nash x li l report nash equilibrium player payment reporting well uniquely report x x report nash scoring affine argument importantly estimator case appendix replace response restrict report characterize trivially absence privacy cost make arbitrarily analyst example budget possible devise report player agree report depend analyst privacy reveal publicly analyst payment differentially estimator result differentially private add differentially private ridge estimator construct technique ridge class differentially private though perturbation construct accurate mechanism replace ridge respect approach score player variable report report quantity concentration ensure grow linearly long grow slowly ensure choice approximate mechanism remain differentially formalize intuition prove mechanism regression indeed attain approximate jointly private private incorporate perturbation add noise mechanism output joint differential draw accord laplace ab jj dp r r formal version algorithm differentially approximate nash equilibrium player accurate individually
likelihood order bf analytically rejection testing wide used number figure give figure well computational estimation importance intermediate rather multiple result point box plot infer sl outperform outlier figure bf slope example work quite bias bf conclusion would affect slope see bias trade evaluate ensure might sl work bf bias limitation bf use highlight sl introduce bias assess abc bias impossible assess sl simple implement wide limited avoid use possibility use form proxy exact find consist point abc highlight sl gaussian assumption appear variable note sl much inference dependence sophisticated possibly sl sl simplicity find obtain exact reason internal place sampler smc take reciprocal tend inexact bayes particularly implementation see theoretical current datum consists ise ise via evidence truth take weight introduce distribution bridge estimator high variance tail expect bridge external since cost total bayes compare stage firstly exchange estimate smc move target employ effective ess fall normal variance run exchange space method bias improve may move take account statistical efficiency regardless whether alternative approach link style sl alternative estimate much attention appropriate avoid abc sl sl unable estimate use essential sl must disadvantage method figure accuracy advantage variance sampler problem inexact smc sampler avoid simulate monte require exponentially practically dimension simulate doubly like focus estimate evidence sampler mcmc inherently carlo beneficial examine estimating dimension section introduce alternative sampler intractable distribution marginal smc sampler smc sampler sequential particle normalise target pf ty z particle reweighte weight represent choose differently alternative mean normalise give resample main smc simply take intermediate distribution idea explore gibbs random offer method approximation weight marginal negligible compare correspond consider avoid calculation namely proposal within smc still denominator try pf p tu py rw weight reversible mcmc choose invariant incremental incremental presence ratio mcmc involve precisely weight update place direct spirit unbiased although precisely appendix incorporate smc target find useful consist point exactly ease add increment smc mcmc standard pz pz spaces eq f move provide become method sensible choice know early aside time may find point every update would add target describe sl approximation likelihood may smc target idea explore abc also provide smc sl explore previously context sequence target even obviously examine smc sampler evidence relative consider precision observation evidence analytically ease cholesky decomposition element wishart simulate use space thus motivate suited smc sampler particle target target pf tt one consist particle choose fm internal sampling systematic resample perform effective ess evidence smc run sampler median summary c st rd evidence example indicate advantageous within weight somewhat analogous exploration analyse biased weight bias u admit deterministic variable generally practice flexible formal setting one section expect weight law estimate compare eq square enough sufficiently assume would bias suggest qualitatively bias bias small increasingly could argument trade bias section investigation importance bias weight effect inexact produce sampler way motivation assumption error principle small level particle monte understood field approximation allow denote time auxiliary random variable space transition combine smc need assume iteration assumption proposal weight naturally finite employ formalism relative exist slightly control error employ inexact weight dynamic forget condition suffice stability correspond approximate iteration appendix demonstrate accumulation error intend qualitatively accumulation weighting potential ergodicity framework somewhat broadly strong allow establish simply justify scheme introduce sufficient together present suggest investigation effect bias smc simulate single estimate alternative smc sampler add target smc internal bridge estimator specifically q v smc run time particle examine bias inexact sampler compare sampler perfect evidence exact inexact smc observe inexact exercise sampler weight sampler present effect improve smc may simulation particle mixing bias inexact inexact observe decrease clearly biased weight useful doubly biased sampler sampler square square evidence inexact sampler compare experience suggest sampler use estimate good theoretical investigation idea worth situation involve likelihood result mix accumulation situation useful intermediate biased decide resample particle choose describe smc estimating outperform previously paper also generally context biased weight accumulate bias commonly accept science network however bayesian use due likelihood develop describe weight monte intractable investigation much interest intractable situation pointwise application occur pointwise example case big consist constant random give model overlap consider previous work introduce simulate challenge methodology whereas depend consider presence simulation specifically approximation us smc complete flexibility counterpart analogue concern acceptance mcmc applicable base sl consider success usually briefly problem outline outline bayesian discuss method metropolis mh simulating look section mcmc avoid evaluation lie normalise arbitrary estimator unbiased appropriately mh interest automatically extension instead q variance strongly appropriate ideally tail suggest reasonable likelihood choose particularly lie motivated importance suggest name make auxiliary low alternative place seem improve present main ratio usually reliably tractable knowledge publish evidence abc approach large
adopt transform inverse transformation image reconstruct image measure degradation structural index spectral residual base sim distinction consistent peak signal noise ratio figure capability image fidelity measure image measurement qualitative compressed visually indistinguishable objective assess transform code embed employ software library video stream employ dct bit bit encode video frame public video library simulation control video step ii agreement metric include logic block flip ff count delay static dynamic final transformation pixel percentage dynamic consumption area resource propose ff ns complexity propose low tool compression compression adapt computational suggest capability transform asymmetric encode several device propose decoder context power bandwidth capability alternative low accord meaningful quality hardware exact consumption decrease field approximation quantization acknowledgment usa cr cr tag pe j des sciences france mail free number image hardware consumption literature image video tool recent several measurement blind verification blind medical image compression transform dct bit encoder quality dct however arithmetic operation device demand consumption exact dct general approximate element possess arithmetic mean bit shift prominent sign dct dct code capable code hardware associate analysis perform transform image video compression scale final remark polynomial matrix aa factorial synthesis x nk entry integer derive require arithmetic consider transform generally less dct matlab language exact dct matrix propose approach discrete seven time large dct less evenly rescale accord formalism detail parametric family aim result arithmetic analytically tractable find satisfied obtain complexity q obtain transformation expression return therefore synthesis equation transform matrix scale may compression diagonal embed quantization explicit transform coefficient
likelihood although use composite various highlight calibrate composite approximated posterior substantially low figure rely eq include summation set lattice overall constant lattice generalise computing simplification arise un likelihood dependent lattice except last drop n represent straightforward exact minimum lag lattice first neighbourhood occur lattice additional straightforward compute summation conditioning true misspecification approximate posterior aim identity gibbs identity gradient moment namely identity express adjustment simply substitution map approximate address note concave however log semi unimodal example optimisation map difficult time find calculated evaluation provide bfgs algorithm carlo bfgs point bfgs algorithm estimating algorithm little despite bfgs algorithm modify remark curvature scalar directly link hessian observation choose block weight everything write identity deal solve lattice compute exactly serve composite computation carry gb computing normalizing take cpu bfgs take second map one approximately minute situation require wang use dramatically simulation simulate bfgs stop monte whereas covariance place integration mcmc burn interaction critical ise exhibit spatial around parameter value right sum evidence constant expression turn plot example clear un posterior denote respectively posterior adjustment provide correction posterior ise variance square average distribution h use law moment estimator figure case adjustment allow option variance correction seem carry approximation magnitude ise kl structure abundance parameter interaction induce figure misspecification evident magnitude adjustment yield see curvature adjustment correction illustrate conditional statistical analysis field likelihood typically concentrated contribution replace composite extend number acknowledgment grateful anonymous insight centre science grant foundation grant play due normalizing likelihood principle approximation paper result illustrate play important distribution lattice exponential arguably popular social include biology physics popularity field parameter trivially replace full generalization refine purpose consider composite inference focus collection variable influence approximate posterior use mis description gibbs composite likelihood especially formulate likelihood issue composite bayesian illustrate various remark finite undirected define adjacency definition directly major due normalize constant depend parameter summation possible trivially pose serious difficulty parameter composite likelihood outline binary lattice lattice normalize write dependency henceforth point index bottom column column order neighbourhood interior along lattice exclude lattice abundance aggregation allow variation
task include metric auc predict order relevance rank hypothesis available belong reproduce product space definite k reproduce work batch scheme rkh formulate q rank metric kernel furth subsection concept robustness despite potential capability deal dataset recently analysis pairwise space establish guarantee almost surely polynomially require iterate restrict strongly associate unconstrained novel mainly operator probability hilbert schmidt paper organize introduce example specific discuss relate present technical lemma pairwise loss measurable difference see study follow learn usually prescribe ball f implement unconstrained generate algorithm sequential access training hypothesis upon reveal iterate obtain iterate functional rkh x mm functional approximation theorem surely implication recall universal universal kernel kx pairwise fractional kl select subsection pairwise f indeed specific pairwise let x positive definite pairwise characterize rkh induce statement gx g g gx use assumption kernel apply equip statement associate define univariate similar proof corollary remain remove firstly using see discussion example author generalization risk estimator sample eq rkhs inner loss definition formulation formulation kx gx gx g pairwise batch bound establish case follow ensemble hypothesis function z fair let hypothesis technique average online project regret rademacher average together f f term side l rate iterate f tx difficulty analyze novel enable overcome characterization mainly prove necessary notation j define derive well theory error j depend subsection term turn attention sample side end establish lemma inspire similar gradient certainly prove equality f ts f k complete denote j kf operator imply lemma hold remark easy inequality variable hilbert difference hilbert surely k fact schmidt see let hilbert hilbert schmidt operator inner usually denote ready accord term j l j first term side k put estimation back observe combine l recursive equality sample hold j apply yield basic notion quantity statement functional b k part prove follow side apply start technical proof appendix ready establish theorem consequently back desire definition prove well g exist equivalent easy q consequently put turn univariate us proposition pairwise n span span section gx gx g gx gx gx complete proposition part online theorem b learn unconstraine rkhs non strongly perform unconstrained aware pairwise algorithm surely rate polynomially decay size discuss form f save improve implementation square particularly result acknowledgement support grant
intersection small correspond assign point outlier let core bind every need cover conclude se se still mistake ingredient conclude definition reliably shape cluster unlike art cluster contrast technique provable skeleton continuously see local skeleton set geometry cluster outlier infinite stream provide theoretical quality cluster massive stream become medium finance throughput community evolutionary web activity email service challenge throughput scenario real typically provable generative force retrieve convex exist assumption heuristic lack need stream use finite stream assume technique difficulty handle effectively time massive stream propose skeleton online address challenge basic cluster skeleton capture geometry skeleton maintain skeleton density skeleton automatically shape skeleton skeleton update procedure outlier recover strategy skeleton allow adapt drift merge split guarantee quality huge offline k mean median cluster radius poorly cluster shape survey variant density combine perform suitable stream another variant technique center continuously point belong rich complex main nonparametric cluster guarantee tune cluster encode split variant center keep purely outlier share exist random arbitrary shape however offline agglomerative size slow aim encode idea intensive store skeleton variant provable inherently offline often pass time clustering shape mainly also assume several iterative propose know online set essentially initial laplacian infeasible shape constitutes shape likely belong datum neighborhood point complicated idea graph utilize via correspond skeleton belong number call clear skeleton correspond weight encode around skeleton skeleton belong skeleton set update skeleton mention skeleton cluster stand skeleton vary initialize skeleton take translate strict provable regard quality maintain skeleton skeleton relatively entirely skeleton skeleton cluster grow skeleton never cluster overall number skeleton generality skeleton variant merge merging splitting cluster cluster extra turn keep undirected element associate skeleton encode denote skeleton cluster assignment iv iw I bx rs algorithm turn skeleton stream create skeleton weight skeleton belong cluster skeleton merge skeleton empty singleton start split turn I bx bx un un merge j vs merge merge v un merge merge min v min w min min merge merge merge assign multiple basically act merge unify scenario cluster initially one combine true skeleton skeleton merge important subroutine skeleton size newly add skeleton independently skeleton seed initialize randomly alternatively skeleton conceptually cell latter correlate far away skeleton number skeleton next consider skeleton skeleton merge merge skeleton relatively merged cluster skeleton set consideration already denote minimum skeleton skeleton merge merge skeleton skeleton point newly skeleton copy point merge point newly contribute weight skeleton close skeleton find increase contribute total skeleton replace skeleton cluster merge skeleton skeleton merge small skeleton point encoding pool entirely conduct skeleton exclude skeleton singleton correspond accord add skeleton algorithm skeleton singleton cluster skeleton aim cover cluster sample form skeleton sequence generate triple complete skeleton initialize newly cluster skeleton create skeleton singleton create cluster update singleton accord new splitting variant handle breaking skeleton weight skeleton point determine skeleton connect mean split component cluster responsible graph newly cluster replace skeleton skeleton merge graph combine newly skeleton close vertex skeleton radius cc subroutine skeleton skeleton x bx I g merge subroutine regard describe introduce dimensional disjoint call core great I arbitrary give rise core probability formally give outside important core due presence word cluster good quality core separability recover nontrivial offline connect component online bring algorithmic computational challenge definition cover ball arbitrary ball express denote ip p kp fp outlier outlier cluster core keep skeleton cluster point core also error online lack enough phase reach reach ready core core see phase least algorithm merge contain upper skeleton cluster practice theorem error come theorem skeleton point word rate produce rely intersection outlier mark green phase theoretical whenever phase skeleton equivalently treat skeleton formulate follow lemma contain skeleton skeleton outlier lemma accord take core merge skeleton skeleton thus four synthetic set contain randomly draw shape outlier shape letter shape shape deviation affect fig cluster quality guarantee art nonparametric dominant comparative produce use cluster dataset handle work fail fail dataset
related rademacher index set sub quantum e rademacher directly independently distance measure calculate invoke depend exactly determine quantum since real relax criterion accuracy cover identify element entropy quantum quantum element speed optimally distinguish discrimination zhang goal given belong concept separable exist separate exist difficult quantum discrimination classify great ask cardinality show dimension quantity function respect generality assume choose denote convex argument level cardinality complete quantum n independently sample hypothesis input ball class collection aim quantum quantum parallel measurement duality discuss relationship quantum code framework linear functional element derive quantum state theorem sphere since embed rademacher however learn rademacher bind series deterministic hermitian rademacher rademacher variance hold concentration inequality eq realization n complete repeat space rademacher assume rademacher q due duality complete proof entropy input proportional intuition unit ball large ball radius perspective fact evident lebesgue measure banach calculate effective exponentially word demonstrate state quantum definition code map bit exist tr th set upper level om ps upper success however relate dimension dimension order consequently om inequality recover integer show directly space coincide previous section constructive way implement quantum ml framework material derivation appendix affine pseudo entropy rademacher paper analyse problem measurement state outcome quantum dimension also show learn quantum entirely tool classical proof also derive summarize unknown reasonable finally sphere show learn ml learn measurement provide viewpoint quantum connection field existence code state quantum area integer linear operator conjugate transpose schmidt inner product stand conventional trace operator trace identity standard norm reduce norm element output hypothesis independently ensemble set dimension cover metric call big constant introduce deviation express sample boolean pac vc provide boolean constant finite provide bound analogous finite absolute n bounded entropy closely et al q therefore sample therefore provide another wherein classical traditional sphere direct gain dot euclidean q state act convex consequently functional operator associate useful algebraic property operator space projection orthogonal mutually rank associate simplex angle kp convention centroid face intuitively interpret kp quantum tr relationship associate input e functional act affine consist bound functional however easily convert behind quantum need reader ref detail basic network simple call scenario sign satisfy condition boundary perceptron adequate parameter misclassifie example compute add termination reach output integer procedure adjust dimensional input datum activation bias perceptron infer perceptron htb kk height title author subject green issue date journal proposition section quantum engineering information university technology edu tw com tw receive significant attention promise progress make quantum theoretic training predict arguably theory setting complexity paper unknown dimension hilbert result solely complexity explicitly able connect quantum science quantum discrimination lastly representation learn quantum mathematically apply artificial intelligence aim devise systematically typically ml unsupervise machine hide clustering supervise characteristic learning machine time determine query hypothesis approximately generalization closely require target complexity trend big feature balance complexity set well without overfitte active capability system recent large integer search feature improvement superposition contrary classical bit combination consequence mechanic wave superposition store give device quantum phenomenon quantum resource result shannon theory feature area broad promising application advance subject classical quantum machine attract substantial classical task totally precisely accelerate computational transform quantum hand fundamental quantum state underlie system statistical certain accommodate value subscript refer set learner accord pure learner access classical ml belong class current quantum procedure consider membership quantum extension quantum classical polynomially process problem memory store method procedure unsupervise additionally quantum execute big verification computation method quadratic optimization problem microsoft approach quantum neighbor algorithm surprisingly number depend rather wang phase hamiltonian ml fitting learn quantum pattern computer important task interested reader comparison quantum state accord fidelity cluster statistical study quantum operation hide quantum statistical model quantum width corner draw center corner mat sep mm block block c wang south north south pos north pos north pos north south pos south c work quantum quantum device quantum randomly state quantum need learn machine decide optimal measurement measurement statistical theory proper quantification propose outcome learn quantum exploiting banach measure rademacher complexity derive require quantum measurement proportional hilbert quantum formalize employ tool solely theorem cover three proportional quantum sphere hence ml apply may relate quantum state physics exponentially point serve quantum surprisingly dimension quantum hope quantum quantum ensemble mutually perfectly discrimination minimize dimension guarantee quantum bad reasoning hypothesis stand bit receiver successful provide quantum alternatively coincide work discussion background theory supervise describe relate function derive quantum addition interpretation cover rademacher quantum code formulate sphere representation network implement paper hilbert orthonormal adjoint inner subscript omit norm trace norm define norm operator likewise finite class operator operator quantum serve yes measurement constant parameter change notation table start mathematical formalism examine error complexity speaking supervise ml observe agnostic pac comprehensive introduction refer reader e supervise aim approximate train performance take absolute convenience easily generalise quantity lipschitz constant complexity measure therefore homogeneity assume square derive sample problem since risk minimum I infimum take possible measurable deterministic almost surely identify collection call hypothesis assign eq effectiveness almost agnostic def hypothesis agnostic pac learnable sample train empirical erm principle assign way evaluate relate risk reasoning algorithm output uniform uniformity respect member measure quantity confidence require class class agnostic criterion interested reader therein train call domain agnostic erm algorithm agnostic result complexity criterion underlie learn agnostic pac fundamental agnostic pac determine rate theoretic hypothesis next complexity introduce size hypothesis far interested agnostic model resource class introduce dimension let domain every generalise vc quantify complexity introduce real n nb bf bx bf bx bf main theorem function dim subset dim domain bf bx bf bx bf bb underlie constant dimension value side pseudo even addition combinatorial quantity measure concept back kolmogorov area mathematic cover number cover cardinality metric endow support fx I fx pl entropy q significance loose technique concentration measure capture sharp bound rademacher bound function variable rademacher associate complexity use ref convenient far measure combinatorial hypothesis eqs sample functional justify quantum practical situation physical aim three measurement state measurement experiment quantum count functional correspondence measurement furthermore unique e proposition every measurement identify subsequently either linear outcome eq space linear effect hilbert determine probability correspondence quantum coincide state matrix quantum banach furthermore value target banach space hypothesis represent functional nonempty interior body symmetric unit ball dual schmidt product duality formalism banach follow set linear functional pac learnable duality operator fix hilbert map trace class operator conversely investigate dimension set functional banach every ball banach restrict core calculate banach rademacher series value formula helpful duality formula upper via rademacher series remain banach measure proceed practical may yes outcome outcome perfect
solution sufficient also say establish existence find solution exist definite matrix proximal operator value g expression observation make metric proximal framework proximal newton proximal quasi choice algorithmic approach starting update size search direction size subproblem n selection prove moreover bad second hand decrease easy verify selection decrease regard backtracking describe ht e use proximal gradient k combine follow equivalent subproblem kf k explain theory backtrack assume inspection maximum many principled way quasi backtrack oppose check far backtracking mild find ff moreover convergence impose evidence need union impose twice restriction probabilistic cf restrict number small hessian become explicit backtrack attractive certain count step newton newton proximal newton newton proximal start q moreover size step whose proof appendix I solver collection problem first denote sp ratio use profile profile self lipschitz fast applicable figure prox operation medium accelerate bfgs update proximal good prox operation solve proximal newton good prox operation via problem bfgs omit exhibit linear fast show variant nesterov dual exhibit perform total respectively surprisingly consistent indicate convergence satisfied practice proximal calculate gradient algorithm record ht number exhibit meet believe form vector form full especially like definition profile proximal bfgs performance good prox operation prox convex modify programming random show mf proximal rather test outperform plot prox prox operation achieve fast variable method minimize applicable usual assumption smooth convergence highlight backtrack new basic local free proximal practical former assumption fast plan version frank com composite minimization function laboratory self like concept minimize analytic property numerical test real composite eq possibly nonsmooth composite naturally arise application science image science loss trade optimally exist lipschitz gradient cf definition development sublinear smooth regularity within quasi newton convergence algorithm exploit principled bfgs bfgs instead focus develop convex self composite minimization function multinomial logistic first metric backtracking operation guarantee proximal adapt convexity convergence great life lipschitz exhibit sublinear help solution form geometric contribution self function descent second pay particular subproblem variant locally variable metric hessian locally
co threshold neuron undirecte present see quick te cause cause method improve incorporate add vice versa instantaneous causal instance insufficient therefore solely pairwise indirect network strategy eliminate indirect strategy derive strength neuron deviation asymmetric standardize complexity neuron symmetric unsupervised eliminate indirect normalize pairwise indirect integrate network capable method average rank sort know refer formulate description supplementary material subsection since indeed conservative correlation discriminate effect experimentally correlation statistically informative plain quantile correlation recover correlation hundred sample capture quantile quantile pair neuron feature complementary spike feature square normalize center signal particular rely quantile extract square influence measurement nearby neuron dataset dataset provide neuron neuron network see either assess receiver operate characteristic roc recall pr propose metric roc auc roc optimistic encounter network recall curve compare method big discard link curve apply good result value small font show observe network cc cc normal normal statistically highlighted use original use direct auc similar conclusion table auc material see additionally unknown area roc rank advantage cpu compute different intel mean minute instantaneous compute proposal improve unsupervised improve art network rely among compare experimentally namely dataset communication technical university bioinformatics biology universit du present neural fast connectivity neuron activity art entropy process remarkably simulate time challenge competition reconstruction elimination understand capability brain cause treatment network neural connect neuron perform circuit responsible well easily group neuron record neural one study population simultaneously neural recover brain neuron
introduce shorthand continuously cauchy schwarz turn hx u hx hx v x z kt x kt hz kt hz kt x z kt kt order derivative eqn directional v u hx lemma guarantee integer hx hx hx v hx hx hx hx hx hz z v hx directional exist directional hx hx v hx hx v hx hz op directional hessian lipschitz uniform solve stein notation limit pointwise hx hx p py dy py dy hz kp compute stein graph stein gx jx b x gx gx gx result graph stein discrepancy stein discrepancy equivalence stein jx stein compatibility imply establish second j g k j g z g g exist z stein inequality fix j b b fix compatibility invoke inequality boundary compatibility property turn establish g b j j z b deduce z acknowledgment share implementation triple early manuscript support fellowship foundation fellowship theorem em em em theorem turning procedure sound rapid create challenge quality stein expectation biased assessment quantify bias target often turn chain hz hx qx target recent researcher asymptotic mcmc procedure asymptotic correctness rapid variance estimate bias add flexibility challenge sampler parameter pool address quality bias sequence stein program design section illustrate application assessment quantification work often gx jk density integration intractable aid point encode mass probability hx sample approximate quantify converge iii computationally starting consider eqn hx large measure converges weakly term many include generate hx distance generate adopt measure computation infeasible generic integration intractable could focus know many class function know track ii practice question distributional return third characterize value act function generator op px gx stein operator derivative computable even normalizing boundary boundary gx gx x smooth boundary suitable domain p study classical stein converge wasserstein set analogous distribution analyze extend reach literature classical stein discrepancy determine wasserstein large include prior stein strongly densitie p stein multivariate sufficient stein stein pz implication proposition px pz analysis paper readily accommodate uniform stein eqn stein gx nx exploit flexibility bound relation stein discrepancy wasserstein well free stein stein stein discrepancy equivalence q efficiently stein discrepancy property discrepancy restrict unconstrained section present domain evaluate function stein discrepancy qx px gx gx gx x stein primary difficulty classical stein stem constraint impose stein way difficulty impose classical smoothness collection gx gx gx classical taylor compatibility remarkably stein strong stein program v j l represent function value amenable prohibitive unconstrained extend coordinate program boundary compatibility b b appendix stein discrepancy strongly stein summarize recommend stein unnecessary complete stein q g sorting point enforce stein computation solve th program bound n qx n x I turn evaluation program simple stein diagnostic scale degree complete stein discrepancy decay student stein stein well notably student function exhibit relatively middle stein stein appendix stein discrepancy compare wasserstein target provably uniform target generator seed point I non classical wasserstein distance apparent graph stein discrepancy classical track wasserstein magnitude separation wasserstein fact stein discrepancy langevin bias mcmc design scalable inference approximate langevin metropolis hasting use grow meanwhile explore space stein diagnostic adopt minibatch sequence select high quality ess diagnostic autocorrelation discrepancy across stein discrepancy select ess greatly greatly slow stein diagnostic resemble posterior ess maximize stein minimize ess stein mh biased control mh qualitatively fewer rapid rapid reduction mh stein discrepancy trade cancer patient whether spread node bayesian batch discard evaluation thin remain stein computational langevin length surrogate normalize jx tw quantification discrepancy sequence appendix stein sequence deterministic dx sampler slope good square metric bind suitable compare bias infinite class functional finite collection functional distinguish discrepancy mmd distributional kernel approximate mmd access ground truth target ex boundedness langevin generator second function solve stein equation equation hx hz hx hz hz hz op hz op hz hz hz op hz op equivalence stein imply wasserstein wasserstein inequality standard follow inclusion smoothed function close integrable h hx hx bound fix lipschitz admit gradient representation tx hx hx tx hx z v hx relation yield h tx b td chosen stein discrepancy pg l z l l l l z objective q stein factor lemma hx hx x langevin useful proof langevin concave langevin diffusion generator lyapunov vx cauchy schwarz arithmetic together q x constant continuously differentiable result growth proximity ultimately smoothness hx op hx hz hz hz z op op op w w op z op x establish taylor hx schwarz yield hx op x difference mean remainder hx hz hx hz hz z hz hz hz hz hx hz h w h eq invoke difference apply cauchy schwarz definition operator norm hz op x cauchy schwarz z z hz op eq hz hx op norm w x op w four proof serve follow term inequality strongly langevin diffusion differential eqn sde p v x kt kt kt x k second order eqn v x z kt v b coupling h op x z ks apply continuously differentiable pz yield ks ks second concavity may desire kt ks ds conclusion order difference v v ds invoke kt produce kt ds x couple together third bind h h u z ks ks ks presentation pz op pz op op x v ks x concavity reproduce conclusion kt ks ds ds langevin diffusion wiener
surrogate three label distribution prediction pt matrix prediction different conditional induced classifier loss respectively represent l ty bayes goal learn close algorithm w optimal distribution function training discrete computationally approximately algorithm approximately e argument optimization seek predictor hold continuous predictor excess immediately consistent w derive vs surrogate excess relate j one loss threshold break favor result proof point previously zhang surrogate hinge predictor however dominant learn predictor instance class manner instance dominant vs hinge surrogate like surrogate conditional domain minimize surrogate real convex surrogate call risk relate loss calibrate dimension assume one onto define evaluate clear partition predictor result excess surrogate suggest good choice surrogate intuitively prediction sense close predict low noise predict make choose via section minimize surrogate set simplicity label norm em ex r md I I block ascent algorithm fix type problem db j projection ball excess bound surrogate loss modification surrogate n em u b generalization proof along omit surrogate hinge hinge extension generalize hinge dataset vs surrogate space class incur bayes randomly prototype vector mean prototype vector generate pick surrogate reproduce hilbert regularization parameter search cs incur risk excess threshold surrogate support cs imply result algorithm less cs error poorly optimize show much ccc ccc cs cs repository class regularizer choose split train simplicity cs algorithms level choose comparable time algorithms run fast option reason speedup function problem speedup cs reject powerful abstraction capture control classifier diagnosis formalize give excess relate surrogate operate small metric direction break excess vector u n prove satisfied surrogate position everywhere else theorem simply linearity expectation rhs equation last equation u equation u u yu also u u n crucial straightforward inequality else f f linearity rhs hence trivial u u u rhs n u also u follow surrogate f follow linearity rhs become j observation j equation thm thm conjecture thm class prediction say reject thereby extend generalize surrogate yield consistent design also consistent operate dimensional generalize surrogate consistent would well take predict make wrong problem medical diagnosis test convex logistic loss adaboost consistent problem hinge require consistent double hinge segment flat segment multiclass double multiclass reject reject seek reject option incur denote call svm binary minimize piecewise arguably widely surrogate like dual
suffer drawback mlp therefore city sentence learn extract structure match drawback architecture sentence desirable sentence meet representation basically slide sentence one e segment clearly convolution segment layer max non overlap illustrated layer perform window eq could pool analogous architecture cc level two sentence high encode dimensional segment among pooling resemble dynamic similarity architecture rich r c convolution pooling preserve information although may segment retain triple consistently gain find correct usual happen choose turn separate mlp actually map devote devoted pair naturally divide filter denote slide rank essentially pool pooling preserve convolution abstraction fully preserve offer capability individual internal abstraction interaction sentence sentence fusion hence object rich structure intuition verify give rank include layer architecture propagation section adopt turn descent batch size easily machine core regularization architecture early medium training less early dropout deal overfitte word embed english learn wikipedia chinese learn experiment tune cope sentence word optimal performance eight three perform layer convolution relu convolution mlp comparable like propose match different nature compare namely sentence tweet match identification natural three task language write prove applicability matching text calculate mlp two document matching layer layer use unfold autoencoder get dimensional sentence mlp sentence sentence sequentially mlp score coherence performance layer correspondence clause basically clause comma match heterogeneous relation lexical hard clause similar original million test positive pair negative negative sentence show convolutional perform fairly well run behind sentence surprising come last cause word embedding split sentence parse sentence original tweet collect major chinese service writing style positive ten million triple translate english tweet select tweet match original negative report task slightly purely negative model loose margin determine sentence language acc object benchmark contain instance early stop state require instance achieve hand significantly design relatively superiority deep structure matching rely raise match whether something convolutional perform indicate importance utilize sequential sentence interestingly experiment train negative sentence surprising auxiliary correctness word response enhance match notice reasonably act composition meaning segment word rarely focus match score product building text largely representation aside nature section fairly embed text mostly convolutional network work pool relatively tailor deep architecture language sentence outperform b chen national foundation china li part china project cb department science school central importance successful matching need internal step convolutional adapt convolutional capture rich level generic nature different language empirical variety demonstrate efficacy superiority object central similarity g retrieval correspondence linguistic level translation english language sentence therefore internal rich towards propose convolutional prove successful natural representing devise hierarchical sentence comprehensive convolutional architecture require tree put part understand contribution summarize first convolutional architecture sentence pool abstraction characteristic architecture sentence modeling architecture detailed architecture section report propose convolutional architecture illustrate take embed backward cc convolutional combine compositional recursive autoencoder example illustration purpose turning element focus segment word offer code layer choose
boost multiply convolution instead stage stage classification cifar scale run bottleneck runtime necessary pc ram peak gb hour cifar percent far convolutional three follow apply pooling stage variety whereas use whereas mark fellowship project acknowledge david dr university dr discussion resource acknowledge present design rapid frequent filter inspire value stage regression stage unit efficacy art classification database times network google house competitive database achieve neural rely convolutional feature extraction rotation memory computing hour gpu periodic able desirable therefore seek rapid even potentially expense image achieve network neural architecture surprisingly entirely select although application filter relatively train show produce classification database set computer apply still result dataset optimisation parameter gain recent review context convexity fact optimisation solve unit pixel train classifier excellent performance yet convolutional architecture together ensure namely classification least square feed output classification database cifar google view house network state near present benefit clearly require competitive hard cifar imagenet result cifar core attribute filter generalize aim remainder organize description obtain classify generic apply describe obtain learn achieve remain specify weight text pixel classifier stage three pooling layer fourth output conceptually divide stage combine first convolutional stage largely exist approach classifier largely stage algorithm image filter describe size filter transformation represent input kk cc ci channel k algorithm suggest image sequentially applicable feature operator construct multiply matrix hence total introduce size concatenation convolution diagonal matrix copy flow mathematically argument matrix toeplitz instead entire pool intuitive explanation receive sum simple response operation often help also nonlinearity nonlinearity classification root projection layer pool instead q effectively l operation follow description whether raw treat feature introduce numerically represent label value stage unit logistic g activation vector employ pooling pooling use describe default randomly non training lead train iteratively backpropagation determine pseudo solution solve overcomplete often follow ridge qr equation mention eqn follow runtime bottleneck multiplication image contain w filter k output inverse constitute close output activation classification decision image value method list database comprise channel namely channel raw pixel convert lp scaling affect preprocesse mnist reason previous convert add rgb contrast cifar convert raw whitening image objective layer dimension filter consider corner filter centre filter imagenet patch obtain randomly class filter database channel channel convert filter implement consequently filter channel dimension filter normalise summing filter convolution using obtain valid convolution remain feature exist tradeoff filter factor point enable filter comparable hyper hyper choose previously superior weight nonlinearity choice sigmoid sufficient nonlinearity good classification strongly presence see choice remain classifier examine vary optimize validation generic parameter layer image convert image fourth kind filter filter marginally cifar
participant final ht compound ccc auc auc mt nr nr nr nr nr er nr nr sr sr sr sr multiple within water actually structure together automatically compound clean run clean routine chemical coded consistently encode calculate describe layer ta layer ta layer ta ta layer ta ta define optimize goal set label eight task validation set add training b avoid sample sample within chemical participant world allow receive well scoring team avg nr sr er average stress panel challenge name save deep performance participant never place total challenge sr nr panel average challenge winner nuclear stress display network show deep highly chemical could previously decade expert field reason also lie representation application set confirm lead research ability greatly environmental health drug become support european union author acknowledge true might exist biological neither performance method chemical state build chemical descriptor decade learn never clearly outperform paper net establish challenge set standard people variety chemical many exposure day drug candidate clinical health environmental future activity effort drug goal develop scale demand testing rely throughput screening investigate whether chemical compound concentration exhibit certain different vary chemical compound reliably determine compound activate pathway interact time intensive typically test several whole many time compound multi effort project thousand highly exist approach compound applicable interact infeasible compound predict effort score density machine feed network challenge chemical compound win multi biological activity protein inspire activity involve whole biological specifically focus measure measure compound compound cause death affect seem well abstraction chemical structure include architecture concept idea depict chemical layer center higher ideally compound investigation several show help chemical boost task integrate task utilize representation latter may fail effective representation task boost task furthermore train one system take descriptor compound input tries predict type act pathway task solve present chemical compound want whether compound property compound property predict behavior compound compound binary later weight entropy binary multiply relu one output different scale try standard deviation nonlinearity bring descriptor scheme filter bring try combine chemical descriptor well amount additionally early cross validation contain hyperparameter consider molecular descriptor similarity hide number yes dropout predefine type determine molecular descriptor feature use layer backpropagation decay weight crucial application different storage format chemical representation connectivity currently perform compound drug presence chemical column produce approximately informative compound ie report literature compound chemical often additionally descriptor compound descriptor group around property count molecular feature extract chemical include van atomic involve quantum area calculate descriptor software median deal hyperparameter deal hidden parameter single gpu gb ram batch since store dense format tb disk sparse storage mini batch gpu convert multiplication validate challenge program challenge collect framework research criterion hard public database compose different challenge different sub challenge split seven nr pathway remain five sr pathways nuclear component control development play stream measure modify gene challenge include
approach second order difficult particular challenge learn dependency propose recurrent network properly entirely possible enable learn initialize recurrent eigenvalue name trick transfer information little initialization gradient backward rather effectively experiment dependency effectively difficult number number handle ability dependency enable classify minimal processing achieving state network temporal connection initialize rise recurrent turn recurrent something convolutional neural hide bias positive rewrite expand convolutional matter keep linear learn recent success backward sigmoid expand temporal gradient identity convolutional balance tendency experiment standard prevent hidden activation equation traditional rnn short gradient search good result forget gate give long set forget gate bias lstm gate add rnns random mask signal mask problem mnist read leave corner image corner ask predict category mnist h rnns fail fully neural convolutional network succeed term treat succeed impact problem input repeat test recurrent language model public benchmark modelling tend per hide confirm par sometimes language need gram give neuron help project term important wide variety ai rnn end page people actor question year birth answer simulate end answer retrieval top retrieval finding keyword document return document wikipedia entity robustness retrieval certain retrieval read top parallel combine unit last answer setup answer token birth softmax year birth token page page train page sure birth set rnns rnns bag softmax birth bag page bag rnns gaussian sigmoid rnns identity rnn dependency network vanish potential overcome identity temporal trick initialization dependency extent short challenge rnn ai problem language dependency relationship event sequence dependency across
find expression since usually call gene filter non biological responsible phenotype distinction effort search small dimension phenotype classification discriminate show informative linearly discriminate couple discriminative exhaustive perfect subset classification weight et propose gene preprocesse highly variability subset likely sensitive microarray limit challenge similarity paper run selection centroid use predictor predictor construct less biological result gene gene group co generate gene year yu similar employ univariate independence phenotype biological reflect exploit gene since account joint cluster classification complex phenotype cancer gene rather genetic emphasize exploration among cluster selection optimize secondly generate yu inactive whose centroid relevant phenotype remain non phenotype classification set discrimination experimental test importantly centroid cluster proximity global advantage researcher decide cluster implement matlab pc windows operate website http ac svm toolbox detail binary evenly phenotype sample evenly divide contribute independently centroid label position depend n centroid expression value performance generate keeps go test jump go truly active cluster truly discriminative gene terminate figure recall gene cluster since train simulate find centroid discriminative model reflect calculated euclidean centroid cluster close centroid distance trend base meanwhile cluster dataset logarithmic transformation reduce gene validate three fold validation randomly third testing phenotype dataset rescaling variance value gene gene restrict discriminative start input set correlate sample metric summarize centroid discriminate decrease generalization sample e f active perfect perfectly voting result perfect roughly use set generate gene highly probably sensitivity select experiment gene appear input size contrast centroid run step reflect stability rather individual discriminative biological active focus biological go hierarchy gene active biological geometric convert score cluster average process associate active refinement cluster third gene four monotonically suggest meaningful one response convergence closely hold biological process activity bottom biological besides dataset follow phenotype study dataset cell breast gene phenotype er er positive contain tumor normal level gene tumor b expression ratio consist expression gene blue cell consist divide class bl dataset phenotype derive phenotype rule nb bl filtering logarithm sample phenotype classification split way last subsection summarize value exhaustive globally cluster apply cluster last l l e breast e bl nb e e among phenotype active increase slightly start execution highlight three indicate decrease ability cluster training separation sample improve set gene appear cluster reflect step range closeness different reflect limited absolute generation value centroid keep keep iteration distinct general show microarray closeness know optimal refinement previous work performance classification table sample version cross direct performance comparison validate validate cross pt dataset yu multi three validation slightly breast however inherently range none sample comparable breast et good match yu also multiple class version method yu seem high nature unstable subset guarantee individual gene subset propose backward gene gene provide phenotype class future study implication regard phenotype performance gene cluster generally exist centroid gene consume seek number gene centroid drawback cluster centroid optima optimum clustering completely require cluster centroid avoid optima overlapping gene reflect gene contribute pathway discriminative certain gene statistical conduct inactive pattern appear test consider improve exploit well provide various computing module way sample phenotype integrate multiple phenotype classification occurrence gene multiple different experimental high occurrence frequency confidence truly subsection execute generate discriminative module well sample phenotype cluster number stage stage involve evaluation discriminative validate use cluster gene average distance gene width cluster width fall width discriminative active adopt find decision hyperplane maximize two class hyperplane vector linear indicate gene base discard single svm systematic elimination reflect biological inactive gene gene consideration partition group microarray active discriminate start gene select gene pair iteratively large euclidean distance near take account iteration group form cluster inactive factor remove add cluster determine linear like eliminate discriminative centroid sufficiently representative expression pattern measure width eliminate gene whose pattern little cluster iteration factor objective multiple adopt popular test split problem remain class construct use active centroid cluster multi cluster accuracy generate gene cluster accuracy correctly however sample average accuracy fold biased estimation distinction cluster closeness cluster euclidean cluster close cluster construction precise recursively l l l significance average distance gene dataset calculate distribution randomly dataset recent microarray technology significantly identification disease phenotype cause individual gene effect insight disease pathway understand disease pathway microarray deriving module gene pathway expression microarray pathway pathway serve fact pathway manual condition activate integrate microarray infer pathway disease activate identify pathway may complex amount microarray data series microarray measuring expression indicate gene cancer dataset cancer module co topology conceptual width eps co similarity previously define low dataset high similarity suggest cluster similarity link activate similar module interestingly order analysis module analysis gene pair share divide within connect component likely hierarchical gene hierarchical modularity scale module module similarity select module infer module activate module apply related microarray cancer relate second cluster network activate cancer identify breast specific module tumor importantly gene play tumor module microarray total disease disease cancer non correlation percentage bend normalize bend correlation gene pair gene gene co network differential co expression scale distribution co topology act highly gene cancer phenotype gene play cancer fall core division organization interaction cell neighbor interact interact ip division ip likely characterization member long serve involve neighbor cancer supplement differential top degree together account degree differential fall main core gene dynamic cell gene movement cell dna l cell degree gene degree node play cancer behave gene tumor differential co summary frequently however tend inactive type cancer expression large connected differential contain connectivity break coherent type dynamic connectivity cancer network module second cluster utilize dynamic connect co calculate microarray term correlation profile distance expression profile unlike commonly pair gene base profile frequently occur relationship dataset likely link co expression module turn phenotype gene connect suggest pair component module cluster fall similarity within connect number test distinction select module show within pathway addition activate certain phenotype retrieve component connect diameter scaling logarithm especially cluster top score remove overlap result comprise module edge module statistically biological annotation cycle division stress mechanism width supplement linear diameter discover cancer module activate module activate cancer involved division genetic represent signature cancer figure module solid gene module involve cell solid tumor next module activate breast width eps cancer solid module network dataset module division genetic stability another module activate consist module locate correlation module tends activate solid tumor expression edge dataset co estimation tumor dataset result order module breast tumor dataset rest dataset correlation module breast tumor breast tumor dataset module protein gene breast tumor module express breast cell increase activity express breast rich play crucial tumor breast cancer induce rich involve survival gene allele c breast cancer gene involve cell suggest tumor module width cancer eps cancer tumor module main module tumor high degree module gene precise encode aa significant indeed cancer sample locate suggest breast cancer tumor gene recently depth decrease breast module find bind growth cell induce similarity recently terminal interestingly pathway direct member bind significantly positively production example breast tumor specific col col gene tumor function adopt tumor identify network module involve cancer tumor member arrange cd line individually breast increase associate advanced breast breast death breast module tumor cd induce production cell factor bind predict bind report expression module reveal breast tumor beyond expression elaborate reader assess pearson correlation connect within activity relevance cancer b module play different process cell division respectively division order underlie member module distinct average correlation module breast module role breast tumor development member gene module exhibit pattern module weak similarity across seven cancer module within second second rise connect module module co cancer pearson module within gene pearson correlation module active bend module pair cross module pairwise pearson percentage bend correlation gene highly module discuss previously module pearson dataset module rapid microarray molecular mechanism disease study utilize microarray derive list specific cancer little characterize et predefine meaningful biological pathway activate module wide pre module limit association study tumor functional growth cancer derive simultaneously cancer study topology characterize module identify module module activate molecular mechanism importantly discover potential tumor network particularly breast tumor commonly advantage simultaneously condition cancer activate thus provide insight complex mechanism characterize cancer approach identify densely module solely incorporate co expression diverse type module regardless network biological necessarily densely module pathway apply molecular beyond available framework module manual intervention systematic way put robustness base observation relationship suggest year aspect network essence exponential integrating provide well accurate module explore future depend sampling different phenotype cancer pair co expression currently impractical potentially imbalance strategy correlation determine non cancer stanford microarray database gene convert gene bend percentage effect calculation number calculate estimate dataset gene expression reduce sample size effect r calculate standard may reality enforce distribution correlation invert gene differentially express cancer positively set correlation I gene pair estimation valid estimation correlation cancer cluster differentially correlation cluster program complete euclidean simple existence correlation miss correspond second cluster gene case size differential process hc pair process separately meaningful module normally edges thus cluster keep module gene go biological process go hierarchy gene directly gene consider homogeneity model significance gene kb scan bind cut sum negative keep per sort predict factor module bind section first like support place want ph thank li zhang dr song manuscript thank go biology stay great friend frank dr zhang dr tu dr li dr xu dr zhang especially wu dedicate modern phenotype approach inherent phenotype make throughput phenotype propose method automate gene similarity robust different aspect technique phenotype phenotype complicate consequently gene subset sensitive phenotype tend novel increasingly gene simulate result perturbation sample phenotype performance phenotype cancer gene network module multiple gene cancer module module coordinate detect breast cancer tumor module gene important module throughput accumulate nucleotide number alternative phenotype association genome project post genome vast still remain largely question gene gene product interact identify interaction various cell gene change biology use poor question address advanced expression transform vast possible functional refer wide make information divide gene dna rna template production gene expression indicate time certain monitoring gene protein throughput several develop understand behavior particularly use magnitude many dna label finally store image process quantify intensity factor variability probe process undesirable effect intensity comparable comparable mean gene reflect introduction microarray principle phenotype observable phenotype activity environmental phenotype sometimes categorical phenotype unify medical language recent year language control science mapping structure give researcher ability translate terminology comprehensive concept network category relationship classify relate processing support tool system accumulation gene rapid large contain genome gene phenotype expression measure value phenotype manually record develop method combine type well phenotype indicate propose automate bridge search provide discrimination phenotype gene primarily phenotype involve study aspect phenotype focus phenotype mechanism difference phenotype difficult quantify throughput fashion lack comprehensive phenotype prevent chapter enable phenotype principle phenotype microarray microarray method cover confirm profile highly true description unique capability phenotype design disease profile factor inference direct comparison different method entire body platform microarray produce phenotype profile facilitate quality phenotype accumulation microarray datum gene phenotype gene differentially phenotype hand since microarray contain gene desirable identify phenotype couple unfortunately sensitive selection training sample gene usually tend overfitte supervise unsupervised increasingly gene combination iterative exist produce combination phenotype backward highly phenotype classification prove stable combination study phenotype integrate consist kind phenotype cluster truly discriminative enhanced technology differentially disease disease well phenotype individual thus important variation chapter cancer develop multiple microarray discover disease module propose dynamic topological module phenotype activate de activate many module consistent annotation module activate breast cancer tumor individual module associate never adopt perspective gene important tumor base provide insight complex characterize cancer cancer incorporate co expression dynamic predict phenotype pair value validation requirement match phenotype phenotype moderately phenotype agree among quantify phenotype measure description share description denote cosine angle map term identify pair threshold phenotype phenotype description highlight effectiveness exploit phenotype select dataset calculate original repeat removal dataset phenotype pp removal demonstrate derive average assign dataset generate prediction two describe various correlate correlation dataset platform predict highly correlate focus remarkably job separate separate capture traditional classification case describe phenotype collect phenotype uncorrelated figure order multi dim eps derive phenotype profile phenotype dataset individual pp train correlate recently merge excess share similar clinical biological regarded distinct disease derive pp help lead improved treatment set design gene phenotype profile patient would treatment serve method novel phenotype dataset phenotype confirm examine whose significantly profile number concept particularly interesting patient mutation profile significantly correlate know affect response profile demonstrate specificity phenotype utilize large microarray due phenotype many microarray thus inference
keyword incorporate kind specific superior offer elegant way keyword knowledge keyword framework keyword signal keyword dependency probability mass express model estimate maximum maximization active keyword joint keyword detect respective student mixture model variable people processing enable aid human home environment wherein multiple machine understand machine able speech recognize speech stream effective solution environmental large etc target hard challenge identify speech recognition speech scenario challenge different recognize target scenario wherein sentence record recognize letter digit signal wherein white although restrict author identify note performance interference sir speech complex recognize segment respective noted vocabulary suitable task drive home environment formulate keyword denote relate home environment create student pair pair contribution sec ii estimation em newly database home environment contain subset give signal passive assume keyword let introduce boolean keyword iff active frame iff p probability active frame jj two assume keyword active combinatorial jk k th jk denote collection k homogeneous one keyword ml jk u jk likelihood data data collection denote one repetition choose vocabulary distinct distinct moderately word inside record directional audio specification hz sample separate system build order albeit overlap sir relevant characterize mixture signal fig available propose category keyword eight frequency acceleration obtain shift keyword assess average percentage pair leave one validation explore clean ml use mixture phrase phrase detect correctly detect correctly detect correctly detect correctly c phrase phrase phrase detect correctly detect correctly detect correctly detect overall phrase phrase phrase detect detect detect correctly flat comparable detecting least correctly keyword pair henceforth experiment sample scalar value yield inactive active mm pt mixture recognition accuracy performance perfect one intuitively content model task detect error task small like confusion outside part scenario keyword answer detect initial expect keyword least
intermediate medical health either drive help indicate almost neither category motivation improve model predict low risk patient know status resource allocation clinical accordingly advanced compare approach patient goal classifier whether linear adaptive regression actual adaptive lr improve application application cutting case justify development effort practice improvement precision specificity general perspective result knowledge combine behavioral operational inference history effectiveness public satisfactory classical equally rely basic high outcome fact outside scope patient practice rely tool massive format medical leave datum contain create serious produce measure simultaneously miss maximization miss iterate public health produce characterize ideal risk assessment machine tool relevant clinical operational comprehensive store database consideration nature medical database continuously data acquisition effort cost predictive progress adapt clinical scalability rarely raw medical problematic entry inherently sparse clinical widely patient outcome motivate severe extent medical datum problematic perspective definition ever short use advanced overcome multiple integrate work project division clinical routine medium scope project combination patient clinical medical health plan aggregate metric patient risk treatment first motivate example feasibility merging claim clinical diagnostic risk addition thus make patient claim base use standard near neighbor empirically weight preliminary investigation clinical patient patient likely predictive logistic pose question would advanced imbalance set feature denote vector solve margin map misclassifie slack penalization soft svm usually transform problem implement tool reliable cope weighting build decision relative penalization class tucker radial rbf performance function bandwidth rbf achieve time consume classification scalable algorithm belong whose system multiple scale final solution combine information efficiently large construct approximated graph ann phase method coverage lead well ann suggest coarse support create number update optimize level refinement project coarse level easily adopt classifier issue prevent create small even majority method classification poor situation prior method domain occur datum completely feature datum occur instance dependent desirable many frequently imputation miss either directly purpose imputation imputation knn imputation imputation regression fit local em miss explore incomplete missing demonstrate achieve storage implement eq em relationship select miss imputation value optimize distribution complete control represented preserve evaluate performance confusion tp class negative tn acc sensitivity sn specificity namely uci rna real life refinement graph framework nearest neighbor typical fold cross validation setup create discard select datum selection base due superiority
entropy distribution typical fluctuation around centre mass replica framework replica breaking fluctuation separate yet learn probability target probability unknown increase move small treating otherwise orthogonality similar low imply must condition originally pure block part get lead positivity stability rs stability mm de de sup universit et paris france sup universit paris france target probability expectation compatible expectation bias give boost increase entropy inverse smoothly measure version space pointwise concentrate replica version corresponding qualitative target vary mean close compatible multi system beyond modelling define number degree possibility interested proceed consistent datum available yet inference parametrize distribution configuration distribution amenable computation largely I I informally possible compatible I propose alternative foundation mechanic illustration boltzmann back I knowledge I information theory reflect property alternative somewhat formulation operator compatible word possible I enjoy valuable sensitivity error I come biology literature history work carry consider take value straightforward entirely sum unity therefore pick hereafter polynomial prescribe admissible set admissible distribution contain distribution I flat admissible objective volume target I inter precise mathematically tractable statistical ensemble precisely uncorrelated realistic indeed reflect space compare adequate hypothesis consider average may reconstruct non negative scalar product typically informative scalar reconstruct correct small configuration distribution boost entropy continuously space infinity amount select I distribution depend bias I entropy apply within replica infinity rigorous check replica replica symmetry breaking fluctuation call mode expect large design sample small remarkably good agreement organize reader sec necessary sec calculation result numerical conclusion consist ise configuration hereafter target purpose lead term convenient introduce entropy curve system denote lie line tangent entropy per bound characterize dominant contribution illustration spin htbp curve mm let system label simply observable admissible hereafter define space logarithm I distribution eq lagrange multiplier enforce distribution consider spin spin I average multiplier coincide field pairwise act realistic perfectly affect tolerance hereafter introduce dirac delta sure admissible vector constraint term refer define probability measure purpose study I shannon entropy introduce define I hereafter spin quite contrast statistical analysis existence entropy since multiplicative eq expect calculate value replica hereafter outcome replica sec picture interest I mass distribution angular version distance center hereafter square distance represent fluctuation mass root square side rectangular lie observable observable mean variance quantify observable fluctuation square fluctuation average scaling due symmetry pure exponential average give dirac saddle saddle locate give agrees behaviour volume calculation replica method presence exponential scaling bias entropy impose non bias report sec sec sec major replica configuration probability marginal close depend later see configuration value dominant later infer boundary hereafter edge separate learn one remain observable tolerance change value ratio transition process htbp learn replica calculation configuration qr switch dominant sl phase separate label value denote phase turn get insight plot continuously reach agreement fluctuation volume largely manner ratio negligible tolerance phase fig grow negative expect become tolerance phase call agreement range take place interpret measurement fluctuation fluctuation performance entropy I dependence quantity importance negligible replica calculation show consequence I close pick distribution version fluctuation measure replica two place observe though transition contrary remain mention phase phase show right panel illustration cm phase tolerance tolerance fluctuation large take large become case negligible behaviour become irrelevant hence limit irrespective I enough compare fluctuation govern large irrelevant replica volume constraint introduce notation integral square replica moment perform make normalization irrelevant discard hereafter rewrite identity approximate saddle assume symmetry replica delta obtain integration assume appear vanish need find consistently analysis outcome meet case eq direct one treat put introduce natural sharp difference argument complementary dominate complementary sign get logarithmic formula expand saddle exponent saddle determine dominant contribution positive imply saddle feasibility compare width q nf saddle justify configuration dominant domain integration expand identical calculation expression indeed approximately decay function equation parameter get eqs match dominant remain meaning contribution probability negligible sec large configuration lead see long configuration accord accordingly ignore case consistently determine solve equation critical separate phase I critical associate iii check distinguish solution large phase case validity rate determine eqs get write hold iii iii stability rs integral call exponent write term change peak saddle approximation apply irrespective saddle mean edge configuration substitution expand configuration saddle dominant sec require normalization saddle edge configuration dominate imply determined correction valid yield trivial derivative eq omit saddle quantify around saddle consideration dominant correspond easy satisfied validity correct compare example p p accord relation apply value imply region solution coincide peak order expression satisfy saddle decay exponential respect large height rapidly peak consequence rapidly converge pure case eqs critical sec replica symmetric rs fluctuation detailed calculation sec stability fluctuation use eqs relation obtain interpretation stability rs become unstable probability configuration eqs irrespective phase marginally stable iii lead order iii tolerance regime sec rs phase marginally simple distribution space linear combination normalize version version instability appearance far place growth confirm calculation insight finite effect restrict tolerance throughout mc method update step random vector satisfy orthogonality fulfil initial condition orthogonality restrict move direction move positivity constraint intermediate may calculate accept reject unchanged along dash see convenient instead wave configuration component mode mode orthonormal basis note wave configuration choose observable random limit choose uniformly fouri mode fouri imply orthogonality soon fouri normalization easily exclude mode carlo markov detailed balance procedure quantity spin correlation target consider ten different plotted figure bar deviation carlo move irrespective show carlo step indicate equilibrium see
propose multi instance histogram wireless detect include medical video situation system discriminate experiment propose construct video belong dataset image detection classify end patch patch instance bag framework learn extract patch texture texture feature instance pool quantization histogram represent vector machine type conduct entire dataset overlapping fold fold set remain fold histogram instance framework histogram image train basic histogram histogram vector classifier please notice parameter turn exclude svm positive measure recall receiver roc value traditional histogram recall figure performance improvement original histogram feature employ function datum improvement increase classification sc sc validate appropriate roc class auc shown clearly method challenging highlight histogram search geometrically protein bind protein drug protein bind site histogram multi protein bind evaluate histogram bind site bind site protein bind site belong site vary select dataset dataset class could site htb query bind site bind protein site rank site site belong rank return bind bind site bag prototype bag feature bind site sparse code rank curve auc curve also measure rank roc histogram roc base discover auc given figure bind site system sc value compare sc histogram htb type function compose histogram reconstruction pool regularization code programming algorithm sc outperform previous future apply security powerful representation method attempt reconstruct sparse sc instance histogram histogram use reconstruction novel performance histogram instance image wireless video bind site retrieval encourage histogram code sc effective representation method try reconstruct new end minimize norm usually impose code vector norm regularization version multi learning traditional could many instance sample histogram sc histogram fact function leibl divergence function quantify error histogram metric paper problem loss replace propose especially formal programming discuss give instance learn vector instance th quantization histogram histogram nd dx ni sparse histogram histogram md dm mi traditional sparse code penalty could impose zero keep however smooth combine avoid norm notice go zero error sum objective trade basic please obtain regard substitute slack release slack directly turn slack slack sparse code optimize optimization iteration optimize fix fixing optimize vector vector code variable turn please variable long
network capture complex relation ibp give accuracy case affect ibp training size good improvement make augmentation improve even ibp aim solve ibp datum much see least ibp efficient dataset possible also plot plot demonstrate curve cifar curve achieve low error clear sign ibp dropout increase confirm add cifar cifar classifier small slope pure dropout give add cifar ibp advantage standard theoretically ibp ibp bp mnist cifar cifar see procedure batch augmentation experimentally confirm cifar shift scale rotation use implement augmentation simplify design estimate fully use cifar summarize far improvement tangent less ibp require structure obtain result unfortunately could significant improvement dataset ibp bp bp tangent bp ibp mnist cifar invariant backpropagation ibp backpropagation learning algorithm require pass derivative calculation derivative around might useful believe useful usage network area lemma vast transformation vector translation rotation variation vector backpropagation incorporate noise extension consist backward apply confirm theoretically establish demonstrate backpropagation network learn relation class also suffer overfitte technique crucially good number preserve label usually variation location image rotation area result speech knowledge process invariance robustness variation vector extension backpropagation combination call simply regularization time act network train training implement increase claim decay reduce learn early implicit regularization employ convolutional layer widely locality fully number give flexibility pure layer tune locality reduce next propose similar nature transformation invariance case cnn convolutional layer follow scale translation invariance variation way transformation sample convenient analytical moreover object also part obtain exist sharing cnns autoencoder variation architecture show object representation able deal rotation sift attempt transformation overview article qualitative comparison present invariant part practical implication part accordingly ibp pseudo code number layer activation map pixel multiplication non empty activation forward pass backpropagation specify layer prediction prediction achieve softmax transformation derivative simply derivative denote derivative differentiable computation activation whole reduce update specify usually time input space move direction classification prediction backward direction loss prediction length specify versa specify pass transformation invariance would need jacobian matrix autoencoder minimization change move class invariant aim good need look derivative output propagate later derivative activation backward pass also quite initialize perform three specifie step minimization find trade crucial algorithm bias notice backward pass propagate third compute bias pass implement contain write sigmoid ii linear iii except cause backward iy filter also immediately iy w dy rd pass layer fully map transformation propagation backward passes loss forward pass derivative activation activation backward linear max pooling propagate position derivative bias rule attempt use achieve author make neighbourhood small rotation tangent initialize network propagate output set vector towards direction vector derivative linearize add propagation appear propose end derivative invariant vector case derivative output rotation loss make robust variation opposite lead huge transformation express combination basic transformation rotation scale list tangent always incomplete general exist implementation tangent suggest require smoothness basic simple operator transformation require vector training repeat tangent tangent rotation perform training experiment backpropagation ibp want determine baseline consider dropout always value within evaluate modification mnist experiment batch exponential decrease function layer follow region within except normalize px maximum scale dimension cifar database epoch initial permutation initial rate benchmark describe mnist contain
pool subsample insensitive rotation global global pixel movement totally subsample metric even mahalanobi belong class face belong class e face people mahalanobis metric semi mahalanobis negative margin metric learning psd maximize close multiply distance scalar normalizing normally minimize constrain convenient later program slow state psd solver semidefinite cone perform solve manner relax psd constraint look solver map function property quadratic use auxiliary everything objective rewrite svm bias prove convert standard form solver solution semidefinite svm objective psd order solve run solver kernel thus avoid look benefit solver vector need store computation far rank many matrix algorithm solver effect include locally vanish rewrite forget finish equality optimal form iv equality first separable easily need add slack second add preprocesse previous run svm solver semidefinite quadratic solver case mnist result size currently quadratic use solve take program exclude semidefinite solver perform slow solver svm shift mnist digit comprise mahalanobis distance negative perform local example compare lead svm invariance add shift metric linear learn negative applicable mahalanobis svm shift intuition shift well variability divide subset one subset training image use align cosine svm shift image shift mahalanobis svm see improve albeit degree show mahalanobis metric query natural insensitive recognition insensitive implication important many tolerance primarily vector efficient mahalanobis metric learn metric algorithms neighbor find appropriate improve successfully face identification metric global mahalanobis matrix psd make distinction psd optimal limitation overcome approach nonetheless svm suffice behind object class mahalanobis try weakly supervise case positive query image bank train face image person unlike metric method need person metric twice look equality minimum taylor value interest mahalanobis mahalanobis costly decomposition dimensional regular semidefinite solver second
must linearly linearly guarantee linearly induction linearly unfortunately algorithm terminate construct matrix occur early termination succeed succeed linearly note choose initialization full co chance select select perfectly variable random variable hold subset exact gram select column column independent precise correspond recovery recall gram equal rank g equal state assumption exact recovery satisfied thm simply combine lemma lemma return terminate return column index column ssc rely dataset provide subspace make subspace reference point matrix union recovery reference subspace theory decomposition combine seed sampling face mnist leverage detection describe evaluation face pixel illumination hyperspectral spatial scene signature class dimension mnist handwritten fire neuron collect perform position target synthetic subspace overlap coordinate add create gaussian vector size point evaluation mnist fast omp fast representation experiment processor dataset run evaluation matlab approximation matrix utilize selection obtain matrix contrast base decay evaluate seed approximation iii iv leverage five achieve equal recovery linearly wide interestingly error seed roughly sample ii behind quickly iii leverage suggest contain structure well make significantly less seed representation signal self ssc consensus see sec dataset subspace sparse seed rectangular spectral think represent bi use method cluster introduce provide elegant relaxation bi co approach eventually find eigenvector matrix lead visualization embed union subspace red dot star feasible efficient seed capable separate seed neighbor weight motivate fig hyperspectral observe improvement sample seed noisy hyperspectral highly seed tolerance omp apply simple subset point point image compare performance seed randomly cluster error mean denoise seed obtain component raw ratio face image cut vs six face illumination face illumination condition full respectively single low dimensional subspace seed behind outli try subspace require contrast try collection signal exploit rank signal lie subspace whether outli self expressive diagonal alg ssc provide tolerance determine column upon threshold column dense case threshold segment bi multi however difficult threshold display ground datum via fig seed dataset corrupt outlier sparse seed subspace omp constrain outlier outlier seed column base determine appropriate rank structure outlier challenge column sparsity modal set explicitly threshold segment matrix seed provable omp solve range approximation outli seed recovery thm real world e obtain column recovery guarantee explore classification cluster contribution show expressive computer column selection approach expressive basis amenable independent exact thm imply select linearly problem prove strong ssc rank overlap direction future implementation sampling provide alg na I sec inefficient step addition calculate result previous formula rank alg vector criterion index denote block note invertible complement non zero case w k b b k eqn fast incoherent alg like thank discussion lee collect nsf nsf mh distinguish fellowship rgb k g aware reduction matrix aim find learn express self alternative introduce scalable computing expressive decomposition seed greedy incoherent form basis seed develop seed low subspace range denoise numerous result seed attractive complexity factorization expressive clustering subset recovery method combination basis simple provide extremely basis often mix geometric express element expression successfully classification provable discover idea expressive approach lrr represent contain zero along interpret expressive principled segmentation apply big challenge ssc construction affinity greedy ssc data affinity development dataset discover upon datum express approach self expressive seed select sequentially incoherent uncorrelated use call incoherence operate subset thus column entire gram second use vector first step compute omp seed detail sec alg demonstrate thm return capture datum linearly ii estimate remain stop aid solve include cluster denoise sec performance seed dataset neural result demonstrate seed scalable ssc fraction organize selection subspace introduce seed motivate study seed four application ii iii iv conclude appendix column concatenation let span say collection th ik dim subspace selection index approximate sense collection signal invertible recovery np force body study propose sample target ii leverage compute svd column approximation computing leverage approximated sample select column pi highly costly matrix computing operate column self expressive q represent reveal representative collection aid hyperspectral group norm within aim form consist method orthogonal pursuit omp subtract contribution atom repeat satisfied target reach residual alg termination either approximation coefficient vector residual signal vi learn ssc follow user expressive underlie ssc coefficient ssc matrix strength edge live affinity graph affinity subspace motivation underlie ssc consideration subspace ssc lead provable least span provide reference subspace sec lie union seed reference set subspace alg complexity seed select termination termination error sparse normalize solve omp stack seed column step form accelerate sequential incoherence column incoherent uncorrelated range machine learn use way find gram motivating image face illumination fig return varied illumination incoherent iteration return highly redundant illumination appendix alg implementation motivate suppose already index loss generality column nystr guarantee select linearly justify one via wide illumination whereas redundant new approximation write q span scalar measure poorly column computing span instead influence current greedy decide approximation correspond denote entry diagonal calculate change nystr approximation select user
remark summary recommender actually highly attack since since far small extract design profile statistical diverse general boost adaboost effectively improve conduct compare technique attack boost recommender variety suggest news book product recommender successful carefully attack term attack lead construct method crucial profile important attack attack formulate user attack traditional knn attack handle kind issue fail effectively aspect firstly statistical attack profile significantly source rating item rating extract possible classification become attack profile profile sophisticated make much easily classify general adaboost feature experimentally classical type specify appropriately comparable type adaboost technique hard adaboost employ weight gradually emphasis concern operator conjunction type learner improve attack experiment conduct alarm rate knn effectively give brief attack attack model analyze focused attack attack profile utility attack knn detect attack et try user detect attack also detection introduce rough theory attack user profile attribute overall attack wu et hybrid attack metric technique zhang capability base nevertheless attack addition et svm profile attack svm create profile decrease speak attack leave desire introduce design distinction attack profile attack secondly conventional particularly attack apply variant adaboost gradually emphasis concerned attack attack bias attack classify way attack attack attack rate low attack high target item attack profile form detail item singleton multiple call target attack attack rating rate minimum value rating function random rating utilize attack attack profile involve attack profile list model attack attack c c c c rating c nan mean nan around item randomly segment item reverse choose nan rating l power item rating item power rating normal c nr item nan l copy profile rating profile nan nr copy user profile nan attack popular item si max mini contain item reverse attack popular entire attack item aggregate score user rate attack centrality item high neighborhood item apply rate discard neighbor item similarity score item top nr item high attack aggregate select require least rate significance power neighborhood every significance discard score select nr select base number rating user overall introduction approach aspect feature extraction attack approach phase phase phase classifier via phase test construct attack profile attack attack attack etc generate mixed profile increase attack construct aim extent imbalance detail attack size attack size form feature extraction employ subsection extract profile characterize set composite retrieve generate result phase testing characterize user feature generally generic basic attempt discriminate attack profile profile type specific detect attack signature attack employ besides employ base additional specific maximum rating minimum rating item attack profile high deviation generic specific difference profile profile item rate score entire item rate user item rate rate rating rate rating rate rate minimum rate otherwise size rate entire rate raw profile transform attack know attack detection formulate conventional e knn handle kind issue prove boost weak learner fit gradually increase emphasis model weak learner model precisely interpret adaboost dictionary misclassifie normalization g adaboost loss boost numerical optimisation add weak learner gradient boost prove show slow lie slow hereafter lin xu gradient boost aforementioned controlling step variant like boost truncated novel improve numerical consequently capability boost good regard certain extent may possess learn boost idea adaboost initialization weight iteration find sum misclassifie tf g introduce experimental metric environment secondly impact compare svm knn attack attack mean set recommender collect researcher field collaborative attack recommender system movie rating mean movie derive website rating approximate besides rate attack profile attack table rate low attack attack analogous attack attack attack ensure item randomly attack attack select movie movie rate attack movie reverse randomly choose movie movie one user whole profile random nr attack diverse size attack dataset mixed training training attack profile exploit attack size size profile profile include attack attack effectiveness detection classification detection alarm divide detect divided alarm attack profile divide numerical ghz ram gb conduct list several diverse utilize maximization create generic specific aforementioned relationship extract feature reverse attack false alarm diagram attack fix illustrate design svm knn test validate follow default profile classify unseen knn knn algorithm fold pearson correlation utilize build week learner fold equally spaced boost fig surface aforementione nr attack example different attack size surface knn present rise attack svm effectively attack attack fairly classification attack also accuracy almost produce little concerned profile detect naturally knn essentially svm attack low much detection false alarm rate knn fail within attack nr attack attack attack nr knn attack knn indicate attack profile profile attack profile knn boost improve iteratively correction adaboost enhance detection false alarm
otherwise general feedforward adjacent unit layer weight activation network label probable minimize posteriori map online arrive sequentially update q intractable store field marginal appendix like marginal likelihood contains generally intractable summation summation approximation assume fan fan normalize neural quite distribution end p ij directly calculate taylor around used expansion pass backpropagation pass pass initialize follow value input layer bayes initialize refer q learn output define alternatively eq value parameterize way process parametrization backward substitute iteration show step weight configuration I l ji h ij evaluate task binary examine multiclass deep fan layer hide convert mnist handwritten digits method method spatial configuration similar cnn unit receive input layer unit connect input therefore cnn map connection implementation unit element whole fan map neighborhood report handwritten digits contain set sequentially identify classifier training recommend backpropagation neuron treat image vector neuron hide add architecture hide neuron layer configuration architecture filter method filter layer unit output select configuration large well hide network configuration configuration lead fan unit size layer second layer layer vector employ architecture dropout network efficiently dropout demonstrate neural backpropagation gradient method investigate hidden unit dropout net presentation presentation deterministic sect sect train epoch performance table mnist observe good layer outperform p outperform grow slightly well although hide one increase unit standard optimize hidden fan fan fan network compare structure table result show dropout configuration hide layer dropout increase besides b reasonable validate dropout prevent overfitte c c unit algorithm performance binary unit bad input contrary performance dropout table input output layer method light extension network block connect convolutional b finding algorithm task performance real fan hundred well improve mnist report backpropagation evaluated dropout spatial weight study classification validate image dataset cifar neural use explore initialization acknowledgment visit
nearly define concave lemma log concave simulate annealing proceed epoch sampler annealing arise convex fy ii ki x j run transformation warm guarantee anneal warm distribution successive epoch round guarantee away warm next epoch concave density support next account impact warm let approximately concave suppose algorithm step prescribe epoch anneal walk round near epoch follow theorem convex prove hold improvement almost instead concentration heavy tail isotropic see bernstein inequalities log concave measure way achieve goal invoke random row isotropic almost surely since isotropic translate log concave isotropic final temperature annealing need oracle informally per epoch corollary approximately epoch unknown lipschitz aim sub chernoff bind decrease repeatedly average observation concentrate randomized optimization value upon visit well precise three return twice query equivalent version optimization problem take box net return information net query parametrize random gaussian call give oracle denote q lipschitz random second affect oracle dependence optimize per oracle optimization union exponential somewhat artificial draw spatial alternatively could union visit function convexity decrease minimum decrease break optimization stage optimize region guarantee convexity aware problem surprising provable consider decrease complexity decrease formalize discussion convex non radius mind property fall order simple target additionally deal constraint handle smooth generic forecaster repeat suppose forecaster observe observe forecaster define mapping vast majority write follow relaxation relaxation call sequential involve gradient might approximately evaluate saddle point proof function view want point function loop maintain either length interval satisfy case essentially two case argue remove still enough hard point support convex fact similarly terminate log view consider interval current continue gx l x l claim terminate soon interval imply gx terminate concave associated level moreover either imply e stationary distribution scheme accord restrict classic reject page error variation restrict quantify tail event sample define shorthand eq bound proof main theorem variation walk start u produce see accord along let truncate distance elliptical tv hold eq compose truncate fold iterate run satisfie induction eq follow closely argument proof device random define tb yx inside suppose assume define q imply follow since although sx hx next contain ball first page lemma q take induction end identify epoch prove claim arbitrary give sake completeness relate arbitrary convex increase linear complete claim hard add effect eq proof theorem time round however close guarantee need oracle complexity term logarithmic distribution log concave distribution implementation algorithm anneal first essentially motivate optimization method produce induce decay minimizer program approximate dynamic particular factor require amount convexity optimum computation access may function oracle value return subgaussian probability oracle know detail motivation study problem derivative essential sketch area readily private programming algorithm present invoke run anneal approximately fast approximately log line sampling concave already early walk discretization motivate central theorem lead walk strategy walk long step motivate somewhat hard pyramid evaluation hence walk present reduce achieve optimal include simple evaluation mention boundary denote negative denote concave log gap long high dimension establish less section section induce run generate step initialization choose pass act sphere line line address cause deviation include sampler search yield build optimization problem line segment function segment produce function e gx method concave initialization show gx gx c gx x fx stop concave lipschitz convex find step initialization find desire distribution furthermore query approximate concave particular concave accord analysis round mix spectral gap markov turn spectral relate call
interpretation mutual show comparable modify optimize log somewhat word hyperparameter embed conjecture corpus eliminate occurrence word less text context count function approximate transform differ form fw dot see try fw information order occur optimize pair corpus monotonically rank advantage rather instead model order fit analogy word find word problem occurrence noisy due word co phenomenon document corpus diverse help effect token use word context occur context word inner convenience embed c product large want rank parametrize sort specific context write follow machine replace popular hinge loss enable upper certainly desirable list minimize quantify monotonically goodness model later assign versa ranking consider monotonically concave function monotonicity concavity imply loss sensitive sensitive list interested context co bottom make either due four alternative interpretation relate reference plug replace sgd sampling around linearize concavity bind tight motivate parameter due also minimize optimize unlike admit sgd c unbiased use put everything detail another line corpus hand complexity two update computationally involve multiplication via step c w converge converge remarkable update another triplet perform key optimization multiple lack include code indicate proportional c monotonically consequently context list give view parameter view modify ranking problem fashion appendix give word introduction work perspective rank score retrieval rank indicator rank observe discount special one increase concave efficiently apply across processor work distribute stochastic impact rank small dataset closely follow combine token consist wikipedia tokens token benchmark token around tokens wikipedia article already plain format processing sentence corpus stanford character discard short token long token token token small corpus extract token must window follow finding symmetric decrease word apart contribute count specifically corpus corpora large window corpus corpus vocabulary k dataset embed boost combination ensemble later interpretation cosine add add experiment word analogy token word substantially employ word analogy six word rare word together human evaluate similarity task google dataset question syntactic question contain five capital people syntactic question nine opposite comparative question correctly select word thus mistake namely score using indicate give vocabulary reporting vocabulary question zero make across experiment argue present function report million dataset table note trend analogy set comparison include unweighted attribute performance implementation text token extension text token test original update google analogy function perform similarity analogy large dataset scale good configuration weighting comparison input occurrence embed dimension train skip art nlp default suggest produce well occurrence corpus small three follow large corpus eventually word analogy consistent finding indicate token case word similarity optimize rank token score somewhat close report token discrepancy processing corpora similarity train word dataset google analogy syntactic result list table google htb token task google semantic syntactic word analogy token achieve appendix result performance analogy achieve loss analogy therefore important emphasize expensive setting extra robust ranking supplementary material worker mutually exclusive exhaustive approximately sized w q begin outer partition context induce context update
passive ep coefficient passive ep illustrate ms ms time ms ms ms ms error passive time vary computational list recovery highlight fast corrupt list linear algorithm suggest ep use improvement bit recover regard hinge many perform robust bit improve recovery bit hinge linear solve ep generalizes propose suitable ep trade hinge improve advance relate quantization implement operate bit compressive attractive model function function hinge bit classification measurement sign hinge binary bit experiment hinge cs observation bridge bit hard thresholde binary hard suitable sparsity true elastic net machine passive propose suitable ascent propose solve prove trade hinge loss improve exist bit dual digital quantization extreme measurement need benefit hardware low signal analog sign number bit find sign q measurement system measurement direction magnitude measurement e lose quantization make assumption meaning bit recovery explain partition hyperplane number hyperplane feasible recovery subset additional assumption unique tell measurement rarely consider application advantage attract cs try sparse sign small measurement cs cs fundamental bit cs count convex algorithm approximately variant hold real accurately bit noise component transmission sign magnitude true analog sign transmission among mind deal sign true empty utilize hinge sign change hinge attempt minimize hinge loss robust algorithm review ii cs attractive condition bit nonlinear heavy bit sign recover regard binary problem hinge widely svm hinge calibration task linear loss rarely enjoy yet linear quite hinge hinge cs closely definition characterize provide bridge loss hinge regular task cs hence expect trade bit norm unit non constraint consider name effectively ep ascent show optimum exist cs section iii introduce conclusion end vi bit introduce attract hard minimize nonsmooth hull bit sphere sphere constraint constraint model reformulate programming become satisfied however solution project sphere project sphere mention binary change application bit cs infeasible feasible feasible classifier sign loss bit signal approximately loss understand replace one sided side iterative one robustness sign improve robustness measure deal change estimation approximately convex propose equivalently put come play model loss rare task rule yet norm hinge observation task motivate special cs hinge bit analog near hinge distance correctly classify correctly contribute optimal measurement sign contribute optimal solution many idea incorrectly classify datum classify example encourage large well influence sign replace transmission detecting measurement sign performance sign counter performance two different number binary well algorithm performance algorithm able detect mention able detect detect analog noise snr quantization flip quantization green dot dash dot solid respectively sign change sign four performance bad main happen mainly sign fig confirm change differently sign main cs leave modification advanced loss naturally replace sided establish robust illustrate solve guarantee global derive specifically convex term model experience task expect suitably may performance great analytic ascent indicator primal problem follow u passive primal get sphere imply small separable apply ascent efficiently subproblem subproblem separable parallel via subproblem let increase u previous discussion give coordinate ascent end analytical passive coordinate ascent ep optimal optimality optimality condition coordinate j coordinate j u j j optimal theorem svm passive matter happen choose maximum number choose stop small though passive analytical problem use ep similar way experiment flip evaluated choose experiment average recovery plot htbp c experiment performance measurement performance contour
htb kl message length analytical calculate derive appendix compute parameter minimum message encode previously table list metric mml across combination mml notice extra compression make mml song mml song mml e e e e maximum mml base reduce mml estimate always mml htb l song mml song mml e e e e e goodness behaviour generic derive score degree compute test exceed critical nan conduct know infer table consequently imply nan reject use incorrect distribution hypothesis htb song mml song mml e e e e e e e test mml behaviour mml size plot demonstrate low variation behaviour irrespective continue value produce low mml actually perform bad song due mml identical approximation accurate limit f depict increase note produce highlight mml fundamentally show note extremely rate limit mml coincide compare mml behaviour dimensionality amount ml mml one htb empirical propose method mixture gradually increase simulation mixture direction true component parameter mixture mix proportion component align axis illustrate variation infer component concentration angular separation become datum correctly concentration concentration angular infer search even different easier whose concentration mixture component present average number component concentration chemical directional protein atom motivate structural modelling generate protein three protein alignment secondary pattern encode protein offer serve varied redundant experimentally protein publicly version coordinate transform directional characterize co measure associated consider precede comprise transform translate origin lie x axis xy yield orientation previous coordinate use coordinate direction repeat transformation successive protein result total structure database protein component infer model comprise protein search mml mixture model structure correspond mml base mixture mml result empirical directional belong frequency protein major mode correspond secondary structural middle direction chain exclude due truncate model point would code protein possible rare empirical mode mml allow compete recently nan nan atom use precede highly compression gain orientation atom surface sphere equal area cell surface sphere uniquely encode encode description state length orientation angle sphere directional protein build mixture descriptor encode angle length orientation correspond unit describe surface equation two compete directional structure htb mml bit uniform divide message length statistic encode potentially various protein modelling introduce robust infer mixture ii von minimum length provide tradeoff model message length search effectiveness directional well current would thank protein provide author like acknowledge l technology involve inference component discuss unsupervise use mml criterion demonstrate effectiveness search parameter key handle fundamentally different type mixture modelling euclidean multivariate address model directional unit contribution addition methodology mml expression von fisher derivation mml test simulated mixture experimentally determine bioinformatic demonstrate search encode state model mixture model offer explain field biology engineering economic amongst compose model kind model aid hide sound probabilistic model extensively machine mixture component respectively sum determine number mixture mixture difficult balance objective determine data parameter fit strategy use control assess mixture ability message length bayesian method prove effective concern model widely poisson von comprehensive summary research partly motivated mixture von sound justification compare decide component traditionally estimation work mml principle unlike invariant unlike mml mml mml estimate scheme cost state parameter continuous precision encoding state map mml process precision message thus length mml mixture challenge model limitation mml incomplete drawback propose mml formulation select formulation demonstrate effectiveness estimation use mixture extend relevant directional use von fisher establish component estimating rely boundary attempt mml simplify matrix diagonal mixture far different number component select good search iteratively begin redundant error search component mml result sensible component component child merge start number component iteration avoid unless require mml expression length estimate discuss model directional von fisher mine significant scenario science physics directional several surface sphere ellipsoid fundamental von symmetric unit mean direction random dimensional kind mathematical use many perform inaccurate demonstrate experiment conduct size evident mml section mml reliable art mml use von circular von demonstrate protein angle text analysis cosine text evidence compare statistical bernoulli widely package parameter mml mml candidate mml mixture version effectiveness implement modelling allow mml framework optimal effectiveness mml protein mml estimate multivariate mml select mixture component depict competitive conduct mml support application section function matrix traditional give maximum likelihood solving equation estimate covariance unbiased give likelihood involve conjugate literature unbiased von fisher obtain equation estimate difficulty analytically solve equation improvement respective improvement provide easy demonstrate well start equation utilize fix conjunction interpolation point heuristic provide perform newton result order taylor series accurate estimate demonstrate iterative truncate iterate propose likelihood govern concentration considerable counter minimum base result unbiased several compete mml mml extend work mml estimator generic exist develop rigorous compete datum mml information theory whose probability length event shannon insight length result compete odd compete encode comprise two take bit bit complexity explain state fit mml paradigm infer several gaussian vector give datum involve choose evaluating length mml lattice quantization mml framework compose part encoding encode mml key mml estimation ignore measure parameter finite precision mml incorporate determine region multiply message encode mml formulate derive von mml precision prior dimension wishart density q computation fisher fisher approximate fisher analytical element correspond c describe due multiply derive message encode substitute mml need minimize mml mml need compute mml observe mml preference traditional parameter explore mml estimator dimensional mml inference formulate generic reasonable support evidence parameter suggest uniform fisher general dimensional equation net message mml equation estimate minimize mml linear discuss comment approximation experimental guess iterate mml appendix mml newton mml mml observe distribution mixture observe data mixture log eq j traditionally standard em standard maximize equation closed gradient employ achieve step fractional membership partial conditional belonging assume expectation log membership maximum likelihood iteration step estimate sum ir update parameter describe methodology involve mml function em discussion describe behind model mml encoding message mml require encode parameter cost encode datum decompose encoding encode message require absence like model belief decrease bit uniform within prior minimal magnitude treat expression encoding encoding state precision precision dimensional encode message mixture note lattice quantization equation parameter update constraint derivation mml update appendix use membership component intermediate mml update ir mml obtain solve successive less firstly optimize correspond include correspond state secondly ml whereas update mml component reasonable initially strategy well amongst converge covariance singular member number numerous attempt infinitely mixture method aim cost associate end penalty certain explain simple add variant aic suggest penalty aic multiple free coincide aic serve model fit formulation adopt type statement associate free cost characterize proper criterion mml wherein part message multiply formulation mml formulation argue task potentially grow proportion mml bic determine mixture aic bic mixture variate mixture complexity mml formulation incorporate discuss mml scoring author gaussians fail method rigorous one length scoring derive tradeoff fit likelihood hyperparameter pre define compute use hessian equivalent empirical identifying variate prior prior assume element assume consider joint p h previously discuss propose scoring density covariance ignore fisher dependent mml gaussians hessian diagonal eigen classic mml component run em within optimal cl likelihood equation arise mean entropy cl parameter score model good amongst choose mml criterion formulate score two message encoding free component weight assume component jeffreys describe density encode mml scheme encoding parameter use parameter detail make follow encoding assume length vector treat independent jeffreys prior jeffreys ignore jeffreys note formulation difficulty compute length greatly notice weight number bic discuss highly mml entire essence goodness accurately em point component consequently component search mml however remark search allocate component ignore subsequent assign hence bind reduce assign consider scenario mixture equal mix proportion infer wrong relevant component decrease optimum update low increase subsequent attempt robust place overhead handle component achieve true rigorously mml adopt simplify approximation mml formulation message formulation fisher distribution section trial elegant comparative superior method outperform base regard alternate heuristic infer limitation infinitely mml mixture overall see mixture namely mixture parameter propose mml unsupervised operation htb current split component well merge current search mixture suboptimal smaller perturb new new retain split great line first two optimize algorithm component integrate optimize reason rather already similarly component mixture record merge match divergence mixture evaluate mixture merge initially start child optimize split well retain splitting component perturbation improve current repeat improve provide observed membership execution membership adjust subsequently achieve optimum membership operation membership component index current generate amongst component fractional membership carry assume membership remain component carry sub state subsequent update goal identify distinct within provide maximum variance parent component side parent ensure reasonably apart serve good optimize membership membership child computed component component maximum characteristic mixture adopt distribute randomly membership mixture initialize update membership belong child child describe effective membership eq child substitute update give equation eq mixture parameter equation update equation since mixture multiply influence child integrate component usually member exploit start upon integration ij new follow initialize membership start estimate maximization component perturbation component result new goal remove current mixture check whether mixture explain component weight membership good r ij mml update ensure see initialize membership possible complete membership membership perturbation result new explanatory improvement join component improve merging runtime another merging identify compute kullback kl runtime constant close match identify merge component involve membership optimize component merge let form merging pair integrate mr merged component merge ir merge component merge membership message length perturbation merging result current mixture propose consider gaussian component simulate point strategy search various detail begin infer split step explanation fig depict dotted mean dot initialize use child length update mean black dot denote mixture post em phase bit iteratively merge show splitting first component show merging first identify close kl merge operation improvement operation initialization operation update fig message length discard merge perturbation total message terminate htb green denote mixture post bit htb blue component merge merge along parameter optimize bit stage intermediate mixture employ consider option possibility getting optimum propose balance tradeoff useful prior knowledge nature way appendix evolution infer explore overlap terminate demonstrate optimum number infer length show length algorithm converge vary length curve drastically initially start length gradually clearly suggest encode parameter complex decrease length mml evaluation criterion message comprise statement state complexity mixture parameter part increase increase illustrate message length axis fitting decrease message mml behaviour consistent sharp error datum overfitte dominate message optimum number metric monotonically component mml illustrate methodology cite information integrate discuss far section experimental true time mixture mixture mml allow length encode message odd mixture give mixture explain methodology score mml scoring infer score mixture search optimize score expect mml mml mml method however scoring demonstrate superior lead define prof use kl metric give two distribution metric relate simulation kl expression experiment mix proportion bivariate infer experiment separation gradually increase percentage simulation determine two htb correctly data b illustrate separation number correct message divergence infer therefore experimental performance roughly apparent similar close separation mean infer table depict fig many overfitte correctness mixture infer comparison difference message mixture propose infer simulation mml mml function result optimize mml mml function bivariate across close divergence kl denote red zero compare vary value kl divergence zero variation kl suggest mixture employ mml scoring htb along line conduct proportion distance plot htb component show infer search low correctly infer quality mixture message kl infer low message fig message length suggest search sub optimal mixture fail analyze kl bivariate fig infer kl value kl component however htb gradually increase average infer component simulation fig show infer average
mp mp generalize lipschitz high compute stop feasible compare trial weight computing consuming consider linear current trial atom svd efficient low storage suppose tensor combination rank store store help break curse mp completion iteratively redundant completion prove generalize generalize select atom nonconvex successive sr sr trial see sr strategie sr either intractable need alternate allow closely wolfe constrain recent cg cg atom trial lie convex significant cg constraint cg constrain counterpart nuclear norm exactly approximation computational derive approximation power type normalize singular inexact pair correspond lead th tensor recursion subroutine recursion although complexity lead singular svd go acceptable compute normalize pair x establish tensor recursion normalize step subroutine also unfold induction v induction k inequality relationship frobenius subroutine subroutine get x obtain di x course subroutine apply subroutine trade computational cost time discuss subroutine subroutine subroutine together subroutine method base space svd require high whose odd always tensor key tensor unfold organized subsection extend tensor convergence l l uniformly value tensor w subroutine l l q l clarity k k k naturally term k r w relationship frobenius multilinear letting form stacking row task represented unfold write convergence positive uniformly large eigenvalue generate subroutine l ls denote r w l l k f naturally hold f km recall numerator f k however every positive definite size follow add rewrite square symmetric tc f assume ls ls subsection conduct function end characterize denote finite assumption restrictive variety huber fair define huber parameter huber nonconvex loss mention still begin analysis completion satisfy assumption say tensor derive hadamard e define loss deviation cost completion completion w w w k k follow boundedness uniformly hand tell k consequence q diagonal diagonal ii unfold q act remove deviation x apply multilinear assume bound eigenvalue mp ls w q k w f recalling recall f last complete make modification discuss x th column linear multilinear learning assumption satisfy define inequality improve synthetic well focus tensor completion numerical conduct intel support loss prior fp solve value convex relaxation tensor completion norm fp tensor factorization use sect criterion generate ten tensor randomly mr report particularly mr perform mr value table show efficient mr optimize use matlab three two three mr fp htbp r mr e e e e e e e e e e e e e e choose hyperspectral brain image hyperspectral tensor rd fp consume due case less particularly hyperspectral mr relative size take performance fp mr might consume slow speedup compare fp useful miss intuitively performance method miss fp mr fp art nuclear latent norm scale nonconvex nuclear nonconvex regularization treat stop tune validation specifically follow available education student student school indicate bias task index school year index therefore jointly learn mse var performance restaurant dataset rating restaurant rating aspect restaurant space index aspect tensor mse compare school restaurant dataset school well accordance method still bad eventually slightly view bar mse efficiency around iteration give desirable tensor scale school restaurant examine effectiveness completion cauchy loss employ robust completion robust criterion previous randomly mr vary entry contaminate outlier fig mr mr increase rapidly experiment method give tensor treat directly consider setting miss image recover g short paragraph seem remove recovered tensor cp tensor th image I row recover entry store recover store speak storage ccccc examine axis iteration stand fig ls l red ls fig plot plot robust curve confirm derive sect tensor completion logarithm axis confirm art storage cost sharing convergence dataset receive european research european fp author contain grant grant project grant project medical science policy office definition subroutine multilinear propose pursuit cost matching type large problem store tensor storage help break curse dimensionality various circumstance provide approximately compute tensor provable help analyze experimental synthetic effectiveness key pursuit learn nonconvex tensor appear generalization vector make represent problem tensor goal rank tensor provide allow pursuit several task represent lie information high rank learn low tensor encourage rank widely mode nuclear encourage tensor low tucker scale many exist main rely singular decomposition category several factor minimization avoid tensor algorithm solve scalable scale motivate efficient mp tensor propose pursuit mp update trial combination namely cp tucker define
equation markov expectation development demonstrate easy simulation adopt bayesian expectation section contains demonstrate conduct future difficulty deal expensive feasibility issue transition mcmc langevin hamiltonian employ finite batch long induce convergence practical approach difficult mixing use hypothesis might confident accept reject decision substantially construction quantification one towards induce original explicitly respectively orthogonal expectation contrast construction require elegant exploit additional computationally due size subsequent asymptotically availability appropriate tight likelihood precisely bayesian investigate promise generality applicability mcmc mcmc complexity mix mean evaluation iteration require chain propose provably sub number unbiased low likelihood several expectation available likelihood pseudo section coherent big exploit device develop approach unbiased expectation intractable infinite expansion attack arise contribution complement monte carlo procedure static target partial target px tn pl subscript construct increase batch whole exposition geometric batch set possible small consider assume integer use experiment present device component transform expectation posterior lemma provide way construct unbiased estimator converge evaluate correct bias different present recently hilbert space convention moreover applicable finite regime truncation variable replace sum estimator require intuitively tail match expectation simple setup along truncation fashion replicate procedure reduce variance copy scheme repeat empirical illustrate discrete corresponding number replication desire tolerance truncation variable tn explain key methodology datum I I compute stop posterior dot dot line connect procedure dot dot list advantage something think generate estimator mcmc chain batch evaluation result l cost reflect amount single core partial posterior expectation order te stay large bound expectation e speed fit tune complexity namely budget result replication work figure see correct asymptotic finite markov overall corrupted practice careful burn run reduce bias way address asymptotically give sequence unbiased expectation unfortunately expression unbiased line partial expectation noise augmentation acceptable sharp exist scheme decade mathematical engineering effort methodology software package expectation mini size decrease posterior often structure independently expectation true replication batch size roughly computational resource speed practice mcmc accurately clear produce sample burn estimate approach fair comparison likelihood evaluation mcmc notable posteriori inferior challenge standard optimisation evaluation example bfgs commonly benchmark iteration reasonable somewhat map estimate one sufficient avoid challenge extremely base indicator variable point mcmc dark obstacle dark point point need suffer compare fair replication take median likelihood sum replication already converge ground stress extremely conservative appropriate computational mean reduce bar additional outli convergence replication behave similarly bar line ground plot replication correspond evaluation extension compare experiment inference large unbiased expectation require eq simply access sharp computable form prohibitive typical gp eq inversion mcmc suffice look posterior cost ap still infeasible practice apply combination induce perform fix mini batch toy map feature choose mapping eq covariate spread add observation predictive mean explore partial repeatedly observation top show mse get zero almost unlike functional corresponding average posterior top prediction multiple mini batch reveal knowledge none sub scheme therefore resort compare inference feature subsample bar note mse eventually vanish compare replication mean cost slice plot prediction size mini batch size give zero approach inference gp combine gps descent huge streaming allow cut induce combination fourier predict delay record involve consist label covariance via fourier sake apply include centre preprocesse essential however adapt covariance match induce gp match computational roughly replication iteration remarkably rooted rmse conclude tune experimental protocol etc instead make achieve highly leave variational convergence mini batch random report hyper average reproduce rmse chosen obtain rmse constant batch bar repetition formalism streaming scenario expectation unable force batch discard still possible fix budget large process hardware restriction replace truncation still stream fully constant mean estimation gaussian biased bar batch make replication conceptual address unbiased need assign truncation result expectation runtime also result due resource limit arise covariance matrix side partial posterior exceed memory allow truncation develop solution computation outperform connection truncation functional dataset aim weight note dataset take figure top partial tend figure behave partial truncation relatively happen around sampling yet reach acceptable regression look around perspective many serious arise employ transition chain simulation statistic partial posterior implement exploit exist parameter furthermore conduct accurately compete simulation methodology likelihood carry experimental stochastic area future explore tradeoff detail deal use iii thorough formal show increase batch variable variance ratio express evaluation truncation tn truncation normalize constant z addition depend convergence partial full partial point estimate almost sequence posterior suffice grow undesirable remarkably slow partial expectation moment remain variance remains bound large rate quickly converge independently fit practice investigate partial comparison among posterior stress small variance figure determine derivation select key bayesian expectation monte consistently expectation average sample feasibility goal scalable full straightforward leading
one see approach semidefinite relaxation maximization semidefinite q anchor estimator give approximation submatrix sensor sensor relax constraint rank via last equivalent formulation introduce enyi uniformly increase performance essentially anchor sdp formulation show superior sdp xy message pass practice would choose formulation pass expensive hide offset outside plant average experiment classic rank inconsistent numerous vision rotation usefulness demonstrate numerous recent graphic sensor localization biology semidefinite sdp round numerical college game propose ranking guarantee synchronization amount ground synchronization aggregation approach art rank truth propose sdp synchronization technique subgraph locally ranking identify small player pairwise comparison noisy identify relate open question angular synchronization semidefinite programming ranking least angular rank eigenvector sdp synchronization aggregation section hide ranking player ordinal pairwise comparison application information especially noisy fraction inconsistent ordering total consistent instance meet circumstance outcome cycle seek rank player rank comparison even incomplete considerably distribute affect procedure noise recover partial consistent investigate possibility clique dense subgraph graph scenario dynamic structural exploit aware citation modern efficient sort main especially modern internet application engine google feedback amazon crowdsource individual human popular netflix college economic system item worth exchange reciprocal matrix reciprocal pair aggregation asset pricing market universal perhaps noisy offset note economic triangular traditional theory prove datum ordinal numerical true system somewhat preference movie movie rather application outcome often via team line reflect intensity propose preference microsoft assign online outcome engine update level underlying inherent assume early player pairwise order perhaps popular date google rank web relevance structure web note website spirit identify user assign page high page weight page high another output order adaptively assume available reveal somewhat like event strength compete ranking mle datum idea explore social pl utility numerous pricing much relate focus refer liu comprehensive tool community another boost preference base boost machine et propose parameter break parameter moment break ranking pairwise comparison synchronization author break together parameter propose adaptively certain recover choose computer science literature every meet encode preference np pair player meet seek rank come huber row et seek rank angular aside work angular synchronization explore yu embed compare embed later traditional formulation come impose additional satisfactory result image utilize iterative rank estimating encodes outcome aggregation rating compare address permutation inference comparison available less realistic recent summarize make similarity item assume decrease along furthermore demonstrate impose contribution summarize explicit connection rank angular spectral relaxation robust ranking numerical rank literature variety graph numerical pairwise addition compare outcome game english microsoft game college game two recently state art aside traditional simple singular independent interest currently investigate art player prescribe adjust approach rank aggregation inconsistent pairwise produce finally ranking extract make angular synchronization sdp relaxation robust ranking semi supervise player also integrate information provide incomplete inconsistent pairwise player single numerical exist across comparison player algorithm aside traditional furthermore decomposition theoretically separate art method graph also outcome english microsoft computer college identify ranking extract advantage ranking stem simple may come comparison exist group translate noise allow almost recovery underlie truth point angle angular condition perfect ranking ordering angle preserve paper serial algorithm angular synchronization literature describe rank eigenvector sdp method aggregation sdp relaxation angular synchronization problem problem sdp synchronization section variation open relate propose extract comparison serial english briefly serial perform compare method summarize centrality et aggregation discuss rating make amenable ordinal singular decomposition svd popular square l player pairwise player player construct rely recover ranking another related ordering goal recover similarity summarize additional rank svd stem observation noiseless one column random perturbation ranking choose singular sign order naive approach comparable recent serial centrality explore relational develop interactive refinement aware svd research svd rank random note enyi method amenable analysis word skew decompose skew zero decomposable rank skew investigation decomposition noise similar entry matrix long limit dependency comparison whenever enough e ranking setting common highly master much investigate additional amenable light random literature relax dependent dependency nearby ranking square vertex incidence vector contain least solution rank et show square graph far reach various area theory graph laplacian systems topology et item encode outcome aggregation pairwise comparison player set member goal rank often across match player player never play word chance frequency reflect rank distribution associate interpretation science centrality design structure within application web dynamic propose temporal human imaging markov adjust applicable rating system equal player player otherwise player assume w associate start player player long next transition result denote node make sure stationary top entry score sort applicable rating system ordinal alternative win inherently win transition stationary average result winning hybrid approach centrality serial matching comparison third reference intuition behind player final solution ij two close proxy difference win player unlikely play match one two player player mind spirit design win meet balance favor assign win keeping proxy eq version centrality rating system well centrality measurement win incorporate score intuition behind win large possible winning large small possible mind win next play find ratio synchronization group estimate matrix ratio represent confidence noisy relaxation solve instance angular synchronization rotation angle eq difficulty amount subset available element ratio realize edge vertex correspond available edge measurement probably use random complete relaxation angular eigenvector hermitian soon eigenvector successfully recover angle measurement available enyi phenomenon encounter soon build hermitian element preserve angle perfectly e assignment exploit frobenius rely non follow replace individual magnitude weak maximization form eigenvector hermitian matrix orthonormal v relaxation large estimate hermitian v angle rotation angle additive eigenvector angular synchronization normalization eigenvector consider previous formalize normalize bottom unknown rounding via semidefinite programming attempt preserve angle good may follow maximization hermitian give diagonal exception angular synchronization remark size distribute alternate multiplier large problem statistic point cut finding difference fact optimize matrix estimator cholesky rank sdp ranking phenomenon noisy favorable sdp program able explain sdp relaxation bring impose magnitude constraint enforce via angular ranking suggest similar denoise angular embedding observe robustness measurement recover rotation motivating compute circular ranking minimize initially pairwise player lie comparison noiseless ordinal angular embedding rank player circle say angle correspond angle imagine player around angle angular angular synchronization around circle play role choose map circle would cause ambiguity end highly rank map unit ideal synchronization player upper circle anti direction however solution angular synchronization problem plot figure post processing step accurately underlie ordering match end circular ranking minimize illustrate step instance plot ranking induce angle recover synchronization show ranking angle angular synchronization sort label position denote associate outer vector hadamard edge offset cyclic shift repeatedly circular shift tuple circular shift tuple circular norm take account sign enyi measurement initially angle angular synchronization angle respectively recover truth rank simple angular synchronization around say offset eigenvector cyclic example wrong figure ranking adjust permutation rank synchronization rely sdp relaxation matrix pairwise comparison noisy comparison transformation build hermitian h e angular synchronization sdp recover angle increase order good circular permutation comparison output induce ranking different l glm enyi measurement comparison outlier level satisfactory able accurate solution outperform respect glm sdp comparison ordinal undesirable scoring accurate rank pair order serial algorithm base offset preserve want reflect offset one think player rank favorable player offset though could also perhaps adjustment proxy small number strongly synchronization denote sup yield initial synchronization set apply twice method rather base investigation somewhat ordinal winner game similarly across measurement record winner frequent e synchronization superiority give second order glm run ordinal measurement record winner rely difference case rate inconsistent ranking item player perhaps usa match question rank player consistent possible setup inconsistent partial comparison nonzero correspond pair item measurement inconsistent individual b prefer vote rich sciences literature date majority become option b aside appear social recommendation movie possibly pairwise comparison rating mathematically aggregation formulate pairwise illustrate slice set however usually approach parallel produce rating eq final player one ranking sort decrease induce rank par par glm par sdp aggregation run average ranking ranking albeit naive would available circle figure svd avg avg avg avg glm avg sdp far result simple bottom matrix sdp naive rank counterpart hermitian I denote graph connect system aggregation formulate q synchronization mind angular correspond whose constraint eigenvector solve constraint correspond long feasible synchronization unfortunately long cast eigenvector simply encode constraint sdp semidefinite one rank eigen perfectly sdp solution practice eigenvalue necessarily piecewise whose support induce accounting circular minimize aggregation illustrate confirm sdp significantly accurate aggregate rating ordinal comparison measurement naive serial f lot redundant subset correspond essence sdp row inner product aggregation eigenvector implicitly write involve sized sdp solve rank eigenvector sdp relaxation aggregation ordinal rank centrality plot measurement version centrality adjust rating simply win compare uniform yield accurate result naive aggregation avg par method glm come good sdp ordinal four sdp par sdp avg yield accurate especially enyi measurement illustrate sdp follow closely par par complete come next performance rather avg glm avg almost especially gap two one avg glm order point centrality l plot l plot glm plot p plot par avg par avg par avg p avg glm avg plot avg experiment suppose readily subset rank player obey impose constraint propose relaxation angular synchronization post able case rank small synchronization constraint dimensional graph application biology latter realization euclidean one subgraph know biology node spirit refer non player shall know priori mathematically synchronization formulate measurement element compose sensor noisy element offset node previously enforce hard sdp relaxation see spectral relaxation section angular synchronization semidefinite programming light sdp angular synchronization real matrix diagonal anchor hard relaxation angular synchronization noiseless sdp return one top eigenvector recover angle ranking rotation circular anchor player search circular permutation anchor induce right illustration truth place half magnitude anchor free player mind propose prescribe keep anchor player apply cyclic player ranking rank rank anchor player stay cyclic pairwise associate ss outer choose circular total denote stay fix solution preserve possible relative player recover alternative apply cyclic total permutation position rank word guarantee happen return program rank hence projection several model ordinal note measurement significantly accurate future direction concern possibility player rank sdp post constraint alternatively may spectral synchronization ordinal encodes pairwise synchronization constraint parameter available pairwise angular could also generalize highlight future believe interesting improve question may denote hermitian pairwise angular similarly hermitian offset user enforce angular one sdp synchronization case offset soft pairwise thus wish maximize form condition constraint absolute number kkt condition remark constrain weight subject sparse respectively cluster work hermitian one matrix recent principled constrain eigenvalue recent laplacian solver explore whether soft allow enforce enforce player instance sdp furthermore sdp pairwise comparison alternatively explore synchronization structural biology interesting variation could concern enforce certain I one game lose word reconstruct ranking order arise genome sequencing overlap read reader relaxation sequence svd remarkably global information h r concern outli setup multiple voting pairwise rank aggregation provide incomplete inconsistent player natural whether derive estimating consider recent van introduce class vote yu embed angular synchronization angular investigation angular synchronization use orthogonality often regime relaxation well computational direction find perhaps oppose consider point player rank south extraction collect longitudinal player roughly speak player rank relative ordering establish consider sphere parallel nearby perhaps also investigate extent extra overall robustness find circular eliminate freedom search rotation sphere result agree robustness remark exist guarantee synchronization trivially noise exact ground perfect angle synchronization necessary perfect enough angle could synchronization svd rank minimum give cost incomplete angular synchronization relaxation exist provable guarantee computationally spectral rely extensive english match microsoft college pairwise relaxation player prescribe consider aggregation rating system inconsistent pairwise produce global aside traditional amenable tool constitutes plant partial ranking player pairwise trivial extract partial fact unlike synchronization preserve order player plant partial acknowledgement author thank institute fellowship support stay berkeley fall carry grateful suggest year possibility apply angular would grant fa serial discussion literature beta give microsoft aside real life one subset uniformly throughout pairwise lot noisy rest network raise whether partial ranking locally consistent empirically preserve partial addition ranking rely spirit exist plant clique subgraph approach extract ranking rely recent multi partitioning inequality partial ranking necessarily residual seek whose union final set expansion plant locally consistent rank player pairwise one enyi consistent ranking start estimate rank result hadamard product product adjacency measurement method instance ensemble position ease visualization consist corner whenever offset induced measurement conversely offset initial measurement expect magnitude ranking player red highlight perfectly method fail note title plot ranking ranking contain correspond sub long identify subset average inter residual know science unweighted concern maximum clique np restrict bipartite graph become problem seek maximize either seminal enyi place clique efficient spectral adjacency
evaluate harmonic reference accord author label model reference gold standard another reference either make accuracy trivial well performance class division zero negative influential reference one influential show ten fold validation baseline reference predict citation macro measure table add feature use four model cf table sign statistically two tail pair level greedy high achieve model semantic add bad hypothesis task remove high title cite core section base influential model count useful combine calculate individually reference svm reference influential reference model high influential reference influential svm svm describe logistic distribution binary maximize gold label weight classify instance value predict influential set logistic indicate matlab conduct logistic assign value select important correspond likely influential regression classify reference experimental svm logistic vs threshold result reference axis curve also table curve percentage equal peak axis table figure test gap perform citation count make count cite design ignore count citation conventional citation counting count type ranking influence citation count citation count direct loop rarely period slightly citation label graph direct edge cite direct edge might influential way citation count citation citation drop paper cite citation count citation add count great try threshold reference e pair network building network cite paper paper citation mention body relative list sort weighting reference citation count cite paper body apply convert citation count function might edge higher citation direct citation paper cite cite twice add cite paper count cite two conventional citation formally let exclude citation stand large least author cite citation author conventional except least author cite cite mention influence receive paper large q stand influence index network citation paper publish paper close publish citation network impact researcher community desirable try identify statistic paper count regular expression precision manual regular line multiple author another automatically distinguish main reference may increment count numerical g use paper manually citation citation count section paper citation count reason network dataset contextual precede differently highly influence cite work citation raw unnormalized process automatically manually manual scale paper versus count citation count influence two ranking paper paper group paper rank count correlation coefficient influence paper highly cite paper accord citation paper increment count paper vector citation citation two highly cite paper count influence citation drop agree rank paper less cite group author calculate correlation index row table trend go count count different index correlation identify association identify select predict future paper paper top index precision divide show range top rank rank find two mark seven equal index negligible show commonly list engine list document case range equal otherwise author otherwise score whereas conventional precision encourage evidence weight identification author paper author paper several annotation could quantify delay paper interesting machine approach human purpose text available indexing restriction limitation extend database another without feature cite title feature overlap could cite cite similar work author moreover view adversarial attempt game exploit count survey manuscript normal review majority report influential reference approach art precision influential circumstance occurrence citation resolution convention ever mention citation number influential fine reference influence bring avoid present two influential reference hard task occurrence metric simple citation influence refinement address citation distinguish acknowledge type research paper cite influential paper investigate issue contribution significantly influential one counting cite influential reference confirm citation fairly scientific reference list alternatively count section cite assess research robust track research reference superior benefit combine grateful identify science grant thank comment importance article cite treat equal variety central variety citation label automatic good evaluate number feature conventional citation determine impact approximation count cite refine yet score article threshold publish cite treat citation hence equally error page number time likely citation reading cite illustration report cite citation create paper cite read count reference raise serious quality citation count write cite hundred reference find reference redundant attribute fraction tool determine reference inspire core contrast journal many make influential reader often citation effort linguistic near citation body pt citation text cite frequency article cite literature reference cite effectiveness identify influential reference four particularly useful cite core secondary purpose citation future author researcher reflect citation frequency well measure account cite give weight ability determine cite substantially influence attempt identify researcher solely publication compare unweighted measure well say degree precise influence researcher know write evolutionary likely influence influence good paper good decide label label influence give influential influential acknowledge wrong whether actually plausible say influence author might might admit nevertheless reliable determine influential author label prediction influence motivation paper citation rather reference alone influential one paper influential potential application citation potential follow long reference could reference familiar field nature read material prove author index citation count sensitivity could contribution survey highly cite far recommend review perhaps thing author rigorous writing methodology influential filter influence impact survey methodology put author journal less sensitive citation organization impact count also impact citation benefit influence track science interested idea noisy way track reason spread people may network link web page link research citation could could improve web page need read filter might help recommender work count field phrase back early day citation indexing reason way reason previous work article physics distinguish class conceptual operational evolutionary reference cite wrong negligible citation indexing similarity document first automate select category machine distinguish category identify via linguistic classify weak neutral positive neutral classifier rely phrase citation self citation citation text manually acquire annotate access text build machine svm na I bayes library rank superior methodology differ significant author identify influential reference use believe analyst seem classification knowledge report moderately inter agreement concern propose measure citation count benefit good assessing arise annotation thus citation provide rich machine characterization relation identify example generalize broad one acknowledge one author journal weight citation publish less paper appear journal intrinsic citation journal cite publish propose importance citation get paper frequency article original contribution closely cite relate reference times least ten common give classify concerned research paper reference create take author paper cite influential create author gold approach generate vector reference manually reference testing standard vector contribution wide pair count feature position however feature intuitively attractive extent follow subsection feature count frequently body likely influential reference five count pt count count introduction core section include exclude already exclude conclusion future section feature reference feature appear even lp similarly applicable useful influential influence originally suggest inspire subsequent expect preliminary report old early update extend greatly drastically weak create category three mean three bad strong represent factor six end end citation label influential occur citation automatically extend label cover word feature citation ten reference cite body increase citation context citation sentiment human annotation association word annotation whether eight basic trust number eight basic sentiment citation indicate citation influential sentiment ten occurrence citation citation body cite influential intuitively important seem base location citation sentence pt pt pt binary indicate citation appear appear cite time begin sentence feature next base location citation pt pt sentence reference include mean total position range sophisticated location reference influential appear solely fit arbitrarily put together pt pt citation paper receive literature occurrence metric estimating evaluate organization journal cite highly cite likely collect raw citation reference accordance convention self citation refer phenomenon know citation citation among old cite influential publication year calculate publication result non negative length reference range paper predict influential normalize raw range normalize kind normalization mining improve time reference cite cite ten reference would cite score cite normalization contain normalization let reference reference pair distinct nf ij f r f correlate achieve gold reference author help create reference direct fill online paper essential reference highly influential influence experimental choice research reference merely believe expert assess reference essential reference know need give reference without different online table usa researcher lr country france uk usa mathematics physics gold standard dataset us benchmark give paper indicate reference convert plain extract influential influential total boundary parse reference document scientific run hand code expression citation occurrence detect paper annotate name standardize publication paper body second manually correct google citation include item manually correct citation explicitly reference mention precede preprocesse annotation paper pair paper datum occurrence reference reference text reference speak influential influential research determine influence reference pearson various influence simple correlation base
ie ie ie ie w j te ie j te ie tw design bernstein w ex ex ex ex ex substituting rhs eqs bind z k k ex inequality recall random interpretation independent triple overlap triple schedule match three triple index consist triple prove twice round triple exist prove order cover triple triple triple contradiction order triple whole triple triple copy triple copy triple proof threshold since f use theoretic low follow packing simplify notation suppose draw uniformly randomly item good generalized denote leibl divergence partial ranking sd since process drop alternative marginal respective jensen inequality kl ranking draw exponential alternative rank choose remainder packing integer pack set bound ex ex last ex hold last last inequality consider note nk independent change exist subset k j b b item upper technique independently fact supremum sum supremum follow inequality inequality ex bernstein inequality ex ex follow j j ex ex ex ex appendix long account take winner directly appropriate row one sampling respectively uniformly always count occurrence remainder rely pack integer positive packing entry lemma imply ex last hold maximize hand side desire claim succeed produce desire prove generality orthogonal random notice ex ex uv ex uv ex uv ex ex uv term bound probability find term exist ex follow satisfy entry event hoeffding th variable draw sphere theorem prove concentration fix xu application recommendation management preference predict logit hide preference low rank reveal preference various form relaxation approach context interest collaborative choice convex relaxation upper many recommendation preference predict assumption success collaborative model learn ordinal collaborative preference subset obtain item tracking activity spent page rate make want user similar unseen item predict prefer discrete choice model describe typical else particular connect significant optimize offer accurately history type capture interact category predict choice item choice multinomial logit describe ranking rank preference provide rank preference item represent low rank represent item match correspond first item preferred user pool user preference whole population noisy true preference category norm minimization ordinal data two context choice provide result finite minimax information theoretic factor interpretation upper interest analyze context collaborative rank pool exist work wise propose relaxation matrix bound statistically optimal generalization similar comparison match result general sense analyze comparison refer rating ordinal guarantee remainder collaborative ranking provide collaborative analyze similar relaxation ex ex ex ex iv inner indicator event integer model collaborative preference widely similarity preference logit capture capture item small rank item simplify number analysis might differ choice give ranking ranking v underlie preferred likely nature capture reveal model describe decision alternative dimensional decision maker rank utility draw intuitively rational proving appendix notable pl special pl widely machine mle centrality quite beyond pl overcome restriction pl rank algorithm provable apply recent advance pl clustering approach heavily mixture additional guarantee mle polynomial time provable solve relaxation observe preference rank negative accord I convex surrogate optimization search maximize nuclear minimization provable extend identify convexity satisfied convex performance guarantee notice equivalent give rank list estimate class ij item way characterize quadratic norm incoherence draw replacement independently draw far treat item I apply technique analysis rank assume provide show term potentially rank hypothesis theorem solve regularization matrix need achieve arbitrarily degree directly match logarithmic range sub scale dependence although linearly sub dependence also simple special paper range advance illustrate choice wide another underlie realistic approximately formalize ball decay relatively optimize get result matrix suppose hypothesis least strict recover factor low exponent panel scale line rescale choice dimension mean scale square plot analyze illustrate actual insensitive broad rmse rmse leave plot versus rescale rescaled rmse broad convex next fundamental limit counting indicate scale degree construct packing accurately estimate true probability generalize constructive argument minimax establish sharp logarithmic prove nuclear universal numerical infimum list theorem provide interest ignore regime comparable regime theorem upper bound factor another scenario interest category second category denote present present fixed simplify notation set alternative user independent accord equivalent class sum alternative relaxation observe compare rank correspond user person subset preferred alternative alternative category draw respectively precisely draw necessary analysis appendix corollary matrix optimization corollary show sample need scale order factor degree fundamental bind ball rank theorem since identical omit suppose sample universal infimum measurable observed term comparable establish theorem factor research still slow want method provable initialization simple model pl general analytical notation conditional position p ji v v ii hessian follow ex constant least interested number hessian restrict nuclear convexity restrict strong collaborative section divided case ex ex least assumption prove desire ex singular decomposition orthogonal respectively projection onto tm rv rv topic form row concavity cauchy inequality ex ex ex ex ex notice u rv tu u rv tr r ex ex ex e absolute ex e
definition analogy vertex fig hypergraph incidence vertex denote diagonal form matrix hypergraph model hypergraph attribute correspond hypergraph correspond share attribute attribute matrix strong break penalty hypergraph provide correlation regard heat clique certainly scheme apply hypergraph utilize model hypergraph regularize classifier call attribute predictor hypergraph cut cut whose hypergraph cut keep attribute relation label normalize relation hypergraph cut denote row attribute vertex sign identical reformulate normalize hypergraph supplementary material besides measure attribute attribute obtain euclidean shift label attribute lead hypergraph introduce hypergraph preserve th attribute relation hypergraph cut hypergraph row actually hypergraph shift attribute space attribute cut find feature align space mapping whose predictor attribute correspond hypergraph substitute avoid overfitte equation positive positive matrix regularize square solve derivative zero solution give prediction project sample span vector encode predict attribute th span attribute specific introduce meaningful enhance attribute section example information enhance exploitation enhance always share attribute approach incorporate first hypergraph hypergraph relation subsection hypergraph hypergraph second construct pairwise encode attribute connect belong hypergraph heat adopt finally laplacian hypergraph equation correspond laplacian hypergraph predictor hypergraph graph denote short capture intra manifold preserve pair shifted align empirical rkh attribute representation embed evaluation associate number equal attribute assign support scenario shoot shot sample predict sigmoid normalize scale probability class label calculate posterior class sample label maximum regard annotate attribute attribute template probability classify sample attribute template class class attribute activity attribute contain annotated image test roughly attribute facilitate rest test video video around video class disjoint class testing report dimensional baseline dataset already feature represent database database already provide histogram rgb histogram sift histogram shot database database attribute prediction accuracy report know attribute approach indirect run surprising class label limit comparison semantic compare discover attribute performance provide feature notice propose outperform gain accuracy comparison approach gap attempt sample preserve manifold use unseen structure unseen enhance attribute attribute complementary attribute ccc accuracy shoot conduct interestingly significantly outperform confirm capture incorporate grouping together quality intra class structure performance dataset one testing take consider common categorization define employ equation near classifier sign accuracy cauchy apply accuracy accuracy kernel algorithm database accuracy attribute dataset originally report supplementary involve complexity consume quite take dataset time second matlab configuration cpu ghz ram attribute hypergraph attribute predictor collection hypergraph hypergraph cut hypergraph exploiting attribute incorporate hypergraph mapping attribute also extensive attribute effectiveness shot shot categorization integrate shoot propose perform shoot paper science china grant program team university grant author relation derivation hypergraph vertex instance element incidence ed respectively hypergraph return prediction attribute vertex normalize hypergraph hypergraph derivation control attribute avoid overfitte adopt control loss aim pay attribute categorization shoot learn shot categorization system pay parameter decide separately choice cm influence tune one tune procedure tune fix value parameter selection learn correspond initial replace accurately value performance database database relationship sensitive conclude value influence shoot need employ obtain aforementioned start accuracy database database big demonstrate accuracy figure similar peak optimal number also database respectively report performance however really conclude uniform database performance indicate get performance university usa edu cn edu hypergraph attribute predictor hypergraph attribute regularize hypergraph projection hypergraph align directly act attribute linear consider incorporate class shot achieve intermediate encode share across play role semantic communication show supervised attribute object description category encode task problem unseen training lot approach attribute fundamental start pay attention learn vs exploit attribute attribute categorization shot category label ignore prediction independent attribute exploit correlation exploit attribute power attribute competition framework suitable describe retrieval annotation exploit attribute preserve natural shot attribute break semantic point water attribute correlate classifier learn separation attribute water cluster although attribute utility subsequent categorization overfitte besides cluster preserve hypergraph general classifier flexible information attribute prediction computational activity attribute categorization consistently effectiveness rest organize work propose experimental evaluation classifier suggest attribute domain svm categorization show
randomize define prove last assume argument conclude conclude initial end definition strategy swap supremum previous upper bound jensen equal let rademacher rest precede expression relaxation lemma bind pick node sort step random irrespective forecaster pick expect forecaster distribute modification proposition shall simple modification adversary pick random adversary construction pick analogous relaxed simple tc condition hold k constraint hold recursive condition randomize recursive conclude q definition split v k statement theorem corollary remark define combinatorial aspect find nature combinatorial burden interestingly compute semidefinite program however enter regret rademacher benchmark trade motivate let prediction evolve social round user network observable type covariate may gender age education system predict user outcome conduct unseen stand type prediction behavior person devise aspect consideration second covariate leverage global computationally feasible edge strength dissimilarity system make binary reveal develop instance roughly mostly encode class side label class side nevertheless information measure therein similarity dissimilarity minimize weight negative laplacian latter propose obtain feasible expense start solely model shall item involve formally real assignment computer combinatorial cut unique conjecture relaxation ratio goal problem online learn allow reveal forecaster sequentially moreover identitie little except evolution forecaster particular constraint graph know ahead take network prediction constraint forecaster distribution identity side stochastic mind situation local global coherence label model appear involve yield tractable improper develop slight guarantee towards develop presentation provable future constraint arise classical rademacher complexity piece value conditional rademacher combinatorial relaxation framework suffer semidefinite prediction prediction two distinct upper minimax obtaining increase solution sense online relaxation relaxation distinction semidefinite relaxation compute relax round improper still effectively quantify increase large rounding procedure crucially multiplicative increase gap regret constant front opt relaxation relaxation statement modularity soon one small gap regret employ tighter base prove weak note solve situation offline extend additional prediction tradeoff remark spirit third tensor level tight relaxed involve framework individual stream fashion individual manner know like allow describe formalism relaxation state random guarantee admissible relax gap alternative lagrangian several low near shorthand observe set forecaster make prediction vary forecaster shorthand constraint assignment unweighted labeling cut rise item induce represent example hyperplane classify margin way indicator forecaster respect forecaster locally recognize forecaster face set information forecaster able conditional constraint edge connect reveal accord draw fix constraint accord property employ average rather nice unlabele play pool example variant generative model formation though much upper horizon concern easy incorporate let mention literature prediction graph node precisely notation static expert node class strategy draw next compute water argument outline regret course many solve might computational use pay bad rademacher topic next previous section randomize employ semidefinite program efficiently constraint depend information good formulation henceforth reason hierarchy interested labeling write ready sdp sdp relaxation vector correspond constraint first constraint program standard label whenever assignment common similar hierarchy detailed treatment semidefinite perform efficiently give maximization obtain think solution hierarchy sdp describe set sdp level regret randomize round strategy solution two program purpose analysis serve bound end define solution level expect main theorem provide convenience guarantee hierarchy via forecaster level hierarchy sdp term value correspond sdp suffice go observe solution feasible cm km use solution second since remark really refer draw improved gap stress gap prediction require round strategy round rademacher already mention benchmark optimal benchmark computationally hard improper nature around clearly thus lp gap weak immediate implication minimizing violate alternatively think constraint round problem consist sdp step sdp level relaxation put constraint us level hierarchy sdp notice penalize provide let solution optimization gap sdp context clear cost hierarchy constraint sdp prediction far randomize relaxation rademacher vector variable end sdp view regret rademacher sdp multiple satisfie inequality example second version problem introduction weight information l generic rewrite sdp integer level hierarchy opt since rademacher conclude small normalize graph well behave like near analyze set game similar problem quadratic constraint randomization incur metric labeling problem aim assign item combinatorial part cost graph cost assign space multiply encouraging item pay map singleton type separation edge otherwise polynomial provable penalize relaxation
respect bind prove supplementary let condition due achieve accelerate incorporate mini like direction mini setting stage cm algorithm k iy z ki x sgd step introduce multi key method give insight fx fy fy fx fy additional require assumption condition mirror fx k fy I fy fy k respect history get fx fy fy complete option moreover lm b b fx bind total solution respect k k analyze method boundedness compact change modification without material modify objective every assumption achieve accurate consider objective minimum satisfying k pn fw fw fx stage monotonicity notation pn fw small outer accurate generate bregman parameter condition acc boundedness complexity sag acc prox logarithmic calculation harmonic outperform c acc sag acc propose heuristic first sufficiently complexity upper estimate easily drawback adaptive technique perform inspire start third idea exceed run description website use perform well outperform mnist r quickly work tendency evaluation mini batch far accelerate gpu ccc propose incorporate acceleration convergence incorporate accelerate descent achieve support strongly strongly minimization optimization type empirical minimization smooth term follow smooth latter assume strongly strongly obvious paper propose effective sag sdca gd acc prox sdca prox acc prox gd reduce linear computational efficiency deterministic sag problem strongly add increase difficulty recently propose accelerate prove insight specific converge complexity difficult decide section heuristic determine present show norm distance generating function
instance input column vector dimension wish series convert segment scalar constitute concatenation segment mask divide trace duration time series delay couple combined recurrent neural recurrent connect state delay role network analogy connection interaction node period low filter self connection interaction pass significant performance affect interaction happen use practice infinite signal always constraint limit bandwidth signal consist fully number convenient convert depict immediately pass filter operation delay trajectory project picture rnn show grey infinitely many dynamical differential compute gradient involve sequential trace costly especially need multiple piecewise approximation combine full adopt replace duration derive combine eliminate lead relatively quick simulate measure datum highly good correspondence simulate one challenge experimentally fit exactly turn slight long period hour consequence turn challenge apply directly apply output next input physical useful feature system train systematic difference setup physical part limited serve input term offset control able cover start roughly accounting simulation argument map back range subtract due delay chance argument fall occurrence rare could benchmark task first mnist classify use essentially input segment time change classify dataset period depend initial practice use digit nesterov coefficient learn duration regularization shift digit shift include output present original example training simulate cross digits nesterov momentum momentum test directly compare noticed figure indeed result signal field group together confirm feature single period new mask ordering also employ scale column difference cause notable optimize internal offer comparison add art mnist comprehensive mnist mention website experimental datum simulate pre process dimensional energy enhance call delta delta delta common arrive wish potential demonstrate physical computer extend art performance addition perform wise informative determine randomly sequence frame nesterov momentum duration far simply meta depicted mask process strongly rescale input emphasis delta delta channel repeat scenario previously optimize simulation random present right optimize result random bad optimize comparison mention though state work value even mention please quite improvement detail dynamic less crucial mnist need suggest reason dataset second provide value act indeed therefore already secondly may pose obstacle recurrence task provide current time way mixing occur physical delay optical physical setup input system fully optimize usage dynamical system input inefficient input relatively digit optimize boost encode mnist directly utilize inherent system resource reservoir give task greatly scale effective good achieve reservoir setup step reality two problem measurement part keep mask physical hundred potential advantage optical would research future current hinge ability mathematically unclear usefulness direction improvement apparent could process could phase current backpropagation simulation end without next direction machine current physical backpropagation reservoir research argue recurrent state reservoir always use dynamical remain restrictive possibility optimize system system recurrent connection relevant dynamical accommodate appear backpropagation currently recurrence delay desirable optimize internal accommodate alternatively loop rich recurrent connectivity backpropagation abstract use design analog hardware perform signal scope realization neural result good processing capability consumption acknowledge office european agreement brain project f acknowledge les european acknowledge les grant acknowledge l develop interesting fast great parallelism digital far employ processing paradigm applicability descent backpropagation optimize encoding system demonstrate obtain work reality system common reservoir may inspire analog computer influence architecture availability computation require magnitude development allow researcher dramatically turn lead major limited effect recurrent rnn processing series account arbitrarily long implication system depend context present feedforward dependency scaling carry relevant update practice recurrent suffer important drawback first feedforward fully benefit architecture recurrent inherently nature rnns acceleration number operation rnn slow learn recent solve hessian promising attempt heuristic idea rnn grow branch employ initialize term series create still possibility physical address find remarkably complicate encode physical paper beyond work experimental strategy physical dimensional validation input
interesting bit odd difference vanish I last differ bit use min confirm format hashing bound bit simply finally reader report keep bit bit clearly good min bias bias vanish conduct svms hashing hash discard keep number generate practice typically often store obtain store bit store effective experimental figure present variety panel dash blue bottom linear curve bottom dataset test accuracy min bit interesting bit bit dashed view feature allow practitioner generate approximate scalable online equivalently nonlinear pay price linear datum might dimension use random resort model intensive interestingly develop way approximate max consist three firstly conduct extensive nonlinear svms answer min max linear kernel secondly surprisingly implementation validate via extensive real finally demonstrate svms min generalization design paper extensive min min use massive linearized hashing practitioner min max svm remarkable work consistent form record hash unbounded building large scale simple discard extensive bit essential approximate validate publicly expect work interest among practitioner like utilize nonnegative nonnegative entry dataset show build via hashing define popular term write kernel soon clear reader vision min max intersection extensively intersection interestingly outperform min max existence conceptually intersection design show affect marginally min max apply hashing hash concerned kernel example combine fashion e multiplying type convenience enforce normalization recommend max literature widely binary hashing logistic issue max mining kernel table public compare linear min kernel intersection min kernel kernel hash max remarkable min max form theoretically effectively implement need provide surprisingly completely discard validate set hash bit min max extensive present kernel machine kernel pre summarize classification kernel svm report accuracie fine figure individually high result figure confirm min typically kernel justify max application min max kernel color min max kernel dash dot linear max kernel expect boost combine multiple kernel core multiply chi kernel projection difficult fortunately consistent good min max cost paper classification figure effectiveness min accuracy mining hashing technique relatively algorithms practice truly scale application click prediction hash min vector alg procedure time clarity vector basically matrix projection say probability conceptually positive basic building approximately kernel clear briefly achieved bound unbounded alg note sample bit space mining bit hashing ignore alg encode information rigorous turn scope try observation call bit call since bit h f min r air job united states list english tailed dramatically application sense bit proposal challenging
become underlie widely perform importance connect define kronecker factor kn along fold asymptotic behavior rectangular unfold nontrivial signal without square unfold tensor conjecture us mode wise truncate unknown rank tensor tensor take form unfold k sufficiently contribution unfold phase fast decay error decay subspace directly tractable base estimator become trace propose increase threshold also previously empirically demonstrate tb recursive unfolding ordinary unfold bind norm theorem prove tb c p p cm recursive unfold ordinary subspace ideal notation summarize number denote square unfold unfold partition part index rectangular unfold denote inner tensor norm norm tensor recover plus direct classic perturbation rectangular desire result ratio view insight decompose lead add wishart gaussian correspond wishart noise speed phase exist figure perturb happen predict tb regime observe decrease unit n slightly recover rank propose trace nuclear norm nuclear unfolding define achieve trace subsequently guarantee ratio scale propose square side translate become recursive unfold k pn square unfold unfold see recover information rectangular unfolding next consider tensor contaminate exist rectangular unfold odd general kn behind show estimate two horizontal product tensor dimension tb k product go around tensor multilinear span leave singular vector exactly recover inspire mixture tensor eq kronecker dimensionality define prove km k norm semi eq old inequality last arise choose kn kronecker rao k kk k noise could tensor mode kn regularize associated input rank f solve direction multiplier n tensor residual primal introduce write ball update step finally solution primal see lagrangian multiplier write singular soft tb notable subspace dominate multiplier let tensor q estimator matrix kk restriction incoherent matrix mode denoise norm q incoherence kk k norm kronecker construction regularization scale projection p scale dimension sum rank square case conduct tensor denoise observation choose spaced decomposition latent approach initialization assume cp know true norm top constructing space norm admm measure initialization select leave measure line show line theoretically confirm sharp increase around place see see grow relative panel small choice parameter addition place plot optimistic tractable cp clearly error optimistic grow reach critical subspace reach subspace subspace hard optimization dataset semi commonly benchmark modeling five sample contain amount emission measure tensor standard spaced fed cp cp initializations space optimistic scaling put number context compare similar synthetic cp behave near cp large regularization overlap norm latent norm observe tensor noisy tensor normal intractable case linear optimistic ignore minimizer feasible nuclear eq spectral reduce cp rank tucker denote orthogonal r r k orthogonal tail second eigenvalue eigenvector derive condition consider first second theorem construct exist universal n inequality inequality last follow bind union
accuracy accuracy sign stable random svms panel panel present accuracy svm sign regularize svms tuning projection high good panel accuracy sign stable projection svms panel panel also present accuracy curve mark accuracy sign projection regularize I addition present accuracy svm mark experiment bit select clarity sign stable projection demonstrate bit mind bit nonnegative row bottom stable consistent curve mark result linear four solid label represent result correspond bit higher conduct require accuracy sign projection weight sampling curve label stable projection dash bit achieve accuracy compare sign sampling curve solid svm four curve label respectively different curve correspond higher conduct achieve accuracy compare random solid marked value curve bit high bit require much achieve bit dataset straightforward min kernel explicitly outside strategy expensive prove correctness easy except would able kernel example random accuracy confirm bit compare sign stable consistent weighted panel mark respectively sign correspond bit provide extensive large application present task mind neighbor search interesting sign projection empirical need line research stable projection work department university usa data tool neighbor sign process thus provide approximate linear nonlinear kernel arc arc provide two stable projection ii bit except practitioner sign ready scale application parameter literature effectiveness sign projection variety dataset sample comparison comparison large sign exceed typically sign projection number bit regardless favor bit consistent nonnegative advantage stable projection type problem bit sign projection core focus sign projection application matrix multiply projection context stream scan bit parameterization stable available definition eq make sense I e dense limit issue largely max variant normalize max similarity efficiently min also sample use call consistent traditionally consist unbounded make much convenient scale machine although sign bit stable accuracy important individually dataset report kernel
matrix class draw obtain example covariate variable class covariance element equal elastic net elastic net package regression respectively net adaptive elastic two performance despite increase elastic elastic net dominate elastic produce elastic net penalty relatively mistake penalty similar example ht ic ic ic hinge make lar algorithm square error time logistic article propose high efficient compute competitive valuable toolbox high classification minimize th power inverse usual second programming problem find readily try tried drive opt leave report paper associate suggestion appendix suffice check derivative e first lead directly setup support penalize art cone dimensional order overcome fine regularization publicly classification svm vector svm widely use modern classification consist seek maximize define th slack margin tune trick produce boundary separate hyperplane extend reader refer detailed explanation notice lie hyperplane boundary phenomenon generalization method discrimination find separate inverse margin point inverse margin replace generalize thereby improve whereas svm svm novel geometric cone programming solve primal interior dimensional covariate much size dimensional affect variable discard one use cause accumulation estimate classifier classifier generally classification svm produce svms svm scad net work consider penalize dimensional solve cone challenge cope associated penalty dimensionality derive combine implement package give quick observation gene panel depict take code elastic observe several sparse formulation standard svm often quadratic equivalent hinge therein poor svm replace norm eq elastic penalty lasso elastic show elastic important grid cross validation refinement elastic penalty replace adaptive elastic enjoy net adaptive penalize far consider adaptive adaptive computed elastic net trivial handle penalty propose strict net elastic elastic net focus sake presentation standardize u coordinate close solve principle form part intercept algorithm summarize cyclic descent iy ix update intercept u iy r warm strong increase implementation path warm stable grid small point warm solution warm start sufficiently kkt scale strong likely inactive set correctly check whether incorrectly discard solution incorrectly discard add survival eq update incorrectly discard back survival boost apply another cycle investigate set finish change active margin update default show strict descent use elaborate
include minimal performance like review routine retain transform retain block method dct assess image common size dct sufficient adopt retain dct q permutation performance dct trade additionally separate measure quantify compression arithmetic complexity arithmetic elementary multiplication bit shift transformation focus attention context video quantization diagonal contribute dct resort assess table display complexity also dct calculate dct employ modify approximation dct low arithmetic shift definition exact dct matrix dct matrix iii measure compression first code efficiency snr next subsection description dct dct total quantify transfer matrix angular frequency per expression quantify energy dct quadrature signal unit satisfy mathematically minimize maintain approximation dct code compression tool mathematical code gain transform uncorrelated efficiency markovian analysis image compression compress assess image degradation original version transform mathematically transformation divide sub block block particular retain employ reconstruction range reconstruct subsequently recover literature regard assessment take consideration similarity employ support adopt quality collection image instead particular robust procedure public bank absolute compression ratio outperform I dct coefficient discard ratio table show measure measure ratio approximate transform method proximity measure compare dct good transform arithmetic complexity modify compression indicate qualitative comparison show dct approximate dct propose instance although could well computationally demand code transform improve section offer comprehensive several figure proximity dct dct implement implementation intermediate result wise transform dct adopt transform block buffer circuit order dct circuit format buffer block transform signal indicate wise transform digital dct hardware flow diagram architecture cb fig modify cb section bold box realize base rapid hardware co architecture digital matlab synthesis option auto transfer hardware description architecture ff device architecture realize increase fast order delay fine realization measure hardware loop verification digital resource varied range adopt word test within matlab physical hardware device time logic flip ff delay maximum operate synthesis tool run flow estimate evident modify cb fast consumption hardware ff dct cb architecture tool environment circuit cycle software architecture convert digital hardware design generator tool lead physical implementation architecture technology lead extensive hardware hardware verification hardware language design contain transfer library verify environment mapping technology guarantee could design environment follow behavioral adopt library behavioral source cell fix consumption logic adopt figure synthesis area path delay ns consumption area display complexity adequate area throughput hand real drive force logic design clear technology area algorithm cb low power dct approximation hardware transform prominent possess complexity compression modify propose good dct speed among approximate examine implementation dct approximation tool nm technology operation much architecture optimize digital library realization way post test dct discrete require support usa engineering research processing energy consumption development approximation dct due remarkable approximate transform offer circuit digital hardware lead consumption conventional integer transform multiplication possess peak dct candidate video several dct digital architecture realize digital prototype circuit technology map nm dct compression consumption year significant system digital video device video internet protocol prominent area requirement field traffic surveillance network hardware throughput well context cosine dct video dct energy image first dct substitute two several imaging h h scheme video employ integer transform operate capability achieve performance demonstrated especially possess operation computationally dct approximation video include literature dct operation consumption operation issue approximate transform introduce dct possess complexity optimization minimize computational cost hardware implementation dct approximate dct transform dct image compression cb dct round modify dct vi dct architecture base successive call take advantage separability kernel dct algorithms video propose possibility video rapid hardware realization dct describe associate fast transform discuss dct quantify assess dct digital architecture hardware field gate array nm circuit conclusion current dct calculation dct approximation meaningful low dct totally requirement arithmetic prominent address transformation nan require dct provide low power design transform characteristic approximation exact dct hardware multiplier dct video provide operation availability fast digital valuable asset consumption driving factor quality reasonably low important system picture device video device demand extended life library algorithm dct engine dct offer master device device switch low dct storage certain alternatively dct picture quality snr video metric dct video intra frame dct information measure metric certain demand picture foreground frame say switch dct intra basis account vary picture clarity digital dct mathematical select dct matrix format contain number positive number diagonal require image quantization approximation bound power nan multiplicative
stop average stop achieve divergence eq optimum dependent pac martingale use union bind carefully choose technical recognize problem achieve follow presentation closely rest stop version favorable property nonnegative infinite whose concentration show exponential would every proof fully e statement kl give desire stop nonzero expectation pe f pd consequently outcome control thm main previous continue write w early expectation condition event go need stop analogous exactly precisely stop time involve calculation e simplification conclude thm thm definition give pac time inequality simplify state flexible usage pattern stop data consideration know determine occur frequently practice descent sgd empirically highly concentrated choose manner fundamental limit law iterate logarithm lose concern instead result uniformly time manuscript focus issue general present large hoeffding bernstein view version martingale induce fair coin repeatedly write rademacher variable discover random walk law iterate logarithm rademacher walk rademacher generalize half rademacher true upper bind capture regime interest regime encounter sense concerned failure examine dominate tt discussion focus rate follow result statement occur occur absolute definition martingale positive martingale bernstein sequence bound iterate logarithm explicit mention allow mixture evolve index dependent martingale find process tailor posterior manuscript method prove though complicate paper none sufficiently low compare inferior uniformly time iterate
strong appear upper rest main outline proof theorem ensure algorithmic approximately concentration gram correct interest canonical basis hull cardinality sphere matrix ic write absolute constant namely subgaussian analog observational subgaussian random subgaussian subgaussian assumption subgaussian identifiable particular next state theorem consequence interested eigenvalue wise approach row restrict eigenvalue state entry minimal understand integer still lasso condition ensure suppose hold matrix vector satisfy give outline denote imply general condition case bound statement hold require increase correspondingly dominate paper large suppose model independent entry satisfy programming admit f absolute outline condition obtain somewhat show restrictive subgaussian regression bind lemma lemma condition modify gram theorem suppose theorem lemma pass lemma essential state immediately reveal regard hide precisely enough fm k low denote curvature smoothness mf assume condition define lemma appear sensitivity jk requirement enough satisfie combine lemma give yield leave choose f x ty prove belong feasible event proof lemma goal section see corollary immediately condition definite let define probability c show norm isometry property estimating gram x f state stay positive probability rank let ai fm b constant depend state fm proof corollary appear eigenvalue correct row across subgaussian newly derive framework work follow definite moreover effect investigation model future extend method measure multiplicative additive study moreover current measurement acknowledgement mark support fa part nsf dms corollary corollary section rest present variation concentration stochastic cone vector let location absolute k condition fact part fact k choice gram negative general bad eigenvalue stay tx state auxiliary may independent interest independent satisfy x hold random copy symmetric copy z lemma f r f define lemma large I c c c state immediately corollary check condition theorem pt pt ann mi parsimonious fitting dependency dependency matrix wise dependency set representation variate kronecker covariance n generalize x w subgaussian analyze restrictive eigenvalue able recover model single observation response vector variate social science become increasingly popular biology process communication graphical structure recent kronecker equivalent stacking call contain column mm covariance stack covariance column row see relate kronecker encode high error decomposition matrix kronecker sum identity measure practice exception work deal match pursuit omp recover case subgaussian entry bound dependency compose subgaussian vector
alignment kernel dna protein kernel graph validate method apply challenge real datum set cancer clinical record cancer patient year brain cancer group together clinical important background knowledge essential alternatively disjoint subset series permutation object lattice partition gauss integer mode distribution sequence forget induce integrate one well define k chinese restaurant crp see instance partition covariance kronecker process invariant permutation reduce b ij covariance eq block define assign joint read one viewpoint partition possibly wishart directly dot suffice conditional assume zero n crucial calculate analytically impose severe problem derive move similarity similarity pairwise transformation assume without access rotation information replication access plausible strategy empirical row dot procedure since mean probably requirement wishart matrix correct replication subtract row vector operate center transformation relate row replication subtract x xx certain might rotation use principal coordinate kernel decomposition project axis direction principal axis estimate highly lead fix rotation contradict column normalization pairwise dissimilarity even solution might avoid invariance constant move vector similarity pairwise depict move information rotation lose translation whole matrix red reconstruction directly dirichlet cluster cluster observe suitably pre process euclidean characterized mean equivalent absence distribution generate wishart jj ij transformation general notation distributional generalize wishart observation transformation kernel follow within covariance generalize copy square eigenvalue argument conditional probability read serve fact infer wishart parametrize care influence encoding versus row conduct present novel evolve dirichlet exchangeable differ different cluster evolve exist cluster static able structure completely identity object notation section cf block size size th chain first markov notation manuscript consideration want infer adjacent adjacent expect result cluster independently cluster evolve assumption observation arrange static x describe left evolve flexible allow distance cluster cluster couple rich obtain cluster object belong element per cluster imply different right cluster centroid general iff ij j prior dirichlet partition prior partition generative sense idea forget partition denote generative point static generate label introduce integrate dirichlet partition point note define invariant permutation wishart freedom wishart distribution change size differ possible cluster case reduce matrix need obtain draw way detail k degree freedom wishart generative model te fig generative te correspond inside matrix tw tw tw cf apply mcmc assignment conjugate sampling algorithm infinite model exist epoch epoch totally object belong prior object belong table c neither e cluster probability variance one whole denote metropolis hasting choose lead old include cluster degree shape applicable scenario crp view process label switch crucial initialize block rate influence decay estimate belong distribution weak effect conditional estimate pre parameter contribution assign eqs metropolis te define complete consume part characterize row remain probability new partition determinant regard work investigate track example grouping article topic topic become popular invariance longitudinal additionally track course pairwise distance object construct definite decomposition project axis axis underlie structure hence cluster need grouping datum point already assign track preprocesse identifiability compute sampling routine require computing routine slow way point generate point per large dimension way sample draw create distance matrix draw point new sample sample draw store pca per correspond illustrative separate te burn algorithm analyze trace block trace plot usual perform take sampler ground rand te te compare evolve te model well linkage single linkage separately static te burn phase repeat tree cut nonparametric compute separately scenario static well evolve expect group single point run cluster te separated cluster te comparison pool pairwise distance single point pairwise distance across repetition cluster object point belong compute rand pool fig explain datum new shift object group together cluster cluster group true te combine pool evolve te linkage linkage separate except pool second experiment overlap computer roughly hour performance translation invariant evolve static state te gauss overlap probabilistic linkage linkage fail te dynamic model te gauss demonstrating yield directly statistical te model h te synthetic simulate highly color significantly outperform baseline generate performance te independent demonstrate repeat highly period point cluster consecutive multinomial sample gaussian large result overlap randomly compute move synthetic te significantly baseline method show fig comparison dynamic model te gauss clinical brain brain patient highly variable depend age gender average first total health record group vocabulary treat binary vocabulary use rank comparison sentence cluster use obtain patient rank patient document window year available compute patient represent patient entry correspond patient specify cluster find ten year vanish year year remain patient cluster decrease patient patient year patient suitable cope kind datum patient differ patient time course death leave document year patient appear occur flexible suited model change change every patient computer take time switch tumor status year analyze cluster detail analyze cluster would scope death rate see sentence treat combination explain ex five cluster cluster describe patient addition brain cancer death consist patient speech vision
lemma open counter pair event word position weight let random know probability p therefore
parameter constraint xy dependence nb follow regression methodology poisson nb carry initialization scheme concern em strategy large explore poisson involve sort count assign observation nb generating model simulate order integer observation come group try poisson score regression right ccc aic score minimize fail produce demonstrate methodology tn summarize ht type education line logarithm median show fm fit proportion significantly count component significant indicator belong indicator respectively count great proportion school low decrease also indicator black proportion indicate proportion white predictor inconsistent conclusion ga numerous however show tell count reason education good size education education investigate public propose novel framework response count systematically arise dataset contribution poisson able select determine count suggest count disease disease number city count finite involve come framework carry criterion demonstrate variable responsible interesting trait datum people treat count probability event give event regression linear variable covariate via define effect exposure incidence subject exposure responsible determine define mean count violate say binomial poisson binomial come treatment observe count count come mean fm modeling within past unsupervised task principle mixture treat concept useful heterogeneous appear model extent largely covariate covariate heterogeneity context throughput sequence online develop literature criterion aic mixing give force proceed information binomial hypothesis statistic assumption confidence level aic prefer choose
root leaf subtree root child compute pos tag th child head word embedding representation embedding concatenation operation relative sentence relative distance map dependency tree pos tag embedding composition matrix pos tag capture syntactic example noun noun activation dynamic child fix information pooling subtree root node figure tree parent model phrase nlp task parameter nlp interaction head child semantic plausibility subtree dependency parse node child simple correctness final terminal parse head correctness subtree pos tag goodness tree sum tree rank output base criterion dependency score count eq set final objective minimize plus score incorrect decrease gradient direction subgradient use diagonal step rate subgradient use discriminative parsing dependency parse third generative ranking list forest dependency parse substantial parsing give sentence combination hyperparameter base sentence train discriminative way base tree approach dataset chinese use score english split development tag development automatic pos tag way pp bar pt symbolic md ylabel limit true style font legend anchor north style sep black pos result perform slightly good base limitation add line initial discount also experiment final previous oracle achieve minimal engineering rank affected base overfitte although result also work large think large increase greatly large need multiply output experiment show achieve significant improvement add list dependency parse neural parse dependency parse tag arc make action transition parse nlp propose compositional rnn image difference node compose subtree parent pooling position parent convolutional vector recursive probability utilize treat tree regard recurrent unlike discriminative base difference compute besides also sentence dependency address level phrase dependency capture syntactic compositional phrase architecture parse tree therefore nlp effort engineering dependency paper regard semantic sequence length fix nlp text research limitation investigate thank anonymous valuable comment national science foundation china program technology laboratory processing school science road china edu cn problem phrase dependency dense convolutional syntactic compositional phrase dependency convolution pooling layer model informative discriminative list parse tree effective improve art dependency english chinese dataset discriminative much dependency parse million feature ability complicated distribute semantic extensively language nlp semantic phrase help address generalization representation parse dense representation complementary focus keep unchanged parse optimize task important unseen phrase vector parse parse recursive neural binary parse parent child node tree phrase dense propose convolutional architecture compositional phrase architecture parse tree dependency parse give dependency first unit interaction child recursively output input parent output length illustrate phrase red contribution paper summarize architecture phrase sentence dependency regard sequence length jointly nlp classification complicated child pool rank parse parsing decision experiment model briefly neural architecture language nlp phrase sentence sentence every context classical layer whole sentence binary structure length leave word recursively length whole multiple tensor product figure illustrate rnn branching triplet triplet either word node give p ap bc p bold font letter compositional syntactic compositional compute plausible syntactic parent high scoring standard parse apply recursive phrase rnn enough
eigenvalue number way eq unique function maximum integrable therefore martingale reduce proof mind proof correspond estimator form change h equation scalar product rewrite auxiliary integral unique get eq integrate get du du outline correspond however case calculus integral fractional fractional integration q constant st kind transition equation help operation get equivalent right rewrite rewrite equivalent h v arrive part formula rewrite eq thank helpful discussion theorem maximum two independent integral equation fractional brownian index require case develop tool mixed process demonstrate likelihood regime fact stochastic calculus support center possibly interval space worth mention also form integral verify step extend define square integrable h beta map isometry hilbert wiener process put well call fundamental martingale square martingale reduce existence uniqueness concern existence st prove fact unique tp fractional integral simplicity formulate let h x brownian motion square observation process wiener drift x combination consider proceed probability topology probability measurable set measurable functional e x give wiener independent also use center measurable q still projection alternatively arrive exist constant change line apply inverse
variance closely call vary model explicit view pose used remain network see test ability generalize pixel make flat rotation represent dc achieve image include previously unseen transition intermediate pose seem sort angle angle unseen flat train deep graphic interpretable graphic static image utilize convolution train use force face encoder latent decoder network never dc component arm see example less handle complex deeply handle large object architecture spatio utilize motion visual move handle recurrent network also replace decoder hope motivate interpretable representation variant access fellowship helpful discussion science intelligence laboratory mit brain cognitive mit microsoft research uk edu mit edu microsoft edu paper present deep convolution graphic rotation convolution convolution train use encourage pose image pose qualitative model engine remarkable automatically hierarchical cnns boltzmann generative successfully relatively little characterize et al consider propose theoretical irreducible come open question work theory representation work representation abstraction represent happen world graphic go compact description graphic typically fine transformation pose compactly identical al graphic code align recent work graphic probabilistic latent et beyond stage encoder domain decoder produce interpretable graphic reproduce interpretable complex transformation rotation variation hybrid encoder transformation object plane rotation direct graphical convolution variational bayes encourage representation train mini batch active inactive transformation value learn function texture inactive automatically create quantitative efficacy convolutional inverse graphic dc encoder decoder autoencoder consist decoder neighbor training produce consist pose texture shape gradient back force dc show mini batch inactive transformation g face light pass etc graphic graphic model propose representation unlike relatively recently et al feed forward neural encoder serve handle grain geometry face relatively extend apply jointly train utilize convolution de convolution encoder respectively convolution massive increase recently use cnns object specific supervise truth directly image task amongst encoder decoder comparison proposal intermediate variational encoder spirit assume representation piece spike comparison encoder interpretable graphic graphic rely work work use graphic depict attempt face camera source target mini batch scene angle source might occur generate batch scene hold face consist many different face pose property mini intrinsic stochastically sample batch reflect identity desire unchanged batch hold neuron force variance batch full change neuron receive reconstruction close likewise neuron proceed representation make gradient figure correspond angle source intrinsic mini batch minibatch representation calculate entire output gradient pass continue backpropagation encoder representation dimensional intrinsic work work encoder decoder neuron force batch change neuron value gradient encoder put variation qualitative capability dc latent smoothly leave smoothly leave unchanged strong face encoder transformation encourage neuron wish transformation mini dc train inactive act encoder point invariance close care care face matter way scale small reconstruction qualitative capability learn dc original light neuron change dc train batch generate face shape texture pose meta momentum decay also perform varied
learn potentially overlap neighborhood denote bias feature produce encoder g w induce pool learn although recover sufficiently reconstruct correspond possible reconstruct reconstruct group sparse activation additionally penalty activation include nonlinearity critical inference convolutional dictionary show architecture convolutional pooling conceptually identical connect described york ny new york ny edu classification rely coherent video datum feature unlabele adjacent exploit train encoder establish connection coherent neighbor likely space example video likely adjacent frame assumption feature introduce temporal coherence feature slowly discrete adjacent frame degenerate degenerate mapping informative input discriminative criterion pairwise geometrically weak high propose term prevent constant act preserve priori like preserve possible optimal extract slow
note last equality z schwarz use invoke b inequality proof use mp precede small sufficiently conclude eq equality next invoke ai invoke entry moment due satisfy e adaptation precede difference satisfy precede martingale satisfying constant subgaussian increase positive constant every j uniform l inequality last inequality concave cb independence across invoke constant precede array triangular measurable plain definition inequality dynamic dynamic regressor fix show one uniformly valid asymptotic allow conditional important time allow band contract dynamic widely economic social extremely differ unobserved repeatedly dynamic however inference model presence lag effect regression seek explain economic growth determine factor panel big explanatory control result explanatory arise control form economic panel access reason decide investigate subset control one propose inferential procedure dimensional progress decade popular lasso research however recently possess estimation independent plain linear treat establish oracle dimensional datum study property penalize gmm high arise panel datum coefficient shall involve panel consecutive individual dependence panel reason assumption panel case correspond approach static panel consider effect nuisance parameter time straightforwardly difference model error correlate stand effect intrinsic interest hypothesis simultaneously involve side explanatory truly zero classical severe impose vector effect reason sparsity instead total magnitude effect control variation expect albeit dependent interpret percentage change fix remain percentage variation control vast covariate sparsity actually variable deal assume structure group dimensional sparsity inferential regression lasso invertible gram context regression inverse covariate suffice inverse weakly entry need contribute group behave differently joint asymptotically three type parameter increase consistent robust conditional panel consider error asymptotically uniform subset show band uniformly rate type size organize introduce next robust seek sample construct contract parameter section carlo defer denote norm unit column entry dimension maximal cardinality index kronecker product fix exist constant maximal minimal give rewrite p np confusion however argument tend often assume observation source lag next write compactly compactly something linear difference heavily propertie block properly gram impose sparsity oracle inequality technique fact get expression instead characterize equation variable hand side may reasonable heterogeneity high dimensional effect logarithm fix effect unobserve factor motivate sparse weak instead infinity weak sparsity strict exceed work handle define equal start point panel differently observation solve weight probabilistic scale different must break step turn need inferential procedure impose data expectation martingale respect error martingale considerable high furthermore need distribute individual rule term conditionally terminology lag introduce scale matrix singular conduct suffice compatibility type tailor define integer eq restrict make effect sense write diagonal really submatrix bound away assumption trivially moment impose compatibility standard literature various version investigate subgaussian plain static common covariate assumption dynamic panel generate completely subgaussian property behave wide defining inequality inequality least q valid well use end inference novel inequality allow grow even upper go correspond oracle panel technique quadratic equation analogy inequality linear inequality finally independence concentration sharp mixing restrict increase fast conduct observe convex belong subdifferential multiply leave q would invert inverse shall oppose term add back define sparse lasso limit consistent presence interested asymptotic basis interested show discussion work regression high importantly row regression properly j z jj j kkt subdifferential shall rigorously kkt write eq z eq inequality require argument needs understand construct diagonal removing row submatrix row remove column th remove except multiplicative row sparse generally weak sparsity sensible population coefficient shall define write replace row respectively reasonable assume I section bound subgaussian imply zero translate sparsity impose row entry equivalent justify dynamic panel reasonable mostly conditionally adjacent conclusion important sparsity part part impose term regression define large establishing asymptotically induce however parameter limit uniformity result limit estimator reduce low corner follow tn motivated following establish uniformly bound need order allow note allow sample hypothesis interested simplify hypothesis correspond assumption necessarily necessarily moreover cardinality provide stress total much allow consistent hypothesis involve inference extension relax inverse exactly furthermore vary dynamic depend relate inference static panel classical setup interested inference gaussian equal exist illustration variance equal hypothesis similar reasoning hypothesis involve asymptotically convergence accordance straightforward usual asymptotically inference restriction differentiable usual even impossible involve weak expense show band contract precise satisfied pi nj nz percentile standard let coincide convergence consequence reveal band uniform band important z guarantee irrespective guarantee coverage value achieve desire clearly confidence optimal particular narrow contract contraction fast contraction show band base band one contract worth non inference investigate calculation carry formula naive monte replication estimator square rmse procedure monte replication construct lag regressor interval involve parameter construct evaluate carry level confidence nominal coverage regard plain lasso report dynamic burn generate datum generating root lag disk toeplitz th entry dramatically covariance conservative report precise form I lx turn calculation reveal construct unconditional finding drive plain change unconditional carry non entry test power follow consider replace zero entry theorem variation consider baseline far experiment eight replace freedom rmse lr lr lr dl b l dl l dl h square variable encourage base see due wide confidence assessment uncertainty size superior least fix effect fix assume actually band superior coverage rate procedure interestingly affect result less expect towards lr lr ls dl l dl l dl turn estimation nominal price band wider believe band due accurate uncertainty narrow band experiment assumption accurate oracle one rmse lr lr lr size power b l dl ls increase compare oracle band nominal rate band test oracle procedure may add rmse lr lr lr power b l dl l dl surprising go furthermore band base confidence band belong left hand variable become narrow fix method similar exact relaxed experiment well band become wide experiment rmse lr lr power l dl ls dl l final add tailed covariate high set table error increase band increase sparsity fix roughly unchanged procedure addition conclusion consider dynamic panel increase test hypothesis simultaneously towards valid matrix inverse next band contract contraction simulation assumption extend subgaussian covariate error allow rule panel uniformly subgaussian assume furthermore subgaussian root outside monotone exercise page f tu p assumption make proceed theorem shall event event valid minimize lasso yield use trivially hold positive bound event compatibility equivalent estimator satisfie constraint introduce compatibility valid hence upon combine tx yx quadratic right root second right minimize desired formula root namely oracle n arrive uniformity enter oracle deterministic oracle norm random usual norm provide let positive entry th l xx lag lag ns l nz l ny kt tt ti conditioning zero martingale argument martingale arbitrarily conclude assumption calculation natural precede positive satisfying definition hand satisfying define thus assumption block I
observation observe shall natural bias serious validate dataset optimistic estimate validate define predictive performance complexity penalty rigorously perspective estimator criterion ic historical information require code although promise problem unknown approximation originally aic encode aic information true green green dot truncate add due content encode aic aic lead consequence aic lead information limit continuous freedom appendix derivation detailed meaning exactly approximation subscript understand result aic expression write play role illustrate encode mle goodness dimension increase encode optimum aic aic plot red powerful difference model parameterization small great complexity simulate black aic estimate initially accurate rather cross aic aic selection clear immediately large analyze computed evaluating eqn invoke correctly predict clearly significantly figure aic dotted model dramatically true complexity clear key assumption aic mle normally context aic fail sensible fail often propose base selection frequentist criterion frequentist analogy aic predictive dataset draw unknown compute approximation parameter complexity index analogous hypothesis estimate complexity construct call frequentist aic evaluate minimize small true expect good asymptotic approximation rather decrease generic feature nest parameter space associate see complexity parameter identifiable figure motivation describe analytic complexity complexity increment specify th write slope figure show identifiable justify identifiable justify clearly justify add predict plot cross entropy parameter entropy understand identifiable regime entropy ambiguity cross parameter complexity analytic equal ambiguity appear nest un nest complexity expectation chi freedom harmonic degree harmonic derivation give cumulative write model nest nest un nest write distribution relate model clear eqn slope slope closely q interpretation use inference large true exactly regular absence approximation accurate aic inference include statistic clear interpret test advantage traditional frequentist intuitively clear assign hypothesis hoc frequentist base specify automatically ad hoc confidence suggest bring acceptable example complexity compute simple generally necessary could result explicit tractable realization resolution beyond note probability offer promising model context exist na seem intuitive strictly decrease therefore probability model expectation dataset cross rule observe rewrite probability replace sum naturally measure cross appeal principal maximize equivalent leibl measure expectation identically think metric since distribution drop rhs equation compute relative minimize information maximize minimize mathematically great insight might significant approximately gaussian mle matrix expression normally fisher approximation independent term eqn distinction information estimator unit modify one derivation analogy derivation goal execute start distinction degenerate model true since minimum degenerate cross entropy understand follow strategy aic taylor expand information order perturbation follow definition nest un model true entropy remain perturbation acknowledge term term drop perturbation replace fisher simplification eliminate cross eq expansion minimize result write dependence explicitly clarity analogy aic harmonic normally precision equal perturbation chi parameter keep chi square correlation adjacent minimize entropy therefore choice chi need chi square entropy complexity define note chi square specify square must compute eqn eqn perturbation product line relate eqn eqn heuristic criterion aic apply frequentist analytically many context aic understand unbiased therefore limit exhibit aic like instance observation unlike bayesian hoc information inference theory justify narrow predict finite observation parameterization degradation mechanism degradation intuitive accurate fit reduce parameter ii high respect qualitative goal realize quantitative minimization maximize upon independent succeed context failure criterion frequentist criterion aic broadly divide description establish connection frequentist approach reference reference application application approximate model probability write absolutely distribution model
agent turn policy heuristic trajectory exploration tree agent accommodate policy rl horizon balance avoid confusion refer primary distinguishing encode exploration denote choose one policy uniform success agent execution evaluation next evaluate large control policy convert episode high reward although vb vb special dp close well dot solid line flexibility explain paragraph cr unknown r time time c box c advantage variable episode accumulate reward average test episode collect episode collect agent follow agent procedure approach use report exact controller node number perform suffer maxima issue see bar level employ uncertainty policy infer controller robustness low value em traffic control reward leave controller algorithmic iteration several monte policy rl approach exploitation reward iteration summarize category optima algorithms policy allow flexibility fix indicate trade time periodic generating model allow trajectory policy outperform produce low near solution produce solution previous rl worth discuss rl number traffic agent control traffic intersection agent locate except code compare traffic direction heuristic generate fair addition examine exploitation initial behavior exploration exploitation produce high solution scalable framework decentralize exploration exploitation reinforcement experimental benefit infer scalable domain problem size allow quality policy acknowledgment support us office research award nsf award variational standard divergence minimize low n decentralize vb update equation follow vb inference maximize joint vb n w r solve give reward allocate step vb k weight compute construct vb gamma issue non maximized way operation derivative form difficult search stick break connection characteristic useful detailed reader stick measure disjoint associate weight stand generalized dirichlet truncation density use keep equivalent indicator argument zero backward ki pz n ki k backward message compute recursively impact sequential batch exploitation five batch update body thm thm definition remark thm question rgb maximization decentralize size converge maxima far consider construct stick prior lead controller variational available demonstrate several showing algorithm art decentralize decision sequential numerous exploration control control make decision stream observation action dynamic decision generally belief decision making make planning horizon optimally scalable infinite continue infinite agent represent scalable em show learn trajectory without know work affect unable sub yield sub bayesian nonparametric controller previous assume centralized execution decentralize accomplish offline decentralize controller stick break prior controller posterior trajectory recognize prior contribution algorithm directly operates shift simple bayes net framework moreover vb agent episode vb agent problem size scalable domain simulator realistic adopt learn reinforcement knowledge policy able propose method describe relate tuple action agent joint action receive world state rs receive discount agent agent observe local observation maintain observation policy local policy infinite horizon belief objective bs maximize tuple n denote controller action policy notational cardinality joint agent obvious thus take history agent controller choose problem transform introduce binary reward pr pr net optimize policy representation vb problem bayesian stick break used specify structure formally decentralize stick representation index notational simplicity define dirichlet stick breaking I specifically base rl nonparametric decentralize domain world plan previous method employ hide hmm controller conjugacy hdp impose therefore storage employ gamma hyperparameter bias among node reader note process encourage compare dp sparse transition allow correlation break always episode agent vb low lb lb use hyper return policy controller govern dirichlet multinomial variational accommodate unbounded node apply prior agent stick break construct prior hyper application increase replace normalize w ki equation theorem vb prior reward reweighte improved step improve
tx x algorithm track particle keep track come past history use appear leave kind kernel particle extension localization guarantee consistent go forward sign yield un p dx semi growth map py infimum expansion tw go direction worst could grow basically particle contraction pf fix frank wolfe translate rate error kernel wolfe obtain particle explicit particle standard filter depend rather distribution rate would translate tb start investigate mixture give different gaussian fw though rate empirically significantly increase method remain application kernel filtering use frank wolfe quadrature monte quasi system detail experimental dimension switch govern series mutually order difficulty filter kalman filter run kalman albeit nonlinear closed density run pf particle reference batch system allow exact filtering fw compare pf resample carlo particles discuss assess compute rmse filter filter along quantile run nonlinear benchmark improvement somewhat difference see upon bootstrap pf mmd section propose algorithm six pose consecutive frame motion estimate pose model comprise velocity acceleration orientation bias position currently number rao filtering extend kalman filter remain evaluation modularity approach simply simulation setup fw run fw pf pf use run method fw comparison run time reference pf average second give assume know zero natural available unstable basically keep fw give improvement bootstrap pf particle error particle role focus investigate gain implementation evaluation update online scale particle pf spend step particle bottleneck ghz overhead fw fw fw experiment practice fw pf particle fw particle still pf filter particle performance quasi monte modular particle filtering filter particle filter future future work include convergence theory acknowledgement centre european project supplementary improve pf rao use rao tractable pf assume system comprise conditionally transform standard normal argue sort discrete mixture accord transformation naive gaussians store nice sensitivity implementation detail synthetic matlab control toolbox observable pair observable pair correspond observable main text report figure plot obtain stay carefully current effective optimize mmd many particle filtering error kernel matrix nonlinear mmd numerical precision ask increase translate reduction filtering error particle fw seem suffer big tb tb py px nuclear q py x f dx dx hx hx dx f hx dx upper expansion gaussian thus big decrease rhs thus take quite norm px conjecture result n ny py nx compute fourier transform representation integrate dx xt xt w dx may w w change variable w w n c w e dx w b w w w perform b w w n continuity small small less un thus linearity related norm scalar multiplication mmd term frank wolfe z finally control normalize repeat argument convention back quantity fw rewrite normalization constant go worst grow back would interested note explicit example really say c hand whether f without disadvantage presence quantity explicit though close upper repeat working quantity similarly get tw tt extra preferred tight remove really unit augment hilbert rkhs frank wolfe analyze analog instead fw mmd vertex extend step wolfe size gaussian experiment give step objective mmd fw vertex r g fw ball radius center eq fw g crucial interior yield giving imply become rhs convex maximize get induction thus strict translate back appendix say fw subproblem wolfe mm proposition radius unlike conclude seem might infinite thus worse previously around arise bad reference definition definition frank procedure integral reproduce rkh potentially convergence monte integration special replace particle filter frank wolfe quasi monte emission additional localization improvement quasi explore idea approximation constitute involve eq space set beyond sequential monte carlo inherently challenge computationally common relate robot vision synthetic evaluate image pose filtering solution reason bottleneck arise improve complicated allow filter leave standard inefficient option contrary nearly acceleration develop toward class filter bottleneck computation upon particle avoid arise simple monte bootstrap build appear frank fw quadrature particular gaussians past convergence preliminary give accuracy particle filter particle integral belong reproduce kernel pointwise reproduce property feature rkh briefly integral associate empirical mass independent use unbiased variance estimator analyze cauchy bounding approximate central quantity quadrature rule act standard rbf fact refer regularity object lie closure hull finite insight frank wolfe quadrature wolfe algorithm iterative algorithm optimize general banach iterate obtain vertex g k iterate suitable high decade old survey hull vertex run frank wolfe yield g k I optimization reduce negative frank wolfe quadrature rkhs g k maintain px include normalize tx propagate normalization constant frank wolfe rule shorthand adaptive quadrature rule pair fortunately insight central quadrature call fw vertex search non optimization exhaustive fw call sample wolfe fw vertex search show material add bad return quadrature choose search hereafter fw refer always weight alternative previously visit vertex wolfe hereafter refer k quadratic simplex min active reader frank wolfe quadrature fw guarantee summarize follow infinite
later construct give inclusion construction paragraph trajectory follow ball surely subsequence rescale choose fix inside example could linearly trajectory force jump recursion conclude tn tn tn q trajectory sequence ball around origin tm mn tt tt rescale noise converge surely sake completeness show independent unfold recursion expectation norm use eq letting rescale convergent consequence assumption claim assumption need begin show rescale trajectory sup l mn rearrange inequality get e kt dependent purpose fix solution clearly lemma inspire follow limit coincide limit sup thus compact let n xt x since map observation early deduce weakly lk nm lk xt xt dt xt lk dt nk xt nk bound sake convenience h nk nk xt xt contradiction subsequence k xt sample ready stability stable nothing hand xt tt assumption ml explain xt origin globally asymptotically stable loss generality assume martingale sequence loss assume work bm bm surely converge possibly compact invariant recursion early drift begin prove map follow lipschitz set xy yy hx nn ny nz n hx ny hx bm satisfy approximate satisfie map cx cx cx h cx let xt lyapunov origin proof n yy h x h nx h corollary approximate almost addition close connect since true even require special happen give previous consistent find list solution value initial refer proving outline prove length point prove lemma proof work assumption bound subsequence non empty every hard dc cx dy containing neighborhood h x h cx upper xy yy xy h exist linear functional convenience claim ax claim true proceed pick ax n contain n nk nk nk nk c nk nk nk nk h nk ax nk n nk fy contradiction ready modified retain assumption let h cx update h stochastic inclusion statement cx early stability iterate prove identical manner invoke iterate converge set explain immediate stability follow question sufficient stability lipschitz x h xt subset define cardinality show recursion limit exist satisfy iterate connect generalize assumption recursive chain nx c c nx h c section show version convergent omit proof iterate stable converge xt relax fall exponentially prove iterate follow corollary set origin similarly nonempty change make statement differential inclusion remain theorem aforementione explain deduce trajectory fall around origin rest extension mean value set recursive inclusion immediate drift problem corollary exist average stability two sufficient stability one natural theorem recursion lipschitz function recursion refer limit asymptotic detailed exposition subject show recursive specifically continuously iterate compact sometimes refer iterate word iterate guarantee stability several discuss show dynamical system extend iterate map reader martingale refer see special cardinality stochastic recursive overlap respectively stability case accumulation present unified take care aforementioned extension relaxed notation connect set prove stability stochastic outline work boundedness map differential inclusion let iterate sufficient iterate
consist modify present dataset patch extract task versus separate two disjoint piece take patch patch patch transform norm binary build subset dataset class image position multiclass image handwritten set image image norm class cifar parameter unique technique low burden threshold quantization run reader understand one one test use build methodology substantially accuracy considerably reduce bit multiplication show b c bit precision point throughput present differently classification accuracy ht cifar error report cause random gd portion mini sample train affect half range highly gpu cifar substantial half bit ht feasibility bit shift instead multiplication slight consumption expensive application technique technique enable single operation precision almost computational throughput worth note nevertheless accuracy dictionary set perturbation training increase generalization unseen investigation learn adjust integer multiplication bit shift dictionary thresholde entry run introduce substantially reduce load cost tested increase technique power consumption partially improvement high education allow access nsf mr mr code last detail von discussion classifier learn soft threshold shift method modify energy dictionary apply soft dataset indicate solely sum shift integer valuable implementation throughput decrease energy consumption cost enable instead bit almost double throughput resource trend feature feature overcomplete dictionary learn dataset classification drawback application computational resource drawback learn approach multiplication soft realize due parallel multiplication hardware much work consumption explore derive reduce four group image raw replace costly operation integer operation hardware accuracy dictionary classifier near multiplication bit shift slight dynamic image reduce quantization range value train vector multiplication slight decrease technique algorithm name soft reduce substantially substantially simulation test technique last sufficiently general different use multiplication extract extreme dnn paper approach learn valuable embedded consumption necessity operation architecture image briefly representation signal identically sparse pure good learn soft differently map simplicity process jointly hyperplane n loss prevent overfitte extract sparse follow classification class return reader deeply understand simplify finding technique purpose operation approximate relative power md show behave begin train atom build multiply open evaluated set figure ht classifier representing include eq penalization solve constrain problem iterate generate compute iterative method computing require dual constraint lagrangian problem lagrangian lagrange solve gd method upon gradient evaluate n establish modification comparing
sample strongly completeness simulate classifier result data mapping predict object old object ground label classical machine vision bioinformatics micro formalize decade work produce many e discriminant analysis lda artificial vector tree attribute instance justify image object g face variety approximate affine camera coordinate point move object lie subspace rise study subspace important branch compression know subspace contain adopt class instance obtain subspace discrimination variant version way show subspace face model handwritten speech classification explore theoretical justification interest justification know variable independently functional whose minimal rule predict minimal actual classifier small spirit fact function converge word large misclassification matter knn boost consistent certain condition consistency linear comparable consideration complete experimental classifier rest description suffice integer restrict classifier find subspace dd form singular class n desirable property rule say strongly consistent rest obtain follow theorem classifier variable center reveal prediction condition weak result svm boost simple linearity important easy analyze order good interpretation limited foundation generalize therefore long point bayes rule partition optimal practice form form observation rule classifier nx q p main proof basis consider plug difference fix therefore show due estimation mle proof evaluate lda svm demonstrate result show serve complementary perspective reason note significance study classification simulate brief find sample experiment subspace disk angle class recognition chemical region determine find machine repository dna find recognize dna sequence retain dna represent three class neither database digit service recognize digits image signal acoustic original evenly impose recognize collect experiment dimension na na na news real repository collection news principal explain reduce subspace carry matlab default toolbox multiclass realize author split subset split record pt htbp lda dna c dna news result know comparable lda meanwhile computation roughly lda lda require covariance positive reason ambient news restriction review simple model subspace prove prediction show result especially
datum fuzzy flexibility binary fuzzy value hyperplane optimize fuzzy hyperplane apply propose artificial world obtain fuzzy machine fuzzy hyperplane idea introduce classification success character outperform precisely bioinformatic version square svm task accuracy suffer drawback fully assign strictly assign class many consider main concern assign importance degree moreover classifier ability approach cope fuzzy fuzzy analyze fuzzy concept operation introduce offer capture inexact fuzzy unlike phase propose treat datum point importance apply fuzzy membership point fuzzy fuzzy bias fuzzy rest include model fuzzy quick review main behind training point follow solve geometric interpretation depict toy group hyperplane hyperplane weight sample geometrically toy hyperplane close class indicate class hyperplane slack one desirable penalty employ equality preserve accuracy constraint theoretically four standard hyperplane hyperplane explain fuzzy improve notation sample sample represent slack equation transpose application belong uncertainty elegant cope fuzzy final degree influence influence fuzzy assign application induce discriminate class vector symmetric triangular fuzzy fuzzy fuzzy component fuzzy define fuzzy fuzzy equation inexact eq slack rewrite rewrite equation would appear rewrite substitute hyperplane hyperplane equation q fuzzy find hyperplane hyperplane definition find point hyperplane fuzzy data fuzzy hyperplane n nx fuzzy distance fuzzy hyperplane membership determine fuzzy hyperplane fuzzy hyperplane hyperplane accuracy experiment environment pc intel ghz gb ram false false negative cross methodology focus first svm record record determine circle circle show result algorithm paper svm accuracies hyperplane responsible line section exactly amount high responsible classify two line fuzzy nature line discriminate data b dataset uci machine repository heart cancer dataset represent detail four accuracy note version two algorithm version propose non version meaningful lc lose heart cancer
fix shape sufficient transpose see univariate univariate identically distribute iid exponential iid standard model family extensively sequential hypothesis widely literature reconstruct example family matrix column matrix row statistic define ty cumulative call statistic short denote give iid belief vector assumption belief reconstruct natural statistic mapping k component third definition fourth involve family family rule k completing suggest belief sufficient statistic determined dimension hypothese interpretation essence prior period dimensional embed statistic space dimension sufficient belief statistic bayes namely assumption reformulate minimal statistic space belief truncation explicitly oppose become compare contrast two acceptance yes apply statistic review however scalable testing simple assume variance concern prior belief one cumulative optimality equation illustrative describe belief belief lie fourth panel clearly path remain next describe sufficient statistic acceptance interval low action decision independent acceptance interval implement desirable increase heuristic draw sharp multiple hypothesis period period natural state dynamic programming conjugate conjugate fact arbitrary flexibility solve belief flexibility collect figure acceptance interval figure increase long require go high dimension chart clear gradually increase observation square suppose natural difference identical still different matrix statistic ty x obtain correspond mean give prior dimension example figure horizontal axis cumulative vertical square improvement provide policy development asymptotically sequential observation identification zero case hypothesis distribution zero series posterior study compare cost combination simulation less average cost percentage table error optimality consistent optimality policy hypothesis difficult another response easy satisfactory range matlab intel core prohibitive application adaptively multiple mode diagnostic power hypothesis decision hypothesis decision maker one hypothesis terminate choose mode sampling true hypothesis generate observation know decision prior family fy k sequence sequence clearly follows denote rank full rank sequence sequence brevity belief sufficient statistic control beneficial use many action systematic observation matrix low sufficient mode account acceptance region discuss yy p dy variable test multiple maker take multiple stop sample period observe average x mf global let become unless take generalization widely discuss structure policy difficult implement devise efficient scalable without assume conjugate sufficient statistic dynamic natural standard belief approach desirable quick natural use variable learning problem often sufficient chapter need sufficient solution become commonly table observe dimension sufficient especially situation policy increase curse draw intrinsic dimension exceed family commonly moderate computation illustrative suggest also extended hypothesis test sequential alternative hypothesis observation maker one accuracy goal quickly translate minimize expect incur incorrect involve trade arise vast include security monitor clinical target recognition study sequential hypothesis test maker identically iid statistical one strongly stop accept observe policy implement hypothesis policy hand numerous asymptotically study policy note review view multi observable identity generalizing stem curse dimensionality dynamic hypothesis scalar one vector belief increase make come exponential family central theory dimensional hypothesis belief many binomial reformulate find moderate even hypothesis solution rise region opposite standard belief experiment substantially suboptimal delay parallel involve mean contradict specific method scalable hypothesis grow reconstruct belief technique observable limitation tie suffer curse see come exponential iid observation function distribution hypothesis observe alternative accept make new stop multiple hope identify desirable quickly respect historical acceptance
bn e converge model sample structural every dag ss subgraph super vertex average degree super around feasibility bn vertice cluster search improvement hamming benchmark report time prohibitive great hybrid cb ss thousand vertex skeleton contain true network extra extra bn attracted recently control e false compare hybrid ss allow fair difference ss max min parent child learn variable subroutine parent child combine incremental divide cb candidate empirical conduct pc min hill various formally bn tuple direct acyclic dag bn parent statement extract graphical note exhaustive intractable separated separate bn converse denote parent common unique bn handle cb identification neighborhood scalability cb systematically independence independence fisher test decide dependence upon rejection acceptance nan independence discrete multinomial represent independence discrete datum function frequency c l configuration shorthand classic mutual define factor degree particularly sample contingency nan increase heuristic perform sample user power structural zero contingency degree brief overview reference combine weak learner attempt pc learner inter computation may think false weak perform extra receives node hybrid combine benefit extract conditioning increase pc run pc learner decentralize search candidate pc true working domain effective restriction neighborhood less severe decentralized significant enable neighborhood correct parent tx add true positive tt positive change xt set dag xx hill try inter incremental receive return rough de pc omit brevity conditioning size de significantly increase reliability two hundred thousand variable restrict relevant relationship ss phase discussion idea efficiency appear candidate unconstraine greedy idea candidate hybrid identify parent hill begin edge direction score score continue similar recursively search adding discover list list keep last good local change list attempt occur score ever encounter search terminate score heuristic enter pc ss range c c c experimental comparison datum benchmark learn algorithm claim possible compare bn benchmark repository investigate parametric reference pc implement integrated package develop pc publicly type pc cpu ghz go ram run window bit investigate quality skeleton return pc cb phase false ratio number output true positive euclidean assess dag report five dirichlet equivalent single sample support learn goodness fit generalize network generalize distance quality dependence require match undirecte add orientation edge network structure performance new correspond posteriori distribution degree hold encounter experiment prior learn rely gold employ several reason report skeleton ss phase benchmark benchmark depict table increase gold improvement bad clarity mention regard quality observe false improve benchmark cb maintain reduce quality dag obtain ss bic improvement goodness clearly dominate ability generalize regard dependence pc significantly rapidly tendency less average overall increase factor grow somewhat linearly nonetheless worth package employ pc efficiency compare code currently allow fair consistently generalization cause maintain couple child alarm conclusion pc promise construct bn performance possibility hybrid keep low focus study heuristic combine dependence cb independence type size applicable large sample structural time slow behavior independence test permutation pearson permutation mutual test shrinkage single bn permutation structure goodness output graph one test test outperform parametric test network structure fit picture open reduce super super structure sound improve learn sound rather skeleton expect sophisticated strategy hc sound super easier learn many extra edge thereby result type burden involve hybrid gain accuracy rate miss independently keep track found previously lead redundant design infer target optimize version super get computational maintain cache store reduce computational author hybrid bn hybrid extensive experiment outperform goodness generalize overhead time currently structure structure find outperform margin experimental edge crucial super like
dimension overcome avoid employ filter dictionary convolution receive attention globally decomposition provably activation draw model work extend decomposition invariance convolution operator denote element tensor denote similarly matrix cyclic convolution vi j convolution operation cyclic cyclic convolution twice cyclic convolution cyclic shift discrete entry important use improve computational extensively stack matrix q concatenation stack additive noise incorporate sample activation active coordinate encourage small extend limit ica simplicity estimate decode criterion map map focus develop estimation paper rd tensor extend tensor I stack slice v v ab column ab cm order multivariate tensor use moment third tensor bx ax ax convolution show third nice form denote column third univariate third activation order activation fourth method manner cp usual decomposition unfold order filter minimize frobenius enforce rest devote throughout paper matrix block non relaxation mode perform compute product efficiently implicitly present computational utilize various rao carry incorporate stack reduce completely stack partially column matrix consist column stack introduce stacked identity thing stack filter appeal notation denote nj block stack diagonal simply inversion need result fix iteration note unfold compute inverting processor take degree parallelism serial computation parallelism time degree parallelism combine discussion decomposition framework activation convolutional recovery error alternate fact spurious increase experiment report order minimization alternate scale sample linearly question investigation plan task scalable long learn extend dimensional signal replace block generally framework lie algebra advantage tensor invariant expect embedding block inverse multi know order entry know shift eq tensor format q therefore rao therefore decompose propose via inversion inversion stack partition r invertible invert matrix inversion invert multiplication indicate inverting matrix inverting processor simultaneous block therefore multiplication processor processor lemma corollary fact paradigm learn component learn cp tensor convolutional projection onto operation fourier multiplication compare alternate minimization map decomposition dictionary learning convolutional ica deconvolution convolutional model generate unknown dictionary unknown activation activation speech sentence spike train activation language processing usually loss employ filter add sparsity alternate activation vice versa alternate expensive modern run optima reading np convolutional compare dictionary shift unchanged fundamentally ill pose impose design solution huge dataset paper answer framework convolutional moment via decomposition convolutional map convolutional ica whose component stack invariance popular act sample operate average moment closed use operation fourier multiplication degree parallelism estimate length
diabetes heart cancer screen moderate size repository book short intercept laplace laplace concentrate require logit challenge ep deal ep contrary probit site update moderate accuracy error versus accuracy accuracy laplace panel fig plot scheme component fig dataset supplement box plot accuracy across plot supplement sake scale ep ep left panel ep cpu intensive standard improved expect course cpu hardware note pass laplace suppose replace student supplement result scenario addition represent ep panel laplace breast complete sampling method dataset produce nearly instant along essentially gold standard nice discuss gaussian laplace proposal gaussian probit particularly favorable next factor cpu cc ef mt mse mse breast heart importance dataset probit ef cpu time spee intel hyper core cpu gain mt efficiency mean parallelization speed implement core virtual amazon ec virtual factor run time report median mse improvement mse dataset gain parallelization evaluation chance shall section sake completeness see supplement scenario order relative criterion eq resp square obtain resp cpu sampling importance sampling sampling term posterior observe fig median across reader difference use ill discrete proposal construct nest regression correspond intercept successive regression mcmc sampler binary intractable approach hand see ep sampling smc thesis smc pp importance laplace valid pseudo argument compare approach sampler cpu minute figure smc sampler estimate also consistent reversible jump sampler estimate covariate box pass hyper generate selection interesting extent type posterior deal prior ep important end routine right near regression concerned recommendation always fast implement author ep logit drawback ep lack theoretical learn however manuscript ep well assess second exact particle reduce single even run alternatively perform well calibrate main message title leave alone serve benchmark elaborate distinguish algorithm gibbs sampler covariate matter even offer generic metropolis hmc amount practice propose novel compare gibbs sampler small dataset numerical properly computation scenario binary possibly covariate seem critical complexity therefore computation perhaps strong approximation ep family gaussians alternatively active covariate finding generalise assess statement hand study recommend account approximation former regard latter binary certainly fast whether laplace smc relative performance use acknowledgement thank common regression moderate discuss extent sound review fast laplace extensive might hard day markov inspire hmc smc nest sampling approximation variational book even approximation methodology thing approach binary regression g probit logit benchmark remark bayesian optimisation practically regression question title suggest finding lead current gibbs dataset well diabetes basic toy large seem remain competitive computation algorithm obvious criterion discuss posterior whether easy change link complete extent require manual tuning obtain performance important fact easier free manually tune discuss tuning computer hour cpu pay require much manual may serve review believe already develop relate criterion parallel method perform core architecture parallel architecture although phrase already certainly get big big dataset fair really big datum away kind encounter bayesian paper cover deterministic offer method discuss part contain discuss selection discuss end computation generic expression datum consist cdf transform linear form probability cdf probit take logistic cdf e accommodate outlier predictor preliminary deviation mean range intercept weakly assign outside reasonable default centre predictor henceforth independent deviation scale henceforth jeffreys prior determine method one cauchy difficult tail explain quickly evidence quantity later particular tune derivative compute two concave probit regression stick gaussian map point posterior iterate iteration approximation log newton work concave variant pass estimator infinite variance mle properly complete hyperplane separate outcome extra inference occur variant newton adapt automatically g determine line replace iterate reweighte interpretation seem roughly stick newton cauchy prior shall cauchy ols ordinary happen section come include ep e laplace ep vb vb field probit may marginal component expectation directly mind fast approximation preliminary method describe laplace approximation expansion minus refer particular laplace phrase approximation discussion drawback marginal simply obtain laplace mode q vector stand laplace deduce see pass connection scheme apply posterior hyper grid improve empty describe cauchy recommend student prior log guarantee prevent converge work reasonably briefly describe em em student implement deduce aim single newton approximate one newton iteration laplace refer reader detail laplace laplace consensus well match posterior ep compute iteratively parametric density natural exponential q natural gaussian family could gaussian natural ep consist update equivalently keep match z gaussian site update turn achieve implement must compute hybrid probit compute supplement link logistic dimensional quadrature simply never course simply ep support result sense posterior ep work many variant determine intensive complexity ep site observation laplace perform ep expensive laplace remark modify ep end may parallel factor processor inversion improve laplace marginal describe perform basic laplace vanish point remain implement result adapt different choice simply write evaluate correspond adaptation ep fair model shall see laplace sampling method small form calibration prior since previous laplace ep compute preliminary generic posterior q marginal estimation restrict ep recommend student ensure variance bit call stress however iid assess auto normalise also advantage importance amenable quasi carlo integration explain offer marginal error cpu trade know suffer curse variance grow exponentially large meaning get negligible hand moderate see smc automatically perform something elaborate assess compute roughly approximate target require simulation compute instead ratio elaborate technique carlo estimator express vector inverse cdf replace sequence vector spread evenly g condition monte background construct possibility conjunction importance sampling mention however often often ability quasi monte way marginally average become unbiased assess repeat assess variance repeat reasonable approach chain carlo markov leave invariant drawback g specify start determine burn period assess chain invariant fair regard b start draw cover assess visually consider augmentation formulation vector variable probit model sampling iterate sample b thank conjugacy stacking hence gibbs drawback particularly switch student scale cauchy well cauchy prior b yet require strategy turn thing derive sampler first augmentation mixture finite second paper logistic infinite discuss conditional since implementation main justification great generic investigate numerical hasting consists iterate describe metropolis generic take approximation practice usually importance input hasting metropolis critical move slowly move rarely choice lead fast exploration close ep validate bad news move tends cite motivation elaborate strategy hmc cover hamiltonian monte mcmc perform determine accept hmc make jump metropolis excellent un normalise physical hmc position velocity energy mass trajectory constant practice proceed new velocity keep practice step perform third accept reject see summary rely volume preserve jacobian probability output output momentum perform hmc mass stepsize approximation obtain rescaling incur correlation difficulty drawback hmc tuning seem currently popular vanish adaptation acceptance optimal hmc take acceptance iteration fix much exhibit behaviour large may long distance come spread already take interesting hmc correspond hmc locally geometry main drawback derivative expensive take account well exploration might relate adaptation hmc instead aim trajectory
envelope computation available successive segment th envelope whole envelope calculation persistence landscape alternatively one intersection line segment bad also different landscape intersection persistence landscape calculate persistence bit paper subsequent end landscape clearly computational construct number point landscape bad algorithm number death empty intersection interval calculate persistence encode persistence landscape number interpolation persistence special n average persistence kt give k kf see summary k n consider death constructing may calculate repeat landscape improved min case death evenly spaced combination simply combination two persistence q calculation complexity distance formula sum consecutive write integral dx ap start diagram since calculate persistence death lie evenly spaced calculation persistence nontrivial diagram birth death lie grid distance persistence combine previous calculate combination birth death pair important distance persistence persistence construct calculate interval grid present experiment dimensional persistent homology uniform dimensional normalization difference higher project cloud homology cycle subsequent rescaled range new back scale persistence distance average explain combine compute equal proportion great difference various believe ccc dim dim dim dim dim dim dim dim scale one dim dim dim dim dim dim dim dim scale dim dim dim dim dim dim dim dim ccc average persistence dimension normalize degree persistence landscape implementation bottleneck distance wasserstein current death uniform birth persistence diagram bottleneck landscape bottleneck wasserstein distance currently present w collection independently aim landscape birth landscape distance calculate persistence landscape birth death calculate persistence implementation procedure library maintain user tool user familiar programming program library plan add program describe program illustrate toy program window os available result program form file birth encode also file contain persistence landscape form diagram sequence critical follow file persistence landscape diagram degree file combination persistence name contain birth pair persistence file consist persistence program file death union circle radius measure uniform tb persistent homology use time result name circle file encode degree file homology particular degree list file circle dim txt dim txt file dim txt list file persistence diagram circle persistence persistence txt point example persistence persistence circle circle txt persistence txt persistence txt example persistence txt circle persistence circle circle persistence txt txt describe input file file persistence diagram landscape calculate dim contain landscape produce file obtain file generate plot software build engine instead create plot program name file persistence diagram persistence landscape persistence remain plot use h ccc circle circle circle file file persistence program compute persistence combination persistence file contain persistence diagram persistence degree circle file file contain persistence diagram combination persistence integer text example file txt diagram persistence output matrix file persistence name file persistence diagram persistence file suppose contain class file suppose indicate try permutation indicate distance please lot time user program file circle expect section implementation near classifier persistence topological distance matrix implement persistence diagram class sequence option return program vector coordinate usage program determine program landscape locate training name file file program file later classification classify create use option integer indicate name file indicate supremum value persistence diagram landscape distance average calculate parameter average one run indicate many name file file file name classify indicate file txt order classifier persistence diagram calculate class half diagram classified file file name circle classification element make interpretation easy work except get wrong reason turn interval correspond circle sort well follow outli classifier persistence persistence landscape birth death construct landscape n persistence landscape sort point next k kb db kb kb kb acknowledgment author thank valuable suggestion topological multiscale geometry quantitative persistence topological give persistence distance average procedure intend facilitate statistic topological topological persistence landscape module average summary calculate topological summary tool provide summary useful method prohibitive implementation tool publicly topology machine purpose convenient summary summary call standard simplest encode filter homology field obtain space turn module nonzero homology homology give basis consider pair consider generalize correspond increase sequence summary perturbation lead perturbation choice successful breast signal tracking landscape birth death birth death th large exist persistence landscape extend persistence kt kt distance persistence persistence persistence landscape various library death pair persistence diagram wasserstein distance statistic necessarily adjust metric procedure testing persistence diagram brain persistence diagram apply obtain confidence band landscape average persistence landscape persistence landscape bind persistence landscape kernel kernel persistence diagram reader also persistence landscape birth death persistence death naive persistence landscape min clarity coordinate element clearly linearly total number value persistent landscape array large equal persistence vector persistence piecewise encode persistence landscape map persistence linear construct landscape list death variation appendix computational complexity persistence landscape evenly grid k landscape list sort accord initialize add kb
solution quadratic grid part million cell simplification criterion analysis simplification grid structure iteratively degradation dissimilarity cluster impact merge dissimilarity grid merge minimize degradation minimal grid w r distinction agglomerative stop choose analysis follow cell hierarchical agglomerative categorical cluster increase representative cluster model evaluate average representative numerical categorical result frequency visualize frequency select interest contribution provide visual mutual select part mutual partition observe excess interaction locate contribution interaction expect visualization highlight valuable part bring complementary add mobile day computation confirm cluster traffic study time categorical apply inactive area hour call record cluster nearly amount datum million indeed distinguished plot ratio cluster hierarchical interestingly cluster study satisfy number partition stay rest dot strong country cluster country four due city phone traffic cluster place locate use cluster dot typical locate main already area influence country cluster city city big city less typical less region recent growth country locate area intensive city part central central business locate city cover neighborhood mainly area previous one separate north localize match party area last two group locate area similar locate city differ dark grey introduce mutual visualize traffic call inter traffic visualize fine segment draw position segment proportional contribution information proportional call map highlight call visualization country capital big capital recent city activity explain excess traffic phenomenon phone west country around area densely flow note track traffic time period introduce section categorical call high call study call interpretation treatment ten segment miss indeed period time segment miss group call color call green locate localize miss short localize activate provide understanding year three describe week hour simultaneously partition week discretization hour keep obtain cluster day simplify fix segment acceptable four cluster display day column segment red blue contribution mutual partition discretization business call pm office hour phone traffic business cluster area pm day cut pm last segmentation interpretation period pm follow pm user map traffic period color partition discretization connect locate east pm day connect pm cluster user people area part area experience pm pm business le neighborhood west city economic sum live area work week localize area aim extract different mathematically make first country network country call difference mobile user live profile usage discuss interpretation besides level country confirm impact economic identify branch em mobile service still grow mobile phone first case call available network answer question quality pricing discount depend valuable information spatio pricing receive much attention benefit system public mobile may g mobile challenge etc process phone million daily spatio equip activity suggest activity health monitor al approach mutual minimize obtain locally cluster matrix significant progress way analysis mining sequence forecast combine free categorical dimension therefore network suggest base provide explore exploit component applicability user datum data model aim joint type categorical numerical take cluster category categorical interval numerical variable multidimensional grid whose cell partition partition grid posteriori minimize bayesian implement trade robustness follow analytic cost combinatorial notice mean data categorical univariate categorical categorical variable grid stand priori constitute categorical numerical stand model close prefer get priori grid high cost nan priori probability likelihood cost value grid indicate simplicity logarithm minimum code length grid value categorical equal bn n obviously bottom strategy pseudo code grain make partition interval evaluate merge merge iterate grid consider case e categorical numerical point implementation greedy grid grid resp grain advanced mainly exploit grain pre sparse cell cell empty contribution cell grid stem hierarchy model part interval cell interval merge perform instead grain concern grid dedicate heuristic locally solution post alternatively partition value across move interval time optimum search meta principle consist consider round allow detail optimization method mix available name several real study cl record consist call reveal valuable human show g paper suggest methodology contain well explore original relevance activity massive mainly service basis mobile generate date duration call exclude initially purpose social interaction activity derive unite sum recent valuable information development purpose leverage country rise application improvement economic indicator population city planning management mobile prove mining technique source temporal sequence source generic methodology retain model technique simultaneously variable discretized categorical group see constitute data result brief lead data principle exploit result experimental relate work conclude come set communication million mobile
unit first mask matrix impose property must sure unit connect input unit rule define intuitive applie mask connectivity last layer sample great minimum connectivity layer conditional model conditional beneficial approach order stochastic minibatch miss partially invoke unobserved secondly ensemble construct exploit conditional slightly easily vector original conditional random order first hide agnostic randomly advantage agnostic train exploit create ensemble order unit train lk uniform l w b choose order hide connectivity imaging training agnostic instead minibatch assume whose connectivity input training absence indistinguishable situation inform additional learnable apply strategy weight also parametrization sometimes useful treat every cycle list connectivity agnostic illustrate layer along value lot work feed autoencoder generative behind research test intractable partition design neural autoregressive feed architecture extension state make unfortunately deep code reproduce tag likelihood respective sgd batch early binary uci evaluation put university california repository letter stanford overview dataset name input valid connect dna letter run hide update varied cycle sample chance validation hyperparameter layer activation relu make competitive otherwise clutter deviation deviation supplementary connect rbm mask reveal winner however mask help negligible letter mnist mnist digit update help single mask second hide make gpu model gpu building uci relu conditioning varied hyperparameter value hide rate report table result make network well forward deep yielded layer add pattern case illustrate layer train vary mini batch rbm cd intractable tractable order mask mask compare near set figure ensure simple use mask modification autoencoder distribution direct autoencoder evaluate high probably maintain acknowledgment compute mask mask dna letter make mask make mask make designing modification autoencoder autoencoder autoregressive constraint reconstruct input constrain autoencoder output interpret full multiple framework architecture implementation fast competitive art autoregressive definition general formulate learn scenario miss imputation synthesis many make challenge essence curse impact grow good fortunately recent progress task great scaling focus attention operation explore simple adapt network make estimator alternative mask autoencoder output autoregressive solely precede preserve implementation gpu straightforward autoregressive explore simultaneously multiple observation connectivity binary description basic build upon clearly example concentrate observation motivation hide representation input reveal statistical structure distribution autoencoder attempt learn feed forward close matrix activation function input connection autoencoder specify cross loss treat take negative autoencoder usually descent paradigm autoencoder layer input disadvantage trivial copy reconstruct perfectly consequence equation since perfect x could autoencoder output valid probability specifically able properly correct autoencoder product imply always decompose conditional thus become negative q autoencoder particular form unit sequentially modify autoencoder satisfy autoregressive computational matrix multiply binary mask autoencoder impose assign integer give hide depend conditional exclude create encode overall encode connect thus connect hide connect output notice rule unit mask construction autoencoder autoregressive connectivity
output hide stochastic sigmoid number advance hide unit sufficient work compact representation feedforward markov organize definition deterministic present tuning bias layer keep section bias offer number string binary negative strictly source polytope form consider sigmoid compute scalar activation affine output probability feedforward layer unit output nc feedforward kernel represent feedforward kernel give feedforward network shape every circle sep dot right cm dot minimum size dot node transform cm dot node node dot l probability eq parametrize feedforward tuple distribution approximate arbitrarily closure euclidean topology property k mr free minimal unit k feedforward weight multiply feedforward network threshold extensively classify deterministic hidden function unit term marginal activity independently hide second bias bias tight tight approximate reveal theorem first well depend idea illustrate arbitrarily row copy finally hull model individually hidden piece compact lemma trick input us unit trick produce flexible layer simply shape style shape sep dot dot shape inner cm distance right dot b transform minimum dot end dot draw sep dots node cm dot node style draw size cm dot l node swap l swap follow bias dimensional cube face cube support hyperplane plug previous approximate kernel arbitrarily divide successive pair vector unit except unit unit hyperplane dimensional n kernel let l l deterministic arbitrarily indicate zero vector lemma weight b n precisely strictly entry entry wise map consider n fig pz eq arbitrary choose appropriate refinement arbitrarily certain mutually th indicator assign make relative sum next input bias entry map consider sufficiently large gradually th continuity nz li claim arbitrarily lemma entry p proof make irrespective make pz pz p z arbitrarily transition continuously value transition one upper stochastic feedforward sigmoid probability free hide suffice suffice kernel boltzmann machine show suffice unit suffice bound feedforward
pz model scientific date historical introduction thus integral evaluate generative least precision solver equation transition say give triplet numerical solver general know perform computation aside differential probability density evy drive stochastic previous deterministic function transition analytically integral exactly markov pz denote integral hide outside yield estimate desirable produce numerical exact tuning produce go infinity go consider exact contrary extend kalman filter systematic return uniformly already arrival new available advantage sequential illustrate introduce uniformly rule case importance draw prior per yield go infinity however problem next approximate satisfy implicit pz model free perhaps plug abc posterior markov give go understand true retain go instance summary abc quantify abc estimator exact abc filter plug method wise particle deal smoothing task since publication introduce integral draw k require weight propagate resample weight without significant describe scheme consist draw independently resample approximate filter approximated manner go filter limit bias remarkable since make particle sense filter tool filter kalman dimension particle filter estimator typically improvement study filter filter constitute product form update particle linearly observation prove particle per observation infinity guarantee particle particle step define define k k go infinity however variance well path degeneracy step population distinct time particle replicate precisely element path particle resolve degeneracy issue smoothing value particle lag degeneracy consequence particle method model initial represent dirac measurement obtain approximation obtain recognize degeneracy indeed transition delta fewer recognize early function random walk introduce monte move move leave posterior well high correlation move early method move degeneracy attempt state year advance filter iterate filtering rely filter model particle particle filter efficient recall particle call particle metropolis hasting pseudo particle path trajectory pseudo compute distribution infinite perfect filter estimator yield perfect thus metropolis hasting proposal remarkable algorithm x particle mcmc method study particle variance observation although informative optimistic assumption independently would overall filter particle one path tx compute eq base constitute practical consistent approximation general iterate batch upon arrival begin design proposal number introduce address take play model step estimator section light particle mcmc go infinity ideal smc markov incremental model resample invariant resample n w structure equip particle difference obtain instead incremental move instead complete simply mention smc distribution number fall particle define consistently go infinity design turn computational us mention evidence retrieve algorithm consistently compute integral ideal smc fortunately slow occur typically assimilation perform happen occur operation overall memory involve thus keep available memory cost error equally across motivate algorithm smc reasoning run step overall smc sequential online upon arrival piece computational effort uniform smc automatically along enough make adjustment stable performance currently exist pose challenge series term scale algorithm amenable architecture algorithm computation year pz use generate algorithm distribution particle resample particle systematic resample behaviour smc run ess ess decrease slow time step whereas ten occur half move end run precisely plot transition particle call reach call indicate call trend cost overall year daily estimate set four since call pz differential minute use occurrence incur collect minute algorithmic five run represent pairwise contour indicate dot explain instantaneous population posterior locate recover sequential ability investigate grey quantile marginal use go parameter accord asymptotic fisher information imagine observation reach figure predictive infer particle successive grey plotted circle triangle predictive region observation expect fall region focus time step predictive circle outside indicate triangle grey estimator introduce another pz pz except term use uniform pz odd py approximate algorithmic bayes factor dash support pz pz observation pz bayes pz keep generate pz bayes show simpler available enough confidence accord bayes criterion particle confirm initially simple pz prefer datum strongly support light review parameter filtering integral approximate online filter exact parameter smc estimator update arrival complete applicable run reasonable hardware thousand number change open area development one difficulty cost filter yield evaluation linear unclear whether likelihood could super moderate prediction dimension state variance particle filter typically scale particle dimension another memory store particle involve whenever memory usage storing path study method reduce hardware adapt play compatible measurement particle filter markovian requirement meet markovian particle setting recent markovian instance motivate direction article hide recent transition put gaussian particle identifiability case infinite particle markov recently space distribution markov constitute current methodology ep thanks taylor useful comment dedicate discussion bayesian time series arise various treat arbitrarily complex smc markov fix review development allow currently object interest illustrate toy open challenge scalability review long state space flexible les mod markov les des de smc des pour du les observation une en des en le les de la angle er pour analyse mod en population de pr plus des plus des mod plus constitute series value countable collection time observation arise latent markov chain specify initial successive state call transition specify current distribution parametrize integer explicitly write collection resp represent daily water omit indicate candidate volatility financial series phenomenon study phenomenon choice synthetic dataset give trial one intuitively hope future observation reliable inference ad hoc procedure connection section transform integral introduce desire computing section review compatible implicit meet desire requirement smc monte method smc mention section methodology open challenge goal hide refer available article refer normalize filtering path trajectory prediction refer give current depend use prediction product denote smoothing refer state realistic lie interest observation prior evaluate marginal bayes normalize useful uncertainty account filtering smoothing account next several compare chapter comparison evidence evidence normalize normalizing introducing account smoothing refer average task assign prior odd
runtime run line theorem use randomness space distribution taking banach exist absolute need bernstein random satisfie find still cauchy schwarz cauchy schwarz inequality degree proof induction schwarz eq equation result singular proposition definition example application tensor th rd tensor decompose decomposition polynomial decomposition relationship matrix concentration tensor represent involve th array outer product th entry tensor application tensor define sum agree correspond tensor hard problem behave survey unlike matrix decomposition tensor specialize decomposition algorithmic idea latent gaussians model dirichlet allocation previous inspire many work limitation although attempt decompose tensor overcomplete tensor machine order tensor base rd preferable interested overcomplete decomposition overcomplete rd tensor understand explicit rd nontrivial circuit rank explicit rd order matrix component also case quasi sdp see recent survey difficulty overcomplete rd tensor unfold tensor matrix unfold rd unbalanced intuitively base order moment allow particular component closely give find subspace closely relate many dictionary tensor almost randomly third close close decomposition close close tensor distribution recover high uniform spherical accurate dependency however refine find refine close prove component multilinear tx k tx x key sum random tensor tensor far unfold corollary unfold rest relate tensor give polynomial rd tensor key tool section quasi algorithm decompose norm spectral norm sum norm tensor decomposition inner matrix block notation dependency throughout high tensor array paper simplicity rd tensor depend goal homogeneous polynomial corresponding sphere hard concept reader section system inequality polynomial define constraint satisfy form easily generalize variable constraint schwarz prove square useful matrix random matrix turn argument expectation think expectation distribution distinguish expectation polynomial degree pseudo polynomial x pseudo obtain pseudo constraint satisfie expectation tx ta concentration random tx net vector close later give observation suffice vector idea pseudo expectation pseudo pseudo though maximize tx pseudo p yes otherwise norm case suffice concentration randomness follow careful noiseless norm hold polynomial rd tensor tensor high unfold algorithm success degree nc basic find find moreover intuitively remain constraint valid pseudo pseudo e kk formalize I use time pseudo satisfy I kk vector intuition old claim average argument detailed prove suppose vector like algorithm follow sdp high satisfie previously find unit follow expectation constant must appendix proof decompose overcomplete rd order rank almost match unfold concentration technique useful tensor decomposition machine although initialization algorithm idea help solve david discussion randomness schwarz claim bind property fy diagonal entry bb apply cauchy schwarz spectral lemma lemma j tm direct consequence simple allow simplicity let ia incoherence sum bernstein sum thm basically say concentration concentration independent exist absolute uniform proceed q n tt ia tt bernstein bernstein spectral matrix spectral individual incoherence variance bernstein randomness two claim bernstein individual probability claim product psd suffice decompose write eq ready return yes randomness guarantee unfold prove return pseudo expectation know tx show return show expectation tensor unfolding unfold term high pseudo exist repeat bound pseudo expectation take expectation assumption imply cauchy
orthogonal significantly outperform individual patch trade capture considerably intensity centroid voxel redundancy independence rotation global complicated post process observe centroid redundant hand patch raw intensity centroid require preprocesse centroid future improve example coefficient plain negative cost future research sophisticated function class imbalance dataset region account account carry volume region try voxel per good training huge train fairly well unseen reason relatively variability brain explain voxel training capture increase unfortunately expensive consider artificial one transformation rotation create artificial distance centroids legend legend legend column college uk ac image assign voxel mr brain capture voxel patch capture context centroid spatial consistency contrary commonly segmentation technique mr model manually brain tackle brain quantitative brain quantitative often brain volume mr image essential segmentation brain require protocol manually consume full enable systematic image acquire benefit dominate patch consist assign voxel neighbourhood intensity deep neural prove art computer vision imagenet contrary traditional feature engineering crucial learn raw input development deep automate briefly review deep architecture dataset segmentation classify voxel protocol segmentation brain segmentation implicitly manually brain consist mr manual widely new query consist query query finally combine strategy heavily non usually perform critical enough accurately map introduce change region boundary identifiable intensity intensive responsible whereby give voxel correspond region particular recent advance learn extract beneficial learning concern hierarchical one imaging segmentation approach neuron mostly patch computation graphical despite increase medical yet brain slice input comparison whole pathway hand side feature merge convolutional layer follow colour share patch scale select layer max windows activate training architecture design voxel corresponding vector network particularly mostly spatial precision first patch local detail patch voxel add capture slightly broad context around patch small amount memory dense patch allow big patch ht input design preserve global segmentation region arbitrarily consistently preserve position subject obvious would simply patch span distant input require add instead figure operation reduce average intensity window full patch size terminology operation pooling patch patch voxel intensity coordinate informative absolute require perform initial generally consume one additional centroid image mass voxel voxel belong voxel region centroid voxel belong absolute invariant rotation distance invariant scale centroid coordinate average centroid voxel precisely voxel increase learn field considerably thus overfitte layer convolutional also impose neuron value layer decompose weight field mean detect connectivity sharing constraint model convolution operation neuron operation feature convolution size field map map bias convolutional reduce merge precisely pool layer neuron discard map field lose windows layer make reduce overfitte connect layer layer apart layer activation neuron neuron unit relu contrary traditional sigmoid function vanish layer activation neuron input output label voxel output reason share orthogonal patch patch lowest learn patch orientation layer experimentally find vector zero one position evaluate dot product space represents carry update weight error beneficial long narrow average momentum scalar rate momentum respectively centroid new directly centroid propose iterative network centroid pathway train mr enable centroid voxel approximated centroid centroid refined refine centroid two time change observe already really poor segmentation slightly long illustrate initial improve even though centroid region lie distance centroid sure network distance distance particular approximated approach competition pixel team require segmentation quality assess mean coefficient image manually translation region win obtain overall coefficient performance learn require computation run gpu gb therefore trade dimension decide randomly brain approximately voxel voxel purpose voxel dataset voxel patch voxel intensity three intensity voxel intensity centroid patch validation early error epoch early stable
cccc panel depict action indicate b various update deviation policy depict move bold cost think boolean hamming distance specification use e sophisticated variant descent pass detail pos use sequence search dependency parse sensitive multiclass search set request gray width text inner corollary edu cs edu microsoft com microsoft com edu microsoft com microsoft demonstrate suboptimal learning learn poor learning compare optimality guarantee unlike enable prediction application structure learner joint variable observe parse output achieve commonly require neighboring solve structured feature capture policy policy structure exist step policy implicitly reference attain typically word pos tag reason constraint say contextual page high website item position item font plausible display page user feedback reference namely web page learn something keep full feedback learn reference goal improve upon optimal core locally operate fashion achieve regret reference sub operate past secondary good variety include whether reference policy reference past algorithm dramatically hand superior poorly hill confirm policy real bandit level distance right bag child bag child bag child right child loss choose new kind regret modification contextual bandit output nf minimize expect search induce consist initial transition pair end structured convenience define clear express input action approach search use agent choose terminal specify policy action start repeatedly length trajectory trajectory reach follow policy accumulation cost state generate state training well internal randomness choose action lead decomposable trajectory generate optimal state trajectory twice grey algorithm deviation bottom reach collect learn top middle decrease I roll initialize loop generate reference ta ic ta assume sensitive predict receive loss perform update online sensitive learner define give online cost sensitive om binary operate algorithm algorithm policy sensitive optimal proceed online fashion along roll roll roll generate trajectory roll decision round reach multiclass multiclass correspond assign difference taking reach roll multiclass learner default roll roll policy batch final across round pick section answer question throughout obtain act begin discuss roll roll summarize use roll roll obvious choice roll roll blind reinforcement hard l roll roll inconsistent learn rl generate marked reinforcement much hard cm node child label edge parent child edge parent end node child child node parent node child right node parent grow node label child edge parent c child child end parent reach policy go bold branch learn policy pick randomly policy choose action please roll roll cause never learn mistake poorly test discussion sensitive whose action use since uniquely reference finally though state take deviation result state learner action achieve cost job unfortunately actually run perform expressive policy pick branch state sensitive generate sensitive example complete roll crucially cost take reach roll policy zero cost regret despite robust mode figure pick roll generate sensitive similarly sensitive learn roll roll make blind hold term local policy change sensitive om arbitrarily suppose one structure depend roll observe policy cost choose depend however well learned instead motivate generate policy deviation reach decision reach notational simplicity take use state act expect action complete roll regret algorithm reference capture regret roll let mixing parameter appear combine scale appendix comparable assume classification formalized arise solely restrict asymptotic gap vanish obtain state correspond average correspond avoid reference individually rather exponentially research result theorem consequence guarantee reference reference policy regret incur reference policy alone reference combination factor evaluate term case irrespective suboptimal term guarantee stay term learn improve poor overall either competitive locally demonstrate reach optimum exponentially reflect equip local optimality establish search policy trajectory feature depth state policy policy index bit trajectory step level deviation bit string distance consider powerful algorithmic give cost deviation learn access cost algorithm powerful reach step deviation class reach policy show compete reasonable step algorithmic policy class loss start construction apply hamming variant contextual bandit structure set round learner suffer search emphasize reference mixture initialize policy step follow end output common partial choose whether recommendation probability perform update policy base round step step average regret early definition algorithm q setup able evidence application multiclass multiclass cost search label recursively label half subset search root leaf end reference result use bad action detection training roll roll reference roll roll reference cm highlight roll roll cm reference reference suboptimal bad learn reference highlight part pos leave prediction loss trivial train suboptimal hamming roll immediately dependency parsing learn generate tree describe syntactic dependency system sentence deterministic lead end policy reference suboptimal apply otherwise arbitrarily choose suboptimal policy prior work deterministic journal
effort difficulty develop rnn easily lstm perfectly learn however general introduce multiplicative connection backpropagation demand long time implementation parallelization difficult due dependency internal feedback loop stream employ parallelism huge memory serious bottleneck implementation parallelization rnn rnns lstm perfectly parallelization parallelization intra parallelism increase mini conventional parallelization stream parallelism result parallelism paper generalize propose derive intra rnn explore inter stream parallelism experimental concluding remark parallelization various rnn introduce rnns basic generalization cover advance lstm forget connection base rnn every propose basically direct consist node layer delay delay amount delay signal connection output weight activation source value delay connection connection state activation gate lstm multiplicative layer multiplication input subscript represent generality introduce function additive wise multiplicative nonlinearity gate connection direct edge introduce multiplicative regard normal structure lstm rnn error convenience derivative q back backward pass layer initialize accord error criterion activation layer minimum softmax output layer indice derivative acquire wise multiplication layer become multiplicative multiplication connection error gradient parallelization rnn dependencies frames rnn determine parallelization intra stream parallelism stream parallelism separate part special direct mini batch feed neural internal recurrent group subgraph inside subgraph find inside remain otherwise node group recurrent ready parallel lstm one find strongly connect sort useful feedforward dag operation frame dependency different isolated node node sequentially step pass delayed connection exclude delayed connection dag compute topological order recurrent quite bottleneck employ multi parallelization parallelism stream mode stream context independent multi parallelism overall execution train gpu mode connect order sequence sequence long apply efficient truncate denote network however pass output error stream output gradient weight pass throughout equivalent feedforward neural increase mini slow easily modify speed simplicity sufficiently lag style font xlabel stream ylabel speed xlabel shift pt ylabel shift legend font label font legend style north west pos align major lm seq txt lm par txt lm seq txt lm par width style font xlabel stream ylabel xlabel ylabel legend font style north west minor minor log pos align inside style major lstm par txt lstm par lstm txt lstm par txt txt gpu experiment since mathematically mean square parallelization language model stream mode rnn architecture network batch amount gpu stream error step comparison forget connection note self connection compare number stream parallelism employ intra stream parallelism stream stream nice advantage stream rnn learn mini batch gpu forget operation
tm tm measure classification contingency versus stop definition true example stop stop set unlabeled nonetheless exist use truly positive truly contingency versus example stop label contingency table versus true label cccc total contingency stop truth cccc contingency stop truth table see example classify truly positive truly count contingency truth convenience show contingency table contingency table truth cccc total iteration truth contingency count learn versus suppose implie meet turn c notational convenience pick notational convenience ad c aa b ba u observe inequality hold used classification assume minimize measure maximize limitation loose tight perhaps expect bad make additional tight note c case practically substantially theorem prove tight utilize insight statement contingency perfect precision stop stop set classify therefore state general prove helpful contingency contingency follow theorem theorem case handle theorem contingency proof classify example stop show theorem scale implicitly precision theorem generalize scale precision factor place theorem difference learn issue connect stop stream generate issue would proportion count simply count stop unbiased select infinity stop probability approach date stopping criterion dominate heuristic dataset widely stop method remain inexact forward mechanic level achieve effective analysis reveal central sp success stop transfer unseen test proof agreement consecutive agreement precision conjunction assume setup serve stop switching region agreement relationship work publish conference learn association proposition em height em em break md nlp bottleneck nlp theoretical stop sp reveal element success agreement successively model impose performance stop result example successive proof relationship agreement difference consecutive exceed bound conjunction active query selective reduce cost create train considerable interest g nlp widely bottleneck new nlp system main effort require learn develop focus annotation effort al knowing annotation process challenge stop early useful wind model generalization lose recently stop al although coarse mechanic therefore crucial achieve effective terminology conservative conduct publish stop behave prediction publish empirical test agreement al exceed three consecutive iteration al heuristic well present stop prediction help deep work class perhaps important enabling stop active paper useful work switching strategy switch strategy similar case proportion instance sp particular stop size relationship agreement sp stop contingency classification learn iteration classification model model place category population place agreement indicate probability agreement chance classification independently agreement chance give cccc contingency probability learn true resort use table classification model frequency proportion expect classification agreement cccc contingency table count learn iteration delta describe estimator variance accord stop stop although work task sp variance stop dna fold cv cv fold course fold fold cv
passive predict whether tweet problem news tweet recommendation formulate content temporal message user social go break comment news et dataset sort future popularity use worth note retrieval propose coordinate ascent learn rank likely also work tweet likely tweet rank unsupervise aggregate previous method number tweet tweet introduce propose also try aggregate performance subsection tweet movie tweet movie base overall extract tweet user movie give opinion tweet movie movie tweet specific tweet contain opinion user movie tweet tweet feature extract extract category use numerical categorical boolean type respectively note normalize analyse exploit elimination retain perform subsection cm cat description user twitter follow tweet user movie rating provide movie tweet user list twitter frequency u day calculate divide membership twitter day frequency day user divide difference movie total movie rate rating tweet user movie people tweet number hash hash tweet tweet tweet age day user twitter tweet average movie hour tweet hour tweet predict tweet predict tweet language method learn build rank algorithm probabilistic basic construct weak rank svm create ranking try ndcg employ inspire use probabilistic rank logarithmic aforementioned fact technique two view totally aggregate increase mentioned aggregation try number final q measure aggregation weight instead number equation rank weight perform randomize cross training consider version dataset challenge contain movie rating automatically user throughout discount cumulative gain top hereafter library hyper randomize search cross learn except exploit source name microsoft report aggregate feature mention retain backward feature category importance user popular twitter boolean none categorical retain categorical difference result ndcg xt extremely randomize tree ridge drop achieve ndcg fs xt learn ndcg emphasize backward elimination demonstrate compare c cm ndcg fs w represent aggregate regression importance consider together aggregate regression far method aggregation aggregate learn rank aggregating method validation achieve statistically cm ndcg rank tweet category user movie show perform user aggregate demonstrate significantly affect method city usa ac ir interaction tweet news post achieve rank medium use recommender paper tweet rating movie focus extract tweet feature movie base tweet category regression learn tweet propose achieve extend dataset provide mining behavioral science social million information social site social recommender let express opinion social give rating movie internet movie website twitter social medium information system recommender study recommender gain comment item comment recommender comment focus movie rating hereafter user add tweet gain contain movie tweet three movie note hidden approach tweet approach globally purpose
view coordinate median contamination multivariate median obtain robust contamination constant determine contamination critical work robustness huber estimator general scatter via estimator various two study robust function give estimator whether possible procedure problem computation provide section dimension relatively great practically difficulty robust location scatter interesting whether problem algorithm discover contaminated write marginally conditioning follow control assume satisfy probability absolute characterize contamination relation real theorem since affine generality two testing elsewhere eq op obvious consider p desire depend thm general consequence following satisfie q constant elliptical distribution combine least least fact theorem argument theorem shorthand union c upper hoeffding finally due relation combine conclusion proposition u u u tx guarantee contradict proof tx u canonical complete characteristic univariate characteristic modify kind depend depend either constant smoothness hoeffding imply therefore sufficiently argument lead probability measure desire conclusion schmidt suggest weak corollary table lemma appendix matrix estimation important accommodate complexity desire procedure outlier arbitrary define concept call depth estimator show huber scatter competitive outlier contamination model covariance last decade rapid development theory covariance seminal covariance guarantee matrix sparse comprehensive work take heavy presence outlier exist outli totally tackle robust estimation high setting arbitrary contamination approximately distribute contamination break huber huber contamination contamination efficiency outlier simultaneously view contamination develop robust optimally concept depth variate semi constant depth parallel depth use location notion verify satisfie multivariate median point robust estimation give use concept covariance may accord one though notion depth definite offer several account structure powerful structured robust matrix estimating matrix sparse principal estimator depth function contamination interestingly minimax unified minimax range matrix classical without contamination quantity modulus continuity work liu contamination given distinguish phenomenon rigorously contamination model besides elliptical specific representation elliptical characteristic scatter elliptical allow naturally elliptical rate claim extra robust besides outlier heavy many work literature elliptical distribution include setting quantify work minimax contamination huber contamination robust small proportion totally robust property affine invariance achieve robustness counterpart contamination suggest huber contamination notion unify discuss depth introduce section structure bind contamination model discuss elliptical matrix estimation elliptical present relate connection proof close introduce notation singular large small singular denote frobenius submatrix cardinality kullback define variant generic constant robust location qp n know average fail sensitivity outlier introduce observation point maxima attain property state absolute high connection valid identity general valid say otherwise long outlier identical median contamination minimax sense contamination usual long achievable optimality perspective characterize outlier natural location inferior via consider obviously upper pn median slow achieve preserve whereas qp depth new matrix covariance observation subspace matrix speak median thus inspire computational reason depth q multiply scalar semi cumulative specify spectra need depth sphere pick cardinality attain state sufficiently computational n square minimax optimal class contamination constant finance order lead notion depth relatively statistical contamination follow state convergence exactly extend robust outlier rate contamination q subset g p q word covariate correlate remain component analysis degree sparsity sparsity depth function define relatively statistical property contamination principal component element row goal robust orthonormal matrix nonzero constant sep absolute sn account case optimal constant minimax contamination model statistical q component analysis whether theory contamination question lie key modulus whose seminal liu modulus contamination quantity measure ability close variation order level interpretation two distinguished minimax modulus state suppose quantity robustness pay dp sparse analysis derivation estimate arbitrary extend set elliptical distribution population scatter achieve depth elliptical property prove gaussian elliptical shape introduce elliptical elliptical distribute sphere simplicity unique secondly random motivated elliptical ec ec always canonical assumption exist elliptical canonical unique object show ec elliptical distribution constant imply constant determined px representation exist special px let estimate scatter contamination require outlier scatter depth induce least estimator small constant absolute constant small uniformly eigen sn ec probability uniformly absolute constant scatter covariance modeling interval subspace elliptical heavy tailed close section elliptical elliptical imply section optimal hard low computation median investigate develop propose adaptation core scatter depth q depth x tu tu tu outline multiple achieve small randomly tie tie back jump back direction back prevent search specify uniform sphere turn present simulation first introduce special correlation th variable define jk correlation scatter scatter ellipsoid find ellipsoid cover ellipsoid estimator covariance determinant estimator find determinant covariance package cover autoregressive autoregressive degree freedom matrix three consider contamination independent package scenario cover error behavior contamination contamination proportion rise though depth three depth behavior contamination compare rise stable estimator show scenario case five contamination table competitive efficiency three perform complete optimal result dependence evidence cancer study great interest co investigation perform pathway tumor exploratory sample contamination covariance cancer since type cancer expression considerably example importance estimator choose mutation disease characterize mutation induce hyper change biological gene involve dna literature raw conclusion create dataset randomly gene pathway outlier dataset matrix dataset difference indicate great
machine use panel support machine cart datum generate learn boolean set rule risk hard overcome use optimisation approximate bad case optimisation find heuristic optimality return training take return empty conjunction select maximize latter favor correctly classify greedy greedy rule conjunction conjunction assign negative force conjunction error consider error cross stop iteration discard utility conjunction negative redundant conjunction effectively example rule conjunction continue conjunction consistent rule lead stop reach conjunction induce stopping reach utility positive differ maximal utility rule small simple genomic number example become example contribute may utility situation likely performance stop mention prevent add rule bad training tb trade early stop stop utility correctly misclassifie u add conjunction represent genome presence overlap least genome training omit discriminate genome boolean rule rely apply boolean phenotype predictor interpret logical fold risk bold cart l propose representation predict grow public concern multi drug start care cost patient world genome combination individually yield dataset often compare regularize cart tree cart pose challenge term runtime make cart necessary filter univariate significance correction fold nested hyperparameter fold include comparison baseline predict tend one svms result svm solution greedy heuristic much difference less cart risk one variant filter preprocessing entire learn high obtaining conduct oppose machine entire require selection result heuristic produce regularizer generalization machine high thank dr loo dr sharing computation universit resource project ad award de pr discovery grants award sup la la sg cover classifier whole genetic exceed three predict machine cart consider feature filter preprocesse biology entire next sequencing lead increase whole
formalize imply exact hardness complete apply whose rank space incoherence passive sample possibly completion q passive subset index corresponding subset input imply passive small column subspace coherent show subset subset know span column error coherent matrix passive passive insufficient notion incoherence base incoherence theorem incoherence passive let column failure hardness column matrix completion limit coherent complexity approximate computing top svd operation take set reverse algorithm remark error selection subset selection target column like enough algorithm fail accurately phase completion excellent illustration phenomenon similar intrinsic probability proportional expected theorem state logarithmic figure plot actual question get bound curve decrease reconstruct sample big practice e figure sampling replacement replacement without replacement column remark iterative estimate project result satisfactory dependency rank conjecture loose believe avoid relative rank make high simulation exponential improve show input low perturb believe sub get input even matrix thank solution group lasso proof direct corollary th assumption apply orthonormal finally complete column represent combination q put bind dominate f f lemma observe r cs ss r span deterministic entry p last small covariance exponentially fraction follow hold put r mean sample randomness r hand projection gaussian rank least x ia lemma note union result prove seminal provide projected term project fix sample replacement th v clear ok desire result prove connect volume simplex permutation ready proof select denote span k k kk inequality lemma q ts f f markov bind lemma cite index bernoulli notation u invertible preserve incoherence incoherence bound subsequently give u r carefully dominate performance consider nonzero position j k n ci j j I ji consequently hold independent eq cs select subset approximated span subset numerous world datum application circuit input propose provably column algorithm matrix propose drawback complexity nice tradeoff employ idea feedback inspire sampling previously task prefer empirical analysis feedback compare input matrix highly aim much specifically compressed norm follow mainly evaluate compare rank large equal form guarantee ideally general error rank perfect remain zero problem example column various population circuit recommendation reader problem excellent class show nearly reconstruct base select column input carefully slight e unify column nearly problem full extensively study hard even genetic variation detection expensive sequence population several include omp explore presence miss pose column subset establish algorithm seem handle elegant identify challenge prevent application theoretical recovery incomplete underlie row matrix weakly selection decomposition effort incoherence obtain column matrix particularly difficult explore possibility gap incoherence need selection set large entry scheme provably scheme ingredient matrix decomposition perturbation fail two infinity norm properly observe complete matrix drastically usually additive strong beyond three entry sequentially feedback drive manner input differ drive science access active incoherent column knowledge column selection coherent theoretical passive error contribution paper provably column via scheme drawback summarize incoherent inferior propose synthetic real sampling error expense expensive addition rank incoherent input however iterative reconstruct matrix distinction entire column norm require entire summary offer column comprehensive accuracy efficiency achieve analysis insight completion comprehensive experimental well modification synthetic world nucleotide image theoretical interesting unknown instance leverage score widely selection regime prefer achieve suggest field spectral unless otherwise specify I generalize definition c type selection select reconstruction output remark always organize knowledge several important proof section complete defer briefly describe miss implementation experimental time background review concept play row three incoherence play decomposition u u always incoherence appear incoherence incoherence assumption subsequently incoherent incoherence assumption row vector norm selection approximate low compressed idea square type algorithm bind pick proportional span volume iterative serve leverage scheme approximation later coherent completion u u right vector relative guarantee approximation column error f matrix employ handle achieve reconstruction input certain structure achieve slow sample table summarize propose observe select active norm sampling column independently norm algorithm construct entry approximation input incoherent algorithm approximation provide show sample dependency tolerance column set x c ti tc u suppose orthonormal ts j ts u ss iterative though input low employ idea select error norm serve scheme relative norm within multiplicative depend exactly eliminate rank resemble completion already column ambient dimension fix span subspace q furthermore step norm satisfied follow bound suppose span column round square probability p mt round accurately estimate norm incoherent defer bind subset pick column corollary fix suppose subset probability volume volume precise cite volume sampling distribution inequality volume error well namely norm volume round strategy eq prove corollary sampling column distribution kk u os c os subsequently apply round e mt take union uniformly deferred theorem note immediately reconstruct coefficient reconstruction recover therefore eq complete note present claim column leverage subset rank right form project project incoherent column consequently row probability subsequently score leverage sampling selection f c previously subset miss theoretical method employ sample matrix mask di index omp use product similar method select manner column project span select nevertheless major difference norm omp product input matrix subspace span subspace exactly decomposition group extension precise propose optimization mask denote standard optimization column consume inexact median report instead leverage quite algorithm p leave plot fair synthetic dataset generate synthetic list first incoherent row space take coherent highlight baseline newly baseline result report number median sampling variant either score replacement result sample exception rate high degradation inaccurate estimation iterative input plus either target high work particularly input column easy gap algorithm block omp lasso considerably observe entry poorly inform underlie highly hand leverage score separation gap sampling bad without replacement coherent column repeat wrong column replacement investigate vary repeat high coherence block omp coherent isometry violate group lasso adapt column coherence decrease theoretical sampling genetic nucleotide human gene select snps capture genetic genome genome capture snp individual apply demonstrate propose entry raw datum selection miss setting omp group
comment discussion acknowledge support research cifar international conference machine true propose recurrent network rnn rnn extend stack control recurrent unit layer recurrent signal adaptively previous propose recurrent short recurrent rnn reveal outperform conventional approach stack rnn improvement adaptively assign different include stack rnn gate recurrent machine reveal rnn promise classification et rnns theoretically long dependency successful promising approach issue rnn e g activation nonlinearity term memory recurrent persistent fast hierarchical pointed help learn term dependency conventional way encode stack multiple recurrent layer recently approach partition hide group predefine feedback partition hierarchy feedforward design rnn call rnn rnn layer stack feedback connection one across fully encourage recurrent layer rnn control strength adapt evaluate conventional stack rnn usual task model experiment conventional approach able arbitrary recursively apply internal state wise transformation sigmoid tangent length let symbol symbol distribution widely model thesis capture term successful fundamental encourage maintain rnn gradient long lstm address learn dependency lstm maintain separate inside update necessary recurrent adaptively input central remainder proposal variant lstm follow consist forget gate gate carry unit control amount exposure memory cell neuron sum previous content forget forget old state current forget unit vector hide lstm memory lstm unit lstm unit gate control similarly output gate previous state memory unit adaptively forget content memory gate carry capture long hand decide forget gate mode happen across lstm unit multiple lstm capture propose lstm content gate gate forget lstm memory content memory content control update gate computed correspond previous candidate content gate memory content content gate compute base previous new memory multiplication traditional transition allow ignore input q long dependency detect feature late gate closed carry content mechanism helps detect necessary capture dependency sequence goal rnn often fast move former dependency ideally rnn capture short rnn partition group implement allow module operate mean module operate precisely connectivity module module b conventional stack rnn propose generalize allow model adjust connectivity consecutive rnn module module evaluation representative train minimize make dataset build english wikipedia character character protocol test vocabulary character token character average bit stack lstm able execute conventional stacking task program include loop logical end character output respectively evaluate sample difficulty sequence target finer grain analysis symbol architecture stack architecture different affine long short memory lstm constrain conduct extra experiment unit unit stack pt stack ex ex lstm stack ex character rnn epoch parameter preliminary result momentum coefficient either case norm gradient update minibatch memory compute update indicate statistically winner different lstm stack feedback feedback encoder mapping output fed encoder rnn state encoder rnn rnn hide rnn encoder feedback connection encoder rnn unit lstm layer unit mix difficulty sample epoch validation prevent evaluate generate length level test contain ht seed stack lstm communication comment increase vi bi pt comment mask material clear feedback architecture try lstm however fail rnn performance fig curve model second rnn train feedback architecture progress number hide rnn optimization propose feedback layer stack rnn lstm validate global layer without lstm conventional stack global confirm importance adaptively qualitatively stack lstm train generate text subsequence character seed read follow probability symbol seed character stack lstm lstm ten seed stack lstm fail trial lstm close tag type stack lstm c stack lstm stack rnn feedback gap rnn recurrent make compare performance architecture rnns multiplicative rnn lstm rnn unit result note vocabulary remove tag
propose encourage science report experience simulate phenomenon often regard ability understand move student complex provide thing assumption check student individual change statistic degree share connection association acknowledgement nsf around calculus thank suggestion comment early cm mm mm em end rgb mathematics college challenge education abstract th look back lead use chance reflect education growth science help ensure datum abundance keyword american history compute education american association mean city besides chapter association ever well exception knowledge ground discuss original agent american education fisher work improve surprisingly since exist somewhat figure american public health survey record record identify health graphical report display survival age four population next primarily neighbor early hence unnecessary cause take year remarkable display display modern methodology algorithmic benefit far familiar rare statistical among many statistical material diverse many formally prediction challenge peak north united population census peak widely mark experience challenging practice turn present stand association internal ensure well encourage development interest analysis lee science progress field ability visualize analyze report hundred thousand encourage growth statistic degree particularly master display student complete master line degree field master level historical program statistic development meet demand distinct add mathematics recent ensure education student complete four year master hundred thousand worker report come small growth decade continue likely insufficient meet demand definition matter raise position formally paper challenge familiar drive broadly big mention page mine big fundamentally traditional remain bag central decision relate familiar development st ensure solid I agree appropriate address suitable analytic interpretation rational need involve connection area visualization datum familiar indicate competitive disadvantage position tend job description cs student tend computation datum easily continue student quantitative topic challenge next generation student think google student course statistic application nan colors ms activity often really mirror world technology hundred thousand school student beyond calculation realistic certainly technology datum address early expand simulation computation I incorporate program issue commonly analyse work understand issue area bring great lack limitation student generally conduct trial make conclusion consider student trial situation student bivariate advanced student statistic program statistic student understand principle statistical student basic principle student mid average test college sometimes figure unconditional association statistically one score ci lower hide behind factor student student teacher teacher tend relatively school student take plan college state act sign average expect ci right student tackle student understand understanding increase principle often another option use student take display group student medium teacher group observe recognize estimate multivariate discrete observational around student tool school without interpretation learn computation develop refine practical theory student analyze aware student need develop mind think work student determine execute implement challenge cycle iterative involve data exploratory interpretation argue load summarize precise visualization report check visualization student technical visualization visualization tool force need beyond manual development decrease difficulty stay preferred student first ensure program excellent place integrate addition help refine student motivate statistic expand master intermediate must new student able life long learner provide answer may provide simulation free aspect allow student answer
context partially enough copy infer fill report highest achieve sampler confident discover start decrease likelihood match accounting find infer infer structure prior complex regular structure recover square express similar starting translate use process summary contribution infer report experimental generate geometric kind generative book group include cut away material product view coordinate specify locate discover understand find meaningful shape structure finite outline definition college google college microsoft model highly symmetric high transform object follow shape sequence compose product compact achieve part process theory shape processes align generative shape propose image term jump chain monte computation feasibility limitation model hierarchy entity notion sciences elementary particle document letter paragraph chapter book music piece etc room may reality exhibit clear mind frequently complex group product shape discover hierarchical representation understanding shape vision graphic long history graphic language create infer history term language account noise ambiguity graphic reality difficult make paradigm focus good graphic geometric tool graphic copy underlie lose full generative consequence preserve structure book generative propose graphic totally idea generative eventually go point far possible shape explain term totally change sub repeatedly complete incomplete shape building super structure introduce stochastic formalism generative give shape appearance process product hence perfectly align hand sketch infer process pixel image jump mcmc approximate view appropriate theory include group concept introduce noisy infer underlie strategy reversible jump framework describe generate geometric characterize shape lead generative object memory set action model algebraic transformation nonempty together property say neutral g gx group transformation act simplify extend third unity affine express matrix multiplication axis rotation origin discretization q horizontal call translate along horizontal direction continuous element copy point transfer location act produce always desirable draw segment translation subset present picture group allow unit discrete often move complex square transfer line side draw square wish transfer group create copy transformation act copy square process start top side imply line segment origin complete history start side cardinality translation transformation map translation generative continue describe element transformation copy orientation construction model initial shape generative history transformation want employ respective operation binary refer act index way let action index implicitly action first correspond history transfer transfer associated description situation shape display integrate theoretical copy application colour part shape like transfer time formation maximize circle square already rotation form origin thus square translate intend want rise characterize trivial origin responsible origin top g origin rotation fold rotation responsible produce abstract mathematic concrete suitable hence mirror fold ng implicit origin structure circle object coordinate multiply deterministic list list concatenation f group model replace add create process account arise generative history intend shape transformation present history account try transfer intend product application generative history exact copy transformation copy receive perturbation noisy thought coordinate subtree act copy repeat transfer transformation big cm cm trivially continuous version translation embed transformation obtain copy I shape bayesian specify uniform follow particular element bound uniformly segment circle vector instance dx noisy sample independently grey infer history shape represent describe generative shape shape belief kind shape leave likelihood trivial domain address use computation abc evaluation follow set simulated live match give reject inference recently specification idea next image define problematic sharp exact match close zero likely discriminate close solution proposal unlikely overcome idea bayesian highlight follow illustrate table generate process image result keep mind although approximated rp pi parameter end assume grey intensity locate interpret purpose reversible refined proposal idea assume upper great impact appearance shape keep level level go level structure u play match look amount keep change scale sample infer might standardize unit implicit interval concern proposal product propose shape keeping change prefer random keep act keep level act nature proposal keep nested structure greatly simplify acceptance
sequence shift parse prediction long principle sequential synthetic practitioner see good structure candidate greedy stanford speech advantage reduction greedy fast label feature map linear high scoring learn index product sufficiently use pair optimize feature model template confident order template score high scoring class aim template template need template pick template group technique compute template scoring label q contain associated label score eq q familiar machine minimum rank ahead notation label rank condition class tb template parameter train initialize test tb ii compute ahead label predict label rank return whole template split subproblem give order template wish learn template want template optimize combine score encourage calibrate stop early encourage single receive equivalent treat simultaneously high template optimize max per indicator define loss add regularization strength speed margin decrease speed large high accuracy three approach template template call norm regularizer encourage template discard strength regularizer learn order group strength technical solution slight abuse terminology induce alternative approach pursuit modification sparse nlp data orthogonal pursuit stagewise stagewise lar linear gradient correlation residual template fit error rapidly ultimately prefer development meta stagewise approach learn order select template development template procedure template hyperparameter empirically find due necessity inspire research cascade score scoring pose whether interact efficiently use requirement incorporate novel estimation separate add approach score add cause overhead cascade use increasingly detection reject sub window early score final avoid incorporation field time budget employ suit e extra pruning offset method confidence nlp stage consideration whereas pruning cascade output context cascade increasingly cascade base parse context nlp template selection test dependency parse technique parse decision graph enforce parse valid employ feature speed prediction dynamically model focus method field general static nlp regularizer template key template test inference problem compare improvement non boost achieve time prediction nlp entire template still require template nlp labeling solution speech parse name entity achieve multiplicative little baseline stagewise x fix base greedy speech employ learn regularization development baseline tag comparable stanford approximately attain divide template template stagewise template start template template add create separately baseline template template pos accuracy provide worth single optimize give speedup nearly could hyperparameter desire learn dynamically template test token template maintain template complicate template hand speedup setting depict template time present template train inference predict template accuracy dynamic template fix demonstrating learn act template group orthogonal pursuit case group correspond experimental setup evaluate group nlp detail good dynamic parse name entity recognition pick template early order pick template parse experiment template pos tag lemma assign head stack token second parse use development stanford pos tag use baseline pos model achieve accuracy lemma automatically label unlabele exclude achieve score speech dynamic train template prediction template dash speech time dynamic parsing label produce list table margin time fast remain time fast maintain template well dash demonstrate predict template successfully fix dynamic greedy right entity template surface feature lemma look token training tuning encoding set tag hardware experiment speedup score test fix fix dynamic learning dynamically select speed structured algorithm fast nlp gain come little art work remove template dynamically score different center fa annotation finding conclusion recommendation express author template discuss setup group nlp template compare pick template pos concern separate prediction rather dynamic successful achieve high template possible dynamic template term method predict template predict baseline speed initially template single instance multiplication template widely different amount time create hash string conjunction take feature behavior interesting template selection help norm naturally bias template include template stage template quickly source arise impact cache weight performance induce template order template run place highly predictive template front template template produce template order unable validation feature order per template offset pursuit pick stagewise order template add generalized template perform attempt find add fit nlp efficiently invert covariance template residual scalable feature problem combine design algebraic group feature matrix template template call template template repeat appear difficult problem template hundred million expensive special nlp feature template hence trivially invertible nlp model pick high template poorly reason apparent notice find subroutine essentially un try template dependent inversion matrix
control slope vary relu learnable relu eqn equivalent relu motivation negligible relu contrary adaptively learn jointly hope activation introduce channel negligible risk variant channel one variant introduce formulation layer represent term gradient deep activation summation channel share sum layer due negligible forward momentum learn tend bias toward relu learn constrain activation conduct deep study e choose sufficient category feasible train relu convolutional conv fc implementation follow imagenet pt c conv pool conv conv conv conv pool conv conv conv conv conv fc fc fc channel wise channel gain relu channel channel share channel introduce compare counterpart parameter critical role gain adaptively shape activation cc top relu channel share channel show coefficient layer two interesting phenomenon table conv conv significantly filter conv texture detector response believe level limited number filter deep conv small coefficient gradually nonlinear early discriminative deep stage network traditional sigmoid activation robust remove obstacle mostly draw distribution deviation difficulty report team experiment pre conv deep local intermediate scaled initialization linear relu follow sound relu initialization extremely deep conv fc converge method forward propagation derivation mainly idea conv response pixel channel filter number connection number filter weight bias pixel activation mutually distribution element product expectation worth relu lead relu put eqn layer initialization design magnitude signal expect proper standard std initialization layer relu layer layer conv layer denote gradient channel vector gradient layer assume symmetric back relu l l gradient eqn eqn relu derivation put eq exponentially zero mean need still overall product sufficient use eqn eqn eqn w w number scale vice versa x axis training epoch center relu activation case initialization lead convergence start main text relu initialization red blue verify gradient converge epoch discussion forward scale factor final signal infinity explain deep network small perhaps appropriate though benefit initialization foundation hope helpful input fc fc conv fc designing table also list modification three conv large feature last roughly unchanged iii spatial pyramid fc pyramid number evidence result augmentation comparable conv feature gpu takes per mini batch k deep extra conv layer wide version substantially complexity b four model improvement degradation depth lead layer speech recognition deep fc similar degradation model extremely layer error run deep degradation depth small dataset suggest conv layer conv severe overfitte attribute training mostly image short side randomly per random color scale fine tuning begin deep initialization help optima decay dropout mini test multi testing dense window pool fc pool slide window average far combine scale multi gpu parallel parallelism conv fc fc perform fc layer fc necessary parallelism besides parallelism introduce overhead fast fc single mini decrease speedup x speedup class imagenet contain rate except rate metric cc relu top top relu wise fair comparison relu epoch also epoch middle relu consistently almost cost comparison next single result publicly comparison relu good believe mainly end well even multi increase width c table improve become factor top relu imagenet team post competition relu imagenet evaluate c team competition net post competition imagenet set multi six train inferior considerable margin top evaluate server publish well winner represent successfully pay attention mini recognize image row col predict label specie display class due intra misclassifie still unchanged human top imagenet train aware existence test class title exceeds report human level human challenge analysis reveal type human come grained class job fine grain recognition specie dataset row grain recognize human recognize human specie algorithm still human require superior particular dataset vision object recognition elementary category believe recognition microsoft microsoft com essential neural network aspect first propose improve computational little derive consider deeply wide net imagenet winner
row index pz represent marginal particular represent pz column pz z way singular vector look projection square sum square standard result algebra third tensor standard path observation draw hmm tuple v calculate eliminate eliminate introduce leaf eliminate continue node eliminate eliminate sum leave marginal structured tuple hmm condition separate form form chain lemma justify recall u I assumption full along root assumption rank imply infinite sample extend path triple tensor full likewise h u u I I u u tree hide hmm generate meta meta equal generate meta meta index meta index I second tree consider index tuple index backward b z matrix contain kronecker second identical note total j px pz pz j pz complete universal following suppose triple input initialization matrix node p h h h h handle inequality meanwhile h triangle handle upon previous assumption follow h u triangle inequality inequality result u triangle h u u h inequality ta u u h u h algebra perturb version tensor iteration input regard decomposition rank completeness universal constant structure perturb q appropriate permutation put canonical appropriate column w w triangle third fact mc tensor permutation conclude provide notational simplicity identity recovery gm I gm I inequality triangle fourth algebra bound z z I triangle fact algebra assumption call obtain condition take get randomness permutation u u p matrix union first recovery accuracy observation final least c item meanwhile u u u inequality inequality equation third inequality follow last inequality equation transition handle already u fact choice u p u u u u u v v first triangle second inequality matrix matrix theorem corollary song chen university efficient spectral motivated mark type markov hmm cell connect main naive spectral cell exploit property hide state spectral current biological sample experimentally biological nine human algorithmic idea variable mark type mark preprocesse bit genome type hmm efficient advantage interpretability extend model relationship type biology relationship type comparative share motivated manner hmm hide state structure tree associate hide tree parent bioinformatics additional node emission biological main hmm genome hmms expectation maximization hmms computationally many hmms exponential property biological achieve key treat leaf improve typically low depth tensor technique full leaf use reveal node key exploit independence emission path version tensor hide model product projection tool implement biological cell result em spectral hmms individually assess spectral spectral rank extend spectral algorithm condition hide model hmms tree model topic mixture involve provide algorithm design hmm parameter hmms modeling sequence probabilistic model hmm position hide comparative jointly structure hide state represent correspond formally represent tuple direct specie represent vector parent denote parent parent structure condition initial iid sequence determine likely typical moderate possible assume node th product whose th symmetric I n j shorthand j j use notation denote co frequency correspond x u u u meta meta meta represent px pz z denote co occurrence meta denote empirical novel tensor learn hmms decompose version third occurrence provide globally solution occurrence co occurrence yield symmetric tensor whitening decompose sequence steady state idea use third co tensor modify learn hmms structured state directly hide implementation design plausible observe generate underlying sample sequence steady observation tree plausible actual instead show achieve exploit idea observation node root maintain correctness naive state naive root occurrence instead node capture root node path project skeleton path occurrence meta appendix even construct take time procedure construction operation take small complexity technique generate individual matrix hmm describe path emission thus store space first therefore construct construct onto dimension could gain biological vs projection co occurrence meta matrix projection would projection differ property efficiently good key observation consecutive hide structure svd denote path root co occurrence h u u recover product technique beyond hmm graphical condition hidden model biology expression sequence successive observation sequence pair triple np assumption generate typically certain parameter parameter hmm tree hide state condition wise ensure root path joint assumption require succeed kind assumption believe future work hold consistent enough learn consistency algorithm iid generate high sample observe run spectral gm follow motivated cell eight mark cell vector poisson background length eight segment combination number hide interpret encode goal discover segment probable cell similar calculate co occurrence successive observation formally use projection round matrix reveal appear tree hmm run memory perform co library application biological compare spectral approximation report structured mean spectral matlab take take iteration suggest spectral variational spectral focus observation matrix row mark condition identify discover weak background state large background mark state interesting biological
nmf smaller add svd identifiable negativity np reference array parallel enhance identifiability represent exist briefly operation unfold th form stack row vector index back example three different appear literature ease number column wise product explicitly q generalize accept two argument help notation exact form rao hadamard eq slight notation loss separable hard incorporated require multi regularize alternate technique cyclic special square iteration discuss especially advance descent consider factorization treat problem become column never calculate cholesky gram compute column forward substitution form respectively cholesky finally substitution take implication least grow grow go least cholesky decomposition nice unconstrained square improve one method qr complexity numerically exploit cholesky decomposition problem cholesky structure tensor efficiently exploit expensive computation j algorithm form rao mode available adopt qr square favorable loss good propose algorithmic factor many possibly st objective monotonically stationary uniqueness exception notice explain coincide improvement update block decrease time commonly propose solving call successive ensure simple put proximal convex strongly strategy formally prove speed unconstrained sub form problem simply sub update everything tensor set ease rao product explicitly problem well want term easy thus omit right appear update introduction universal admm solve optimization follow iterate update equality user review admm reference therein admm split yield update distribute constraint incorporate set outside define indicator admm strongly admm step start least reformulate introduce variable admm important save computation cache dominate forward substitution complexity around become projection proximity operator constraint especially often update constraint reader non negativity wise handle induce ij ij element thresholding want impose structured simplex constrain wise negative projection column smooth add super involve inversion set admm square converge fast approximation obtain naturally take admm reduce alg see cholesky decomposition computation alg dominate update complexity alg essentially calculation plus admm alg order r termination general termination residual admm terminate alg adopt solve introduce notice follow scale dual constraint constraint corresponding lead also interpret least proximal adopt element update list define miss case index fit uniformly corrupted corrupted outlier noise heavy laplacian function resp huber fit huber loss wise leibler adopt loss leibl proximity operation wise k family divergence proximity try fit therefore square loss detailed admm summarize alg termination alg everything least square loss close must minor albeit drawback dual computed admm pay admm scalability consideration become common scalability mind develop big datum usually store regard fortunately commonly become portion huber regard corrupt sparse array close early memory efficient routine propose multi summarize alg alg cycles iteration far away update warm strategy mode operation alg suggest sub iteration precision want actually copy approach adopt store readily without adopt save depend store completely copy early big instrumental proximal fold help loop accelerate convergence admm need stay work initialize use alg previous necessarily alg appeal alternate optimization plug play per admm practice include dominate distribute line preliminary alg fed alg disadvantage alg admm save amount computation initial stage problem derive alg alg admm perform ghz gb discuss world store application treat netflix movie rating movie sparse customer movie customer movie unseen rating recommendation relaxation involve nuclear norm provable nuclear tensor tucker rather tucker value uniquely preference aforementione incorporate netflix movie bias recommender easily equal one movie type tensor completion formulate constrain traditionally handle one problem inefficient even unconstrained case use expectation start zero iteratively fit rank prediction rank recently toolbox use variable relate miss treat loop however admm despite similarity illustrative generate contaminate impose negativity emission loading three chemical use toolbox fig satisfactory cause systematically indicate movie rating consist rating movie include split factorization evaluate correctness absolute error mae attain mae available rating kullback fit fitting compare impose negativity negativity bias seems rate evident negativity reduce rank fitting criterion play role bias prediction mae report admm recommender right believe admm matrix quickly signal represent possibly example fourier compression relaxation procedure recovery resort dl well benchmark clustering dictionary thus svd patch hundred dl formulate q sparsity induce g cardinality norm conceptually scale ambiguity inherent impose well atom th column sharing solve optimal cyclic admm sub routine alg separability handle separately previously cholesky warm ensure sometimes study focus large share alg replace thresholding train least square accelerate
section work fix vx vx help require submatrix z ex kt x write discrete notation similar true need result ex element independent sample theorem discrete I bound binomial note look term assume remark conjecture exercise science pa need norm main gauss main main main norm start review width characterization provide expectation process widely study generic analysis gaussian random one width expectation dual value dual property hull scalar property orthogonal property width connect gaussian net cover every small inequality number norm eq q low converse generality analysis constant choice converse inequality width e lemma move analysis full symmetric net frequently sub random sub exponential random sub value gaussian ex ex ex sub exponential follow center variable center variable center constant gaussian ex gaussian variable norm exponential sub restrict establish relation restrict width start rsc theorem characterize belong optimality q far inequality r z rearrange complete size recall atomic g isotropic construction mm r translation invariant clear noting complete ex recovery compatibility lemma rr rsc triangle inequality sub add q n rsc r add norm compatibility imply plug inequality back section theorem r width square design u sub least length gaussian xx independent isotropic linear mapping equip isotropic convenience need upper expect satisfie lipschitz concentration variance element ex ex ex ex similarly bernstein convenience gaussian gaussian eq combine immediately upper inequality g z apply choice determine ex let assume ex ex quantity norm gaussian show width time gaussian set k ex ex absolute direct complete ex isotropic independent p design isotropic sub sub gaussian ex isotropic gaussian pt bound derivative respectively conversely ex net previously follow result along denote ex result interestingly constant em lipschitz application inequality change inequality instead type design matrix analysis theorem modification sake independent e start center u v chapter form form q theorem proof process eq direct unit condition vector ex design sample literature case sharp gaussian rely constant row note na converse next net spirit lemma lemma subsection matrix conversely ex prove result complete matrix isotropic interestingly continue scenario intuitive p allow problem net cover ng function integrate g require x proof isotropic net cover choose converse convert extend result use proof eq conversely eq follow along gaussian design loss gaussian matrix design sub isotropic random ex ex lemma problem suffice ex ex covering complete pt set ex ex ij x p maintain previous ex apply capture constant norm ij suffice n center sub covering denote ex design isotropic dependent ex show dependent sub correlate na constant ex ex aa entry ex number surely diagonal quantity random minimum maximum
popular key rank recommend list bandit approximate regret recommend bandit rank independently propose whose decrease rank bandit combinatorial learn bandit bandit feedback non parameter bandit recommend observe context study variant whose reward know study optimization whose study action learn finitely many environment unobserved environment bandit view partial outcome hypercube base definition rule b te te together counter happen event optimal fact moreover gap decrease value sum tt tt te optimal item upper confidence complement thank suboptimal argument suboptimal optimal second suboptimal sum lem modular induction claim suppose induction factor decompose expectation product definite integral decrease analytic note minimum proposition observation engine list web page attractive cascade attractive item formulate two prove upper algorithm derive low match upper problem violate web recommend list list click user item user optimal list item maximize user find attractive cascade click cascade agent know time list observe item user click user click agent receive agent attractive bandit feedback feedback rich reward know item attractive attractive five contribution formulate learning cascade sample algorithm semi motivated expect probability web gap fourth bind regret upper finally perform cascade model like regret addition prove relate page engine rank web historical click scan page item web page many search click user differently focus web user attract click item user item attractive maximize attractive item user click practice click item cascade explain several extended explain click cascade cascade attractive reasonably click towards understand online cascade cascade solving simplify write bold problem formally tuple hypercube recommend bandit item time item click user item attractive q particularly eq click click click since click list weight recommend particular note say distribute independently mean item assumption consequence attractive express fa item evaluate item list maximize list simplicity exposition set two solve bandit motivate recommend list large eq discussion probability et te recommend get te te te compute probability number step confidence since compute initialize list specifically algorithm regret suboptimal prove regret discuss result item ground set setting first item item probability suboptimal item item hardness whenever convenient list main difference list choose instead exist attractive define follow item way entry right first event outside bind fourth bind base suboptimal step n four step item apply suboptimal extra finally derive contain item product parameterize suboptimal problem consistent suboptimal without generality suffer polynomial bandit achieve logarithmic instance algorithm regret time condition result observation counter event unable instance thus get conclude attractive agent sufficiently attractive practical close bound item attractive item item bound number recommend item recommend proof position extend upper problem asymptotic low bound p p pn ucb factor eliminate order validate upper item increase even violate rank qualitative value tight accordingly recommend item decrease motivated observe four trend number regret recommend item trend bound outperform surprising payoff arm confidence interval tight bernoulli get close rr order ucb recommend rank item attractive lot feedback could reward depend arbitrarily report decrease future satisfied model datum cascade recommend list recommend last user item
mention instability erm result heuristic stability make incur erm expectation erm exhibit excess dropout problem convex glm precise generalize detail privacy concern machine potentially sensitive medical health record privacy notion effective privacy r change one insight private convex algorithm expectation dropout eigenvalue convex function erm assertion erm training stability random adversarial removal interestingly work complementary dropout adversarial removal adversarial perturbation test feature experiment stability cross regularization uci repository theoretical see performance enjoy desirable stability implication strong dropout preserve privacy privacy setting regularization require aspect setting dropout complexity organization dropout excess lead use empirical provide rigorous class descent actually dropout optima gradient procedure training neural approximate polynomial exact measure w distribution g define neural hide layer underlie architecture assume polynomial point perform f dropout local perturb effectiveness heuristic stress section instability dropout perturbation entail procedure minima help minima reduce reach one link neural represent weight fx constant dropout perturbation degree proof error polynomial polynomial identity notational purpose polynomial along need concentration z provide bind ab problem polynomial complex use polynomial network comparative complex additive oppose multiplicative dropout modulus polynomial ensure oppose either dropout help encounter gradient section model neural descent excess risk x value convex risk excess risk excess descent vector exact dropout algorithm analyze variant capture regularize setup analysis strongly dropout variant enable excess function excess risk tb dropout sgd initial sample normalization ii I excess heuristic dropout draw learn randomness sgd excess ip gb outer randomness provide section excess bind dropout dropout plus use argument dropout square notice even scale risk general strongly hope demonstrate dropping node neural theoretical understanding dropout heuristic behave regularizer underlie problem rate asymptotic taylor dropout behave regularizer rate risk assume come underlie modeling problem polytope parameter provide precise heuristic essence provide fourth open pose aspect rate covariance show differentially private convex private optimization significant attention allow privacy differential privacy privacy information neighbor private measurable think privacy induce output randomize one intuition consequence medical record adversary learn absence privacy output provably good generalization fit dropout detail example linear simplex brevity refer later framework convert privacy guarantee tb set simplex nx ij loss dropout level change possible output slack sample privacy since minimizer refer neighboring differ closeness relate binomial non coordinate exclude ratio bound chernoff happen binomial closeness analogous closeness closeness ensure differential direct implication differ privacy differential laplace differentially test along dropout differentially private function direct tail long p exposition tune optimize linear treatment cccc evidence support dropout result fraction capture belief network dropout perturbation form removal remove removal randomly select fraction absolute refer result remain dropout misclassifie show result value dropout dropout outline exhibit stability version even dropout datum regularization provide appendix adversarial removal complete set removal adversarial removal dropout least study observe dropout decrease rapidly tend similarly regression
million label image object huge convolutional tool cnns significantly medical report store modern picture communication diagnostic mapping hundred thousand create volume remain primary goal extract associate semantic datum mining deep scale database image knowledge report mining scale report document patient history contain imagenet manual google accord prune crowd amazon meet label demand annotation task privacy categorical semantic modeling allocation lda interpretation patient provide patient labeling categorization specificity report image feed forward cnns train work building scale database text image please yet medical interpretation effort learn image object attribute crf annotation computer vision feed cnns recurrent network text cnns medical show benefit domain feature significant conventional sift medical key categorization label describe sentence medical make publicly medical image annotation image convert pixel label local intensity bag block ct study work model vocabulary recurrent image label embedding minimize language train imagenet correspondence imagenet reasonably high dataset dataset predict describe annotate unlabeled dataset present annotation medical semantic document cnn comprehensive semantic use available store national health center year instead patient two dimensional writing notable finding correlate diagnostic semantic unique show occur report leverage exploit make automatically patient mis extract report process nlp sentence list nlp g image basic nlp total retrieve match whereas rest extract modality document k k see mass k k small note propose categorization label text report imagenet often mostly ct show high intrinsic ambiguity define assign semantic sub million report mining correspondence originally propose article method document extract topic among report lda flexible learn coherent study regard special hierarchy common hold unseen document hyper topic document score generally fit score evaluate document although balanced image unbalanced specifically topic primary variety body contain image address result lastly topic mining sentence adjacent sentence score keep count small beyond figure categorization label demonstrate good semantic coherence among list key review validation document topic topic disease primarily disease brain disease mix modality mid concept part diagnosis low level visually image topic imply heavily sub disease mass tumor meanwhile document many sentence semantic image imaging imaging associate semantic addition include image association refer figure supplementary text investigate plausibility via cnns category framework split validation cv test divide cnn learn normally rare imaging protocol topic different document level sub topic map systematic diagram semantic topic image cnn challenge reference slight million consist five follow max pooling layer fc final cnn significantly deeply convolutional imagenet number softmax level sentence tune pre imagenet semantic fine imagenet pre medical modality ct help additionally cnn cnn document level topic less imagenet document topic trace cnns initialization learning cnn imagenet transfer image finding different modality verify imagenet annotate medical date quickly iteration unbalanced among deep already initialization via imagenet closely relate fine less cnn part newly initialize new low rate layer rate set rate key image spatial resolution image level learn classify lda induce different shown skip mapping word train hierarchical softmax slide window frequent word diverse set article learn well keep hyper finding robust learn diverse query close term cosine article show report word cosine list variety mostly disease highly diagnostic disease exploit disease term sentence word disease relate gram description trade medium complexity word show l report reference digit bi extract train cnn vector multiple bi gram per sentence image one bi gram bi ignore annotated sentence bi gram relate representation detecting object configuration employ map bi match disease vocabulary cosine bi gram convert cnn minimize cross line form gram recurrent text vector formulate regression softmax output adopt cnn simple tune cnn predict modify cost text classify category newly layer bi gram bi testing topic document level topic cnn top word map bi lastly topic second half cosine similarity key word high cosine bi disease match actual word k k example figure categorization report score work r word nlp describe sentence truth association association description cnn automate patient generate sometimes generate mining sophisticated nlp parse specific disease section aid interpretation scan nonetheless analysis disease automate added mining disease semantic predict cnns softmax describe disease disease rare exactly disease unified medical language terminology associate resource service sciences create maintain library base comprise concept concept name incorporate control organize define link semantic type consistent categorization concept represent choose structure medical unified retrieve imaging report medical record vast concept word appear find disease mention detect assertion algorithm detect absence clinical determine text scope occurrence find disease detect assertion disease find derive occurrence unchanged occurrence unchanged occurrence find occurrence decide disease occur report disease absence similarly challenge match disease sentence softmax function softmax normalize exponential generalization value softmax among cnn imagenet assign disease image disease absence disease number occurrence show c c mean per mean std std transfer helpful tune topic cnn fine level testing cnn absence top mapping word generation match originally disease specific detect high disease top infection automatic label assignment image sometimes statement possibly would unclear present apparent sentence derive test match originally figure four six contain prediction coherent high top figure label visible characterize support failed detect nonetheless label assign unclear statement due challenge unclear image predict second visible figure second high automate mining enable predict compare image modeling labeling loose couple disease image strongly less find image pair probably mass tumor detect detect image unchanged big loose label rather loose label us word help
rnns rnns formally state previous linear eq non conventional easily use train put aspect trying finally value compute illustration rnns neural time generative define input write distribution state function practice back generative generative recurrent neural generating phase unfortunately al optimization issue train model dependency problem back propagate gradient exponentially zero vanish large hoc restrict go trivial decade tackle rnn speech exploit unit flow state element gate candidate sigmoid matrix show h compare problem rnn capture long dependency vanish problem activation try period attribute time previous input hide unit show two input solid dash clear target signal dataset simple motion mit motion dataset generate position consist information orientation generate row unit toolbox visualize datum frame train recurrent neural unit layer generative fashion feed first feed frame real average generation phase figure solid target dash line use step unit recurrent neural dependency dataset mit recurrent network conventional reference motion rnn recurrent problem machine translation learning model temporal
receive passive remarkable indistinguishable normally implication leverage algorithm like passive bag ever since statistical dominant paradigm learning discover blind partially leverage passive inspire treat visual experience activity learn process visual cast term stream sensor semantic signal correspond sensor video camera feature mapping figure convolutional neural cnn exploit recognition image learn sharp visual feature hand target input seek transformation orientation operation sift cnns rotation powerful representation balance much loss representation capacity impose space exploit observer like signal prior learn learn scene furthermore unlike transformation oppose consider exist whereas application explore application jointly apply three public dataset pure recognition challenge learn accuracy disjoint unlabeled car datum task dataset bag favor special wherein transform invariance valuable descriptor sift aspect cnn like hand design invariance shift rotation image instance preserve operator slow learn vary slowly video gradually adjacent frames temporal cnns dimensionality recognition metric supervision perturbation idea method exploit video couple signal achieve general feature design learn operator design explore descriptor plane rotation observer learns aim sort pose illumination transform auto encoder explicitly object part similarly graphic train autoencoder supervision bottleneck layer variable novel look like method limit sense pose encourage impact unsupervised recognition quantify cnn response specify transformation affine adopt use exist descriptor space train transform infer bilinear multiplicative learn content motion encode autoencoder video neural combine future frame tuple individual transformation whereas abstract make recognition interest computer vision though none signal pixel foreground person solely apparent image exploit robot object reinforcement robot movement exploit learn respect transformation associate pose pose capture may encode observer camera pose position roll subset read sensor pair algorithm pose j j frame multiple video category sensor translate pose pairwise pattern annotation sec define precise seek enhance recognition neural first want pair discrete motion pattern motion pattern might collect control camera prefer approach leverage video end discover pose difference frame typically apart detail simplicity apply though motion pose motion pattern choose speed panel video dataset move angle camera consist center cluster car primary turn forward wish motion motion respect exist feature correspond pixel correspond movement particular direction outcome structure encode fully layer work focus motion pattern observer movement world appearance observer motion motion alone depend depth scene object scene frame depth maps observer motion appear difficult especially newly even target discrete limit preserve every atomic atomic right motion pattern turn head motion diagonal map motion among restrict pattern train design map naive translation would decompose optimization cluster correspond summation deal pair annotate problematic perfectly trivial evaluate learn simultaneously negative motion pattern mode apart mode belong bring close input location highlight temporal hyperparameter encourage nearby motion frame coherence target like passive hold coherence explicitly describe generic representation task addition pose annotate together map denote softmax softmax probability case unsupervise bottom tie stack supervise softmax specification map architecture layer optimize sharing parameter network pair feed initialize identical gradient pass epoch weight tie stack encode wish train array represent stack map motion pattern input stack motion addition softmax replica stack weight label fed stack fed softmax depict implement sec recognition dataset view atomic center accuracy repetition right view outperform baseline dataset selection sec softmax loss adjacent set distance motion combine baseline like popular feature unlike passive video pose recognition additionally access similarly baseline receive pool unsupervise datum dim convolution fully layer bottom two dataset image clean systematically camera image pose vector camera discrete pattern cf sec respectively pattern yield create pair positive negative pair treat neighbor video sensor car drive city road select training validation pose consist position forward velocity output discover pattern frame apart retain turn pair positive equally train apart validate camera frame see know select object split select image split recognition categorization associate category scene scene recognition image unsupervise beneficial label follow convnet recommend cifar architecture architecture relu nonlinearity nesterov accelerate base rate select result repetition motion normalize denote perfect evaluation feature validation atomic training us pattern sec atomic composite invariance absence supervision tend nearly novel transformation optimize easily atomic composite small test transfer first improve classification previous worse obtain unsupervised gain three significantly nearby frame see epoch neighborhood validate effectively space offer exploit frame pattern exploit first require perturbation systematically result challenge noisy pose dynamic traffic object enter scene class road task web image category mostly indicate generality achieve trend show preliminary good view robot move help recognize object neighboring view object uncertainty fact behave identify task detail feature easily baseline qualitatively pair near retrieve pair relate example variety top neighbor pixel difference exhibit transformation turn rotation pixel distance perhaps wrong change box large foreground motion decade method focus almost bag image reflect ease crowdsource though valuable informative physical visual experience approach learning generate show learn beneficial great intel would thank work show substantial domain camera road camera content around belong diverse scene category nothing image color subset text main processing nesterov initialize select identify base fix regularizer retain loss loss uniformly objective include loss set margin motion scalability margin equivalent optima scale
rmse rmse tune good extra tuning search rmse study investigate efficient determination rbf mahalanobis base kernel geometric observation map predict introduction traditional forecasting trend feature normally solve forecast many area uncertain use rbf kernel overview grid search pattern search directly extra pre exist searching overview practice form obtain consider indicator acceptable tolerance natural map build x kx kx j map value map mean space deviation te norm observation describe difference map help te convex reach n n k x jx j mahalanobis kernel gx cx x k use balance performance accuracy deviation function fit see figure value begins map rbf bad x finding solve define reduce number critical big value figure table rd c th th experiment company contain description determination rbf r scale calculation employ rmse error h rmse tune show result solve time area
binary generate build independent input similarly difficult find htb boost boost table boost generate tuning observation term generalization far concerned svms propose toy simulation performance eight diabetes set diabetes ten independent body e create contain instance per business average e concrete create I age etc strength fourth cancer cancer al receive predictor eight clinical cancer etc logarithm specific one measure one response benchmark spam uci repository spam attribute use spam diagnostic breast feature datum identify whether contains attribute instance boost boost diabetes spam select build remainder set evaluating performance randomization time number table iii parameter type simulation easily outperform boost algorithm among real coincide experimentally numerically comparable idea road improve boost lemma completeness proof positive ii contain eq let mean derive contain proof lemma fr differential aid divide two step deduce taylor expansion mf k mf mf mf mf convexity use fact along eq note mf mf mf b follow assumption derive arbitrary mf mf k f mf mf mf mh f f h select later hold get v applying obtain aid ball span origin q radius result proof inequality function q q z x mf confidence assumption q c set direct kk numerical rate variant boost name boost focus alternate scale theoretical outperform theoretically numerical property numerical outperform boost imply reasonable boost boost follow throughout essential reader highlight due theoretical may outperform conclusion partly perform second totally direction improve boost boost guess variant boost work issue report progress future publication accord introduce parameter degree facilitate choice boost recommend zhang yu naturally ask good answer paper practically important slow theoretically think select via strategy leave concrete role reveal support foundation china grant national cb lemma remark boost scheme combine accurate prediction rough one aim develop accelerate rate consequently improve show possess numerical sense tackle problem tight error show boost rate common explore response model loss state activity boost combine produce underlying rule combine boost regard wise fitting additive connect boost problem correspond minimization overfitte problem bit prove rate lie slow hereafter boost accelerate capability include boost via linear truncation specify fix purpose propose accelerate near call boost always one idea greedy essentially yu shrinkage impose composite weak learner help type scale boost classification experimental verification classify performance four aspect deduce numerical result show near secondly capability essentially boost justify build restriction provide flexible experimentally modify outperform boost paper organize compare algorithm study behavior error bind section verify conclusion discussion observe consider eq q expectation learner regressor boost weak update repeat although show slow nonlinear search make show walk exist angle walk comprise strategy control obvious scheme appropriate small numerical aforementioned strategy control rate consequently capability idea iteration regard extent impose operator strategy main idea set shrinkage step compare scale hereafter call shrinkage find greedy relaxation relaxation name relaxed brevity mf mf mf norm exist depend easy verify widely focus rate assumption constant depend therefore least function certain orthogonal logarithmic relaxed optimization slow however well knowledge whether loss exponential logistic cost convergence check give select shrinkage degree definition follow may shrinkage particular al different algorithms check correct finite integer constant select brevity turn consistency boost whether approximate arbitrary depict show consistency certain stop approach bayes impose absolutely derive give stop proof constructive simple stop fairly slow converge speed play role estimator kk consistency remark firstly secondly simple method truncation note computation truncation widely boost however drawback usage entail element truncation indeed estimator help estimator show number deduce study boost regression modify version look vc highlight
operator lasso assume covariate lasso coefficient theoretical view several lasso communication use output inaccurate outlier replaces bound huber mean instance penalty require easier propose novel modeling outlier add corresponding penalty robust linear outlier outlier optimum penalty good robustness result local penalty thus contribution penalty avoid optima estimate optimization recover true coefficient statistical remainder organize theoretical analysis estimate coefficient output performance throughout symbol norm respectively nc cb cb na n linear tn correspond outlier vector correspondingly coefficient match assume zero introduce penalty tuning encourage small outlier type tw use algorithm initialize l converge stop return optimization rewrite express smoothly scad penaltie explicit soft scad penalty recommend pt output thresholding yield penalty q cost solve fast computation first reason relationship state penalty huber penalty scad penalty give characterize property go infinity penalty scad non penalties mcp penalty yield good non suffer able minimum global avoid problem directly computable outlier provide first property solution notation doubly exist row shall provide depend require error identically sub satisfy assume error model gaussian thresholding handle theory negative scad mcp introduce concrete order lasso type simple however preliminary accuracy calculate reason parameter e go analysis matrix detail see condition bound correspond may become exclude threshold jj version hold meanwhile sufficiently simulation examine carlo tuning tune practically tune candidate generate design impact scenario various consider draw true outlier draw independently draw k scad fp fp tp tp error fp tp tp show simulation support coverage preliminary percentage outlier threshold support coverage percentage increase hard fp tp fp fp tp tp tp fp tp square positive positive tp preliminary soft hard never vanish investigate outlier exclude preliminary outlier see moderate quite support interestingly would error extreme preliminary preliminary well opposite true error bad property positive carlo simulation outlier various magnitude positive omit procedure high magnitude come low would hide draw magnitude note magnitude pz g condition correspondingly subscript simplicity I c preliminary go l k k doubly imply q q solving recurrence
leverage score condition matrix equal form eq aim decision minimize make unobserve noisy somewhat approximate sgd computational stochastic monte sa approximation sampling sa follow point idea erm exposition describe regression stochastic recover sgd result basis form randomize suggest apply either sa sa choose assume subproblem compute randomize algebra alternatively sa sample fashion descent sgd solve choose follow subsection sa exist solver simplicity constraint generalize nontrivial choice basis effect erm stochastic problem original require naive choice row equally work toy undesirable first element row sampling require fail large leverage score put row row leverage immediate score recover propose algorithmic leverage regression formally include result result produce sample independent obtain hard show update weight update constant depend affect restrict regression follow undesirable sgd optimal uniform ap ax z ax increase add sgd naive distribution matrix grow condition basis I leverage optimal idea distribution leverage score lead main section main see step implicitly appendix refer find algorithm remove randomization consideration exactly norm alternatively norm assume leverage row accord call sgd phase improve domain instead notice simplified iterate last tb weight sgd determine regression output problem condition range score pick update return objective regression corollary regard result mirror proof result condition computed satisfy computed algorithm return suppose iteration eq x x return dx rf suppose exist traditional algorithm corollary error score factor distortion apply reason expect inverse magnitude whereas empirical important role original choice error consideration subtle simplicity restrict vanish error per sgd dense matrix norm show factor norm section typical algorithm compute condition provide overview condition conditioning score without time provide bound complexity sgd per return approximate solution evaluation real synthetic design diagonal response corrupt gaussian since convergence assessment convergence rate include implement different full diag detail leverage score rate phase diagonal mirror implementation done grid search run upper leverage score apply leverage recover appendix gx rescaled since broad combine sampling assume satisfie conclude connection comment exist sample deterministic stochastic extend problem translate sample size hinge exponential dependency sensitivity approximately result idea need similarly type develop randomize algebra sgd uniform sgd empirically preferable medium precision bound regression point direction work extend would like office advanced project department energy providing condition implicitly norm condition implicitly qr exposition high full embedding well solely way recently short nearly sparsity plus low distortion available compose matrix see ccc type dense cauchy cauchy sparse qr implicit compute form reading row norm norm multiply case lastly additional main corollary theorem f x algorithmic leverage central row state enough result failure independent leverage output approximate well condition pick theoretical text equivalence first observe write follow v equivalent hard verify thus cast stochastic suppose condition range define base estimation run sgd pick generate bregman working apply rule later algorithm perform hold simplicity pick q update weight algorithm satisfy relationship know p solving notice particular unconstrained case actually update let show simplicity use rf see relationship proof rhs rearrange complete equivalently employ rule thing hence complete simplicity hence theorem like hard rf case rhs condition basis zero rhs proof corollary putting recall size choose evaluate satisfie e consist view follow item know intermediate desire old condition basis definition also establish q r binary vc subset lie intersection n less next element fix sensitivity case brief overview mirror sa main sgd exposition convex dual distance generate convex continuous continuously common distance function next define bregman sub r norm stochastic composite mirror descent iterate result analysis appear lemma step q condition side complete output particular fy finally convexity expression minimum rhs fy fy complete stanford edu recent year gradient sgd machine applicable applicable paper regression construct distribution sgd process maintain computation effectiveness consistent theoretical finally also need similar problem descent sgd attention strong performance practical sgd new unconstrained case deviation least unconstraine eigenvector iterative depend unconstrained formulate interior scalability algorithm theoretical thus flexible subproblem solve implementation method precision size two algorithmic approach develop take strength hybrid consist step construct weighted preserve sgd quality sgd convergence low run objective empirically perform medium quickly lp lp sgd sa node online well condition q descent sgd sgd fast q leverage sgd result solver potentially us structure view perspective formally I e answer approximation draw deal sa stochastic approximation mini sample weight useful basis solver weight sa algorithmic improve exploit strong capture construct leverage condition immediate
candidate close possible aspect fit reference need model reference reference model reference calculating project individually approximate explanatory power large relative explanatory explanatory effect predictive general reference discrepancy cross validation outside search search leave find leave indistinguishable precisely denote utility choose estimate utility reference suggest report drawback model time demonstrate may useful investigate predictive ability behave formalism uncertainty specification give list model obtain model speak form average adopt discrete integrate procedure empirically variable context combination review posteriori generate optimal zero utility otherwise type ii propose name median contain marginal include variable sum variable define choice close average squared author admit define prediction mean median problem discuss term utility think fix utility depends observe lead k unbiased biased often beneficial model successful utility selection generalization even black dash due get bias variance utility prototype grey represent model different realization true dataset red due utility choose maxima become far away true optimum overfitte fit demonstrate though optimistic maxima grey selection bias utility increase close true optimum though beneficial model approximately overfitte induced bias important concept little attention discuss validation idea utility depict plot utility considerable become practical selection discuss model show illustrative regression binary involve apply gaussian probit cumulative normal include intercept term analytically integration perform probit markov monte carlo problem convenience binary denote variable specification dimensionality construct bayesian reversible jump adjust belief description section important concept difference correlate rest weight irrelevant get approximately l p cv optimization map maximum posteriori marginal probability median ref discrepancy small explanatory power coefficient regression parameter perform test realization generalization predictive performance negative bad variable respect perform poorly model bad worse comparable dotted line conclusion cover green albeit result high utility method reasonably ref ability ability small empty intercept realization htp replication line denote choose average htp training grey realization black dotted insight closely cv cv row start find high bad word gap two curve empty cv utility average overfitte selection decrease grow error term visible save posterior probability much cv select model especially able variable predictive close result projection selection cv still predictive ability figure selection model function reference model reduce variability substantially close another appealing grey line exceed large main predictive study truly irrelevant versus average ordering seem seem necessarily vary figure model unobserve help explain predictive different dataset summarize deal regression zero log normalize negative target population output value discuss dataset refer list relatively uninformative weight reversible jump mcmc first uninformative last favor posterior plot idea decide include effect prior cancer perform l problem dataset due replace cv importance cv loo cv reduce search model repeat leave measure observation set training cross time sample performance validation time time result mean htp htp dot intercept fold credible dot cv row carry cv loo cv htp select perform point test utility remain grey test split black dotted vertical line variable summarize result select demonstrate cv overfitte overall tend desirable perform marginal division projection performance close tend red choose selection bias black conclude measure five probable three seem indistinguishable small input sample performance projection figure able variability large searching variable kk use outside line utility estimate dotted method also difficulty model choose utility variable size discuss section size estimate validation outside search induce bias estimate independent searching perform projection fold performance fold cv hold small satisfying kk variable merely organize differently final credible apply sense utility worst credible remain suggest substantial case small suggest stage despite effort believe search highly give searching recommend superior review selection performance binary select good predictive real overfitting phase may happen variance utility especially cv relatively reference well complex observe simplify less selection probable variable combination retain complexity despite place automated come good incorporate well correct uncertainty regard prediction projection seem robust search reference selection make problematic predictive minimization discrepancy find solve outside search allow study informative way produce utility estimate emphasize depend input cost control thank helpful manuscript acknowledge science compare widely practical selection highlight recommendation prefer classification perform several optimization relatively utility uncertainty project outperform variable demonstrate benefit cross selection predictive model statistical adopt identify true useful usefulness ability future would concern ability still try assess numerous assessment review review qualitative compare preferred believe study give insight exist article present subset regression discussion selection popular bayesian literature posteriori maximize selection ability loo cv widely unbiased predictive information error none model construct uncertainty simple give nearly prediction selection reference average reference kullback
machine master node parameter solve master carry step classical master local vector multiplication communication master form overall never master vector close small cg eigenvalue base closeness symmetric positive identical suffice implie classical terminate hold terminate choose bind eq terminate norm usually upper bind local iteration discuss complexity master equivalent equivalent end round communication communication precision everything together study efficiency round machine bit machine aggregate back master proposition communication algorithm communication ignore constant round communication notice communication round call corollary call bound extra one compute give good condition adjust inspire describe bound communication self satisfie let large q call q call involve round expression desire call roughly twice follow variant newton algorithm instead simply latter choice direction slightly inexact method two communication round stepsize size round communication work converge still complexity define know eigenvalue inequality imply inequality responsible complexity small result corollary since two round corollary algorithm inexact newton satisfy total communication round q bound quite require slow relatively achieve depend objective upper norm may scale bind global erm scaling quantify local improve minimize assume refined space compact derivative bounded formalize constant w w regularize imply proof analyze erm assumption constant choose respect generate high iw w iw quadratic iw w iw f lemma hessian generate hold assumption apply establishe remark radius analysis concentration instead vector satisfy conjecture especially dependence ready stochastic bind communication optimal take generate self require reach ignore constant suppose terminate iteration denote event respectively law event appendix algorithm return depend eq separately obtain order combine convexity additional remove inequality corollary specifically c put everything replace communication specification formalize regularize loss self minimizer constant communication round require parameter automatically tune least proof ignore desire practice hard replace inexact inexact newton replace distribute bound round communication give combine two consequence local sample help make similarity without need obtain loss rescale self convexity rescale scaling factor grow rely lemma balance scale expect round binary hinge loss q least square round strong result self need initial constant find algorithm input manually choose stop manually tune parameter admm scheme size gradient rule calculate progress communication end follow progress ccc note communication per admm communication l consist round communication another search stepsize communication aggregate solution iteration loop round begin inner interested efficiency plot progress reduce round admm bfgs converge fast bfgs speed comparable grow convergence become slow coincide iteration proportional regularization sensitivity relatively exhibit study standard sketch composite minimization convex simple nonsmooth admit minimize composite use k newton reduce since inexact quantify error follow minimizer set need remain devise inexact minimizer long modify master eq auxiliary condition round locally master solve propose equation accelerate utilize similarity algorithm converge composite inexact newton implementation accelerate proximal state erm cost machine evaluate distribute round often grow due regularization cause sample propose self linear classification inexact method inexact conjugate number communication round grow popular empirical self self function inexact characterize implementation inexact round practice theoretical consequence initial require objective experiment confirm superior communication addition equivalent hold nesterov use function value bind derivative combine inequality relation convexity upper inequality inequality inequality combination assumption sequence fw fw suggest come tolerance need proportional tolerance inequality right eq bound tolerance w arrive part immediately combine combine apply inexact asymptotic occur agree since conservative associate assumption hold whenever eq denote small integer iteration decrease constant side finally suffice terminate terminate v left side side desired prove recall imply inequality population risk sample empirical minimizer population suppose modify replace empirical sum ii fact lipschitz condition q terminology function fact value symmetry also combine result error independent first convexity therefore gradient average zero eq used w variable variance sum equal inequality ii fact v finally inequality last fw di w fw consider loss define assumption iv origin ball radius ball center belong eq point cardinality hessian matrix eq component upper hoeffding triangular union side yield desire number upper bind simplicity assumption similar q give corollary claim microsoft optimization minimization empirical communication overall local loss inexact inexact conjugate gradient self discuss ridge regression smoothed hinge supervise size slowly many learning need access whole process grow happen involve optimization storage machine solve optimization distribute rely inter communication generate whose probability support set distribution deterministic large identically suppose sample machine refer erm linear predictor label loss hinge stability term make loss hinge locally alternate procedure communication simple map operation computation communication speed consumption goal communication round sum iterative two communication per therefore twice constant smooth call quantity characterize ill condition descent master master gradient step iteration accelerate technique alternate direction method multiplier admm assumption strongly admm complexity turn accelerate admm accelerate iteration machine iteration complexity scale admm communication rich distribute high complexity similar distribute method assumption thus obtain look note complexity exploit optimization local zhang et approximate simply communication efficient achieve allow regularization stochastic seem ill motivate propose distribute iteration logarithmic paper propose communication efficient minimize newton convex continuous derivative g point average write gradient master node prohibitive use compute cg specifically master due cg direction newton communication loop outer method loop cg inexact use newton erm linear predictor report cg iteration communication first consider outer newton method analysis reach still depend problem function loss iteration inexact second cg take arrive inexact accelerate admm overcome exploit cg method spectral master depend characterize similarity erm show general effective converge bring overall algorithm self optimization popular loss include logistic erm list communication require several show weakly exclude logarithmic compare round ridge binary hinge accelerate admm deterministic except I improve review self popular empirical loss either self analyze inexact compute inexact use distribute communication linear classification list finally extension distribute theory nesterov interior call derivative derivative third limit convex self q self self book self follow lemma self rescaled self self regularize double parameter binary convex self exist self fw q need appropriately additional since strongly convex bind substituting immediately rely long point freedom broad next take loss smoothed loss concrete example logistic third conclude self self favorable hinge loss smoothed hinge segment smooth lemma derivative second mean accord self scale function initial nonnegative sequence w w satisfied analyze inexact newton minimize loss standard addition exact newton computation newton inexact separate perform approximately budget centralize machine newton system characterize detail strictly
whether spurious necessary selection equation collect verification j covariate avoid identifiable impose statistic regard nan level reject distribution multipli point distribution spurious depend one critical associate q analytic quick validity typically fit residual view nan adjust I nk multipli bootstrap approximation carlo employ analogously eq n ns require intensive large avoid compute note find commonly maximize infeasible quickly hundred thousand trade computational intensity branch pick variable say screen procedure implement rare application computational cost standardized vector covariate dimension take take let upper quantile spurious direct characterize extent bootstrap summarize sd simulate quantile calculate replication simulate bootstrap spurious correlation multipli approximation table good spurious correlation isotropic case sd c identity focus matrix number rescale definite vector covariate simulation table summarize deviation simulation bootstrap fairly heterogeneity covariance estimate sd examine multipli bootstrap serve benchmark discovery spurious spurious rp pr pl post lasso covariate fold select sdp model ii replication compute sdp simulation depict dependency collect highly correlate add difficulty lasso severe reflect sdp fit discovery section life whether technique spurious correlation individual chinese international project introduction think study take response particular ten parameter fit lasso pl pl estimator observe fit quantile approximation bootstrap replication though pl decrease discover discovery direct probe disease correlation value fit response empirical multipli bootstrap approximation sample ccccc pl r observe solid observe correlation dot percentile median bottom percentile blue bootstrap indicate red residual employ nan time multipli bootstrap value evidence depict collect prove proposition line n constant condition fulfil tc ns p b every view term pd concentration upper expectation apply absolute every least I every argument prove take c ns bind semi refer introduction tight note successively lead eq hold previous display k prove anti concentration supremum index anti inequality respectively proposition let random satisfy q j imply take completely deal investigate standardized counterpart ip mean induce ns ns center index variate aforementione consistently estimate supremum prove k ns ss k hold c ns follow ns ss sp p hand side reduce happen divide discretization prove net denote ns notational q display imply c ns obtain right side together suffice apply combine together note p follow display complete theorem n inequality maxima sum play important analysis three argument aforementione discretized finish anti establish approximate net let metric together coefficient ball eq yield carry approximation discrete nd v v g g g j j g corollary random lemma eq three imply turn direct put together c meaningful far put least ns ss follow borel subset p u variant take write n event c together identity deal lead q cardinality observe modification imply least sufficiently side multiple least theorem absolute take proposition theorem section remark section mathematics chinese sciences chinese decade find group covariate mine spurious need validate need derive correlation namely response combination covariate possess process hence approximate unknown residual fit spurious testing covariate mining test model result bootstrap false discovery technology change massive store dimensionality characterize statistical science machine response economic finance statistical method behind datum heuristic example base explain group impossible thousand model consistency restrict homogeneity property despite rarely false scientific set mining spurious fit observe green numerical international ac uk take expression gene alpha expression dimensionality fit tuning cross validation value fit sample post fit response remarkable fit spurious diagnostic covariate residual lasso figure assumption spurious covariate subset correlation random noise sn spurious maximization linear fit maximum block correlation identity top importance spurious recognize distribute random point use demonstrate spurious grow quickly simulation study pp asymptotic analytic depend bootstrap tp e certain correlation approximate center hold ps pn p increment ns ns p establishe size condition ps ps nature establish limit statistic chi freedom integer integral express particular last vanish proof proposition place carlo simulate multipli n normal random observe variate gaussian covariance distribution triplet theorems spurious correlation approximate multipli practically relatively proxy
demand target target storage meaningful content layer separate branch layer multiple resolution section explore dataset thus primitive class detail make branching learn target dataset keep l graphic os hardware hardware layer branch class popular make build corpus represent histogram occurrence word vocabulary disadvantage working vocabulary thousand text lead poor use dense tool skip group similar output entire remove language stop represent word vector post average layer branch create hide bias layer layer unit function vanish relu branch softmax logistic well loss target give figure training cost generate involve situation loss experiment cost minimize also start depict simultaneous training training box plot hide final target hide target testing table box simultaneous branching appropriate level cause meaningful improve training show simultaneous minimized branch enforce information branch flexibility modularity scalability exploit meaningful meaningful representation branch convolutional work computer vision ideal connection desire vision break detail output branch hide exploration neuron directly target branching work output output useful computer practically useful electrical technology neural branch hierarchy final target branch provide enforce help final shared layer modify branching layer flexible inference level situation level result accord paper level target make use branch neural network multiple number make raw effort pre network abstract character layer deep easy problem get optima vanish hyperparameter properly tend proper play important deep layer belief network unsupervise pre component restrict boltzmann rbms tuning model tune supervised training regularizer initialize idea network target level target high learn meaningful layer helpful content activation present deep conclude scope future layer branching target arrange hierarchical fashion learn
mnist decay dropout structure testing weight standard th reference implementation pass dropout mc dropout literature model stress different error none fig fig seem ip cifar convnet dropout suffer passes dropout tendency change randomly subset small evaluate dropout fit mnist network alone ip blue dropout later mc convnet ip dropout fig seem perform dataset fit comparison well well dropout dropout probability mc convnet model publish art convnet fully follow suggest mc two achieve year convolution operation follow interpretation extend function effect encourage name imagenet assess cifar replicate paper evaluate standard mc pass dropout potentially report also section standard average mathematically monte average forward argue dropout mlp dataset follow suggest contrary augment convnet significant achieve might exhibit standard pooling considerably explanation prevent convnet kernels mc optimisation converge thus possess fitting additional explain section lastly worth note long training test average forward pass application hardware allow dropout almost trivially could gpu mini dropout bernoulli unit matrix network generate multiple pass average weight convolutional neural robustness fitting placing convnet intractable approximated bernoulli exist tool require include interpretation convolution might relate exist convolutional furthermore imagenet datum use affected probability would mr comment google european remark identity ex ac convolutional convnet offer place convnet filter approximate require interpretation improvement classification finish art result cifar interpretation neural network extensively literature offer offer small mlp design lead however usually amount kernel also convnet vast commonly perform approximation example approximate try leibler model follow past bayesian fairly computational expensive approximate parameter make increase costly bernoulli computationally surprisingly field interpret gp extend dropout bernoulli variational allow principled convnet dropout layer model layer weight layer implement convnet dropout layer derivation forward mc contribution work numerous dropout connection practical convnet structure approach additional reduce fit small technique lastly dropout convnet literature dropout approximation improve state result insight follow briefly discuss implication convolution review relate inference bernoulli approximate variational denote softmax loss matrix optimisation term add often weight result objective often refer input variable take value layer drop binary binary tool bp psd covariance function layer approximate map gaussian write treat parameter treat variational randomly set binary indicate layer drop encoding value linearity element approximate distribution gps map deep gp explicitly hide unit gp obtain bernoulli precision parameter relate derivation convolution operation equivalence extend beyond operation model interpretation one placing layer matrix mlp eq interested tractable need define approximate variational approximate define layer vector distribute optimisation objective log kl divergence kl divergence encourage explain keep fitting carlo distribution extend development th convnet dimensional extract input input weight convolution matrix k arrange w k pooling nn model variational vector bernoulli set kernel dropout element pooling convolution bernoulli bayesian study insight implement bernoulli considerable improvement attain dropout mnist
equality zero row skew e ratio row row skew smooth analog transpose dataset generality take digit argued long confirm condition fail yes yes yes stock various set last establish sampling find sampling hybrid well plot various clearly digit stock datum via truncation sampling truncation turn truncation however show sampling threshold control noise good threshold reality control threshold hybrid predefine threshold sample produce iterative table list use figure restriction digit various set one pass memory compare sample I incoherence value make list hybrid sampling leverage average respectively leverage align result approximation component us work maintain variance rescale benefit parameterize hybrid superior quality show score distribution suggest optimal distribution align accuracy hybrid leverage digit digit principal component preserve digit category digit rank hybrid digit data c c leverage finally superiority optimal hybrid sampling size hybrid sample support pca digit runtime figure pca digit visualization top digit category take intensity project digit show visual list digit visualization principal component column compute visually close visualization pca project onto approximate similar actual pca h data visualization onto five leave project actual onto projection quality pca produce sparse digit pca digit respective finally matrix element wise leverage score leverage score element wise I score ii hybrid leverage average increase get gradually improve quality large produce variance hybrid bias towards regularizer maintain sample rescale need counter rescale structure show benefit parameterize distribution superior hybrid leverage distribution suggest hybrid align datum require achieve turn digit principal preserve digit category digit datum superiority hybrid sampling leverage sample size superiority hybrid rank h hybrid align table overall present indicate superiority extreme wise also usefulness analysis task fast synthetic real element randomize new sampling recover pca data hybrid ability strictly well performance give recover pca datum entry one user preserve want recover provable matrix order principal subspace sketch top quantity control pca sketch effectively address partially principal additional top iterate benefit ij ij start bold bold g column denote n ij k standard basis ij p ij ij ij sample sublinear product receive parameter identically distribute replacement hybrid return purpose addition noise argue strong linear trend capture top remain top equally low approximation low show respectively svd note problem ideal want structure preserve fundamental time dataset element heuristic properly element probability principal pc project pc pca via provable fast approximation pc synthetic promise along situation element construct estimator svd reconstruction surrogate norm quality pc enough bound matter absolute sampling reason small rescale would huge rescaling result poor keep e entry entry remain simple elegant proof approach could truncation argue would bernstein hybrid wise essentially propertie element bernstein main truncation wise sampling balance flexibility arbitrary parameter desire result generalize sampling know stage sampling pass sampling datum discuss one streaming pass algorithm hybrid sampling fix arrival stream give implement hybrid note element produce indeed sample element f note respectively event I ij j ij ij ij clearly ts pp element event pp note correctness theorem need estimate one case additional parameter twice require sampling requirement obtain triple create proxy need provable computation pca unbiased svd reduce consequently theorem show algorithm sparse sketch pca theorem pca surrogate follow theorem last inherently wise preserve original result derive various element wise let score become rank datum mix various derive sampling accuracy measuring hybrid sampling score respectively pca approximate denote also nr matrix also computation experiment binary noise specifically construct whose
need reduce unique depend increase contain element add signal recover call th without generality optimization problem eq reformulate impose constraint nonconvex subproblem converge alternate subproblem objective function formulate subproblem programming subproblem standard form c eqs definite subproblem problem optimal exist qp solve problem implementation scientific package physical present problem old old old extension line section perform figure signal distribution obtain multiplication mixing generate noisy perform matlab follow three label nmf multiplicative build iii nmf nmf normalize signal respectively perform comparison nmf note signal matrix nmf one three along figure scenario reconstruction capture kind rank however increase one fail notion monotonicity source noisy extend negativity quadratic framework illustrative nmf well exhibit behaviour indicate mm mm definition remark remark thm cccc institute technology nonnegative factorization mixing suffer separation increase mix nonnegative effective suffer order approach nmf alternate assumption relax nmf nmf nmf source nmf source nonnegative factorization blind separation widely area sciences environmental systems biology blind separation text analysis wide applicability fold negativity nmf interpretable seminal lee nmf extract signal give blind bss objective source noisy matrix since nmf suffers order incorporate semi impose negativity constraint nmf multiplicative several algorithm alternate project investigate improvement nmf signal scenario monotonicity constraint nmf semi investigate resolve monotonicity nmf signal semi semi demonstrate datum future incorporation entry matrix factorization nmf monotonicity signal illustrate compare nmf bold capital letter letter quantity denote column permutation column transpose denote indicate nonnegative number dimensional identity vector zero element indicate bss source contain number consider factorization often corrupt noise write contain give signal however bss mix matrix bss
california la ca usa ensemble play unlabele encode identify single suppose binary classifier space prediction accord expectation consider label unlabeled marginal assume bad correspond erm motivate test predictor know prediction apparent classifier case word rule improve rule case distinguish without label give characterization minimax prediction give unlabeled development matrix matrix make minimax example paper organize introduce formalize intuition adversary characterize minimax side slack solution link interpret slack minimax providing give build section computational discussing conclude main paper context hold probabilistic ensemble unlabele denote allow rather change intermediate interpretation extend predictor knowledge development constraint use nn probability simplex column ix h j formulate game predictor first play randomize adversary play predictor equivalently view summarize study follow model label infer apply game linear equilibrium side strategy let weight vector magnitude slack equilibrium strategy strategy slack duality minimax minimax predictor study completely characterize partial purpose specify weight weight convex desire accuracy prediction prediction alone slack test depict function analysis label uniform add h correlation unlabele datum test correlation concentrate specifically classifier e failure thus p subset intuition ensemble make seem type strategy adversary equivalent predict learner predictor increase spirit adversary match adversary hold equal margin qualitatively continue learner perspective subdifferential slack slack function differential weight geometric interpretation equation give weight five taking difference sum negative obtain exactly sum category prediction simplicity prove improvement algorithmic unify automatically see label c unweighted majority vote well vote algorithm consequence closely optima slack definition slack simply guarantee slack directly guarantee adversary label force flip solve game give know error noise weight us noise prediction predict example always statement asymmetric majority vote present step simply optimal weighting minimize slack slack straightforward treat programming storing memory unlabele typically without exploit sgd slack converge convergence suboptimal particularly intersection hyperplane piecewise slack memory convex play theoretically practically slack function limit duality one impose since essentially without multiclass expert weight vote nontrivial receive focus theoretical boost form classic vote purely margin label bad formulation bound slack margin among unlabele bound purely vote hypercube general benefit statistical classifier notably formulation moment thereby handle rich dependence moment long universal optimality linear demonstrate set emphasis set statistical stein new unlabeled individual show characterize function call slack slack computationally tractable streaming gradient descent ensemble support rather combine convergence rate programming requirement sgd bayes limit adversary classifier increase ability systematic core duality argument support describe adversary
appears rule quadratic entropy compose purely expand v naive follow section method value impact sort choose discard variance computation function assume locate close point ignore happen acceptable add value sort discard speedup densely valuable locate approximated work densely previous us interval bx ix preserve width need use bin width variance lead x x q assume small bin projection satisfie inequality similarly sort discard q behave particular change expect actual discard reduction htb go show function add enable vast exist technique adaptive bfgs constraint modification add additional corresponding maximized advanced sphere bfgs complex previously sphere norm stay suffer worth function additional affect state software able evaluate uci repository repository approximation code gradient cg cross validation randomly select across comparable optima acceptable accuracy balanced reduction computation report ratio exp cl bin diabetes easily sort discard pair denote much optimization projection distribute projection obvious theorem increase decrease heat figure original isolated error level introduce approximation significantly higher rare phenomenon accept error color approximate fact many notice consequence rough evaluation act like regularization table significantly simplify number number function conjugate especially line search speed cl bfgs b name bin diabetes heart claim small surface removal element summation technique sphere rapidly cg result order simple approximation fast objective
reason application approximately adjacent reality adjacent syntactic noun may semantic consideration embed view embed reason add option learn unlabeled information complexity simple predictor feature due analyze word triangle regard convolution layer adapt word layer region interpretation cnn illustrate view layer layer unsupervise adjacent region word skip gram train study semi appropriately categorization publicly internet review amazon sentiment label categorization training set unlabele unlabele disjoint review disjoint set make publicly internet article month unlabele unlabeled set semi cnn short type cnn fig base region unlabele data vector cnn learn unlabeled unsupervised minimized z I prediction output objective adjacent though concatenation adjacent unlabeled control purpose side word balance absence theory view concept relevant adjacent region syntactic relation undesirable sentiment meet assumption simple heuristic effective word proposition target vocabulary often lead word use word list provide rest seq seq perform activation convolution multiply top layer character convert meta portion data model r gram gram gram cnn cnn w convolution exclude word neuron view embed fix indicate meta tuned table meaningful comparison model convolution layer exclude seq cnn max cnn pooling report review cnn multi embedding show dimensionality embedding thing w clearly cnn confirm effectiveness framework sentiment relatively outperform w region large cnn use predictive suffer sparsity supervise poorly perform supervise cnn contexts word word help superiority cnn vector cnn word embed cnn embed explore integration embed cnn word embed multi view latter text replacement add turn add chosen replacement except illustrate view really g really view due hard reach training advantage w indeed learn appear combine table region layer triangle concatenation feature receive result nb gram nb lm gram seq cnn k seq seq unlabeled sentence lm ensemble nb lm unlabele unlabele k lm gram seq seq comparison previous knowledge paragraph use produce combine independently non ensemble table good seq convolution neuron layer seq seq cnn l micro macro extra unlabeled micro macro average multi split category comparable entire room disjoint entire test cnn many supervise cnn test compare supervise meta repetition cnn label extremely consume could co performance stop co clearly demonstrate difficulty datum method due focus insight text influential neuron neuron contribution view activate top negative sentiment neuron one poor view though negativity prominent multi show present explain word semi supervise embed cnn experimental decomposition rank matrix denote correspond relationship obtain eq third use x equation ph follow define theorem present feature embed unlabele usefulness explain word embed new learn multi view embed text convolutional sentiment categorization learn apply nlp supervise require train large amount therefore semi supervise semi implicitly svm produce performance via contamination another learn unlabele preserve function additional degradation generate mostly supervise nlp empirically embed learn unlabele additional often supervise nlp alternate structure unlabele auxiliary task improve task name entity often intuitively expectation insight development implicit shifted justification learn limited case paper present analysis useful task allow embedding theoretical supervise framework view availability view come definition view build model internal essence cnn learn region particularly multi cnn exceed art text categorization cnn view unlabele demonstrate categorization first theoretical observe view assume relax conditionally rank sentiment assumption concept concept view reveal informative relation view embedding multi exist predict word everything multi view exist view embed decomposition ph say hold produce view target original arise make task predict paragraph view independence propose supervise cnn learn small text come view build cnn suitable image internal structure adapt cnn convert word view learn categorization use word classification categorization superior categorization explore embed cnn word cnn one cnn cnn forward equip layer pooling layer layer token document region view later associate share unit training region concatenation one representation large region train seq center vocabulary save apply generate convolution embed dim dim max layer feature focus
straightforward note suffice verify eqs begin condition verify throughout lemma hold expand along note claim sufficient eq enough large enough second term eq dominate eq state condition complete proof calculation proposition follow expectation component specific combine result establish proof eq q firstly dominant equation suffice already prove negligible claim third first note negligible term negligible argument n w dominate dominate suffice dominate contribution proof straightforward fourth eq eq claim adjust moment tool control random edge order vertex first couple vertex w bridge vertex set I order face edge way remove face type keep face follow happen subgraph induce tuple show brevity write length pair face star label graph vertex valid exist labeling property label occur abuse write valid matrix constant integer enough probability rescale tr x hand claim one interested suppose exist eq rescale suffice expectation center vanish summation wherein twice vanish term prove rv rv r yield expectation permutation decomposition irreducible permutation overcomplete straightforward check decomposition matrix follow eq verify define eigenvalue prefer derivation span hold large enough span fourth claim claim otherwise spectral understand seminal os proposition could self contain reason note ab tr dm assign summation index cycle suffice dm every obtain vertex label connect unique labeling constant inverting yield claim complete recall expand sum compactly follow term pair product exactly tuple occur instance p g face vertex label class slight abuse vanish intersection useful consequently proof intersect every arrive first one concern eq claim bind r label wherein obvious labeling claim occur connect connect twice recall valid contract couple identify length hence complete inequality I suffice firstly symmetry take conclude proceed multiplication therefore finally proposition j mr mr couple r criterion edge repeat twice show induction every repeat twice label labeling follow happen vertex exist second unique neighboring identify induction range complete proof firstly definition define prove moment claim union graph isolate arbitrary iv v suffice label map unique union isolate isolated contribute label hence suffice prove bridge contain vertex vertex claim contract bridge length map label neighbor identify neighbor bridge length induce labeling induction unique hence unique induction finally term eq since vertex recall star term vanish summation length star note path obtain union path since label vertex deal difference eqs tr mr subset satisfy every couple pair generality vanish satisfy every label argument isolate unconstraine decide vertex consequently triangle yield prove intersection event lemma event bind ba application triangle x prove lemma hence prove ba prove proposition deviation recall define follow let bridge claim claim argument apply minor modification eq bridge length isolate vertex decomposition proposition complete favorable intersection favorable event proposition enough equivalently matrix expand correspond develop explicit yield guarantee eqs claim given consider determine submatrix distribution plant clique plant unbounde suitable despite substantial time succeed fails present improve unless change proof use spectrum os complexity challenge research focus bound study match assume unbounded resource fail clique submatrix category submatrix convention law give random estimation whereby special challenge machine clique plant attract dirac hidden correspond whereby presence absence induce otherwise hide clique section statement graph size exhaustive clique solve hand significant effort polynomial gap performance understand theoretic motivate hard rough clique unfortunately imply computational low instance careful preserve instance specific typically hide limitation statement rely completeness call somewhat change complementary attack prove unconditional low broad chain query formulation prove algorithm close semidefinite remarkably prove hierarchy clique write hierarchy hardness clique strong establish analogous sum hierarchy hierarchy relaxation similar close connection conjecture idea broad class many naturally within hierarchy whose treat propose construction solution bound moment unfortunately contain hierarchy spectral fail unless notice guess hierarchy whenever fall argument present impossible improve except remove logarithmic factor provide apply construction submatrix entry distribution combinatorial certain positive semidefinite matrix os subset depend subgraph degree relaxation consider positive absolute objective claim position derive formal degree maximum clique probability immediately clique far set variable probability large mention introduction generate hypothesis unnecessary technical entry order motivate nearly combinatorial look principal submatrix average straightforward eq state least exist eq fix adjacency matrix ij ij os graph gaussian density choose suitably far hold subgaussian standard small hence hidden hypothesis distribute note scale factor suitable constant high distribute restriction index abuse hence control random I key triangular weak decompose state cf essentially adjacency consist
tweet produce game n n vs vs games european english collect tweet game twitter streaming tweet stream relevant tweet guarantee meaningful involve popular filter game collect least tweet team hour processing yield dataset turn baseline final include produce ability evolve design capture algorithm language opinion mining previous sentiment capture prediction market sentiment analysis library system sentiment framework score collective framework social effectively predict unbalanced analyze twitter relative potential reflect sentiment crowd show around yield achieve negligible strict rigorous odd certainly large great odd grant margin reason part focus unbalanced game unlikely high increase estimate unbalanced game crowd towards enhance odd unlikely offset loss incur likely upon result decrease balanced plan extensively future support award gray media stream forecast outcome political stock fluctuation increase social access crowd forecasting power medium media stream automatically match focus highly argue offer systematic testing medium baseline despite strict baseline system sentiment exploit collect informed prediction behind framework twitter full stream monitoring match major european fa social media twitter source understand phenomena outcome yet happen political office market much failure social predict surprising collective crowd expense hand issue bias opinion directly arguably correlation predict twitter movie generate week g fluctuation market potential leverage medium real event unclear systematically address issue team event offer possible outcome match limit occur continuously lot collect medium million expectation future game social medium systematically implicitly reflect opinion crowd continuously reflect regarded opinion discuss validation occurrence unlikely odd leverage twitter game make popular recent twitter make online platform mention unlikely reason upon importantly arguably odd offer successfully leverage medium game six twitter six game representation discriminate game whose outcome odd unlikely odd translate margin crowd social medium properly odd present procedure specific introduce twitter implementation assess economic yield use collection work summary game la world collect historical repository contain entire monitor twitter dataset refer second live monitoring consider odd odd odd odds hill together odd game define course coincide outcome unlikely upper practical odd experience score correctly game turn generate score exceed arbitrary report finding game likely likely odd likely purpose world outcome game play game definition latter constitute subset former depicted fig game game table immediately see consider note whose odd team game potential final interestingly possibility observe table consist game extract twitter game discuss detail collection turn employ section information predict extra game france usa usa exclude penalty minute seek understand analysis match twitter occurring represent duration minute resolution try event event analyze noticed penalty exception event event decision spike traffic annotate event fluctuation collective sentiment sentiment consistently drop drastically immediate team twitter real high level idea media signal live rare user one user tweet contain team interaction interaction user group dynamic interaction group illustrate one precede useful outcome game exploit sentiment twitter predict potential argue sentiment help collective support social behavioral team express team working game turn sentiment produce precede tweet sentiment score range see retrieve twitter occur hour window minute sentiment tweet separately single sentiment team table live window pass windows minute small suggest significant discriminative turn sentiment minute minute start reader minute game news line weather etc outcome game window total window pass baseline baseline france baseline baseline baseline usa baseline hull city west city city qp city city baseline baseline baseline baseline baseline baseline describe potential discriminate odd collect result game draw consider classification approach explore classifier available library performance well good parameter feasibility advance deep even well performance live monitor train classifier perform twitter sample live monitor twitter streaming stream decide keep separate exhibit tweet twitter stream separately illustrate potential constitute dataset classifier near twitter promise full stream table show performance major european national live improve framework game randomize randomly label game game exhibit presence predictive prediction game solely upon sentiment specifically minute start significant prediction determine potential strategy inform medium precision recall precision recall score determine return systematically odd expert continuously adjust crowd systematic less achievable combine contain potential live monitoring european perform round validation strategy predict game turn team otherwise half half draw dataset realize therefore payoff respectively mean rather average deviation marginal bar surprisingly explanation wish exclude consistently classify correctly single return offset classification therefore analogous odd relative surprisingly high bar odd adopt fold potential game equally regardless systematic advanced strategy amount base proportional odd increase risk fluctuation systematic independently game team
turn satisfied may ergodic mcmc scheme satisfy avoid geometric drift give incorporate scheme last decade despite effort adaptive mcmc behind theory numerous development adaptive routine need valid counterpart establish deep stable mcmc rather modify numerous form instance model intractable equally solution standard simulating dimension grow technique may difficulty mcmc issue hide eventually filter chain algorithm directly doubly term acceptance evaluate exactly algorithmic field ising limit applicability suffer poor acceptance rate resolution metropolis replace intractable idea pseudo evolve iteration estimate proposal analogously metropolis accept otherwise extend enjoy specific construct present expansion offer likelihood approach understand current construct importance sampler thus improvement interest study term gap investigate efficiency derive scaling alternative intractable hasting accept ratio let dropping reject term monte preserve noisy quantify exact development likelihood big computing infeasible approach pseudo marginal filtering indeed filter target simple chain associate mcmc iteration transition since return estimator hasting q general notational demonstrate assess notion gibbs greatly particle follow use complex dynamic dynamical probabilistic approximate distribution base exploit resource seem possibility reach far beyond couple chain construct create adaptive gpu monte even contribution towards massive general explore direction approach ahead simulate next regular hasting back branch high deep path path create move reject evaluation acceptance fix costly computation processor high obviously instance helpful actual chain end approximation sub likelihood see capability find chain consist component seed augmentation auxiliary trick gain metropolis hasting index uniform pick select rejection decrease evaluate proposal ess acceptance rate jump free result processor investigate compare parallel therein namely difficulty handle dataset parallel computer outcome sampler asymptotically sense justification previous result accounting subsampling sampler produce convergence decomposition induce issue restrictive iid subsample contain author suggest final closely product mcmc approximation reason curse curse tail misspecification way mix drawback device avoid operate behaviour estimate likely behaviour true transform help tail finally seem target constant simulate could outside modal transform transform use kernel random related separately compute rejection merge spirit parallel mcmc break run independently parallel chain act converge parallel chain shorter set importantly target mcmc partition sampling regard list author need set chain move thus contribute integral evaluation unclear ergodicity depend e lead region chain stand partition challenge modal area mode indeed explore actually unlikely huge gap go partition hard last comment adaptive wang motivated highly complex method inefficient approximate situation detail variational likelihood version quite may feature come statistical induce balanced budget level evaluate depth wrong computational technique kind intuitive researcher simulate infer forward merge mcmc drawing summary play loss model gate field intractable model first quick elaborate incorporate toolbox nonparametric handle partly therefore reason everything else fail small summary summary sufficient statistic imply information formal rely raw raise wide application strictly likelihood genetic indeed intractable sense get intractable include auto exponential pseudo intractable deviation statistic location core abc see inverse concept conditional value true similar actual far abc actually involve acceptance simulate accept tolerance exactly posterior rarely achievable practice h prior accept acceptance calibration select realistic setting never rarely noise dominate example consider curse candidate leave abc algorithm statistic stress quantile simulate prior order distance draw constitute convergent approximation abc outcomes interpretation decrease precision universal second perspective output connect indirect purely nonparametric yet pair median plus special sample conjugate create compute component properly impact inference discrepancy completely eliminate abc equivalent distribution top sampler reference algorithm local regression central simulation abc summary statistic tolerance rate connect nature already early tolerance precise near ergodicity lack abc replicate involve simulation replicate repeat subsequent gold around mcmc ergodic almost technical zero geometrically ergodic condition prior ball result auxiliary ergodicity efficiency simulating replicate however variance select model rely hypothesis operate demonstrate population choice hypothesis validation primary motivation address impact ik statistic normal fit laplace show factor converge phenomenon naturally summary mean intersect counter expectation summary help achieve show abc prevent model factor consequence simplify approximation beneficial variational bayes decade substitute family exponential sometimes close kullback distance considerable gain term difficult assess would meet past five year approximation operate laplace approximation availability ep start target likelihood group observation term member give current value propagation go select marginal hyperparameter kullback leibler stop stationarity understand propagation practical substitute avoid use simulated actual evidence default implement ep empirical one time tolerance candidate look multimodal posterior program avoid complex drawback amount extend pos te map area receive lot particularly signal processing optimisation computationally integration map estimator space compute calculate major development relate optimisation discuss also useful optimisation concentrate powerful method exploit monotone operator mapping carefully design refer excellent book mathematic optimisation processing time fail element bayesian vision treatment treatment uncertainty optimisation need express decision nature lack capacity fast parameter space focus analytic numerical challenge isolate shape grow bring possibility along code another area use algorithmic tool convex optimisation mcmc computational proximal first decade context posterior density high formulate take proximal subgradient satisfie define subdifferential ng ng proximity q gain analyse opposite operator proximity mapping proximity g g solve sequence mapping long type relaxation relaxation converge backward subgradient point forward construction continuously little place notice convex advanced proximal optimisation either proximal operate splitting proximity mapping find ng possibly algorithm involve linear operator positivity admit compute algorithm converge lipschitz unknown remarkable accelerate class introduce notice several optimisation forward backward project proximity onto convex interpret iteration taylor around case dual ng proximity backward accelerate imply lipschitz computational separable proximity overview implementation proximal viewpoint forward differentiable limit efficient proximity mapping backward many proximal proximal proximal arguably bayesian operate optimisation augment unconstrained saddle admm proximity tailor specific way g exploit decomposition proximal interestingly admm interpret therefore characteristic proximal optimisation parallel architecture invertible express coarse system fine e processor gpu also closely proximal notice lastly optimisation main topic optimisation gradient proximity complexity allow adaptive riemannian speed problem connection proximal development mcmc one connection modern optimisation greatly motivated matrix underlie combinatorial optimisation hard relaxed tractable original development modern optimisation statistical dataset show proximal bayesian recover noisy nh represent spread low power ill pose difficulty bayesian image process improper discrete compute vertical horizontal arguably typically assume associate marginal unimodal concave subproblem optimisation example implement h modern compute mapping use mapping implementation algorithm image enhance estimate algorithm vs compute popular power signal ratio db show solve admm observe remarkably sharp region pixel measure quantile langevin continuously mainly concentrate contour detect presence sharp uncertainty location finally figure produce map second conduct computer matlab certainly fast dimensionality concern apart obvious requirement mistake area statistical community decide become reality science rapidly huge human science part argue four article accuracy easy simply fine change count set spurious big solve old mistake ever mistake think bayes development us carlo exploit environment prevent potential modelling modularity big paradigm entail illustration potential develop bayesian tool answer meta problem reliably apply methodology theory performance limitation start example attract create centre mass away area ti computation time work regard complexity modelling factor often development support technique technique seem justify ask extent cut answer extent cut edge answer answer probably agree entirely something agree fail completely goal explain communication statistic computational people community keep statistic arise good community aware perhaps enough people work interface interface easy research problem write language strongly encourage develop way least without successful somewhat parallel past decade language meta language intend handle solution towards user language successful extent proportion fairly often fail concept bayesian like unable validate gap unclear go modify picture still model cover sound towards locally globally similarly address population program outcome area discuss emphasis progress need past year bayesian technique thousand application application bring constraint inference constraint hyperspectral explore validate approximated reveal therefore boundary algorithm develop simulation optimisation able handle part simple complex handle difficult new way also library technique quick fail practitioner retain inference uncertainty strength might fellowship intra fellowship distinguish france universit paris p decade see evolve move proposal langevin drift theoretical practitioner even dataset address difficulty handle ever dataset likely tool dramatically reduce raw capture aspect next reasonably computational start involve computational something raw incomplete past state future algorithm bayesian turn medium computation obviously challenge long mixture normal dataset algebraic derivation follow hard computer certainly towards computer decade tool need monte hard version back surprisingly much later early bayesian toolbox tool em despite availability computer definite cause partly lag become statistic community surprise significant flexible tool medium integral calculation toy conjugacy provide answer discover mcmc offer chance statistic quadrature develop special analysis precisely quadrature moderately appear paper issue relate sampling year extended generalised focus artificial intelligence drive ode tree aside research methodology branching parallel processor long appropriate formulation long indeed believe incomplete central massive computation abc tool section discuss progress issue approximate highlight modern lack less impact think justify discussion section raise science relevance mcmc could day become tool ergodic researcher rather reality traditional carlo perspective output regular monte carlo attention say development handle process advance accelerate parallel cloud computing within monte carlo take certain reach community compare group sound use output require remove lead close asymptotic meet tool monte drive reproduce metropolis hasting kernel machine return operational compute otherwise flexibility choose curse choice arbitrarily date efficiency complex limit access mala proposal combination metropolis parametric generally transition adaptive markov chain conceptually allow search transition available thus whole hope essentially difficulty practice markovian kernel variation
rd sigmoid rr static rr propose system identification toolbox matlab train remain evaluation quadratic covariance total art maximum take computer model know seem sensible trade load particular likelihood consider energy consumption day daily dynamical approximate converge wise exact gps converge follow converge q expansion follow coincide whole random argument converge proof reduce gram computational load fundamental hold right justified step metropolis within article matrix wishart prior inverse degree pdf sample allow specification bayesian form project set family conduct carefully efficient system identification competitive reliable quantification uncertainty powerful probabilistic behavior gps dynamical topic contribution year state novel formulation superior property variational gp induce attempt instead optimal nystr om underlie induce resort perform linear supplementary gp model define dynamical nonlinear n explicitly parametrization assume tackle amount infer strength gp framework systematically poor figure gp sec world utilize probabilistic nonlinear dynamical include state expansion state literature nonlinear model tool force present dynamical phenomena gps encode dynamical function oppose focus learning replace recent interest relationship propose parametrize pseudo induce expansion second use implicitly base procedure algorithm uncertainty learn expansion posterior weight em probably computational load small magnitude within minute load variational approximate rank favorable property prove outline introduce make representation gps correspond theoretically computational load reduce use synthetic contribution extension gps prior state represent model truncation provide homogeneous represent fourier employ relation pseudo differential operator isotropic operator proof feature hilbert hyperparameter gram matrix interpret parametric function gp function weight also give weight element infer posterior clarity model however extension well infer sampler although involve sequential rely asymptotic thank invariant accord present straightforward iw inverse wishart prefer different quantity brevity rigorously minor four algorithm technique infer smc along state form markov smoothing within trajectory na ia k na I weight condition parametrization available material qp problem inverse covariance function posterior follow analytically sample possible sample q sampling hyperparameter easily utilize hyperparameter metropolis hasting mh hyperparameter predictive result load define approximation basis provide convergence rectangular gp tend mean equivalent follow provide scale oppose furthermore sound property assume mh support formalize expect particle monte prove ergodicity associate example comparison method
human feature common neighboring image tf ic propagate information diagonal matrix incorrectly unobserved operation image code validate strategy participant system make interface build summarize participant shoot choose small intra class dataset expert discriminate leave specie collection capture year dataset annotate extract publicly convnet dataset imagenet challenge tune fully truth produce separate convnet dimensionality convnet perform fine tuned convnet additional supervise produce smoothly space well align student similarity benefit crowd ranking user however convnet balance reduce student code project conduct replicate result setup image ask interactive image ask label interface correct answer phase provide use purpose test image choose exclude set length test feedback student delay encourage drop worth note crowd annotation worker possess concept participant prior participant task knowledge moderate student image always avoid worker reject response testing encourage effort discard participant participant baseline outline baseline centroid student present represent centroid choose intra shot student expect baseline similar batch offline student correctly deterministic student offline directly operate interactive summarize table depict image correctly strategy well performing vary table second offline often outperform random performance dataset acquisition datum oppose imaging control laboratory show participant participant baseline strategy calculate table compare strategy indicate level centroid bad value statistical nan hypothesis method statistically testing indicate bad strategy score calculate student along snapshot trend improve recognition image unlike base strategy relatively outli show challenging give variability student five chinese class due answer previous incorrect answer finally explore understanding unable adapt focus student poor begin reasonable end dataset unique modal leave class look learner assume unimodal would species entire currently attempt past previously similar region incorrectly label still label propagation allow incorrectly present image behavior machine learner human task human human expert automatically adapt perform difficult costly teacher take propose interactive multi informative focus representative image introduce student improve future plan pairwise region part discriminative category investigate concept ensure visual categorization automate information extract student suggest version different explore annotation finally assume student estimate student ability would thank development history assessment project institute token extremely classify image possess hand supervision image learn people first follow e ability discriminate class student vary work propose interactive enable computer human image show student ability progress correct incorrect answer real human varied world manually dataset visual category annotation complete crowd label internet service image begins ask incorrectly specialized acquire training potentially multiple designing challenge group improve collective tend vote weak learn trust expert pose expert improve family offer human computer learn model supervision label rather oracle human help outside automatic domain education image biological crucially needs knowledge focus possible boundary human learn boundary show student classification time unlike computer human limit human generalize unknown majority focus interactive offline feedback interactive student visual learner unlike computer human student student regard model student instead attempt uncertainty knowledge amount take experimentally human participant baseline interface explore strategy encourage development human relate concern task categorization note explore sequential additionally interactive receive feedback feedback adapt free bad predict uncertainty learn teacher truth use assess perform sub optimally seek currently uncertain regard informative interactive visual concept classify linearly separable category label student investigate feature exploration student interactive multi feedback correspond class label goal subset interactive teacher image represent learner teacher student refer teacher teacher ground class class teacher student believe belong student teacher reveal ground truth proceed teacher ability teacher trivially know teacher seek minimize student
make local minima initialization possibility property progress relax note weak constraint weak ensure minimize new satisfy valid two scenario construct scenario g gb tw goodness large scenario valid construction mechanism em first mm instance bias maximize bound progress tend rapidly nearby svms fix concave objective attract condition match explain empirically pick valid sensitive hard performance model mm restrict require progress valid mm exploratory progress g mm simple complex non mrf pairwise enforce bias towards preferred imagine drop mrf unary easy efficient minimizing enforce preferred would simplify procedure mean structural ease demonstrate latent mean belong category worth mm fully capable handle index cluster convex upper fixing minimize quadratic popular repeatedly assign construct center mm exhibit desire issue design bias encourage balanced appropriately generate latent lead valid specifically solution walk configuration configuration change extend structural svm prediction example df control negative variable specifically fix l z bound case configuration lead connect configuration correspond program solve sgd cut plane cut g mm progress bound progress cluster conduct different cloud gmm synthetic cloud reference dataset gmm create place apart dataset three initialization replacement center assign cluster experiment note solution g mm initialization dataset recovers fact gmm dataset initialize well suggest initialization mu mu g mm mu mu cloud em gmm mean em cluster method center assign implement standard objective value trial report use converge progress quality mm line cover deviation average dash indicate solution trial progress allow small g mm sensitive run diversity g coefficient possible b three solid represent area dash line represent good trial color different initialization find near perfect merge incorrectly assign color code truth center code permutation axis white cluster color match permutation axis detail objective detail object l six category level annotation provide class setup gradient report initialization strategy latent reasonable initialization initialization adversarial try example z keep increase fold divide fold fold train fold avoid dimension formal c standard error fold three location center corner random bias inspire fold baseline variant consistently outperform latent initialization location thereby rarely occur top expect validation well test g leave g mm g biased cross fold code differently five fold latent location g mm average perform scene mit scene segment mu regular grid cell part part use describe region pre hybrid convnet region record neuron feature l latent variable assignment multi label reader ls sensitive initialization generative version generative model l svms several part region discover discriminative cut node unary dot score extract assignment encourage specifically neighboring node differently assignment use initialization assign initialization require filter bias latter coherence labeling specifie assignment function system recall cell grid neighbor coherent compare performance mm biased bound bound repeat five initialization seed mean initialization converge pick remarkable boost bias attain attain initialization c c l mit initialization control initial correspond initialization correspond region bound correspond c biased ls mit dataset acc image control mu coherent bound mm bias progress coherent initialization mm random bias generalize g generic framework minimization mechanism sensitivity initialization adopt progress g mm deterministic stochastic way enjoy rich modification significantly counterpart generate sequence converge comment solution converge assume initial must positive must ii optimum stop progress mild get mm let bind objective solution exist eq specific latent svms localization issue fold update fold formal guarantee function g mm repeat structural training order sequence usual latent variable loo latent loo optimize result g behavior equation latent bias multi fold unconstrained define make generate regard support estimate practice fold offer cluster bound mm solution standard mean merge incorrectly origin mm white indicate center code match permutation image result latent object mm mm update object image category annotation bad adversarial initialization section paper discover part space limitation use proxy measure norm likely sample discard apply remain lemma machine mm systematically optimize convex upper function optimizer constraint unnecessary mm
network use application edge perturbation action privacy condition use nearly match produce stability state sufficient match achieve much good approximation guarantee bad hardness stable instance cut perturbation max latter improve perturbation perturbation objective median center objective optimally solve perturbation instance optimally perturbation also perturbation perturbation optimal show algorithm solution nash equilibrium theoretic return give alternative finding approximation structure dense min optimal et optimally protein stability al separation instance low heuristic traversal condition onto center close center center assumption axis beyond notion range cluster privacy point close center minimize formally dp I center instance metric symmetry unless otherwise asymmetric instance introduce change small perturbation perturbation perturbation formally satisfy perturbation unique change partition stay small distance perturb end clustering cluster differently two clustering formally center center strong approximation point satisfie partition close use hard approximation stability cost original function perturbation partition cost stable objective objective mean throughout radius show two third cluster hardness asymmetric show asymmetric center surprising one hand instance absence hard center well come cluster condition center behave together asymmetric hard involve deal asymmetric close center point speak behave symmetric explore define property ai ap ia introduce lemma perturbation lemma perturbation similar result approximation stability stability center satisfy use claim assume construct perturbation contradiction construct follow distance increase formally otherwise optimal center partition equal perturbation similarly define partition say previous must subtract arrive structure contradiction construct replace center increase except formally must center center know therefore center start create graph point apart close structural asymmetric instance candidate dp ep ap ig return instance point distance follow subset structure perturbation stability exception representative asymmetric instance stable asymmetric center asymmetric center return polynomial center even stability perturbation perturbation similarly symmetric center center prove two tight center dominate perfect dominate reduction perfect dominate dominate vertex proximity proximity perturbation perfect dominate yes instance translate least dominating establish instance find perturbation perturbation distance unless introduce center establish property perturbation second call cover ball close outside linkage closure repeatedly create formally closure closure linkage empty subset inside close outside h start singleton internal corresponding perform minimum pruning move require aforementioned crucially property merge readily consider bottleneck indeed arbitrary center median p q dc dp maximum arbitrarily close incorrect form merge partially form challenge us property center next formalize appear al property case perturbation establish proof proximity entail directly perturbation perturbation center contrary dp j dp c next contrary cluster root cluster contradict uniqueness perturbation contrary cluster double exception involve case determine ms dms outside partition lead contradiction dp dp ic contradict unique property structural property lemma us correctness remainder closure center closure ba cluster definition condition closure cluster condition closure closure second hold fully merge partially exploit impose show merge fully form partial center large center effectively create contradict perturbation proof correctness closure suffice show every set current first iteration show merge union include otherwise cluster part merge case lemma optimal contradict perturbation ic lc different imply similarly induce contradiction also closure linkage radius form partially work long show identify near optimal clustering symmetric stability strong necessarily insight behind result cluster otherwise make center leads create point graph contain return cluster define add approximation return cluster edge first direction cluster point b n pa ap ap ap perturbation center find linkage furthermore cluster necessary problem core perturbation large different apart outline distance perturbation keep remove center would score indeed center perturbation consider guarantee far center bad case reaction would challenge create challenge present deal approximation partition challenge construct perturbation case reaction reaction close show point contradiction chain reaction power point center perturbation close perturbation cluster instance linkage return linkage start implication cluster instance perturbation optimal cluster must center perturbation center perturbation often perturbation together use argue perturbation center outline perturbation center center center create cluster close exist assume dc dc dp dc perturbation set center score center must close center similarly leave capture point contradiction distance capture idea close chain reaction capture center exist center center formalize ccc half ccc close unique capture ccc point intermediate argue cluster ccc lemma appear cluster center follow majority exist ccc pair together intuitively reaction become center always center exist careful perturbation ccc cluster cluster close point cluster handle center distance statement statement satisfy center point center ccc say center apply statement statement p x dp I r ds r point close half reach problem polynomial pair close stable perturbation possible center simultaneously theoretic format element rank without rank ahead element optimal hold tie fourth need express rank element rank uniquely prove cluster satisfie point show construct valid perturbation close majority contradiction pick majority let rank tie fourth tie rank high ranking size easily linkage start singleton round merge find correspond return stop center cluster linkage set set last set therefore component merge exactly linkage set need find center perturbation np hard analyze center partition close furthermore give cluster show property point center cluster recognize soon form define example cluster include instance size median cost center weak proximity consequence rely weak analysis perturbation closure linkage center discuss obvious property show enough impose weak linkage outline graph linkage cluster linkage edge continue away prove never point add put instance center proximity polynomial high suffice add induction assume iteration towards contradiction figure merge furthermore proximity imply nonempty subset must merge center contradict add suffice step cluster proceed contain different round edge denote respectively need else connect component proximity already add add nonempty call inductive center contradict add search polynomially many edge condition optimal perturbation constant value hard worst thereby demonstrate power limit approximation stability perturbation work result center clustering require open whether asymmetric show asymmetric solve optimally perturbation interesting handle formation extend asymmetric perturbation achieve contradiction lemma lemma stability asymmetric center show exist eq partition replace center use contradict three give towards center center switch contradict stability assume argument center approximation contradiction contradict property ensure representative proof towards contradict assume asymmetric center radius connect representative connect assume call optimal center center contradict stability algorithm prove center stability call balanced dominate np front matching dm set triple find pair match dm dm dominating give integer balanced dominate set dominating dominate set vertex variant show version another hard establish reduction parsimonious reduction element verify np hard nice crucially point appear mention observation strong david dm every element dm easily parsimonious dm map edge element dominate dominate dm give verify parsimonious reduction center must center size satisfy approximation stability create distance center dominate set
fashion without inversion stage divide group correspond partition product equivalent make approximate regard second diagonal justify careful notably justification experiment possess rest gradient kronecker fisher describe furth inverse efficient inversion describe estimate quantity window process mini batch curvature batch fisher obtain practical robust little manual careful various theoretically optimization local implicitly nd k establish practice type momentum k computational cost way point characterize transformation consider result appendix feed presentation closely network output series bank unit neuron receive output unit via nonlinear sum layer unit output activity precise element matrix additional value bias coordinate think consist stack denote prediction make loss training pair proxy actually predictive objective parameterize minimizing follow backpropagation map forward pass output define network datum input model expectation distribution input perspective geometry gradient direction large gradient natural classical idea equivalent generalize newton case semi definite approximation particular define linearize represent logistic sigmoid cross equivalently network light fisher nd method importantly gradient optimization conversely point bring vast accumulate make book highly method fisher discussion natural gradient challenge natural compute large million impractical initial ingredient computable w q w w I block rao layer deep scale version achieve reach network architecture use exact line block purpose plot linearly kronecker kronecker major limit asymptotic seem successfully capture coarse fisher later section computational gradient arbitrary weight network entry I k g k g interpret approximation interpretation consider approximation error generalization order nd covariance intuitively measure interaction order oppose arise upper high I loose due variable error aware practice network particular error weight roughly whose tie high eqn inverse computation well efficient inverting rao computable follow subsection reasonably approximate make computable restrictive cost product compute give suppose associate inverse say row optimal th inverse usefulness subtle informally equivalent degree equal variable independent linear simply variable setting fisher apply derivative try likely regard reason entry block layer predict forward reasonable indeed undirected graphical potential depict figure reality adjacent accord stand joint reasonably fact approximate graphical basis recent inverse figure average inverse predict inverse exhibit note look block visible inverse factored technique section iteration purpose absolute white level differently due extent inverse fisher block subsection diagonal efficient vector present approximation network approximate block take inverse computing associated approximation block block block block sophisticated deal develop subsection agree block block imply establish efficiently assume assume gaussian depict whose graphical graphical model dag moreover also direct efficiently block low one yield nonetheless distinct label mapping source node whose given simply block generalization cholesky precision give perform follow compute subsection amount multiply correspond product straightforward fortunately invert figure examine approximate exactly diagonal must definition block well likely approximate one arguably interesting picture proportional block approximation meanwhile accounting approximation diagonal neural approximation compare block top right due actually plot note factored approximation network difference compare compare subject factored take purpose plot linearly j fisher would perhaps approximation quantity fisher compatible perform break rise finding detailed discussion poor curvature mini adapting seem efficient entire matrix require batch matrix multiplication multiplication arguably acceptable monte backpropagation target average outer various usual pass target cost additional average quite good gradient backpropagation maintain exponentially decay average scheme particular new old weighted averaging equal depend time proceed estimate kind decay scheme commonly involve diagonal diagonal small ng much process batch notably like deal fisher product scheme implement exact independent seem would amount big practical must process follow riemannian imply fisher matrix view tensor take step space method path objective one follow objective discrete understand traditional theoretic large family experiment matrix negative give pair hessian density view approximation nd taylor expansion whose replace approximation taylor whose kind negative natural update argue natural approximation notably sufficiently optimum traditionally optimizer arguably important practical behave help way apparent strong mathematical think nd sophisticated road arguably reason machine understand take sophisticated technique available crucial role reasonably well adaptive technique adapt adjustment impose spherical region trust model book insufficient update proposal seem rise update comparable method exact unlike equivalently guarantee accurate nd intrinsic model small represent curvature direction fortunately able subsection stage update apply slightly fisher inverse technique add curvature account mf approximation amount add individual modify kronecker use invert try sophisticated diagonal long work block kronecker computation replace expand give work slightly principled expression efficiently b b b negative describe subsection stage scheme exact fisher add one produce use input mini mini batch mini backward predictive forward pass describe adjusting version current modified factored technique describe section compute proposal multiply approximate formula layer efficiency final update product loop gradient invariant way parameterize mean follow small direction natural gradient tend local invariance case fortunately invariance direction respect curvature update negligible affect invariant broad affine transformation network version transform various still main result section invertible updating update immediately characterize transformation path default network assume initialization negligible momentum rate fix relax allow invariance smoothness quickly invariance limit transformation interpret replace nonlinearity sigmoid transformation immediate corollary sigmoid activation function initialization negligible affine transformation normalize activity choice smoothly addition invariant go similarly strong variance diagonal approximation elegant particular end equivalent network transform activity center formally gradient free method linear conjugate cg optimize eqn subject solve main avoid costly matrix secondly curvature lot average oppose fix batch course inexact curvature network optimization block large correspond roughly per therefore despite arguably accurate fisher many approximate diagonal introduce center modify dynamically unit wise activity typically skip connection layer preserve expressive efficiency transformation center network argument use quantity activity notation assume plus center skip connection center interpret transform center intuitively whiten account correlation gradient optimization fisher correspond incoming bias method except approximate discussion difference fisher deterministic spirit deterministic use find basic unbiased stochastic nearly closely fisher feed neural similar factored block approximation approximate accomplish basic factored technique add hand factor adapt combine fisher constitute merely course something crucial observe optimally method neural network dimensional output accurate inverse fisher momentum see additional element kronecker factor numerous include use experience section factor scale momentum maintain matrix kind perform section develop potentially insight diagonal difference determine reweighte effect approximately translate expect specific one basic wise block kronecker fisher term measure quantity change distribution assume condition respective kronecker self intrinsic technique obviously produce investigate deep autoencoder mnist curve face due high difficulty momentum include regularization problem sgd nesterov accelerate calibrate autoencoder schedule approximation curvature experience tend well baseline improvement task engineer hardware batch typical average use technique describe use matlab gpu computer ghz core intel cpu gb memory initialization help use iterate average averaging take average multiply optimizer multiply associated optimizer iterate sometimes mini report paper reconstruction actual function almost perfectly error oppose generalization capability size progress make plot per k tends examine baseline rate progress linear slightly sublinear appear factor momentum optimization baseline extent would seem sgd gpu gpu progress rapidly design exponentially increase schedule mini note neural involve autoencoder schedule stop block diag version indicate version plot row begin plot stage axis last plot vs second experiment autoencoder exponentially schedule momentum good per progress mnist face reflect gpu optimize algebra allow efficiently process parallel mini batch result per mini schedule without partition mini batch computation involve mini batch experiment figure progress order overall progress mostly large mini increase experiment increase although expensive solution noise importance use momentum significantly momentum sgd slow include without appear axis plot recall type momentum allow build define fisher across iteration strong fisher update proposal momentum one might momentum responsible sgd conventional well mnist autoencoder problem mnist face version per progress typically block block cost block overall per moderately diagonal multiplication cost differ increase list perform face unit significant average version sized suggest diagonal great simplicity comparable per progress situation implementation version approximately sgd implementation far synchronization step virtue acknowledgment acknowledge google would like constructive comment early eq lemma scalar independent intermediate lemma end use expectation network say intermediate quantity pass uncorrelated various compute provide instead come valid choice expression accord general relate moment term st order correspond similarly eliminate remain require fact know compute v less likewise rise efficient matrix product stein example numerous recent survey involve simulate multiplication stein equation use stein stein particularly application overhead cost root evaluate multiplication symmetric always application eq inverting side symmetric unitary matrix sized compute multiple application need compute cost future avoid computation considerably simple identity eigen jacobian predictive mini latter operation compute correspond forward pass linearize forward pass rank vector multiplication compute inner additionally compute product similarly obtain reduce cost various
minimize call sparse hard point dimensional position subspace negative independently unless quasi seem recovery basic sense relaxation structural rip compressed matrix allow recovery solve similar checking possesse rip motivated pac seek entry sparse suppose produce upper norm indeed greedy ascent base non unit seek solution find co problem powerful importantly without large additional style property rely approach optimization combinatorial inspire multiplicative cover describe weak suffice formal statement negative learn arise lead paradigm lot success speech mixture model receive attention gaussian gaussian extremely start heuristic maximization em give rigorous mixture albeit separation use method show gaussian component recover time separation exponential show recover surprising mild degeneracy thus mention parameter pac give mixture gaussian find success state class gaussian f weak often improper arbitrary context estimation often hard know proper mixture exponential improve complexity give meanwhile improper know general distribution monotone unimodal run improper mixture unimodal concave extend modal hazard improper component gaussian mixture polynomial usually list one algorithm axis gaussians learn remove gaussians dimension learn dimensional gaussian learn learn something proper improper learning suppose mixture axis gaussian running time ok function try mixture bound careful discretization section propose gaussian time component tradeoff efficient conjecture optimal factor weak obtain approximation polynomial algorithm get unless say necessary cover plant beyond algorithmic technique complexity plant require denote entry integer denote know cover element indicator precisely cover equation additionally connection cover q modification potential multiplicative weight potential precisely vary significantly try increment keep potential sparsity key increment progress intuition ax satisfie next appropriately furthermore scaling maintain may start drop normalization since negative maintain co start potential quantity I easily going check least increase get connect want obtain convenience ax normalize take last inequality index seek q last step eq apply proof know succeed find satisfying conclusion lemma completes add go index check allow suffice variable discretization add align respectively density context interval column solve direct feasible secondly guarantee kind difficult avoid carefully rectangular p denote column potentially infinitely finitely sample generate estimate multiplicative gaussian require coarse partition fine solution continuous partition gaussian apart rectangular grid carefully suffice use partition use rough estimate formalize continuous interval subscript coordinate induce notion give vx algorithm algorithm rectangular group bin coarse much mind run restrict mixture ensure obtain gaussian contain gaussian clarity error tn remain sample ns ns da remark theorem chernoff bs concavity however bs bf hence rw w furthermore bs f w w total close gaussian correspond partition unchanged prove depend interval pi tail since since taylor expansion eq summation add every flat close triangle coordinate inequality tool mixture follow last follow heavy lemma comment section lead complexity lower towards nonnegative sparsity nonnegative ax gives plant problem prove unless plant cover solve inspire hard cover disjoint union outline theorem reduce column equal yes universe hand know set cover word union
statistically direction effect combination loss new large multi separate objective thank financial von electrical engineering university place create overall combination introduce indicator alternative gradient adjust gradient weight require priori enable inner like use high loss multiple new direct mean loss provide self adjust divide describe fits improve error express obviously relate capability depend part current machine create different chance minima rarely attribute use building statistical model frequently although sound problematic involve extensively achievable cost reflect behavior consider choice classify incorrectly classified prefer incorrectly nonetheless place high optimizer choose although toy representative want possible boost incorrectly noise moreover achievable model objective optimization perspective transform gradient high loss like single important highlight although gradient hill conjecture place pressure may remove surface therefore allow minima reach provide provide overview characterize self respectively conclude future multi objective traditional optimization compose decision objective minima objectives minima trade objective optimal impossible objective increase say pareto counterpart pareto multi problem combine linearly become weight although combine mean achievable illustrative linear combination objective go properly non pareto form transform allow standard one objective resort frequently candidate expensive case logarithm call many objective improvement cause point pareto property combination maintain since concave maintain solution expectation previous equally indeed find close would equal solver confident hard weight may prevent straightforward mean technique set merge one take weight incorrectly sample place pressure predict sample may self go currently desire loss logarithm find weighted loss minimization automatically priori moreover high high weight pressure maximization weight loss worst become gradient similar infinity uniform single control objective set mini batch evaluate try version reconstruct digits mnist uniform able use problem already adjustment divide weight gradient order normalize objective gradient epoch slack constant pressure bad number epoch mini batch point increase epoch way define remove requirement absolutely allow arbitrarily experiment perform batch epoch function total generator add equal black corruption hide unit evaluate loss corruption epoch test plot loss maximization loss cause objective provide baseline corruption indicate fit moreover large generalization metric weight impose improve overall align conjecture propose sec corruption influence behavior always achieve baseline difference favorable occur set mean cope noise epoch part part bad median
intermediate square root optimization yield performance mix linearly combine suggest much study frobenius reconstruction reader remain interesting question target decomposition singular eigen psd orthonormal index project rescale therefore n n verify relative good rank difficult present follow ready theorem spectral score remark error successful depend facilitate understanding result small present proof defer supplement remark sampling square root sampling minimize square refer root equal e leverage equal scalar value quantity write flat bind skewed law theorem achieve minimum q skew I always l due quantity flat sampling discussion flat vector come tend uniform exist insight especially flexibility score adjust subsection quick strategy indicate result achieve issue ensure impose minimize easy verify optimization next problem slack search feasibility discussion could variance present generate different allow vary ga ga moderately generate freedom refer synthetic empirical evaluation norm different spectral time average nearly three reconstruction fast iv randomize combination value demonstrate well finally constrain regression synthetic generate bias estimator distribution demonstrate sampling u compare optimization mixing include versus size reconstruction supplement detailed supplement work term chernoff apply bernstein chernoff let finite psd q inequality last inequality utilize verify ki utilize bernstein k prove theorem union clear matrix bernstein derive dependent chernoff randomize frobenius interested norm remain nonetheless include similar phenomenon comment least square structural inequality condition term bind combination yield good sampling bring leverage exhibit bound develop constrained algorithm find lemma thm consider subset novel simple depend sampling probability understand tradeoff probability exhibit insight specific distribution square root uniform leverage score bind demonstrate benefit compare state give compressed select column svd yield maintain close recently problem identify population select minimize reconstruction pseudo norm particularly target randomized select build advanced chernoff bind bernstein inequality novel norm svd quantity dependent sampling scalar quantity leverage inherent knowledge dependent bring several benefit allow tradeoff well uniform well skew iii motivate attain well analysis efficient solve constrain well probability reconstruction establish bind exact closely work section empirical rank approximation spectral column deterministic algorithm randomize select criterion representative category qr variant numerical set fall define sampling representative square known leverage allow bound review achieve qr bind run use make linear target stage select exactly bound sample related reconstruction rank qr work time require show
main analysis response underlying splitting induce purpose partition parent operate repeatedly leaf splitting child require partitioning contain least parent terminal example terminal node observation implement g meanwhile child incorporate parent analysis satisfied induce tree leave kind valid fit infinitely partition name tree forest grow tree general splitting splitting comprise correlated show forest comparison individual forest average forest partition ambiguity proposal splitting depend rely structure paper review tree forest practice recommend original allow evaluated hold bootstrapping effect study concentration show concentration present promise adaptive analysis consistency forest asymptotic still uniformly cube requirement simplicity assumption distribute grow magnitude size principal establish adaptive regression tree apply recursive partitioning adaptive decision tight assumption assumption must polynomially strictly allow ignore sample valid partition expectation practical fit good support valid random forest use analogue directly dimension depth analogue generalization implication guide role cart rule regression cart good intractable problem cart something help bring valid imply compute empirical minimizer valid forest greedy tree may repeatedly question forest datum author true low split concentrate cart split comparison forest forest dimension example actual imbalance partition forest lebesgue rectangle process begin generalize leave hoeffding concentration yield section complement concentration bound guarantee ensemble forest proof give appendix ir r support define finally leave partition bound cube detail constructive construction generalization scan hold volume q approximate set tend leaf lebesgue sense approximate jointly recover thus job volume construct approximate length build form ab integer get guess reason volume become every geometrically cut example immediately exploit generalizing set th observation coordinate specifically establish tail independently whenever coupling tight follow generating lemma follow choose finish generalization forest tree thus forest understand variance us ensemble rate ensemble difficult order analyze ensemble forest far insight us uniform forest surface motivate post approach treat support seek guarantee consistency hand consistency require unknown practice assumption especially learn forest know true prior would forest prefer take shape split reference concentration allow surface conditional theorem apply thank imply condition corollary leave plug sample consequence see meaning decay leave corollary eq everything small immediately remain recall tend meanwhile tend combine desire conclusion complete approximate choice yield desire suffice conclusion directly let denote child parent meanwhile recover choice meanwhile complete show analogous recall guarantee analogously choice define construction proof converge rectangle q event q converge simplify tree construct triangle inequality eq individually last know eq q term similarly theorem mean parameter hoeffding combine bound state chernoff variable binomial desire hoeffding calculus meanwhile q q fact strong leave union multiplicative require apply find must converge set conclusion recursively feature split feature specifically study partition meet partition split index rule node round node event terminal combination split occur differ eq overlap hope comparable within always know partition proceed employ corollary get tail readily eq simplicity standardized variance small detail argument q lemma n together provide write leave determine establish leaf super write approximation reason denominator point fall leave leave verify tend simultaneously valid meanwhile tend arbitrary argument tree define q meanwhile q quantity conditionally variable hoeffding tend bind note prop prop conjecture prop prop prop surface introduce pick split model split formalism forest forest bound perspective predictive whenever forest estimation need consider predictor forest use machine variety field surface especially surprisingly compete network forest stability beyond draw forest believe convergence procedure tree theoretical describe forest pointwise provide tight forest concentration view occur stage stage find split treat tree fit worse fit tree adaptively split tree tree split jumps fit affect position split relate post provide estimate main tree splitting practical give promise asymptotic space training leaf regularity show regression tree forest universal forest split whole tight within modification routine cart original proposal size comprise
outli mle drastically tune present difference full outlier suggest nature equivalently nan outli removal fail reject nan apply et al paper divergence figure nature clear case theorem composite trivial result restriction impose hypothesis illustration composite hypothesis robustness test encounter science restriction perform test tool inferential classical test utilize restriction robust misspecification model outlier well develop robustness test density power divergence paper case minimum discrete continuous focus robustness base test composite provide online supplement introduce density power pd several minimum estimator study restriction estimator parameter restriction measure density parametric family sample space kernel description use x case replace countable f g sa sa supplementary material sa respect restrict divergence equation asymptotically supplementary restriction g independent dominate measure bring curse condition asymptotic help kernel derivation smoothed version density function use divergence subject divergence fitting exist consistent root divergence asymptotically definition supplementary definition independent smoothing restrict sense discrete testing et approximation statistic level give quantile nan p routine help desire composite functional corresponding restrict contaminate contamination contamination statistic statistic simplify influence correspond functional see unbounded consider asymptotic contiguous contiguous tend neighborhood huber contaminate material level let derive general expression asymptotic composite contamination freedom chi degree freedom centrality nan asymptotic distribution f equivalently p supplementary put alternative put asymptotic coincide independently discuss finally power divergence test present clearly whenever nan bound al influence test statistic p illustrate propose mean know population univariate unknown base sample want specify assume know
recommend comparison book instance rank computed test algorithm rank situation pool comprise algorithm pool comprise irrelevant compare algorithm point yet ignore issue ignore equivalent compare whose discuss illustrate example etc comparison post understand regardless specific adopt comparison correction powerful even drawback nan hypothesis test counterpart compare discuss organize denote algorithm performance rank column dataset row depend statistic degree freedom hypothesis establish significant perform perform family wise least comparison control correction also rank valid regardless claim algorithm different mean correct derive probable assume rank configuration probable yet post hoc nan hypothesis present analysis show test test five algorithm accuracy eq rank algorithm compare dataset two difference side sign test rank case favor assume compare nan post standard quantile adjust comparison rank reject result post hoc rank compare reject set simplicity want need comparison sign test numerically power rank statistic significance q claim significance mean test rank compare mean rank experiment finally accuracie seven average naive j locally forest assess validation accuracy alone first rank nan comparison pair quantile rank rank statistic small significant decision mean rank consider run claim five classifier differently different claim significant pool rank classifier clearly alternative classifier assume b z symbol comparison drawback guarantee small equivalent algorithms aspect issue recommend rank test hoc perform compare rank robust assume power sign recommend sign rank make sign sign symmetry datum regardless adjust control discuss adopt sign value always report decision less corrected significance level matlab rank post comparison instead recommend adopt pool bring counterpart overcome drawback nan example htp data machine
panel completely isolate edge isolate risk entity link concentration completely isolate isolated increase concentration subgraph involve american international whether provide augmentation alternative inclusion column variable v eq inclusion indicator involve express tr ij v holding conditional density kind supplementary material matlab implement frequentist use site proposition graph two bayesian literature spike use discuss produce efficiently hundred variable statistical inference among type use represent dependence variable learn refer graph carry follow undirected zero graph problem model induce approach wang model estimation model always bayesian impose sparsity prior positive definite fix prior determination stochastic graphical inherent nature permit theoretical address characterization modular integrate graphical model progress model make year adapt grow publish small day ghz days problem matlab report second edge ghz improvement necessary large call search concentration idea behind use prior characterize normalize update graph continuous shrinkage prior prior exist motivation come successful development shrinkage substantial attractive concentration prior loading handle nevertheless fundamentally distinct estimation estimation continuous shrinkage little know structure positive matrix pose challenge contribution two shrinkage undirected bi graph learn minute dimensional normal let graph model briefly review section concentration model concentration encode undirecte represent pair random except paradigm conjugate wishart prior bernoulli inclusion indicator inclusion degree freedom normalize constant choice directly posterior pp sampling matrix type share framework inefficient large feature manner mean loop iteration feature normalize non decomposable monte unstable complexity slow graph work avoid remain paper would problem computer concentration impose encourage penalize likelihood penalize shrinkage interpretation maximum posteriori exploit efficient hundred bayesian help propose treat consider lead run prior avoid constitute treatment eq maintain scalability insight covariance graph encode dependence bi bi full undirected graph covariance theoretically learn rely hierarchical density specifie structure likelihood b unfortunately quantity normalize decomposable graph carlo importance infeasible beyond later investigate class prior normalize constant decomposable graph portion advance framework early general motivate lasso thresholding likelihood ratio testing approach estimate bayesian derivation excellent report similar gibbs although graph report upon request strength combine approach denote covariance use set small represent normalizing depend component diagonal connect familiar symbol element component variable ij integration constrain proper distribution behind concentrated close appropriately come zero miss graph view indicator control indicator reflect inclusion prior imply inclusion probability chance reflect belief expect edge approximately inference consist whose focus inclusion truly knowledge relation comparison help turn unstable iteration yet concern prior intractable normalize dominate inference concern appear problematic hyperparameter parameter involve depend instead dominate show different reference curve display curve suggest bias introduce introduce fact constraint specify see large impact fix vary panel display imply function different reference plot continue definite force never extremely reflect concern lack challenge incorporation prior example suggest configuration support choice substantially regard practically zero large precisely explicit imply illustrate aspect setting essence aim small density long difference lack calculate infeasible numerical method estimate markov mcmc another perspective choose close mass edge sense mcmc issue standardize mcmc long element usually assign entire plausible experience contain structure insensitive primary scalability one indicator inclusion monte carlo intractable require carlo evaluate generate graph joint generating manner generate depend graph model proposition distribution symmetric zero diagonal full conditional diag last bernoulli conditional column correspond something look indeed imply view information regression coherent fashion interesting hierarchical proof hierarchical symmetric conditional eq kind surprisingly proposition normal speed scalability block sampler evaluate empirically standardized implement hyperparameter computation implement core block gibbs sampler across element column improve solid display minute minute generate approximately minute graph matrix inversion update measure calculate lag sample burn lag suggest efficiency experience usually reliable monte mean far time htbp evaluate scenario real world pattern daily analyze ba exchange website concentration model use true assess positive positive fp evaluate benchmark adaptive graphical wishart prior wishart classical fold validation adaptive seem well model hyperparameter belief iteration graph tp pattern observe compare tp fp positively tp fp fp especially concentration fp partly positively relate treat imply inclusion table benchmark competitive except favor inverse wishart method htbp c htbp wishart tp scenario dependence understand biological relationship cancer gene scenario two graph estimate matrix base panel display correlation nonzero correlation correlation within repeat benchmark propose benchmark graphical thresholding take day evaluation wishart worth experiment little original wishart expensive importance normalize thus slow numerically requires greatly implement website adopt inclusion hour substantially minute require fact fast posterior inverse surprisingly wishart element strong distribution empty follow gamma eq rapidly distribution graph conjecture well estimate support implication bayes factor might truly largely reflect word concentrated allow edge standard fundamental strong perhaps space depend standard wishart allow thorough beyond scope paper safe call hyperparameter relation scenario great benchmark concentration suggest
comprehensive know basis represent control show provide multi resolution representation consider covariance transformation see give three basis scale merely fine resolution capture bt like use conventional conventional form center location basis conventional basis ccc propose function form expression consider ml k practice aic function random tb cc effect estimate give consider six among aic comparison estimate ml cc c resolution performance various predictor krige perform poorly third quantile aic true daily dataset package year weather average daily consider identification q smoothness use obtain estimate apply smoothing select validation know covariance bt divide part consist datum datum underlie sample validation exponential mean surface predict surface b na kf submatrix kf column f na k shall twice trace equality minimize k institute spatial model flexible modeling computationally krige appropriate class thin function degree function small detail lead consequently basis commonly first select total function resolution require variability considerably estimate fourth basis function location effectiveness method fix krige spline process dimensional covariance z tn mutually impose effect w kt uncorrelated modeling spatial depend parameter estimate include commonly radial discrete basis advantageous basis center estimation basis support radius spatial spatial function situation approximate set f matrix non shift control parameter poorly cause significant seven extract thin spline term spatial need model class several advantage commonly select number computation estimation considerably reduce precise estimate fourth location take location location space organize introduce derive simple example daily present develop thin datum observe distinct penalty
ten plot display performance ten ten display agreement add individual sufficiently achieve condition classifier adaboost first since obtain eventually iteration lead robustness environment rather severe overfitting label localized differ rule localize final still average average bagging overfitte increase additional overfitte empirically forest provides far illustrate increase localization result form sample direction force close proximity training rule error interpolation localize proceed quite label comparison continue steady rate even practically yield disagreement illustrate average adaboost prevent performance signal decision small boost ten analogous row show classifier decomposition respect hold represent classify differently bayes rule point incorrectly along incorrectly vary considerably display mistake ten adaboost wise weighted location classifier classify ten still set exceed interesting follow easily theorem non start mass shift recall also I already establish either must completely determine latter probable large proportion b prove adaboost least successful noisy dataset datum continue iterate fit boost adaboost regularization size number contrast adaboost optimization well stage adaboost forest light provide novel intuition example adaboost forest forest classifier evident adaboost ensemble way adaboost actually forest tree average surface nevertheless hope adaboost argue average adaboost behave similarly forest interpolation couple hold belief interpolation argue neighborhood prevent average serve prevent fit random forest desirable interpolation deep tree hope average aspect broad success extend margin rgb adaboost margin however point forest method substantially forest propose classifier predictive procedure rather forest average create adaboost adaboost forest mechanism justification conventional conclude like forest regularization stop boost approach powerful ensemble weighted realization fact conference adaboost world adaboost early success follow effort wise realization lead boost estimation adapt view success adaboost computer science bound margin cast fully understand adaboost implication perfectly wide situation statistical suggest perfectly fitting build decomposed traditionally model smoothly balance extract fit fit hand automatically classical noise irreducible error hard consequently classifier huge g cart create community prediction iteration maintain algorithm claim analogy forest ensemble regard unlike accept boost create tree average canonical example completely exhibit self property generalization forest clear contribution adaboost weight contribution interpolation combine averaging create effective classifier turn interpolation kind extremely locally classifier couple average influence fit become localize adaboost adaboost demonstrate point decrease adaboost demonstrate effect demonstrate adaboost discuss view adaboost main conclusion interpolation correctly provide fit presence simulation discuss forest namely decompose classifier datum perfectly implication run adaboost deep deep allow component classifier bag section theory explain self strength emphasis focus statistical literature development boost briefly adaboost also variant review round version misclassification rate update round final weight datum weight n iy tw iy mf attempt adaboost predict generalization adaboost increasingly overfitte attempt resolve margin think confident label produce margin possible generalization error margin increase observe adaboost iteration decrease margin demonstrate apply maximize suitable separability appeal margin boost margin would hard optimize margin arc lp boost adaboost provably reduce margin adaboost yet error margin loose qualitative crucial large generalization margin view certainly investigation yet provide great view heart familiar program approximate exponential place search combination learner base classifier explanation article recent review boost seminal activity dedicate mathematically adaboost although statistical optimization adaboost surely problem fact minimize exponential classifier introduce variant beta boost except exponential despite beta boost able adaboost present exponential adaboost boost algorithm overfitte avoid learner one overfitte regularization opposite deep deep many recent work tree generalization although one suggest able extract fit maximize however boost fit one excellent job quantile summarize performance cart increase explanation zero continue minimize exponential loss smoothed unlike statistical optimization view perspective iteration adaboost deep tree allow draw adaboost component classifier noisy environment think draw grow reach variable output let rf fit without poor classifier forest serve argue achieve careful perfectly match general semi smoothness quick forest analogy adaboost forest gain popularity often achieve performance respect highly tool many application algorithm review procedure forest cart design bootstrap one average tree far reduce across fit close one bootstrap forest point label vote wish success forest optimize independently index construct fashion adaboost surface analysis hard justify leaf size next interpolation label training fit perfectly generalization come mind near show environment noise fact binary class easy asymptotic error neighbor high generalization problematic perfectly forest claim modify datum else measure insight prevent classifier single ensemble fit smoothed goal create surface small neighborhood everywhere constant fit influence generalization average mostly fits prevent region away average classifier fit continue localize conceptual illustrate idea poorly local point fit wrong relatively neighborhood generalization second sense rapid neighborhood process obvious forest many fit region crucially concept conceptual interpolation help try heavy two blue line locally line influence blue robust spike boost robust extremely red substantial fail little bit strength htp return classification distribute independently far suppose conditional pure general view approximately bayes possible closeness bayes data red red evenly essential htp result fitting learner predictor convention restrict throughout boost sub close vary small classification differ expectation set dimensionality evidence noisy environment large fact rule spike vanish measure obtain consistency stand contrast conclusion necessarily lead would classify rule htp b classifier consistent many display result allow tree bayes near neighbor boost one even rate example illustrate fact additive classifier differ combination flexibility class spike increasingly demonstrate superior performance vary considerably forest ensemble robust forest individually final smooth extremely visualize forest nearest neighbor classifier forest data nn classifier forest less point generalization adaboost subsequent section algorithm tree maximum terminal adaboost take point hypercube choose nn adaboost colored light colored training classify expect adaboost substantially well classifying long boost noise classify visually forest adaboost near sensitive adaboost forest fact noise neighborhood seem degree region forest adaboost visually follow similar dimension iteration adaboost practically still b training interpolation desirable forest previous crucial display forest six decision tree forest visual show majority forest light region region thing reproduce contain bootstrap region localize thin apparent five tend fit decision tree nearby classifier majority vote tree tree poor relatively get small indicate agreement bayes easily imagine wider fitting reduce noise average surface affect point iteration adaboost serve fit localize htp
periodic ar important dimensional application market apply ar type ahead european load conclusion basically possibly covariate stationary contain model autoregressive ar I weak covariate process allow huge class popular error response furthermore situation ny arbitrarily arrange obviously concentrate directly base classical reweighted least square literature g small joint higher impossible non loss many fast motivate resp thus receive common unfortunately never replace perform process practitioner sometimes multiple first computed result part second priori receive n new repeat end sense resp increase use within n v penalty reweighte adaptive special choice w w require usual estimator case worth show different tuning parameter crucial might demand optimal tuning parameter time lasso time framework subsequently optimum almost option information criterion bic discuss generalise criterion amount establish consistent initial option elastic net ridge subsequently ng n n n n l n however process reweighted adaptive estimate compute new l value information criterion reduce computation convenient lasso adaptive optimisation eventually stop plausible resp suggest algorithm n difference asymptotic shown get depend link estimator asymptotic vanishing getting prove achieve basically behaviour complicated point process infinitely n take account concern property notation n n correspond covariate standardize exist nn partial coordinate n n assumption adjust require adaptive make error state restriction grow behaviour weight unweighted property precise sign consistency normality nk furthermore option residual decay laplace replace polynomially decay residual possible variance discuss maximal growth impossible argue possible rate slightly fast linearly polynomial like get relevant growth small last require normality parameter want without choose stick long help several growth observe estimate clear asymptotic help parameter information mention process deal several extension periodic ar threshold multivariate autoregressive index lag process ii jj enumeration everything correspond regressor detail common process recently process restriction absolute moment j I enumeration everything multivariate distribution fix estimation size finite j jj j n parameter I adaptive estimation estimator precisely restriction l provide estimate resp suggest plug common square estimation high positivity parameter residual weakly slightly advanced require non restrict adaptive normality stationary however act shrinkage give computational aforementioned effect mean parameter lt lt j lt lb l periodic weakly stationary weakly periodic factor periodic spline periodic wavelet good mention general nevertheless periodic another application one break ar equation periodic capture periodic take option build triangular lasso particular modelling receive change powerful use inference study past ar switch finance call threshold option threshold lead introduce ar type covariate process threshold option volatility popular regression general interaction weakly full quadratic give eq popular ar ar likely method process lasso approximation idea large residual regressor contain autoregressive regressor matrix automatically iterate receive well principle principle take variance model possibility framework specify every monte model restrict analyse reweighte close consider dimensional ar subsequently aic bic generalise give aic either furthermore freedom replacement monte lag fix uniformly replacement simulate process ignore propose simulation additionally graph sigma range close information bic parameter hence small bic aic propose conditional analog criterion expectation robust tail result distribution satisfy distribute analog clear conduct another bic ahead absolute define h h forecasting reweighte additionally calculate forecast oracle oracle structure autoregressive simulation b dot line correspond sigma additionally estimate namely resp basically relationship well worse one remarkable setting structure usually setting dimensional market application propose ahead price european exchange consider autoregressive lag criterion take lasso iteratively reweighte iteration ni also forecast step ahead forecast conditional influence forecast consistency quite application ar market simulation additionally unknown specification case underlie observation impact series every exhibit finance behaviour analyse penalty parameter high research concern show work sometimes tail situation see absolute
low extension mapping generalizing generalization nystr om propose generalization coordinate symmetric coordinate objective extension function initially generalization many reason general supervised kernel version determine class nystr om apply method generalize supervised function performance likely suboptimal learn application aware make exploit jointly extension embedding measure concentrate around support manifold minimal euclidean interpolation classification class possible error deviation projection manifold practice avoid training datum regularity property magnitude meanwhile likely linearly separable nearly give embed confirm experimentally boundary class ambient especially separable dimension px px correspond direction like interpolation direction coincide derivative magnitude average denote directional induced nn tx normal direction element normalization directional aim boundary relatively derivative different separable directional achieve along meanwhile variation strong directional boundary enhance class interpolation arbitrarily average gradient magnitude along boundary embed formulate interpolation optimization exist explicitly label near euclidean distance let unit direction near neighbor counterpart q expression denote directional x ix average along direction nearest come manifold usually set embed compute supervise embedding sample lie class simplicity embed decompose may preserve learn formulate extension problem supervise learning estimate class label rest focus interpolation interpolation radial basis common rbf property smoothness adopt dx equivalent determination class algorithm construct interpolation alternate construct rbf interpolation kernel center select term gradually n solve compact attain minimum iteration interpolation class close low assign within near neighbor decrease assign near iteration selection center iteration center score compute center high stage score center ff x df k c k ty matrix center embed optimization choice optimize regularization parameter numerous meanwhile experimentally variation regular scale set across simplifie reduce propose solve decomposable scale theoretically meanwhile monotonically underlie ensuring separation directional along separation boundary fast highly interpolation localize around center well sufficiently strong thank underlie separate condition impose training coincide general attain increase overfitte function sufficiently boundary lose strong direction overfitte manifold represent class select concentrated manifold embed sample respectively interpolation fx directions red figures rbf plot display region correspond respectively figure cover manifold accurate interpolation separate two strong derivative show meanwhile overfitte strong directional observable direction overfitte g red yielding find configuration objective rather embed well link objective represent function separate gradient manifold yield scale indicate examine iteration class calculation label estimate iteration manifold rely manifold onto convex near update project onto employ mx mx coincide denote index mx approximate continuity mx ix dy x embed give parameter fitting yield order without iteration propose employ learn extension semi supervise interpolation assign rbf df k high manifold scale subject nn interpolation f I initially method essentially loop determination complexity projection overall step require value x class directional neighbor point k throughout iteration training repeat step throughout complexity section discuss extension supervise rbf interpolation ridge know linear dy modify square adjust weight dual problem product sample product since permit high kernel feature translation invariant family link kernel ridge dimension set I interpolation define write coefficient interpolation give kernel ridge rbf manifold embedding ridge model embed kernel make vector class meanwhile supervise coordinate allow geometric sample concentrate manifold separability preserve performance present extension manifold compute evaluate embedding objective embed remove provide manifold embedding test neighbor dimensional semi interpolation method rbf fit interpolation compute iteration embed map adaptation sample point near add embedding neighbor nystr om nystr om discuss modify nystr om coordinate formula gaussian sum neighbor original field regard classifier semi label first face image individual face database take pose illumination convert subject supervise laplacian embed enhanced separation class separable total directional large due overfitte optimize minimizing avoid final interval deviation meanwhile compute objective embed pair parameter become biased laplacian choice linear reliability may monotonic slightly procedure parameter ratio set embedding display misclassification label experiment early rule apply curve high interpolation add center method well repeat image first image database object category manifold object show figure image normalize learn normalize object embed scale previous experiment obtain object misclassification sample classification database unlabele outperform graph semi regular consideration extension learn supervise building figure differently purely use neighboring vector meanwhile approach function coordinate rely representation experimental confirm attain good performance respectively exploited kernel center learn interpolation classify unlabeled test way iteration assign label nearest classification low generalization image next test highest estimate completely embed computed laplacian well embed compare interpolation extend vary confidence score assign image contain misclassification rate throughout ratio term well propose iteration term strategy may throughout strategy embed influence consequently next iteration embed dramatically even kernel preserve interpolation throughout iteration inaccurate assignment note propose interpolation effect regularization classification accuracy embed construct rbf interpolation function scale parameter rbf fitting sequence interpolation compute misclassification regularization misclassification figure smooth scale resemble regularization objective coincide permit capture parameter learn extension manifold manifold construction rbf interpolation interpolation optimize sample estimate show regularity interpolation control optimize regularization encourage sufficiently strong direction separation ensure effective separation outperform solution application along method classification would fr map sample ambient space low preserve separation supervise manifold available embed known problem learn become especially propose interpolation provide supervise manifold algorithm radial embed embedding manifold smoothness interpolation class interpolation
approximately result parent parent influence target prominent parent avoid parent parent practice indirect improve parent explain distribution decision tree state tree compact learn clarity modification unnecessary complexity key scalable greedy exhaustive well parent subset intuitively influence small influence may detect parent parent detect non parent conditionally strong little influence conditioning parent parent bind cause parent c dynamic states explanation omit possible parent variable contain conditional ensure target strong attractive parent variable long prevent case due large actual parent quantify information parent parent illustrate action implicitly consider matter perfectly determine transition every ensure parent parent add return exist conditioning infer influential parent together useful detect assumption prevent form hardness may implicit dependency separate belong initially true finally assumption beneficial subsequently output hold exist return satisfy divide material derive mdps state mdps function transition realization trajectory inequality subsequently consider evaluate accurately parent policy likely visit trajectory realization infinite capture greedy parent evaluation depend probability trajectory visit constant indicate effect hardness large arbitrarily wrong structure multiplicative parent realization estimation transition must advantage lack multiplicative effective decreased exponentially parent characterize policy target policy policy realization never behavior infinite unlike approach difference parent realization behavior behavior policy visit space domain randomly domain compare normalized furthermore error refer evaluation behavior monte model construct partial free sampling use heuristic ratio flat pair build pair base parent need probability table behavior know wrong parent sample efficient free dramatically evaluation target greedy approach drop select uniform solve discover return problematic trajectory policy policy resolve problem modify return action select useful benchmark true thus rd quantile scale policy exploit structure require evaluation evaluation achieve normalized scale trajectory take structure large trajectory low approach line rl free trajectory similar trajectory adapt high variable parent parent table uniformly ensure distribution sparse return last return horizon randomly policy derive linear episode discount stationary domain modify policy return ensure action state least generate flat flat scale state compare performance trajectory policy fail task artificial slightly well use trajectory behavior probable target evaluation verify factor present challenging bit ram horizon behavior experience linear episode require extract action select trajectory achieve evaluation show evaluation average trial much artificial trajectory complexity sample exploit dramatically large small trajectory analyze three restrict imply weak parent weak significantly relevant parent believe world satisfy learn combinatorial correctly parent structure knowledge say effectiveness structure encourage adaptation l time horizon index number factor notation process mdp previous next variable subset parent parent big high break part derive trajectory need evaluate notice learn everywhere visit likely visit behavior visit never parent number trajectory proposition tell need perform high outcome least proposition outcome q visit event notice infinite never notice l distribute bernoulli least trivially outcome random variable x complement eq completeness use factorize state action v random variable give receive denote target k observe sample realization distribution estimate trajectory apply realization action pair trajectory notice hold directly realization action score realization automatically discard contain parent meet number error hence probability simplify add add parent break distinct parent parent realization realization parent enough hold trivially want sufficient applying probability hold probability use assumption stand realization proof least assumption sure strong alternatively enough realization add parent least combine stage iteration correspond strong strong parent include iteration second weak parent add stop twice hold bind algorithm least union variable add observe parent parent assumption probability inequality since add specific everything trajectory satisfy lemma set less probability induce mdp mdp proposition verify bind equation result corollary problem essential provide superiority policy evaluate policy computationally sample exploit factored dynamic environment sample well high reinforcement rl algorithm rl choose action quickly leave business understand problem customer successively maximize click click ad test company management test unless company exist obtain company exist general generate call generate policy try policy reason construct complete trajectory construct complete little internet technology million transaction world generally million want extremely occur look cf analysis
datum dense noise thresholding let invertible ssc satisfie nc optimality ensures triangle guarantee let triangle use step sufficiently convergence date dense execute thresholding invertible ssc level iteration obtain accurate tail bound chi least q prove satisfied appendix detail give corrupt execute invertible ssc level see obtain vector follow ssc convergence large design discussion give establishes rate least constant rely less unity translate translate make turn challenge get establish general matrix union modify rsc act isometry reader detail along modify give step denote something invertible shall gaussian sub design satisfie get general level corruption improve tolerance result readily vector permutation magnitude e claim accord chi freedom trace concentration inequality purpose perform exercise involve corruption establish exponential chi freedom moment triangle bind norm center second third step markov give repeat give complete ex microsoft com problem robust regression corrupt specifically underlie corruption vector coordinate solely formulation impose strict assumption hard thresholde mild recover exactly sub result generate propose extension sparse fast recovery solver sized corrupted response variant call fast good solver ex error experiment mean give error square least address regression economic computer goal corruption set clean optimization jointly admit efficient solution indeed exist exist provable guarantee observation adopt follow wish corruption value assumption corruption recover penalty result corruption severe restriction either sample incoherent universal less recovery amount importantly extremely unable guarantee recovery result albeit clean error intuitive seem long adopt good knowledge rigorously non setting despite appeal contribution guarantee thresholding mention fc provide select non global dependent convexity ssc smoothness definition rate recover ssc satisfied h allow adversarial value admit universal hold stress rigorously formally completely hence recovery robust fc well large solve address issue design gd geometric rate recover fc hard popular sparse geometrically constant sub experimentally thresholding algorithm significantly fast solver recovery property white corruption solver error organization goal estimate allow represent regressor generate perturbation potentially enforce example corruption clean point unbounded generate clean response sphere compose vector eigenvalue key ssc satisfy strong resp strong uniformity definition ssc sake necessity face adversary precisely ssc bad ex perform hard operator magnitude thresholding operator regressor update regressor fit three try regressor fc regression minimize active progress gd perform update single gradient objective active beneficial noise present along prevent fc expensive execute gd progress adaptively select fc gd active hard thresholding convergence variation also applicability technique setting subsequent sake ease exposition dense analyze fully fc convergence step carry fully regressor set regressor corrupt execute parameter invertible ssc constants algorithm sketch residual set whereas xx c performing give guarantee design actually ssc condition high let least similar prove large ex readily accommodate addition sparse unbounded direct reader setting would like requirement analyse mild assume assume satisfy gd perform gradient rate execute fc make distribution fc hybrid algorithm gd adopt advance pose problem problem fc execute fc gd policy enforce iteration solution reader assumption readily satisfied sub design albeit shall attractive fc approach response shall dense exist objective would recover recovery alone fc fc refer rsc define sparse recovery analyze rsc convexity smoothness shall say constant rsc level convergence satisfy constant see execute thresholding solution particular sample n complexity high constant corruption index fc see solver corruption fc problem solver carry dimensional high experiment offer statistically fast dimensional regressor choose sample select uniformly diagram repeat plot robust augment multiplier implement solve fine grid result fc solver solver ghz extensive comparative study homotopy outperform counterpart recovery extend study solver approximate message pass amp problem solver non phase cpu require compare present indicate run
expansion write linear consist part remain note thank computation v n n n leave side equal follow proof ib b c I ic ic ic I sake completeness herein auxiliary argument conclusion remain condition notice continuity boundedness derivative inequality likewise satisfied therefore corollary multiply additionally gamma euler formula p obtain rewrite j multiply side jt kt jk jt h j jt p jt jt p I apply laplace side equation side let f p f p multiplying multiply get additionally second last calculation denote di apply moment side obtain di divide side equation tt obtain tt divide side tt argument continue side tt change variable similarly compute generating generality minimum index firstly divide side let divide finally divide side argument theorem rewrite c kx calculation iy iy different pairwise multiply side iy therefore let side equation j continue prove appendix integral notice xx x rewrite equation tx dx turn integrable odd turn l c z rewrite indicate element simply argument multivariate find hyperplane tt tt sequel u l uv v uv obtain j multiplying l side fashion uv uv tt paragraph readily formation formation yx ib dx multiply side lx lb lx lf lx dx dm multivariate hyperplane tt rt u jt uk apply equation multiply side multiply ic lb l lie outside l l argument student appropriately symmetric jt jt generalized I multiply equation db put hyperplane tt kt assume distribution minimum side eventually high dt bt tt bt dt tt db k la lt l conclusion assume half circle I follow rx direct calculation additionally notice achieve prove continue care handling variate sequence I j nr p ir ip uv uv fx f identity fx equality convert rewrite proceed equation measure kp n uv j k n kn k k uv du w b extend hellinger contrary argument sequence nk k ij n assumption g g combine choose combination x satisfy p index regard j j likewise divide let system scenario polynomial equation trivial contradiction n divide numerator denote absolute ne combine notation theorem result immediately continue argument without loss p j j easily na p divide numerator case n n n v p I contradiction one eventually imply h p n p exactly p n cc know imply mean h polynomial obtain term positive admit equation admit assumption demonstrate yield h n p h yield polynomial finite happen n thus p case p k nh nh nh n n n n n j k nh np n j nh nh n k n line consequence generality p n n divide numerator denominator scaling argue case divide numerator denominator contradiction argue case divide numerator denominator happen equation admit consequence overall conclusion important get contradiction hold nm n treatment simplicity sign follow equation part b theorem theorem n np I result j notational assume obtain divide numerator denominator obtain result second get two system nz nu ip np n j nx large constant therefore fact assume applicable third order denominator fourth system p admit assertion argue way taylor expansion part contradiction odd number odd without assume divide numerator denominator odd odd already solution generality choose nm j j g n obtain part good scenario low minimax minimax skew calculation appendix divide far infinitely loss dp c np I sum contradiction infinitely hold n dp n np dp n dp contradiction nc dp lemma section support grant nsf nsf dms nsf thank several valuable type study identifiability behavior type fit broad variate show rate applicable include scale shape scale mixture devote demonstrate class structure role determine parameter fit determine rapidly simulation demonstrate identifiable kernel pose mixture density datum understand convergence practical one assess structure work chen metric cumulative line measure use wasserstein major advantage wasserstein measure wasserstein compute effectively popular distance total result well wasserstein distance mix support coefficient practical indicate wasserstein allow people rate paper mix therefore variate popular tool model unobserve associate underlying issue note recent research back continue nonparametric deconvolution beyond mixture contribution chen identifiability mix fit finite mixture oppose fit mixing specification mixing know chen scalar restriction wasserstein provide natural mix convergence mix model study focus per se show condition mention computer science efficient procedure cluster fit mixture carry vary location covariance considerably gaussian scale skew skew gaussians goal variate arise variety mixture euclidean belong know mixture measure location family elliptical family distribution shape skew exponentially location shape matrix rate enable rich heterogeneity among type behavior addition shall setting fit fit later complex consider class function elliptical covariance exponentially shape include modify rate know rate determine mix rr infimum take converge rate atom happen atom atom vanish generally rate mix sharp wasserstein variational distance mix wasserstein distance sharp mixing g extension result dd attempt give power variate space entail amount independence quantitative density several identifiability definition type strong take order involve identifiability criterion worth note tend primarily order identifiability something along additional range actually admit euclidean turn fit mild regularity condition establish sharp method mle induce minimax mixing addition converge rate type develop exhibit kind identifiability identifiability family include student second model type proof characterization theorem insight draw smoothness express characteristic vanish infinity cm n identifiable student exponentially identifiable location exact fit exact generic exact generic generic generic unknown logarithmic fit modify exact fit fit order identifiable covariance I fit gamma generic generic unknown logarithmic skew generic dependent fit dependent fit unknown term point common satisfy either identifiability family identifiable family skew setting location order theory describe exact fit mixture fit gaussian mixture turn separate novel treatment throughout family weakly weak identifiability lead extremely shall able precise non weakly identifiable class algebraic smoothness determine covariance lack identifiability due entail take order minimum value trivial emphasize bind sharp fact convergence find one precisely mix explicit determine algebraic geometry keep use standard addition convergence quickly component shall describe class density one positive order identifiability identity combination true parameter prevent class exclude case measure gamma identifiable class identifiable terminology convergence behavior class estimate fit mixture fit fit happen location bind location logarithmic among generalize skewness skew exhibit family really identifiable somewhat consequence measure generic true measure accord admit exact skew carry behavior subset convergence tie certain polynomial equation turn mixture skew identity second derivative skew manner dependence long adequate brief description general strongly specialized weakly identifiable class sharp choice strongly strong try force taylor vanish inequality sharp resort careful taylor continue key taylor expansion derivative independent back original process equation exponent desired link problematic behavior convergence establish popular gaussian mixture comes assess quality mix mixture mix theory redundant expect redundant complete spectrum logarithmic mix measure deconvolution gamma mixture wide within class useful way identifiability favorable convergence identify avoid organize provide preliminary present strong identifiability address provide devote weakly treat density separately easy consequence maximum many case theoretical bound contain available behavior likelihood mixture use family mixture difficulty traditional deconvolution fail let find bad use le approach wasserstein metric p p n identifiability student identifiability multivariate fit first identifiability general exponential skew exponential fit singular case p strongly weakly identifiable study li g derivative g kn large borel sigma algebra take kp define restrict addition point kp p ij ij tool analyze adopt wasserstein optimal wasserstein matrix relationship wasserstein clearly quantify establish notion probability fp p x via composite regard wasserstein take family combine display arrive wasserstein tx g g multivariate g p g g exponentially modify let independent combine bound I g p g g density multivariate product combine vector g develop accord density wasserstein identifiability essentially need notion identifiability order variate model advantage range allow wasserstein measure hold mix location gamma skew general fundamentally distinct exhibit interested skip variate positive notion fx identifiability fail many instance assume x clearly elliptical anti say r cr first identifiability derive support identifiable uniform depend impose boundedness nonetheless sense close distance varie extend globally measure mild property positive constant g g c verify identifiable thus remarkable result class wasserstein fit move lie interior give strong class identifiability family hold different uniformly r establish fit set family second large element bound suppose fx b fx bind distance sufficiently b elsewhere remark counterpart mixture exact fit setting fit attribute hold set method continue apply part version mild address small parameter away remove impose eigenvalue b estimation hellinger induce iv boundedness boundedness meet bind establish vanish fast subset fit case result wasserstein identifiable fit identifiable uniform k tight proposition private communication mistake correct add note condition valid strong fitted establish simply subset vary subset density family identifiable order admit proposition mix measure set lose identify broad identifiability hold also continue certain transformation vary qp generalize fx fx von distribution modify variate space variate generalized distribution identifiable multivariate odd degree freedom odd freedom modify gamma theorem chen proof however nontrivial conceptually somewhat straightforward demonstrate establish infinity common establish strong interesting next shall meet identifiable class structure theory later class identifiable c fix transform identifiability transform preserve assume order class identifiable modify jacobian conclusion first identifiability strong second sharp bound wasserstein distance useful discrete measure continue class function derivative derivative old boundedness first old relaxed old old identifiability develop class family identifiable class rise gamma skew quite specific algebraic density role determine identifiability mix covariance belong broad gaussian order identifiable multivariate density within broad class location parameter weakly identifiable family multivariate identifiable immediate thank identity whose satisfy x x identifiability mixture eigenvalue obtain sharp equation value equation solution choose check k n n get choose equation show sequel trivial therefore determine case appear difficult method deal basis basis appear equation inconsistent admit rescale precise relationship mix measure gaussian equation p g k sufficiently investigation part together mixture yield interesting convergence estimation mle density fairly moreover entail extra mixing actually see place restriction fx gamma identifiable thank choose identifiable fix vary strong violate neither estimation estimating setting logarithmic obtain lower collect identifiable class consider fit convergence identifiable location covariance gaussian convergence rate fit minimax convergence convergence logarithmic condition logarithmic location logarithmic skew normal fit even convergence rate k hellinger centering set capture entropy fit admit take ingredient rate heart wasserstein low identifiability interest particularly equip variate exact family generalize multiple constant depend give degree fit multivariate generalize gaussian shape part big location gaussian minimax obtain mix skewed density behavior finite mixture skew class skew positive skew distribution support cl g n g give generalize univariate univariate multiple involve achieve identifiability criterion difficulty entropy densitie detail defer iid true mixing likelihood algorithm em algorithm maxima multiple time obtain distance size repeat obtain panel convergence establish confirm finally gamma even difficulty converge maximum question open b g iid distribute accord mixture density density fit mixture exact accord asymptotic mle boundedness regularity smoothness one hellinger distance n pp verify paper sense instance part immediate le cf constant supremum infimum form mle conclude optimal logarithmic mixture gamma skew gaussian fast logarithmic summary obtain number mix collect convergence minimax fit identifiable rate fit fit rate condition generic location exponential rate logarithmic exact convergence satisfie convergence rate condition hellinger center integral denote identifiable non universal fit admit main ingredient lie heart distance weak identifiability able establish well low number variate type exact gaussian positive multivariate sufficiently student family odd multiple depend multivariate shape big part big g cnn cl scale gaussian n class density function like measure skewed class mixture skew assume constant skew fit cl k w mixture univariate depend mixture generalize univariate class sufficiently hard difficulty condition entropy mixture defer illustrate theoretical true obtain possibility obtain wasserstein vary experiment bar panel distance establish panel metric confirm mixture even b rich behavior fit mixture choose class skew accord location distribution figure support scale uniformly bound skew illustrated simulation generic p mix support upper match previous b mixture gaussian exactly mix mixture wasserstein plot sample simulation agreement theory rapidly turn generate iid sample carry generic g remarkable within even finite achieve logarithmic take inequality precise specific characterization present representative theorem fit set identifiable weakly identifiable insight organize gamma skew proof spirit interest proof infimum hold k g n g ng n converge notational replace sequence application p dx g g x replace plan w g j point easy observation sufficiently ij inequality observation follow important taylor nx k nb nx rewrite element argue opposite tend entail p g nx tx td nx g nx nx nx lemma display vanish identifiability criterion establish follow manner find sequence tend nx g g k g nk k fact write ng g ng limit notational replace subsequence sequence ip strictly potentially singular may assume semidefinite use fx shorthand multiple possibly non plan achieve mass couple probability find plan check k sufficiently p n order nx ij lipschitz clear derivative respect order evaluate pair coefficient associate depend shall vanish indeed take summation lx uv therefore p absolute nc nx ix second identifiability coefficient g ni n ni element equal taylor r expansion remainder n dx conclude c w c exposition univariate scalar employ proof p ij point n ij ip fx ip ij enter identity n f f v derivative sum proceed prove trivial construct give k k observe arrive bind hellinger distance w n fx verified fx fx v p g x second form cauchy v g claim turn suffice pass nx combine extracting summation I v n nj j differ one likewise equal divide numerator denominator obtain least element atom trivial entail solution equation contradict vanish hold converge ne differ limit odd linear combination function even coefficient employ entail proof solution I write equation divide b consequence trivial trivial without generality side basis polynomial polynomial equation retain display choose verify basis additionally choose check basis appendix theorem characterization regard strong transformation fit exponential mixture fit skew gaussian fit skew mixture proposition corollary already defer k p I p g p w contrary n g n c w g g g p almost surely identifiability g complete proof wasserstein mean case order identifiability appear guarantee impose condition ii tuple k xx possibility finite hyperplane tx j tu tx first choose finite distinct hence hyperplane tx find hyperplane I j multiply side f e jx jx result tx repeating argument get equivalent rule rewrite follow uv uv uv entail imply identifiable assume contrary modify jacobian matrix possibility complete multivariate x direct check conclusion proof defer assume result ip v
child root th tree path label plain base rademacher class value random think dyadic quantify form norm class small decrease consider parametric sequential covering set cover form nonparametric following rademacher absolute lipschitz however sequential scale yet base contraction give loose regret introduction offset minimax control offset rademacher random value term value notion minimax offset end first sequential rademacher class bind obtain offset rademacher take advantage negative offset rademacher recall conjugate conjugate controlling supremum finite offset rademacher calculation infimum achieve technique beyond lemma cover rather lemma yield bound sequential minimax offset rademacher dimension bound minimax smoothness subset singleton carefully crucially smoothness point fix low offset rademacher match constant long exhibit match constant square loss quantify covering say exist tree depth q value tree lemma entropy rather combinatorial notion closely depend behavior corresponding involve hidden suppose statement statement suppose lemma class exist arm upper ready lemma detail regret assumption growth rate match upper growth combinatorial dimension statement class furthermore factor dependence size devote upper recover absolute loss convex yield match logarithmic properly convexity examine see discussion loss growth cnn n cn pp finite logarithmic factor class bound lower modify growth loss assume check convex third derivative remainder strongly smooth truncated technique universal correct modify minimax function parametric function convex sequential bounded combination function follow pointwise entropy scale scale banach sequential rademacher upper yield forecaster relaxation enjoy specifically ds sp sn problem literature phrase expert possible sublinear regret expert randomize loss picture also define front infimum bound call inexact oracle lead pac yet expert repeat beyond beyond case correct bound discretization bound emphasize obtain equip supremum norm entropy logarithm aggregate procedure net give logarithmic amount aggregation capture indeed conclude slow obtain entropy phenomenon learn concept paper relaxation sequence mapping observe condition one prediction relaxation specifically condition recursive say admissible forecast eq version admissible bound relaxation algorithm relaxation enjoy offset hold rest restrict x tb schema relaxation ty schema proposition closely concrete decrease likewise monotonically decrease happen within admissible base give b outline provide schema set bound regression regret alternatively basis knowing predict thus loss round readily schema design algorithm enjoy bind include limitation partly finding consider space remark distinct exploit aggregate unified technique learn algorithmic one algorithmic aggregate aggregate beyond lie pointwise independent cover covariate difficulty arise bound end notably difficulty recognize cover empirical complexity measure behavior characterize optimal understand covering log partition ball radius aggregate combine regret integral similar statement empirical sequential number phase match logarithmic even rademacher phenomenon notice phenomenon given convert statement sequence thus recover technique many statement relaxation provide improper denote distribution regret third last step observe term outside linearity definition eq view jensen step jensen function function pass upper replace subgradient variable jensen fix jensen bound conjugacy pass far bind step q proceed upper claim statement appear except account bad along path q fix sequential sense e construct cover include soft thresholding tv upper denote let specify later write time bind term term bound choose arbitrarily restrict set possible optima depth choice result observe interval suppose sake contradiction depth least must label path clearly order leave easy leave leave complete subtree size tree tree sign bind choose particular expression definition stay close tree obtain low q inequality since free choose delta function suffice bound growth cover balance change next give turn obtain use second derivative first low bound symmetry binomial give point q grant dms research loss loss curvature affect entropy match online regret forecaster enjoy computationally finite linear predict forecaster abstract encounter observe response literature former fall series form base past set probabilistic instead predict well benchmark strategy latter term
however optimization like get beneficial design architecture well optimization technique identical stream combine bottom meaningful representation roughly resemble vector aid matching perform patch two channel feature width height channel compare patch path patch patch identical multiplication patch combination involve computation result pass intractable maximum map globally produce dimensional combination patch respectively backward pass implement cnn good extract level shrink aggregation image refine pool main ingredient layer consist convolution layer refinement map map preserve pass feature map low twice time small input refinement improve result full image bilinear bilinear approach term start use coarse fine scheme iteration full resolution additionally boundary detect boundary smoothness expensive simple bilinear add field variational refinement see frame truth per frame cm ground unlike neural require task ground overview training ground truth train special motion real stem move observer distant capture optical truth dataset ground truth special realistic version clean effect image ground truth magnitude train cnn provide training image retrieve category city landscape cut image multiple background result view per figure generate sample affine relative interpret camera object move second image optical image number position transformation sample adjust distribution supplementary dataset background arbitrarily strategy neural overfitte augmentation online transformation well quick operation gpu augmentation increase image variety flow accordingly field image sigma sample multiplicative channel image additive change use gaussian sigma result fine refinement keep network nine convolutional form relu nonlinearity connect network size size deep layer layer start fourth deep roughly use error error optical flow euclidean cnn modify show descent momentum fix parameter recommend pixel fairly batch divide tackle problem increase iteration overfitte tuning input although optimal dataset factor dataset term type fine network flow tune clean tune use performance define table fine ft table show public well additionally train realistic real network outperform compete cc cccc train train test cpu gpu ft ft ft one ft even average often smoothed solution interesting qualitative result figure raw optical predict two truth figure show visually error net especially region partially refinement projective transformation encounter network fairly additional fine tuning variational probably outperform aside various net interesting thing bad variational realistic set well cm ccccc cpu layers gpu art time cpu leave aside thank alone enough optical fairly pixel tune fairly augmentation answer augmentation allow draw conclusion strength generalize clean motion suggest though training training heart datum though current setup datum become problem discuss pixel pixel ft px explanation increase computational recent train optical realistic affine synthetic optical flow natural accuracy prove capability cnn perform acknowledgment start grant grant cr ec project visualize field use provide color magnitude color intensity motion illustrate code flow vector pixel magnitude pair main independently normalize pair apply pixel foreground view angle image image set type uniformly size deviation transformation translation coefficient angle translation aim roughly match simply gaussians cc family distribution contain precisely gaussian interval overall bernoulli show flow cut pair pixel histogram translation translation observe sampling gaussian accurate filter filter filter structure converge coarse visible filter correlation layer direction magnitude http supplementary gpu capture pixel show life video http frame video fig false h edu van technical technical university de convolutional networks cnn successful computer especially optical cnns paper appropriate cnns capable optical flow train cnn competitive accuracy many field prediction flow optical precise localization involve image representation optical flow fundamentally differ previous application solve cnn capability train end level scale abstraction help find actual predict surprisingly way optical competitive generic help optical flow dataset art optical material trading
experiment involve effective collapse perform particularly sample count subspace integrate assign time take cluster e j denote collapse lda mixture integrate beta sample produce prediction accuracy comparable dataset interpretability subject experiment involve require show performance compare output visually learn feedback important dataset unsupervise depict digit dataset per document dataset value rescale bin picture lda initialize label incorrect iteration use section learn subspace incorporate sample technique depict assign cluster lda accuracy cluster lda depict capture dependencie full acquire dataset comparable produce digit indicate achieve compute measure prototype prototype generate sensitivity introduce learn sensitive within range reasonable verify interpretability perform incorporate require require name six choose dataset subject require incorporated subject allow effect possible balance half subject participant age answer question question accurately break four question representation I top cluster prototype number number prototype run lda initialization ground truth visually identifiable statistically ground truth label manually code expert author analyst produce ground label statistically spend average per spend style produce statistically difference preference insight experiment demonstrate participant degradation p prototype stick water illustrate learn function later show characterize cluster prototype interestingly highlight make one showing absence loop among cluster initialize tend cluster near refine show third digit tend share sparsity prototype set cluster highlight box subspace prototype cluster ingredient case base prototype come prototype set defining show quantitative interpretability neighbor base reasoning historical back intelligence offer topic balance accuracy interpretability predictive mit edu framework prototype bring framework represent cluster simultaneously play important role prototype interpretability preserve subject statistically participant compare art look people make recommendation make amazon instead amazon customer customer ignore medical large patient favor medical example individual numerous reasoning involve fundamental effective strategy decision decision service usually successful human leverage source decision provide decision make intelligence case reasoning approach rely situation solve world fundamentally limit complex fashion dataset discuss prototype cluster powerful neither situation regardless model model feature model bridge human produces verify human subject subspace meaningful important aspect result participant understand dataset compare output people ai reasoning insight learn result solution provide alone maintain complex challenge backward cognitive load case require manually mixture discover distribution intuitive point interpretability reduce per feature proxy interpretability problematic interpretability model present machine unsupervise learn important subspace preserve interpretable interpretability prototype view three prototype interpretable intend third type explain focus neighbor intuitively generate observation important piece relate movie profile movie subspace prototype observation feature indicate indicator generate bernoulli feature describe wherein row discrete outcome length particular g mostly important consider prototype outcome consider irrelevant look generate next large prototype feature take select agree prototype copy prototype within cluster subspace rest mix wherein important piece prototype dirichlet parameterize hyperparameter index feature observation assign allocation begin mixture though necessarily hyperparameter q extend measure modify square set hyperparameter mean hierarchy plain classifier illustrative subspace compose feature color assume ground two subspace shape define cluster face h cm p cm
contrast often architecture dataset range tf rnn term lstm architecture provide benchmark task human labeling release empirical version briefly exist architecture exhaustive list due space survey resource state use management system interactive formalize agent attempt predict resource allow significant progress field though compare neural architecture contain extract twitter al million extend take long context use triple et however public service human stream message room twitter datum believe room closely micro aggregated resource thus investigation traditional two party dataset interaction seek study c c type word pre topic computer restaurant tracking system human hour track twitter post micro generation twitter twitter triple b micro twitter human extract micro generation focus approach attempt leverage development neural notable et rnn initialize denoise tackle twitter idea superior performance retrieval near approach al exploit structure recurrent decoder achieve poor twitter triple translation al encoder decoder one rnn rnn generate study overall highlight potential neural architecture interactive large research system human neural network ai turn corpus refer internet network protocol participant room channel channel channel technical support issue free channel question potential address avoid confusion day simultaneous channel never occur extract user stop problem continue nature constant stream fairly message room extract intend message clear user old er question perhaps comment corpus extract room party message time tuple tuple easy separate rest intend message trivial sometimes locate user correspond word stop false positive order english user intend message match assume message response frame minute presence name recent identify question say duration along far process standard nlp initial multiple people user user treat dataset issue hour rare researcher filter axis word min turn per avg avg per median property corpus crucial architecture characteristic turn show see turn per approximate aside rest corpus process extract pair triple boolean correctly identify create triple contain actual triple elsewhere test example move response set wrong response l well guess get copy format tag part context stochastically use simple maximum context unnecessary length context select short context often break medium long turn task metric language task agent ask response language perform classification improvement classification lead neural architecture benchmark baseline tf memory pre use library tag word category name location lstm architecture full test triple start rd overlap minor issue rest datum word document retrieval put appear elsewhere word appear context appear classification tf response cosine select response return neural allow time state time previous state diagram rnn tie last rnn context context maximum diagram rnns primary current language rnn encoder rnn encode response upon question answer utilize consist rnn tie embedding embedding embedding respective rnn token tune rnn learn generative measure similarity dot convert label frobenius simplicity training response draw elsewhere layer neuron initialize orthogonal initialize optimizer gradient critical rnns rnn architecture change unit unit sentence embedding dependencie lstm unit determine input old retained error feed lstm overcome rnns otherwise layer configuration number neuron optimize rnn use tf rnn lstm model train evaluate ht tf rnn lstm r various recall lstm tf rnn case likely due ability rnn context overcome correctly classify ht context I response n transfer file rank processing training confirm importance training increase research describe construction availability several possibility research rich preliminary rnn lstm selecting obtain lstm several sophisticated separate trying replicate retrieve support subject interesting difficulty control move false response
close peak mode shift magnitude shift computation current position expensive shift mode perform cluster base merging mode neighborhood increase distance mode neighborhood merge experiment image density use locality lsh find point choose implement lsh lsh available lsh hash per picture lsh present indicator cluster kde result kde areas fine tail capture pick far apart htp different ccccc cd cd image breast heart e e diabetes htp accurately approximation mean recognize radial desire may work kde kde proportion present advantage bandwidth cluster kde less lsh approach perform index hausdorff order magnitude acknowledgment department provide part definition remark mean frequently estimator issue single estimator computation sparse establish incoherence radial construction gain look problem proportion mean quantity form rigorously kernel estimation mean statistic context kde kernel reproduce kernel hilbert motivate kernel review concerned word kernel sparsity seek accurately kernel problem kernel prohibitive efficiently regime scalable argue exist slow primary approximate algorithm radial sparse rest review kde definition formulate approximate sample relate work establish incoherence apply rely demonstrate preliminary appear matlab two review feature address sparse problem although kde ingredient plug anomaly detection detector commonly employ shift kde reach hill test kernel evaluation undesirable prohibitive kernel evaluation shift kde numerous demonstrate kde symmetric definite square semidefinite hilbert reproducing think closed span rkh property reproduce mean rkh mapping mapping derive fact kernel permit distribution hilbert size base n reproduce product embedding mean pairwise inner product evaluation computational approximate sample motivate satisfied density estimation say estimation equivalence view def common radial kernel form q student normalize depend illustrate indeed symmetric definite rkh radial property x x write bandwidth parameter although closed need generality consider abstract h k z form later develop dictionary element hold abstract sparse hard effort time overview matching pursuit originally pursuit atom capture magnitude inner atom iteratively portion note require quantity nz undesirable pursuit cluster point minimize cumulative heavily kde think simpler pairwise computational task speed sum make kernel question yield efficient point effectively reduce evaluation effort rapidly quantity query case construction kde seem concentrate effort computation problem problem calculation discrepancy original consist approximate complete nystr nystr compose detail connect nystr om scheme tailor nystr one algebraic propose coherence base context main atom quantify large absolute atom complete involve coherence however minimization contribution summary contribution sparse error novel radial solve center center approximate running kde important computational subsequent calculation perform address particular automatically select demonstrate kde proportion shift kde separate part find finding value nonzero index pose eq inner optimization unconstraine quadratic z rewrite approximate briefly highlight om ik q nk express nystr nystr nystr om nystr om norm space commonly frobenius solution strategy find orthogonal cauchy confirm existence equality reach basis minimization think incoherence establish q begin inequality due since p z approximate mean radial strictly definition note radial g j minimize translate product pose let ik approximation describe linear algorithm htp choose first choose element output base find determine coefficient burden freedom time possible kernel I example symmetric definite kernel semi conjugate approach advantage evident tolerance value stop record compute overcome indicator let use increase element algorithm max ok max avoid te complexity make simplex k k alternatively account constraint negativity solve complete sparse specific machine task reduction class estimation finally explore result description probability visualization consist create similarity similarity among reduction case embedding notice kde kl divergence obtain start difference distribution symmetric distribution empirical accord construct matrix induce visualize distribution need distance yield computation assume sample inspire flow cancer patient range analysis translation procedure evaluation respect factor small large run htp htp computation total htp range kde kl divergence kde function project indicate construct kl kl two half divergence criterion result f resulting embed also furthermore subsequent negligible htp
degree vertex connect objective separable connect component rest form connect graph sequence precede walk begin circuit edge visit thm decompose algorithm operate penalty consequence representation turn decompose rewrite grouping together slack visit slack vertex admm routine edge along first concrete case crucially fuse efficient combination make efficient dominating reach primarily influence question experiment type strategy approach half first connect walk every description odd degree graph graph decomposition appeal find intensive decomposition phase fix instance little extra time yield open dual quick well fairly maximally leverage solver trivial subgraph add unlikely tractable graph potential instance add set remain generate split merging creating result unlike provably subgraph set ts clear decomposition strategy vs impact build delay present motivate simple node immediate much short conversely grid row dash lr pseudo trial display pseudo impact large speedup substantial algorithm different goal mean high path step require randomly graph preprocesse massive dna algorithm ten evaluate unable adequate slack repeatedly laplacian form benchmark conjugate acceleration hierarchy add subsequent highly efficient available straight spatial example stack proximal split solver benchmark flow idea strategy compare naive treat demonstrate median heuristic heuristic decrease long decomposition update programming package report trial trial create precision vary acceleration exception spatially real world synthetic create square grid adjacency node immediate increase however network often see grid uniformly structure many scientific application computer vision notable exception well understand dna acoustic collection adjacency exhibit dna regularity simulation dna result benchmark example strategy perform perform overall structure significantly iterative preprocessing section light cause performance similar balance aid highly structure grid median balance size example balanced cost short median length dna balance note recursively break half length directly convergence pseudo grid versus random grid approach log random graph little strategy underlie fast note simple row competitive specialize max flow work none solve fuse well empirically method make immediately drop replacement solver currently use regard approach property decompose contain intuitive balanced answering leverage admm convergence work heuristic decompose graph row proximal generic lead adapt lasso lasso nevertheless recover htb tolerance smoothed truncation smooth stepsize determine via backtrack line search practice saddle grid fuse lasso recover substantially smoothed particularly background axiom conclusion theorem exercise author g solve operate predefine technique lasso close flexible set
nk disjoint metric metric cluster representative cluster distance center center minimize distance cluster metric give location optimally problem assign np polynomially approximate kp ip distances location problem calculate new distance ki n dimension become particular proximity query discrimination jj large curse applicability distance space dimension high discrimination plan metric case center continuous previous center power running dimension space prove section result comparison purpose x median necessarily define weighted median median minimizer consider weight rational real rational relax assignment add eq point center square serve seminal paper lagrangian eq zero add k result minimum recurrence together four area inverse fuzzy principle work estimate point function far lie convex center interpolation harmonic harmonic analysis specie harmonic confirm hundred specie importance harmonic establish zhang harmonic distance axiom contour original hard fuzzy simultaneously strength c compute update add center calculate convex combination control assignment assignment classical mean square widely dimensional g choice contrast justify idea classical membership distance decrease well necessary condition probability seem analogy may circuit parallel current circuit optimization well define analog circuit equal instead separable center separate problem cluster serve relax produce membership normalize high exponent close assignment element assignment numerical experience iteration iteratively use exponent update increment assignment w increment arbitrary center mm distance assignment center operation iteration calculate dimensional time assignment center great b iteration close well stop bind iteration c generalization center update center subsequence decrease center minimize center probability exponent increase short distance become probable synthetic cluster datum consist take use rule step record percentage misclassification vice table give misclassification test mean mean mean algorithm table r algorithm mean mean algorithm misclassification algorithm mean datum use mean show r
formulate dual polynomial vanishe vanishe let optimally primal hold z point development inspire formulate dual lp sdp research law space allow g optimal differential atomic solving sdp dual algorithm assertion sdp hierarchy compact translate yield feasible polynomial optimal satisfie place similarly polynomial context er exploit paper er theorem also negative write square polynomial hierarchy priori small speak conjecture theorem standard convex close cone compact cone let matrix semidefinite set kx kx kx jt jj xt precisely moment measure interior differently imply invoke conclude sequence jt coefficient wise prove mm mm mm remark domain semidefinite sdp multi frame algebraic standard focus relaxation gram matrix representation exact primal super square overcome notably super great domain algebraic keyword sign total domain problem pass filter application image theoretical baseline want reconstruct equation enjoy affine study ten super idea property restrict isometry property stability reconstruction lead knowledge super early paper super appear matter invert discrete fourier separation spike inversion apply negative entropy role non negativity regard separation clear understanding role measure subset measure frame point member concentrate deal positivity matrix select member similar quantitative size solution evaluation point family function chebyshev moment negativity one equation frame assumption I unique solution minimize total norm super differ dramatically analysis sensing compressed sense recover super differ formulation see reconstruct e line point paper compress sense exist line continuous prescribed frame super compressed problem go infinity analogy program variation super property primal program polynomial phase issue super resolution frame important discrete satisfy condition fundamental paper huge indeed sharp dual phase support henceforth dual compressed deal resolution frame finitely last point severe practice matter deal formulation extend algebraic domain cope parametrization author infinitely relaxation involve proved solution guarantee recovery inaccurate prediction robustness recover solution often equivalently gram toeplitz representation constraint negativity domain rely programming sdp globally nonnegative er toeplitz toeplitz sdp form relaxed sdp version great sphere expand super resolution implementation frame semi operate tackle norm parametrization decade super contrary approach focus primal truncation infinite minimization univariate sdp standard optimality primal rank extract algebra optimality attain point attain taking polynomial measure er consequence dimension limitation sake knowledge e solve sdp moment measure numerical relative represent polynomial indeed polynomial attain attain red measure super resolution spike deconvolution harmonic investigate spike perturb use real appear naturally formulation sake sphere spherical support purpose consist variate degree relaxation pc solution moment use algebra indeed attain value prescribe prescribe blue appear greater close algebraic whose degree assume compact algebraic sufficiently linearly family index banach sup norm sign banach topological equip supremum disjoint measurable respective nonnegative nonnegative respectively eq decomposition q rewrite lp cone eq mx problem maximization lp gap close compact bound weak banach subsequence weakly element ir dual lp supremum feasibility lp close contain lp attain exist duality condition atomic atomic equality equality counterpart result discrete support converge hierarchy dimensional programming extract key construction paragraph semi algebraic define conversely value identity say represent cone sequel approximate cone degree kp functional acting kp p symmetric gp p construction notation finite semidefinite resp resp index primal lp primal moment
space output message advantage specify distribution characteristic tractable fourier novel distribution approach give fast update quadratic advantage approach query sampler pair prediction update regressor informative forest proceed expectation propagation importance training message operator overview kernel message message embed rkh message describe three artificial regression four demonstrate learner correctly regression incoming change implement distribution j form deal black box paper variable give thus assumption stochastically value expectation propagation ep iterative procedure belief pass message factor see project distribution message message send neighboring variable q numerator evaluate leibler divergence non distribution factor analytic often require approximation expensive integration box fully nonparametric technique alternative integration member qx compute statistic numerator base carlo forward draw support parameter projection could use algorithm suffer sample reliable running variance computational message map tuple incoming fm f ep employ forest message uncertainty function indicate exceed threshold require message importance cf new f via look forest leave mechanism problematic uncertainty uncertainty forest consideration demonstrate heuristic forest inaccurate move initial forest newly initial drift leaf split chain bad storage training finally factor specify tree traversal notably potential incoming expert forward sampling present employ term prediction forest traversal representative tree traversal importance point cost tree forest tree internal involve traversal one cost leaf make tree representative use forest gamma incoming message regressor typically income message around mode incoming characterize thousand leaf propose operator tuple message apply message incoming purpose belief propagation message prediction regression whose inner product directly tuple random feature evaluation incoming message tuple variable tuple tuple reproduce tuple us respective distribution embed product tuple product q kx alternative tuple message embedding dot product bad supplementary incoming message would employ suit grow set even moderately sized prohibitive map yet close kernel transform compute kx yx translation invariant inverse inverse fourier transform expectation approximate monte average frequency follow similar stage fourier expand exponent embedding invariant kx write mapping show kernel feature complex vanish invariance analogous material way store need gaussian input transform translation outer outer x nn incoming node sufficient output message notation feature tuple via estimate message close kernel estimate prior capture importance treat sufficient statistic separate treat incoming arrive sequentially n express covariance cost function uncertain make used evaluate two gamma second capable quality mapping message third fourth ep approximate compound gamma experiment reliably quickly adapt shift message encounter regression problem infer interface default otherwise numerator calculate straightforwardly ht f logistic factor dirac delta fig deterministic incoming p dot sample dataset run iteration message pair ep infer logistic message predict leave cross observe significant improvement feature z q I truth belief numerator prediction histogram kl error message kl evident relationship incoming message ground percentile supplementary ep uncertainty base variance forest kl prediction fix incoming message operator set crucial estimate part message use forward incoming message incoming message plot evaluate pass operator random forest move densely smoothly high set robust importance key forest e uncertainty nearby point check reliable divergence experiment approximate ep loop logistic generation sequentially present generate keep scenario common practice observation share number mini operator initial batch initial batch operator estimation simplicity accord heuristic full heuristic supplementary material output variance uncertainty log predictive setup importance sampler net implementation show predictive predicting upon dash problem shown observe drop stable incoming uncertainty display collect incoming message drop ep infer engine lead repeat zero message show classification error posterior generate infer net logistic factor infer good alternative importance sampler compound tail gaussian follow compound gamma shape infer normally distribute infer two gamma factor specify compound one directly default rely quadrature sequentially parameter fig infer run infer fig plot show agreement net pass factor indicate change income subsequently present learner uci repository learn message point point minibatch parameter early show fig essential rapid first message diverse sharp follow steady indicator message adapt learn incoming message place computationally demand uncertainty efficiently additional train mapping mapping far novel topic current hyperparameter selection adapt anonymous constructive financial pass propagation supplementary kernel inner mean outer embedding depend merely validation likely yield heuristic mini l incoming mini batch tuple let product income tuple tuple message subscript consider tuple kernel tuple message kx lk r x l r define I review relevant random contain invariant correspond infinite feature map approximate feature equality feature cost translation feature generate invariant properly transform complex exponential cosine approximation transform dx suggest extend product draw incoming message message kernel variance kernel unit width message incoming treatment reduce familiar euclidean random straightforwardly mean translation product sr feature consider vanish immediately w rw rw I I rw definition rd c translation draw compute sr equality expectation analytic kx randomly feature trial randomness trial entry wise define tuple one income rkhs kernel income
mod arithmetic acceptable network undirecte recover sign choose one negative node true get consistent reconstruct show reduce start completely indistinguishable equation plausible trivial identical water economic biology overall expensive singular value reconstruction entry acceptable structure compatible network realization need steady state safe steady realize steady causality determine direction every edge plausible acknowledgment dr constructive department chemical institute solve reconstruct network steady column state capture variable comprise incidence estimate fundamental pca circuit time illustrate water flow datum key pca graph identification many broadly effectively predict explain improve capability lot direct identification structure incorporate structural stage open technique reconstruct network steady polynomial structure largely process include energy balance simple write describe accumulation mass term reaction action hence accordance topology connectivity write balance interesting possible topology note connectivity steady evolution insight redundancy centrality understand reconstruct wide infer connectivity flow water relevant include topology measurement involve fit appropriate time model connectivity coefficient however steady rotation ambiguity steady series method ambiguity enable reconstruct topology closely distinct measurement iii realize limitation steady wish matrix steady row full network draw brief algebraic pca fundamental limitation pose transform describe recover realize limitation steady reconstructing involve structure approximate different equivalent relationship network thereby enable look equation cut set tree minimal possible set linear overall polynomial arithmetic graph loop multiple allow incidence structure trivial verify steady essentially structure column entry enter rank working network terminate incidence entire contain represent particularly deal example external assume connect external entry idea illustrate exclude environment sum circuit fundamental tree construct circuit say fundamental since circuit represent fundamental one matrix analogous fundamental circuit cut far span mind fundamental circuit cut matrix characterize ambiguity later mod trivial cut circuit reader book singular widely multivariate statistical provide matrix variable assume suffer rotation ambiguity datum stack corrupt general scale co scale matrix orthonormal singular value orthonormal eigenvector remain entry root eigenvalue principal sort decrease order matrix require computation objective suffice store hyperplane hyperplane hyperplane original recover hyperplane hyperplane due dimensionality pc retain absence criterion percentage idea well interested sub admissible sub find readily retain pc rotation ambiguity multiply invertible row incidence wish recover impose structural case pca orthogonal basis adaptively input indeed develop total square problem shall exploit constraint structural recover denote get denote order option upon e approximate like show pca step flow flow label knowledge network flow equation independent table true add flow computing flow diagonal co sensor apply value close lie constraint assume bad svd specialize remarkably obtain partially clearly true element space adaptively row subspace matrix vector constraint angle angle space span example angle project row sum wise difference projection comparison frobenius calculate experience clearly indicate constraint especially criteria another method experience motivation procedure purpose flow partition completely partitioning set invertible admissible since rotation invertible obtain eq due useful interpretation come correspond act incidence constraint construct discuss forward q fact exactly make thus reconstruct span equivalently old graph circuit discuss former every full pca gauss arithmetic unlike multiply straight forward must permutation column always identity order equivalently get suffer row column know round
methodology context organize health context methodology dedicated engine acquisition equip sensor physical pressure speed measure instance quantity analyze engine anomaly detect diagnostic send engine depend monitor sign engine status methodology introduce engine health experience see survey concrete early sign suitable temperature variation value result variation behavior typical computational remove practice measurement transformation generally record measurement instant record software recognition automatically reproduce segmentation indicator indicator learn actual indicator statistically vector signature identify origin generally specialize mainly diagnostic currently diagnostic incoming information difficulty sign failure discover perfectly decision level furthermore false lead least operator failure issue potential failure long health monitoring automate precise optimally drastically reduce operational event partly current box understand gray partially reasoning help propose integrated decision design indicator present propose methodology engine health focus health event datum temporal engine whose sampling individual record say sensor expert series turn high dimensional outline engine dissimilarity observe quantify anomaly transform anomaly anomaly detect somewhat characterize interpretation feature know major informative interpretation minor failure guarantee numerous variant indicator explain approach field experience feedback coverage monitoring problem indicator decide engine anomaly responsible construction typical univariate anomaly display world identify variance figure case instant roughly window signal fix anomaly detect change illustrate multivariate shift anomaly delay design generally explicitly test aggregation technique calculation indicator variation detector compatible expert expert quantity early normally student test coverage time shift compare summary expert window case expert rough idea inclusion recommendation test possible indicator indicator generally value insufficient training choose indicator rejection hypothesis case point reliable difficulty balance specificity anomaly detector order difficulty indicator test anomaly number nan period reject turn parametric knowledge explore range numerous possible indicator link specific easy temporal apply e classification class possible discriminate anomaly engine explain gray paper choose forest adapt binary indicator high robust performance interpretable cart measure indicator bayes classifier also independence decision naive bayes easy understand thank probability reference finally hundred indicator important anomaly redundancy appear unlike feature projection interpretable expense operator therefore detection technique redundancy maximum excellent dimensional difficult sign sign expert anomaly whose fix look huge amount propose production real world section begin simulate present accord univariate time time e point use time generative purpose actual could potential say follow slightly case noise may seem goal paper evaluate methodology choose distribution test plan associate test observation sample anomalous precisely anomalous length choose anomaly model figure type shift I change parameter change change slope case shift slope sample ij ij procedure correspond anomaly type anomaly shift c explain slide window shift center different position create indicator precisely length series signal window sort position conduct half extract series test notice parametric mean parametric distribution hypothesis indicator significance window window length different level test fix transform window extract level give whether nan hypothesis reject lead binary turn raw indicator produce simple one window obtain way binary classifier indicator process window consecutive window namely binary compute equal window consecutive nevertheless window window experiment derive last indicator indicator value derive indicator length binary indicator addition expert smoothed signal indicator indicator explain several smoothing indicator note used indicator illustrate possibility see summary practice indicator cover anomaly test kolmogorov test average balanced signal proportion full keep classification percentage prediction regardless fitting subset indeed observation constitute prediction observation aggregated decision classification detail confusion insight indicator curse acc acc acc table report forest indicator forest neither curse fitting report bayes performance one confusion perfectly confirm adequate already anomaly variance slope anomaly slope forest satisfactory human operator operate box indicator interpretation review addition bayes significantly low forest one favor forward acceptable performance notice add never remove redundancy issue indicator random forest naive evaluate classification accuracy subset black accuracy bag bootstrap estimate classification indicator give classification accuracy white detail summarize indicator random indicator indicator test tend estimate performance procedure tuning procedure datum confirm moreover reach performance forest set performance naive slice natural classifier observe conditionally estimation practice indicator performance respect accuracy jump indicator leverage efficiency bayes meanwhile detail indicator case difficulty shift decrease classification shift trend indicator maximize indicator maximal indicator many classification naive classifier binary indicators acc acc performance good one indicator show probability get table indicator indicator positive naive variance contrary easily interpretable human htbp ccccc type change variance f f ks paper introduce diagnostic methodology health monitoring expert automatic build expert parametric anomaly plausible hundred cover space aggregation classifier diagnostic binary technique useful indicator allow human operator understand automatic classifier base methodology expert model parametric anomaly methodology
tensor tucker possess unique community tensor hypergraph conditional membership membership identifiable original assume resource community learn approach pure natural expect assumption consist project top rank connectivity resource detect tensor decompose efficiently community pure prove recover applicability dirichlet tight community resource tag resource identify pure construct star tensor use pure pure resource triplet tag tuple tensor require membership carefully moment accurately recover perturbation analyze test inequality connectivity connectivity community impose set strong employ exponential form get regime efficient guarantee community social establish success size scale correlation intuitive make act require constant mix hypergraph graph decay bound body detection popular cluster convex handle membership belong approach community graph world times art star subgraphs star subgraph leave triplet dirichlet membership modify star tensor cp extend beyond distribution star count decomposition tucker learn general distribution beyond membership hypergraph membership learn relationship hyper membership incorporate mixed community densely pure un normalize membership limitation present normalize community membership model node lin towards co cluster co belong community consider community guarantee wang discover mostly modularity provide anchor west communities anchor outer anchor west ml machine outer sep dash corner white label north ml draw corner green fit user south corners north fill fill name anchor west anchor name ml anchor theory guarantee draw thick corner fit north tag green thick west green thick west joint thick thick white thick color blue east color thick node tag occur tag resource tensor denote resource soon become edge hide community community belong community membership similarly provide hyper user tag resource membership community tag resource belong tend resource mainly comprise tag dependent contextual category resource intuition formalize denote community membership tag resource similarly membership resource tag user select tag draw community membership triplet basis u hyper group membership resource resource tag resource tag resource order happen explain eqn resource resource comprise tag resource resource tag select resource context resource convenient model resource tag resource dependent resource resource hypergraph resource edge community membership contain conditionally community membership context resource dependency beyond membership community connectivity give user resource tag unknown community look resource tag topic draw community fraction pure I community stack operator reverse denote hadamard restriction subspace singular top paper realization adjacency clustering usual distance show pure resource community resource tag resource resource community resource look hyper pure resource tag tag membership node issue project xx project classify different pure mixed consider project u whether pure role pure pure resource community learn membership node nod subgraph partition subgraph set pure resource node process membership vector thresholde weak value set membership connectivity community membership resource node threshold vector partition resource part u x rr x u role pure node factor constant membership resource community homogeneous merely remove hence resource resource separation satisfy intuitively imply enough connectivity connectivity across test need intuitive role act tensor alternatively intuitive sparse membership estimate hypergraph recovery membership community tag reconstruction community user tag appendix row membership node guarantee via assume case dirichlet represent guarantee thresholding norm well obtain error guarantee case rao define kronecker vector rao eq rao products kronecker number rao preserve recall user community resource tag tag simple connectivity star membership u r entry vector count particular relationship detail column correspond mixed node big succeed identify node correctness moment full learn community form column test succeed moment perturbation analysis outline define exact upon divide part commonly refer perturbation subspace perturbation begin perturbation analysis define perturbation appendix dominate perturbation subspace perturbation success threshold q node pass pass correctly detect pure eigen pass pure I eigen error control star construct pass novel probabilistic approach modeling propose detect present realistic constraint impose system scalable tractable draw parametric note dirichlet membership limit membership impose fraction resource single expect practice community assumption learn spectral consider note extend award award nf supported award thank detailed discussion extensive visit aa microsoft new membership model acknowledge bernoulli depend contextual variable r chain establish community thus take definition star prop community pure modification connectivity learn membership community resource community connectivity tag resource community pure resource w ab c nk k eigenvector estimate community initialization eigenvalue compute eigenvalue denote entry let denote column respectively absolute proof guarantee without need call combinatorial entry conditionally applicable term dominate product assumption sufficiently lemma decay moment every randomly partition pass test recall line hence q combine pass almost eqn make orthogonal act guarantee perturbation perturbation substitute condition assumption succeed line due hypergraph set w j moment eigenvalue exact subscript whiten impose requirement q initialization dominant error line whiten perturbation whiten whiten perturbation tensor substituting resource dependency reconstruction community vector dominate adjacency random conditionally bernstein zero entry wise indicate hadamard entry wise vector inequality thus u ty enough concentration homogeneous bernstein column apply bernstein p I follow low rao involve
play work paper parameter algorithmic estimation set estimation approach principle apply common panel estimate panel tend infinity purpose implement application author via related consideration reader point call correspondingly adjacent correspond adjacent node correspondingly four connect e path nod r short path accordingly r obviously disjoint quantify later formulate actual spatio temporal rectangular domain e integer noise may interpret field lattice random impose fourth moment non empty assume assume eq set partition noisy statistical non temporal mention time infinity rectangular choose already extended recall partition change quantify change change formally convenient reflect terminology brevity motivation extension think digital array sensor sequence represent light intensity e affect noise article establish cf aim whole sequence record assume across simply rely average obtain recall consistency simulation show moderate assume change scenario observation common change slice set series interpret panel horizon original sub change situation illustrate whenever fit restrict consideration weight q usual attain make sub slice theoretical normalize slice dependent identically cf notice model assumption rely behaviour slice fulfil consistently optimal cf slice overlap slice vertical slice sub slice r dr resemble rd think sub slice vertical counterpart treat series assume slice slice aggregate context w slice restrict admissible require reason step identify induce straightforward critical point identify boundary tend one set critical overlap rule w slice overlap choose assume single asymptotically critical tend exclude theorem sub slice estimate tend consecutive slice th intersect slice condition fulfil line violate contain slice restrictive reliable overlap rectangular spatial sensitivity demonstrate vertical slice perform horizontal vertical pool critical point connect asymptotically contain try identify many carry separately class theorem horizontal procedure pp analogous vertical overlap slice indicate select tend rectangular common connect horizontal intersection horizontal vertical horizontal furthermore ratio slice ensure occur contain sub slice consistency change fact estimation procedure parameter domain noise normally consider usual rectangular shaped round shape shape relation due repetition overlap clear preferable inspection g accordance former latter notice figure base spatio large improve small hand table
combine step give similar note h os rearrange os q calculation strictly increase convexity expansion decompose combine case h give section lemma useful first short consider spherical q dominate mf n pn mp n pn basic find fix column q equality write event pn come appropriately event let equality apply apply symmetry claim definition b consider eq mn b mab calculation least combine give claim moreover second q least j jk z jk pe z pe cn j jk jk cc therefore inequality least proof lemma exist claim write q come np l calculate write q integral infinity moreover result eq dy dy dy pp ct ct pn e dy dy appropriately since ty dy tr pn determine utilize enough dominate theorem datum estimate former rare allow successful pca feature remove whose classical pca aggregation method cluster possibility yield limit colored closely interest test whether model limit insight discover transition threshold partition region pca successful threshold post low interest microarray subject two facilitate class assume dimensional standardize useful signal paper study dimensional strength fundamental separate signal rare impossible correctly label region enough successful cluster yield successful statistical except possibility statistical computationally occurrence three objective discuss important fundamental iid model closely spike especially cancer class label quantitie scientific motivate view normally latent factor great interest recent focus selection e grow scientific increasingly directly validate cluster relatively easy validate result real spike model validate find model extend spike direction help bridge interest spike cancer motivate future application result simply possible feature impossible opposite figure right denote aggregation special sa stand tune determine denote estimate optimize cluster aggregate important aggregation sa mean target case np pca flexible wang particular gene microarray adapt pca select use wise q post consist q skip selection step need effort wang propose screen however pca moderately sparse yes yes invoke weak contrast vector eq mass drive asymptotic fix mostly modern large really extend permutation take performance eq respective fix achievable aggregation cluster tractable tractable eq fix consider model sa q pca successful selection column wise score region phase counterpart boundary separate possibility boundary computationally boundary see limit tight match remark monotonicity monotone le new column column become see hard monotone le comparison colored address sure moment conjecture tight fact mention normal spike tight aforementione theorem challenge fall boundary interesting never flexible clustering wang investigate challenging fall heart screen either trivial range either screen fail signal strong screening calibration slightly calibration screening introduce let transition pca pca singular otherwise theorem random matrix independent entry deal entry conjecture tune version post column screen screen tune design cluster reason unclear white panel impossible screening cluster possible version replace colored matrix generic vary occurrence occurrence color continue theorem prove construct current view add inference hard setting perspective independent row column correlate section bound color replace pca desirable way attack plug estimate deviation say statistic moment discussion far test support selector two fix interior region limit goal global hypothesis alternative model space fix interior region type type ii procedure test limit see figure statistical strength need signal interestingly phase segment statistical limit recovery address recovery limit moderate argument test bottom right panel relatively raise investigate idea section much broad setting error error consist tumor class well study report difficulty apply modification mx mx normalize j post selection leading mean well different feature pca work choice determine fdr control fdr ideal classical skip work suggest associate sample classical apply suggest pca address ht ccccc method error study signal strength separate tractable bind spike scientific perspective scope different model phase phenomenon color dependency among column propose first reduce dimension g idea penalization approach highly reveal interesting preferable simultaneously see recognize cluster recovery testing signal insight case easier rise two cluster recall statistic signal recovery randomness think vector reason ham low problem prove upper signal recovery sa recovery pca study study generic constant normal independent sample q theorem convenient grouping associate together statement theorems theorem follow statement hold aggregation consider sa op sa aggregation sign sa sa pca computationally concerned op bound op claim elementary statistic direct lemma follow restrict restrict claim continue hold adapt therefore item b define sa sa op op negligible hamming fix respective b suffice realize nonzero replace zero b z n l range solve write b op elementary solve gs combine prove ns goal aggregation test aggregation third test recall idea sort hc p sa design unknown complicated newly procedure measure sum curve lower consider see either hypothesis fix testing problem sa sa testing consider hc similarly way test fixing theorem show aggregation sa sa hc omit claim consider elementary sa op op sa n sa versus equal type testing f suffice short drop calculus denote leave three triangle recognize leave side nothing else affinity x hellinger note h p x basic algebra p time x iid entries elementary compare hypothesis joint pearson fundamental follow remain simplicity hx fa equal hx fa df r two connection ii error desire drop let law calculus e impossible signal proof bind write j fs p f p eq second condition theorem q respectively conditional sx pa sx f pa pa schwarz sx copy independence equal recall entry geometric write variable jensen rearrange hand k algebra p p x p small algebra follow exponent negative hand combine claim signal etc p pg p pp lot effort investigate theorem hold leave method fold dimension reduction algorithm dimension hundred screen pca dimension implement motivates array great interest edge singular investigate theory primary problem include related focus hypothesis without careful cluster paper signal paper boundary section also reason theory especially overlap surprising idea interesting problem set paper closely recent computationally low plant recovery model recover recover closely transition closely primary theory bernstein random theory property prove let respective fix notation simplicity omit q least eigenvalue
path connect valid transition exist canonical transition function provide condition motivate follow binary string hamming distance construct thing well end attain kk claim obvious distinct lemma canonical path specify first influential influential hasting flip transition form influential covariate q transition backward flip mh form add influential explain variation transition forward flip update mh form replace influential covariate influential case involve double flip transition valid give rise unique set canonical consist state canonical path state path ensemble specify path connect property two remain valid define understand construction path compose state canonical path notation path lemma important canonical path ensemble pair path canonical path eq take reversible satisfie hasting substitute yield suffice inside make event first guarantee state intersection high side combine lemma bind length inequality complete prove subsection separate part therefore constant matrix corner imply claim block inversion claim return define event least quantity concentration function claim somewhat worth auxiliary characterize select transition second depending case intermediate ensure specify expression integrate determination q posterior give normalization display posteriori penalty parameter conversely pseudo conduct base example profile log correspond aic bayesian criterion prior bic regime high regime interpretation equation focus cp transition randomness moreover imply consistency example need bayesian mcmc mix follow argument reversible markov chain spectral upper analyze mix example notation choice theorem equation equation since projection sufficiently tail display combine q denominator combine yield claim transition influential covariate schwarz fact c combine display complete happen index transition write appendix c last add influential current combine two display obtain combine satisfie use prove q guarantee precede definition step argument construction incur replace influential covariate condition matrix step follow assumption precede apply precede eq claim subset simple linear algebra ii claim jj square last optimality since two express ratio since covariate cauchy schwarz projection ii obtain follow display obtain bound choice consider lemma consequently large step I begin eq display obtain q section assumption ex em ccccc california berkeley department electrical department statistic approach variable relatively mild imply rapid mixing truncate rapid hasting time control spectral markov path ensemble greedy algorithm area science lead exceed size address ill pose address impose covariate optimization incorporate yield nonconvex design placing posterior one report subset posterior oppose single provide theoretical understanding moderate scenario mean influential grow spike variable consistency resemble well dimensional evidence conjecture snp genome wide association confirm theoretically widely fitting design match posterior computational efficiency lag central object markov characterize initial algorithm must bound mix dimensionality interest grow polynomially exponentially hope sample reasonable large positive mixing sampler regular parametric converge normal result reach stationary genomic exponentially length dna goal paper metropolis dimensional selection analysis hasting broad regression marginal past analysis include order move neighborhood motivate search contribution bayesian posterior chain rapidly mix product counter illustrate necessary rapidly assume challenge characterize chain posterior complex object distribution physics markov mix probability property characterize chain condition tendency generate even distribution particular motivated examine procedure greedy use decide covariate overall bayesian concentration property benefit rapid markov draw rapid efficiency somewhat theorem bayesian along simulation proof technical detail conclude markov chain analyze mix response recover absolute weight induce selection think indexing covariate index shorthand active associate subset identification understand similarly use submatrix index analogous notation define specific past work mcmc hasting walk update hasting walk local involve neighborhood uniform move stay choice metropolis analyze randomly select new j empty scheme understand model either opposite switch value metropolis describe reversible e detail describe condition reversible vertex distance give chain difficulty sample polynomially mix exponentially variable give turn consequence sufficient hierarchical vector three maximum number control dispersion model covariate make remark first namely analytical simplify realistic would however difference choice show regime posterior behave popular substantially view indicator mixture motivation theoretical lead likelihood essentially proof et covariate impose vanish many sequel additional rapid mixing response linear rough term formalize fix quantify minimal requirement influential consist index relatively namely coefficient zero magnitude let indicator influential without generality zero regime large involve regression component satisfy simple true trivially hyperparameter consistency necessity rapid letting onto low eigenvalue mild requirement selection projection choose projection matrix neither unit information bayesian consideration motivate establish utility hold set sparsity b exist literature e posterior true characterize consistency truncate sparsity analyze hierarchical reader influential concentration assumption require influential covariate magnitude consistency procedure rest due exactly allow nonzero long noting cover regime exclusive possibility snr regime snr snr hyperparameter completely low snr conversely snr characterization snr regime intermediate regime mcmc exhibit appendix satisfy metropolis grow exponentially seem counter intuitive sharp rapid mixing distinction sufficient condition scheme mixing ensure converge difference assumption sparse involve rapid either snr low constant upper theorem characterize mean initialization state though impractical upper understood number iteration require period state iterate iterate mcmc algorithm match characterize intermediate component inequality base statement conduct simulation example correct succeed mix choosing noise case design signal explore behavior boundary simulation snr size set model figure typical log signal receive stationarity iteration prediction behavior nonzero intermediate fail stationarity within follow design poor intermediate signal regime design h b design grey chain initialize perturbation nan model signal true order performance hasting walk initialization perturbation nan choice empirical nan near markov gr scale chain stationarity gr determination span stationarity effectively gr scale within grow linearly covariate little mixing compare sp h sp n sp successful trial gr difference high n difference quantity simulate dataset h sp sp proportion gr probability model compute gr six see selection posterior high find markov probability model receive correspond weak signal regime regime show ensemble exhibit fast weak regime however markov design among suggest characterize chain slowly interesting see characterize fundamental efficient selection leave question open future reveal oppose relate fairly restrictive condition lasso whereas succeed high necessary theory model condition speak receive negligible covariate
remarkably efficiency retrieval slightly art rotation formulate analytical trade relationship code hashing build analytical nearest ann retrieval database recent accurate vision achieve ann vector quantization study ann high quantization high space small subspace representative product case rotation dimension calculation binary hashing technique efficient retrieval recognition transform feature highly favorable task memory lot propose approach vector base hyperplane method hash eigenfunction derive locality hashing nonlinear hashing propose hash ordinary deriving lower compare uniformly kernel base recently spherical nonlinear hash hyper euclidean calculation hashing get high bilinear hashing treat k high fold unable special orthogonal high paper new binary hashing inspire isotropic hash natural analytically develop efficient isotropic produces yield isotropic main previously state art reduction accurate remarkably fast main cost calculation covariance whereas iteration practically fast although measure hashing quantization bit criterion study analytical naturally mainly applicable distribute expand gaussian discuss low hashing extremely efficient along hash quantization original property binary hash hashing method translation transformation assume center discussion sum error generally center calculate deviation straightforwardly binary purpose group group transformation minimize isotropic hashing quantization measure angle rotation transformation range treat eq angle axis axis symmetry probability plot quantization quantization minimize mean quantization trade relationship compatibility depend heavily trade relationship quantization quantization determinant variance covariance propose algorithm minimize invariant subspace rotation subspace dimensional investigate code balanced interpret minimization entropy maximization possible trade quantization minimization hashing transformation substantially encode yield transformation projection encode reduction treat encode treat reduction work gray permutation variance large apply behavior variance multiplication isotropic graph sort state isotropic rotation variance pair transform basic transformation variance isotropic isotropic isotropic rotation matrix fill two isotropic correspond rotation variance isotropic isotropic step sort dimension take rotation set process denote permutation sort basic isotropic rotation follow variance sequentially apply make isotropic isotropic apply completely isotropic finally factorize highly sparse dimension sparse substantially decrease factorize precede however quantization retrieval balance entropy keep accordingly balance hereafter one simple fill fill pairwise rotation isotropic fig pca angle rotation tuning range pca degree balance isotropic definite basic rotation necessary rotation accuracy application fill matrix sparse rotation isotropic pair rotation pca rotation discuss rotation attain retrieval little introduce factor pairwise rotation hash quantization discretization gaussian property function low expansion viewpoint regard approximation datum consider high analytical considerably toy sift evaluating propose sift evaluate top recall hash gaussian query sift sift protocol k database create original sift data k pick dataset matrix sort rotation rotation set keep nearly counterpart transformation make variance completely isotropic isotropic retrieval differ counterpart isotropic lift algorithm hashing use opposite isotropic trade quantization quantization assignment center thus kind hash nonlinear parameter bilinear like like gaussian sphere sharp sharp gaussian sharp log low row artificial gaussian behavior create covariance eigenvalue log sharp retrieval little notable lower completely
function manifold maxout corruption autoencoder understand like problem broad biology understand related brain learn brain equally inferior aspect effort bridge beneficial away biological neuron salient biological dynamic purpose would know spike nn implement biology energy dynamical system appear naturally suit state machine experimentally bridge gap understanding spike system connectivity neural upon instantaneous fire many statistical framework deep rule activity neuron neuron dynamic method require statistical addition activity connect distant assumption activation function period learn demonstrate similar deep specifically feedforward architecture connectivity utilize deep directly digit collect train classification learn supervise reference operate continuously delay signal model system also network operate discrete architecture without delay encode artificial perform static activity hide neuron connection specify activation general consist external source neuron connectivity pair neuron include delay connection neuron receive need learn may calculate set quantity connect activity self temporal terminology vary delay quantity history event dirac integral allow neuron allow spike rate spike simultaneous strength spike one alternatively strength value simultaneous spike description quantitative unchanged aside spike combine value spike vice fix use sensible instantaneous training output neuron activity integrate period spike operate discretized implementation artificial code interpret spike give activity spike occurrence output learn produce strength spike allow output important focus trajectory use connection delay history activity network different finitely particular range perturbation neuron network neuron four event within neuron spikes ii neuron iii supervise spike supervision neuron spike dynamic activation could also make time periodic showing operation learn neuron spike indicate rule dot vertical dotted line indicate suitable hardware dynamical spike within focus hide pattern indicate neuron spike ii spike conjunction modify occur prediction spike occur occur combination event fig series iterate occurrence relation suppose clearly require meaning cause strict equality nature value occurrence approximately require q current supervision unique way fractional number time single necessarily fix eq require alternatively noise spike poisson supervision target divergence stop converge single mean therefore adjust eqs total require choice restriction unsupervise deep feedforward layer successively autoencoder activity autoencoder neuron direction connectivity causality layer cause hide feedforward function neuron activity vertical line dirac delta height dirac delta learn activity begin connectivity act like encode memory layer input specify period supervise activity layer include train neuron see neuron equation neuron machine typically characterize neuron choose commonly use fig approximate modify calculate need product across delay connection advantage though future wide could sum continuous convolution act convolutional modifying include convolution simple piecewise piece modify delay delay spike model delta multiply spike weight update rule eqs choice update implement implement supervision hyperparameter choose rule within useful period specific could sigmoid change rule predict activity learn future dataset collect dynamic vision type camera camera event light intensity upon pixel change upon event camera output coordinate intensity mnist view demonstrate learn mnist handwritten record camera light intensity camera primarily edge digit camera scene illumination result record across digit contain event relate movement digit relatively number handwritten digit last testing pixel map neuron intensity training selection digit event duration five parameter delay width ms neuron classify current digit layer initialize range initial corresponding connection divide classification neuron connect time decrease half repeat value decay period heavily prediction hide idea training demonstrate layer active neuron threshold learnable cpu operation learn number event
pixel model raw range conventional computer optical straightforward vision visual optical video datum constitute common visual attention background visual optical depicted object category model background one robustness critical situation background intensity component pixel mixture take student outlier method introduce user threshold component drawback dirichlet author aforementione drawback affect illumination object different address aforementioned work author contour extract foreground initially utilize background exploit extract foreground unimodal usually capture background uncertainty mixture gaussians predefine number make asymmetric generalize predefine limitation environment propose exploit foreground make foreground object sufficient dynamically handle aforementioned specific pixel account unknown component advantage change address correspond functional inference incorporate adaptation heuristic accumulate strategy framework avoid issue know utilize derive analytical memory inefficient selection make time analytically derive describe discuss present background task experimental formulate framework reason section behind introduction yield estimation gaussian linear superposition component eq satisfy propose modeling infinity recommend less cardinality I define term marginal distribution component introduce transform exploit observe dataset distribution transform q avoid allow yield analytical solution prior denote set latent estimate maximize logarithm purpose coefficient gamma uninformative prefer dirichlet subsequently model order express preference introduction uninformative prior hyperparameter goal approximate evidence bayes equal minimize exploit show section inference distribution factorize expectation initialize formula initialize associated initialization initial value dirichlet gamma value change em algorithm guarantee convex respect least component describe fit mechanism permit adapt rule sample two successfully successfully close component mahalanobis close new mahalanobi stand denote x upper respectively proportion increase upon arrival estimate around new sufficiently updating call associated close observe model approach model create coefficient standard parameter see mix equal value uniform new create remain unchanged new mix normalize present adaptation use observation deviation initially column new fit generate distribution mechanism overview capture history initialize section training frame classify foreground utilize background initially create observe use train classify pixel foreground represent model threshold classify classified foreground dynamically define threshold define use university international european project camera raw widely illumination weather totally capture frame use capture view narrow perspective video camera height testing background method interest pixel wise figure visually observe satisfactory apply contain high precision low recall pixel classify foreground foreground misclassified seem suffer outperform particular score per frame present good term recall score among frame examine regard load em converge require time requirement method applicable em optimization value use logarithm substituting get logarithm factorize
grow researcher try improve diverse evolutionary reinforcement mention extend examine action fitness anti opposite actually simultaneous entity capable difficult exhibit significance publish clear actually scheme algorithmic begin opposite context year thing exist calculate bring ii opposite suppose meaningful mapping contrast look receive unknown justification intelligence instead sort reward fitness etc enable assess true train learn could contrast propose meaningful accord q priori know continuously evolve manner present relationship available another via inference system evolve depend application whereas evolve review verify correctness section evolve system introduce find initialization guess solution name population genetic policy reinforcement random away exist location search considerably time case become direction opposite chance function random opposite learn continue evaluation measure optimality compare fitness reward report past know guess begin calculate stage evaluate differential seem one successful among early work paper opposite type opposite fuzzy complete machine intelligence volume also notion opposite neural survey overview et al provide compact base publish xu type ii paper independently possibility ii et idea deal look action agent take opposite directional opposite instance environment discount accumulate opposite train episode accuracy approximate interact stochastic environment action value take extract relationship opposite term world mlp due computational expense suggest substitute mlp approximation type calculate focus centroid base landscape boundary unknown look help value interpolation type initially fuzzy idea evolve extend add rule one approximate capture online calculate center recursively data one decision make exist point certain threshold close old cluster evolve instead initial update become however fuzzy inference system way fuzzy preference perform type fuzzy propose online fuzzy rule expand iterative pose approach rule use model comprehensive report advance evolve fuzzy system find compatibility play simple ii fuzzy emphasis find quasi datum adjust fuzzy consist work input expert however world extract available fuzzy operate fuzzy general x x nx iy fuzzy function w nx membership represent logical type evolve find provide assume become increase mining representative generation priori type opposite select continue reliability increase approximate design look calculate straightforward diagram calculate opposite diagram understand diagram find opposite figure produce ii course might validate describe training boundary datum process opposite look domain average opposite sometimes scheme necessary emphasize opposite opposite line may employ fuzzy output cluster rule perform fuzzy ii unseen input report superiority type htb initialization determine opposite iii train fuzzy save conduct demonstrate usefulness evolve fuzzy examine verify correctness algorithm know inverse relation reliable testing matlab implement fuzzy matlab number iteration fuzzy enable formulate exception test mining comparison error error opposite clearly exception function run htb difficult enough demonstrate happen system block one base generate accurately approximate figure standard decrease reach observe evolve long due nature evolve step begin point extract result htb online rule respectively htb commonly without test two low error superiority ii ii conjunction whether guess guess optimization generally simultaneously look continue good result depend continue type exist literature propose question benefit column type third reject simultaneous consideration benefit error run column distribution
clearly location directional derivative independent cauchy univariate rewrite straight dimension marginalization occur distribute provide equation coordinate variable integral clarity bivariate deviation step additional require sum truncate write normal cdf acknowledgement research part nsf provide methodology directional formal gradient previous focus directional follow extend additionally accommodate continuous covariate whose directional also surface assess explanatory include explicit hence theory enable occur post proof methodology set illustration point surface adopt log cox relate point pattern species forest cauchy process directional gaussian cox mat ern location conceptual view realization surface finite covariate explain substantial portion response spatial structure unobserve krige enable interpolation regression coefficient unit provide sensitivity expect may vary locally study local study spatial enable across sensitivity vary covariate weak strong across specie vary spatially within specie approach aim derivative address surface give defines directional derivative enable interpolation region post derivative well researcher consider mean covariate accommodate desire assume interest spatially smooth stochastic behavior response associate process derivative explore contribution spatial accommodate model covariate gradient surface joint distribution significant relationship sensible gradient behavior surface marginally surface response surface covariate surface strength accomplish give derivative another introduce directional sensitivity angular discrepancy inferential tool directional derivative process well mat ern function form mat covariance ij g sd e another directional uncorrelated follow dimensional work ern parameterize n overall conditional ij ij equivalently form implement informative prior prior observe distinguish difficult straight forward result direction correspond environmental response value covariate surface effect exclude unconditional surface directional sensitivity multiplicative overall serve directional mention dx sx directional derivative ratio directional directional describe adopt spatially deviation directional directional ratio call cauchy well simplify clarity involve kn gm cauchy scale normal response surface change normal chi degree freedom process isotropic freedom direction function draw fine surface treat parameterization describe package assume ig ex summary ccc mean truth region contour line produce location indicate location gradient estimate since figure gradient predictive direction gradient gradient denote angle angle section provide region suggest direction small towards surface opposite illustrative forest site exhibit range road site focus regard respective location year tree stem treat give location record specie observation region roughly pattern species cox place point likelihood divide realization gaussian elliptical sampling describe bivariate sx analyse difference response unobserved model posterior sample provide fit table fit specie estimate suggest facilitate identifiability fit credible scale intensity intensity value ccc parameter ht ccc intensity gradient us sd sd independent process center vector consider directional ratio specie result surface majority close suggest recall fairly region region directional derivative ratio coefficient namely occur rapidly large intensity absence tree rapid intensity ht surface discrepancy region close suggest surface opposite surface surface pattern support strong relationship pattern quite intensity nearly opposite everywhere relationship highlight weak region methodology perform sensitivity analysis bivariate associate directional derivative multivariate parent parameter
control account relationship create successful treat twitter behavioral agent receive base valuable service user research oppose learn collect collect social action allow cause seem popular examine contribution formalize contextual encourage behavior execute month agent control live evidence reward signal past aggregate result accurate prediction contribution development define control twitter system agent round reward goal reward formalize multi bandit need answer learn entirely explain express character perform like increase action traditional armed bandit assume content try choose learn valuable unlikely model armed observe tweet status update describe feature describe choose receive reward learn decision reward reward predict reward select correspond enforce enough action user lead time hour round first agent request twitter specialized tweet receive twitter hour request information change agent calculate reward begin immediately exploration try uncertain reward multi armed want agent action learn select action action reward execute machine learn twitter random account control account hour collection tweet twitter tweet status extract one hour signal agent agent twitter experiment tweet tweet coefficient align weight number tweet string flow status language filter make divide three uniform signal extract tweet feature encourage status update update mse utilize introduce algorithm tweet agent hour observe tweet change receive make tweet generate hypothesis signal step hour generate new agent ol instance reward weight mse select one month may generate status account statistically average fold demonstrate strategy raise question experimental gain deviation notice offline ordinary result analyze ridge regression elastic support regression method oppose generalize agent prediction mse test remainder plot day collect indicate comparison user user user examine data reward non probably satisfactory action seem sort small divided instance select yield testing take agent mse train datum result predict train sample sort test datum hypothesis finding prediction oppose examine learn specifically value median
pre unsupervised learn steady assess base replica consequently unsupervised field successively implement learn label pre follow formulate simplified analyze process section nontrivial generalization amount datum summarize label dimensional vector function binary vector margin resemble label joint probability datum number follow p w py consisting label vector meaningful become flat dimensional multi dnn simplify aspect actual perceptron assess nontrivial aspect deep reader sketch conduct label maximize structure utilize hide unit redundancy aspect simplify coarse picture term step precisely classify thing architecture training gradient classify newly often employ e adequate computation reach weight employ weight weight randomness dataset characteristic namely derivative kind quantity compute overlap estimate weight follow spin apply replica energy partition calculate replica replica evaluation introduce simplify calculation energy give solve expectation introduce representation multivariate b ab replica solution impose symmetry replica write auxiliary random unit rs saddle point saddle equation rs variance rs bayes saddle disagreement label datum output newly train perceptron generalization quantity combination unsupervise fig logarithm plot show value tb fine error nontrivial nontrivial solution remarkable nontrivial state several decrease point move difficulty vast attain label however become need nearby hand cause value error high pre architecture achieve origin contrast cause drastically incorporate error exponent characterize decrease generalized integral exponent perceptron supervise saddle solution error label achieve difficulty reach multi neural dnn unsupervise noted study verify numerical message reference modern iterative th component weight numerical gradually label tune independent start water phenomenon label fine tuning tune remarkable performance difficulty require good tuning deep reveal technique degradation cause tb combination unsupervise supervised behaviour classifier hybrid unsupervise remarkable increase sense pre deep essential label datum ordinary iteration optimal deep pre crucial reduce initial condition state specialized nontrivial behaviour involve low role label confirm behaviour water phenomena existence reach semi efficiently technique work simplify essence aspect hope author thank discussion
cubic spline convergence differentiable show grey practical side mean fact thin top constant seem quadrature rule good model probabilistic collecting viewpoint principled procedure procedure might size anneal bayesian quadrature benefit parameter quadrature much smooth associate show thin grey leave smoothness grey quickly converge bar qualitatively comparable convergence arise wrong spline statistical calibrate confident inefficient column identify regressor start calibrate quadrature computation show quadrature calibrate question computational quadrature precisely capture inherent require rule arise strongly restrictive situation precisely check assumption perform computation principle formal form code describe test differentiable future design quadrature know quadrature actually incorporate salient information lead tailor numerical well framework tackle quadrature solve determination node optimally clear upon grid quadrature evaluation stand monte carlo additional uncertainty generator uncertainty worth random generator computational overhead pseudo principal eventually disadvantage evaluation technique improve quadrature common discard advantage course robustness quadrature integration star aim recover produce provide datum analytically intractable modal quadrature smooth secondly beyond achievable classic quadrature strictly uncertainty final contribution permit informative select evaluation carlo metropolis quadrature use quadratic node take probabilistic drastically fast estimate plot computational cover projection arbitrary exact cg convergent improving involve one operation produce cg performance projection numerical measure py x condition policy describe collect work exactly cg element stack dirac width perform strict move x ax choice unit positive assign every computation sense course cg derivation also provide something ph particular observe projection quadrature width scale mean numerical method cg member uncertainty estimate design answer equivalence covariance match cg degree freedom multiplication typical runtime construct scalar number cg computational overhead quadrature valuable formulation calibrate helpful relate however require challenging derivation viewpoint extension regressor universal enable distinct expert progress statistic slightly describe blind deconvolution imaging provide task sequence vary noisy truth spatially vary blind deconvolution estimating problem large I run cg iteration less start mean precede rank low rise computational loop cost increase computational equally method problem art conceptually clear inference method rule increase repeatedly xt xt collect crucially xt xt h sp point computation amount efficiently project tractable perhaps interpretation numerical suggest share strong method probabilistic analysis kind application marginalization prior method capture wide ode filter regression notion recently show connection conceptual make family gauss process match thank property implement give new result identify formulation classic numerical highlight general approximate number yx I require measure tractable structure eq e separate encodes class assign explain tractable number basic role describe classic rule collect precede could independent regular grid markov type previous ode rule associate theoretic example case grid quadrature integration opt ode gradient bfgs post est est aforementioned result classic fundamental numerical problem cast posteriori description often quadrature greedy linear scientific tool viewpoint suggest practical formal stable implementation development various scientific hope reader issue motivation scale computation science uncertainty opinion foundation two complementary goal implicit prior hope either algorithm convergence strong example conversely conservative might cost regard statistic ode severe checking bias increase effort secondly reach uncertainty would challenge interpretation may mass function distribution prominent optimization machine subset problem figure motivate classic shape research gaussian role numerical ideally suited parameter calibrate fix reproduce analytic fit elsewhere algorithm computation perhaps flexibility fundamental insight solve new recent distant motivating field adapt interact conceptual environment action choose encode sequence step marginalization fitting prediction area base solution numerical automate give receive perform computation model accumulation unnecessary convergence input rough machine inference rectangle north north edge edge north quadrature north estimation south prediction north sketch collect numerical inherent uncertainty across target effort allow input output turn along message management chain computer dominate error truncation reach sufficient available algorithm uncertainty outcome explicitly classic abstract analysis estimation problem give rise method task cast establish structured posterior calibrate uncertainty interpretation yet rigorously establish lead remain formulation uncertainty thus active effort hierarchical modular computation rgb draw size width sep black rectangle text black thick rectangle draw minimum sep black corollary institute united university united linear algebra equation return uncertainty uncertainty arise hardware much science decision lead management seminal numerical suggest provide benefit probabilistic scientific open framework calculation e potentially diagnosis source theory scientific evidence validate probability language uncertainty quantity mathematical statement make sense assign remove noise deterministic mathematical rest notion randomness quantify arise solely web page deterministic computation os show factor integer statement factorization integer concept uncertainty quantity early introduce deterministic interpretation gauss formulation add generative might assume useful generative add interpretable capacity become area dynamical krige square loss maximizing add gauss elimination conjugate direction regression language uncertainty computation method call arise solely lack intractable quadrature access finitely answer integration optimization proceed iteratively provide run answer place note approach recent growth automate article connect application show numerical probabilistic inference procedure measure arise interpretation improve performance conceptual clarity draw point computational may discuss overlap statistic pose deterministic problem identify degree computation uncertainty bayesian identify degree pose uncertainty differ stochastic quantify uncertainty computation investigation projection randomize framework already analyse create generally structural probabilistic classic naturally analytical runtime frequently similarly magnitude locally mostly criteria termination internal algorithm thus embed usually algorithm diagnostic tool add hoc argue formal framework uniquely suited construction perspective extension chain directly computable readily available computation even exist numerical inference quantity result show make establish various posteriori class observation picture one classic prior member amount
hellinger obtain response measure reliability ratio ik space entropy let denote probability uncertainty large I k h kk consistency denote discrete proportion answer theoretical answer limit maximal e item corresponding item realize response approximate answer represent occur answer answer early entropy analogue alternative setup reliable replace show reliable number alpha less htbp ex theoretic internal consistency demonstrate capture purpose refine large simulation study measure strong plan power alpha thank ever definition mathematical institute technology statistic university propose alternative popular alpha reliability context strictly function entropy response reliability index track alpha advantage great I represent response level neutral agree neutral agree agree crucial alpha internal let tuple represent initially late thousand practitioner like alpha sum item variance item coefficient item depend variance group sum positively increase correlated px version alpha consistency I refer response item tuple alpha fortunately contribution whereby heart categorical define appropriate possibility type many author develop scalable data concept entropy variation measure kullback leibler hellinger name theoretic concept create several measure consider nj jk jk contain relative question probabilistic essentially response specifically imagine counterpart correspond give value agreement student item
frequently occur segment circle radius hence figure frequency locate right clearly number lie real tailor numerical technique reader result occur search real life derivative point flat gradient differentiable equation measure differentiable alternative parameter smoothness reason equation divide distinguish previously distant margin call devoted differentiable define l formalize pair distance great identical formalize equation sure achieve case frequency smooth version pair depict right plot diversity construct task achieve maximum objective maximize frequency zero range overall learning preliminary converge b plot match upper plot match show plot per smooth score function frequency diversity decomposable derivative derivative derivative derivative step detail start length critical hyper step h input rate smoothness k j c l return start direction dynamically square partial order per segment distance line positive line explain accumulate consequence time illustrative overlap however form conduct maximize learn updating maximized diversity minimize left execution worth note diversity crucial preserve diversity experiment figure experiment diversity title maximize diversity cause similar demonstrate plot concave gaussian demonstrate non concavity use eeg generate value plot axis another frequency maxima dataset term frequent sub sequence could also force sufficient demonstrate superior searching naturally translate superiority approximate force qualitative protocol comparison across learn compute length percentile illustrate percentile segment distance way force search distance way pick value drive neutral manner ensure convergence rate segment slide window threshold percentile frequency keep force execute research code htbp lm vs width width width r pt c width pt width lm lm lm lm lm lm lm lm eeg eeg c threshold percentile high display optimal learn denote indicate lm experiment lm frequency plot assess show dominate lm difference lm deviation zero significant difference frequency well alternative claim learn small minute dataset hour intel processor ghz extract file discuss repeat extract audio file channel coefficient illustration measurement original reading force blue extract length method percentile segment distance value series segment plot find frequency totally improvement investigation reveal k translate concrete segment segment ground indicate specify find word match correspond word important accuracy channel sound series contrast art technique principled optimization maximize objective avoid optima combine ascent optimal form center ball segment experimental qualitative search european acknowledge university california university suggestion improve lar schmidt frequent practitioner interpret phenomenon sequence frequently sub find pattern propose demonstrate frequency contrast search able discover pattern occur sequence life dataset find threshold arguably type domain medical financial video monitoring sensor intensity dataset know practitioner inspection infeasible reason expert understand phenomenon diverse source attract considerable research brief repeat current art discovery segment sub sequence candidate entirely treat variable naturally task formalize principled optimization devise derivative high match theoretically superior case search domain discover pattern real match search exactly threshold discover ambiguity initially define occurring stream paper close segment paper frequently close close series sub variation task force segment effort devoted force length stream rely find line geometry repeat sequential beneficial useful behavioral concept pattern term discover probabilistic distance discover furthermore unsupervised searching involve include learn order exploit find search tree try every segment sequence segment symbolic agglomerative scalable approximately resolution scan utilize symbolic version pair wise discovery large mining versus symbolic discovering another alternative graph implement employ detect multidimensional since previously pattern little length reality discover optimal instance addition inspire attract length classification computing principled optimization lead improve quality real measurement slide pattern point generalize number normalize segment count matching iterate slide check interest segment formalism overall sum concept segment match optimality frequency match optimal achieve important constraint equation
stage unify stage propagate clustering infer stage large tree hierarchical cluster insight exploratory motivate unified series infer index region e census region strategy discover dynamic share stream allow model individual framework share price throughout census model house sale census census may house time sale noisy census accounting house index capture census since focus transaction compute trend datum aggregate focus model census market trend simplify house function compose represent house sale price global market refer autoregressive ar census choice observation census eqs illustrate census sale price intrinsic census house discover group idea cluster stream generate school census increase estimation pooling group stream seek mechanism stream relate series house price price neighborhood price intrinsic cluster dynamic census census treat independently intrinsic price within k census block block jointly census challenge task infer discover order census block add matrix factor membership infer membership particular latent load tp ik ia element matrix diagonal define clustering stream cluster infer dp drive allow related domain specification specification first utilize dynamical cluster intrinsic price dynamic cluster dp distribution infinite probability draw base measure weight break break dp produce discrete multiple identical cluster unique integrating stick break membership join cluster chinese restaurant crp census detailed stream dynamic cluster latent specify nonparametric latent census indicator weight cluster cluster indicator serve dirichlet base autoregressive loading variance place prior parameter hyperparameter supplement emission value supplement house sale price intrinsic census transaction month accounting effect census ar marginally dp flexible prior census latent factor induce price specification bayesian house summarize draw stream census cluster loading I box realization abuse notation strategy let k th stick specifically model house sale transaction weight intrinsic conditional census factor crp exchangeability stream marginalization upon stream covariance eq include ar observational variance upon compute condition integrate kalman include detailed supplement census belief crp prior becomes specify kk backward intrinsic common latent process ar conjugacy exist follow derivation loading loading membership derivation supplement conjugacy normal cluster assignment ar coefficient provide supplement conjugacy census hyperparameter provide supplement sample variance detail find supplement hyperparameter sample provide supplement consider sampler reduce explore marginalization induce crp intensive evaluation census kalman aggregate census serial adopt trick collapse dp similar conventional emission order mcmc processor auxiliary auxiliary assign processor prove remain representation processor indicator price auxiliary framework machine mcmc describe supplement sampler evaluation derive simplify exploit specific stream house sale adjust sale price simplified kalman census change runtime census kalman filter utilize identity simplify sufficient take kalman likelihood evaluation supplement validate simulated section stream sale census census respectively generate price eqs intend observe sale price sale house city house sale price sale cluster census run simulated hamming assignment set label demonstrate cluster converge census blue smooth per census demonstrate posterior latent track house level one metric assess repeat sale include well house price order fairly compare regression effect estimate price sale localize index level available hierarchy examine city computable serve resolution house house price house case city include city city index use case city code census census use index predict house price use denote represent mcmc analytic use house table predictive metric root mean error house sale importantly house sale prediction largely regardless notable improvement compare index test improvement break deviation trend global trend measure latent trend improvement decrease percentile reduction tail well capture neighborhood trend cluster enable improve fine scale important capture deviation trend index census improvement l rmse mean rmse median th city code sale lower improvement th percentile tail might expect city go fine city code suffer result index ill construct lrr improvement city code top sale l point sale examine impact overall comparing treat census one intrinsic census embed em summarize table coincide repeat stage improvement index smooth kalman smoother rmse p central truth home proxy mention section index form house fine scale region bayesian treat bayesian census analysis code dp index construct average census dp component code nonparametric significantly high volatility supplement start dp especially align dynamic census global census towards informed index information shrinkage end extremely repeat sale study middle difference case show tailed method particular look cumulative figure tail case hierarchical however code error code baseline code importance approach region c c index fine transaction significant challenge dynamical utilize approach flexible share region multiple shrinkage individual trend region dynamical avoid repeat sale ability track change local market observation sale house classic sale estimate code city may specificity sale describe imagine longitudinal trend house long process autoregressive side information road school cluster induce heterogeneous acknowledgement wang discussion fellowship work award center grant fa series condition consider observation use sufficient observe house sale simplicity kalman multiple derivation calculate obtain census filter filter belong observation observation distribution multivariate distribution follow multivariate distribution observation sufficient statistic kalman effect remove specific removing variable observational observation operation transaction use recursion sale response include provide derivation kalman filter work forward time implement sampler draw backward jointly autoregressive vector multiply likelihood conjugacy summary write rearrange conjugacy ar parallel condition assignment assignment datum give processor machine processor processor within use hasting assign accept reject current processor cluster machine derivation show material stability multiply choose flat tail hyper prior hyper flat tail distribution variance performance examine figure parallel figure city display intrinsic price plot paper raw space trend clarity additionally estimate p home city level kk estimate price trend without adjust trend comparison figure sale time together market subsequent form dp dynamic variance time value evolve construct coarse region mask price market neighborhood census example census house sale observation aim address challenge leverage multiple census discover bayesian nonparametric build enable infer cluster correlated census scalability computation yield level census code basis dataset market united states service statistic economic analysis change important policy individual nature composition period report price necessarily reflect consequently meaningful price sale sale house house sale serve surrogate house house level sale capture intra sale period largely cause composition body extend original repeat sale modification repeat sale home index core medium sale single sale transaction discard sale area repeat sale transaction house detect repeat sale single property precise propose hybrid combine repeat sale sale autoregressive repeat sale sale without need lead code level sale transaction perform relatively large despite house sale aggregate fine neighborhood census region localize sale census sale per month average transaction make stable repeat sale stability limitation local coarse scale average sale house aggregate house home house appeal weighted repeat sale period addition census nature try house house recent homogeneity particular home history transaction may need problem need adjust house house create index fine method indice valuable house introduce census level index although idea finer inform individual house census include detailed sale price house sale census latent sale census similar dynamic unlike model since spatially neighboring census census adjacent dynamic nearby census adjacent house
overcomplete transform present step sparse operator image processing patch image base enforce coherence overlap patch high field bi level heart unconstrained operator stage row simultaneously update gradient sphere rank penalty except integer serve bind solve operator update stage inspire work decomposition sparse scheme among overview drive proportional multidimensional datum I non drive argument inequality argument rademacher offer discuss co co operator learn find operator avoid additional enforce constraint theoretical result several field self contain f distribution minimizer rely complexity introduce define differ absolute supremum definition coincide closed propose empirical rademacher vanish single definition generally encourage predefine possible bound another bound furthermore random base rademacher surprising us rademacher gaussian rademacher bound noting jensen sample briefly constraint achieve goal require matrix structure refer employ operator learn want operator separable operator subset equation pose form concrete require ability realization task eq ready result draw accord sparsity lipschitz finally hold consider variable bind value change vary within empirical state argument detailed bound last ingredient main center q preliminary take care sample within let g previously separable result consider separately r norm apply lemma cf inequality learning expression right separable previous version signal manifold similar q hold separable remain fact product q row due sgd scale geometric discussion connection excess result derive prediction set optimization provide excess empirical average quantify strategy closely discuss complexity corollary via minimizer f subsequent seminal sgd attract solve iteration computation computation involve accordingly assume independent notation denote batch represent index follow geometric optimization classic introduction optimization manifold manifold euclidean euclidean riemannian projection tangent keep notation simple denote riemannian manifold point search follow straight geodesic topic provide update read sgd base iteration terminate f ki ki ki ki ki ki cost batch total iteration ki stopping terminate fall threshold optimization selection typically continuity lipschitz advance author line minimize cost predefine heuristic disadvantage require propose backtracking algorithm separable filter sparsity compute respect batch come average update problem utilize iterate average fulfilled initial step length successively condition cost calculate function ki respective batch via slide window eq denote fulfil include least step goes stop trial proceed filter ki max ki ki b purpose serve norm property enforce operator rank penalty hence training sample weight impact incoherence handle allow comparison scenario numerical demonstrate filter train learn extract filter separable version enforce thus separable weighting constraint fix initialization visualize value iteration dot framework enforce separable structure separability impose separability cost second terminate update filter offer learn structure separable kernel application analysis operator employ inverse imaging task conduct investigate truth measure recovery permutation ambiguity filter absolute possible denote row column deviation I build confusion account permutation ambiguity method confusion accumulate constraint visit end accumulate sum serve measure low pick twice prevent retrieve match filter recover procedure ground choose separable previous predefine co sparsity truth filter response gaussian standard signal original signal approach framework without constraint separable conduct varied ten I ten generate box correspond separable accordingly right horizontal box trial mid dotted indicate achieve truth indicate see separable fast small optimization framework utilize full learn separable filter conjugate generate ten predefine synthetic stop fulfil summarize result sgd converge execution time sgd reach dropout enforce separability operator trial upper filter avg error avg iteration time sgd sgd separable cg operator structure signal processing denoise rather exist denoise separable filter slightly compare scheme filter impose analysis goal analysis operator extract image include goal couple white operator regularizer formulate utilize denote image represent overlap weighting factor operator weight noise reduce great extend competitive please optimize task couple within ball assume utilize prove co bound depend structure impose furthermore allow incorporate separability aspect design average confirm add present separable operator sense compare respect sample benefit characteristic sgd sparse addition bind common draw accord deduce convexity jensen supremum step complexity rademacher achieve like support foundation european project black thm lemma corollary definition electrical tu mail de co co sparse signal class interest response inverse adapt reliable purpose operator separable operator sort response advantage contribution evidence provide operator
rate compare art introduction paper aim develop status svm learn status statistical svms current status assumption status exceed format common include field mention example infection tumor determine exposure tumor order absence tumor difficult failure importance topic advanced analyze status expectation investigate measure version function censor current order decision current version analyze status datum gamma non cox hazard accelerate survival cox hazard proportional exponent combination failure time assume include cox status cox include status suggest algorithm parameter parametric demand assumption reason estimation decade censor neural splitting mostly censor suggest survival svms survival suggest censor insensitive justification propose censor svm censor optimal suggest survival far svm status theoretic study svm censor simulation tool censor development status development censor prove rate current status discuss quadratic svm correspond dependent theoretical sample contain conclude remark code notation throughout briefly risk I triplet z z take failure censor status indicator contain example testing example exposure presence tumor tumor develop censor definition risk space subset risk df quadratic reproduce kernel rkhs characterize rkh decision current identically function failure censor respectively risk incorporate identity r pf respect function convex loss function loss positivity constant I decision current status censor replace c f denominator status knowledge censoring estimate note censor estimation consistency svm construct censor true censor use additional condition bind svm bayes finite sample svm family learning consistent case censor density novel svm censor learn know quadratic nd censor l pf b sample assume censor r bb h proof definition mean consistency survival identifiability limit consistency prove pf bayes decision theorem great choose converge clearly consistency measure sample censor censor estimated sample utilize bound svm rkhs kernel old function mx mx kernel du mm concentration like minimize include setting difference cox reference censor censor censoring estimation matlab library fit cox status r monotone estimate choose choose knot follow suggest knot knot evenly matlab server r toolbox kx exp choose cross find risk risk iteration time censor generate obtain result cox ph failure ph cox cox though svm ph comparable cox especially sample size however cox produce risk superiority reduce grow coincide failure censor variable generate time variable figure superior datum space svm rbf size multi uniformly failure parameter z risk compare size risk significantly superior converge quickly size grow whereas cox produce svm produce smooth conditional covariate generate truncate present rbf case sample svm compare base svm kernel comparable failure status certain appropriate true comparable approach assume form additionally svm dimension conclude remark svm property believe demonstrate current open remain study estimate expectation quantity status censor failure censor covariate consequence assumption third extend censor censor believe censor great interest please file file nd unique svm decision q h f df eq q error pf thus r pf pf l df p r l df pf p df l df df eq lipschitz thus obtain compact exist first n hz I l hz n nk hz nk hz c hz n k remark hz hz b combine obtain pf bn pr sup pf bn pr sup r ph bn bn l pr last l ph cardinality conclude zero nh nh cauchy schwarz e nh show bandwidth proof theorem thus part hold r n
tree method explain section pair query rank relevance partition datum use compare validation score set apply combination ndcg position select ndcg task optimize gain ndcg check ndcg significant gain position put improvement perspective ndcg fraction tree slice available repository histogram create ct infer dimensional value involve cross fold train set every good rf restrict dropout rate furthermore rf require low loss leave lowest comprise leave fraction per per challenge dataset image face select ensemble prediction get main difference rate significant difference skew label random exhibit accuracy task forest accuracy network additive find accurate task notably web motivated add tree propose ensemble create contribute evenly towards accuracy study several direction adaboost simplicity direction algorithm drop contribution tree target drop exist tree conduct author diverse task practice suffer wherein tree iteration tend impact negligible contribution remain performance unseen certain issue explore address employ recently context network novel regression use dataset outperform margin also issue considerable algorithm shown achieve task well make help sensitivity ensemble adaboost iteratively achieve learn add predictor increase model instance boost ensemble tree negligible towards prediction instance algorithm increase make significant make initially call subsequent prediction regression contribution nevertheless propose neural connection therefore rely example successfully case use feature training phase approach employ wherein different novel employ tree employ call dataset show outperform random margin yahoo learn encourage reason improve address balance boost issue impact instance impact employ example data description similar tree ensemble contribution significant contribution contribution ensemble inherent leave tree tree extreme case algorithm row ensemble color gradient leave stand negative start presentation gradient derivative loss formally loss generating every typically every current denote prediction current loss predict add minimize loss make variety early square logistic define ranking task order evaluation ndcg result summation detail discuss gradient style employ employ address leaf multiplying shrinkage observe describe algorithm place gradient ensemble iteration random create create place new ensemble step close predictor drop try introduce drop tree drop ensemble roughly time magnitude tree drop order drop drop added scaling factor
solution set discrete repetition boundary boundary simple real surface nevertheless datum interest lie low manifold embed ask approximate normalize one begin appropriate closeness eq relate close approximately mass approximately problem normalize actually far ignore essential concerned making normalize one scale tradeoff satisfy self achieve pair iv ax like asymptotically may explicitly construct scale logarithm class guarantee choose likelihood necessarily desire low thus self normalize suitable assumption construction achieve characterization conjunction class good prediction whenever dependent result obtain represent taylor mle likelihood vanish remainder time eigenvalue favorable indicate easy whether nearby normalize analog remain progress develop class natural equivalence say respect denote associated finite arise multi identifiability parameterization input precede summarize predictive low entropy normalization relatively likelihood mixture distribution likelihood gap generally associate experimental initial weight label smoothly high introduce normalize addition tradeoff gap kl self relaxed gap quantity gap gap self procedure basis understanding characterize construct self normalize self normalization address question self normalization community provide open else approximately correspond perfectly parametric relaxed accommodate construction involve relate normalization rise suffer exist low existence variance fall quickly distribution self insight translate inherent theoretical answer question apply provide toward complete understanding sketch lem computer division california berkeley berkeley major computation attract learn analyze introduce unnormalized score extremely theoretical work largely distribution normalize prediction difficulty fit validation prediction general include generalized offer flexible tractable modeling expensive machine translation self normalization cluster unity surrogate particular zero ignore paper aim understand normalization appear applicability spread include prediction expect input finite order million input rich nontrivial seem challenge enough good seem much input choose gap practical experience open normalization seek answer question generalize much distribution believe characterization self characterization self unconstraine provably represent self difficulty conclusion survey feed forward softmax turn log proportional sufficiently vocabulary prohibitive probability must class arrange output product along lead limit learn appear practice output provide slightly measure define formalize normalize normalize example either hyperplane solution upper variance nonconvex eq easy normalize distribution normalize motivated robust improper generalize approximately conditional statement either example make inherently involve input carefully informally self large occur instead concerned
hard analyze use perceptron update application adjacent improve combine perceptron lipschitz perceptron perceptron algorithm dimension sim low learn reversible hard also moderately sized practical empirical index sim consider propose extension use minimize solve follow problem everywhere else perform sort shall invoke routine point dependent g motivated pose underlie goal unseen like provide unseen sim learn fit perform learn loss eq euclidean fit monotonic need learn efficient ignore overcome drawback eq alternate ideally proximal size require derivative perform proximal perceptron spirit perceptron unity q keep keep track estimate hold keep tp square estimate transfer take gradient propose sim try learn let q transfer standard squared logit penalty take sim via sim optimize integral solve minimize update step problem calibrate unlike proximal bring challenge design minimize program efficiently satisfy h qp follow derive calibrate interpolation non alternate procedure initialization achieve initialization demonstrate empirical book step approximate solve line step evaluate calibrate loss loss via fix competitive sometimes glm size superior risk hypothesis excess list technical main excess excess logarithmic notational assume satisfied lipschitz inequality convex see replace expect deviation standard deviation upper bounding equation property maxima collection q excess hold run run initialize first since hold claim hoeffding inequality small get dominate output theoretical guarantee get sign uncorrelated bind prediction result analysis standard multi variate predictor scale high setting dependence need batch sample run competitive number refer test real squared test dimensional sim via mac auto dataset additional result supplementary since dataset dataset consider except well encourage poor sim learning introduce sparse parameter index dimensional excess employ novel superior compare hinge plan sparsity general definition though use concern lemma x following immediately let normal eq similarly obtain equality zero result paper x g universal constant least l large deviation similarly concentration inequality q put follow notational shall follow set know max value get try empirical eq mark monotonicity lipschitz square marked positive lipschitz lemma reasoning upper know g run assumption restrictive practice result large class g shall case rademacher establish w hard bind rademacher consider run run hold validation iterate validation
impact merge dissimilarity partition sequence grid dissimilarity merge merging minimize simplify minimal degradation merge good partition without distinction start agglomerative build keep grid dimension partition partition total number partition negligible optimization choose reach agglomerative hierarchy dimension increase thousand cluster grid point cluster modification intuitively evaluate impact representative typical cluster close cluster visualize heat visualize suggest namely additional valuable frequency dimension mutual mutual partition partition grid cell observe excess cell event conversely cell finally either mi nan quantity expect partition jointly explain w positive highlight characterize set former say excess locate cell highlight interaction visualization cell cluster bre exploit add value grid available validate real experiment follow successful clustering find event underlie discover kind result exploitation tool let event define htbp l e e l pattern pattern e average triplet event value value choose various point event pattern htbp vary adjust rand ari evaluate two discover significant pattern per point per average pattern necessary discover increase hold per increase discovered discover increase datum set increase set per lead value lead similar underlie pattern discover cluster detect w per pattern pattern insight valuable pattern bring cluster pattern relate excess conditionally cluster characterize light blue stand interaction noisy event cell whereas rate dimension relate whole hierarchy event organize view interpretable human figure look correspond science research field communication typical branch label communication recurrent publish event significantly notice also research repository many sub field computer cluster event author visualization see red relate diagonal whole period cell indicate almost singular cluster scale group observation make consider cluster former mainly relate research focus htbp field hierarchy ai db reason field mining intelligence recognize field intuition terminal cluster dm thousand author characterize typical trajectory rest characterize db whose xu yu db highlight db author activity semantic yu w respective birth sub discover cluster big picture discover typical characteristic series base measure exploit categorical series cluster distance fr distance datum tuning classify branch attribute attribute al theoretic grid cluster bottleneck ib stem theoretic paradigm ib aim group ib mutual wang build upon ib extend ib two categorical agglomerative multivariate bottleneck construct interact interaction network exist segmentation free method interpretable summary go progress multi instance mining event forecast future event cluster co evolve mainly dedicated approach efficient know build recent relate suggest effective exploratory categorical sequence group dimension event group whole form grid probable posteriori free grid dissimilarity representative interpret find finding illustrate real plan forecasting direction plan customer customer define communication interactive sale call service period cl exploratory temporal dimensional grid model temporal define three model partition discretized event partition cross multivariate nonparametric event dimension distribution along suggest agglomerative characterize real grid discover categorical application mining temporal discovery nature temporal event mining mining classification datum time g explore medical find behavior time life literature dedicate mining annotate mining receive attention discuss dedicate sequence without annotation biological popular suggest methodology exploratory technique propose summary show along highlight event segment whole result propose valuable trade facilitate analyst computing involve methodology expert knowledge exist cluster methodology suggest co sequence event suggest categorical grid interval partition trade robustness grid method exploit derive criterion information measure contribution discuss conclude toy compose event interval cluster toy event find group event event find two regime event opposite htbp variable variable cell grid interval point event point number value resp categorical length order categorical set define three categorical one object point force co grid generic goal event time say time interval notice partition set however simplify incorrect partition grid posteriori explore minimize bayesian trade accuracy cluster interval result grid event cell belong event analytic datum hierarchical model hierarchy mean minimal partition subset way stand grid term number cluster choice group specification specification fourth line stand multinomial resp locally line stand locally intuition behind priori whereas close prefer priori grid high likelihood grid posteriori length description length principle criterion plus code grid combinatorial partition bn exhaustive optimize use greedy
frame decode scoring time encoder scale parameter vary standard recurrent weight train norm constraint norm low development negative adaptive constraint low fine development batch use attention mechanism direction activation top recurrent layer maxout unit generator treat purely attention eqs network convolutional sec decode stop token start network fail produce show fig wide c baseline conv conv convolutional net model achieve competitive convolutional relative smoothing baseline repeat demonstrate repetition also frame begin slightly visually baseline lead repetition sequence irrelevant apply remain fig help proper help aware network inspection middle long baseline repetition within window happen support sequence use alignment obtain force baseline fail narrow constrain aware network wide window resemble condition attention wide aware mechanism modification decode propose novel end speech hybrid attention mechanism combine order position decode recognize much train recognize speech may language model architecture normalization smooth extract feature apply speech instance attention neural improve generation conduct library acknowledge national center cifar focus h h rl plain keep conv universit universit e de universit cifar generator mechanism show translation image generation mechanism need speech recognition reach competitive error per recognition roughly qualitative explanation failure add new long prevent frame recently variety synthesis translation object relevant extend applicability construct introduce applicable recognize sequence speech perspective synthesis attention compare differ sequence distinguish attention architecture capable noisy input attention speech recognition system hybrid system acoustic hmm dictionary multi hmm recently work leave acoustic benefit aware recognition task mechanism select signal produce extraction potentially speech weighted condition rather mostly second exist translation baseline seem entirely competitive reach per quickly adapt track absolute content short inherently modify attention explicitly add input attention weight convolutional significantly well consider convolutional feature training contribution fold present novel purely speech architecture attention propose attention modification frame example recurrent recurrent network stochastically generate often process output attention work context feature overlap audio frame recurrent step generate focus state recurrent neural attention call call generator memory lstm recurrent recurrent activation content base hybrid attention mechanism g score separately normalize score similar equally regardless sequence convolutional element capacity always location attention alignment generator previous mechanism synthesis speech location attention would limitation hybrid natural candidate informally like use previous alignment short select one confusion content mechanism describe matrix extend mechanism make extract previous alignment potential issue normalization sequence likely focus attention frame input less translation ms acoustic frames coin softmax aggregate frame straightforward inverse temperature frame according normalize issue narrow propose mechanism subsequence predefine median alignment frame indeed sense bring replace unbounded softmax eq bound logistic sigmoid speech rnn follow early signal perform
theorem section start bound arm draw ip iw ix choose proportional explicit name armed bandit side observation performance algorithm due arbitrary regret satisfy eq become arbitrary bind corollary probability least union set armed bandit round tn tn tn corresponding learner sequence propose ix estimate drop weight arm proportional follow establish satisfie eq action similarly switch times learner prove besides fix share know provide guarantee rely enforce exploration update guarantee improves previously track regret become turn environment armed bandit observe arm correspond presence arc imply learner observe show learner variant use start outperform arm evaluate ensure compare actual range theorem algorithm rate varied multipli regret game loss deviation multiplier several type outperform setting perform regime performance identical rate eventually get behind notably respective performance around shift mean suggest control respective fix first elementary inequality define notation jensen definition result w let right jensen inequality put equation last proof build arbitrary sequence straightforward modification z let apply sequence argument g combine last fix hoeffding bind remain difference thus combine lemma surely france address arm problem focus performance prove modification intuitive come guarantee modification learner time essential strong without undesirable estimation ix remarkably clean multi framework robustness technique armed bandit probability armed bandit problem formalize round interact pick environment loss subsequently incur observe solely goal literature measure environment irrespective take learner base past action multi armed bandit call loss adversary choose sequence learner round reward formulation guarantee grow know goal regret need sense concerned bounding learner prove actual regret hold hard serious change make analysis confidence guarantee learner repeatedly easy performance loss even focus force grow steady result high guarantee tend perform sake consider expect regret confidence variant conduct bandit conclusion online inferior version hold well plain still demonstrate high confidence reason online strategy conservative nature arguably elegant previous particular ix strategy propose side loss strategy range expert tracking bandit bound know another work prove reward reveal lead tight survey aware game avoid respective analysis coherent previous advantage game organize regret exploration strategy precise concern concentration estimate conduct simple benefit implicit exploration technique principle online learning exponentially forecaster perturb box appropriate estimate challenge construct base observation follow traditionally reward easy call draw one enjoy pseudo bind around loss bound keep fluctuation appropriately compute help keep enable less standard discussion confidence reward paper short ix exploration reward true allow prove high variant multi bandit bandit bandit
gain score phrase mt decoder outperform ir despite ir response ir capture message rank candidate triple rank mt directly pattern message mt mt improve relative ir penalty even learn ir ir list support formulate previous paragraph since ir solely capture mt gram match gain gram match may select inspection reveal good provide highly mt ir ir outperform respective ir baseline improvement mt albeit low space limit diversity mt gain model hand gain hence ir well retain contextual gram match difference observe architecture gain ci mt mt mt mt ii ir evaluation score mt ir preference ranking investigate mix exact gram overlap information hope hope extremely mind walk month tweet send armed nuclear response corpus mt response system short token response set token overall plausible contain common illustrate book response message nonetheless long content response likely example mechanism maintain component topic especially response incorporation extensive context help yield representation outside likely remain formulate neural generation medium generation contextual extraction metric context sensitive consistently independent baseline mt ir albeit drive self improve system work direction room improvement network word message context interesting potentially promising greatly thorough automate thank ray helpful discussion pt sensitive ji universit microsoft usa ai ca ga usa system train end twitter neural architecture address integrate contextual classic allow dynamic generative consistent gain sensitive baseline open domain vast medium twitter generation system construct statistical translation status twitter translate address broadly speak context linguistic physical virtual linguistic ability key building active contextual mt contextual phrase table increase skew rarely statistical approach share intrinsic semantic occur twitter context sensitive generation phrase compactly encode syntactic argue embed transition dependency word alignment present sensitive utilize language past response relevant typical orient modular easily end without require annotation drive train massive application come introduce automate generalize et paradigm shift work traditional apart management generation task response generation track many typical code particular attribute model completely drive code continuous phrase retrieval ir recommendation translation mt lm successfully embed refine rare phrase translation traditionally affect crucial extend language representation function natural language generate work construct topic generate stop word commonly yet overview recurrent architecture extend sentence sentence parameterize w w initialize dimensional vocabulary token token embedding keep process project vocabulary pass proceed recurrence logistic sigmoid recurrence apply softmax activation recurrence back propagation gradient accumulate distinguish linguistic entity message sequence context generation generation condition straightforwardly give encode useful subsequently response comprise multiple modelling range dependency difficult open present next context encode vector single bag word representation length bias recurrent decoder encoder minimize compact representation leave ii decoder encoder encode bag word representation preserve feed propagation row vocabulary distinct decoder follow eq context force useful help information response address issue mapping bag feed represent prior common strategy representation encoder concatenation token computed efficiency burden restrict sentence cover month triple extract select triple frequent appear corpus twitter triple additionally approximately triple scale yield triple score randomly triple response supplement human pairwise challenge automate response generation reasonable extremely diverse describe status reference ir potential response towards multi ranking align future corpus triple avg min ref tuning triple minimum reference cover triple context first ir ir calibrate triple response original message formally triple bag bm response score formula provide diverse plausible candidate within triple human retain result response token response evaluate parameterize typical mt generation system base phrase mt decoder mt include forward translation phrase distortion twitter response translation phrase million filter long select phrase fisher mt decoder specifically distance phrase exact ir feature triple whose match ir response iff dm mt ir traditionally contextual match match match capture
size sep font node f node b e fill node ab ac node node factor node node factor node ef ac ab ef e circle minimum size pt node factor ab node factor ac node node cd ce ef node ac ab cd ce ef terminology graphical model refer graphical intelligence constraint function graphical describe constraint realization software tool describe seminal complete function constraint finite integer rational hard encode soft realization minimum weighted stochastic translate use exact mode stochastic rely firstly multiplication potential joint potential secondly subset identity basic algebra offer product potential function x elimination function ix ix describe count graphical simplicity insensitive order similarly successive task involve sum include marginal associate define variable compute conditional restrict domain probabilistic linearity use evaluation graphical probable mode require normalize counting task elimination elimination artificial intelligence max see solve exploit generic elimination deduce solve counting model operator problem logical elimination logical combination elimination operator include variant describe elimination recall chain variable elimination formally variable elimination key choice ordering notion graphical elimination elimination variable elimination markov hmc hmc define two realization unknown hmc hide specify initial classical hmc ph ph font circle fill gray sep font hmc realization equivalently q realization exponential viterbi maximization likely variable product operator elimination equation potential create maximizing involve section formalize trick elimination property indeed since elimination one task also hmc elimination counting optimize consist expression elimination first involve operator potential use rewrite follow elimination new potential neighboring except connect together clique edge neighbor instance potential right elimination distribution graphical elimination successively single potential value scale circle fill size scale draw fill gray sep sep b c b circle fill gray inner e e involve create vertex call edge log transformation normalize evaluate graphical obtain elimination change require mode bx I take mode elimination keep successive state space entirely complexity eliminate variable complexity elimination successive elimination elimination decide order elimination illustrate lead two subset efficiency elimination illustrate formalize viterbi hmc shape elimination eliminate unique neighbor graphical first eliminate order elimination order vertex one single generally elimination cardinality model possible elimination neighbor graphical associate length connect adjacent create edge vertex clique elimination fill clique equal clique graph clique elimination scope create elimination require storage quickly exceed depend maximal scope view elimination inner vertex vertex c c g draw inner font node vertex bend leave bend c bend bend bend draw fill font node vertex f bend bend bend bend bend bend bend bend b bend lowest achievable perform call discover graph front equal graph graph follow orient orient follow edge go toward elimination game graph least order call tree graph cluster link edge vertex build elimination tree decomposition tree cluster cluster contain cluster contain connect intersection gray inner font node vertex vertex gray sep vertex node vertex label definition induce graph trivial edge vertex induce tree large width decomposition create order equal hmc tree equal establish equivalent task question drive variable elimination graphical variable elimination apply exact equal elimination elimination solution rely computation sub give graphical counting task eliminate elimination rely tree characterization root elimination toward root eliminate cluster leaf cluster close root start elimination leaf parent assign rewrite expression eq eliminate eliminate cluster x elimination continue elimination scope complexity perform elimination tree decomposition elimination vice elimination elimination eliminate easy decomposition order clique identify define maximum span vertex identify combinatorial elimination relate fourier elimination elimination benefit linearity formula elimination repeatedly serial david boolean elimination backward hmm many elimination impact computation gauss elimination h upper vertex insight vertex quality guarantee time allow guarantee reasonable dominate empirically instance algorithm work well though approach broad greedy algorithm elimination elimination optimize select minimum fill edge vertex add fill fill edge call implementation minimum g graph fill heuristic slow approximation practice elimination graph linear heuristic break tie break random done find four minimum fill randomize iterative fill five mrf disagreement linkage linkage benchmark surface scene decomposition vision characteristic minimum fill second respectively maximum tie iterative version htb problem nb nb nb type mrf mrf randomized elimination randomize iterative exploit first branch good trade memory search effort elimination allocate hour gb ram report able two heavily category fast total gb fill amount second cpu measurement cost search solve instance category version option combinatorial solver option benchmark vision benchmark provide fill fill exploit fill order red blue message pass make external structured message extend elimination efficiently marginal mode elimination compute apply conceptually perform parametrization modification potential external tree describe extension elimination marginal elimination leaf function send parent always subgraph handle message unary variable potential marginal unnormalize elimination define root root subtree turn formally tree message leave first leave leave processed send neighbor message jx x process illustration leave mark marked message link follow c function incoming except message message pass way decomposition previously handle cross approach exact computation expensive intensive exponential typical example algebraic exact alternatively belief another message update termination meet return approximation marginal probability however marginal deep message product max exact approximation logarithm algebraic define compute exact mode compute general graphical message pass pass graphical tree structure joint graphical interest marginal read behind parametrization conceptually simple multiply involve preserve graphical suitable pseudo see possibility potential equal structured say fact pair variable advantage calibrate incremental update cyclic exact decomposition intersection parametrization potential message inside cluster joint calibrate agree exploit jensen incremental calibrate locally set parameterized estimate marginal read rule convergent maximize typical schema reweighte linear normalize seminal publish exact loop potential convexity also use deterministic consistency enforce consistency property define calibration transform desire calibration consistency arc consistency consistency exact maintain exact logical point local closely pass map always convergent calibration may structure submodular problem mainly useful elimination available interaction shape pixel applied start algorithm priori opposed call approximation nevertheless distinction perform sometimes marginalization heuristic principle belief numerous decade understand variational approximation class leibl application continuous variable principle cast variational choose nature variable variable low independent field work class distribution numerous remark remainder key component approximation leibler divergence tractable explain variational constrain accord set distribution marginal normalize constant vertex field fully factorize mf qx ix mf correspond joint respective namely term minimize respect fix relation conditional expect distribution approximation trade opposite guarantee contain focus bethe class enable heuristic liu model factorial approach multivariate state state markov chain apply use gaussian posterior ep minimized field among mechanic cluster variational leibler energy mrf associate qx j qx qx qx coherent tree bethe free draw sep gray draw em width color gray example couple hmm hmm hidden couple hmm assume display tm tm h tp em em tm tm tm h tm leave bend leave tp tp bend tp em em em h tm tm tp involve aim maximize likelihood achieve stand kullback leibler perform observe encode markovian dependency coupling write merge graphical value variable mode compute either forward recursion viterbi
learner solution use solution typical solution construct representative cluster cluster large solution eq likelihood belong auto experimental useful learner denote expression feature solution evaluate first p k also learner make less alternatively benefit common learner help guide also enable automated error early step demonstrate efficacy automatically learner roughly course solution feedback solution dataset response mathematical question set include college level signal question detail statement question pre extract full expression solution randomly sub solution solution similar sub rs sc affinity run gibbs iteration burn ap early mae average absolute estimate mae number sc consistently almost performance likely cluster ap e auto question take compare second computer cpu learner accurately mae moreover eventually enough belong cluster tune achieve balance maximize effort automatically balance around represent require demonstrate efficacy learner incorrect expression occur feedback early carry line become tool learner mathematical question solution learner eq learner enter compute solution expect expect full processing open mathematical potentially solution cluster provide indicate substantially reduce effort large enable visualize help common group learner track step solution enable indicate outcome research currently platform thousand planning extend account ordering expression solution cluster make robust would predictive model multiply answer question discrete time omit calculate discrete fouri simplify answer summation process discussion visit website rgb g f g h thm n z sparfa mean scale aspect education weak kind mathematical technology engineering mathematic language number learner learner language comprise convert response question series develop generic automatically potentially solution track correct enable indicate learner world demonstrate reduce scale capability education provide learner include massive system communication effective mean scale learner stream reading web interact simulation via user online assignment weak link way handle learner substantial restrict language nlp program mathematical verification paper response mathematical science technology education tool education binary correct incorrect knowledge open question shift burden learner need learner obtain develop solution correctness solution mathematical assign partial score likely scope involve mathematical expression include science engineering solution correctness focus evaluate success nlp method analysis short answer comprise main response derive symbolic mathematic canonical solution correct partially incorrect solution two cluster approach numerical affinity propagation ap group learner bayesian solution assign track assignment indicate likely learner develop tackle challenge analyze response mathematical notation mathematical learner refer question admit path yet possibility correctness solution especially question answer learner recognize correctness computer program formulae specify different checking accurately certain kind feature extract language nlp automate computer program response lack structure correctness logic logical operation kind open mathematical calculation involve science engineering cluster question short use visualize program use cluster feedback structure differ answer correctly simplification extract expression feature assume learner extract yield solution expression column numerical representation solution frequency encoding frequency illustrate solution expression unique let four expression observation mathematical share learner true instance suggest limited solution incorrect conclusion particular effectively section solution affinity ap matrix mathematical detail dataset appendix node estimate proportional correctness outline solution define similarity similarity solution entry informally expression two solution figure development future cluster solution similarity cluster algorithm solution sc ap sc specifying cluster ap ap identify figure correspond processing node solution observation answer although able identify correct require able simplify final identify solution connect visualization extremely easily lack adjust accordingly learner solution solution assign automatically construct set break randomly demonstrate auto via outline interpret extension short key cluster assignment denote assignment learner matrix solution implicitly learner analogy learner question assignment cluster consider parameter impose chinese crp cluster assignment crp characterize partition
task require experiment run corpus collect build tf create category include category sub category positive category category error positive writing digit include grey classify digit average overlap result dimensional digit class digit digit positive word rate run size dataset see figure always achieve significant first achieve note regression accurate black power nonetheless similar increase difficulty consume train allow regularization achieve novel transfer purpose classifier ratio test general improve distance linear divergence attractive normalization term decompose relatively later generic law totally verify universal nn universal knn bound absolutely nn probability jensen right side show new apply z n assumption se utilize boundedness se g converge converge via fold validation criterion change iteration gradient tuning mathematic mathematic learn frequently already acquire writing enhance write focus purpose probabilistic transfer sparse parametric model obtain multiply show method achieve promise world text hand digits classifier testing slightly pattern write train recognize user due background large classifier refer label task similar may general purpose task natural idea assign assign contribute target target rise function function risk minimization sample parametric must simultaneously regularization utilize enforce closeness learn task face happen build massive since costly writing come rapid build patch transfer consist separate stage general second light weight translate model intensive involve general purpose dataset sophisticated key difference minor since task handle may handle much simple change especially contain crucial build sophisticated machine software end hope adapt device mobile key idea probabilistic already train multiply classifier composite completely totally still error later efficiently parametric convex conduct motivation algorithm datum yy enough focus scenario relatively desirable boost previous later stage go clearly valid normalize q however approximate normalization introduce ratio log substitute term evaluate pointwise sufficient pair practice sample may pair observe sample may approximate estimation quantity posterior feature therefore together cross notation posterior approximation nn estimate fy reasonable completely make expect transfer method follow absolutely continuous neighbor rely state approximate step model error cause obtain multiplication divergence identifiable q different mean train create kl blue line figure comparison sample plot figure show divergence help improve bar red bar however transfer introduce baseline logistic regression sample modal previous would use ii indicator ratio center show predict transfer dot dataset truth mark
analysis however bad subspace e combination improper although supervise attention drive unsupervised solid foundation development leverage model massive novel product hermitian random streaming generate rectangular identically transform variance empirical surely q eigenvalue inner radius z outer h acquire law spectral show management general standardize past model vector system certain arrange form map far consist n decide length historical big raw area form rectangular hermitian I introduce column z transformation depict single law equal eigenvalue propose transformation raw step raw single conduct visualization unsupervise aim conduct dimensional raw visualize status estimation arbitrary historical time focus drive model topology addition control principal component grid select select pilot orthogonal pilot n train step choose n pilot illustrate scale train historical tag system eigenvector method hard still six package manual white fluctuation sample plus change grid work move event par demand include time area play dominant closely curve decrease event decreasing conduct detect achieve raw datum statistical orient decide node latter high parameter whose decide conduct visualization combination analysis visualization interpolation plot respectively much area change influential conjecture demand system visualization important raw whereas status trend whole trend statistical htbp pdf pdf scale pdf sets set datum realistic node work day node reach size control matrix acquire figure curve data window working detect meanwhile load mathematical visualization conduct status trend make learning method realistic validate effectiveness drive unsupervised highlight even towards power system big unsupervised research radius long goal reach year age grid big pdf pt blue rgb rgb rgb rgb ai xu event become increasingly grid feature vs handle one conduct system hand extract directly propose raw one dimensional statistical visualize reasoning speed traditional spurious bad realistic effectiveness grid unsupervise grid law radius map become resource grid readily various device communication technology measurement unit device acquisition volume velocity vs curse aggregated grid require massive multivariate pattern mean signal grids generator fluctuation error generator execution aim streaming datum dimensionality big resource hardware resource handle tool base extract multivariate multidimensional big establishes inherent system gain rather build assumption already numerous phenomenon quantum system wireless data scope system grid produce label complex power original datum impossible event
even though optima performance likely performance rnn well explore compositional long memory include kb rule rule path kb certain modern tune threshold train feature perform kb extend rule simple multiplication compositional phrase language network successfully phrase neural task language model translation parse recurrent neural like parse classification answer logical semantic rnns attention path connect entity final shot drug shoot neural develop compositional completion recurrent baseline prediction prediction modify ability shoot acknowledgment answer numerous question thank stanford nlp retrieval agreement award google part reproduce notation recommendation express material necessarily cs knowledge kb previous relational relational path atomic z multi rnn path unseen training capacity compositional zero relational triple leverage train embedding construct knowledge reasoning natural triple also binary relation incomplete many fact usefulness infer already kb automate triple leverage triple kb completion focus symbolic learn clause binary relational path threshold triple entity triple entity triple rank greatly efficiency exhaustive walk use relation binary learn imply infer fact later greatly available raw material kb schema relation connect corpus symbolic million treat put aside relation limit applicability modern relation rapidly type relation obviously generalization operate representation relation semantic learn representation relation vector perform prediction kb kb embedding tensor universal schema cast completion likewise embedding kb entity entity evidence propose advantage embedding reasoning generalization network rnns semantics path represent output vector represent extend input step consume relation entity path consume path vector compositional unseen neighborhood atomic allow million path kb separate predict type alternatively composition relation perform shot composition work continue collapse path substitute original relation type path map nearby embedding shoot pre train tuned completion completely additional new large million triple preprocesse kb entity path relation million entity pair k experimental compositional outperform feature statistically compositional substantially strength shot predict unseen background path connect entity employ composition obtain connect entity kb use path connect entity training entity path walk target million path per type improvement neural rnn phrase composition linearity concatenation operation phrase propose kb connect entity relation representation path length kb l binary fact p vector randomly backpropagation thousand without directly describe previous section capable fact type shoot modification composition fix relation unseen relation beyond capability kb composition predict vector relation general composition initialize vector representation update training prediction type never see vector relation sigmoid softmax contain relation unchanged predict unseen training composition use irrespective see compare pre tune hold development iteration use regularizer batch triple avg path avg fact avg instance avg relation run publicly link dataset create node kb object triple sentence entity phrase entity relation length great keep last consume dependency parse relation type triple fact reduce relation relation fact statistic path relation learn rnn quality rnn generalize book book write author person language people language people person book person language location location location bridge location location location contain near people person child people parent people parent target relation example unseen show relation relation mark train create path method replace relation membership path find exactly create pre type simple wise linearity composition rnn initialize rnn use additionally path extension feature compute prediction prediction assign score sort
introduction limit maintain obtain eq variance free fact penalty term final introduce additional cost hdp hmm generative let state transition last state alternatively integrate take logarithm asymptotic expansion hmm hyperparameter cost kl hierarchical bias derivation small asymptotic derivation asymptotic hdp hdp infinite hmm crp crf numerous bayesian model year problem series analysis however mean extremely simplicity large view assignment equivalently dominate asymptotic generalize derivation bayesian dirichlet process obtain straightforward derivation nonparametric subtle aware dp dp hdp probability reference hdp material hdp model hmm serve introduction nonparametric reader also chinese restaurant crp chinese restaurant dp hdp iv dot tensor convention crp observation exchangeability crp parameter restaurant restaurant count repeatedly customer restaurant eq
vector computation track past stochastic gradient refer special case consider requirement set balanced algorithm neighborhood pairwise explicitly purpose sharing gradient neighborhood motivate regard define modify intuitively compute observe generalize exploit neighborhood correction neighborhood practical overhead find well yet plain variant candidate drastically way subsample refined core set g remain sensitive even hash cost affect throughput recurrence imply recurrence utilize e continuity apply parameterized note asymptotic immediately contraction yield contraction factor relative gold sgd thing complicate parametrize share bound pair regard iterate sequence result contraction bring main challenge store e constitute require somewhat complex inspire lyapunov conceptually initialize maintain valid iw quantity expectation lyapunov recurrence without uniformity expectation increment iterate convergence main sharing state sharing provide lot motivation insight see may depend q suitable collect contraction theorem investigate bound optimality factor fw recurrence work get simplify bound identity good maximal result apply claim case case taylor approximate rate note nb q improved yet error vanish control neighborhood neighborhood guarantee share g straightforward get contraction behavior progress towards enough reach ball optimum walk ultimately switch sharing effectively minimizer desired behave sensible neighborhood requirement year sgd paper uniformly superior almost superior albeit sometimes regularizer commonly occur namely logistic million song uci repository logistic obtain website dataset compute exception practically relevant improvement correction everywhere roughly plain sgd schedule axis express epoch really gain significant start plain counting step reference expect actual somewhat gain typically solid speed start difference difference sgd state run asymptotic experiment cross epoch gain epoch streaming present although analysis purely loss also take set proxy generalization somewhat early epoch c novel sgd demonstrate insight role gradient evaluate correction speed remarkably knowledge corollary inf stochastic descent machine know relative sublinear iterate recently overcome vanish step maintain result either full correction disadvantage employ streaming speed neighborhood structure gradient across significant phase epoch investigate family thorough support variance regularizer finding parameter minimize empirical descent straightforward repeat computation become prohibitive new select
problem ease exposition system augmentation develop technique uncertainty drift demonstrate affine dynamical state drift assume lipschitz control control online functional dynamic origin denote eq since invariant xt map accumulate feedback compact controller eq characterized admit continuously solution constitute e differentiable derivative argument control open express solve infeasible actor replace denote replace objective actor controller learn motivated actor maintain substitute bellman eq actor exact approximate system give approximate establish basis function require instead operating domain intractable basis aim obtain aforementioned development cx universal weight compact select cx rx denote function note change system change system change dependent change ideal continuously continuously learn ideal evaluate form actor weight system start initial feedback controller constitute serve indirect close ideal hence gain along system trajectory simulation achieved select nx base law matrix eq constant factor lyapunov improve brevity time hereafter origin facilitate subsequent eq lyapunov select constant strictly regressor regressor regressor hence regressor hence strictly frequency make make function state computationally infeasible high trajectory enough simulation instead establish upper gain law ensure tt q furthermore tt lower positive lyapunov subsequent lyapunov sufficient provide enough kernel kernel small result almost identical approximation select meet assumption gain controller ensure ultimately derivative weight lyapunov express sufficient candidate lyapunov use conclude exact demonstrate simulation perform dynamical system cost select optimal approximate three c tx vertex shrink triangle state shrink point uniform xt ix matrix trajectory generate control estimate unknown ideal ideal trajectory kernel weight system state maintain show control compare signal control show origin ideal figure show decay uncertainty rl uncertainty drift system ideal unknown value xt dt fx approximate select form identification implement origin stack ten record cf computed order initial value dynamic stable simultaneous figure track nn periodic weight converge periodic drift dynamic converge value h nonlinear nonlinear system steady system analytical tracking compare system converge ideal analytical tracking available ideal value unknown drift dynamic dotted error point develop time steady move technique develop tracking simulation run hz simulation run table develop controller resource know quadratic quadratic since basis use inexact generic tracking ten polynomial basis trial since unknown inexact result steady sense use result small trade computational inexact estimate trajectory table develop horizon control solve function aims maintain good value neighborhood efficiency rl allow selection number simulation horizon art horizon problem online entire operate completely value art state aim maintain value kernel sense ideal lose leave unlike memory lemma remark optimization recommendation view horizon online compact stability optimality achieve significantly global popular online system challenge rl action open rl via employ employ challenge controller decay sufficiently determined rate introduce achieve nonlinear system generally basis sufficiently require recorded history rl increase increase rich cause increasingly undesirable experience rl demand rich cause store stack number respectively grow hence drive find novel base rl achieve sufficient undesirable effort like traditional computational technique effort decrease basis key online optimal controller current approximated facilitate summarize reproduce kernel use continuous ideal small would prove facilitate rkh set continuously kernel
language computer double format implementation computer format sign bit exponent bit group integer indicate number exponent plus minus infinity infinite bit bit symbol exp mask exp bit mask bit sign two type available computer implementation arithmetic quantity processor old processor addition slow carry finally order bit field allow exponent bit store mask bit sign computer big little intel past arm architecture summation first summation small preferred method moderate component detail design contrast primarily hardware implementation obvious design sum store bit carry propagate bit loop furthermore sign magnitude handle sign change additional represent carry propagation somewhat save representation high bit low carry defer whose determine sign complement start low order denote eq range code symbol high exp bit bit along represent capacity bit format bit within sum produce advantage arithmetic due sign however canonical produce carry propagation happen whenever correct need propagation start integer high end order whether positive negative carry describe carry negative require modification ie avoid set bit next produce carry absolute value add carry routine remain carry propagation actual subtract necessary per carry justify even procedure mask treat sign sign sign bit point value infinity storing indicator inf highly inf fairly nothing add zero otherwise exponent exponent neither normalize shift mask separate exponent bit bit exponent bit subtracting seem subtract modify bit give exponent order bit operation modify subtract bit modify add determine number quite overall sign operation show array code expand loop function carry routine loop loop check carry define type inf inf inf nan propagate double u value mask exp exp mask exp exp bit else else inf inf exp exp mask exp exp exp mask exp low else add sum carry propagation inf must examine proceeding note round number obtain high point look converted operation operation could replace bit bit highest examine potentially round exponent somewhat round present commonly round round implement round straightforward summation fix zero scan carry final roughly naive operation operation term compare add term modern bit processor probably processor exploit parallelism nevertheless motivate per summation term summing eliminate summation inf checking term decide accomplish bit possible exponent bit still sum initially start view integer shift fill bit bit define exactly least acc holding add remain use use bit bit large count large ix else count count ix exponent entire sign exponent mask add index done keep hold routine check inf check index initialize check index small circumstance routine exponent index pass inf handle associate add process indicating must indicate count two proceed add partial add final add adjustment sign bit time multiply bit remove bit transfer add shift leave bit bit magnitude would small modify consecutive bit conceptually position low bit next however shift instead find left ie exponent implicit beyond top bit appropriate bit top great initialize count small many application typical number sign obvious look count overhead keep array word word skip region scan bit whose bit maintain slightly inner summation operation figure sum array twice fast summing use small evaluation performance sum element array architecture processor speed lead assess summation would run ideal processor parallelism allow depend computed insight assess question perform computer measurement computer assess insight summation characteristic conjunction limited assessment serial many processor core apart implement careful attention obvious attempt branch straightforward might superior decision manually version similarly optimize version order routine add turn double summation use index add allow level supplementary information reasonably implementation improve sometimes good choice appropriate among sum summation seven size try ten cover processor test effect summation divide always fold array fit cache memory cache cache vertical line cache need assessment generator rand avoid term mirror element perform test array performance six colour solid dash line processor year intel bit intel processor high end end intel processor span family x e v processor different pt qualitative picture processor sum term combination superior processor fast array less advantage sum processor nothing array memory dominate advantage method summing array intel core slow slow summation cache memory dominate summation add order summation summing array size summation large intel processor intel distant intel intel ghz intel iii processor processor parallelism case processor summing array large method combination except array overhead cache dominate intel slight sum identically small processor intel intel picture combination almost array small array difference reflect integer processor actually ghz whereas summation perform substantially bit intel ratio time exact summation though cache reflect somewhat specialized processor support operation bit computation modern bit processor perform processor slow summation modern show processor processor bite intel plus optimize overhead look per term processor assume summation model fix model perhaps overhead fairly I array affect stage little effect performance processor one perhaps branch row sign processor branch method much method sum reduce summation produce modern bit fast term I vector square vector sum product method product usual rounding bit implementation intel figure time inner loop method multiplication execute parallel sum dot cache large method processor intel sum computing norm order indistinguishable summation length picture reason expect apparent manner parallelism would however exact great summation supplementary modern processor implementation two summation introduce dominate two result improvement obtain summation less slow order three slow sum vector reduce eliminate method return advantage summation unlike situation summation probably loop small conditional positive eliminate conditionally sign none code improvement summation likely expect processor would allow implementation summation limit discussion exact summation purpose processor way array parallel add together partial sum write routine add straightforward operation comparable produce course integrate might parallelism core eight current core limit bandwidth intel processor memory ns summing array take ns limit impose suggest core sum possible probably maximum may suffice v processor issue cache core evaluation investigate limit regime distinguish achieve wide perform slow summing term ten order summation scan cost dominate term limit summation three two arithmetic overhead keep bit array become bit quickly non produce increase current method fast moderate exact summation improve
follow integer iterate function domain iterate smoothly invertible inverse set verified operator use integer function operation concern continuously since exist inspection equation th integer choose non iterate property hence understand likewise respect logarithm continuous real necessity logarithm argument derive substitution gives examine constant interested restrict z branch fix function infinite plane since contraction mapping arbitrary applying behave affine expansion exponential circle around becomes give repeat application logarithm point circle iterate solution r derivative c z contour application circle radius rotation complex plane branch part iterate application half plane instead thus avoid branch n map half disk series disk thus schwarz hand nn application iterate lead plane give think plane call complex plane plane calculate exponential plane grid cross origin map circle whole integer composition z case range maximum two method net operation adjust straightforward neuron summation value neuron behave like input dependence neuron output neuron compare net real computational necessity output sensible unit avoid neuron accord interpolation transfer tie iterate occur neural net namely computational complexity unchanged conventional neuron function neuron consequently framework replace complex matrix weight neuron value ij task consist binary vector integer output element index efficiently implement architecture fourier dft inverse factor shift definition amount one else neural one output nm encoding pattern shift cell shift multiplicative additive neuron transfer compute dft fouri compute dft shift pattern see automatically neuron net solve continuously addition integrate neural mathematical concept integrate net behave efficiently work technique exponential first sound iterate functional composition monotonic interpolation multiplication landscape ad hoc transition introduce transfer allow back drawback implement area pdf ai combine neuron either inefficient resource procedure present transfer mathematical concept non integer functional neuron smoothly importantly addition multiplication decision integrate backpropagation procedure neuron sum input illustration multidimensional neuron function sigmoid layer context network transfer function arbitrarily sufficient unit though architecture acceptable alternative weighted replace weight use law layer element incoming exponential apply product hybrid summation pose additive neuron solution stack alternate additive skip additive
author interest classical topic world setting paper resolve instead assign fix topic stochastic binomial three author document document mix account multi algorithm model real world capability interest topic mine artificial traditional text mining topic commonly regard mining interest side corpus include author tag label incorporation side lot among topic interest author jointly topic interest variety scenario recommendation recommend interest author surprising author paper rank exist fix number normally dirichlet limit exactly application author topic relax distribution specific gamma three level capture hierarchical vector process simply measure normally gamma process author document introduce gamma closed distribution hide number topic new fixed author gibbs sampling get briefly describe preliminary knowledge propose section direction work second part nonparametric probability originally mine task discover topic good powerful representation music author propose interest document suppose interest detail attract lot work elegant author document incorporate side tag time mixture suppose infer idea multinomial limitation multinomial mp pp infinite property task dirichlet probabilistic model one process gaussian mixture gmm mixture hide extended infinite hide infer process binomial summarize nonparametric apply application model propose gamma circle thick draw black solid pt draw font rectangle rectangle model use author hide paper interest interest graphical interest topic one topic need predefine real scenario process measure product indicator improper get process parameterize process negative also point integer binomial normally count counting variance mean cm black black thick draw fill gray right edge edge normally parameter kind binomial beta gamma base binomial make document base gamma process equivalently augment poisson parameter augmentation tb description word document infinite model author binomial successful fundamentally document three document add process level capture gamma hierarchical author binomial process however thick draw solid black circle gray font rectangle right edge edge edge edge assign topic interest summation however topic assign author author see set multiple author gamma document combination process weight paper gamma interest document frequently around truncation truncation accept potential express good approximation infinite appropriate mixed resolve various distribution far split topic independent assign inference help resolve gamma author gamma author due conjugacy make binomial compound poisson distribution logarithmic logarithmic restaurant distribution theorem finally formulate gibb design list number author ar ni na kp da kl conditional implement procedure summarize n burn stage number output latent infinite topic finite author topic description author paper area database mine artificial description find document training selection two document specialized column requirement requirement sure topic predict author interest widely assess document document well different normalization influence comparison understand big training fig table comparison adjust topic propose hyper rest
node paradigm raise question made round need influence probability learn manner probability assume cascade seed influence probability price seed spread spread perfect minimize spread market structure learn phase exploration knowledge gain phase achieve spread exploitation regret network come mab influence maximization application firstly status seed observe live definition status refer observable seed activate fail observable feedback try face assign neighbor whereas real paper challenge focus ic contribution I influence combinatorial mab motivate regret aim efficiently influence section combinatorial exploitation prove round probability learn online set various minimization conduct extensive dataset effectiveness intend summarize direction graph seed maximize spread cascade threshold variant ic discrete seed get chance inactive neighbor activation succeed probability attempt say neighbor parent inactive multiple activate time parent capable parent ic expect node seed distribution show I nice seed spread gain exploiting et show seed large gain increase spread seed use research around diffusion scalable algorithm multi multi armed bandit mab paradigm reward arm play generate reward continue round reward natural goal bandit goal result play suboptimal regret much hope well learn reward al pure mab bandit paradigm paradigm together arm contain reward reward et al chen et play round consider individual combination al upper ucb algorithm obtain al pure chen et al ad page arm briefly combinatorial armed I consist base arm denote trial support identically unknown play round play arm possibly reward number base mean well play take accommodate attain I arm either live reward arm correspond specifically seed play become live live number diffusion process linear arm maximization oracle constitute approximation output mc serve mean update context notion play role characterize play update world underlie arm reward application bandit I basic assume status live edge possible network arm frequentist formula assume know active key attempt realistic precise status e call compare feedback weak realistic flip estimate challenge node feedback consider parent parent active status edge node level parent responsible overcome adaptation frequentist approach specifically scheme whereby assign assign active assign follow edge feedback inherent uncertainty infer live vice probability feedback bind ultimately achievable verify failure establish effectiveness feedback let denote arm edge probability influence resp use feedback relative learn influence learn situation infer edge live infer live assignment live recall condition fact parent live world hence characterize failure follow pr pr v pr cascade ex pr pr k pr u u pr u j pr u u pr random pr u k theorem use feedback influence edge error status level tx round successful let reward arm play section empirically typical verify feedback cascade input adapt method employ develop cascade cascade action reveal status perfectly align feedback online cascade stream cascade cascade view describe cascade influence probability neighborhood function show maximize propose make need learn improved probability describe propose make offline mle characterize likelihood individual cascade number cascade network l write follow attempt term activation attempt notice concave node separately gradient contain mle maximize likelihood stream online function advance reveal function objective cost advance offline choose minimize online compare offline algorithm depend greedy updating function correspond likelihood cascade round game mle exploration frequentist cascade seed probability within edge correspond round round arm play ic cascade get activate number cascade learn cascade arm follow edge live cascade e game give time probability within number cascade seed choose seed eq follow result cascade learn within relative need combine cascade feedback work round select pure seed spread edge round round combine exploration seed explore intuition active cascade write edge cascade frequently edge define exactly expect spread count activate add seed spread seed intuitively seed large edge call exploration se dynamically se across round effectively result first formally tailor I intuitively give seed seed correspond seed play I spread ic true p hard optimal seed monte mc spread seed know greedy alternatively recent reverse rr use motivated assume notice inherent randomness output randomness rr set true probability know spread seed successive exploring improve seed probability improve choose seed lower quantify randomness output strategy round invoke seed seed size maximize current update budget select seed budget return seed cascade update accord combinatorial logarithmic confidence bind network initially maintain estimate find lead implicit regret feedback initialize exploit cascade arm increment value strategy involve exploration select achieve budget parameter initialize cascade update achieve level proof explore algorithm exploit every knowledge implicit pick seed cascade learn influence oracle influence recently act run briefly diffusion ic obtains spread adaptive near optimal runtime operate generate random reverse rr node reach enough bind rr rr seed node node rr seed seed next cover maximum rr seed al strategy enough idea range probability may method frequentist initialization beta conjugate prior beta update tx act like pseudo formula technique laplace nlp put prior goal various regret regret true generate probability initialize k diffusion sampling world purpose diffusion graph budget influence oracle notion reverse lack refer reader obtain greedy mc iii readily serve runtime value verify run algorithm round find actual spread generate value seed process also learn plot varied pure network edge algorithm feedback exploit mechanism r dataset exploit cm within edge exploration exploration strategy couple feedback base mechanism exploration choose feedback try avoid sample decrease frequentist compare fast node moderate probability typical frequentist quick error depict roc fraction whose probability learn quick round percentage cascade network exploration sample algorithm try learn cascade minimization experiment pure exploitation explore seed regret achieve worse omit completely likelihood learn runtime use minimization greedy set initial set result back feedback cm average increase slow explore edge decrease control amount use see become low end round probability estimate well enough comparable spread know probability almost exploration phase pure exploitation end round reward initially probabilitie lead observe feedback us node feedback failure find number parent big previous cascade vary vary level well plot goes expect choose seed true pure exploitation lead seed initial however bit exploitation explore network seed omit plot observe probability lead spread show spread observe spread always edge feedback irrespective seed study influence cascade adopt armed paradigm formulate interesting regret spread suboptimal set various bandit performance real sampling extend continuous model influence interesting theorem assume paper propose availability diffusion cascade learn offline tackle ii cascade adopt combinatorial armed paradigm formulate spread suboptimal round problem
discuss submatrix localization em theoretical submatrix localization problem noise determine transition boundary hide detection focus submatrix entry relaxation signal noise require submatrix regime wise thresholding well submatrix small snr relaxation improve snr boundary dense scale dimensionality low literature boundary similar message pass snr submatrix submatrix diagonal emphasize localization combine treating achieve set however crucial utilize denote submatrix vector induce precisely latter surrogate norm inner exist universal bn equivalence factor iff sub gaussian random universal random scalar run submatrix answer localization minimal read respectively terminate restriction combine statistical submatrix localization submatrix localization boundary clique hardness phase diagram consider case boundary eq separate region theorem correspond separate snr entry snr small hide clique submatrix furthermore require submatrix run sdp clique mention early localization investigate result provide finish phenomenon figure localization result submatrix intractable region plain beyond localization statistically possible appear distinct principal computational statistical region boundary computational upper introduce extension technical proof defer addition since localization detection boundary submatrix bind upper bind computational hide clique computer science identify hard sense hard problem hard deal seek token difficult clique quasi polynomial describe work precise clique clique clique instance way clique connect connect clique plant tend possibly detailed clique location location nn q uniform clique precisely clique hypothesis recently claim localization solve also localization low localization theorem defer difference construction localization bind ensure submatrix tuple within localization sec pf key idea insight bootstrappe introduce randomness clique field us mixture submatrix submatrix localization need clique support exactly plain reduce clique still exact clique technical please introduce solve localization graph localization submatrix top respectively calculate separate thresholding return several appear secondly require automatically submatrix lemma spectral guarantee spectral succeed exclude exhaustive search boundary submatrix submatrix localization clique hardness proof spectral relaxation theorem analyse multiple grow number single submatrix statistical introduction establish expense begin theoretic accuracy theoretic submatrix search aggregate statistically optimal unfortunately computational algorithm introduce analyze extend gaussian noise localization submatrix sum submatrix report extract large sum greedy fashion submatrix provide achieve submatrix algorithm exist universal return probability boundary theoretic analyze submatrix statistical submatrix high minimax tend boundary section submatrix perturbation combine know thus learn line form sense cut clustering recover summary spectral succeed eq computational localization low detection polynomial algorithmic relate instance proof k n quantitative submatrix time solve localization probability contradiction chernoff bernstein let suppose time stage graph stochastically hide clique easy analysis property clique bernstein least clique long hide clique node clique clique take submatrix set independent subsampling replacement weighted generate matrix q bootstrapping lc stand clique clique n l lm edge condition index clique clique rademacher position matrix submatrix sub estimate submatrix signal side q bernstein inequality position submatrix parametrization slack precisely return submatrix correct go correctly identify clique node correspond clique quantitative lower boundary submatrix amount power correspondingly submatrix introduce clique node tend algorithm rely clique contain clique subset adjacency restrict clique mr lk correct clique recovery complete n induce solve clique problem coincide true go hide hardness time proof testing compose similarly cardinality corresponding measure probability uniform prior space invoke bind leibler q binomial coefficient condition submatrix translate testing pick inequality localization total unique reach need trick row cardinality set use satisfied constant thus pick submatrix sum overlap go theoretical justification submatrix multiple submatrix singular n thus lemma canonical angle observation observation singular use lemma thus invoke basically know learn metric succeed term explicit problem eq feasibility set function although bi relax course need solution relax exact high condition submatrix singular g low relaxed unfortunately estimation simultaneous exploitation relaxation expand original term time nk localization submatrix denote top singular implement multiplier admm disadvantage theoretical hold remark require knowledge submatrix guarantee submatrix universal relaxation mn cm pair w relaxation satisfy find primal q mean feasibility lagrangian associate kkt expand q choose hold thus hence need q see q succeed q achieve boundary upper submatrix tuple hc submatrix going build hide submatrix model q several graph stochastically clique graph property probability clique long hide clique clique node connect otherwise equal take right submatrix index ns ni q case note clique inside variable clique clique n variable thus construct submatrix element clique two side order bernstein computational detection clique school accuracy draw increase computational boundary submatrix localization submatrix contaminate establish threshold term correspond boundary threshold
standardize regularization lasso put imputation fold cross predictor multinomial three death set coefficient vs normal death vs normal penalize likelihood criterion initialization step size backtrack backtrack stop parameter procedure minimize yield discusse exclude consideration outcome individual age outcome age age size associate point nonzero multinomial logit death vs ht vs else person last else far early gender yes white yes heart difficulty arm could substitution task symbol correctly code trust probably else digit substitution symbol code stand gender ever else per day current status never gain lose exercise major status diabetes yes pressure one yes else detail subset validation relative importance longitudinal estimate meaning categorical coefficient display less majority proceed result keep logit outcome record predictor future cognitive status facilitate exposition formal vs odd age lack odd early early range death death variable analogous interpretation low diabetes health status difficulty contrast age digit account odd death odd death show intercept coefficient increase death age predictor broadly risk gender heart disease lack increase provide multinomial fuse demand multinomial fuse cross selection parameter incorporate variable plot typically interesting concern convert categorical selection tuning parameter important yet parameter multinomial fuse illustration consider cross validation aic bic misclassification likelihood cross divide fold usage say pick simplest interpret nonzero aic score fuse tuning number longitudinal degree denote loss multinomial typical aic close cross training bic computationally validation fold set aic log misclassification selection entirely fold fact division evaluation repeat fold rate positive positive rate identify standard selection ht seem favorable balance evaluation model accord majority death yield rate death rate cross true class concern cs simple degree rule absolute g fully multinomial dimensional operate assumption contribute persistent effect fit fuse respectively profile highly proximal demonstrate applicability discuss practically issue selection reach place group categorical variable may encourage sparsity variable complex fuse trend penalty polynomial trend choose application penalty assume variable mostly time trend effect concern fit longitudinal example band select profile interpretation stability unfortunately quite regularization penalty inferential development relate dimensional light work classification regularizer piecewise longitudinal adaptively select gradient descent propose disease health motivate tuning assessment study longitudinal record individual place predict measurement allow employ e extensive regularizer problem lasso encourage active fuse regularizer encourage persistence work cs factor disease ad cognitive people year old matter increase age incidence per age later examine age predict ad age assign longitudinal clear matrix element determine lag generally denote partial indexing multinomial coefficient introduce extension basic number individual index outcome array separate logit time eq eq multinomial fuse notation fact g broad encourage persistence generally generally point piecewise coefficient trajectory kt independence rather longitudinal setup partly role fuse tie across help example normal coefficient remain coefficient trajectory leave relevant outcome multinomial right intercept likelihood dynamic see multinomial fuse estimate pick underlie overall toward prediction multinomial advantage repetition estimate outcome regularize fuse lasso idea extensively community interesting fuse lasso genomic association author setup primary motivation age measure multinomial death individual age mainly complicated factor death death category alternate cox death traditionally cox naturally scheme multinomial hazard depend hazard relate instantaneous failure predictor multinomial death would determined maximize log lasso odd cox describe similarly routine cox minor modification comparison beyond scope current manuscript topic future development next cs discuss estimate coefficient relate discuss numerous approach tune fuse lasso conclude future descent compute lasso regularize multinomial algorithmic implementation direction multiplier proximal simplicity fused regularizer describe number choice size differentiable repeat denote obviously long precise variant descent share proximal descent generalize routine repeat strict convexity criterion define provide computable descent apply map perspective minimize plus quadratic current iterate optimization statistic typically nonsmooth trend somewhat though mapping history community term proximal enjoy convergence rate descent amenable acceleration technique proximal gradient applicable regularizer encounter mapping fast exist compute rely elegant propose negative multinomial lasso fuse lasso formally rewrite convex nonsmooth describe tuning respective computed intercept coefficient penalize proximal consider g evident proximal operator intercept proximal map identity intercept term consider proximal arbitrary predictor fuse specialized problem make compute proximal practical issue arise return rewrite proximal generalize gradient rewrite resemble choice iteration proximal proximal know easily appropriate backtracking line search straightforward algorithm constant shrinkage backtracking routine start initial guess satisfy proximal current lasso map define hand backtrack proximal refined terminate meet common stopping iterate stop meet tolerance outline proximal procedure multinomial backtracking criterion htb predictor c backtrack input ks j k practice individual longitudinal mean outcome predictor individual predictor outcome observe time issue arise penalty another experience sum effective regardless sample effective modification indeed end hundred cover descent interface author website broadly generalize rich thousand cognitive past
orthonormal basis chebyshev power odd I span inequalitie property plug invariance frobenius norm rotation fall span column span orthonormal row expand il eq norm l give necessary return span give proof singular henceforth statement extend guarantee top singular prove align existence ensure perform intermediate span odd contain power fall vector inner inner outer inner outer outer fall span outer remain singular orthogonal outer inner outer inner outer bind first unit vector slightly contain use outer plug span column frobenius norm apply choose polynomial necessary property already inner give ml w w ml argument single outer give place iteration rank probability return satisfy value additive frobenius matlab column explicitly use improve algorithms principal eps algorithms frobenius guarantee tail outperform case error confirm rapidly simultaneous often possible simultaneous long justify convergence gap comparison rapid begin rapid confirm gap comparison frequent singular gap take constant insufficient low approximation special prove svd n draw full rank show k f property note f gaussian distribution row orthonormal gaussians concentration k result high logarithmic chebyshev polynomials actual chebyshev gap exist construct chebyshev recurrence chebyshev polynomial simply degree satisfie place rule suffice eq claim hold derivative chebyshev verified chebyshev recurrence xt xx reduce note prove show chebyshev polynomial decrease also prove equation give thus additive satisfying follow exactly completeness di ji kn technology usa analyze simple value gap iteration approximation within norm give first provable runtime simultaneous give guarantee substantially experimentally despite history subspace method practice furthermore accuracy benchmark issue minor modification simultaneous give nearly far finally fast take advantage improve use singular r r analysis principal want kk vector principal component direction great great principal component denote singular expensive typically time arithmetic inherently iteratively traditional include qr obtain rate environment take hence research randomized seek nearly quickly become practice popular library like learn contrast classical depend gap gap small due identical difficult distinguish inherently depend singular value gap approximation goal randomization avoid need find subspace singular distinguish close fast randomize svd run lower satisfy often insufficient analysis singular noisy multiplicative number suggest remain tail guarantee decade simultaneous subspace achieve become choice practitioner classic simultaneous fast runtime performance section even though discuss improvement simultaneous subspace method highlight decade theory randomize excellent accuracy gap issue error rank output kk strong svd gap note intuitively strong spectral square small singular leaving mind guarantee practice svd amazon co eq principal significantly similar phenomenon popular additionally rank svd address introduce require singular capture I singular classical numerical analysis stress converge singular gap post return guarantee contribution runtime satisfying approximation pca exclude time simultaneous qualitatively weak give gap decay modification large justify start intuition simultaneous power gap progress considerable back full svd achieve rough factor randomization fast paradigm dimension either top leave singular project onto approach refine os recent type reduce multiply frobenius error approximation consider dominate sketch solve efficient regardless pass processor set furthermore small typically norm error seem limitation sketch singular value impossible extract svd singular word inherently norm symmetric simplicity apart accordingly lower much small top singular see specifically effectively method approximate frobenius singular provide xx high value high approximation iteratively repeatedly rough suffice simply column knowledge analyze gap technique gives achieve iteration start develop paradigm mention numerous paper possibility simultaneous experimentally variant none paper bound accelerate simply put well polynomial allow power iteration shift chebyshev nearly run polynomial block subspace polynomial scale lie match find approximation nearly surprisingly efficiently lie subspace span challenge recent frobenius post understand simultaneous block spectral low rely block singular much singular top good unfortunately gap singular lie gap guarantee break separate nearly low fall towards compare proceed full linear use compute note singular value let k spectral norm square j work k r r svd globally simultaneous subspace span svd rank obtain svd k kk norm perform simple norm span column orthonormal orthonormal orthogonal writing k sketch frobenius proof lemma appendix take outline polynomial tail chebyshev appendix chebyshev briefly runtime simultaneous modify way choose column give accuracy achieve per pca however change
approximate eps assume upper left panel obtain upper panel match blue linearization approximate lead predictive distribution approximation desire x control compute expect compute example polynomial gaussians minimize moment control analytically follow describe analytically gradient search j repeat move shorthand tp controller obtain p eq partial derivative I depend representation derivative linearization sec detail chain derive gradient match difference overview evaluation expensive evaluation analytically analytic base bfgs optimize policy prediction parametrization policy evaluation approximate gps detail computation covariance predict gps linearization mean matching moment predictive linearization computationally advantageous iterate integrate input law contain independence start p iterate expectation obtain multiplication diagonal eq th covariance pair iterate represent integral summation integration unnormalize remain integral determine gaussian predictive see obtain desire submatrix approximation exact alternative x gp mean compute gp around moment describe policy control real ideally take account detail implement control deterministic control moment interesting force plan control limit amplitude account limit differentiable function amplitude final policy eq wave normalize analytically moment multiply illustration tb figure convert figure eps convert unconstrained preliminary policy constrain policy although periodic within half wave preliminary initialize produce single period matter practice constrain control signal execute preliminary unconstrained require close control appendix control present representation preliminary computation covariance preliminary policy offset dimension combination row plus offset predictive drawback flexible however controller equilibrium play target center axis function representation deterministic gp rbf gp signal see support variance right additionally regularization contain squared scale gp eq determine preliminary covariance see uncertainty describe u possess parameter control matrix one offset parameter function per target control center summarize compute p gaussian approximation distribution preliminary distribution u x u cross exploit independence integrate lead distribution p fourth analytically gaussian function gp dynamic representation policy see see learn use distance task naturally lead cost quadratic unity large control width reasoning validate typically however cost target predict early predictive describe sec become therefore unnormalized unity analytically matrix unnormalize either immediate partial x state require analytically exploration illustrate fig eps eps convert pdf wide likely substantial tail region early uncertainty largely uncertainty forward automatic state region subsequent state situation tail region region case automatic prediction far target expect lead control hardware applicability outline alg computational burden collect datum assess term speed light approximation state bayesian apply simulated task cart important framework speed bayesian applicability briefly experimental double link see inner measure length zero control controller double eps convert challenge two experience double parametrize deterministic basis position outer choose width immediate cost unity cart cart run cart velocity measure angular controller balance position middle track controller capable nonlinear feedback see sec learn cart length cart prediction horizon constant control order require move horizontal evaluation moment match sec linearization posterior demand sec learn sec dimension evaluate dimensional computed need computational expensive determination uncertainty input scale dimension convert pdf eps convert pdf linearization moment illustrate effort linearization gp exact generate training datum derivative graph train approximate linearization fast eight trajectory controller initialization cart episode correspond double learn learn policy line start controller successful average success cart experience approximate linearization gp fig computationally demand match computationally advantageous linearization average experience reliably cart relate speed rl method bar solve task figure eps convert figure convert pdf moment matching linearization time cart task without use currently convert task linearization learns successfully linearization posterior function success reliably inference match linearization learn reliably reason linearization reliably task get minima largely predictive confident horizon problem focus solely rely evaluation sec linearization match circumstance convert eps convert approximation suboptimal learn fig inner stage gaussian ideal trajectory multimodal deal model controller trajectory unimodal gaussian explain wide require variability however cost uncertainty lead high choose marginally multimodal minimize expressive approach good approximation rl greatly model close look strictly necessary well necessary successfully nonparametric bayesian discard uncertainty long prediction kind policy gp whereas deterministic probabilistic degenerate propagate state tb success track take long term rl uncertainty appropriately learning learn small close target therefore region uncertainty choose prediction iteratively end predict trajectory essentially vanish policy realistic show eps convert eps pdf human longitudinal maintain toward apply describe controller balance use differ conventional learn controller control account separate keep trial quickly trial run policy colored bar position depend angle controller balance desire configuration sometimes due successfully policy challenging task modification besides define tb eps convert convert eps eps convert pdf control interaction control cart cart cart learn sufficiently dynamic confirm cart delay figure eps convert pdf pose feedback learn controller precision arm learn stack block purpose possess six base three open close six duration make radius system additionally joint configuration use camera external visual tracking block robot camera sensor provide structure camera useful object approximately distance continuous value signal comprise learn center object state initial choose camera coordinate robot similarly measurement camera policy e end camera approximately depend robot desire frequency control discretization block split build individual target b bottom top fig robot train share gps deviation comprise robot arm synchronization delay etc movement learn noise level slightly camera signal ten initial generally stage b learning block close sum controller speed learn insight distribution tb eps convert convert eps pdf eps convert pdf learn pay attention cause relatively system coordinate soon collapse video stack robot forward bayesian gps forward expressive inference control application control engine predict state give policy inference e moment exploit gradient approximation long search efficient require compute gradient exploit instance gps straightforwardly exploit high parallelization trajectory sampling limit small discuss exploration result encourage exploration bound type function also get minima approximate gaussian cost deviation key benefit incorporate planning instead treat transition dynamic suffer gp access approach model could transition control location planning robot obstacle function find path ever initially uncertain conservative stay framework learn minimize kullback distribution robot task base gradient advance rl state success principled reduce model long planning rl initialization prior knowledge demonstrate nonparametric hence fundamental role avoid explicit lead receive ec fp grant agreement grant intel process blue rgb decade since datum drive engineering knowledge reinforcement learning system consume approach typically task expert extract learn explicitly incorporate
eigenvalue norm measure chernoff quite powerful numerous application submatrix fix analysis problem closely basic question bernstein inequality mean chernoff bernstein comparison main tail independent explain chernoff spectral random submatrix matrix chernoff result scalar set nonnegative upper result study sequence independent bernoulli trial success chernoff behave drop fast phenomenon semidefinite meet scalar chernoff chernoff hermitian common minimum furthermore proof easy expectation help average term ambient plus reach decay subgaussian second tail decay variable receive chernoff matrix concentration especially large deviation chernoff derive expectation inequality dimensional spectral content next present refinement practice regard observe bind exceed valuable unbounded especially tail chapter sum semidefinite demonstrate precisely identify constant may sharp every side indeed justify display side right obviously suffice always remove term natural q entry elsewhere binomial representation eq logarithm logarithm come mean tail accurately example arise necessary satisfactory expectation appear estimate concavity hand form nevertheless chernoff numerically sharp situation let positive semidefinite chernoff predict determine event occur instance within draw transition verify chernoff numerically sharp occur chernoff sharp chernoff extreme singular deal linear chapter th express refer elsewhere context consider submatrix define include independently remove column obtain expectation submatrix row positive singular random semidefinite eq zero determine vice versa weakly matrix chernoff calculate lk chernoff next row column random submatrix th entry get share total reflect size large ambient common calculation submatrix need extra column refer positive hand calculation analogous omit reach deterministic still control examine eigenvalue first identity positive right apply reach simplify slightly last chernoff direct close spirit treat norm sum random diagonal chernoff eigenvalue therefore submatrix share plus logarithm combine reach expression enyi basic random whether connect vertex address recall element call simplicity vertex include undirected symmetric matrix indicate distinct diagonal equal zero whose degree matrix convention positive semidefinite eigenvector zero modern second small strictly walk connection enyi os enyi vertex mutually os enyi enyi graph nonzero entry possible permutation vertex adjacency diagonal reflect explain os graph adjacency expression translation definition random verify edge entry reflect degree reflect enyi near enyi goal second eigenvalue laplacian strictly chernoff second small eigenvalue random need coincide isometry mutually ensure semidefinite minimum eigenvalue coincide second eigenvector show follow partial isometry direct expectation apply linearity linearity multiplication inside come diagonal note identity displayed arrive smallest bind small unlikely rearrange os seem worth toward semidefinite establish linear chernoff convex lie connect particular q q eigenvalue interval thus imply second applying simply preserve semidefinite eigenvalue bound eigenvalue begin trace monotone substitute master matrix eigenvalue map observation identify infimum admit make variable argument change tail relate minimum eigenvalue state step state trace exponential respect focu reduce fourth line piece minimum eigenvalue introduce master change infimum tail usual continue reference date bound identical proof combine version chernoff inequality case matrix equivalent form substantially bound bound expectation extend chernoff contain inequality information theoretic tail establish proof problem eigenvalue sum character reason closely phenomenon rank one refinement statement work stress result easy elementary matrix closely different principle mention study long history theory provide natural paper functional clean random column submatrix fix literature inequality tool study random study arise submatrix use chernoff sophisticated random graph appear application matrix concentration om develop paper concentration analyze compressed laplacian subspace eigenvalue compression coincide eigenvalue bound report development concentration concern spectral finite satisfy condition sum matrix inequality expectation statistic bind matrix bernstein inequality powerful tool give researcher approximate introduction chapter several technique randomize replace dense proxy approximate multiplication approximate become bernstein effective study approximation nevertheless chernoff happen often matrix explain bernstein matrix bernstein scalar label bernstein inequality apply large focus scalar tail show zero tail subgaussian deviation analogy simple concern sum demonstrate much last paragraph introduce appear moment consequence variance coincide reach law hermitian coincide variance weak ambient feature explain tail scalar bernstein bind appearance inequality informative well helpful moderate tail decay tail whose comparable tail decay fast bernstein result sum mean dimension sum statistic sum immediate ambient example corner everywhere achieve circumstance bind tail behavior inequality relax weak growth moment hermitian set discriminate tail end annotate matrix bernstein strength insight present require appear omit jensen natural also essential suppose symmetric distribution right side comparable always summary match version appear contain matrix random representation ensure remove logarithm justify natural therefore chernoff correctly appearance iterate rely heavily poisson appear bernstein inequality research object structure simple whose target average independent copy number become tradeoff structured example may need construct obtain offer tool assess subsequent explain let identify desirable example decomposition need probability quantifying approximation choose specific insight construct ensure unbiased lot copy linearity small number approximation complexity incur essential obtain sampling write distribute eq relation triangle control statistic identity matrix second expectation semidefinite semidefinite follow definition fact calculation relation positive likewise matrix approximation matrix bernstein suffice bring error examine inequality per reveal aspect proportional achieve phenomenon ultimately central note valuable error involve linear bound use good achieve tradeoff drop construct analogous shrinkage target family abstraction fundamental suboptimal substantially let approximation practical highlight reason suppose singular construct random sampling satisfy j incur q accurate decay quickly bad paper frobenius acceptable interest occur illustrate approximate desire purpose independent spectral error approximation large require relationship approximation satisfie noise datum subject question decomposition approximately freedom approximation capture dense elegant identify proxy randomly select number entry retain several advantage expensive store dense operate recent due analysis immediate end history frobenius easy matrix convention therefore unbiased estimate nonzero copy linearity nonzero challenge quantify perform short replace placing sparsity determine always exceed assume discover replace achieve error norm nonzero entry proceed randomized bound key appropriate one therefore matrix semidefinite q second reach invoke numerical algebra part computer science basic multiplying system focus develop deterministic operation unfortunately become architecture computational heavily communication resource challenge contrast execution useful modern computer randomize fail concentration end task linear multiply compatible complex matrix product form algorithm divide cost approach consider outer dimension set matrix lot row proxy usual column sum method pick frobenius norm use easily cost probability operation product row separately q require unbiased quite large variance combine copy linearity expectation approximate operation determine inner achieve I multiplication beyond sampling express form way represent approximately simplify spectral equal compute spectral reasonable factor rank mean obtain randomized multiplication substantially matrix method error corollary need bind invoke mean since spectral express kind second moment calculation hold one increase matrix semidefinite order reach estimate parameter let technique sophisticated matrix propose david return value angular point write angle vector convention generalization gram euclidean matrix semidefinite positive definite kernel product replace kernel evaluation regression feature advantageous domain sometimes major insight modern big contain entry construct kernel require universe nevertheless datum redundant kernel redundant replace proxy measure approximation write empirical construction sigma assume random pair want matrix set relation form feature definite usual construct value deal modification randomize corollary sampling matrix construct q let furthermore corollary apply multiplication small appeal number norm state challenge refer find start early intrinsic intrinsic dimension relation finally identify stable quantity consideration imply analysis replace dependence ambient stable require proceed concentration inequality intrinsic challenge chapter ambient early transform origin weight resolve transform tail mind adjust real laplace hermitian nonnegative line require indeed tail spectral indicate eigenvalue exceed return discover bind trace ingredient allow semidefinite term intrinsic intrinsic interval positive semidefinite connect eq fall interval intrinsic extend intrinsic begin extract version transform bind state semidefinite identity strong conclusion side reach q intrinsic recall notation line step assumption tangent follow concave semidefinite q relation inequality concave mapping proposition eigenvalue trace semidefinite increase substitute increase argument intrinsic bound usual hermitian set concentration statement hermitian hermitian intrinsic sequence hermitian value intrinsic bind appear sum whose proof hermitian transform imply hold trace right side examine bernstein hermitian matrix intrinsic dimension intrinsic identify depend substitute discover consequence side latter define convex minimal attain tail four develop finally bernstein hermitian point intrinsic induce complete argument obtain integration control integral argument select combine improve concentration concentration random first reduce similar matrix zhang theorem suboptimal essentially somewhat bound zhang easier contain dimension intrinsic chernoff bound intrinsic bernstein bind argument calculation constant marginally well approach sophisticated algebra derivation begin short explain fact relative devoted way core matrix logarithm serve advance trace function exponential chapter matrix hermitian map convex hermitian contain require support result proof chapter symbol positive number convention capital letter symmetric vertical hermitian matrix letter always refer capital letter unless compatible involve include implicit two shorthand formula involve relative entropy relative relative entropy related arise statistical mechanic entropy definite matrix map chapter concave let variable concave fix concavity see supremum take entropy theorem fact entropy formula trace definite relative side formula desire relative proof latter compactly hermitian side concave ensure define concave observation establish deep set extra structure relative relative positive arise measure discrepancy distribution show entropy matrix maps vector entropy ultimately easy measure nonnegative entropy nonnegative function obtain complete rely idea proposition detail establish elegant bivariate univariate perspective perspective define perspective interpretation ray origin perspective point paragraph analytic convex convex interpolation combination combination another interpolation quickly determine identity write identity definition property remarkable extension construct bivariate perspective reach jensen develop idea represent calculation express entropy substantially involve argument investigation establish adequate relative subtle argument convexity relative construct hermitian matrix trace trace function hermitian contain eigenvalue first trace weakly induce function preserve recall relation dominate semidefinite hermitian result range domain convention semidefinite quickly monotone weakly hermitian weakly increase concentration special monotone monotone hermitian continue write argument hermitian eigenvalue list order family orthonormal inner interact complicated way scalar let hermitian eigenvalue contain k matrix linearity identify inner scalar prove state matrix nonnegative real lift formula sometimes nonnegative toward relative entropy logarithm demonstrate convexity semidefinite along monotonicity logarithm decomposition presentation base formula representation logarithm integral logarithm scalar simply integral integral inverse seem thin air motivate monotonicity logarithm introduce abstract monotone hermitian contain fact operator monotone point easily weakly line operator monotone line monotone interval monotone somewhat fortunately monotone inverse monotone define definite rule inequality finally semidefinite relation direction combine logarithm monotone logarithm monotone matrix demonstrate logarithm preserve positive next investigate abstract function real hermitian function monotone cone somewhat family operator convex definite lemma suppose calculate essence elimination bring original block leave positive extract result proof apply top fact finally verify logarithm argument base logarithm line definite invoke integration preserve order statement average automatically average content jensen spirit hermitian matrix whose relation remarkable call combination average rich general average hermitian decomposition form call average hermitian let property convex preserve positivity combination convex though function actually jensen inequality let hermitian identity introduce lie fall apply combination unitary unitary unitary matrix omit label restrict diagonal express convexity work unitary key apply inequality return identity block average look block equivalent block line semidefinite relation interval complete first relation function unitary identity formula preserve block diagonal block entropy perspective go perform perspective property semidefinite scalar perspective function bivariate perspective definite refer denote root remain make perspective perspective matrix hard perspective definite operator jensen convexity perspective interpolation parameter scalar combination eq interpolation observe decompose identity construction express convex give access jensen definition introduce inequality reach relation finally matrix argument entropy matrix simplify restrict attention simple kronecker hermitian hermitian first property fact construction kronecker zero kronecker q kronecker product bilinear usual calculation identity kronecker product importance note kronecker two hermitian matrix hermitian matrix kronecker positivity positivity let definite observe usual unique hermitian must semidefinite discover invertible discuss logarithm role elegant logarithm kronecker valuable logarithm kronecker product exponential equal formula rely apply mixed kronecker complete simply choose identity trace kronecker hermitian pairwise preserve semidefinite valid hermitian column argument course operator evaluate rule arithmetic introduce kronecker calculate kronecker represent relative let tell perspective q preserve conclude relative entropy draw variety range article source book recommend major result paper resolve conjecture concavity certain motivate quantum state uncertainty system control presentation idea corollary difficult concavity complex paper approach author prove concavity trace function implication relatively easy see matrix differ usual quantum mechanic quantum include change proof adapt directly paper nevertheless idea date paragraph relative construct relative divergence entropy divergence say construction suppose bregman divergence square divergence common introduction bregman divergence see ar divergence recognize relative paper recent divergence divergence classical result mechanic monotone characterization operator monotone quantity monotone interval fact scalar derivative monotone function pick develop theory monotone somewhat develop operator monotone convex formula operator write write nonnegative integral closely relate formula monotone logarithm motivate recommend book book monotone jensen establish decade book unable identify idea important direction pair pair prove entropy author kronecker integral construct definite matrix particular perspective year introduce quasi entropy involve influence presentation convexity operator jensen derive relative analysis remove argument kronecker product operator identity interpret consequence definition kronecker right presentation heavily skew chernoff version matrix hoeffding bound inequality value convexity relative still deep convexity chapter contain argument thompson inequality establish roughly martingale bernstein advantage study random show type article tail eigenvalue sum independent matrix chernoff semidefinite type sum establish stein pair inequality matrix sum moment arguably simple markov thesis lead satisfactory logarithmic exponential stein inequality primary estimation roughly bernstein inequalitie unbounded matrix extend hoeffde combine paper combine chernoff chernoff bind random positive semidefinite replacement role establish inequality derive inequality consequence inferior work concentration bound depend ambient develop variant argument control matrix rather ambient contain semidefinite dependence control essentially describe dimension intrinsic variance argument adaptation author present refined obtain intrinsic ambient dimension parameter important development matrix laplace thompson inequality inequality identically concern bernstein completion bernstein concern matrix overview moment concrete literature strong exponential inequality unfortunately typically abstract difficult recently von algebra finite equip power version inequality prove inequality sharp establish random moment growth describe fully argument appendix sum thm thm proposition thm thm thm thm recent mathematic classical expert decade arithmetic aim describe result page matrix discuss herein present coherent exchangeable omit seem broad researcher interested reader article annotate describe influence researcher great friend would like people improve reader inform I manuscript include anonymous suggestion give feedback final acknowledge office air award fa fellowship complete apply mathematic like institute foundation motivate begin connection computational mathematic application examine result assess presentation lie group orthogonal multivariate statistic another appear study behavior algebra solve system linear von arise take estimate procedure typically recent year nuclear early limit energy spectra slow random matrix appropriate quantum reaction random book lead distinct random field os model random theory arise throughout mathematic branch science several distinguish computer serve phenomenon reflect author interest mathematical problem compute develop fast multiply dominant singular input appear explain practice aspect accelerate matrix replace proxy elegant produce proxy entry rescale entry paper contain related idea play fast learn one subsample nystr approximation analysis template dimension many dimension random paper mathematical computer science appear variant idea combinatorial optimization relax constraint solve back round compressed object relatively freedom compare ambient able object refer central multivariate study property typical statistic area typical signal analyze identify column submatrix matrix structure analysis problem orient discriminate stochastic block one describe community assume individual community individual refer quite algorithm extract hold generally random model statistical procedure high theory key wireless matrix recognize may coincide reality allow sense generic intrinsic field phenomenon area random role edge vertice expansion property adjacency argument numerical bad elimination solve numerically however stability arise phenomenon probability matrix condition elimination state banach slice close ball turn slice dimension property quantum information information theory channel property hold capacity result random random theory challenge experience take intensive effort almost everything beyond arithmetic working variety special attention familiar maximum hermitian eigenvalue maximum hermitian eigenvalue probability eigenvalue something spectrum one distribute ask question act geometry three attempt bound application result expect relevant problem branch random fundamental principle describe introduction hope complicate main positive semidefinite star refer transpose operation position zero elsewhere word record covariance statistical imagine eq unbiased estimator know adjustment incorporate fit paradigm type tool theory substantially parameter express variable address substantially question elsewhere establish study extend explain kind return simplify variable center subtract second random sum piece usually together bernstein inequality independent sum proof refer yield substantially truly scalar bernstein inequality independent sum bernstein develop present give detail interpretation bernstein uniformly introduce sum denote proof appear version three salient mean hermitian coincide factor scalar limit include quite challenging expectation bind aspect far contain interpretation mean random introduce study depend assumption bound hypothesis relaxed variant appear random identically apply uniform variance uniform norm triangle hermitian coincide determine fact preserve drop reach square hermitian arrive need estimator invoke bind attain case qualitatively sharp argument corollary building showed essentially overview analysis result little every suppose introduce appear bernstein sometimes significant come factor give describe behavior distinguish tail useful tool practice collect precise book available concentration bernstein familiar exponential many admit focus independent describe random rectangular toeplitz entry chapter rademacher rademacher write fix independent thing random sign sign arise treat chapter chernoff bounds chernoff decompose whose subject uniform submatrix laplacian bernstein concern matrix spectral include multiplication feature paradigm chapter material inequality spectral depend ambient dimension illustrative concrete extend concentration quite useful mention annotate establish concern independent subject inequality extend tail bernstein improve positive semidefinite dimensional interesting obtain concentration martingale hoeffding bind bound norm view martingale bernstein technical explain bound matrix martingale set moment heavy tail reflect matrix simple form polynomial lead inequality annotate include moment inequality another establish argument exchangeable chain concentration moment inequality reproduce stein stein exponential random advantage exchangeable build exchangeable elementary approach take effort inequality material modify unfortunately seem student researcher mathematic random minimal algebra classical elementary beyond good chapter material tail sum matrix significant early example concentration major work concentration inequality practical random small eigenvalue appear literature include optimality necessary phenomenon basic result ease theorem concentration depend annotated contribution hope organization chapter contain background need develop concentration matrix chapter concern chapter introduce chernoff application chapter bernstein intrinsic conclude resource concentration make presentation smooth follow article almost appear chapter elaborate main overview extent concentration inequality careful cross reference able proceed discussion material concern behavior review especially us field write field array complex part depend specify really matrix none require essential consist complex equip complex symbol complex transpose product induce equip inner write complex space entry multiplication space algebra multiply frobenius frobenius induce topology matrix n symbol eq open define norm topology notion open write vector matrix add subscript instance identity basis linear vector write equal standard basis zero elsewhere letter unitary reader prefer prefer orthogonal analogous write hermitian hermitian multiply frobenius bold letter symmetric around represent hermitian hermitian number unitary matrix completely unique permutation eigenvalue denote algebraic eigenvalue extreme map homogeneous careful pass scalar map rarely hermitian aside arise usually eigenvalue weakly confusion prefer set read hermitian term square denote trace existence eigenvalue trace hermitian matrix valuable relation connect frobenius norm calculation semidefinite hermitian equivalently hermitian role nonnegative positive hermitian family form subset combination positively homogeneous geometric describe consequence semidefinite close convex begin consideration definite matrix form cone real semidefinite nonnegative preserve importance hermitian matrix general dimension semidefinite property semidefinite trace extend hermitian eigenvalue let matrix entry matrix confirm sensible fold hermitian whenever power logarithm logarithm definite matrix non power matrix immediate important consequence function theorem hermitian eigenvalue power expansion series real eigenvalue generalization value fail nevertheless inequality extend semidefinite transfer rule hermitian whose decompose immediate allow invoke hermitian equivalently series expansion hermitian always exponential monotonicity dimension establish logarithm logarithm functional q valuable logarithm preserve definite dimension stress monotonicity decomposition admit nonzero value order square decomposition square conversely extract eigenvalue decomposition expression expression property singular unitary hermitian definition hermitian matrix vector coincide q identity application lower express rank stable rank continuous power hermitian matrix hermitian hermitian clear hermitian discover coincide invoke repeatedly justify first column unitary appear calculate norm coincide construction identity inequality eigenvalue derive depend norm matrix singular norm schwarz inequality focus connection prefer abstraction unnecessary frame sufficiently justify expectation limit valid broad circumstance particular position help helpful clarity scalar letter gaussian variable letter take measurable think letter hermitian letter subset simply expectation linear matrix identity far comment markov nonnegative random obeys central tool concentration inequality jensen convexity eq form cone matrix preserve deviation extension concept hermitian value interpret column write value use number eq quantity mean wish rewrite matrix statistic random finite hermitian familiar statistic sum inside statistic relation identity suppose random dimension semidefinite hermitian variance hermitian valuable reduce variance scalar define definition coincide original deeply hermitian direct calculation value second definition matrix maximum diagonal block hermitian interact independent repeat calculation lead summary sum arise everything establish discussion reader treatment matrix analysis excellent book book reference book introduction two process book book comprehensive useful book aspect modern extremely survey work classical comprehensive treatment offer book matrix theory solid chapter core reader concentration may move concentration bounding recommend satisfactory extension allow transform argument concentration elementary purpose chapter application fluctuation hermitian inequality eigenvalue hermitian develop independent challenge arise present idea overcome result allow develop abstract chernoff bound bound heart laplace present hermitian note expectation may varie aim expand formal coefficient refer matrix moment hard set laplace transform start hold tail eigenvalue hermitian extreme classical fix eq hold monotone last identity depend hermitian positive eigenvalue achieve prove monotone identity inequality discussion convexity trace adapt hermitian analog scalar eigenvalue hermitian matrix q positive eigenvalue state relation jensen proposition draw final inequality depend trace positive laplace transform set laplace method sum decompose case subtle indicate thing sequence satisfy multiplication relation sum scalar relation variable imagine perhaps unfortunately hope subject hermitian situation improve result thompson physics analogous eq consider real relation extract logarithm look sum hermitian hermitian q identity fail nevertheless admit satisfactory scalar proof deep consideration discover convexity trace exponential fix hermitian definite analogous result describe complete let consequence remarkable valuable transform section involve hermitian let random hermitian interpretation logarithm functional expectation generalize fundamental approach matrix finite equivalently rule expectation remain random matrix hold iterate invoke formulation follow substitute finally general sum laplace develop apply master matrix sequence hermitian furthermore eq proposition nevertheless sum rectangular hermitian present hermitian include historical establish trick researcher work concern transmission information version substantial number major appear appear generate technical repeat thompson argument hermitian case coincide identically fundamentally weak bind worst unnecessary detail beyond argument appear develop effective thompson establish bernstein analogous bernstein value specialized bernstein constant concentration approach base introduce article recognize content lemma master tail article detailed discussion thompson see get appear seem weak certain advantage however matrix set subsequent research martingale version work report tail independent matrix closely old theorem moment contain extension control fix variable researcher moment scalar random admit overview mention moment play concentration tool quantum mechanic seem distant area study via quantum systems thompson inequality major quantum mechanic book physical book thompson three combination spin first establish important trace main establish quantum system chapter present set fix variable formulation surprisingly range simple precise finite fix independent spectral matrix look express use matrix gaussian represent build attack classical toeplitz idea treat sum recall take new problem spectral sign entry begin overview series bound subsequent describe concentration example substantial part conclude sequence standard routine scalar transform demonstrate turn directly rademacher dimension finite variable appear message subgaussian tail whose decay follow coincide formula bind reduce case nothing subtle scalar see illustration mark dark red vertical coincide tail decrease subgaussian variance answer section claim series argue quite norm available especially begin indeed spectral jensen step use integration tail integral explicitly complete ask exhibit construct series right correct see dimensional gaussian lead logarithm sometimes major computable factor chapter moderate type series remove technology
deviation generate value increase use call pair rank induce comparison improvement select sorting improve noisy small active furthermore seem active keep survey refer aim website image option date million outcome collect purpose result outcome item knowledge generate pairwise despite comparison remain item proceed bt comparison outcome model passive approach experiment passive use realistic figure systematically observe outcome average fitting poisson considerable fraction express opinion difficult probably look preference type large comparison result outcome outcome rare compare active bt necessarily hold bt fit bring quickly passive example reach collect assume order denote observe inconsistent ranking element pl formalize wrong decrease observe bound q di operation item compare misclassifie misclassifie item correctly sort represent precise next operation misclassifie constant misclassifie therefore argument consideration partition proceed main proof partition operation cause chebyshev inequality remark know need comparison convenience different constant value maximum consider operation recall chernoff let call tree randomize step item middle least partition subset condition though middle unlikely small least leaf recursion therefore least item lemma never exceed total item exceed theorem survey pairwise noisy active black sorting enable efficient practice perform systematically centrality recently rank aggregation provably convergent iterative item collection comparison diverse range rank player game preference recommender arguably comparison simple human attempt question aggregate million cope inconsistent noisy outcome bt model bt item noisy item distant item already distant strategy agnostic strategy explore active sort recover basis efficient approximate run sorting induce item rank sort repeatedly budget sorting ignore set pair rank outcome develop likelihood estimator bt start work centrality aggregate finite bt analyze generalization first interpret likelihood ml enjoy essentially provably convergent compute induce bt choose advance investigate strategy sort label bad mistake finally theoretical finding speed outcome sort notation use throughout without loss generality denote preferred preferred define say informally bt observe outcome inconsistent rank decrease intuitive define particular way make parameterization probability prefer multiplicative assume sometimes grow number parameterization model give w parameter parameterization enable make intuitively uncertain outcome concave tractable algorithm centrality aggregate comparison bt square comparison select result mse insight centrality relate among contribution way extend comparison centrality estimator present minimax minimax optimal recover observe minimax instead novel analysis estimate way guarantee mse bt far parameter include impractical dependency ml bt unlike ranking accord ml estimator variation justification comparison noisy model sufficient one noise development provide bt consider datum bt set bt ml pair provably convergent iterative estimate ml centrality towards preferred consider factor ensure stochastic walk diagonal node transition represent unbiased intuitively towards stationary k interpret contract probability reason error interpretable proof bound stay error move relate estimator one centrality satisfie balance compare hand likelihood claim satisfy ml consider approach lead directly expect scale approximation centrality alternatively absence knowledge fall centrality furthermore stationary factor ml w markov essentially let key insight error stationary suffice comparison theorem additional factor recover alternative completeness satisfies w q show pair vanish ml factor newly enable markov via equation error available procedure maximum adapt centrality centrality describe chain irreducible proof mapping soon self irreducible ml estimate define lastly arc similarity maximization solve relaxation however mm whereas gradient comparison random think seek try rank bt consideration change let assumption range grow bt dependency induce easier distant outcome noisy well even actual grow ultimately easy bt arbitrarily make realization bt measure rank one noiseless necessary sort ranking question characterize produce sort operating section sort select uniformly two always rank outcome claim algorithm comparison non ranking read pre traversal notice plug bt theorems average produce bt model ranking sketch proof sort comparison mistake
conjugate pick latter optimization refer shown convert dual high solution allow indeed yield clearly unique verify coordinate point denote point generate minimize coordinate sequence variable start compute hypothesis expectation process govern eigenvalue order normalize eigenvector obtain distance great showing rate generalize denote continuous strongly comprise smooth strongly quadratic minimizing minimize observation since stationary canonical b correspond initialization form far assume precise seem restrict respectively inversion evaluation comment equivalently correspondence soon play polynomial optimization matrix radius sake brevity characteristic inversion lastly minimizer consideration assumption consistency say initialization say broad characterize rule linearly first instead precise descent newton also degenerate vanish ball detail descent algorithm rewrite act repeatedly minimize row popular mainly stationarity requirement fail extension cyclic piecewise future conjugate similarity descent let x clearly form g stationarity imply minimize assume arithmetic computational scale iteration computational inversion time notice coefficient inversion execution inversion level accuracy rigorous fact analogy respect accuracy measure employ theorem provide characterization root long dp root characteristic polynomial evaluate remark constant may consideration minimizer second derivative sake clarity initialization subtle issue regard like point say however employ fortunately although strongly q combining depend distance logarithmic setting equivalent bound imply adequate section presentation case present useful polynomial case finite explain inversion conclude use establish efficient specification see reveal turn extremely characterization condition hold matrix inversion consistent recall consecutive side yield rearrange hand consistency generate convergent limit point formalize system hold precede extensively illustrate significance simplify deterministic free characteristic modulus root ideally like consistency condition maintain seek imply ingredient choose inequality q plug bx apply lower bind e simplify optimization generalization accord derive low root b brevity dependency upper triangular denote order root characteristic polynomial consistency derive modulus root end find section otherwise hold subject obtain design presence see assume consistent root eq reasoning optimization arrive let motivate effectiveness state relate inversion bind compute know super quadratic possible root q consistency regardless strategy balance approximation iteration put differently various restriction inversion turn polynomial rise low quadratic inversion bound optimization scalar meet much wide inversion depend derive low smooth case already hard matrix turn positive meet sake eq maintain scalar inversion split range range low q condition x good scalar matrix overall exist low tight turn suitable reach spectral decomposition minimizer restrict attention coefficient efficient attain see optimization namely inversion disadvantage guarantee convergence satisfy imply since algorithm extension radius coefficient radius us characteristic factor consideration inversion take follow scalar maintain function path natural economic polynomial hope root hold follow equation substitute get linear multiply remarkably enough table equation extension heavy polynomial relate consideration strongly could argue recover specification say necessarily fortunately coefficient reformulate substitute original sequel briefly property canonical extension path optimization nx denote scalar thus rearrange plug q function extension behave answer essentially minimizer unlike guaranteed initialize close enough converge smooth investigation precise principle kind unconstrained sequel employ show theorem tight end lemma modulus root polynomial uniquely attain eq matrix admit state fix need factor characteristic eq accomplish orthogonal positive exist coefficient accordance theorem whose characteristic optimal optimization parameter decomposition use grow yield iteration spectral structural impose polynomial analogy state polynomial spread derivation numerically optimization oppose employ coefficient eigenvector require function vs rate comprise close theoretical gain counter initialize quadratic imply optimization must x eq b somewhat bind optimization algorithm nesterov consider regularization term case shape spectrum preserve demonstrate nesterov low space consequently eigenvalue order derivative relatively distant quadratic fast coefficient shape spectra applicability real simple idea give application analyze allow prove elementary lemma determine recurrence application space converge series note part seem algebra matrix eq modulus exist trivially diagonal index respectively may magnitude row follow equality plug h inequality derive scalar derive bind equation term tend entry zero equal precede constant equation yield u sum power exist claim define direct implication establish update equivalently express new euclidean q mapping convention equivalent regardless initialization improve functional coefficient matrix follow discussion furthermore expression iteration n realization iteration matrix convention multiplication order product multiplication expectation side r h update suppose thus consequently convergent question fast need x previous initialization exist satisfy complexity eq note space sufficiently q govern radius suppose f x f x radius equal radius characteristic combine corollary proof characteristic iteration apply elementary determinant polynomial sake omit precede algorithm implication immediate consequence characteristic polynomial initialization I sake omit dependency moreover verify eq plug equivalently denote degree prove fundamental root root q eq write second yield conclude part reverse triangle remark whereby eq conclude real prove convergence rate inversion optimization whose inversion general dimensional dimensional mx definite note use argument positive satisfy precede rest carry scalar ne develop novel smooth strongly algorithm deterministic focus quadratic recursive turn reveal connection whereby whereas low natural lastly polynomial novel systematic nesterov accelerate rather one solid motivation descent descent mathematical interested solve form eq problem science economic readily express far reach say solve approximate various along convex precisely continuously e wide interesting fast kind problem say answer otherwise address nature computational accept theoretical analyze seminal propose information regard impose resource show employ receive oracle obtain e start view query attain descent seem trivial query exceed moreover class attain e require intensive quite common gradient variant simple indeed quadratic use oppose optimization utilize accelerate heavy ball see sag cyclic piecewise accelerate sdca inspire boundary admit stationary rule canonical algorithm interpret algorithm linear essentially magnitude bound analyze procedure analyze lyapunov mathematical however primarily derive magnitude eigenvalue work low bound merely root maximal modulus absolute root polynomial condition polynomial polynomial correctly rate purely theory modulus root although vast bound root g good bound bound radius adequate consequently new tool argument derive heavy polynomial chebyshev polynomial g formally optimization iteration execute efficiently obtain partially early whose
distribution c generate albeit additional constraint constraint addition normal point associate b broken part overlap obtain alignment synthesis overlap decide point simulate identifiable slight use affine evaluate across alignment accord base outperformed emphasis region component propose variance scale measure propose elastic within measure alignment dataset neighbor bandwidth tune base error clear observe specific dataset outperform scale offset difference trend affine discrimination ratio propose elastic contribute call compare difference elastic distinct generate classification quality elastic difference include subsequence move merge penalty compare elastic overlap measure outperform seem offer perspective offer specialized advantage competitive elastic loss elastic suggest dataset even elastic evaluate dataset dataset good ensemble classifier demonstrate series ever literature propose difference evaluate incorporate ensemble difference measure figure classifier elastic difference dataset measure base nn error method rank dataset rank rank note difference compare elastic describe figure nn absolute exceed might far scale regardless appropriate offset iterative manner update accordingly normalization simulation normalization difference mean evidence well classification want property alignment scaling offset dependent occur motivated demonstrate width crucial sensitivity width sensitive tune ever dataset extract subsequence delay wave medical interestingly tune region width roughly offer handle baseline motivate example thereby nn outperform alignment runtime complexity share recall convergence alignment compare dataset database dataset database alignment calculate randomly select series computation largely follow tune nn exceed almost actual differ multiple dataset require reflect cost large alignment affine emphasis variant set pointwise alignment reflected match variant offset amplitude emphasis combine advantage affine apply globally locally outperform dataset nearest neighbor associate alignment recognition alignment arrange match analysis series finding point match point financial stock market gene activity match time assumption illustrate purely alignment variation connect line visually focus series comprehensive survey demonstrate produce context result context two time amplitude scale offset bias well offset alignment simultaneously fall temperature variation environment alignment temperature subject offset variation figure series peak series normalize extent visually consistent scaling offset rotation mapping impose offset alignment model mapping time transformation arise desirable accommodate scenario substitute pointwise distance interest potential determine characteristic shift degree overlap alignment desirable affine emphasis local one local manner affine offset emphasis subject temporal location place emphasis section heavy amount emphasize short offset temporal variation time scale offset series slight movement objective correctly align rest review method conclude paper easily interest point match assume mention otherwise match time subject search optimal alignment minimize subject monotonicity boundary monotonicity step refer difference programming reduce find alignment constrain problem scaling offset similar way assume run compute time place weight potentially accomplish substitute pointwise measure width distance summation programming manner formula turn observation applicable correspond time width discuss crucial achieve alignment alignment offer target global model scaled offset time emphasis goal minimize subject finding solution em apply convexity set derivative v gs c gs g c well alignment emphasis h scale offset find minimize eq q follow match eq constrain dynamic utilize manner eq backtracking apply appropriate attribute element update element follow observation update thus furthermore complexity introduce bandwidth detailed bandwidth length evaluate reflect reflect stop parameter section unless complete evaluation tune tune
q next rate produce compare coordinate project coordinate even consider asynchronous point old method benefit match assume geometric rate author rate coefficient linear coefficient precise result worse depend broader strong several define project employ project define ready admit relax begin follow structure strongly recently show satisfied convexity hold estimate global property prove fx theorem gap auxiliary eq iteration depend coordinate compute otherwise optimality know q hold remain know x therefore auxiliary imply give hold conclude fx fx x proof optimality use projection plug expectation easy ready choose latter read x w vector expectation observe function q notice minimize q combine gives theorem usa engineering framework descent feasible sdca lasso problem framework linearly duality sdca dual interested convex feasible produce hold every yx include cyclic fit randomize first become question extend randomize give randomized version formulate inexact projection algorithm gradient corrupt fit notation enjoy convexity vector satisfy weak convexity property weak convexity smoothness wise continuous denote continuous lipschitz projection operator coordinate learn literature framework also detail goal label find minimize svm hinge loss smooth j j smooth special strongly double lasso reformulate machine parameter use loss fit square error solution prove feasible enjoy convergence begin locally global property show rate author asynchronous weak recently show smooth global convexity property contribution framework show fit randomize previously expectation duality gap sdca duality gap section compare result briefly review use duality converge linearly sdca apply dual svm brief summary cover gradient cyclic inexact classical method good knowledge framework popular minibatch method fit r randomized descent random property exposition r difference r whereas r weak must iteration deterministic hold framework see later relaxed theorem option r minibatch analyze describe input matrix minibatch option x k option function coordinate strongly problem dual neither assumption hold coordinate descent remark compare cyclic rule coordinate f w nr however
reproduce system capture feature fine picture decompose color distribution equivalently color construct color optimally color color dynamical system statistically sample prescribe distribution consequently equation material lyapunov imply sensitive dependence dynamical system investigate particular convergence mcmc compare sampling hamiltonian mcmc multi modal dimension metropolis hasting require proposal prescribe arise htb supplementary text trajectory generation dynamical system prescribe dynamical trajectory track visit dynamical dirac delta integral rd rd spend coverage domain region distance spherical fourier basis satisfy belong rectangular describe constant easy control describe fouri control maximum equation dynamical evolve trajectory color produce computation potentially multiple different lyapunov signature lyapunov spectrum initial divergence trajectory trajectory rectangular region equivalent picture identical pixel outline equation give eqn note computation lyapunov jacobian analytical jacobian eq q compute wave particular dynamic dynamic jacobian determined along approach compute dynamic lyapunov lyapunov exponent note display dependence initial condition however final statistical invariant hamiltonian investigate extensively rare extensively machine form research hasting slice modal compare compete provide high computation construct fundamentally traditional chain successive pick history trajectory brief description metropolis hasting hamiltonian mcmc comparison hasting popular accept reject hasting tuning propose base distribution much trial error pick distribution modal example hamiltonian hamiltonian hamiltonian resample perform metropolis hamiltonian explicit proposal momentum dynamic typically perform hamiltonian metropolis sample avoid hamiltonian underlie target momentum variable hamiltonian one momentum sampling graph uniform essentially point constitute slice advantage one implementation slice software metropolis hamiltonian pick normalize dimension second gaussian distribution deviation correlation see pick high analytically metropolis hasting proposal eqn mcmc eqn respect simplicity use predict figure explicit dynamical lie reject trajectory pick rejection error modal significantly hamiltonian slice additionally x metropolis hasting hamiltonian mcmc approach require proposal markov monte sampling primarily address additionally development rough integral switch present evolution run euler trajectory trajectory produce additionally correspond evolution capture movie htb vs time lyapunov imply modal use comparison hasting htb convergence sampling hasting slice htb frame superposition trajectory evolve htb sampling first superposition blue frames red green frames compose evolve htb evolve frame superposition frame trajectory evolve frame superposition red green blue red green compose trajectory evolve movie superposition green frames green single trajectory evolve big first frame superposition green compose single evolve superposition blue red green frames color evolve cm united center ct united research center berkeley dynamical da water system aside aspect picture novel reproduce control dynamical property color picture capture refine beyond reproduce expect quantification scalable develop acceleration monte design dynamical availability seem manner modern fundamentally human picture speak pixel determine appropriate level intelligence human objective human capturing randomness prescribe relate theory ergodicity system time average function equal spatial average individual trajectory color desire dynamical exhibit supplementary picture irrespective lyapunov potentially robot applicability challenge lie heart probability mcmc likelihood tight slow complex modal machine big supplementary manuscript present hamiltonian slice low construct euler scheme beyond capture fourier rectangular computed take wave picture force converge great give scale dynamical achieve weight fine one fouri fix maximum higher additionally run burden due fourier underlie mathematical reader supplementary image decompose yield value color obtain color prescribe time color demonstrate
detail completeness implementation simplify equation shorthand quantity start macro remark wolfe community theoretical property investigate application learning focus fw suggest study promise medium benchmark svm classification frank wolfe hereafter fw researcher fw enjoy powerful iteration execution improve rate basic work iteration advantage super complexity result suitable learning statistic bioinformatics classification fw hundred thousand thus provide promising medium focus previously kind benchmark show able accelerate average speedup dual tolerance parameter strict running consider random elaborate advantage drawback general overview fw modification theoretical property examine fw numerical assess fw continuous idea fw exploit iterate direction fw algorithm define k either fw sake primal sublinear optimal algorithm satisfy give computable criterion motivate primal iterate bind well furthermore tolerance clean endow theoretical standard fw exhibit tends become orthogonal date consist algorithmic variation add variant kind frank wolfe method modify fw alternative eq good descent pairwise swap fw iteration value prove enjoy analogous algorithm extensively discus refer literature option improve fw iteration conjugate fw fw hull include paper due variant gap obtain iteration another fw case advantage would possibility explanatory reach approximate observation successfully fw solver motivation mainly fw solution approximation fast large svd prohibitive motivate svm significance allow comparison effort propose traffic fw iteration good scheme investigate basic incorporate average fw iterate via inner cycle south anchor anchor south circle fill fw black dash pos color thick dash fw thick swap current fw direction use obtain look iteration fw tend orthogonal figure consist extra fw case depict directly apparent approach advantageous function scale red dot circle black label pt acc acc pt fw gain dataset cpu count due employ comparable appear issue accuracy possibly capability fw promise solid solver experimental preliminary provide example application fw machine variant
evident hyperparameter automate search community believe benefit greatly project g circuit cancer grant fellowship medical project van cancer via plan bm image rgb frame language title hyperparameter challenge capture element common feature choice hyperparameter complex theoretically sound strategy focus capable capture give coherent within ability predict give discrete range neural parameterized appropriately user learn various result hyperparameter manually hyperparameter predefine approach impractical number briefly inherent combination search current available software key balance learn choose datum poorly unseen overfitte low trade strongly biased hyperparameter control instance via network learn task include mean construct involve parameterize parameterized hyperparameter formalize tuple return give algorithm often split hold obvious train ensemble subtle instance affect training exhibit inherent randomness initialization estimating sometimes typically function usefulness setting optimum fortunately many search probe optimum good post smoothness usually hundred step many case usually type integer hyperparameter architecture ensemble task highly complex space hyperparameter conditional optimize hyperparameter layer induce number particle genetic couple anneal algorithm surprisingly randomly establish relate sequential base use variant
object ds discussion propose crf object characterize sequence connectivity dynamically base inter frame optical flow adjacent facilitate spatial context probabilistic ds crf model facilitate pixel object time target experiment real world capability crf exist track propose ability change inter establish optical connectivity layer deep meaningful temporal change inter frame optical connectivity adjacent maintain prediction factor incorporation object also handle shape ds crf incorporate inter optical via descriptor inter connectivity within video sequence furthermore explore extension crf relationship boundary aim explore ds crf video energy work support science development author z ds crf w work formulation derivation ds crf cm cm department computer mail abstract field crf purpose tracking ds particular adjacent inter layer connectivity dynamically optical flow incorporate context dynamic structured ds allow accurately change greatly well within scene experiment surveillance multiple ds approach tracking tracking introduction predict structure interesting important number object video challenging dynamically change drastically early tracking state kalman filter state observation motion follow behaviour filter address make filter kalman resolve non behaviour motion address behaviour lot use particle filter arbitrary filter difficult especially track appearance change drastically dynamically time recently significant object track generative probability state relax independence assumption generative method study base et al within video must predefine spatio model object object component correspond temporal constraint propose purpose tracking image pre segmentation temporal dependency conditional estimated frame subsequently refine subsequent frame exist segmentation video foreground time different crf similarity spatial continuity motion crf resolution crf approaches limitation crf limited great frame increase limitation object recently concept structure facilitate modeling without complexity incur crf deep crf state improve inter layer et crf linear chain replace sum yu al structure crf compose layer layer track video contribute predict frame efficacy crf model concept deep structure discriminative introduce ds crf state spatially characterize inter dynamically inter optical spatial ds develop efficiently stage situation large change material object classify pixel foreground object goal characterize discriminative ds framework tracking object level frame characterize frame form structure conditional adjacent tracking follow detailed ds crf field amongst use crf crf measurement require sort independence commonly formally undirected vertex crf markov except neighbor respectively crf normalization essentially call partition clique potential maximum crf relationship amongst clique number feature vision segmentation classification undirected crf although early relationship amongst track play role crf inconsistent importance appropriate crf predict base frame without movement two frame see prediction result poor crf object poor tracking tackle appropriate track al incorporation optical crf modeling relationship visual approach promise crf motion frame position motion dynamic change handle motion tracking benefit manner address issue crf crf state model along motion dynamic scenario deep structure random propose detail layer ds crf establish dynamically temporal observation incorporate ds present graph representation ds crf characterize model conditional normalization intra clique feature inter optical target object inter state spatial clique state correspond motion inter connectivity temporal mean inter layer clique inter gray object movement frame feature inter connectivity incorporate crf optical flow target appearance crucial velocity adjacent frame optical optical motion upon sequence specifie move motion optical flow pixel inter frame optical observation utilize unary appearance feature describe appearance appearance unary shifted velocity shift base target scene imply strong target word target frame add target change rough segmentation enforce segmentation frame ise utilize problem incorporate feature function estimate training propose ds maximize log concave global derivative parameter respect exact iteratively ds crf training belief train ds crf graph decode determine state maximum eq ds crf tracking crf across frame annotate velocity two ds crf starts frame crf optical flow optical frame crf rule context object motion dynamic spatial within ds crf model describe base learn ds crf computational complexity crf tracking utilize object field indicate object pixel temporal observation correspond binary issue label suit object appropriate object issue target accomplish determine component field determine detect target matching evaluate ds crf purpose object tracking perform understand analysis different involve motion capability ds handle motion set video human move ds crf handling drastically shape object acceleration ability track object different time simulated acceleration motion object move constant velocity acceleration well frame motion motion track uncertainty motion object appearance life video target ds crf scenario object drastically shape different move used evaluation illustrate capability sequence scene bottom capability sequence scene become person scene illustrate capability top third top scene bottom exist tracking shift tracking maxima achieve previous base measure track tracking detector semi
coincide finally coincide specie sample py us genomic datum express tag est obtain free widely biological consider library cell previously constitute sample c library library count discover step sequence estimator complete specification mention clear estimator exhibit value rely model jointly estimator note exhibit diversity already fix additional sample lie basis library basic py discover gene step decade frequentist consistency major generally accept see study suitably exchangeability frequentist term shall asymptotic kind asymptotic achieve modify among start fix concept first datum assume term type natural neighborhood achieve neighborhood support ensure furthermore desirable study gibbs mixture model gibbs process condition hold suggest type discrete come distribution serious really generate design must w compatible gibbs primarily interested investigate type come strategy show weak say checking identify investigate asymptotic behavior explicit allow guess neighborhood quantity section sample exchangeable choice clearly sure hand discrete henceforth shall stand turn asymptotic discover gibbs type constant eq comment regard order type prior expression regardless mild distribution guarantee trivial exclude discover converge assess consistency limiting think bad concentrate guess learn place visualize case py already predictive distribution imply unless correspond see concern prior focus occurrence phenomena mixture py sufficient state behavior show sufficiently extremely mild assumption mixing require ultimately meet commonly measure one gibbs always discrete tail state lead satisfied gibbs characterize specific mixing focus conclusion hold ensure consistency prior characterize heavy tailed admit bound admit limit second positive integer light since consistent sub prior therefore range increase large limit assign guess identify tail conversely large finally minimal inconsistent behavior even close extension previous refer random object temporal transition stationary least coincide gibbs type important distinguish area random prior outline former drive inferential purpose analytical community concern extremely front besides contribution back generalization dirichlet framework less restrictive dependence covariate date dependence simulation slice factor inferential framework root relate nonparametric underlying construction dimension depend specie allow species rise yield population frequency marginal poisson dirichlet clearly prior author opinion highly possibility interest diversity process constitute concern dynamic hand activity start vast literature concern law use nonparametric prior choice two instance neutral typically since conjugate right conceptual prefer conjugate conjugate convenience prior thing make assumption value depend assumption empirically quantity gibbs counterpart consider exchangeable sequence request invariance nonparametric type exchangeability rule obtain dirichlet turn made nothing implication result concern importantly logarithmic dirichlet spectrum going linearly increase flexible component distributional expression derive appeal retain intuitive relate learn instance concern prediction require frequency sufficient specie frequency accordance key reinforcement mechanism scheme implication sum review provide answer question title confident see well bring foundation obviously prevent gibbs type e concrete drop european fourth associated infinite partition product equivalently depend categorization species model iii specie amount first depend gibbs process case frequency require namely prediction exchangeable hence side coincide side coincide contradict nk iterate step mass characterize dirichlet nk cm remark pr universit di universit di de sigma mx exchangeable induce key address popular surely induce exchangeable elegant dirichlet prior appeal view admit characterization use term precise assumption learn stand iii special besides unified treatment highlight implication nonparametric frequentist concern serve idea intuition inherent class prior dirichlet phrase bayesian exchangeable partition law act several recent review cover use completely measure concept one trade far inference concern analytical represent parameter review also especially terminology introduce adopt py reduce parameter nonetheless distributional py process fundamentally equal class give prior briefly motivate inspection apparent distributional crucially novel prior serve use moreover gibbs analytic issue nonparametric stage highlight allow simplification relevant follow simplicity admit prior notable process inverse generalized gamma prior paper provide survey gibbs type account probabilistic literature point flexibility inferential beyond retrieval survival among bayesian exchangeable framework focus ideally assume distribution use bayesian inferential dimensional typically inferential generally topological desirable ahead distribution prior distribution py say broad gibbs unit concentrated variable henceforth iid general terminology follow sample generate predictive conceptual view observe distinct include namely dirichlet py process whose admissible value integer apparent recover exchangeable another paper essential extremely interpretability induce observe py form py identification lead predictive predictive combination p interpret guess observation prior particular suit address inferential specie sketch framework specification apparent describe structure type proportion model proportion integer correspond nature differ reveal connection terminology adopt name virtue adopt typically distinct specie detect number frequency draw consist specie frequency rare diversity interest biological linguistic play important answer basic building model dependent keep thing estimation define sequence real introduce bayesian date serve cluster distinct group inference great gibbs section provide suitable specie follow overview distributional emphasis discuss type mixture gibbs prior deal frequentist asymptotic discuss extension gibbs prior dynamic context conclude remark title interesting specie generate new induce characterization gibbs type result term probability generating past associate specie specify classify denote distinct value frequency denote k kk model classification generating n px n otherwise prove conceptual view seem generate new suitable specification scalar indeed depend distinct since summarize heterogeneity virtue gibbs specific want increase correspond later iii correspond general information principle operational need hand rise serious case typically quite complicated expression specify observe reflect opinion mechanism opinion finite mathematical type due simplify prediction appear flexibility motivate state predictive provide exchangeable type specie sample negative recursive light reason product accordance mixture dirichlet mix base ii denote factorial depend generally obtain respect still lie gibb clearly specie although preserve conditionally induce predictive distribution guess weight predictive intuitive observed consist value conditionally determine old value past observation weight reinforcement mechanism place see ratio probability assign cluster proportional size cluster cluster frequency represent appeal context reinforcement mechanism mechanism work besides exchangeable element determine size introduce eq limiting term worth dirichlet parameter class nice structure share light aspect important assess property essence nonparametric candidate topological natural nonparametric gibbs requirement full priori positive unbounded possess measure whose prior guess space outline introduction application discrete type prior occur within hierarchical correspond exchangeable density eq particular ingredient model random set index inference numerical mass allocate reinforcement mechanism look induce cluster determination factorial different let eq denote number display suit highlight difference fix component dirichlet control distribution large right imply essentially process role controlling play interesting concern display straight line simplification evident allow distribution yield flexibility py htbp dirichlet type characterize value simple suppose expect number nonparametric fix five process figure clearly informative large variability dirichlet imply specification imply information cluster often furthermore py latter produce k five implication specification end well separate clearly nonparametric specification process distribution hierarchy setup prior opinion wrong whether possess towards namely iteration burn adopt algorithm acceleration depict posterior thing posterior py towards py see value strong mechanism prevent completely wrong prior htbp c py py finally consideration parameter beyond toy represent structural understand g concern considerable heterogeneity inference component well py depict already address prediction specie compose individual biology economic one inferential purpose yield piece information specie label specie last alternatively reformulate specie must typically summarize form induce resort prior characterize novel estimator unobserve sample overall estimating specie correspond frequency thought overall specie hand quantify estimator specie estimate realization unobserve sampling either consist discovery rare specie population compose display specie suit tag serial complementary library rna population goal consist identify compose frequency gene library population due portion whole library overall characteristic framework take example identification status estimate contrast population limited specie type model deal py one display finding carry topic frequentist relevant estimator probability namely proportion specie quantity framework discovery term good estimator coincide sum yield numerical instability arise moderately greater enough appear illustration hand frequentist discovery new distinct
fail able remainder summarize accurate efficiently linear complexity present format format level thing provide evaluation superiority algorithm compute resp submatrix interaction algorithm column basis inner compute non rank storage scheme rank approximation latter vector thus full research scientific aim accelerate partition box interaction potential grow exponentially dimension fail lack scalability transform offer multiplication growth polynomial moderate growth via gold low cost work randomize onto subspace approximation nystr om sampling computationally well matrix uniform score version nystr om column leverage score variant leverage without entire improve random method choose alternatively qr factorization nystr om use cluster real feature euclidean inner problem matrix rank datum cluster block wise approximation require avoid formation method stable deviation trial briefly difference vector block accurate sampling selection rank cluster desire give memory fast product algorithm capture keeping partition denote define th approximation j submatrix submatrix rank memory cost independent store htp linear give detail selection similar fashion point partition shift invariant state shift kernel ft continuous lipschitz partition q proof strategy radius bind reduce appendix differ superior tend evenly cluster already matrix cluster compute basis submatrix restrict randomize svd however form block construct entire impractical massive cost restrict column row u u randomize appendix construct ir achieve satisfy low cost interaction block diagonal achieve reconstruction bound pt therefore rank look low memory proceed memory observe enable minimizer small vector matrix maintain ir inner list include vector benchmark art approximation method ghz memory census house exact computed approximated matrix stand frobenius approximation method deviation behave memory close approximation time addition implementation cost require memory cost carry time demonstrate show value also spread dotted indicate standard speed small increase memory comparable memory exhibit observe deviation approximate low approximate fix get include property get become less kernel go portion memory bad kernel error matrix diagonal increase part entry vary result demonstrate scheme work kernel parameter kernel dark blue cluster roughly describe rank gaussian vary accuracy ptc growth dataset comparison increase enough require behavior performance synthetic dataset behave matlab plot vector total plus case behave differently examine consistent complexity work comparable higher compare method memory randomized appendix center quickly optimum algorithm center centroid centroid assignment cost iteration svd svd approximation briefly describe rank denote apply qr get svd matrix work space seek row apply index denote denote apply qr simplification get side code author column replacement author code matlab interface claim theorem li scientific storage kernel parameter radial mostly case storage idea scientific computing situation construct block approximation block factorization work extend applicability demonstrate deviation superiority dimensional role machine compute
primal reason algorithm avoid optimality slack allow conditional insight operator alternate smooth penalty elegant close fashion dimensional break small subproblem retrieve solution subproblem quick splitting conjugacy simple eq lagrangian lx hold primal point dual idea ascent solve problem ascent therefore iterate appropriate take lagrangian lagrangian add ridge like lagrangian problem q primal minimum convex mild dual ascent iterate dual update ascent upon notice rewrite augmented formula residual shorter augment lagrangian problem bregman iteration compress recovery signal know coordinate observation augment step usual step bregman algorithm appeal divergence idea admm direction augment problem admm similar ascent lagrangian individually jointly pass alternate dual problem correspond nesterov regularity convexity convex bregman divergence induce vertical guess multivariate everything carry bregman family parameterization sometimes repeatedly context envelope recognize dual value value generating parameter induce use variable splitting parameterization expect parameter still canonical bregman split penalty poisson simplification split divide optimisation form add slack divide together use split admm bregman admm divide break hard split global section envelope envelope generate step envelope relationship framework extend possess constant assume proper quadratic envelope possess property pick symmetric definite stationary envelope original satisfy p lx establish property envelope function backward envelope evaluate gradient envelope produce algorithm way iterate class envelope apply norm convex see rate comparison quadratic envelope variable usually understand convex need newton iterative mapping diag lx v b gr bregman objective make envelope divergence attain bregman law establish descent mm insight generate envelope add smooth penalty exponential bregman divergence composite statistical structural making term construction statistical address mapping broadly combination together broad art start summarize function split duality motivation primal lie problem refer formulation primal joint exact exhaustive view formulation dual proper convex specify exact sub form computationally critical related form envelope representation objective slack proximal especially proximal construction proximal like move objective worth efficacy depend proximal connection literature proximal quadratic lagrangian application proximal similar augment squared composite lagrangian admm eq like operator imply appear contain effectively purpose impose approximate produce argument exact see solution backward problem include linearize split inexact context primal demonstrate framework proximal algorithm proper convex lipschitz give proximal namely complete leave sub take minimization next proximal problematic imply iterative two step basic alternate inexact demonstrate proximal q arise second around necessarily split symmetric assume positive like together proximal form equation reflect involve approximation objective solution term point quickly mean per general like attempt use forward objective cholesky decomposition cholesky imply simplified exposition start linearize inspire z proximal base split forward q contraction x note restriction scope illustrate vector non trial number composite envelope commonly use quadratic lipschitz adjust acceleration nature illustrate logit fuse problem inspire quadratic envelope multinomial loss bound envelope perform fuse consist pre decomposition svd thus provide illustrate fused non composite operator since poisson loss function still convex replace accomplish track em likelihood plus convex penalty bridge develop problem proximal form proximal involves find norm norm value result inexact value proximal interestingly kl backward appropriate solution choice map affect property variational convergence half cyclic fashion cyclic derive problem remove similar descent q signal match give plot mean square penalty consist contour plot interesting relationship clinical volume age seminal percent common lasso elastic net exact proximal operator hard figure path major difference jump solution proximal classical descent property arrive implementation mm statistic provide mainly lagrangian originally recently convex broad apply algorithm construct optimization envelope close evaluate numerous demonstrate efficacy advantage proximal fix advance speed approach nesterov acceleration smooth help modify nesterov provide scheme progress couple mirror progress couple convergence help direction statistic explore relationship proximal splitting research combine proximal algorithms r p p p dx x x x p tc strongly accelerate proximal gradient frank wolfe newton l forward semi recall translation operator operator satisfie provide sort quadratic continuity quadratic want fix functional convexity improve step decrease finally compound improve momentum derivative bregman divergence law bound operator w intermediate subscript yield update momentum convex empty global minima non involve range level generate lx lx critical nonempty neighbourhood subgradient trajectory kl alternate typical relaxation function kl one possess kl kl figure proximal useful machine obtain exploit proximal operator envelope half envelope optimisation function smooth objective illustrate logistic bridge fuse provide discussion descent direction future keyword bayes shrinkage splitting fuse regularization divide large solving step function precisely canonical problem together regularization modern statistical curve fit map prior survey alternate multiplier divide dc frank fw split processing tv thresholding maximization mm iteratively fall although machine general work iterative fix banach space useful acceleration smooth gradient reverse descent illustrate proceed section notation operator operator envelope extension rely envelope gradient consider optimisation compute exact proximal general envelope methodology poisson fuse lasso bridge commonly document half list case nesterov acceleration conclude measure impose favorable bridge induce parameter interest vector covariate encode structural trace composite view index function continuous vector pay penalty fused concept useful exploit equivalence constrain slack introduce linear envelope dual quadratic envelope lx supremum dual hold conjugate say function convex satisfie use algebraic take extended line tool semi scalar envelope conditionally least ridge regression normal general proximal proximal operator one provide start advanced differentiable encourage suit iterate take differentiable operator differentiable intermediate lipschitz allow q equality algebra optimum value hand evaluate descent mm convex lipschitz modulus convex ensure optimal obtain meet
supplementary material response vector except carefully ol dimensional need question ol ol view property ridge regardless observation notice hand side consequently ol dimensional dimensional version essentially orthogonal projection projection observe thresholde small exact meaning demonstrating row sample dimension standardize design diagonal visible obtain include variable thresholde far obtain refine term pre selection denote use refinement ty replace tx refer ridge thresholde concrete form need row draw distribution allow various correlation widely illustrate rely restrict condition assume second contrast need bic parameter tune however detail consistency straightforwardly imply exist bic bic state result choose material guarantee rely particular term appropriate preserve magnitude computable alternatively nested form ranking coefficient adopt select well stage ordinary tend assume identify tend ridge condition parameter note constant I algorithm see take form threshold general rank true stage least priori I result condition imply consistent require extensive assess penalize method include elastic biased stage figure respectively lasso noise experiment structure support set component equally correlate factor compare simulate synthetic record iv actual use bic regularize huge computation mc find first predictor know set fold tune ridge finite hard thresholding comprehensive rmse htbp htbp see plot table good well mc time rmse rmse iv htbp ex ex ii rmse iv collect gene week old responsible select sufficient reliable fold variation expression link eight study assess fold offer regularize fair extend record reference report cccc cv average runtime scad error follow select parsimonious interpretability prefer hard performance method exist regularization appendix recall estimator nc c ic follow proposition zero finite variance absolute corollary matrix variable invariant transformation I fact uniformly conditional orthogonal e magnitude know distribute iid notice q put piece accord second use z pz sl match ready prove theorem q great cn obtain x dt define assume singular argument number p dx yu q coordinate submatrix eigenvalue therefore least tx want eq argument proposition dp definition result need quantity much complete singular matrix expansion hold ni know obtain chi bind exist least n proof immediately imply definition applicable sample version ridge propose novel algorithm fitting thresholding intuitively computationally implement consistently compare penalization analysis potential long mild ol widely model unfortunately ol model exceed penalize lasso exploration methodology
specify threshold false discovery show sis pc discover discover large appendix whose unit give expression fix limit limit random variable xy grow rate xy xy note prop remove distribution give role identify transition assumption prop pn undirected fig part th entry denote denote value prop thin match match match dotted dotted dotted dotted dotted vertex connect phase exactly critical decrease increase satisfy xy expression prevent bipartite section screening use I pc recover pc recover prop prop show proposition proposition correlation correlation inactive correlation inactive assumption proposition thm recover thm prop prop function sis recover probability prop sense prove prop accommodate heavy tail sample allocation mse rule online sec w constant increase function stage allocation budget skip prop stage use prop prop stage ol result demonstrate world pc sis pc screening pc compare simulation row dimensional mean sparse exactly entry independent inactive response importance magnitude fast tune minimize screening stage truly sis pc small sis select important inverse variable pc sis evident regularization set cross pc sis propose sample selection compute choose evaluated figure regime sis instead pc stage suffer rmse low sis predictor stage value side pair difference pc sis different value h c c pc sis rmse experiment pc outperform pc predictor sis sample stage coefficient similar reference inactive ar w leave computed pc sis fig leave note estimation sis increase figure independent evident pc sis improve solid rmse pc sis respectively plot pc stage pc sis dash plot marker pc pc support recovery pc coefficient get bernoulli bound sample prop h sample consistent prop predictive health datum set score subject collect subject become ill level clinical record number point include post measure predictor accurately predict measure gene consider scalar response apply predictor consist specified gene previous time take value use logistic ol predictor likelihood logistic coefficient performance evaluate leave cross leave validation trial pc perform predict score l l rmse sis pc validation predictor regression correlation predictor variable stage high throughput selection use dimensional false discovery stage experimental advantage work subsection explain sec proof sis discover discover representation correlation exist column norm show sis entry square calculation define u u u u lie diagonal matrix obtain u u vector discover least discover pc discover notation proposition proof present define p screening proposition represent upper converge dependency define complement neighbor complement play key observation f permutation expression independent von fisher sphere parameter parameter gamma exchangeability feasible write assume f yield n n desire precision intuitive follow prop discovery rate screen know model unit value throughout entry magnitude j vertex n union anti yx yx ni xy conditional joint yx summing conclude second chen method b n xy pn b j b summation give b op combine bound prop weakly proof proof prop prop moreover average dependency satisfy ok xy xy x score correspond dependent large number sequence g converge grow entry hence u thus p conclude p x xy moreover pf conclude partition dependent relation x noting proof prop jointly sample correlation exist cn uniformly cn generality v entry area spherical obtain incomplete beta dt n dt dt obtain cn form f z z z variable em c b independent g detail score without assume score u score z c z r ab e ac ac ad bc ac ac ac w ac ac ad bc ac via let cn increase auxiliary k r constant complete x u x x u u screen prop sec stage stage asymptotic hold outcome expect outcome variable mse constant one wrong ol bias mse select correctly mse rate c depend use ol expect mse minimum since become draw fill black paper propose adaptive budget predictor high dimension screen experimental finance engineering illustrate instance cost linearly stage tradeoff collect small collect pass low regression sample variable online implement false select variable well asymptotic convergence allocation estimation stage much effort objective engineering estimation wireless communication internet gene science predictor difficult normal equation overfitte computational complexity number two method elastic class sis offline screening predictor principal common budget well perform sure screen sis generalize ordinary ol response predictor poisson false specifie phase discovery total establish ol third establish multivariate offline implement try regularize costly large lar interior develop lasso differs regularize objective costly via min norm regularize ols offline correlation also independence wherein thresholded paper transition discovery among sis perform recovery discover absolute large ordinary square ol define min determine min regression inverse matrix coefficient ol pc method pc value
interpret frank refer reader discussion point boost arbitrary limit furthermore lead wide provide small consider scheme excellent boost profile coefficient profile modify approximate herein produce produce predefined grid generate grid warm start statistical sequentially update predefine fix value r k identical structure stagewise step describe rigorous feasibility specifically satisfie boost every hold right theorem recall error boost error specify generalize notion family solution theorem quantify boost approximate part along value respect would ideally guarantee spectrum value guarantee quantity sufficiently produce vanish control learning error correspond summary flexibility control complexity regularization come weak provide value nevertheless error entire regularization parameter array example explore property herein dataset matrix multivariate diagonal entry take choose control signal specify generate take example consider four publicly describe process create package process artificial response take process artificial analyze process dataset form second interaction third form detail refer reader standardized column norm run herein well finding figure discussion appendix explore test iteration regression good optimal know method good performance sparse good play reasonably important role acknowledgement preliminary herein show profile regression c fidelity versus profile synthetic correlation zero run panel profile dataset run run panel panel highlight profile axis normalize horizontal axis scale interval express norm fraction semi assume exist optimal qp hold algebra generality qp write zero column qp equation straightforward qp since establish whereby furthermore root gradient rearrange utilize k j equality seek function yield rearrange invoke I term similarly iii l simplify iv eq q inequality elementary derive q last second note ie complete vi simply derive coordinate status zero present additional coefficient follow hold take root therein iteration iteration complete apply describe indeed link characteristic step value progress rapid progress term progress amount training change residual informally minimize loss towards quantifie decay similar theorem rate dominate point rate begin correlation sample decay slow correlation square long whose variable furthermore eq inequality jensen imply note equivalent normalize follow give scalar elementary hold indeed elementary arithmetic follow rearrange substitution descent size substitute simplify convex value show instance subgradient subgradient descent examine proposition context cm norm covariate standardize run inequality provide iterate residual imply square model iterate index attain inequality use present convex quadratic format eigenvalue substitute minimum attain rearrange prof follow note chain iii simplifying term vi structural completeness show certain fidelity formalize relative prediction pose allow run iteration study guarantee theorem compute fast regard run iii follow shrinkage boost give whereby tolerance need achieve denote bound model produce relative ccccc equation target prediction panel shrinkage relative panel summarize primary goal prediction appropriately determined achieve also shrinkage run reader iteration boost profile profile small small suggest accomplished possible relationship problem square useful function subproblem optimality set result direct scale proposition optimality closely dual weak feasible eq duality q associate solution r let construct least function min drop dual q scale feasible direct yield equality q imply last cast problem convex case duality linearly optimization apply feasible follow tr tr strong ii eq since formula residual recall j tr r j follow rr update finally g g therefore k precisely update write k hold induction exactly descent apply residual translate residual second x follow since fourth follow column norm uniform hold elementary obtaining side prove item quadratic set arithmetic equality whereby inequality take square item essentially boost item iv theorem similarity profile general profile explore research algorithmic lead dataset stagewise incremental backward approximate accuracy produce point model importance sparsity well variable square degree freedom often collection processing proposal apply coefficient machine explore proposal adapt square regression author boost herein study coefficient unchanged coefficient loss holding recover soft thresholding stagewise update propose subgradient frank wolfe analyze perspective subgradient interpret frank update primal wolfe subgradient interpretation algorithms parametric author similarity frank wolfe feasibility parameter induction feasibility j something strong residual give format elementary specifically similar translate iterate iterate namely logic leave right side l third feasibility prove second fourth assumption normalize combine q complete ii feasibility prove consistent last iii coefficient describe perform real dataset four publicly microarray binary covariate approximately process subsample covariate package covariate covariate retain take matrix sample retain generate create create enhanced note example unit run study herein l error bottom panel monotone start reach test panel panel describe c forward error exhibit performance seem marginally predictive sensitive run limit version suggest sensitivity learn array real decrease iteration however sensitive reach furthermore method namely performance well predictive correspond value proper lead superior perform evaluate predictive accuracy know method take run find good obtain predictive excellent solution predictive performance play crucial boost iteration important role obtain quality e model zero well boost dense herein section regularize indeed solution table run value regularization path regression compute reach unconstrained term boost always good automate worth investigate excellent heuristic version achieve good snr achieve snr snr cc ccc example snr method limit boost large sparsity coefficient minimal norm run instance interior well similar term remark corollary author fa mit cat author research mit cat analyze perspective method classic boost incremental stagewise algorithm modification may easily compute may also interpret master maximum loss guarantee several modern computational guarantee statistical boost description fidelity rate method weak learner powerful adaboost boost develop classification particularly form influential boost particular adaboost instance stagewise fundamental tool yield crucial insight underlie boosting provide additive method viewpoint function greedy reader usual py center regression residual regression lead model attractive property popular least fix covariate eq current regression coefficient k k j k know stagewise essentially repeat start iteration find maximal decrease fit residual residual coefficient unchanged evolution root slow strategy describe shrinkage rate qualitatively speak learn compare increase training eventually attain fit reach training shrinkage empirically lead short point tradeoff quantification present freedom non pursuit closely incremental stagewise forward stagewise initialize k j iteration covariate regression coefficient residual factor strategy fidelity shrinkage fashion qualitatively refer reader evolution herein first description quantitie lot contain subtle difference firstly covariate lead choice difference rather plain residual step amount successive differ across square note factor successive loss herein equation difference absolute sign gradient least square expect precise term qualitatively speak update behave shrinkage progress converge globally hold necessarily converge fit operational guarantee square predict albeit sublinear sublinear sublinear difference step size call replace depend version unify view special instance difference versus run draw correlation learn discussion classical tool model forward regression sequentially variable identify absolute residual predictor update quite additional known regularize nature implicit difference implicit explicit regularize regression especially dimensional far exceed parsimonious regularization regularization explicit square solution contrast boost wherein control although boost explore certain profile profile upon profile exactly panel profile albeit fairly strong effort understand angle correlate residual move towards aspect unified version coefficient profile coefficient shrinkage coefficient profile bottom panel cancer profile see coefficient profile probable modification boost path one study algorithm inclusion backward path nice understanding aim view master instance view special algorithm problem regularization term residual residual interpret residual determine assign describe dual subgradient algorithm apply lead new almost incremental stagewise first coefficient coefficient become call path quantify depend learn therein derive herein precise fidelity obtain along compare model contribution boost method exist boost aim residual viewpoint operational characteristic boost computational estimate algorithm towards produce respective demonstrate slow sublinear least square computational amount fidelity boost iteration rely upon distributional generating show subgradient descent regularize rescale every increase evolve towards derive guarantee quantify algorithmic subgradient present naturally boost subgradient descent residual error implication expand computational experiment improve place notation vector vector ball coefficient ax q pp denote subdifferential convex positive matrix small implication generate converge square solution characterize fidelity global linear square convergence coefficient describe shrinkage coefficient change useful gradient equation guarantee shrinkage eigenvalue least linear training l square solution prediction square shrinkage part linear rate write eigenvector large assumption column whereby holds let immediate remark iv error geometric exponential square counterpart least factor part behavior depend pairwise correlation function fix different dataset matrix zero correlation fast rate linear convergence converging confirm behavior boost slowly theoretical justification empirical make stagewise widely least procedure discussion tend correlate whenever updating thereby covariate correlate compete update algorithm progress decrease contrast bring sense line explanation attempt convergence value phenomenon observe reader justification illustrate fidelity present new boost rate show herein algorithmic interpret subgradient residual interestingly result strong section section guarantee theorem herein fidelity consequence guarantee counterpart develop motivate briefly subgradient briefly motivate descent differentiable close function satisfy lie linear intuitive formula kx lie outside feasible onto differentiable virtue subgradient subgradient subgradient generalize state denote point descent generalization differentiable replace subgradient guarantee bound hold objective obtain right side norm subgradient correlation residual predictor residual cm important namely optimization residual absolute predictor therefore problem residual predictor residual least square solution whereby residual vector nonnegative square cm view optimization boost subgradient method cm initialize iteration instance subgradient non iteration I recall k tr choose select k k g projection precisely ii whereby iii interpretation especially traditionally translate light fidelity shrinkage characteristic easily run algorithm show subgradient viewpoint algorithmic new minor solution consider total index hold error I exist square solution hold k versus fidelity theoretical shrinkage regression versus left panel theorem part describe related quantity least counterpart rate shrinkage evolve sublinear decrease dramatically fast iteration theorem difference limit demonstrate theorem nice square unlike towards limit part imply distance solution computational idea figure multivariate gaussian appearing theorem train much value validate three decay carefully explore shrinkage two follow tradeoff tradeoff tradeoff curve level unlike tradeoff range shrinkage error shrinkage shrinkage correspond examine shrinkage alternatively shrinkage shrinkage large enough also shrinkage
metric event diagnosis information measure highlight decomposable metric focus distinct characterize confusion average infinite population class decomposable thresholded practical direct held contrast understand classification aware metric exhibit simple square count sec bridge classification analysis decomposable metric many comprise sign first formalize retrieval family monotonic subset fractional family special case sec population performance study sign thresholding quite evaluating involve computation aside fix test show light complexity computation propose run case full scope manuscript key accuracy optimize could misclassification work surrogate decomposable metric analyze expect chain optimal population equivalence go infinity recently give analysis multi label analyze efficient let label label iid focus decomposable confusion positive negative false negative label confusion simplify sometimes depend utility manuscript instance generally compute utility also pointwise remainder utility population confusion entry utility utility utilize principle identify classifier relate class respect immediate iid sample give decomposable counting sec design difference positive sec n meaningful property consequence metric identify sufficient consider u eq consider tp monotonic tp monotonically increase first tp verify easy tp guarantee satisfie provide tp monotonicity sufficient hold consider performance metric metric metric satisfy sort p v example recover family study metric thresholde contain equivalently result prove tp monotonicity three study easily metric simplify empirical monotonicity property say monotonic monotonically list satisfy metric admit familiar sign thresholded performance tp monotonicity monotonicity satisfy tp monotonicity monotonicity admit optimal familiar monotonic tp monotonicity weak monotonicity consider tp monotonic pf algorithm decomposable performance hard consequence possible light top sort trick evaluate cubic general principle top nk nk compute note evaluate via invoke tb sort nk fractional metric focus attention fractional family decomposable fractional linear efficient solve give generalize rational consider family fractional step I implemented line algorithm v nd j nj k k tb sort j u v ns r nk k k j j k consistent fix consistency estimate depend consistency estimate tp metric show algorithm applicable empirical prediction experimental synthetic serve principle metric second benchmark comparison optimal minimization table namely harmonic pr theorem simulate conditional sigmoid sample standard objective plot probability synthetic verify optimize threshold ii conditional metric pr pr fractional achieve hold aforementioned metric classifier metric optimal training datum select thresholding baseline report news article topic training article article handwritten character letter consist scene consist web page test instance dataset optimize test choose suggest utility baseline baseline refer compute individually average class tp pr tp pr first indicate individually goal gap show metric principle sign literature monotonicity guarantee principle monotonicity large subset
also tensor column row large go proof observe entry th method j column q nn q fashion vector
regularization serial different manner figure parallel eq q new way computation core go multiplication proxy b w b w tune follow u p take expectation get nonconvex yield repeat recursively il ig ix obtain sum get obtain convexity ic smooth give w remark exercise develop new risk enable free sdca moreover define erm allow mini come even convex able mini risk minimization erm successful paradigm learn erm throughout shall assume smooth constant always well effort put realization long state erm idea algorithm belong include sag sdca gd gd cd sdca prox sdca analyze arbitrary alpha accelerate involve computation typically pick uniformly design dependent computation gradient typically equivalent reading work develop regularize erm free sdca mini scheme method example arbitrary flexible scheme useful reason development ii importance aim obtain elsewhere utilize access iv processing mini mean assign reduce setup example lead well dependent primal allow arbitrary characterize update value I example mini sampling define probability example maintain store pick mini scheme gradient follow relation maintain w maintain convex indeed update write sense believe converge circular word describe reason iterative tight formula instance inequality exposure decay b tb tc assume function hold decay moreover equal decay ic tc potential decay moreover erm analyze mini dual cover non illustrate method variant method importance adaptive marked disadvantage mini choose subset random parallel processing processor differ cause
product approximate accuracy alternate parallel transform sample auxiliary core distributional induce address solution constrain make optimization aggregation variational family f descent sgd dimensional sgd variational bayes objective writing optimize derive objective decompose kf segment power jacobian evaluate proof inequality supplement denote jacobian supplement approach give crucially thereby possible without entropy concavity tighter low put everything together relaxed objective paper pose exponential family derive great generality decompose concavity treat emphasize aggregation concavity partial hold arbitrary aggregation supplement family individually concavity entropy density u decompose aggregation variational concave individually assume aggregation decompose concavity concavity relax kf f concave concavity broad setting objective individually family simple capture preserve aggregation meet simple impose semidefinite psd cases psd aggregation suffice general lead therefore sophisticated aggregation preserve orthogonal far crucially da da dd spectral guarantee psd simplex w posterior algorithm mixture gaussian accommodate global center introduce alignment indicate associate partition one worker cluster center estimation baseline gain experiment posterior place moment uniform gaussian average baseline large assess three moment moment mixed moment model spectral aggregation rule restrict diagonal computational point one sgd running optimization normal inverse conjugacy supplement baseline great partition uniform partition run poorly portion bayesian focus variable variance assume sample implementation hamiltonian carlo hmc figure drastically estimation test analyze show boost estimate serial mcmc partition unlike previous indeed method error across method second cost moderate serial batch operate speedup serial marker marker factor schedule estimate gradient factor scope active area assessment probit mixture size sample show negligible sampling second approximately corresponding second error convergence serial sampling number achieve optimization moderate speedup serial bottleneck optimistic increase technique number serial particularly optimization batch marker marker previous parallel also aggregation analyze variable sophisticated lift aggregate recall useful building approximation multimodal posterior overall acknowledgment university support institute california berkeley fellowship nsf fellowship award award award fa amazon web services intel microsoft intractable unfortunately mcmc scale typical recently consensus remove limitation drawing partition combine sample consensus monte variational optimize aggregation function approximate objective relax mild advantage literature demonstrate superior approximation moderate overhead relative reduction measure serial mcmc compare consensus task estimate probit expectation error mixture component gain runtime serial speedup modern inference scalability innovation distribute asynchronous variant achieve similar success estimate chain carlo within former stochastic bayes successfully optimization adaptive apply achieve operating subset advantage architecture motivate parallel asynchronous variant belong communication avoid subset core core sample q centralize combine efficiency procedure algorithm monte proceed intuition aggregate correctly one instance full clear aggregate obtain partition aggregation motivate gaussian raise numerous wide stand first aggregation achieve close covariance weight modify paper consensus monte parallel algorithm bayes possible adaptively aggregation achieve flexibility likewise support aggregation applicable structured aggregation appealing lead approximation assume data partition map global parameter flexibility view possible alternative form within particular posterior place cf scope paper evidence circumstance
explanatory code level factorial keep point factorial typically fraction keep factorial remove carefully refer design factorial composite factorial factorial factorial fractional axis control side origin design factorial may verify way satisfy literature orthogonality eq coordinate vector x hence quality estimation property property diagonal imply vector component make factorial design orthogonal fractional design keep orthogonality design even hard refer factorial fractional factorial design factorial distance origin unchanged rotation orthogonal design order design design determinant justification computer design estimate minimal infinite space experiment multivariate design want design orthonormal n r appropriate appropriate constraint context fourier spline sample drive pca orthogonal norm take orthogonal pls permit interaction procedure pls basis context reference therein functional orthogonality consider see subject order nothing order square directly design verify functional adjoint linear design cause cause circuit incorporate water circuit cause nuclear inner risk pressure heat transfer explore physical develop reproduce behavior temperature heat transfer figure represent evolution depend ccc temperature pressure heat pressure transfer temperature minimize margin failure increase aim factorial pressure heat transfer good consider temperature pressure choose maximal fractional design pressure point around initial heat transfer curve take heat remove point result curve design obtain pressure heat count ccc initial give figure curve direction solid line temperature dot estimate response alternative energy atomic want thank final
observe fx prove martingale counterparts dimension q coordinate bernoulli variant use perturbation algorithm present normalize mse normalize define measurement iteration measurement measurement measurement difference fact use simulation outperform fact iteration measurement result mse comparable c consider direction newton perturbation use aforementione perturbation simultaneous scheme newton know perturbation newton analyze observe asymptotic mean square numerical computationally efficient newton scalar term diagonal term diagonal ij fx expectation rhs rhs simplified dx combine w equality fx observe rest hessian comparison gradient vary bias technique proof consequence fact apply martingale proposition near claim proof asymptotic normality scheme cf segment connect gradient write amenable q fx n recursion estimate gradient establish converge identical imply imply verify order limit c pt height white coordinate thin symmetric perturbation optimization newton perturbation unlike simultaneous perturbation iterate simulation mean achieve compare incorporate perturbation par sometimes newton newton provide latter engineering research network involve system performance concern paper minimize measurement search gradient measurement finite pp require regardless randomly direction unit particularly computationally inherent perturbation observe perturbation cauchy distribute perturbation independently derive convolution see integration convolution regardless parameter perturb parameter direction finite low sf amongst perturbation simultaneous perturbation largely ease observed approach perturbed distribute study square perturbation simulation hessian estimate use iterate hessian objective gradient year considerable aim adaptive newton optimization propose perturbation involve symmetric bernoulli gradient iterate four perturb update project definite perturbation one average objective incorporate approximation certain simulation balanced perturbation hessian estimate hessian inversion procedure effort see similar except computational geometric half parameter certain feedback newton book perturbation first newton contribution summarize benefit perturbation newton simultaneous require perturbation second simulation procedure distribute unlike surface sphere perturbation first two simulation balanced perturb newton also nominal simulation newton scheme gradient sure rate convergence perturbation asymptotic perturbation achieve gaussian propose significant scheme require evaluation rectangle height width em fill white distance coordinate coordinate black cm xshift cm label sim right height yshift width update sim update sim corrupt scheme random estimate illustrate noisy measurement denote I e fx fx denote sequence sigma symmetric uniformly whereas perturbation uniformly distribute algorithm follow continuously bound second assumption ensure ode pose comprise ensure analysis convergent next govern provide refer sketch appendix gradient indicate order perturbation surface computationally perturbation scheme asymptotically differential xt xt compact require hessian th follow project onto positive definite order move hessian happen basic algorithm hessian estimate sa noisy respectively noise direction henceforth second would effort per former require generation perturbation variable measurement require require loss sigma assumption na nf fourth f nx I I si almost surely require system simulation assume c converge ensure recursion assume govern almost surely next additional label assume sketch appendix show asymptotically perturbation define certain setting optimally unknown obtaining average employ employ couple correspond perform adaptive iterate denote obtain scheme dependency
insight measure reliability correspond precisely among avoid combinatorial experiment sec ratio whole fix accuracy measure conditional probability randomly set class fig estimation together white deviation gray plot approximate measure depict vertical ball radius bin expect drop cardinality us insight report remarkable increase suggest effect new confirm trend investigate near extend object class principle actually constant physical explores environment notice cardinality trial high randomly would accuracy perspective expect typical quantify capability report minimum accuracy within level well understand implication relate pass predictor guarantee least accuracy visualization use perspective depend discriminate expect fig human low curve decay fall great could remarkably improve regard change observe object interest train combine frame rule returning occur classify principle beneficial accuracy would evaluate describe stream consecutive classify select frame previous sort label actual since consecutive stream always confidence fig varied frame system clearly benefit probably discriminate misclassification filter boost accuracy achieve accuracy fig confidence gap regard account aspect principle improve day report test separately day line mix observe day predictor train day redundant predictor curve day day day day assess day pt day ex day visual motion around actual compare hand whole perform benefit manually gain strategy consider day system accordingly strategy introduce motion boost suggest presence actually consider large image imagenet often background manual fine approach eventually manual segmentation manual example recognize visual recognition describe interested reference value return recognize recognize visual actually answer ask far ex pt em object pt completeness close brief comparison test architecture visual word implementation convolutional network due reader deep architecture cnn layer remain method carry notice trained cnn clearly outperform gray day day avg ex use test recognition capability robot analysis address currently recognize formulate accurately visual recognize human investigation collect scenario comprise class first would confidence result multiple recognition capability empirically adopt weakly strategy reduce amount extremely principle result hand visual representation architecture cnn visual setting hand extremely challenging far ability visually recognize fundamental indeed task involve interaction accurate scene good visual bottleneck system convolutional remarkable performance visual regard remarkable would generalize possibility take step develop vision platform particular benchmark question recognize experience computer vision approach arguably channel operate human environment visual major agent behavior imply planning progress especially mainly classify image layer architecture recently rapid acquisition train tailor kind ultimately test natural ask system task see offer platform question start collect experience preliminary confirm propose highlight challenge pose lack supervision paper conduct study aim answer object recognize robot interaction acquisition teacher speech label day broad question sec empirically representation application acquisition research many object recognition capability address question divide recognition acquire generalize observed measure expect hold world contextual information offer deal information incorporate recognition observe different robot contextual vision setting unclear employ sec address question artificial able rich expect robot benefit incorporation visual preliminary perform sec life internal supervision ideally robot communication channel speech supervision teacher robot human image manually around object rely strategy motion segmentation eliminate object evaluate fine image vision recognition categorization identify representation datum ideally hand discriminative different separable hand invariant scene rotation class mapping patch optimization loss separately jointly propose local map learn map least square nn availability parallel make cnns accord evidence architecture rich amount use description particularly appeal paper effectively high computational least setting context line research recognition matching often employ suited supervision previously robot supervision hence acquisition protocol collect detail employ employ briefly outline human front object annotation detection track hz localization image pixel fig process representation module information descriptor set comprise distinct object evenly organize category acquire second reduce acquisition frequency image
sensor bs dynamically suggest journal equivalent energy employ incoming intend energy intend algorithm controller amongst compete controller allocate consider comprise system instant controller decide instant amount algorithm controller optimal controller try efficient computationally expensive numerous find allocation follow subsection management policy sensor network multiple sensor discrete buffer sensor obtain energy buffer generally buffer level node buffer slot bit source queue level upon let unit node slot bit data function non decrease shannon channel particular non concave optimal energy sharing form function consider power queue length evolve buffer queue evolves give denote sensor evolve noise spatially correlate arrival evolve depend assumption energy independent x x tuple single controller move prescribed probability state term formulate share mdp set run action tuple comprise buffer buffer note k max k remark tuple simplify state action arrival ks ns energy bit e split refer stationary policy choose set single stage formulate energy mdp generalized jointly arrival arrival easily satisfied history computationally infeasible policy evolution k evolution mdp chapter noise form policy augment sensor transmission long string data string buffer discrete model discretization discretization discretization discrete energy store queue discretization generation energy sdp map energy split run include hence take enable form I policy stage average sdp stage give energy long infer lemma minimize average interested stationary policy deterministic optimally share energy aim minimize cost stage minimize sensor cost delay well deterministic class policy provide optimal small state space large computation problem find optimal tuple tuple update adequate play finding nevertheless buffer tuple amount computation energy increase share energy scenario tackle curse complexity threshold fundamental feature prove source differential easily manner buffer queue buffer queue buffer level differ buffer level monotonicity differential justification nearby aggregated cluster state value policy close property state group arbitrarily aggregation yield policy action go combine guarantee policy representation combine aggregation sa continue aggregation previous formulate aggregate quantization buffer space buffer buffer quantization represent buffer quantization buffer prescribe energy buffer I instance energy buffer partition range x l data bit range bit hold buffer node buffer controller energy two aggregate controller bit node let aggregated action cardinality reduced instance cardinality partition cardinality aggregation explain although information respect counterpart state indicate aggregate action tuple proceed schedule facilitate exploration employ mechanism describe every present buffer level aggregate storing aggregate computational describe dependent aggregate sensor grow energy buffer iteration six buffer compare aggregation must controller I aggregate aggregate aggregate action state buffer level require since aggregate state action optimal indicate level add hold choose energy division sensor system partition suppose energy bit energy bit belong partition aggregate action bit bit remain bit buffer order bit energy node exact buffer advantage aggregation manually scheme result increase partition show sake implement greedy heuristic implement method bit allocate energy available greedy unit share node let action space kt energy need distribute decide upon find learn policy action space action proportion nk bit greedy consider jointly markovian arrival buffer buffer size noise mean keep buffer accord simulation arrival node varied describe markov markovian energy arrival process buffer buffer evolve noise variable component keep buffer buffer buffer cluster partition experiment stepsize greedy ucb I case fig simulation carry fig arrival energy arrival arrival I vary keep energy consider fig data arrival node keep fig policy mean arrival learn design policy algorithm fig policy combine distribute split poor compare mdp energy total max max fig cost policy obtain method since free irrespective arrival fig near plot method average node occur h max variation average aggregation increase well policy single stage cost effect action take tradeoff queue collective observe stage derive take importance queue combined method stage buffer fig indicate rate fig indicate queue value plot queue transmission relate show collective queue average fig occur learn well combine node importance queue cost policy learn energy usage albeit queue length combine q learn aggregation greedy axis aggregation perform method aggregation find greedy available instant storing compare one devise energy requirement naturally incorporate moreover thus naturally early cite work bit require node combine greatly I state node without aggregation learn fast node learn suboptimal fig poorly combine tradeoff policy sharing consider energy sharing sensor scheme problem understand greedy method certain regardless form algorithms sample share action k k buffer buffer share evolve slot must system model observe incur choose require rl computed method exact decide share comparison state
theorem I often give portion time simultaneously runtime th suffice symmetry compute rest find partition size master perform require carry optimization simultaneously apply runtime nonetheless master speed modification give slightly statistic still size master grow approach variable every matrix cd bn time simultaneously appendix proof bind runtime analyze separately compute equal set contribution simultaneously correspond equal size summing give bind analysis corollary undesirable however strategy work typical asymptotically fast heuristic demonstrate difference translate substantial realistic rest paper algorithms simulation data curve square bias noisy bad comparison light present examine simulation I space relationship simulate distribution variance statistic two respect reason usage substantially well instead median perform relationship fact whereas heuristic second result exponent variance expect regime general regime lead regime lead negative regime word detect distinguish among strong value cause detailed recommendation practice extent standard analysis equitability show paper equitability lead analysis sample size conclude respect vast majority examine pearson pdf xy along legend pearson perfect equitability describe th th percentile relationship estimate allow visualize statistic interval plot correspond red high equitability reflect question relationship maximal equitability total coefficient contrast maximal information rather nan supremum estimating supremum characteristic set variable number one procedure undesirable positive population characteristic intuition behind entry characteristic expect small independence power away avoid sum entry property relationship coefficient fit conceptual characteristic prove consistent power define version characteristic whereas denote ordered lead independence test analyze care define quantity grain part grid characteristic sample statistically matrix behave dependent understand bind translate quickly proposition technical information yield jointly integer either give understanding part independent exhibit therefore grid entry without generality argue imply definition element strictly finitely entry column prove claim grid size since allow empty satisfy word last distribute guarantee proposition proposition tell immediately convert write meet tell last grow yield test right test independence jointly sample distribution characteristic suffice monotonic therefore show alternative strategy translate know bind alternative lemma lemma take choice nan independence say alternative hypothesis exhibit proposition grow grow section matrix calculate affect prove turn independence base evaluate pearson correlation choose consider examine simulated estimate right independence analysis compare quite examine three unlike relationship term detailed type relationship size marginal wider increasingly accurately achieve measure dependence independence test latter relationship strength two goal contribution way maximal smoothing supremum infinite consistent computable coefficient statistic arise independence consideration detail show noisy equitability respect quite equitability independence across dependent hypothesis allow quickly enable contribution importance reason characterization limit know converge tell see result smooth mutual uniformly constitute toward role normalization continuity ordinary mutual latter third alternate characterization grid large finite significant improvement approximated evidence basic normalize mutual achievable grid characterize interestingly possible grid excellent theoretically computationally specifically little cost either individually imagine non significant association rank remain strategy leverage reduce burden utilize art equitability effectively course useful nevertheless exploration future sample grid line sophisticated discrete second approach joint infinite seek supremum precision finally characterization representative promise theoretical would advance equitability like acknowledge r constructive useful reference discrete respective whose q entropy rescale analogously proceed proceed proceed bound bind together fact expression fact consequence convexity term complete number function variable result let variable variable eq bind hx hx hx hz hx hx z z hx h fourth line hx hz hx let mass rescale let let normalize magnitude total add go previous lemma remove mass k k h finite independence total result characterize versus jointly distribute speed optimality cd cd entry partition grow therefore supremum partial sum jointly fix grid size quantity state show grid quantity size jointly parameter error rather outside distribution every parameter consistency follow sketch distribute proof th axis restriction contain grow become fine fine fine fine optimal convergence grid pointwise seek abstract lemma index size sequence output sample sequence hold almost converge show exist definition know convergence observe I coordinate analogously straightforward state show ij notation converge sequence sequence begin pair jointly sketch entry zero identical exist take column index finitely large entry together consistent tail algorithm jointly distribute monotonic analyze nan moreover adapt argument apply entry since may follow theorem depend represent number invoke parameter sub partition relationship value compare average finding size introduce size mean grid find grid high sample effect grid regardless exponent use need search grid curve alpha alpha curve alpha alpha david pair retain score statistic systematically assign relationship type avoid assign equally noisy regardless paper characterize population measure way statistic optimal marginal know simulation well bias high set functional nan statistical equitability testing excellent focused depth evaluation dependence show equitability together valuable tool exploratory grow hypothesis science whereby help formulate one among practitioner evaluate pair sort variable scoring low manually examine popular analysis whose value asymptotically trivial relationship utility dependence assess independence relationship dependence exceed explore biological relationship manual study human subject yield powerful direction systematically assign score relationship linear crowd score paper second assess measure equitability informally statistic measure assign equally regardless equitability useful dependence equitability respect strength introduce new bivariate dependence toward goal equitability power begin introduce dependence distribute finite grid cell gives view introduce smoothing sense regularization formal small supremum infinite sequence term dimension time grid greatly simplify associated quantity estimate algorithm call know computable fast hand property guarantee reveal tune toward general study demonstrate equitability relationship though use relationship detection ranking goal equitability independence coefficient later aggregate via rather maximization distinguish signal e distinguish relationship independence arise naturally free computed compare practice addition equitability measure equitability together result maximal excellent one remain computed measuring dependence extensively grid distribution grid analogously distribute mutual information sample case grid row mass column way row column establish matrix norm section maximal information coefficient jointly population way later jointly variable maximal coefficient see regularize grid ensure fall characterization view population matrix denote characteristic relationship additional characteristic useful presence absence rather quantify strength population large limit statistic review result hold hand trivial proof order characteristic define order let define statistic estimating theorem statement recall projection uniformly pointwise since supremum realize yield intuition sample might characteristic turn reason consideration consistent supremum ni suffice characteristic converge particular slow always individual entry characteristic eventually technical heart quantity quickly theorem lemma build grid consider master grid contain bind much grid seek difference seek require entropy contain central idea prove grid impose allow grid cell grid let respective define cell distribution column taylor whose find extend grid grid master grid rely prove later difference let random let cell resp mass cell cell provide away grid replace every horizontal close line resp moving show argument use term bind grid distribution mass sample integer lemma characteristic continuous technical continuity later non normalize mutual without apart mutual pair relate movement prove separately straightforward conditional entropy line magnitude cell column mass leave cell lemma p appendix binary entropy bind result mutual information depend ready continuity characteristic map uniformly step function family consist since consist argument ball remain tell distance normalization respect desired define map therefore continuity family within supremum give continuity map corollary characteristic include fact entry characteristic normalization characteristic continuous function line generality cell uniformly column see give information supremum continuous correlation equal mutual enough sufficiently cause continuity canonical smoothed lack favorable property supremum alternate characterization computing precision second foundation introduce later boundary observation empty know corollary true population characteristic characteristic characteristic show entry fix corollary therefore supremum exceed characterization stem fact express maximization one partition grid equal wish show observe column q q give variable partition bin expression maximization partition rather grid compute characteristic precision utilize dynamic programming brief overview set optimize axis description algorithm optimize paper master return mutual sub algorithm work exploiting condition store subproblem black box mutual theory develop boundary characteristic computing boundary resp computable error numerically mutual claim partition maximize sub master set row get close enough prove restrict partition partition omit since show ok move close
outcome endow validity obviously apply paradigm include produce pseudo helpful comment reference conference conference hold help universit paris paris france fr paris bs calibration france uk paris paris conditions work justify base construction prior analysis arguably central piece since inference define datum lack bayes construction automate extent reject argue non prior via moment maximum even involve consider jointly propose perspective hard long fix infer term truly motivation justify method difficult analyse introduction fisher one author integrate construct within specify produce equally arbitrary transform around freedom happen dirac variable theoretic difficulty analyse somewhat exclusive alternative hence conversely induce joint line define pseudo among define far remove determinant depend reference moment moment condition derive entropy besides produce inconsistent se incoherence specify require possibly valid whether set priori compatible seem highly must transform consequence joint unlikely likelihood likely select fisher example contain borel inference part deduce fine prior hence borel algebra compatible agree regular borel algebra solution abstraction axiom one define derivation reason small density likelihood remain situation consideration notion analytically verify approximation make sufficiently completely distribution regular intractable much alternative bayesian variational suggest suit exist induce distribution item irrelevant wants rely solely observation distance aside author harmonic mean rather poor potential close unclear I stage characteristic available expressed convert adequate available comment aspect true scientific
corollary proof one perhaps way environment gb ga environment list form construct must environment contrary gx gx contradict environment different please list want theorem definition create might think would file discuss environment first title include appropriate form expert name start document file run twice resolve create file source comment alternate file helpful moderately expert read useful please usa format lars rv circle p rv email email document alternate look author page file try include sort title mention produce pdf version software record conference seek conference quality appearance requirement document specify balanced specify size body copy live page specify margin top bottom leave right width news another balanced page class remainder concern rigorous description hierarchical section subsection subsection hierarchy use around name example appear paragraph sentence separate paragraph character already see sample indicate phrase text code change instance handle document take care begin text symbol character english character complete display three text environment symbol structure display display text center display equation environment use symbol available give couple first equation notice environment enter equation handle article conference book list occur automatically citation key item cite proper file key word title item file detail file exhaustive detail specification citation format support split across table table align properly row horizontal vertical material sentence file english comment common wide table live area
like extend discussion formalize argument theoretical claim building layer representation advantage gradient learner hide synthetic allow early computational extended modelling develop modern use transformation mean implication strong promise expand discussion dependence theory probabilistic derive top multi node demonstrate equivalence optimally structured dependence novel relate finally conclusion cast serious consideration provide outperform percentage point dataset aside advanced hundred label publish present variation model dependence underlie rarely implicitly explicitly literature marginal dependence co complete intractable pairwise ensemble incomplete dependence particularly intensive inherently need model measuring entail classifier among scalability much model negligible range time dependence outperform dataset good forest ensemble improvement prediction multi author development year model predictive modelling gap ground truth make risk effort could advance implication label irrelevant word lack one performance classifier illustrate take dependence multi expert human visually classify know contain modelling dependence one label base crucially mean guarantee amount training change change ideal toy guarantee factorial complexity model full limited linear learner perform equally poorly perfectly scenario human conditional base dependence summary perform method dependence condition method due insufficient dependence multi outperform classifier inspire deep remove dependence among top powerful top later base learner separately come result popular lie advanced label considerable success option special consideration neither removal time truly parallel label chain scenario number scalable learner serious division pruning fast cost propagation additionally huge family practitioner go default model basic relevance inter linkage option degree measure also accuracy derive classifier scenario purpose illustration toy imagine represent subject imagine latent event affect composition various event behind add desire case generate independent aside basic illustrate independently independently identify relevant expert piece place write text document label domain expert potentially illustrate impose e linearity recover latent recover knowledge redundant drop structure I interaction unobserved label document bias alternate external combine generic layer output equivalent dependence construct optimally energy improve nothing since even learn build universal approximation approximate label mean linear suffice layer inner radial return though learn restrict rbms study layer find label reverse variable remove overlap novel method rapidly field challenge learn rather unsupervised layer divergence recent adaptive dropout become predict discriminative prediction feature representation multi available scalability reduce subset degradation respect discuss layer inner fashion stack look like middle stack middle combine stack form chain leverage label unit must respect supervise fashion family probabilistic perspective directly approximate probability scalability employ decision tractable create vote get vote st label cast imagine create subset vote relevant label node weight word mention project space generalize function label eq return index use replace representation internal however cascade could could equivalently skip network augment middle flexibility stream machine remove function example stream review deal relate already introduction linearity otherwise expansion technique either suitably arbitrarily high polynomial degree learn input rbms sigmoid cast stochastically stack radial rbf review build linearity unlike basis typical method neural formulate error rbm early label box consistently art pick widely multi text implement dropout competitive compare network even disadvantage multi back learn input back adjust weight adjust stack rbms use tune early propagation layer error idea call hide parameter choose build hide layer obtain huge typical closely fact rbf unlike another difference label feature implement framework library collection statistic type logical audio biology medical multi evaluation empirical predicting carry ten iteration train randomize effect make scalable training disjoint vote propose relu logistic base learner case relate split node obviously trend predictive split average music medical avg music scene medical avg dataset music scene medical avg overall highlight propose explain relatively relatively huge already albeit naturally perform function dataset another proportion label correspondingly proportion synthetic label label incorporate several run parameterization average final slow dataset scalability increase concern drop former time latter extra chain internal split around factorial chain occur second time obtains indeed lead improve include underlie frequently dataset general perhaps ensure label overfitte ten scene classifier ensemble latter simply space thus among avoid ensemble improve mechanism rather give explicitly implicitly use ensemble take ten similar explicitly literature evaluation contribute great depth explain modelling advanced help model dependence inner layer label independence outer show simple base learner create benefit flexibility series novel method exploit space label classification instance cascade find advantage label create combination output ordinary create inner layer concept advanced competitive datum typically beneficial dependence fundamental separate construct hundred effort build discussion deep develop dependence base rather inherent exhaustive analysis inspire remove leverage middle unit base also
experiment active active focus active strategy follow drug interaction measure base round measure batch accuracy stop reach high enough learn entire label dark red interaction represent drug determine truth interaction drug interaction lack denote experiment denote interaction drug target drug approximate drug target kernel prediction product project bayes pairwise provide semi classification use initialization batch greatest sigmoid predict stop active predict point along process predict characterize unique unique uniqueness value interaction measure independence difficult purpose current feature row average prediction fraction drug compute pairwise learn uniqueness parameter kernel distance trajectory learner measure ground truth perform fit validation prediction al drug interaction matrix nuclear evaluate outperform strategy channel reach predict introduction decide stop good active learning therefore uniqueness perform learning simulation red fig datum guarantee true least true stop accuracy simulation rule adopt threshold active great fulfilled occurrence predict accuracy experiment interest begin active amount prediction confident reach peak predict accuracy naturally drop drastically lie range terminate active predict stop evaluate fold set choose reduce training perform make experiment purpose active strategy modify size uncertain prediction target predict acquisition accuracy stop auc result five report stop four datum list rule rule auc channel overall uncertainty adapt base describe predict difference time experiment evaluate define reach pa threshold average time fix threshold maximum uncertainty sa stop well criterion pa pa pa method drug building kernel accuracy furthermore drug limit world uniqueness achieve good matrix dataset basis please limited target produce drug convert replace strategy diversity learn trace simulated feature accuracy accuracy calibrate stop rule achieve drive undesirable presentation abstract conference institute advanced study biology department pa biological sciences engineering usa need drug actually save active crucial evaluate decide criterion actually simulate drug analyze regression stop unseen previously drug effect apply criterion result total highly prediction conduct search affect become find successful require search meet exhaustive selective drive machine refer guide drug method grain exhaustive verify manually limited expert perform time expense exhaustive limit trace drug rule little criterion unlabeled pool consistency instead use stop describe active trajectory use train simulate adopt drug target prediction major
efficiency estimate decide amongst initial stage collect choice repeat feasible low refer choice step typically complicate simple technique suggest behaviour motivate regression secondly regression vary ensure zero regression suggest efficiency choice efficiency use asymptotic see note abc omit return perform subset regression kernel bandwidth manually repeat training relative abc magnitude ess study spatial make ess abc abc include calculation second time core review abc introduce simple tuning provide comparable spatial potential extension include version focus simulation interest costly simulation use appropriately abc process tune approximate bayesian give possible interest define conditional standard sampling version equation w iy l n iw g estimate surely correct inference trade evaluate act since map dimensional statistic vector define typical accept reject normal match produce significant tuning choice consider aspect abc abc split stage refer require simulation considerable simulation stage operation tune introduce function behaviour simulate save division htp algorithm continue simulate relate discussion sketch argument follow assign randomness act weight due theory importance sampling ensure algorithm argument target abc decision converge
result non decomposable recover previous metric natural force suitable respect inefficient alternative provide efficient cg avoid cg feasible form plug consistent confusion knowledge algorithm provably family decomposable also unlike plug method metric novel technique cg solve approximately regret bind smooth metric metric confusion metric mean classification plug apply class metric learn prediction know consistent metric sensitive classifier close optimize multi setting method apply set seek decomposable additive standard performance interest multi express sum loss example loss example potentially decomposable label metric section give multi class metric decomposable non design alternate efficient conditional decomposable metric appendix shall denote simplex yy component closure appropriate space tie break favor set training classification model interested deterministic classifier simplex setting evaluate decomposable metric expectation loss denote class py sd py py metric h fourth column hold confusion linear metric yy yy n concave differentiable us confusion correspond confusion classifier express bound eq macro measure widely text retrieval contain example theoretic look decomposable seek throughout use randomize eq value sample say consistent probability draw classifier loss exist give decomposable method sx regularize empirical understand decomposable metric optimal binary monotonic place certain specific measure geometric median exact also classifier assign empirical suitable statistically decomposable thresholded cost sensitive fractional decomposable metric see little extend sensitive minimization binary metric single generalize class tune class proceed convenient rl r dl start decomposable find classifier decomposable matrix framework characterization classifier decomposable metric satisfie mild maximize decomposable gain give decomposable confusion knowledge general generalize decomposable performance pt begin confusion confusion classifier probability give cast viewpoint useful characterize design multi label random distribute uniformly simplex say absolutely w metric metric n differentiable non classifier maximize decomposable construct h optimal multi metric simple application fractional linear coefficient also metric eq confusion remain show achieve first necessary equation along g give statement decomposable continuity assumption assumption indeed metric unique mean metric classifier worth certain restrict characterization fractional min max decomposable min max sub characterization classifier metric simple plug decomposable search show decomposable also explicit metric like alternate conditional algorithm consistent family decomposable confusion matrix decomposable metric plug suitable class natural approach perform force search gain pick plug classifier see algorithm reduce threshold class probability perform linear number hold exact search maximization grain continuous force method statistically give h n ss g g satisfy using guarantee metric couple lemma fix gain probability entry second give confusion set learn gain sx ij md sd proof theorem optimal let n g g h h fourth matrix one special metric certain like force plug n dl c c c classifier draw constant bind metric non table suitable smoothed metric indicate n performance dl norm learn draw independent theorem metric concave smoothness constant sample draw g mean prescribe table learn concave smooth see form metric fourth find n c nn guarantee plug discuss make strictly high performance randomize classifier high h plug learn fail return algorithm ensemble enable handle general consistency performance derive metric consistency result involved approximate long also point cg objective method hence result general metric show without special concave performance fractional micro linear provide unify decomposable loss metric decomposable performance binary metric learn optimization consistent concave technique novel tool literature particularly thank helpful discussion helpful discussion support fellowship thank technology fellowship c randomize lemma I randomize classifier classifier x f jx x third random f p p integer affine vector take q let entire let denote set hull affine hull dimension denote set ir r g b dr inequality dimensional radius dimensional sphere radius width finally entire space argument small absolutely r base element among component arrange monotonically monotonically obvious scalar column p cc c argument empty constant absolutely continuous singleton maximizer unique maximizer proposition g maximizer equation prove n uniquely lemma complete shall go virtue confusion maximal strictly bad confusion away c f f b f apply equation p f p contradict equation thus two learn g sx x e argue value xy sx implication gx gx sx sx equation g x gx gx py approach thus zero gx r md sd mc observe b bx bx hx g concept intersection correspond vc hold gain I let h h gx em gx h x satisfy assumption dl c let x theorem g p thus g p g lc fourth step assumption last result curvature c observation entry sum approximation h n classifier construct train j due inequality second equation distribution dl norm learn probability distribution simplicity assume exist derivation lipschitz performance constant bounding parameter maximum hessian c n lipschitz norm n n c u n u c c u c n c c u bounding matrix c hold entry assume n table c c c u c
input pooling pool branch branch max fc softmax cifar relu compare activation relu variant baseline observation experiment surprisingly normal relu relu relu lowest relu relu suffer severe overfitte superiority significant cifar cifar dataset cifar big activation test relu relu activation relu relu relu relu relu relu htb htb analyze finding popular activation relu type relu consistently outperform relu reason superior performance justification aspect activation scale type convolutional standard unit relu unit relu randomized unit activation suggest incorporate slope finding belief performance relu overfitte counterpart success various classification object characteristic non relu counterpart g lie aspect vanish second accelerate convergence activation one relu briefly desirable activation pass relu commonly superior relu come sparsity paper want ask broad class activation interest relu contrast relu negative totally drop relu zero slope part predefine author imagenet variant unit relu linear relu illustrate comparison th pass subsection unit variable linear restrict formally activation q mathematically fix set like small parametric relu classification eqn propagation randomize randomize relu dropout suggest competition winner test activation search activation cifar cifar image rgb image pre augmentation ensemble
component despite promise progress pca due whiten tb randomly pick x u u u stochastic summarize reduce per minibatch empirically fast batch version prohibitive fix power show due accuracy fully solve risk unnecessary contrary level efficient accomplished ok concentration try direction simultaneously else whiten present dimensional canonical dataset description mnist image lexical annotate challenge shoot annotated handwritten cca learn leave image extract journal million token vocabulary successfully build embedding cca word sample uci repository million first million ip lexical cca feature lexical feature bank define two sum canonical estimate canonical lead proportion correlation compare well cca relevant capture second cca subspace pose correlation capture truth dataset number amount numerator w algorithms draw normalize hold adaptively regularization add gram matrix people compute practice performance dataset bank capture cca much version big thing notice reveal less denominator numerator mention classical algorithm fail usually prohibitive huge advantage cost generate reveal iteration capture correlation amount correlation description whiten matrix also whiten invert whiten cca compute lead subspace subspace top cca subspace cca back cca dominate heuristic svd randomize correlation increase heuristic incorrect approximately datum correlation paper tackle large cca nonconvex optimization free scalable prohibitive regime concern far incorporate canonical sparse hard cca whiten therefore well corollary remark canonical cca technique structure multi algorithm usually huge matrix computationally storage cca scalable canonical thin matrix usually require store whitening propose generalize especially huge batch prohibitive property suit effectiveness introduce characterize relationship multidimensional fit low unlabeled supplement refine label semi supervise fashion improve prove dimension cca association genetic variation cca algebraic matrix xy np recursively xy u diag identifiability singular vector imply lead dimensional cca computing whitening truncate x xy large common natural corpus computational algorithm whiten huge computational decomposition even whiten dominate truncate svd top svd classical whitening qr perform indicate difficult factorization compute besides whiten dense number bottleneck consider capacity ram system communication ask avoid decomposition huge matrix multiplication online e scalability begin play important role modern attention progress scale technique directly cca whiten several author try devise scalable cca cca thin recently proportion product formulate square exploit consider pca date back compare empirically fast apply streaming setting comes sequentially pass whiten step contribution tackle cca novel advantage state fold multiply width small dimension free eigenvector pair gradient converge proposition proposition leave failure nonconvex project nonconvex domain normalization contrary decrease maintain scheme guarantee alternate multiplication require extra input seem nice behind similarity subtle compare auxiliary unnormalized iterate scale turn fix characterize relation among quantity proof v xy substitute second equality third complete proof connection minimization intuitively close square informally second newton step solve sequence square pair give characterize lead canonical unnormalized counterpart insight intuition estimation actually least truth approximate update therefore enter pair achieve reveal broad contraction randomize capture correlation tb n spirit one compute normalization matrix u multiply thin width
vertex vertex total triplet vertex cluster triplet vertex error perfect clustered uniqueness suppose clustered triplet positive neighbor contradict perfect vertex thus vertex perfect particular perfect cover triplet cluster triplet triplet pairwise triplet force cluster together contradict conversely disjoint index define q perfect vertex namely ground triplet vertex vertex edge triplet error error edge vertex error cluster repeat keep self contain imply total cost total cluster lp tx follow eq pay cluster type contain cluster split reject time cost suffice second accept consider factor time number edge lp definition edge accept low cluster edge proposition definition agnostic label partition cluster try within cluster unlike minimize error enforce partition vertex bipartite problem graph give polynomial algorithm round cluster form object represented label whether go optimization cluster whose cluster cluster error np complete obtain unique approximation theoretical focus special nevertheless approximation graph van approximation approximation complete bipartite van recently cluster outside chen xu classical many science recommender bioinformatic rather cluster np bipartite introduce technical difficulty problem minimax social member community alternatively view constraint enable cluster application recommender quality recommendation organize factor approximation algorithm minimax version counterpart appendix minimax bipartite graph detail proof version minimax cluster integer integer interpretation classical throughout vertex neighborhood positive likewise interpret weight measuring vertex enforce objective classical clustering think minimize scale seek view limit relaxation relaxation round main introduction arbitrarily thus st ts u bound total generate optimal time pay error cost solution thus lp produce otherwise main difference pay error lp cost error complement solution proof follow vertex feasible error depend split safe mark safe x algorithm incur first pay incur total edge singleton edge positive lp must pay edge safe pay mark safe include grow bad pay bad make rigorous singleton inequality rearrange obtain bx uv uv k pay singleton total lp bank pay singleton mark safe bank pay bad safe bad safe bank pay pay uv pay time cost call edge appendix time lp edge outside also time x xx time pay instead cluster cost edge uk positive edge lp possibility inside clustered vertex lp pay bank times lp pay lp leave receive lp also make lp bank lp pay edge singleton k output singleton output singleton lp may pay bank pay lp pay singleton receive time ratio wish plausible optimum incur lp cost edge factor edge still pay bad argument incur lp lp bad triangle yield vertex set hand output q combine rearrange cost independent negative pay singleton cluster total create bank pay bad safe safe bad safe bank describe pay negative edge pay pay edge time still must rest edge must inside factor pay inside vertex inside pay outside lp let vertex xu xx jx u iw lp pay cluster lp cross cluster thus lp cost cross pay lp edge inside cluster cluster cost pay bank pay leaving receive type total lp pay bank times lp pay times pay receive lp times pay bank account lp pay lp pay cluster second lp cluster complete originally state attribute set induce triangle van van triangle np complete np every maximum minimax complete tolerance admit mistake perfect completeness proof regular vertex label graph partition triangle mistake expand optimal cluster vertex essentially augment every clique vertex vertex vertex every vertex clique clique belong clique mistake tolerance exceed tolerance perfect cluster one contain vertex clique cluster mistake exceed cluster four contain cluster triangle partition triangle cluster vertex exactly mistake among cluster cluster neighbor since triangle mistake vertex outside vertex cluster restriction
model event pairwise model distributional group base distance careful determine cluster suffer error robust combine discriminative signal distributional generative correctly mention mention connection gain room think key problem model information distance compatibility repeatedly mention external resolve sampling scale corpora resolution feature discriminative agglomerative generative note easily linguistic exhibit contextual effectively resolve extend operation university pf edu hierarchical incorporation use guide document distance use learnable prior corpus show document resolution identify describe share partition resolution document crucial processing task track extraction answer broadly applicable entity resolution deal noun phrase entity resolution extensively exhibit entity single event associate event decision event example event across document event ht resolution operate extend resolution g pairwise indicate agglomerative merge major decision base decision document propose nonparametric event encode guide toward well limitation dependent resolution rich pairwise bayesian cluster allow automatic determination event rich preference build dependent chinese process dependency extend incorporation learnable cluster encouraging event hierarchy allow group cluster group effectiveness conduct experiment large annotation integrate promise document extensive event less explore address event context extraction specify type group agglomerative cluster entity mention feature extension hierarchical across document consider linguistic likelihood nonparametric linguistic feature learnable similarity iteratively entity regressor quality level merging operation preference dependent chinese restaurant sequential limited work explore model distance topic encodes distance document assume document map segmentation novel employ document inference adopt terminology extend use something situation occur happen involve happen corpus action south location refer mention refer participant noun phrase phrase actual involve particular combination location note text may figure mention contexts document event problem group relation consider document resolution improve within cross extraction event extraction extract text event argument location one action reasonably argument semantic labeling predicate noun phrase capture event extract action participant location event head word matching produce high sentence annotate employ semi markov augment objective boundary rich include pos tag semantic phrase e np hold crf identify participant location head semi event associate mention training event detector argument event heuristic intra action event mention associate customer self affinity observation crp say sequential sequential customer mention refer mention start event relation exist document cluster model employ first second link distance form large cluster process describe restaurant imagine collection collection customers global corpus serve customer configuration structure circle represent customer color customer customer restaurant table restaurant customer customer restaurant table first use calculate path table undirected customer distance compute posterior link configuration carlo chain sampling sampler sampler generative customer calculate choose gibbs iteratively customer customer customer link customer start factorize denote customer belong marginal observation associate customer mention prior compatibility rich set event head pos cosine similarity head word embedding use train dimensional embedding event tf window mention participant tf vector time role logistic regression order document collect document event document tf across document conduct experiment corpus annotation event resolution annotation seminal event divide training train document within chain corpus c cd bl hdp event bl hdp pairwise event extraction annotate participant within cross previous merge document seminal meta seminal metric gold predict merging operation recover b proportion overlap gold gold standard predict bl group event head word pairwise single agglomerative resolution pairwise merging threshold document hdp within hdp feature hdp perform base measure hyperparameter development variant crp cross document reveal incorporate dependency
trust make circle similarity commonly inspire propose domain verify common relational datum structural relational mention oriented level whereas technique factorization trust meanwhile social trust reality extremely hundred friend besides people similar behave trust rank trust problem suitable paper recover matrix np relaxation order solve minimize row svd certain recovery sample rapid matrix completion mention whose recovery world recent utilize notably collaborative nuclear encourage recovery completion formulation handle inspire study completion establish minimax frobenius practical constrained section notation formulate max problem meanwhile accurately expense study dataset conclude paper discuss future capital letter row respectively denote element norm minimum simply explain tight nuclear eq row thus nuclear regularizer row max give approximate eq q expect projection otherwise nuclear surrogate nuclear uniform motivate norm max solver definite element solve obtain solver scalable max combine conduct trust link link dataset link label table link user individual degree form dataset trust baseline tune method choose let metric average error mae square rmse c c observe entry entry cm cm cm mae mae mae rmse mae mae rmse randomly measurement range split run sample deviation mae rmse detailed table result indicate obtain much small mse comparative baseline observe entry mae observed mae comparative term method superior comparative life mse rmse study examine application average entry illustrate accuracy efficiency also mae second second enjoy mae rmse compare baseline order magnitude three trade influence dataset little rank one know actual local minimum optima trust special trust observe uniformly completion solver utilize examine formulation consistently outperform observe achieve accuracy comparable year collaborative filtering empirically theoretically superior popular nuclear norm encouraging promise max practical clustering etc technology mail edu usa edu national university edu sg lemma corollary social address significant problem explore user formulate indicate however challenge trust problem bit sample non handle motivate recent propose since optimize utilize project superiority benchmark popularity network friend recommendation information etc trust negative relationship social user trust friend enjoy naturally social within matrix bit code imply th fraction goal bit pose solve social category first similarity individual people completion
control low rank differentiable strongly constant x deviation e distribution constant resp line frequently uniform sequence zero one resp furthermore notation state assume term depend hand gradient constant take constant large match gaussian completion sampling provide upper error q satisfy bernstein control uniformly logistic bernstein instance exponential noise estimator show coincide oracle derive tool allow prediction risk true belong w integrate bregman divergence leibler optimality provide convex detailed previous inequality imply hold use assume sub state suppose consider uniform satisfy factor depend treat frobenius error appear union combine proof nn u tc exponential bernstein rectangular see start inspire matrix zero integer subset cardinality distinct element difference element leibler xy ny give ia cauchy schwarz imply matrix consequently triangular duality q give sharp event define apply union bind argument let argument give eq ny apply inequality plug eq note duality subdifferential easily subdifferential cone exist divergence subdifferential resp right singular x w r side bound inequality x q holds see first argument therefore bernstein ensure independent n z q bernstein inequality solve q conclude distinguish completion consist reconstruct class nuclear estimator sum distribution family minimizer term penalization upper risk improve completion exponential r leibler translate upper frobenius risk rate completion exponential range collaborative filter much total whole sample index assume conditionally class nuclear estimator nuclear extensively decade prove setting additive sub unknown recover efficiently low rank prediction denote frobenius prove actually factor although discrete first later consider prediction nuclear case belong rich discrete exponential provide upper see match suggest room mild logarithmic penalization provide control noticed exponential additional inspire additive family kullback leibler order inequality minimax factor rest background exponential family distribution give completion inequality scheme finally defer throughout notation integer hilbert schmidt bregman
accomplish standard neural layer embedding feed let ground cross help neural network explanation since embedding sentiment secondary syntactic lose analysis embedding capacity capture word semantic supervise approach task sentiment small space irrelevant use small embedding protocol analyze subsection namely memory consumption setting code reproduce please refer website test sentiment aim sentence categorie sentiment weakly negative dataset testing training sentiment phrase sentiment approach leverage art convolution vary range well may teacher remarkable teacher additional knowledge dominate student dataset teacher express additional nonetheless encode teacher embedding cross complementary encoding embedding fashion straightforwardly long latter compare largely reduce representative recurrent neural rnns convolutional network cnn channel deep short lstm rnn extent indicate notice architecture lstm dimensionality nlp word embedding experiment superiority network combine exist complementary rnn rnn lstm cnn lstm zhang software institute china neural resource address problem embedding nlp task dimensional embedding retain well directly encode knowledge attract past year address probably extract large dataset aspect memory consumption appeal train small aim retain particularly neural application scenario device large much literature feasibility neural feed forward recurrent teacher classification guide argue truth feasible background review despite generic focus embedding nlp specificity bring knowledge word represent multiply table multiplication column retrieve particular embedding supervise apply encode specific original one fold address embedding propose phenomenon embedding notice encoding teacher complementary exist like say training teacher teacher guide student depict typically class variable teacher valuable student class impose training
near neighbor work give nearby stay increase table statistically tight rate perform well datum unlikely assignment several neighbor incorrect label c c randomly draw cube leave half example permutation exactly sample use reduce boundary change role assignment increase stronger increase weak bound distant play ratio neighbor assignment generate characteristic class cube cube cube eight cube improvement rather classify improvement even score likely score improve blind choice produce accurate produce bound example future fast speed test may permutation great challenge avoid assignment working programming likely explicit partition produce partition produce bound challenge friend join testing apply develop interesting concept bound well development new permutation validate bad assignment effective statistic bad assignment especially accurate classifier classifier permutation bioinformatic apply ideal result test bad permutation bad produce set example input work know develop classifier working example validation produce work bad work evaluating cause work example assignment rate assignment statistic assignment evaluate assign work neighboring follow term section review assignment present score work learn label random label input work classifier predict output w associate input eq labels indicator produce probably sequence example among ranking map draw sufficient entry equally likely working assignment likely rank unknown work set compute score score draw draw uniformly eq equally assignment bound highest consistent ease influence design scoring outline develop set permutation random permutation scoring function improve nearest nearby score scoring permutation last nearby among work scoring typically produce strong bound score nearby neighbor contribution emphasize neighbor among set breaking scoring q indicator scoring specify much nearby distant neighbor nearby scoring follow work favor assignment label base solely assignment example neighboring example near bound adjust influence distant affect setting refer scoring figure disagreement much disagreement neighbor example figure increase one neighbor role accept reject baseline scoring near neighbor scoring equivalent equivalent result different value parameter vary amount plot average row table error datum nearest neighbor work subsequent show cell show cell subsequent show mean standard difference mean standard deviation bounding vary quite bit error note cell value estimate trial size uncertainty mean trial deviation
behave factor ensure probability ensure optimal underlie max regret good regret see five estimator trial uniform perform estimator step underlie generate perform poorly furthermore dirichlet prior look figure lemma university california estimate distribution sample symbol learn kl decrease alphabet size min view limit know underlie know symbol equally show competitive reduce uniformly every alphabet essentially advance nearly natural also incur competitive natural demonstrate effectiveness term kl intuitive distribution generate distribution evaluate distance bit encoding estimating achieve estimator namely min min view know quantity collection alphabet simplify alphabet specify redundancy distribution measure consider appear motivated bioinformatic modern fair achieving example english alphabet english vocabulary whose corpus alphabet size constant show alphabet size several modification reflect estimate probability symbol show regret alphabet appeared give time restrict unimodal regret whole consider max competitive regret tight address collection derive drive well reasonable collection consider datum estimator assign symbol symbol exactly every sub person two relaxation come person compete person know restrict particular arise oracle consider know interpret competitive low regret partition part see appear context compression call competitive collection keep know natural question keep know permutation example permutation clearly relation partition let bound partition contain another restriction still exactly force appear time observe assign nx nx since estimator oracle compare drive low eq estimator good estimator knowledge regret estimator q show give permutation partition estimator organize state result section min competitive logarithmic incur estimator nearly however equation fact imply hold every eq big equation consist combine auxiliary denote appear sequence class show natural symbol ny follow negativity kl distribution hence ny ny estimator max
prediction comprehensive comparative feature task computer diverse traditionally technique art quantification tool quantification survey address encode information concern utilize feature texture example type classification several unsupervised style signature encode adapt typically texture statistic purpose highly affected resolution researcher feature base histogram sift sift discriminate comparative evaluated sift sift semantic encodes semantic low conduct al inconsistent texture pattern et al metric influence wide style cover five conduct bar convolution neural categorization lower optimize style preferred indexing large collection explain methodology appropriate combination metric accurate base style image low e object importantly metric learn raw visual meaningful additionally scale collection fine explore find explain type feature represent similarity art publicly knowledge large collection collection fine range abstract etc landscape etc collection close collection target automatic visual feature extract task limitation variation visual specific large intra visual challenging task select ensure testing use subset date restriction least total similarly use subset style follow first depict extract image prediction optimize style induce project optimize feature learn classifier prediction vs svm focus two follow use extract visual feature prediction project separately final training classifier fusion want information vector intractable separately project third project metric optimize obtain vector learn individually take account criterion computer vision unsupervise way learn categorization cnn dimensional characteristic high descriptor mean feature notion object low design scene categorization provide implicitly capture dominant semantic purpose image represent confidence presence image capture basic comprehensive visual apply feature vector follow extract convolutional remarkable categorization cnn layer follow three fully bar et al show output performance style cnn vector find zero function optimization eq trying adjust accuracy avoid overfitte result depend unsupervise mahalanobis distance decompose distance interesting dimension importantly significantly reliable metric supervise differ form objective near start project correctly classify classify member decompose choose rectangular matrix implement happen next solve optimization mahalanobis distance optimum solution involves locally instance target class apply sample stand lead popularity variation introduce call gb experiment assume poor root visual extract fact combination learning find metric combine mahalanobi shown learn merge metric final metric theoretically perform find similarity style rather mahalanobis involve information use measure I via aim preserve indicate start metric set similar euclidean iterative minimum perform sensitive objective leave distance weight minimize leave result variety explain extract visual level feature dimensional base representation descriptor sake fair task analysis eigenvector order type drop eigenvectors cnn feature eigenvectors metric l cnn boost version http www implementation adopt implementation www uci edu smoothly follow implementation author nearest neighbor regard metric fast one computational parameter metric feature category style randomly follow style section aforementioned concept learn metric however metric impractical highly toward list partition find penalty term fold l cnn dim boost percentage metric row metric style quantify similarity metric feature raw boost style great improvement baseline gain boost boost represent finding paragraph big abstract first art verify abstract characterize much active process confusion happen column row effect scale early capture system confusion color agree member color lastly acceptable art note post row th synthetic synthetic later act color perspective get reasonable ten vs feature row different achieve performance boost metric generally classifier less show confusion classification boost investigate landscape nd rd confusion element hand landscape similar daily despite landscape l dim boost vs produce confidence combination boost metric improve except cnn gain good confusion reasonable case row rd interestingly history friend paris interaction one column confusion acceptable american influence look report improve classification independent perform bad boost image perform rooted amount supervision cnn training unlike design bounding box style metric feature individually next aforementioned goal give project project together vector three task table early show fix bar compare report perform style half image achieve variation achieve classification metric project table feature work happen dimensional outperform art image reduce representation gain classification qualitatively feature learn fusion output right fusion close learn base across boost name six row six six investigate applicability metric meaningful metric measure metric supervise put close far learn significantly conduct publicly available aforementione task superior style superior learn work individual information metric across
fairly satisfactory precision estimation enhance efficacy cost flow generalization numerical assume generic pair wise one infer homogeneous precise estimation recent comparable belief empirical future study apply essential perform support grant foundation science massive continue importance rapidly boltzmann physics hamiltonian field interaction learn volume demand big require study boltzmann involve construct even amount substantial amount relevant quantity acquire selection science capture essential generative identify characteristic origin impose wise successful regularization norm pair employ seek number component overcome property parameter wide enable model study resolve lack implement technique often namely minimization reduce interaction problem mechanic exact point optimize organize boltzmann development area resolve fourth summarize ise bias represent sum adjacent component spin configuration gibbs boltzmann ising standard estimation word kl kl divergence evaluation therefore technique approximate partition pseudo function maximum term problem approximated quantity type field pseudo amount implement minimum method inspire impose flip balance q intractable due remove update rule instance metropolis heat probability flow follow flow computation tune expect change divergence combination elementary algebra master lead order monte impose balanced cost minimize estimate likelihood satisfy precision exceed straightforwardly assume assign component interaction bias originally unique equation impose condition estimation approximate description statistical mechanic law mean field validation generic powerful boltzmann pseudo likelihood minimization follow minimization let iteratively technique reach recursive derivative pseudo derivative yield backtrack gradient spin th th configuration backtrack acceleration technique minimization condition conduct several experiment spin configuration generate markov carlo spin interaction bias bias zero interaction restrict neighboring pair know lattice know selection priori estimation likelihood estimation iteration blue top flow fast convergent use convergent estimation correct bias interaction fig interaction confirm estimation interaction characteristic wise bias parameter observe nonzero wise bias small estimation threshold probability panel estimation minimum red slope guide pseudo iteration minimum interaction lattice precision lead estimation wise bias profile interaction show fig comparison emphasize prior sense snapshot indicate generative truncate
unless monotonically decrease happen trend consistent monotone change proportion population hazard big treat patient trial short hazard dominate effect trial baseline hazard combine effect big compare harmonic mean parametric sense confident control group trial harmonic trial effect underlying covariate need survival patient clinical time follow proportional hazard xt h p trial follow hazard baseline hazard variate baseline hazard hazard need hazard parametric subscript stand harmonic hazard aggregate patient realization maximum mle fit pool record follow try true overall treatment treatment concern condition get treatment establish assume proportional hazard true mild regularity prove mle solution converge probability detail asymptotic derive side mix third discussion maximizer minimize kullback mix ft mix mix ft version log parametric kullback distance proportional hazard hazard hazard interval still group patient text formula remain unclear discretized consecutive event history ignore hazard beyond estimate fact patient specifically interval consistent hazard survival true trial patient consider pdf xt pl patients survival law guarantee hence estimate hazard within interval formula investigate various definition patient go quantitative relationship definition hazard follow hazard hazard trial trial always equal follow hazard iii harmonic define true harmonic trivial eq achieve equality le le e equivalently algebraic definition imply comparison rule practical small typically small observe close indicate demonstrate plot three estimator always basis comparison ht pl lc pl lc assume great hazard ratio effect depend hazard ratio various combination l l l liu treatment effect clinical patient characteristic account real practice necessary development hazard cox proportional hazard challenge combine clinical trial formulate treatment effect estimate hazard interpretation analysis involve survival trial trial newly develop variability patient clinical trial efficacy overall efficacy population cox proportional proportional hazard hazard hazard vector log specifically patient interested inference base pool trial analyse comprehensive clinical hazard derive patient collect th trial scheme popular option obvious minimize though researcher recommend ratio covariate value effect effect meta mis baseline method require interested trial far develop meta set hazard unique log hazard trial vary trial hazard ik attribute randomness outcome however effect essentially genome treatment subject true vary patient population heterogeneou inclusion appear align impossible adjust hazard factor detect currently technology treatment pool statistic report patient investigate effect hazard survival method cover li provide another good necessity trial trial advance solely screen reflect patient profile intend capture either difference treat result combination treatment opposite life ready patient overall treatment cover patient concern effect calculation derivation guarantee correct amount significance despite usage completely exact measurement treatment hazard cox hazard obviously regard survival time matter coefficient treatment concern hazard later natural counting event hazard artificial concept cox hazard hazard overall cox population ideal case may performance shape hazard ratio say treatment readily functional address typically paper statistically drug efficacy challenge patient aggregate trial patient clinical survival patient x ik covariate function many accommodate covariate independent censor patient I hazard definition pdf hazard hazard trial collect assume hazard independent censor patient second trial trial matrix setup underlying value indicator arm concern convenient discussion pool line consider pool patient record estimate effect good answer first question obvious require patient originally overall define limit worth effect clinical statistic vice versa different treatment valid discussion provide example impose cox pool hazard hazard care establish true pool hazard control hazard formulation limit develop lin hazard patient pool convenient notation convention lin lin hazard effect censor hazard hand formula hazard eq calculation f xt expand notation proposition nf xt mf nf xt te te te te notation covariate identically order plug definition pdf survival substitute let censor true treatment either trial know denote respectively overall hazard define covariate treat numerically need patient nonetheless aggregate available solution henceforth subscript stand baseline hazard likelihood procedure know regard condition censor definition performance censor noise impose survival trial underlie hazard ratio censor always overall treatment matter censor assume sort randomization analysis use arm hazard hazard calculate pool patient record simplify transformation side strictly overall hazard superior small alternative look overall log hazard combine trial actually censor censor survival censor limit solution mixed population censor length censor hazard trial cox indicator clinical trial censor unity expand definition variable calculate censor simplicity fix equation define pdf integral unity fraction otherwise pdf turn define stochastically cdf make censor maximum indicate accepted treatment recommend report overall log hazard multiple former unbiased censoring provide chance researcher aggregate illustrate censor round patient trial randomization e patient control group survival times standard patient efficacy survival time trial patient censor confidence hazard censor various censoring try report censor trial trial censor pool pl
performance structural broad regime may bits less distribute non previous work good study communication communication machine problem however specify agnostic distribution one one optimize moreover convexity proves distribute split uniformly flip emphasize distribute study setting architecture e machine shared optimize study bound matrix norm denote exist parameter lipschitz g strongly correspond eigenvalue computation communication round computation communication complete computation clearly machine solve measure bit typically constant factor logarithmic domain gradient object model accuracy introduction distinguish relation natural correspond split different statistical often sometimes assign statistically quite similar example context point machine rank early situation typical randomly partition datum essentially communication one function remark relate typical randomly partition datum however construction actually randomly partition dataset partition gradient aware multiple round set initial statistical independence condition quickly seem assumption function impose certain mild operation involve gradient vector involve communication round algorithm sec initially round iteratively compute point every provide span explicit part include machine machine weak lie previous subroutine computing satisfying span incorporate optimization plus previous gradient well assumption satisfied technique aware restrict complexity perform requirement break bound namely origin convenience easily starting accordingly technique optimization construction dynamic roughly necessity communication round progress machine present smooth even machine quadratic sufficiently f round purely convenience decay implie large whether local method gradient round clearly feed result communication round iteration use descent smooth convex yield round constant identical remain utilize round low paper scale total communication round complexity somewhat function open suppose two machine local verify smooth optimum average function diagonal point coordinate progress additional result statement assumption simplicity informally discuss extension smooth therefore adapt namely state even satisfies lipschitz continuous unit round thm convenience together thm convexity smoothness communication round emphasize allow machine arbitrarily many operation assumption matching implementation subgradient actually function smooth aware algorithm performance wide thm idea fix round machine smooth argue result compute non exist satisfy machine without communication round finally result set related multiplying kind structural single communication round still capture realistic interactive distribute fp group feasible point infinite sequence lower lead principal half main q assumption communication recall point round odd machine hold state invertible admit yield absence machine function machine point imply machine progress without subsequently local round consequence communication prove main first smooth realize root choice verify range find machine round bind last must vanish fact strongly smooth bound pick round eq computing minimizer lower depend similar communication sufficiently construct provide type statement explain extract specify later ball q eq convex due lipschitz euclidean subgradient function moment linearity lipschitz linearity smooth case matter choose gain point communication modify differentiable machine point analyze round e execute must machine local machine similar line standard machines eq diagonal therefore mean contradiction assume exist absolute case contradiction remain depend odd valid subgradient satisfy rearrange zero well contradict hence thus generate hold repeating absence round whose assume initially therefore generate hold note contradiction odd term involve case absolute term odd depend subgradient satisfy rearrange term imply q implie eq contradict machine whose function round hold machine repeatedly get corollary round turn namely optimality communication round dimension employ ingredient derive corollary round must triangle bind output flip minimal putting take low well communication round strongly bind apply plug round consider must must construct two machine order provide receive let constant whose symmetric equal establishe definition indeed smooth strongly fact spectral eigenvalue indeed invertible inverse lie upper optimizer strongly show eigenvalue lie eigenvalue smooth hold machine produce machine construction communication smaller formalize follow theoretic lemma bit communication define return satisfie expectation convexity average theorem prove column thought algorithm base return operation get statement expectation uniformly random value eq let show lemma random exist symmetric symmetric whose letting plug random norm principle randomness remains choose e independently speak much information much probability choice recall entry sign bit hold providing recall independent send machine hold conditioning kullback leibler negative recall jensen e root upper root equal information variable compose ii eq second hence corollary institute limit efficient distribute bad room thing objective statistical datum otherwise communication round machine
recall class highly reasonable incorporate cost misclassifie misclassifie misclassifie assignment reflect class penalization boundary cost obvious weight interestingly strong constraint aim discriminant oppose much leave enough generalization performance uci learn usefulness public use input include output median make quality excellent new become combine category process leave randomly remain testing conduct preserve ht nature principal whole illustrate principal plot figure difficult considerable appear scatter plot align weighted misclassified point type misclassification unweighted plot omit omit bar compete ordinal observation three severe article version classify binary simultaneously extra ensure sufficient qp condition integer turn without nk j maximize example dual possible maximize objective ultimately share boundary flexibility usefulness fisher validate ordinal ordinal binary binary interesting include comparison binary denote first product suffice boundary possible consider q write two class positive complete acknowledgement support college foundation thank statistical spend write wu liu theorem condition pt treatment ordinal inherent class due issue ambiguity machine reality area disease diagnosis national security quality tumor iii security five category green blue red order severe quality randomly excellent good ordinal classify ordinal base actual importance ordinal ignore one ordinal one body versus versus ordinal sometimes suboptimal treat equally relative superiority reveal desirable approach utilize available class ordinal sequentially conduct classification combine meta liu simultaneous idea classifier simultaneous well many parallel classify classification boundary maximize pool binary formulation regression optimize parallel separate hyperplane properly ordinal problem ensure fairly either kernel lack framework property example study conclusion remark use ignore ordinal ordinal introduce ordinal lastly principle ordinal classification class prediction py aim classification mc classification rule opt ignore ordinal however suggest wise united state vote party least vote receive north blue much large latter color home classify state red great blue state x seems correctly identify simply break appear underlie root ordinal herein make ordinal appropriate simple lead next randomly state conservative state relatively less conservative conclusion furthermore binary separate combine combine negative label rule observation subproblem aggregate classifier intersection htb bc bc bc table classifiers compare meta observation second prediction reach ordinal prediction pool binary classifier first ordinal top length block class k inside objective kernel svm th minimize objective boundary however rich cover indirect approach call subsection kf slack incorporate wolfe duality come variant multiplier constraint tucker kkt kkt condition item eliminate full whose top primal problem kronecker dual nothing qp equality solve party qp implementation beyond scope k classifier nk lead implementation except lagrangian kkt invertible rest identical sufficient monotonically decrease ultimately condition discriminative need monotonically aim sign regard monotonically constraint I specifically condition logical implication involve number integer seek article exactly impose training vector train datum rich enough sign rather especially norm penalty svm difficult objective aware efficient solve show article simplicity mixed package capable deal available solve qp linear probably statistical integer integer aspect bayes fisher ordinal second normality binary py binary intuitively fisher decision rule fisher classifier function kk
independent build word predict linguistic text however restrict predict semantic benchmark propagate concept never abstract semantic derive corpus apply variety semantic purely symbolic ai linguistic modality lead linguistic often automatically induce collection text visual still drawback generally build linguistic visual concept context visual knowledge modality linguistic coverage apply vision labeling issue upon skip gram construct linguistic context relevant image present corpus human word predict representation jointly linguistic encourage propagation visual representation direct available enhanced achieve remarkably semantic benchmark shoot indirect representation abstract word break cognitive study literature multimodal distributional semantic representative straightforward induction text construct singular decomposition concatenation advance visual representation rely annotate visual attribute multimodal fusion stack autoencoder concatenation visual skip system approach derive multimodal unimodal concept image jointly topic empirically weak propose incremental recurrent focus acquire realistic scale focus affect class less effectively integrate feature word thus linguistic extend model proxy incorporate easily recent address image common induce annotated linguistic retrieve word multimodal follow et al implement subsampling option randomly discard word part softmax context target vocabulary normalization equation considerable actually word fix identity take concept like linguistic often produce visual visual gram multimodal equation ex skip force representation account note visual systematically e generally objective result distinguished multi way force visual representation try directly linguistic representation dimension linguistic induce second linguistic move maximize similarity max use connect margin enhanced word visual advance range visual sample visual act encourage word currently uniform sample control visual linguistic include linguistic representation sketch linguistic onto jointly linguistic straightforwardly substitute modal mapping induce overfitte l regularization corpus wikipedia comprise multimodal visual imagenet occur accord corpus associate imagenet convolutional correspond activation word derive representation facilitate comparison
graph misspecification graph adaptive weight compare final fuse difference correspond elsewhere consider per c penalty select performance clique weight improve clique similarly even clique information c subject group show star star graph star particularly group unbalanced version seem phenomenon behavior group probably theoretical value balance around evaluated computational set variance well penalize bic group bic compare present effect present difference detect parameter enjoy fuse select theoretically fuse nan difference fuse select include effect fuse l real drug stroke reach mainly due anti p improve clinical trial conduct gp incomplete three way interaction pool subject alone alone subject alone subject subject plus patient drug correspond elimination volume individual suppose drug pose small add fix g group penalize algorithm suppose group adaptive use ensure comparable inclusion penalize compose concern certainly low low among de gp experiment concern effect variance estimate fuse allow difference parameter variance iteratively penalize simulation theoretical study future variance volume correlate prediction penalty tackle suppose equal introduce concern bic validate especially dimensional since one subject receive modality could spurious association inter I mixed analyze group usually group group among allow comparison group parameter fuse fix effect penalize couple alternate multiplier solve maximization illustrate compare real drug mix mixed field especially clinical trial clinical modality drug drug clinical trial two patient treat patient population trial assess significant group categorical influence study intractable version em combine criterion group drawback group state reference allow select combination group group consider group difference difference encourage estimate fuse coefficient induce mixed model effect variance effect variance complex difficulty likelihood intractable paper deal semi selection penalize genetic variant penalize maximization square recent apply work use objective fuse jointly several detect variance effect maximization variance sum absolute suggest admm direct penalization covariance admm use group introduce bic criterion section introduce fuse penalize tuning section simulate cross clinical trial study drug drug observation th patient two measurement vector decomposable group patient transformation normally suppose diagonal explain estimate non linearity penalize introduce similarity algorithm maximize reduce criterion close form em divide two simulation individual simulation belong stochastic statistic explicit numerically except joint group separately characteristic similarity penalty within encourage detail penalty penalty encourage objective study potential maximizing calibrate except fuse penalize effect fix tuning parameter h penalize eq random usual update effect fix update expectation least extension alternate problem piece rewrite equality split solve iteratively solve primal lagrangian lagrangian box update z include algorithm tune replace vector expectation complete group depend penalize admm lagrangian primal dual augment lagrangian generally box box variance initialization convergence update solution numerically approximated tuning replace tuning infinity group difference bayesian criterion optimal return minimal q bic define q distinct penalize particular lar ols hybrid relax unbiased select constrain solution nan weight difference high finally correspond behavior simulated subject path depend parameter present impact variance estimation subject benefit weight influence penalty simulate set group joint variance compare variance parameter normally set implement last stochastic step equal evolution
label positive exploit label score negative surrogate bind importantly optimal separate sequel surrogate margin form bipartite select subset position happen item irrelevant scoring assign rank implicitly positive loss surrogate moreover scoring set defer conditionally notion weak condition scoring simply margin iff substantially negative strictly notion binary classification negative seem natural notion weak margin dataset surrogate tight optimal scoring upper bounding imply due surrogate surrogate upper notice rank upper way different surrogate score low rank positive highest rank surrogate well consistent strong margin label margin strong actually much incorporate relaxation tight replace negative consider convex ex q reader proof notable recover original label scoring set margin margin condition scoring margin strictly weak definition fraction assign score assign negative margin weak separate negative positive negative demonstrate three surrogate surrogate formulate mistake condition perceptron maximize performance setting batch gain popularity go away individual instant point top update update negative sake depend perceptron first positive negative score point fail rank score note I extension let receive sort score k perceptron enjoy mistake state loss defer appendix suppose cumulative mistake execute batch mistake also simple score batch mistake become easy hyperplane margin margin binary technique negative mini raise update slightly high dimensional design rank negative score false negative positive negative enjoy let mistake algorithm simplify situation separability definition norm exist condition batch mistake exactly classification bind strong perceptron outperform latter tight mistake suggest fails exploit sgd scale minimization erm pass optimal estimate notice problem mini processing sgd optimize mini batch surrogate make via gradient crucially unfortunately bias perceptron sgd novel theorem online batch guarantee generalization version surrogate score divide convergence predictor population w surrogate appendix well exhibit uniform establish result establish partly term surrogate manner positive labeling nevertheless strong batch compose point I u feed random stream batch generate model bt mistake ensure ensemble return stream c rt b perceptron surrogate well establish execute sake convenience us ks ns ns define thus scoring prove condition label identify positive ns k k case last margin claim mistake suppose let mistake define mistake tell far repeat application start p desire sake brevity step obviously last negative negative positive combine proof update convenience mistake fast mistake mistake prove part however modify prove lemma two sake update since sake convenience note false negative position rank I p crucially utilize help pointwise lipschitz norm establish additive top rank list nature situation analyse universe population arrange without replacement arrange least thm surrogate exhibit uniform convergence four separate subsection replacement sample notation label shall tuple first score let population arrange q application fix uniform fix define large well sample write vc convergence thresholded vc argument establish recall surrogate uniform convergence cover give require convergence identity residual hoeffding inequality tell residual follow order order establishe conclude uniform conclude least sample involve surrogate true label point define true take every two give surrogate definition measure population achievable respectively classifier achievable positive assume step third since fourth application union lemma set element optimality similarly write crucial give far last optimality proceed proof ease simplicity q similarly simplification assume last step since analyze term analyze nature define assign negative label suppose maximize clearly top negative rank positive formalize point arrange negative arrange fact proof exhibit convergence average score thing contain use least q show pointwise lipschitz evident pointwise analyze compose sort negative separately position list score list pointwise uniform lem sup fix analysis lipschitz application lemma tell inequality gives result lem rank universe total item arrange decrease let replacement population arrange decrease assume sake round element bottom sort population sort note bernstein replacement step complete surrogate generalization technique involve prove version thm conv execute stream batch length proof theorem closely theorem confirm precision top find relevance learn severe imbalance popularity significant gap notable lack stochastic optimize heart family bounding surrogate surrogate motivated principled natural margin surrogate novel perceptron provable devise scalable provable bound rely novel convergence structural conclude experimental state cut stochastic maximize relevance several life anomaly rank event rare spam rank accord importance performance average ndcg top rank list informally relevant item widely classification ranking learning remain knowledge reveal general rank aim develop classification agnostic notion distribution give deep framework setting margin condition appropriate recall top notion call relevant item separate irrelevant margin restrictive notably much restrictive item irrelevant notion margin suited perceptron surrogate performance surrogate key firstly secondly surrogate satisfy mention early consistent condition discussion reveal surrogate lie hierarchy gain analysis design perceptron extension perceptron perceptron mistake margin mention early mini style prove bound batch perceptron surrogate novel result require measure surrogate establish sgd algorithm perceptron algorithms organization present formulation surrogate margin condition reveal consistency perceptron mistake
decision fact well match make side conceptually learn interpretability netflix interpret assignment hierarchy movie movie split assignment assignment etc figure induce restriction branch cluster together beyond look branch come root tv range collapse guarantee additive cluster accurate diverse achieve publish domain lead model modularity explicit actor interpretability like valuable support google fellowship national foundation research fellowship grant national science conclusion recommendation material reflect view national foundation thick minimum draw circle draw sep inner black fill blue sep blue minimum google view google com j pa google view usa completion popular tool recommendation take drastically co allow learn surprisingly clustering modeling suggest capture latent preference decision make present classic cluster collapse sampler guarantee excellent efficacy art netflix netflix compare state art rating movie user preference item recommend like refer netflix research propose top item win netflix art large combination amount memory interpret integrate large drastically previous collaborative filter start assumption high completion study competitive conceptually interpretable netflix combination clustering user preference matrix factorization movie weight movie user movie neutral assume partitioning movie might part like correspondingly cluster partition pg rate rate take combination co clustering benefit movie user group instance movie certain age rating shot certain actor take combination attribute motivated order encode combination regardless template row column co nontrivial co partition magnitude small compete require user per number match human decision significant say well aim model real completion finding minimize residual geometry row banach space present devise collapse confirm efficacy completion netflix interpretable hierarchy network believe offer promise direction outline begin discuss related recommendation parametric simple mean co subsequently define bayesian collapse extend bayesian direction probably factorization svd success recommender ensemble relate bayesian ibp assume movie binary cluster somewhat overall similarity ibp simple parameterization co strength within intra improve beyond factorization capable factorization closely membership originally primarily row formulation suit biological wide variety ensemble far co clustering body mainly projection aim recommender parsimonious seek aim combination subset possibly scale per column note interpolation mining find way understanding lead pattern item database discrete rather dataset key template md nm st small simultaneously store sum already indicate general minimize compression trivial codebook store element store bit hold storing error dynamic basis nonetheless accomplished factorization approximation linear combination clustering co solution co cluster subproblem hard inner proceed column replace approximation find essentially new good mahalanobis distance stack case quite assignment exist exist obtain correspondingly entry assignment likewise coordinate count outcome row assignment vector alternate row column cluster objective minimum step approximation round dt ensure minimize loss additive clustering key approximated appropriate sake obtain cover unit ball incur element key relate entropy number singular scaling scale coefficient bound via rapidly decay value one dimensionality harmonic ball cover radius operator multiplicative clustering mean convergence apply guarantee denote singular cluster row singular cluster approximate contain ball unit latter accuracy guarantee residual matrix constructive set submodular bound practice manner penalize regression counterpart gaussian begin bt text height text gamma beta sigma observe thick thick alpha c beta r thick thick sigma gamma sigma fit membership draw chinese restaurant template rating begin co basic template belong draw chinese restaurant movie belong cluster analogously additive draw normal variance via conjugate pick primitive combination flexible formal inverse joint q user movie characterize conjugate likelihood accelerate considerably membership form conjugacy efficiently discuss collapse effectively membership family take additional statistic see movie denote user movie express encourage formation collapse need shall see keep rating integrate rating additive normal distribution vector rating likelihood expression determinant determinant assess beneficial assign user movie operation sum collapse gibbs likelihood assign offset log add fairly rating user movie purpose infer variance check normal parameter note play role classic term stein distribution denote analogously gamma movie equation implement efficient sampler cache per sum movie new matter check operation initialize partition movie n n update assignment analogously statistic beneficial iterate initial disk provide movie compatible possible model column regardless respectively nonzero partition column separate bin entry obviously crp unlikely retain fit rich cover piecewise block correspondingly noise unchanged gaussian inference note though jointly indice tractable overlap intersect expensive tb residual residual instead instead algorithm pass modified capacity modify I follow pass operation setup result real netflix run later practice yield hence implement range avoid simplicity infer assignment proceed burn period assignment many available dataset netflix movie standard three split approximate face black image node create learn matrix treat real value hyperparameter depend result quite factorization model factor svd user movie co cluster assignment column calculate size conservative row contain row since primary motivation filter discuss classic netflix measure rmse avoid divergence number using report publish simple conceptually parameter contextual bit training rmse
unlabeled cardinality introduce density label proof framework probability state extend multi originally object density adopt theoretic multi object suffer compatibility involve product power inner product density object weight normalize geometric fusion rule label object chernoff fusion propose subsequently show particular solution geometric mean derive necessary implement fusion rule result hold weight q l w summarize agent l note quantity eqs thus fusion chernoff fusion pdfs normalize geometric x rule apply theorem find summarize proposition agent agent share birth I dx bernoulli independently eqs overall fusion indeed chernoff pdfs time distribute scalable iterate average subsection agent iterate argument per follow primitive doubly consensus iterate converge global unweighted multi infinity stop consensus counterpart review subsection object represent gm q involve provide gm preserve gm gm depth approximate x b ab jt ab jt ab b ab x jt ab jt ab ab jt ab jt ab j b ab ab jt jt ab ab j b p fusion agent pairwise rule property order pairwise irrelevant notice fusion result fuse b separation component common approach represent single pdf combinations dirac delta kernel square burden resource demand gm recursion paradigm describe scalable multi algorithm along consensus propagate code object tracking order unique index distinguish object object kk clutter finite respectively omit k k density density motion birth death concept convenience label omit object state continue exist step evolve distribute accord birth contain element superposition object k k bi ii x ix bx w il detect generate likelihood multi superposition detect intensity clutter object mapping specify track measurement track track track one instant multi update posterior cardinality update take I p iw kx p ix gm algorithm sequentially carry locally agent operate interval gm produce object outcome description gm gm find consensus receive carry rule proposition perform merge location pdf reduce burden step procedure pdfs extraction gm object forward object predict density posterior k kp x z w k ir reader efficient gm reported sequentially carry gm gm section operate gm produce operation consensus gm gm max r describe object track surveillance wherein sake consensus track e velocity motion model nearly velocity model interval arrival arrival measurement respectively linearity aforementioned sensor order update three different clutter parameter therefore clutter severe mention surveillance scenario surveillance area birth birth summary c birth location state region birth clutter false cutoff average monte object independently measurement realization gm gm hypothesis describe merge gm survival maximum merging threshold truncation birth intensity consensus simulation display mean gm gm distribute algorithm merge lose track correctly algorithm localization gm gm cause gm filter drop track generally able object propagation gm gm similar fig deviation number gm gm gm cardinality gm fail track object become snr fails properly set track factor b clutter lose full cardinality extraction track fail even density show scenario case gm fig gm point gm exhibit term current present tracking sensor use consensus fully scalable way collect multiple admit multi object efficient gaussian implementation test initialization sake density get sum delta delta turn normalization exploit w x holds consider density instead straightforwardly evaluate apply theorem consider density get l prove induction proposition proposition ba edu ba edu address track heterogeneous communication capability theoretic distribute dynamic novel filter namely consensus tracking multi mixture confirm scenario label bayes consensus individual challenge track numerous literature fall major filter wireless lead agent capability net technology picture e benefit call agent operate knowledge flow consideration sensor network central scalable respect size operate operate information combine reconstruct scalability requirement fusion iterate cause presence loop suboptimal fusion ci generalization require multi approach state however uniquely identify hand refer jointly object paradigm explore specifically theoretic together filter filter formulation object track work multi generalizing develop consensus base label trajectory principle moreover conjugate prior development analytic tracking filter filter amenable tractable solution key object suffer like filter rest paper necessary fusion theoretic term leibler multi object bayesian present novel evaluate scenario end conclude notation object h convention generalize delta adopt inclusion generalization indicator shorthand whenever letter letter g throughout consensus unweighted iterate compute satisfy relate property square omit fusion consensus primitive stochastic column primitive doubly consensus collective unweighted primitive path vice versa graph whenever node receive receive information primitive doubly I unweighted carry methodology surveillance application vary association extend consensus methodology general trivial proper dealing object review next
patch patch exchangeability assumption extra beneficial prediction originally study house dataset task interpolation pixel choose random imply heavily initialization run initialize argue decide different see mcmc mf average preserve variable initialization except take gibbs consistently outperform outline study epoch mf dual among reconstruction mf gibb believe appearance reconstruction independence subset consequently much structure test mild extra cm cm cm cm c house mf breaking dependency experiment mf gibbs algorithm let mf need datum mf local optimum optimum gibbs predict mf prediction place little probability neither encourage similar encouraging would otherwise clean possible vast genomic crucial importantly inference cope dataset sparse experiment cancer sample include gene drug focus model around set feature rather interpret biological pathway understand characteristic drug profile randomly capable measure abundance thousand cell second control protein heavy effectively spectrum cell analyse consist heterogeneity result gene converge predictive mf algorithm beta conjugacy natural find preserve significantly intra dependence evident significantly mf mf interpolation model genomic mf suggest maintain dependence global mf variable multi bernoulli type local sensitive gibb also dependencie local variable care need ensure encode variational consider appendix detail variable approximation text paper clear bernoulli mf mf mf optimum optimize parameter maintain descent global analytically conditional gibbs sampler design scale probabilistic assess primarily derive genomic dataset investigate lda specifically demonstrate picture sampling certain dependency effective important lda intra perform decade see development flexible diverse nonparametric powerful enable adapt might adapt interest feature appeal scalability parameter analytically multimodal particularly norm along concern mcmc sample summarize simply give typically sufficient performance computationally inference gradient descent improve mini batch theoretical continuous unbounded influential modeling wang say factor continuous drive availability huge text corpus generate still time thousand high sophisticated analyze simple heuristic pca capture advanced analysis great variational one trade complexity evidence work global vector document vector suggest contrary case maintain dependency ingredient bernoulli increment measurable algebra generate consider simplicity beta may set form beta function measure follow z ik ik process stack dimensional matrix infinite difficult derive chinese restaurant dirichlet process scheme process ibp introduce strong dependency derive variational crucial rather reason dimensional parameterize modelling point belief induce encourage hadamard idea beta bernoulli parameterize beta draw generative model beta gamma gamma separation global crucial sake brevity I n dependency take compute draw simple shall mf spike method maintain spike dependency variable lose analogous mf local use local conditional p I mf mf gibbs latter variational approximation initialize step unbiased idea variational first propose stick ibp promising limit use ep scale ibp parallelization submodular perform limited positive field stochastic scheme interpolation task meanwhile model great develop dirichlet availability text idea initially propose refine million book idea learn parametric though sampling improve negative optimize deal conjugacy change conjugacy improve quality variational approximation exploit conjugacy finding carry stochastic variational inference carry denoise apply genomic transform hyperparameter rate schedule question variational answer
term specific problem poorly behave poorly behave cover imbalance runtime imbalance sensible behave bit term must entirely recursion individually query quantity amount bad rely branch bind technique work prune away success pruning depend show bound sufficient enable runtime hold hence runtime intuition answer query take constant independent large theoretical dependent independent plug get runtime dual tree require analysis constant denote imbalance dependence difficult maximum query reference respective whole follow similarly whole calculate dual bounding already build tight imbalance sublinear closely reflect behavior follow show utility simplify runtime dual tb distance near neighbor simply describe query reference study numerous approach cover near neighbor due algorithm sense respectively compare query point subtree improve neighbor node depend definition eq traversal store current candidate array array represent distance possibly query query candidate correspond notational convenience following take set cover tree prune tree traversal algorithms set c n clearly runtime thing remain reference encounter property r qp rp last point hold true neighbor hold eq r r step center point imply contradiction dp every separate yield behave sublinear represent runtime base kernel assumption adapt tight give kernel accelerate thus attention turn towards absolute dominate theorem search also gaussian bandwidth additionally g note dependence bandwidth demonstrate runtime reasonably choose approximately runtime approximately algebraic give exponential exhibit dependence bandwidth understand dependence bandwidth intuitively consider bandwidth increase thing reference scale allow less pruning level effect opposite gaussian kernel give regardless estimation reduce less relative relative division quickly tb query node kernel node combination q contribute create given prove approximate assume assumption theorem size possible approximate satisfy condition time expansion slightly rule tree relative approximate tree node calculate prove approximate estimation range practically identical size work begin range require understanding sufficiently solve array tb qp tb query reference difficult bounding query define expansion slightly set notion may run reference expansion dual traversal also run runtime pruning query reference reference ignore ball lemma produce necessarily subset assumption conclude take size obtain dependence runtime simplification sufficiently runtime simplify easily exponent get reasonable runtime necessary expansion tree retain parameter bind node framework tree tree traversal imbalance theoretically construction tight bound accelerate play runtime bounding show bound approximate theorem count tree involve maximum reference numerous algorithm core computation input implement approximate bad runtime prove problem runtime dual cover plug derive entire demonstrate plug first guarantee tree tree density estimation search surprising computational iterating near every reference close one require answer size subsequently prohibitive pairwise compute require reference accelerate favorable upon intuition physics dual tree extremely reference query separately query query simultaneously traversal dual algorithm easily understand recent query tree reference prune traversal traversal combination consist reference point hold reference single branch child node tree exist numerous tree kernel span search theoretical neighbor search approximate runtime guarantee span calculation combine generalization develop dual require introduce cover theoretical reader familiar cover tree symbol center tree cover hierarchical originally propose adequate description slightly cover level index tree node level associate parent consequence definition exist node scale child child contain within radius center note cover may easy child node remove child self take node tree construction explicit representation property expansion definition metric dp eq heavily literature dimensionality scenario dataset draw distribution converge see generalization expansion small fx fx close easy however empirical speedup existence dimension small distribute add origin whereas small single add origin encounter much convergence child lastly introduce convenience packing argument expansion subset may trivially sp sp point separate bad pack perfect imbalance lead degradation performance child effectively neighbor cause sort sort imbalance understand imbalance formal imbalance performance measure imbalance another aim imbalance utilize tree already cover index leaf level child need strictly low cover child balance hand child refer cover cover nothing number branch happen practice imbalance far graph dataset outlier away happen node outli point dataset easy outlier structure top illustration chain way motivated imbalance imbalance cover miss parent small root imbalance write calculation cover imbalance easy calculate imbalance tree cover imbalance level perfectly miss imbalance figure entirely like imbalance case imbalance cover reference imbalance near point imbalance leaf parent cover tree specifically aim imbalance perhaps imbalance cover tree actually cover originally intend neighbor approximate neighbor calculation dual algorithm abstraction tree node node pruning pair tree pruning traversal later node reference r r r qr cover traversal traversal cover implementation library traversal originally initially contain depth query reference end recursion maintain node maximum line query tree tree determine aim keep query reference combination reference combination checking line query prune strategy significantly node child node possible combine node pair hold separate reference scale suppose exist implicit child node child implicit representation argument hold set fact runtime notion traversal inter alternate contain recursion reach converse notion scale dependent nonzero pairwise pairwise define top minimum minimum node scale leaf scale reference extra reference recursion recursion situation happen let reference tree cover traversal extra happen query recursion happen recursion reference thus result apply
message contrast message variable incoming amp give behaviour evolution treat appear measure bit decoder analyze exponentially allocation tt remain step q termination amp decoder terminate sufficiently large main follow amp lemma lp block decay allocation error amp decoder measure outer rs one symbol rs guarantee decode rs section modification performance length allocation improvement error rate second hadamard gaussian amp mention consider amp decoder couple hadamard hadamard spatially couple modified power characterize parameter let normalizing ensure power recover section increase increase allocate turn help amp little correctly want amp gets start track large intuition limit correctly must exceed proportional need threshold decode power section decode performance flat allocate power compare decay allocation objective assign ensure final enough analogous limit allocation recover evolution top curve rate predict evolution point different give determined amp exponentially decay curve rate allocation rough guess solid curve describe use constant decoder specify exponential allocation design concentration around flat allocation improve rate bottom fig dash curve prediction power fraction correctly step show step evident yield question good rule allocation allocation section limit section step go proof essentially check good finite length challenge investigation computational decoder multiplication run time remain operation finite decode scale linearly gaussian memory requirement proportional store bottleneck scale amp decoder reduce decode memory generate hadamard design matrix pick uniformly result matrix column norm multiplication denote constitute compute length keep extend equal vector keep hence improvement decode infeasible power performance hadamard ingredient amp system accurately particular show almost ratio limit comparison similar recursively recall vector zero first define sigma recursively distribution product involve ingredient lemma column denote projection projection sigma algebra imply equal almost zero large recall mind determine quantity include function lipschitz statement say statement basis constant define gaussian independent section jointly exist finite lipschitz jointly convergence convergence almost limit strictly ingredient prove act limit convergence zero termination index decay power obtain taylor c know converge surely summarize consequently b orthogonal operator onto fix law triangular array array mutually second pseudo order jointly gaussian invertible jointly strictly positive constant let constant depend stein variable exist ni tc ij c element wise due exclude similarly due exclude exclude ai bi index contain expand taylor series argument z derivative alone ignore keep term side analogously amp give eq e need decay exactly vanish fraction section write write inner expectation jensen inequality get use variable recall rhs least kn eq small positive lm expectation bound recall cdf proof statement simplify geometric formula become simplification obtain use expression induction equal entry cf cauchy schwarz expectation non index jointly marginal side line imply n z together thank amp rv acknowledge support grant leave proposition capacity superposition approximate pass sparse superposition code channel rate codebook combination pass superposition linearly design rigorously achieve appropriate power allocation finite demonstrate matrix paper construct capacity achieve code white channel generate input input channel require goal channel capacity give superposition code algorithm decode decay exponentially despite achieve decoder soft decoder guarantee improve finite approximate message pass amp decoder performance prove decode grow decode proportional design polynomial class belief propagation algorithm dense amp proved particularly reconstruct small commonly compress describe measurement reconstruct though algorithm strong theoretical guarantee fast amp find cost dense infeasible implement pass message complicated value function amp difficulty scalar mean amp approximate approximation equation demonstrate could rigorously evolution hold constant sensing problem comprehensive relate propose amp paper rigorously decode go block tend infinity decoder length demonstrate simulation allocation scheme significantly improve close decode design build follow directly go limit rigorous analysis amp ratio section decoder probability decay probability amp decoder go give compare add allocation role exponentially decay allocation use decay section discuss length design know encoder communication begin length denote specific index scalar size receive generate successive message denote zero equal function variable q component contain brevity understand amp iteratively offline via monte relation follow terminology finite decoder iteratively compute maximum obtain statistic amp follow statistic message property presence reader refer term amp algorithm
birth year birth birth year birth year birth year year birth birth htbp ex description mean intercept gender interaction gender interaction year interaction year year ex abstract challenge survey propose survey relie available several national conduct consist disease mechanism survey generation model rate affect health survey rate way first rate participant non participant participant high death participant participant tend participant economic status education trend rate trend indicator decade trend indicator look mechanism ignore deal non make joint data sensitivity design longitudinal may recently subsample full variable non follow survey linkage naturally health correct illustration datum survey decrease decrease utilize detail survey analysis compare trend approach conclude pt national study project setup health education five risk key disease public health north beginning survey conduct survey area sample systematic people simple draw age sample balanced sampling age extend old north ex design event balanced area balanced area balanced age area gender age gender balanced age group answer daily elsewhere seem investigate period indicator page non htbp care health cause link follow contain date death disease j follow non indicator person background person age area gender variable survey survey people survey sample observe participant participant participant follow consist age diagnosis disease cancer participant participant follow person cause event censor censor age censor death date diagnosis date death person death person disease diagnosis structure concept causal model design represent causal bottom background affect vary depend area gender belonging case age affect people person background q cumulative censor survival baseline north risk describe different baseline stand area differ difference year particular study year variable indicator north area area area study indicator logistic gender study indicator eq gender study area gender year reference participant year birth north non participant lose imputation imputation trend monte regard eight iteration first discard remain store eight realization convergence diagnostic chain model two figure shape mixed autocorrelation cause coefficient good summary appendix posterior predictive population augmentation censor draw obtain participant draw imputation censor censor generate straightforwardly event survey sampling treat imputation full level estimate utilize utilize use rate comparable obtain trend imputation trend consider trend correct trend trend adjustment trend trend decrease difference difference correct correct estimate percentage difference comparison original trend study present ex gender credible north htp approach overcome challenge miss apply population non factor cancer potentially participant level provide cancer event participant model design bayesian utilize knowledge availability follow decade follow cancer unclear extent apply directly
likelihood estimate persistent seed simulation marginal sl sg non persistent seed persistent similar trajectory show walk abc mcmc scatter plot trajectory limitation persistent seed predictive seed extent posterior three sl mcmc row middle bottom right trajectory sl sg sl trajectory sg class persistent random sampler regard stochastic gradient variance abc greatly introduce issue repetition interactive statistical mcmc determine abc one monitoring ensure sampling example noise class work usefulness surface similarly abc surrogate minimize call yet still benefit hamiltonian random mini batch langevin perform start momentum necessary drawback momentum update update current limit hamiltonian dynamic prevent error dynamic hmc avoid directly author naive sampling full avoid b address estimate introduce scalar dynamic act update equation summary practice hamiltonian abc plug gradient implicit simulator gradient keep track random seed allow treat simulation function outside control generator part state synthetic particularly difference gradient high choose hmc abc representation produce close dim problem figure part around sl simple sl input x r x x function deterministic outside emphasize acceptable abc abc approximation free set gradient simultaneous perturbation stochastic work free wish optimize mask name entry estimate side call estimate side two side sided maximum estimation blue circle simulator deterministic smoothly simulator sl limit gradient due gaussian smoothed heavy tailed sum previously work side gradient make exploit step would analogous mini batch side simulation computation seed explore gradient hamiltonian landscape additive dependent stream mcmc use proposal persistent seed seed say seed internal persistent seed chain hasting randomly propose seed time metropolis transition location seed seed propose seed independent uniform ratio leave target distribution qx acceptance simplify could fix still sample keeping noise seed carry step persistent seed persistent seed histogram posterior persistent persistent sg achieve posterior problem let posterior shape simulator generate explicitly simulator seed vary reveal simulation blue circle horizontal indicate suited fixing sl likelihood estimate density gradient sl estimate analytically sl exhibit low sl quickly start remain possible leave persistent seed right persistent seed distance persistent seed walk persistent seed optimal seed persistent gradient consistent result posterior chain sl mcmc version sl marginal sl mcmc give identical space limitation seed gradient step persistent set run experiment table report posterior average sg seed persistent persistent seed sg trace single chain leave trace persistent seed
discovery approach score base constraint test construct skeleton exclude oriented arrive constraint ic tc score assign graph score score np often disadvantage inherent instability structure estimation change outcome discovery conservative develop structure new score causal discovery finite advance subsample search structural model causal scientific exploratory incorporation background constrain produce attention describe base discovery simulate world conclusion graphical section graph node arc e reciprocal relationship cycle direct acyclic four undirected graph edge order triple adjacent causal way state relation drawing equation represent variable cause error assume mutually typically follow hypothesis fit modify typical hypothesis add arc exploratory search literature address search genetic optimization prefer fit data objective propose make optimize objective multi optimal solution dominate bad objective model well dominate front dominate call dominate sketch dominate sort multi several procedure objective well preserve diversity model explain ii develop complexity sort lack ii population mutation form sort dominate sort set front sorting generate form create fast dominate sort new combine forming sort member front next population aspect instability change describe robust subsampling method yield sample wise estimation infer non tackle loss overfitte parameterize estimate objective determine identical end probability randomly subsample element traditional concept cutoff sensible similar model indistinguishable represent derive model particular also belong dag direct dag reversible direct arc undirecte reversible edge relation cause effect undirected members arc method phase phase search combine exploratory search iterative process return pareto front come phase combine ii stability output relevant return compute graph stability model stability complexity divide stability path pair level stability regularization parameter threshold occurrence correspond relationship stability threshold minimal model bic path causal relationship intersect top parsimonious call phase combine node edge edge background visualization interpretation knowledge example denote extend work translate dag specification perform measuring convert outer constraint may violate convert undirected edge preserve constraint return reversible dag produces order edge impose edge reversible return dag edge dag transform fully connect dag transform direct start stability path start path end loop represent loop graph j subset dag make population inner form initially else previous sort use mutation combine sort pareto front outer loop start size contain convert pareto dag stability graph consider edge implement modify handle package handle propose set mixture discrete ii stability ii loop population mutation rate initial represent predict model contain parameter predict minimize objective objective threshold effect equal minimum bic iteration loop subsample continuous compositional difference income light emission emission variable cause filter toolbox set roc vary path approach e region actually show well roc curve explanation stop curve stability value higher tend roc curve approach able find high stability end stability roc curve corner effective causal lie entirely method disease relationship development dedicated causal threshold minimum occur example graph subject originally longitudinal slice focus treatment assess individual strength assess measure patient physical sf implement e range treat continuous variable add eight causal show eight stability line correspond figure causal path path graph accord eight second orient background causal path reliability maximum direct path cause except cause study literature activity sense measure result change control physical sense control self focus whereas consider subject variable exclude instance insufficient remain variable subject assessment use treat knowledge variable
mark mark solid fill black forget crcr mark option solid row sep blue mark mark mark mark option black forget sep crcr mark mark mark option fill forget crcr color mark mark draw forget row sep mark mark mark mark option solid black black forget row sep blue size mark mark option solid fill forget crcr mark mark plot sep crcr mark option solid fill black plot table crcr color blue mark pt mark mark solid black forget sep crcr mark marks mark fill black forget sep crcr color blue mark mark mark solid draw forget sep crcr color mark mark mark mark solid fill black plot sep crcr mark mark option solid fill forget row crcr mark size mark mark solid sep crcr pt marks mark solid forget row sep crcr mark marks mark fill black forget sep color mark mark mark option fill black draw forget row sep crcr blue mark mark mark option forget sep color mark mark black forget table row sep crcr color mark mark mark mark solid fill black forget sep crcr blue mark mark solid forget plot sep crcr mark mark option solid fill draw black forget sep crcr color blue mark mark solid draw forget sep crcr mark mark mark fill black draw forget crcr mark mark mark black draw forget row crcr blue size mark black forget sep crcr mark mark solid black forget crcr color mark size option solid fill draw black forget plot table row sep crcr mark marks mark mark black forget sep mark mark mark mark draw sep crcr color blue mark option forget crcr blue mark fill black plot row sep crcr color mark pt mark option forget crcr color blue mark fill forget crcr color mark mark mark option black forget plot crcr color mark fill black forget plot sep color blue mark mark fill forget table crcr point locate center unit coordinate widely applicable sigma radial explicit utilize scale intersection axis dimension sigma refer building upon integration formula integration polynomial sigma sigma refer order sigma sigma filter sigma sigma filter construct gauss method sigma product dimensional weight disadvantage exact grow exponentially dimension exact formula omit brevity see grow gauss apparent polynomially number evaluation point symmetric formula rgb log sigma legend anchor north fill align table crcr solid row sep crcr color sep crcr color crcr black sep crcr forget crcr forget crcr black forget sep crcr point filter smoothing place low square cholesky covariance k k smooth equation likelihood equivalently three filter smoothing likelihood sigma log algorithm conjugate expectation em lower iterate bind update direct function sigma note maximize unnormalized log unnormalized log low log likelihood side measurement compute filter assume density equation evaluate evaluate recursion sigma equation enable evaluate gradient marginal base equation recursion k obtain filter predict prediction jacobian derivative q jacobian algorithm omit maximization find unobserved log sep crcr solid forget row crcr forget plot crcr forget row crcr forget sep crcr width leave south axis bottom line legend style north column align xshift crcr row sep color solid pt sep crcr solid row sep crcr solid forget plot row crcr solid forget plot sep color forget plot sep crcr color solid forget sep color forget crcr article sensor speed rate noise eq angle gaussian measurement assume separate sensor independent measurement tangent measurement variance sensor covariance know noise ground truth measurement sigma scheme sensor truth value sigma scheme sigma already state implementation toolbox initialize investigate uncertainty coordinate deviation component original initial consistent prior high sigma amongst accurate median mle compare close essentially identical sigma sum sigma negligible addition sigma trajectory simulate smooth simulated trajectory try em practically converge couple sigma rather direct mle estimate xlabel xshift align inside minor legend font xshift rgb rgb xlabel initial location sd ylabel median mle solid option forget crcr axis cs mark plot sep crcr e axis cs solid mark forget crcr e color mark mark option solid forget row sep crcr e axis solid mark option forget sep crcr e cs color mark x forget plot row crcr axis legend font rgb rgb height xlabel initial coordinate ylabel rmse trajectory color mark mark solid forget plot crcr axis mark forget plot sep crcr cs color solid mark option solid forget row sep crcr solid forget crcr solid option solid forget plot crcr color mark mark option forget crcr mark mark option solid plot row crcr axis cs conference various parameter maximization smoothing algorithm point well paper focus sigma transform order gauss particle filter sigma point filter extend kalman particle converge filtering particle assume kalman however high computational satisfactory nonlinearity sigma filter approximation integral cost claim integral beneficial test method case univariate maximum scheme similar gauss order conventional transform suggest utility legend style font xshift rgb height xlabel ylabel forget crcr color solid forget solid forget row sep crcr rgb rgb rgb width height scale xlabel ylabel solid forget sep crcr forget sep crcr forget plot dash row sep crcr dash forget row crcr forget crcr axis cs sensor uncertainty target sigma difference amongst method uncertainty parameter nonlinearity variance since sigma gauss scheme scheme consistent high sigma produce well filter sigma integration derive polynomial degree guarantee high integration rule guarantee filtering well hand approximation gaussian integral accurate approximation tracking mean smooth sigma point experiment increase rapidly initial uncertainty demonstrate local linearization sigma scheme approximate sigma sigma target sigma evaluate considerably suggest dimension produce essentially computation close sigma point tracking vary sigma point use iteration affect scheme sigma fraction sigma measure rule converge obtain reasonable approach sigma accuracy could refined scheme direct support grant acknowledge resource file intend serve file journal produce wish subsection go appendix one go like thank text text publish international conference information fusion sigma filtering consider space optimization em give expression approximation filtering sigma require transform quadrature simulate univariate tracking compare filtering transform accurate article parameter expectation sigma direct filter interest filter kalman filter gauss filter surface form k compute k resort sigma bayesian lie estimate static marginal joint state help equation see directly via linear maximization iterate computed maximization bind parameter require smoothing problem smoothing aim extend show gauss sigma use em model maximization linearization extend kalman computing base although easily sigma derive gaussian enable kalman gaussian equation sigma arise integral sum multi sigma present discuss approximate integral assume density approximately covariance covariance consist prediction result k compute covariance equation expectation take k k iterate smoothing gaussian k k density follow q expectation respect k kp maximization q evaluation smooth need solve gaussian integral integral follow form weight multi generalization refer gaussian sigma weighting sigma unit sigma root choice weight sigma stem trade sigma require quantify high method exact axis blue mark mark mark draw forget row sep crcr mark marks mark option solid black forget row sep crcr mark size pt mark mark solid forget row crcr mark mark sep crcr size mark forget table sep crcr scale x bottom blue mark fill forget sep crcr blue mark mark fill draw black forget row crcr mark mark mark option solid forget plot crcr mark marks mark forget crcr mark mark option black forget table crcr color size mark mark option forget sep crcr blue mark size mark mark option forget crcr mark mark solid black forget row crcr color blue mark mark mark fill black forget crcr height size option fill forget crcr mark pt mark solid fill black forget table row crcr blue mark mark option fill black forget table crcr mark marks mark mark option fill draw table crcr blue mark mark mark draw black forget crcr mark mark fill white forget sep crcr mark mark solid fill draw black forget sep crcr color blue mark mark black forget table row sep crcr color mark mark mark solid draw black forget plot mark marks mark forget plot table sep crcr mark mark mark black forget sep crcr mark marks mark black forget blue pt mark option solid fill black draw black forget crcr blue size mark mark mark forget table crcr blue pt forget row crcr mark option forget plot sep crcr mark marks mark mark option fill forget plot table crcr height axis line color mark size option black draw black forget sep crcr mark mark forget sep crcr pt mark mark option black forget sep crcr blue mark mark solid fill forget plot sep crcr color mark marks mark mark solid black forget plot crcr color mark mark fill draw forget mark solid sep mark solid black forget crcr marks mark fill draw forget row crcr mark marks mark forget plot crcr mark size mark option draw forget crcr blue mark mark fill forget sep crcr color mark option solid fill black forget row sep crcr mark mark option solid fill draw forget crcr blue mark option solid fill forget plot crcr mark marks mark mark solid fill forget plot sep crcr mark mark option forget crcr mark fill black forget crcr mark mark option solid black forget plot row crcr color blue mark option fill forget sep crcr mark mark option black forget sep mark marks mark fill forget crcr mark mark black forget crcr mark solid black forget plot row crcr mark option fill forget sep crcr width line bottom axis line leave blue mark mark forget row sep crcr mark mark mark solid forget crcr color mark mark mark mark option solid fill draw forget plot crcr mark mark option solid fill black black forget table sep crcr mark mark mark option solid forget mark pt mark mark option solid fill draw forget table crcr mark mark solid black forget plot crcr mark option forget sep crcr color mark option solid fill black draw black forget sep crcr blue mark mark mark option solid fill draw forget sep crcr mark mark option solid black draw crcr color mark mark solid fill draw forget crcr color blue solid fill crcr color size mark mark solid fill forget crcr mark mark mark option black forget crcr mark marks mark mark solid fill draw forget sep crcr color mark mark mark option forget sep crcr mark mark fill black
neighbor hence motion locally pca iterate mnist digit smoother anti reduce read vs preserve aspect digit sophisticated denoise average filtering loop remove digit orientation several robust reduction classification completion extreme tangent recover another spectral dimensionality reduction vs extracting cluster degree c shift applicability small converge focused technique subsampling structure numerical segmentation practical combine seem study mean benefit parallelism iteration finite structure know priori segmentation ms without iteration arbitrary neighborhood vary particularly large try accelerate ms ms accurately mode approach mode close datum run typical million pixel attack keep really convergence newton modify hessian newton reason low suited system compare gradient ms sizes effective hessian newton method ms ms ms often suffice move iterate mode newton reduce strategy datum close iterate ignore point introduce update em require near neighbor unless bandwidth portion weight another one datum however run point structure involve grow iteration predict simply ms assigning close differ near neighbor cluster cluster discretization fact trajectory mode keep trajectory soon iterate close mode image code cell iterate converge since cell end reduce shift assign mode discretization run shift r paris accelerate shift spatial classify iterate clustering notion topological persistence geometry enough approximation search neighbor compute truncate become bottleneck near neighbor locality hash lsh acceleration essentially fact point suggest soon replace point happen size iteration original iteration pn section nm nm pn prove replace single component method indistinguishable stop contraction segmentation arise constant bandwidth unlike dominate iteration merging mean speedup ms discuss early mode shift meaningful address categorical take centroid nk assignment kde move centroid mode separate combine cluster assignment suitable bandwidth high bandwidth becomes assign close shift centroid kde define robust homotopy algorithm means gradually optimize objective mean thus iterate colored projection middle plot cluster colored fig towards centroid slowly low linearly shape denoise stop iteration consist cluster keep eventually merge cluster show example location intensity result note ms generally gaussian clustering arise size horizontal although vision sometimes bandwidth patch fig illustrate kde manifold small mode manifold mode estimate reasonable kde handwritten digits image pixel feature ms produce mode uninformative mode outlier panel ms tune nonconvex centroid distant digit look laplacian look valid digit representative c c clustering mode circle right corner mode contour kde red centroid laplacian mode mnist datum assign input classical depend respectively represent shift advantage automatically find mean mode error hessian kde also mixture learn mixture density panel learn mapping robot forward mapping give angle configuration particle learn map speech shape inversion american english sound sound mode square illustration arm mode represent multiple correspond x reach mode conditional whose contour dot black sound shift also video image preserve track surface denoise iterate ms smoothing denoise shift nonparametric shift accelerate popular nonconvex image segmentation mode remain perform neighborhood reduction alternate datum iteration modify eigenvalue future high direct design walk laplacian cluster e distinct mode acknowledgment wang undirected edge subset connect connect depth recursively edge adjacent repeat remain runtime provide vertex threshold define vertex point md nm ball connect component give unless clearly separate reliably step ideally converge stop shift iterate tight mode separate tight corresponding mode connect cost separate tight exist large diameter small connect fig iterate heuristic accelerate dimension soon calculation since typically segmentation pixel pixel cluster boundary computing try average entire pixel compute usually interval segmentation feature pixel need mode must least pixel meaningful produce tight naive apply efficient component assign assign cluster number dataset cm lemma prop example conjecture conjecture remark false n electrical california characterize region contain density nonparametric estimate find practice behind algorithm shift tracking denoise mode laplacian acceleration strategy large segmentation intuitive assume datum whose contour suggest cluster step parametric maximize elliptical optima finding global dependent user initialization selection find shape segmentation optima kde mathematical crucially enable maxima require user bandwidth nonconvex shape need cluster try focus mode maximum kde iteration average one converge review kde convergence discuss extension mean shift denoise respectively disadvantage cost mode mode mode image area categorical radius kde contour kde bandwidth mode indicate rescale jacobian hull follow start cluster shift cluster belong kde contours bandwidth indicate mean shift cluster input multivariate scale call point gaussian weight bandwidth isotropic full covariance presentation scalar bandwidth commonly easy analyze rise simple formula rearrange discuss average simplify elegant bayes pn probability mode mode smooth give rise filter shift ms pt pn connect component mean pn pn stop connect ex ms nm nm stop nm nm stop criterion point assign shift bar give covariance firstly mode happen examine step second stop number would mode numerically merge mode connect convergence lie small graph user mode shift smoothed iterate mean shift iterate filtering processing bandwidth quickly meaningful tight cluster move relatively quite reliably ms component although shift produce specifically bandwidth clustering produce quite fast ms particularly accelerate approximation ms independently practically large iteration asynchronous soon make pick unlikely fundamental bandwidth way bandwidth kde set loss rule point bandwidth estimator give give sense exploratory range ms scale point set log vary clustering adaptive shift probability reweighte inverse bandwidth track object use scale scale operator scale filter space efficient rise shift mean computational efficiency since involve neighboring pair still find finite generate piecewise spurious behavior mode minima bandwidth scale vision image sum pixel equal isotropic gaussian scale give kde components image gray modes kde merging take kernel never create structure reflect mode mode increase single unfortunately gaussian kernel create scale practice rare create point large pixel track location running mode tree agglomerative clustering topological persistence explore hierarchical shift tree tool visualization sensitive cause tree construct indicate pn nm nd nm affinity define nm graph establish regard hence turn use filter scalar one parameter result filter possibly runtime involve involve iterate iteration costly consider find fast runtime sense walk slow ms dataset old new costly kde run ms assigning reflect fact ms whole space point view advantage shift kde nonconvex shape intuitive physical convenient cluster explicitly cluster determined initialization affect kde cluster change successful application pixel color mode depend g medical shift one costly sometimes monotonic meaningful mode create mode mode nonconvex remove mode finally mean computationally discover time derive kde ascent prove convergence stop observe use reduction locally centroid line bandwidth ff therein n early shift attention thank demonstrate success image follow many theoretical shift cluster anneal remarkable regard converge fast character convergence domain surprising area include computer vision attention reference cite sum unimodal mode decrease motivated paper show create mode appear mode need occur create mode see nonetheless two example bandwidth qualitative gm mode even isotropic diagonal far isotropic equal study consist gaussians vertex narrow extremely construction simplex vertex gm component simplex show mode large simplex narrow range mode peak slowly decrease towards perturbation prevent isotropic apart isolated understand location mode gm hull isotropic work shift scale gm identically isolated indeed nonempty repeat etc taylor triangular create ms step make give convergent valid integrate ms devise mode line search ms kernels ms mode minima modes ms gaussian kernel occur iteration sum piece possible subset within piece would ms coincide newton gaussian sublinear convenient relate kernel generalize define dataset probabilistic maximize likelihood gm model point bandwidth translate consist solely locate origin thus maxima occur coincide index origin compute eq ms non update correspond likelihood consequence ms apply em iterate increase unchanged indicate ms jacobian jacobian large along eigenvector associate large eigenvalue always case ms close smoothly principal component hull focus ms covariance initialize far mode ms progress property optimization agreement ms iterate follow property path iterate lie hull datum consecutive shift centroid fact surprising nonconvex ms desirable aspect boundary iterate occur define flow prove shift kernel broad support convergence see set kernel small enough several cluster iteration multiply eigenvalue eigenvalue point coincide broad result cluster converge easy rather integral solve towards axis converge cubic specifically deviation evolve reason extremely fast increase thus component slowly explain show fig iteration bandwidth linearly shape cluster apply generalized update standard axis function depend linear put spectral product pn nm walk posterior normalize commonly spectral eigenvalue structure considerably enhance keep eigenvector second however change iteration collapse cluster slowly remain thus block constant trivially extract piecewise constant eigenvector generalize implicitly without compute rely several give bandwidth computationally spectral solve run perform product connect operate intensity filter diffusion shift basically iterate operate jointly describe range update appear whenever square weighted weight one center another laplacian objective nn nm affinity laplacian optimization mean laplacian objective appear reduction note mean example cluster predict track video histogram simple initialize next current histogram histogram note kde histogram differentiable also kde shift maximize linearize location find pixel region enough track object time robust partial clutter camera et kde tt consider give importance mode mode scale define euclidean space sometimes cluster lie low iterate mode diffusion imaging
describe review art design global statistical redundancy summarize combination applicable obtain audio signature plus minus france email audio name hash powerful audio identification basically concern audio signature usually quick offer wide real art survey audio discuss audio thank increase mobile device internet music recognition song place want tv people want service tv require audio identification match audio signal store database direction signature medium consider monitor audio widely investigate technique recently content synchronization repeat detection live identification audio identification extraction derive relevant audio follow audio collection audio name etc systematically store store audio label match htb world audio signal kind distortion error derive save memory resource property audio requirement short numerous attribute structure centroid continue e gmm though diverse propose paper remain author knowledge comprehensive review eight year survey date extraction audio presentation particularly benefit researcher audio architecture various extensively model conclude depict purpose summarize filter amplitude normalization fast fouri dct wavelet frame input major since directly affect system diversity investigate reduction invariance summary approach first map length frequency coefficient centroid etc derivative feature variation audio signal localize peak top spectral multiscale pursuit feature previous recognition amplitude audio weight triangular spectral algorithm find audio video sequence user implementation music peak amplitude sep argue discrimination sound frequency coordinate peak describe sparse exploit peak automatic audio occurrence together spectral peak widely coefficient audio frame bin range binary difference neighbor wiener relate aspect audio often eq indicate power low concentrate frequency similarly feature also factor promising audio measure audio mass spectrum sc argue addition sc audio experiment world background sc result recognition base form transpose additionally characterize temporal pass post figure adapt statistical reduce redundancy spectral feature characterize ensure discriminative gmm gmm audio signal identification etc audio spectral model multidimensional density conjugate transpose determinant respectively k ml sense global log via em state characterize gmm kk n however gmm explicitly amplitude sound source similar template
leave error incur rather population mmd give synthetic adversarial net illustration generator mmd training noise gaussian generators mmd generator expect mmd bottom dataset distribute distribute datum parameter generator decrease regular space produce cast generative g measure discrepancy optimize adversarial net function net discrepancy incur origin point maximize exist every generator whose close shannon minimize generator mlp minimax propose take alternate along note composition mlp permit gradient paper adversarial net balance optimize suggest maximization step overfitte step bring desire unclear sensitive regardless gb ram potentially tractable adversary sample introduce replace family architecture adversarial mmd net solve reproduce hilbert rkhs carefully rkhs function product reproducing express closure kernel rkh nonempty compact space borel let mmd p fx choose kx underlie rkhs purpose mmd achieve desire detail access mmd n unbiased mmd define proposal input transform let function comprise depend minimization descent carefully rbf depend use propagation net operate mmd find empirical may mmd despite input mmd mmd sup e q fx complexity dimension bl nh ix estimation w q x rp rp c p proof appendix slightly restrictive hypothesis dimension generator take bound translation invariant length training generate appear mnist despite mean test report adversarial net several density second mmd understood optimize subsequent clear connection might explain suffer rbf kernel specific shift invariance image generate train mmd parameter adapt mnist use mlp architecture unit rbf batch generate sample log figure kernel density adversarial net outperform adversarial set digits mmd mnist kde task mmd worth costly acknowledgment discussion rademacher independent independent also fx c pp q take probability p state training least know therefore h p prove eq prove begin fy bl assume fy constant n p independent fy jensen fy fy introduce conditional sum expectation fy n fy random add therefore identically triangle e e fy q p f x assume ny ex proof apart state expand define kx nn x kx n therefore supremum also x unbiased eq split kx kx next define p g w hx h fw fw since bound expectation apply inequality theorem write f difference w hx w rate split q obtain look get finish proceed imply exist eq rewrite rp given depend ball induce infimum training neural maximum university convert theorem lemma generate frame statistic informally speak generator network nan hypothesis statistic unbiased discrepancy nonparametric kernel sample compare al game generator incur optimize empirical mmd draw fix close despite
methodology bayes space successfully statistical due enable proceed density bernstein polynomial natural logical spline density guarantee theoretical property spline reason spline provide mathematic devote spline smoothing interpolation conditional smoothing continue example concern spline spline knot denote spline degree dim b call write q derivative spline write notation upper triangular eq square weight parameter spline smoothing spline denote semidefinite stand spline want sufficient condition minimum linear inverse I matrix class solution minimum solution unique symbol require smoothing spline obtain transform function fulfil spline spline spline knot coefficient g vector simultaneously system perform consideration solution eq reasoning multivariate component introduce spline back transform counterpart visualization real survey national institute lc interval discrete version value obtain proceed approximate spline study option discuss furthermore I cubic knot spline coefficient fulfil spline also function transform nuisance variability effect middle one lc n n h finally present cubic spline proportional knot result smoothed consequence knot negative tail although numerically avoid logarithmic original subsequent spline ignore capture relative high lot respect local policy status behaviour proportion monitoring enable huge amount analysis become statistical spectrum need solve paper handle one smoothing tool purpose spline interpolation example advantage approach clearly cubic spline advance provided mention enable analyze dimensional functional g avoid compare polynomial concept nevertheless similar character paper challenge concern concern knot hand set direction development field science r economic via proper represent challenge task usual fully account character feature space constant constraint reasonable analysis function provide space issue aim spline transform density account reasonable discretized distributional development transformation spline function borel function frequently database individual preserve intrinsic enable meaningful usual seem inherent feature density density contain borel way space pa hilbert density order finite support reference stand integral without density popular transformation compositional carry obviously need
combine process create surrogate separate parameterization uncertainty quantity epidemic actual keyword quantification epidemic polynomial intrinsic parameter mathematical quantify reliability sophisticated quantify interaction parameter force quantification computer carlo carlo comprehensive processes variation create intrinsic reconstruct method simulation characterize range sufficient generate statistical process sample surrogate model parameter uncertainty variation uncertainty model parametrization need prediction range know rate disease typically observe epidemic population input parametric prediction create refer stochastic variation observe epidemic become infected nature community contact input connect specific recover specify input epidemic know uncertainty epidemic epidemic source combine statistical allow intrinsic uncertainty show statistical intrinsic uncertainty gaussian presence intrinsic sample fix account intrinsic variation perform variation parametric analyze add variance increase far situation unimodal distribution satisfy unimodal separately eliminate simulation fall thought study simulation event mode response carlo surrogate model parametric reproduce quickly simulation behave though exact remain introduce epidemic simulation account uncertainty sample throughout surrogate keep mind come epidemic lack epidemic general due use epidemic system sde represent disease epidemic sir differential sir outline derivation individual time infect constant recovered evolution sde eq q wiener infection population represent infect individual recover completely individual recovery rate infection rate run stochastic sir indicate variation sde studying least distribution unimodal transmission recovery uncertainty experimental datum distribution histogram plot recovery account sir quantify uncertainty draw solution record repeating form ht sir infected series effectively h kernel estimation sir detailed paper able reconstruct sir simulation able denote x represent quantity discrete index indicate discrete controlling know absence sir notational translation kl uncorrelated lack aid kl decomposition benefit possibly reduce first separate remove correlation kl find eigenvalue euclidean vector onto basis kl problem independent control eigenvector eigenvalue effectively show compare figure kl coefficient implementation reduce effectiveness kl sir distribution kl reconstruct approximation depict uncorrelated construct decomposition two goal allow sample distribution parameterization although random variable polynomial basis variable show choice polynomial sir basis scheme orthogonal polynomial respect reason basis degree dimension polynomial q standard normal density high tensor multi index polynomial multidimensional order indexing dimensional vector correspond q variable purpose space reconstruction three sir polynomial smooth truncated equation formally formal since live space monte expectation domain standard probability common explicit cumulative jointly onto cumulative uniformly random variable cumulative likewise map expectation numerator note inverse important carlo sufficient equation characterize intrinsic explicit practice fix finite denote conditional cumulative estimate kde univariate tensor product univariate denote denote derivation follow kde equal kde goal build conditional function density easily accomplish increase compute process repeat coefficient decomposition equation approximate simulation pc pc expansion important property uncertainty define probability intrinsic quickly realization intrinsic uncertainty gaussian intrinsic depict sir fine scale preserve well increase sir form kl pc decomposition depict maintain cut due effect truncation kl bandwidth kde pc surrogate parameterization pc arrive gaussian process set review statistical highlight process form coefficient coefficient gaussian fit series way away uncertainty lack realization simulation parameter coefficient k coefficient sample value subtract dividing decompose use truncate construct keep singular build standardize r r independent truncate decomposition nk independent construct come truncate q w length vector p I apply process evaluate point standardize k matrix principle vector relation define normal step sample hasting univariate walk independence sampler one hyperparameter distribution prior ts base I ti model pair respective apply matrix prediction relation coefficient variable
france currently ph degree university technology research interest hyperspectral engineering engineering receive ph security france laboratory university france research interest wireless signal nonlinear system good award signal year review paper france email fr fr nmf powerful extraction apply field technique single propose bi feature account weight bi nmf pareto optimal instead single front study approximated hyperspectral confirm bi nmf art nonnegative factorization pareto hyperspectral factorization nmf provide become oppose principal discriminant nmf decomposition yield tractable interpretation datum recognition blind name approximate rank nonnegative input two consequence hyperspectral cube scene light reflect range available pixel pure material extract recorded estimating abundance pixel interpretation view euclidean product either frobenius generalize kullback leibler intuitive interpretable decomposition temporal smoothness dispersion g bregman divergence oppose research activity nmf work nonlinear extent scope several kernel nmf employ high perform trick map hilbert call know nmf consist product input scheme additive combination map note residual matrix factorization map severe disadvantage basis reverse difficult obstacle difficulty space optimize directly nonlinearity section detail either conventional essence dominate vice datum real fusion nonlinear reveal close ground nmf chen former nonlinearity depend post study bayesian residual share one nonlinear process conventional vertex separation elegant framework estimating combine paper stem define objective exist decomposition deal method literature integrate pareto front multiplicative organize difference optimization propose bi nmf demonstrate hyperspectral work variant nonlinear equivalent consider separately represent scalar frobenius square wise minimize measure coordinate alternate keep straightforward nonlinear variant study suffer pre study consider column norm inner call machine example function apply wise entry residual error analogy descent scheme difference linear sample namely minimize euclidean consider feature two trivial attempt bridge gap estimate mapping pre problem detail literature investigate coefficient implicitly worth framework machine see several nmf extend provide optimize input next investigate combination additive nonlinear nonlinearity outline nonlinearity additive relaxed post nonlinear study residual nonlinearity investigate width nmf approximate nmf solve simultaneously simultaneously two feature see optimize simultaneously function namely ill indeed find dimension bi bring space objective belong beyond optimization multi objective widely literature take advantage bi optimization dominate j j inequality pareto dominate objective improve degradation objective pareto pareto objective successfully evolutionary weight normal name reference respect argument kernel descent scheme differently guarantee easy sensitive lee nmf without generality restrict presentation valid multiplicative procedure derive update rule stepsize additive rule yield become gradient method e nonnegative stepsize page multiplicative page division multiplication element wise dd pt distribution nmf hyperspectral problem initialize column randomly image stop attain iteration reach stop iteration aggregation predefine threshold mention multiplicative unity update imply tucker kkt however kkt concern kkt guarantee multiplicative nmf conventional show provide relevant art propose bi nmf pareto front propose hyperspectral digital sensor top part pixel original image raw channel recommend clean accord build acquire contiguous spectral band removal band band area dominate material employ vary gradually iteration initial objective input determine pareto pareto pareto front strict solver minimum mention nonlinearity nmf problem pareto refer pareto front operate abundance matrix evaluate single point approximate pareto front objective evaluate dominate front outperform objective good pareto pareto maker dm worth pareto solution dm make final dm specifie generate detail regard pareto original multi problem pareto front objective dominate surprising could scheme due solver solution pareto within objective interval obtain solution pareto dominate global pareto point nonconvex global solver generate pareto optimal front pareto solution pareto front nonconvex front attain use case nonconvex front probably result drawback nevertheless pareto pareto dm single objective underlie nonlinearity hyperspectral two reconstruction define denote regardless lead comprise abundance exist estimate connection extraction estimation technique enable estimation variant extraction vertex existence seek large technique abundance constrain use consider sum nonlinear abundance call linear nonlinear bilinear factorization jointly constrain dispersion regularization nmf minimization
w w nx nz partition entirely integer partition addition suppose maximal inside whenever solution g nj allow repeatedly transformation let suppose suppose follow update keep fix clearly partition always ensure terminate step convention proposition k mutually exclusive alternative addition step supplementary algorithm check sort nonnegative project onto algorithm preprocesse appendix supplementary material initialize g nz go previous immediate outer loop z update maintain every iteration pass terminate pass terminate naive bad sort ratio update get implementation relational implementation consist tuple ratio notice assume tuple order view constant boost library remain delay fista computing quite intel cpu matlab b table display generate solve apply level e paper algorithm make available website thank read suggest introduce discuss implementation issue intuitively split partition I I g g jj g jj z I w z w eq expression w constant p z ji gx eq n proposition n eq whether optimality part proof n n w w contradiction argument z g identity imply prove claim g g z group strict contradiction conclusion onto radius weight pass tw z pass tw pass finish example integer nonzero solve unfortunately nonconvex np hard convex instead recently researcher beyond predictor motivated yield nonzero identify take account feature structured predictor select group feature prediction mathematically replace regularizers regularizers include fuse regularizer invariant desire know en grouping development weight see include recently investigate discover atomic norm characterization computation atomic apply frank wolfe cg however nonsmooth fitting include frank wolfe long ball proximal cg get root project onto terminate introduce code project onto norm proximal projection devise currently compute proximal norm compute unlike proximity norm arise evaluate operation introduce partition alternative proposition theorem
par variation I en les si une application optimisation dans cl es analyse energy de abstract evaluation adopt uncertainty optimum measure entropy come negligible propose solution several evaluation integration energy electrical keyword analysis computer energy electrical minimizer construction mu problem end certain adopt classical constructing shannon quantify approach minimize evaluation equivalent mutual reader criterion point approximation carry gauss quadrature need entropy plug range turn sample moderately path noise moderately criterion essence going carry since single limited yield little progress dominate first left idea evaluation evaluation iteration contribution suggest build residual result minimizer minimize suggest large variation carry evaluation iteration evaluation moderately expensive processing possibility update evaluation idea evaluation artificial motivated fact numerical condition k consequence simply respect conditional second path simulation condition available development branch toolbox cb cb ct ct ct ct ct ct ct ct ct ct ct ct cr cr cr cr cr cr represent corresponding sample leave gauss integration strategy energy electrical describe operator connect strict economic requirement value cost fx denote characteristic connection expectation scenario computer program assume evaluation scenario scenario generator identically I evaluation noisy sake simplicity evaluation without initial spaced batch evaluation perform iteration reference iid kriging firstly estimate initial evaluation adjust batch depict entropy fast iid budget evaluation suffice accurately cb cb cb cb cost ct ct ct ct ct ct ct ct ct ct ct ct ct ct ct ct ct ct ct cr cr cr cr cr cr cr cr cr cr cr cr cr ct ct ct ct ct ct ct ct cr cr cr cr cr cr cr cr cr
combinatorial ibp importantly scheme process see miss ibp measurable process indeed exchangeable direct process ordinary due characterize term describe ibp analog chinese restaurant produce bernoulli generalization stable show ibp weak definition theory probability one assume equip algebra lebesgue measure completeness aa note require partition element measure algebra projection value e alternatively iff distinct measurable onto measurable general eliminate agree clear measurable function random say intensity point measure uniquely result characterize poisson process q increment completely random fundamental completely reader ensure space characterization completely note compactly completely random measure purely atomic poisson call q eq uniquely specify component evy latter encode position intensity evy measure intensity let support independent eq evy functional bernoulli beta process mean purely atomic measure evy fix independently measure poisson sx bernoulli ordinary simply evy simple bernoulli process bernoulli process straightforward let ordinary countable variable evy independence increment suffice process point theorem evy bernoulli law characterize mean consider measurable completely purely atomic evy q ordinary implicit refer remainder insight state merely ordinary component infinitely beta process break later due g describe conjugacy censor result observation completely hazard combinatorial absolutely continuous lebesgue rely implie claim omit conditional introduce follow terminology let say condition bernoulli every comment monotonic convergence nonnegative measure direct concentration combinatorial induce chinese table proportional generalization space countable every copy process concentrate measure whose atom atom among atom measurable sequence randomization elegant would work work minimal unique extension measurable extend us law arguably decide scheme characterization equally conclude independent increment rule expectation increment recognize law bernoulli parameter scheme concentration unique moreover parameter condition give agree determine distribution exchangeable bernoulli argument hold countable generate simultaneously parameter ibp space measure sequence verify equivalence denote sequence functional maintain enough every point atom opinion characterization equivalence induce propose follow ibp ibp define later introduce relate exchangeable random characterize generalization exchangeable sequence combinatorial structure partition may empty class complete event nonempty token structure part class among take token integer sequence integer sequence finite exchangeability construction satisfy know completely show scheme hold say de know tail measurable random characterize limit relative token depend sequence way combinatorial arrival token token appear transfer argument scheme bernoulli interested mark bernoulli mark converge scheme every every exchangeable measure independent random version event moreover characterize q pp precede extend define whenever first token extend limit boundedness understand term characterization b measure mean measurable measurable hazard bernoulli rule complete may continuous characterization component let satisfy sum characterize law write binomial distribution variance every particular restriction measure eq claim note poisson probability measurable satisfie let count sum identity complete measurable rule suffice equal identity much focus n k k column let f n eq straightforward verify establishe claim verify law distributional average average continuity suffice nonnegative measurable boundedness continuity measurable dominate complete p fs partial average partial average let q straightforward verify q surely intensity proof note intensity limit average long appear converge limit support singular measure identically support almost surely follow borel characterize countable collection measurable take distributional limit average sure sense development direct bernoulli x existence limit partial version development completely infinite yield completely last equality corollary note following develop stick break like identity calculus measure scheme measure measurable identity argument understand component one take introduce right bernoulli complement support total mass ordinary component upper chain expectation eq final claim immediately see independent study combinatorial structure poisson process lead generalization process special entirely cardinality poisson measurable fact randomness recall appear time indeed distribute trial eq statement multiply identity exchangeability q immediately corollary exchangeable course underlie indeed another note combinatorial exchangeable sense consider exchangeability permutation array exactly equal correspond sharing determine array order develop ibp write order except order adjacent equal view atom label correspond measure atom allocation leave realization order form informally array uniformly random permutation sequence column label array obtain sort distinct column denominator fact indistinguishable copy nonzero symmetric dividing play exchangeable exchangeable induce characterize homogeneous describe generalization ibp exhibit understand similar discount chinese crp exchangeable direct measure characterize law ibp parameter crp deep ibp correspond induce two crp crp define measure vary homogeneous law purely atomic I frequency token chinese absolutely continuous ordinary component recover show stable beta ibp perspective connection parameter crp ordinary atom see define recall mean beta crp multiple dirichlet distribute eq distribute trial even absolutely beta hence absolutely simulate exactly stick break characterization know demand theory make precise particular computable distribution rule work pr provide absolutely continuous characterize line describe simulation produce appearing time ordinary mass know beta structure ibp function special k n token appear st equivalently allocate chinese appear new calculus model make exchangeability n new token st admit additional copy token lead agree appear connect note informally speak condition exchangeability appear summarize parameter ibp discount worth g propose law conjugacy might consider governed binomial make correspond kernel ordinary component beta imply law weakly together imply beta work exchangeable investigate limit purely measure strongly let suffice show convergence complement generality partition countable restriction fix satisfy claim locally map distribution intensity suffice establish complete weakly contrast established follow special counterpart thank author college international fellowship transfer translate distributional claim variable extension underlie measurable space space random measurable may whenever borel space measurable exist measurable space iff measurable proposition var atomic beta scheme describe combinatorial conditionally process share beta measure show beta combinatorial ibp beta scheme parameter measurable scheme ibp exhibit power idea probability rise generalization beta dirichlet chinese restaurant sequence beta process generalization process change introduction ibp characterization relationship extend ibp direction despite beta beta conjugate beta dirichlet subsequently consider measure scheme stable ibp give rise article combinatorial structure de combinatorial structure collection informally allocation component subset element distinct locally countable hausdorff set algebra generate equip generate cardinality recall part simple fix purely atomic atom partition call process informally every block partition independently atom take atom partition permutation carefully equivalence relation exchangeable sequence exist purely atomic give completely mean measure hazard measurable partition induce crp highlight sequence limit necessarily purely characterize chinese appealing argument refer allow vary across outline agree construction measure
information multidimensional decomposition interest decade successfully apply range area vision body tt decomposition similarly tucker decomposition td context contain greatly reduce original core tensor previous mathematical frequently multilinear algebra multidimensional array contain tensor scalar letter letter capital letter tensor tensor I vector third general multilinear sample nk classification supervise category define td single core tensor space tensor far alternatively core tensor tensor core tensor paper common firstly tensor index introduce vector sequence decomposition index arbitrary position core need ensure form rearrange canonical identity position orthogonal extract subsequently core tensor core regard training use classification contain necessary odd order core feature may sound exploit core reduce number core pattern obtain rate image divide two test ratio datum structure average trial high hold several accuracy ratio maximum accuracy htbp database originally order trial high hold ratio hold ratio hold high accuracy ratio respectively htbp database pose illumination image area pose size tensor fourth test core tensor machine v nn nn accuracy plot versus feature obtain high classification accuracy respectively tensor direct affect classification test high necessarily feature large increase ratio seen decrease outperform method color convert image face accuracy multilinear discriminant discriminant multilinear discriminant propose mention classification show use multidimensional supervised require needed classify high tensor recognition problem need efficiency detail multilinear
leave deal nk employ km n nk km n nk km put computation universal constant ii super exponentially case union deduce divergence kn substitution give error nn n universal putting inequality apply reveal c c denote hypothesis recovery output class employ contain let take global offset non trivial l cardinality cut homogeneity recovery ability restrict hypothesis know definition union one k finish establish special without like alternative begin set produce hypothesis denote associated cut fix necessarily mind far cut attain inequality q finally consider alphabet support compose hypothesis guarantee type suggest pick hold put result establish theorem vertex black determine put colored vertex nonempty cut degree repeat argument repeat color scheme eq claim begin expression measure dirac point remain hellinger divergence elementary identity mp mp mp mp indicate mp since recognize apply chernoff yield sn sn union assumption union sn chernoff sn n sn sn employ union sp inequality rely assumption union indicate put bound e p p np complete obeys l represent index quantity cut clearly together rise quantity separately cut ensure empty together q exceed exist secondly total feasible constraint cut size exceed eq put together k inequality hellinger divergence say immediately establish known inequality paper concern jointly measurement imagine take pattern represent measurement channel transition tool decode problem general structure alphabet channel family characterize corruption homogeneous almost irrespective general application outlier lead order recovery case improve random geometric graph various directly pairwise relation pairwise include cluster relative rotation pairwise pair sequencing later substantial consequence joint recovery feasible soon pairwise pass paper explore imagine graph accommodate nature solely channel represent channel uniquely difference ji pose receive field social biology list exhibit community group share feature aim observe similarity member simple represent assignment encode two belong view single angle position rotation several view view pairwise application include vision biology reference image shape physical across input cutting pairwise match aim refine globally numerous graphic people mostly nucleotide position single nucleotide snps associate snps cause various develop sequencing method particularly sequence reconstruct disagreement pair pairwise recent primarily motivated consideration spectral develop provably synchronization choice study manner limit application despite develop instead account similarity motivate graph fed channel information perspective distance success measurement metric graphical insight feasibility exact understand application pairwise recovery turn benchmark evaluation comparison paper towards unified characterization tool measure kullback determine feasibility channel super polynomial coincide broad homogeneous case fix alphabet characterization possibly different rate increase asymptotic tend illustrate effectiveness theory concrete consequence application investigate prior outli problem recover regime side focus characterize theoretic limit determine information theoretic limit genome random graph graph condition general order preliminary recovery structure develop tight characterization measurement aforementione special graphical refer channel whose probability edge previous graphical channel quantify residual output investigate full might interesting notion grid step general let degree two vertex another complete graph denote vertex connect edge introduce widely reference depth way vertex edge eliminate edge effect connect vertex edge upon divergence hellinger unnormalized define particular abuse p divergence hellinger elementary qp vector support mean paper remainder organize describe formal setup develop non special present graph structure emphasis homogeneous graph framework develop general theory specific finding direction proof defer imagine alphabet operation broadly define pairwise operation stand additive partial list modular addition integer multiplication stand represent rotation hence case multiplication capture belong measurement pattern undirected illustrate fig pass conditional py ij illustration abuse observation symmetric mapping say contain corruption oppose code employ across center distinguish shifted version light introduce zero offset factor offset regime vanish proceeding separation development kl hellinger divergence channel minimum reflect channel see various measure constant close hellinger rest see suppose qp qp appendix part quantity specifically self determine number intuitive measurement channel output sufficiently separate quantitative start begin likelihood ml decoder well minimize error prior develop recovery probability characterize tradeoff channel sufficiently universal decoder achieve cn essentially asymptotic limit regime tend infinity exactly degree condition read mn develop setting decode herein concern channel bottleneck minimax present output distinct hypothesis rule code separate say output information distinguish highlight information contain measurement quantify divergence information capture distinct ground truth call bit distinguish exceed interpretation cccc pairwise realization show blue constitute ground reader hellinger kl shoot measurement remark technical unable develop recovery divergence fix divergence grow hellinger divergence stable convenient analyze sufficient hellinger recovery condition account one measurement minimum shot measurement continue replace examine analysis continue hold parametrize py ij I j metric place precede sufficient recovery generalize recovery continue hold probability assess two necessary condition recovery hx p residual concern specifically investigate asymptotically vanish specify hellinger convenient demand exact various datum term alphabet surrogate arise interest complexity tight sequel pay two popular divergence kl hellinger regime case j read multiplicative irrespective alphabet characterize super way total obeys carry see scope exploring rely widely encounter graphical vertex degree vertex degree cut subsection introduce quantity crucial presenting comprise cut particularly cardinality define sequel k factor I cut important homogeneity illustrate ne size little kn interestingly homogeneous interest bound shall constant uv denote vertex uv one highlight concrete graph homogeneous geometric property connect share geometrically close share fraction example worth graph vertex connect away another concern expansion lemma n exceed order aforementioned depth helpful simplify divergence metric channel characteristic begin characterize connect achieve universal hold size cut homogeneity exponent irrespective metric distribution state probability replace fundamental low admit recovery kl divergence characterize alphabet dominant form connect bridge mainly directly hellinger become weak investigate success aforementione emphasize homogeneous turning graph recovery widely adopt homogeneous n n either super arrive fundamental recovery guarantee extra concern distribution truth whose identical vertex determine define logarithmic specifie bit cut hence information theoretic broad include limited various homogeneous cf condition coincide must bottleneck constitute nn apart hypothesis broad oppose rely homogeneous error precede regardless scale discuss full generality distinguish homogeneous graph cut graph edge summarize fundamental geometric graph theorem even gap contrast separate sense differ instead challenging form accommodate variety scenario alphabet decay category testing hypothesis set minimax contrast upon hellinger divergence unify enable characterization minimax limit literature see wise exist block sbm generative partition pair whether fall infer produce interest considerable attention regime structure treat output encode suggest corollary transition exact cluster year precise condition theory factor remark begin accommodate broad regime leave technical interesting observation match fundamental precise imply square distance right recovery recent characterize fundamental cluster determine hellinger depend cluster certain recovery bad situation imagine cluster nn definition theory develop accommodate several include alignment measurement eq rate word act outli consequence present concrete limit model ease restrict consider start comprise evident mp connectivity component apart illustrate precede sequel range alphabet alphabet comparison apply general b bound fall configuration adopt regime g bound graph notably accurate imply hellinger distance quantity recovery give simplify depend substitute respective checking compatibility one immediately say feature interpretation small alphabet increase limit information nm nm increase regime measurement regime fundamental connectivity bottleneck connect single useful measurement hence isolate regime alphabet measurement formulate represent sequence minor allele employ certain sequencing obtain pair stand snps denote assume read realistic sequencing snps geometrically close dna typically median adjacent denote l separation number read fix read nevertheless simplify sufficient capture additionally geometry consequence suppose universal sufficiently condition obtain follow r ii whereas
business term diagnostic deep structure divergence link network company company multi view social network mit student message close evolve snapshot view view citation show whether two keyword title abstract netflix record user record category record review operate ht truncate value kl dataset row matlab corresponding figure describe structure different figure use frobenius norm kl divergence rank seem acceptable large result seem model use number show good component dataset number many kl discover structure two line week minute try however able detect perhaps extremely record pair amazon first product figure reasonably recommendation dataset seek find coherent people recommendation suggestion purpose extract product component remarkably due kl divergence fit similar tend book book book detect anomaly extract complete basis evolve view continue extension cluster forecasting mining bioinformatics text towards number tucker community evolve tensor finally bayesian automatically towards automatic mining algorithm minimize intervention propose mining heuristic provide evaluate method show superiority well real dataset discover meaningful support foundation grant conclusion recommendation material necessarily view valuable sharing observation tool unsupervise mining tensor exploratory extract quality quality interpretation novel automatic mining minimal intervention extensively variety dataset automate tensor mining practitioner decomposition exploratory aspect popularity largely aspect henceforth multi aspect mining growth application citation network name powerful analytical datum tensor decomposition tool challenge attention make tensor decomposition scalable facebook write ever grow decomposition small facebook big category turn e facebook hundred highly scenario exploit scalability al introduce exploit scalability distribute solve problem attention quality baseline enable another assess tensor portion exploratory sort seek extract concept datum crucial extract model datum especially quality variation always tensor seminal exploratory mining component manually link entail measure g validation select generalize label truth hope lose minimum length cost depend heavily application boolean additionally deep operate intuitive decomposition independent require exist influential literature introduce heuristic determine rank decomposition comprehensive aspect contribution mm propose comprehensive methodology mining multi aspect manual trial intervention solution assessment assume divergence effective highly count real explore hide pattern well apply discover meaningful pattern synthetic encourage code publicly available notation subsequent section scalar multiplication kl frobenius efficient ii ii negative decomposition henceforth entry usually represent accordingly decomposition tool admit intuitive latent component see soft use cluster tucker tucker compression super useful motivate diagnostic expressive hard outline introduction heuristic name model imagine fitting tucker tucker tensor restrict tucker super core possible q least f f I ic element core rank high rank model chemical mining application valuable case acceptable reasonable quality e g high mention introduction introduce suitable dense contradict area vast mining application derive behind avoid rewrite problem kronecker product vector potentially big product henceforth refer achieve far extend recently beneficial poisson natural first next exploratory piece usually expert expert provide data process completely ground label impossible tensor ground label provide guarantee human intervention trial attack describe decomposition unified mining user intervention quality frobenius norm least problem kl close iterative apply prominent mm hard minimize minimizer employ use k store use throughout performance exploit break eq compute give decompose expression numerator particular efficiently structure product rewrite respectively kronecker sum property kronecker product conclude put everything end equation iterative kl efficiently tensor diagnostic htp n order minimize human intervention mining automate tool box two offer follow user provide reflect neither require whether count say frobenius norm fortunately equip handle case follow drive let whether capture structure grid measure diagnostic value quality informally problem intuitively objective maximize however get front subset dominate end effective intuitive datum c essentially select max maximize enumeration maximize extract hard example axis intuitively investigation show step point select kl choose discover rank select aim quality expense acceptable select contrary previous component extract acceptable however perform closely depend preferred mining run component maximization output seek good combining
monotonically relu j dirac delta everywhere except poor lack flip relu convex hence encourage suggest relu produce zero leave majority unit close experiment word sigmoid purpose sigmoid corollary sigmoid learn suggest ask show objective term propose aim minimize reconstruct corrupted version usual choice gaussian objective corruption taylor overcome represent gaussian corruption sample though term exact straight gradient monotonically activation practically epoch ignore jacobian aim input auto encoder loss coefficient form hide learn order suggest property corruption dimension derive form iterate corrupted sample loss apart goal hide decoding activation require note value except encoder form enforce intuitive result notice sigmoid activation separability maxout hence guarantee sigmoid satisfy property encourage latter individual drawback relu poor corollary sigmoid hard gain propose activation drawback individual activation j note monotonically increase relu discussion share sigmoid two handwritten digit value train world image cifar size cifar real image objective optimization rate epoch batch size hide unit cifar unless train loss decode zero unit say perform confirm linear sigmoid explain sigmoid sigmoid sigmoid increase record negative percentage activate unit sigmoid activation enforce activation empirically model study dataset bias form attention possibility relu bias term order taylor section corrupt mathematically analytical marginalization practice optimize corruption batch wise manner vanish relu mnist cifar protocol sample activation contribute towards discuss gradient relu generally slow hence conclusion corruption advantage marginalization capture order together lead drawback relu sigmoid encourage evaluation effectiveness learn activation apart mnist randomly choose background digit background pixel image validation dataset train unsupervise hyper extract candidate mnist relu believe trade well relu capable zero opposite relu sigmoid across relu sigmoid produce perform activation consistently relu strong mnist relu relu sigmoid relu relu sigmoid relu sigmoid relu sigmoid relu sigmoid relu sigmoid neuron exhibit sparse establish auto fold encoding encourage pre monotonically increase activation encourage theorem c form learn representation insight activation drawback exist convex section representation absence advantage conclusion combine yield activation whether supplementary es e th j j j monotonically increase activation monotonically f j monotonically convex activation monotonically increase j j increase extend fix j monotonically increase activation chebyshev recall corruption nd nd corruption process approximation yield identity rewrite expand order get square nd h jj cyclic trace operator become upon encoder decode sample decode square j edu mm auto explicitly learn encourage study regularize auto regularization activation play role provide de encoder activation like sigmoid learn together activation insight gain activation sparsity produce par auto sr use heavily distribute representation observe former focus distribute main representation manifold separability power investigate regularize learn distinction distinction encoder decoder aforementione sr follow researcher empirically unsupervise behind convex function efficacy activation since try sr encourage activation hide analyze multiple activation desirable auto auto encoder encourage analyze learn besides comparative tool use exist predict deep understand auto network minimize intermediate encoder part encoder map encoder h back decoder basic motivation repeat though map invariant manifold encoder generally formulate eq fix drive force objective force analyze encourage show activation role achieve j th j proposition go reduce training course gradient practical interpretation term pre activation hide sparsity property analyze gradient activation optimal every iteration monotonically increase negative implicit exist set finite initialization monotonically finite thus long aforementioned set low pre length easily guarantee widely simply constrain lie ball update
computationally fundamentally hardness several g cloud service train produce ideally notion individual differential service output might individual private perform produce beyond address comparison work machine complementary actually correctness guarantee theoretical pac let instance space boolean sequence increase dimension accord target learner select approximate target hypothesis hx cx concept concept choice learner learner sufficient sample polynomial parameter call proper otherwise improper pac learner private set neighboring q privacy call identifiable use hardness private release adapt choose replace succeed identify high showing differentially private formalize follow property I sx cx ni cx ix cx find obeys identify learner efficient trace may relax condition hold learner scheme properly differentially private eq typical satisfied differentially pac differentially private q contradict tuple public key deterministic procedure key output special compare must separate requirement correctness say succeed correct require succeed succeed namely word output correctness notion weakly informally security particular correctly security parameter message strongly comparison informally require length fail security security notion security security key block trivial attack always right challenge message challenge generality message sort message scheme event output experiment message scheme sequence run ib definition security single challenge prove many challenge security hybrid differ message hybrid differ message hybrid single challenge security hybrid first indistinguishable security hybrid security adjacent indistinguishable moreover identical moreover hybrid hybrid challenge security imply indistinguishable actually scheme strongly define class scheme throughout discussion space ideally concept efficiently pac learnable learner comparison public address way example support public use binary string pair produce concept reasonable pair length sequence string let output coin notice concept efficiently description work pac learnable include public example pac work stage determine significant public exactly parameter mas good heavy set learner apply learn pac request ib j observe learner pointwise correctness coin case place place least r f therefore suffice least long receive tf pointwise hardness scheme concept class recall concept example attempt one learner take polynomially close challenge scheme example public need space produce advantage distinguish adjacent distinguish message natural separate completeness e succeed immediately discussion example sample security security reduction sake efficient natural security adversary sequence l nm I nm lb adversary distinguish unfortunately subtle distinguish lose overcome instead security differ message adversary message message agree suggest message guess guess force actually target query receive rx rx place example latter bit search key strongly scheme cc hand conduct search place I rx place answer set next threshold yield good strong correctness strong apply repeating argument correctness strong correctness apply yield contradiction explain satisfy notion correctness protocol multilinear construction noisy term grow correctness multilinear compute multilinear give answer operation compare introduction give generic weakly strongly correct scheme modify add interactive proof result correctness underlie probability protocol correctness protocol multilinear introduce protocol correctness protocol eliminate error gaussian unbounded result center statistically security protocol truncate security protocol un truncate straightforward candidate build multilinear weak correctness scheme map additional perfectly binding randomized perfect bind computational indistinguishable message function protocol string take input reference output requirement security perfect adversary quantity valid satisfy bilinear map perfectly sound assume perfectly bind run rr c cc c b notice component value moreover completeness valid correctness proof valid mean verification c b bm bb c c c simulator namely key game security reduce security security adversary valid distinguish hybrid indistinguishable advantage hybrid negligible break security l lm lb ib bb hybrid probability negligible break strongly advantage lack perfectly proof follow random p public check fail output c k bs result thus security construction rely hardness reason apparent security suppose suppose security show security also prove message adjacent moreover security suffer loss adversary adjacent adversary produce guess adversary advantage adversary security case requirement adjacent challenge receive modify public let skip bm bm bs x compare would particular result program never correctness indistinguishable show hybrid indistinguishable hybrid change security change rely security argue concept separate private representation way learn syntactic place learner want arbitrary circuit follow elegant idea suppose adversary concept concept concept without identify representation actually target differentially infeasible produce argument try properly infeasible hypothesis hypothesis good construct base way signature derive private proper analogous triple private public output indicate signature correctness scheme digital signature scheme adaptively oracle obtain sequence signature game iff super signature function super digital signature scheme fix convenience hypothesis representation pac achieve request ib represent fix k learner place place weight receive place weight get otherwise hardness properly learn properly class polynomially let super digital signature learn class follow mf none find completeness example succeed oracle scheme message produce acknowledge helpful discussion suggestion rgb true theorem fact proposition support fellowship public pac private concept pac learnable fails differentially question al j prove generic enable comparison construction differential privacy learning yield great differential aim enable give strong formal individual note speak differentially learner label presence absence positive concept since require learner work show inherent class sample privacy address complementary differential privacy initial work efficiently learnable concept efficiently learnable limited progress negative question exhibit learnable plausible prove private may interest pac universe draw label accord concept output class approximate different run pac think randomize learner neighboring dataset differential privacy substantial gain gave show properly description size toward private al make powerful observation efficient efficiently simulate differential et concept elimination differential elimination efficient pac fact lead learnable class also efficiently al progress toward pure complexity learner inefficient show generator proper proper substantially proper hypothesis assume learnable learnable approximate privacy detail version learnable concept improper learn powerful differential element privacy consideration unless efficiently learnable plausible resolve al improper learner existence correct efficiently efficiently pac learnable differentially private hold improper learner relax approximate differential remark understanding efficiently learnable different overview construction concept pac class admits learner positive example hypothesis concept minimize error underlying example suffice domain fact totally efficiently learner still example learn nothing modification example efficiently pac learnable learnable distribution place correspond condition correctness message fashion technical contribution weakly scheme strongly able efficiently pac argue differential example security scheme ensure essentially thing give trace back produce build conceptually connection differential adapt learn motivated answer publicly sort order precisely public take input reveal correspond requirement give learn ordering multilinear keep scheme privacy multilinear map construction insufficient purpose issue arise learner distribution include achieve weak message compare probability compare specifie work correctness message valid cause completely learner fail correctness guarantee strong scheme program scheme perform incorrectly comparison wrong multilinear multilinear much argue give generic weakly exist scheme interactive form key comparison procedure check proof comparison
modification state generic define cardinality word functional width fix distortion scale distortion begin rip property combine state theorem distortion successively cover inside generic instead rip mapping near tradeoff vary embed begin state lemma embed belong distortion defer obeys level sign identity q place ready main aim close neighbor fix conclude conclude q conclude complete also identity follow aim hold rip hold together apply every complete imply note prove hence sum identity apply inequality conclude complete proof fa fellowship institute computing nsf award nsf award award amazon web services google blue energy facebook intel microsoft reading manuscript ex ex pt rgb rgb theorem proposition subsection subsection rgb depth reduction structure similarly gaussian matrix multiply provide efficient dimensionality embed optimal obtain via embed certain embedding find engineering perhaps state preserve factor modern form nm ni prove project point dimension later could normal recently multiplication implement efficiently storage please recent detail improve dimensionality arise embed finite preserve precise sphere state measure define mesh unit matrix special point minimal allow matrix albeit constant continue certain ensemble entry dimension factor characterize paper analogue structure efficient heart analysis preserve norm multiply preserve euclidean state transform provide distortion embed also distortion rigorous justification replace scientific sharp connect begin define isometry state rip preserve vector multiplicative distortion isometry isometry sparsity shall restrict lie dependence purpose refine rip simultaneously sparsity distortion level rip distortion isometry distortion rip sparsity inequality require satisfy rip low reduce rip vector look proper satisfied ensemble reduction suppose obeys rip sign obey good embed pattern random ensemble commonly purpose sparsity distortion scale theorem unitary orthonormal uniformly measurement ensemble diagonal focus row choose matrix orthonormal please ensemble isometry matrix resolution rip distortion require matrix hold probability long state complex matrix bound constant logarithmic analogue tradeoff distortion utilize numerous allow sample problem paper theorem result establish analogue hold set mention interesting perhaps restrict isometry set sparsity level et spirit distortion also characterize use ensemble low significant loss distortion establish result suboptimal tradeoff state require sample rademacher one establish logarithm cover relate pseudo incur two requirement
equal measure nonlinearity simulation pearson nonlinearity dependency cloud used generating band eq pdf four type dependence spread different dependency pdfs capture pearson correlation different pdfs pearson characteristic functions pearson depend largely nonlinearity mutual correlation monotonic transformation nonlinearity pearson information remain undesirable incorrect monotonic theoretical generating figure exist show several property pdfs describe efficient algorithm random variable joint variable dependence marginal good distance know hellinger marginal call mutual mutual almost irrespective nonlinearity measure symmetry measure partially quantify random extreme normal first show axiom pdfs maximize band pdfs pdfs addition structure estimator suit avoid numerical integration briefly bl cutoff frequency mx monte pdfs linear quadratic cubic carlo pdfs datum cubic row pdfs work equally square theoretical use carlo fast irrespective nonlinearity generate equally linear normal convergence fast second convergence variance cubic see bottom convergence pdfs showing square different different pdfs cut compute bin contain advantage require pdfs also estimator invariant strictly monotonic transformation correlation achieve mutual showing slow pearson bias show decrease increase time bin fast mutual building paper cut band limited pdf approximate cut band pdf band limited frequency analysis need normalize lie modulus unity correlation case bivariate metric measure distance strictly transformation otherwise computation institute engineering md com edu section satisfy axiom measure marginal date paper parametric band limit pdfs mutual mutual standard pearson know pdfs rate ability nonlinear dependency require converge theoretical capture science several quantify mutual pearson distance mutual thought benchmark quantify pdfs pearson correlation directly estimate correlation directly datum nonlinear slow often reflect dependency correctly enyi axiom strictly transformation axiom table axiom popular pdf product dependence six axiom invariant strictly mutual copula property
corpus experimental parse lstm stack parse tag pos stack lstm parsing model language stack lstm parsing use head stack representation parse lstm classical recurrent rnn exclude symbol comparable test cc lstm pos h cc lstm pos rnn l lstm pos rnn cc lstm pos composition pos rd substantially exception pos chinese parse baseline gold pos tag parse note predict pos tag english add value suggesting parse directly also compose dependency head word implication baseline rnn capable good structure sequentially condition approach first supervise recognize stack shift make top stack decoding finally understand toward large parse exhaustive cube discriminative chart decode lp relaxation parse include feature randomize hill enable feature global discriminative sensitive part stack approach arbitrary stack recurrent art dependency stack possibility learn observable I supervision final far give device learn g observe parse alternative external memory machine supervision stack stack technique reinforcement make early chen office grant support european contract h project edu cs edu dependency innovation stack stack parse element top stack addition maintain stack efficient parse unbounded look ahead buffer income complete iii stack build tree backpropagation parse parse series read sequentially buffer syntactic structure build projective base parse computationally challenge parse action encounter development alternative simplify modeling make recently last line state state history complete partially syntactic global sensitivity parse parse state representation incoming stack although step construct sentence technical variation recurrent neural unit parse three stack represent stack syntactic one history stack syntactic token syntactic tree compute learn chinese english parse section brief stack follow write letter e write letter e scalar letter letter refer input discussion defer cope vanish gradient inherent rnns rnns step apply concatenation pass sigmoid nonlinearity rnns long range difficult repeat nonlinearity result address three control current input memory proportion previous forget update follow sigmoid hadamard product lstm control gate nonlinearity cell improve capacity rnn architecture layer input layer differentiable conventional multidimensional innovation stack always add position location stack lstm new addition sequence stack lstm stack extend stack never add back stack stack stack operation stack must efficiently maintain queue control stack middle box row lstm ever middle cell affine transformation nonlinearity refer vector stack continuous summary stack available refer stack stack influence stack lstm flexibility extract stack knowledge novel recurrent stack stack stack rnn problem preserve structure base buffer process stack construct element stack augment space syntactic additionally introduce third history take stack lstm architecture illustrate stack lstm buffer word stack history take pass relu nonlinearity embed transformation pass softmax layer distribution parse decision representation first word symbol stack time compute stack take update stack symbol contain tree symbol history operation define lstm encoding buffer stack lstm stack lstm encoding pass component relu nonlinearity finally embed stack buffer previous decision valid input arc standard transition transition indicate stack buffer result stack buffer state bold symbol leave stack partially build syntactic buffer keep incoming parsing choose score arc parse construct bottom right head recursively compute construction syntactic structure modify algorithm another head compose strategy token learn type neural representation pos token provide auxiliary pass relu datum lm vocabulary limited parse present lm word ensure parse stochastically singleton parse token iteration create option skip name skip word model skip define window rate epoch recursive network enable phrase stack challenge syntactic arbitrary simplify parameterization combine head expand syntactic syntactic relation satisfied embedding head apply nonlinearity pair triple recursive branching eq construct computation sentence forward computation parse
natural classified embed experimental cr cnn softmax fair comparison cnn softmax embedding cr softmax get convolutional softmax tune validation value cr cnn softmax convolutional cr cr improve et embedding similar softmax word embedding softmax embedding less data c ccc net cr softmax cnn cr cnn softmax cr present classifier fed rich traditional result et neural vector distribute vector method name pos use present cnn softmax employ lexical yu compositional embed derive sentence embedding embedding utilize cr sentence embedding report reach remarkable external resource nlp tool name cr play role informative reverse direction relation towards meaningful various class variety task recognition natural processing successfully different nlp sentiment role deep yu author tackle recursive assign every tree syntactic name recursive sentence embed position lexical extract lexical fed softmax performance yu et compositional embed sentence word utilize dependency name high difference cr cnn wise use softmax top cnn rnn cr effective artificial embedding approach tackle use perform work new state art costly classification use embedding rank deal artificial cr effective extract cr cr cnn relation acknowledgment author suggestion research com ny usa us ny com relation processing system rely tackle relation classification task perform rank cr cnn artificial perform design classify mark sentence outperform art costly additionally softmax representation precision use embedding target nlp task question base decade interest apply availability task classify mark sentence introduction book summary text focus network aim reduce lexical resource nlp dependency entity cnn cr tackle classification propose network learn relation segment convolutional layer produce compare reduce impact artificial extensive cr cnn outperform cr cnn follow representation word embedding remainder neural network detail evaluation previous deep network nlp cr compute embed input step cr transform word value convolutional finally cr compute dot semantic sentence consist word word convert value therefore input word vector vocabulary embed w position classification need determine relation come word al keep role label et instance respectively word embed concatenation embedding use embed word position embed input convolutional w step nn create representation input main challenge sentence variability appear convolutional tackle create size use convolutional vector representation convolutional produce sentence combine use max operation vector sentence convolutional layer apply matrix size successive window concatenation embedding th word order overcome word special begin convolutional convolutional sentence vector convolutional context window hyperparameter choose note vector sentence network compute dot w class c dimension embed size sentence representation network round score generate logistic train cr cnn q difference error term side score class incorrect use minimize loss function like rank task large classifier significant impact learn negative number small experiment sentence choose sgd among incorrect max class backpropagation gradient cr backpropagation group relation relation nine relation group characteristic cr cnn make easy artificial embedding omit benefit prediction step cr term right relation classify actual score otherwise annotate type belong nine main type nine consist predict take consideration cause cause water pressure cause instance macro average nine relation take consideration initialize unsupervised perform pre skip gram tool snapshot english wikipedia corpus removal english substitution character text stanford pos removal less character word substitute digit result clean corpus token tune range cnn configuration show hyperparameter decrease training epoch epoch
alignment right artificial intelligence vertex label xshift fill yshift xshift yshift xshift cm white extend cluster entity relational cluster entity group also entity resolution entity propagate model social community relational discuss apply entity relation assumption triple incomplete entity proceed name scalar letter letter bold letter bold letter stack e n kronecker background formally define graph entity relation knowledge triple entity relation variable existence triple tensor array n whose cf interpret world derive interested triple triple knowledge graph adjacency tensor triple depend close assumption triple exceed type number valid triple actor store actor star movie important issue relationship efficiently ideally graph e linearly linearly relation triple discuss presence absence certain triple correlate certain triple conditionally give relation additional mainly existence triple triple independence write sigmoid form criterion maximize margin triple desire probability via many define discuss proceed general triple triple triple denote parameter strength problem possible loss question contain fact emphasize notation triple generalize way triple negative triple understand valid unknown triple false would generate negative type irrelevant assume type actor triple event reduce encourage focus plausible negative extraction run triple due extraction good plausible negative triple miss close valid incomplete precisely triple triple functional triple triple really general score triple false triple margin first assume example triple likely objective sgd scale well square optimize alternate square pairwise specifically triple triple less likely world close world local world world depend cost use closed often model discuss presence way node latent characterize degree call factor directly latent node latent space factor compute edge distribution sigmoid grouping parameter analogously plus specify detail alternative amongst directly treat mrf variable variable various connectivity markov field case relational mrfs product network triple detail kind whose control loss triple convenience sometimes log occur unfortunately graph observe usually heuristic fitting model specialized loss function get train world indeed false set relational interpret miss open triple exist triple indeed triple assume triple denote occur object triple edge heuristic discard triple set pairwise triple exist triple less use present train world world close world cost graph deterministic rule locate usa infer usa typically pattern true nevertheless power pattern tendency entity characteristic star movie usa relational one kind refer property divide group group might actor star movie science actor consist science movie entity star chain triple involve usa depend city dependency involve entity usa relational able create domain variable conditionally independent latent explain triple via feature instance explanation receive award good explanation entity actor observable award call latent follow latent model award award vector note infer hard behind entity derive interaction possible way derive mean entity number relation number entity l relation positive triple negative triple mean partially triple score triple slice mean vector relation sigmoid relation feature entity size layer layer entity bilinear entity h relational explain triple pairwise triple entry specify interact th bilinear vector model magnitude anti correlation efficiently negative compute triple entity interact property representation entity subject furthermore entity triple subject triple object k entity since share propagate information triple dependencie embedding entity similarity entity representation entity similar latent entity similarity representation act non relational access recommendation tensor factorization compactly illustration tensor efficient factorization adjacency explain entity product triple derive via composite representation information share representation entity compute gradient stochastic assume second eq via efficient update triple non zero iterate update arrive current update tensor parameter runtime update update relational moderate latent scalable product triple capture global relational three provide dataset markov logic relational model cluster factorization prediction aside link task entity cluster instance state predict author publication publication database semantic create clustering entity embed relational factorize adjacency cp tensor web page web predict interaction tensor triple graph dataset recommendation graph number tensor fact apply boolean discrete tensor decompose factor algebra adjacency tensor subject column relation cf unfortunately formulation object lose complexity compute entity require parameter interpret creating triple rewrite equality product feature representation predict existence triple create composite representation please explicitly require layer triple predict let composite alone reason add hide final via difference product approach require disadvantage mlp lot call mlp entity please er mlp global relation project reason show train semantic embed er representation relation compute er mlp put near close parent birth birth parent edu hold job job edu job neural lr lr lr mlp r h h bilinear h bilinear neural precisely slice slice combine bilinear additive mlp bilinear use paper additive interaction latent latent model social derive probability relationship representation entity relational propose social network refer se extend idea relational feature representation entity relationship loss entity relationship se translate offset instead multiplication triple vector note unit euclidean follow h rewrite eq experimental diagonal version prediction er mlp comparison leave future existence predict extracting predict due social parent person could triple existence child reasoning triple via observable directly triple kind neither art relational art superior knowledge strength latent complementary aspect suit modeling computationally triple model suit local graph computationally triple neighborhood entity theoretical factorization inefficient fortunately often via difficult model however easy existence existence edge strength model promise graph training kind model model optimize observable pattern allow result increase solution either logistic square gaussian gaussian efficiently least update relational various joint pattern simultaneously reduction require improvement runtime learn learn relational latent entity representation object composite representation subject object pair efficiently triple triple entity kind include model spirit rate additive information factorization machine allow observable input way stack output er mlp layer stack advantage kind disadvantage need jointly separately bag stack mlp scalar employ mlp flexible kind ensemble interact interaction interaction logic template potential dependency arbitrary relational mrf logical formulae mrfs tool suffer difficulty estimate rule paper general compute hard gibbs soft relaxation system fairly show estimation cast convex quite call pseudo cf dependency flexibility relational mrfs schema predicate content web source annotation second serve compute extract fact extraction train mlp predict combine discuss score fuse derive de page illustration bernoulli label drop mlp mlp employ set mlp system achieve auc roc roc subsequent combine auc neural model observable aspect prediction achieve combined score slightly classifier achieve triple integrate google probability name logistic fit million triple approximately triple reason triple low perform triple predict million triple substantially big structured repository give extract triple triple include extraction triple multiple indirect fall accept cause former belief final fused combining method extraction triple calibrate probability plus perhaps date handle unary unary relation statement property entity person row column tensor approach unary modify say see high relation graph relation via express two actor star movie loss auxiliary actor movie character entity auxiliary triple format relationship without transform related truth change google page schmidt fact correct fact annotate begin construct represent auxiliary however duration fact necessarily usage auxiliary easily order high neural impose hard useful powerful language language formulate computationally demand fortunately machine face evidence deterministic triple relation dependency usa north triple add knowledge triple constraint knowledge graph apply entity domain limited modelling manual constraint consider scale relation greatly reduce relation type although range triple induce city etc mutual deal entity mention knowledge consideration current newly latent representation entity calculate approximately explain relationship relative current calculate relation already probabilistic triple might quantify expensive involve handle review relational conjunction machine read build show massive machine memory many application represent kind human possess notably miss representation fact water thing knowledge email etc represent reasoning ai expert relation type triple label nsf award technology grant mt rgb rgb rgb rgb center base xshift yshift name north east arc cycle lr lr study method relational structured paper predict world discuss relational massive dataset latent second graph observable decrease finally discuss information automatically google project characterize categorical main relational object datum form node label relationship goal node pattern arise analysis social biological pathway far de logical article review community relationship entity knowledge google discuss cause application relational grow automatically
helpful mathematical california technology plant flat introduce strongly difficulty plant flat flat rapid field increasingly naturally notion complexity motivated algorithmic aspect consider jointly pose hardness understanding algorithmic attract lot successful link arise abstract extensively theoretical computer hardness random hardness approximation primitive hardness improper hypothesis plant primitive detection subsequently high understand come randomness computationally manner investigate treat detection show flat flat formula clause exclude introduce problem instance testing plant unknown make flat rate minimax base various able inspire successful sample discuss plant significantly affect detection plant solution flat focus flat detect plant phase instance transition compute successful plant describe dimensional flat determining whether alternatively whether take independent j flat random yield constrain coordinate clause flat flat satisfy flat exist element asymptotic underlie plant uniform denote independent identically distribute uniform dimension linearly independently plant uniformly identically denote generate linearly contain satisfying assignment consist transformation fix contain particular contain uniform particular procedure bit result description description tuple require allow invariant confusion representation actual oracle flat base flat purely make membership oracle list basis flat formally uniform uniform contain q write v flat study flat result collection doubly compute suffice element derivation together flat flat exponentially small sharp phase regime probability transition equivalent choose independently among interpretation independently correspond term flat flat equally behind jensen variation consider lemma yield approach variation plant sided observation problem distance converge powerful flat regime since view check cover detection flat constraint multivariate flat equation hard lift system equation obtain quadratic general technique take embed equivalent instance flat solution intersection constraint intractable order relax solely constraint flat associate equation flat solution equation consequence always recall kn multivariate multilinear exist element distribute uniform eq aside tight obtain take test remark go time linearization analogue plant sample linear benchmark plant vertex clique greater primitive hardness plant assignment study bound type come statistic let consider sum tuple show behave differently great typical deviation powerful typical sign sum triplet sample version light show nature suggest modification successful plant flat hypothesis alternative happen dimension uniform define therefore v tackle statistic maximum flat hypothesis n variable hoeffding union q hoeffding result prove powerful direct consequence second point bind divergence similarly optimal still
yes yes c average fan adaboost wang adaboost diversity deal majority work different combine final learning cite regression keep online wang testing procedure last accurately interval record classifier adaboost select record run scheme scheme comparison deal feature dimension set context context belong list reference mis classification set importantly among technique accurate accurate poorly accurate datum expert case prediction presence drift concept slide scheme w slightly able adapt quickly change context fig decide classifier send fig refer decision exploit context time instant select equally context relevant select expert datum learner learn instant drift irrelevant automatically decrease time exploit obtain context help reward window quickly bottom decision make affected drift w concept scenario c c set achieve information label low scheme adaboost adaboost two time label adaboost horizon good happen q ta ta variation ta p ta regret exploitation reduce order bind regret contain happen level level happen level level must eq maximum level hypercube happen pair interval context consider counter select upper bound original scenario scenario scenario type multiply maximum possible exploration bad interval great interval bound achieve exploitation level exploitation hence level event analysis ta ip ip tuple inaccurate action tuple td ta tuple contain relevant action candidate ad sp chernoff tuple imply happen happen tuple variation ta ta ta ta conclude exploitation small regret bad context active interval contain arrive happen guarantee contain maximize since maximum hypercube level hence hence need contain consider update counter select interval tuple regret due tuple type tuple let configuration level different type different interval regret tuple interval tr subsection equal corollary recommender medical diagnosis security require go example difficulty present big available valuable integrate efficient learning curse formalize maker dimension advance dimension different action relevant contextual armed add exploit bind number relevance absence good contextual exploring observe outcome suboptimal action arbitrarily breast diagnosis news article contextual recommender drive diverse source diverse include document transaction file big surveillance health monitor stock market etc stream continuously dynamically evolve way decision stream tackle online big challenge exploit application embed know advance relevant action decision build bandit formalize stream perhaps process characterize process act e receive vector take generate context context action reward meaning depend security context attack context gender reward indicator item reward reward context context application reward relation relevance relation advance decision arise naturally practical treat disease patient image medical often treatment drug close indicate patient care characteristic past strongly home relevance avoid curse bound relevant dimension dimension summarize phase general bind growth gd incur phase observe exploration phase perform control even observe reward costly arbitrarily confidence select exploitation provide medical organize formalize learn relevance action type numerical summarize c c contextual similarity continuity bad always contextual bandit relevance contextual bandit problem paper lipschitz contextual similarity reward action come stochastic good action context regret achieve cover compare work regret dimensional contextual bandit problem consider linear context learn corresponding assumption generate context arm reward assume context process regret space work arbitrary reward lipschitz lipschitz take reward decompose reward function directly reduce bandit reduce graphical bandit action ai imply bandit problem reduction find contain approximately instance project onto low adaptively representation base work action different relevance relation work observation efficient consider function bind online prediction consider lie predictor derive similar work desire never however assumption form expect continuity special datum stream special relate costly assess label stream provide unlabele instance active deal sublinear exist base combine reward combine expert update goal take ensemble analytically expert reward hence expert contrast benchmark context bandit action choose denote give dependent variable subscript vector unknown relevance di kkt ta choose maximize cost consist ti ii element infinite notational value lie correspond ta generate process know priori l learner know need algorithm result show aa due compare oracle choose denote learner learner denote choose learner observe learner benchmark definition relate work cost call active learner reward balance cost incur able two observe reward regret achieve sublinear growth rate depend section well simultaneously relevant reward context way estimate control operate active learning parameter type analytic perform know enough operation summarize adaptively compose type vector estimate tuple tuple estimate explore observe reward cost reward current action tuple action variation hypercube interval similarity dt I explore ta ta ta ta ta ip p ip I ia exploit slowly sublinear reward action good relevant tuple type big type relevant action adaptively space disjoint interval denote let arrive small tuple type past observation vector lie form mean reward create balance sample mean reward due past calculate form keep counter counter exceed duration level create example otherwise remain duration next describe keep keep tuple interval determine exploit tuple tuple let element tuple tuple value type let tuple assign depend cardinality active reward reward action guarantee action context close type expect action expect contain ensure within hypercube exploration guarantee enough sublinear compute explore empty select explore observe learn cost reward eq ii estimate form action tuple type reward fail tuple select nonempty compute variation ta ta relevant mean calculate tuple find tuple type mean select reward tuple interval know hence compute reward tuple interval different learn action high reward tuple however learn high form tuple greedy type sample reward set tp show arrival process independently explore sufficiently many exploitation contain context problem pair form reward relevant action subsection derive sublinear bind prove online regret exposition simple numerical section randomness action incur incur flexibility learner objective minimize label cost exploitation learner trade exploitation run control number instantaneous reward select relevance total exploitation tr context arrival dependent exploitation choose zero come reduction step exploit I require exploitation take stay nonempty regret value state theorem relation proof give appendix duration get give sublinear regret increase run state exploration exploitation balance order exploitation order exploration contextual focused balancing exploitation context vector reduce proved contextual bandit relevance result say reward pair require relevant however comparison action prove work case action great never impose explore begin reward relevance duration control level relevance probability tr sublinear duration parameter control eq gd gd match relation independently sample reward reward equal average know select action average reward estimate type type action relevance relation develop relevance give figure general regret keep mean action tuple tuple tuple time similar explore newly reward estimate tuple similar maximum mean tuple relevance relation action sublinear regret know bandit algorithms breast cancer diagnosis iii simulation learn accurate prediction classifier
cognitive business web specific user click three ad query describe entity relate precise view learn view wherein complementary naturally view learn basically usefulness correlation approach however fail explore paper advantage feature view contribution high combination order allow different error logit list basic use view click user ad query description aspect click click represent view click predict array second index index definition tensor mode product tensor index value mode product k multi view view wherein complementary interaction extra w mi view view kf overfitte assume interaction rank w I basically factorization element wise transform order multiple view tensor factorize th vi denote flexible order interaction interest learn sometimes intuitively redundant scenario overlap view construct full interaction order outside investigate view complexity largely equation time interaction reformulate model risk overfitte importantly choice conduct popular choice loss number differentiable least etc logit model otherwise possess property gradient independent iteration loss eqs moreover eq initialize deviation accord eqs learn search hold improve much hard memory training mi convergence discuss extension multi include vector svms machines factorization svms margin hyperplane essentially svms integrate hinge loss view concatenation shown obviously explore restrict remove svms svms implicitly interaction nonlinear svms interaction exist nonlinear enough reliably instance either estimate interaction svms svms factorization eq instance allow interaction effectively train investigate interaction view machine high interaction decomposition estimate g reliably higher critical low interaction high rank latent achieve machine advantage svms order fm j v interaction include multi redundant correlation within view thereby group achieve pairwise interaction I p qx robust svms instance main order completely
capacity free beneficial appear triple regularization capacity encode section kind validity different datum introduce previous control way way capture approach training embedding way different embedding space capture embedding pre combine stage work benchmark also systematically scheme add embed add thorough strategy besides benchmark prediction embedding insight behavior organize follow work scheme benchmark discuss art modeling embed method one simple hold tail close label hierarchical asymmetric relationship knowledge basis modification entity hyperplane translation idea except shall additional performance kb dramatically hard perform high context link modeling assumption one dimensional however represent bilinear correspond tensor criterion parameterization criterion neural tensor combination way way share entity entity embedding combination embedding improvement difference interaction parameterization argue maximum degree parameterization embedding term linearity nonlinearity embed overview difference recently purely embedding entity relationship framework explicitly extension blockmodel entity entity similar relationship entity way share discuss symbolic way path go multi relationship weight represent relationship project conjunction symbolic also embed fix entity relation triple head index label relationship type learn score set triple receive triple unlikely learn low triple embedding head entity triple canonical dot appropriate term embedding entity even dimension embed dimension constraint relaxation translation special entity unit exactly basically entity besides parameterization embedding interaction constraint way image leave relation score embed space function result magnitude strategy indicate train strategy depends jointly denote sum fine tune training accommodate train directly without separately strategy combine fine tuning parameter unchanged hence follow combination rank later bfgs additional version discuss parameterization classification good netflix item bias contain embed embedding depend bias see mode factorization parameterization play collaborative play bias analogue collective factorization matrix b exactly little argue factorization bias space way type interaction motivate choice score embedding add hyperparameter add hyperparameter reasonably remain collaborative filtering critical feature bias collaborative rank square kind leave aside idea singular value rank pattern bias exist weakly strong capture allow offer control capacity translate useful end capacity closely add entity quadratic form gain ensure useful control capacity regularization parameter well turn effective admissible embedding lead conclusion really embedding absence clear concrete capacity believe embed space less expressive useful prediction hand motivation term relational basis relationship like capital head country tail huge entity identify type entity person filter prediction term correspond bias head embedding entity feature predict head predict natural share embedding first two b last h intend entity capital city connection france country city link position respective type embed feature use type object diagonal change keep reverse rotation preserve regularization come intuition direction choice triples paris capital france rather france capital paris invariant direction relationship inversion direction task invariance replace letter h w e ex stochastic design triple fact express kb fact suppose triple provide kb positive triple corrupt one carry discriminative approach create replace triple may create wrong negative triple rank set triple application h define gap stochastic minibatch set disjoint triple triple keep whole pattern lead initialize model disjoint pre tuning stop validation weight stop convergence margin rate initialization entity embedding normalize subset metric deviation connect subject relation extract wikipedia act subject object rank metric proportion rank raw triple random triple example epoch validate epoch validate validation criterion validate radius determining fix radius validate among apply parameter validate among tuning rate select learn margin regularization training carry way alternate experiment version impact shared share soft hyperparameter grid hyperparameter performing method extract main hyperparameter dimension compare model head label occurrence head experimental configuration model c soft hard soft soft soft top variant bottom perform bold filter raw l mean rank c hard hard hard soft soft soft recall ft lc combination provide model alone significant combination bring basically somewhat one potential impact constrain head tail entity regard come triple kb entity head totally uninformative highly rely interaction well irrelevant interaction may poor turn r kb side reach completely conclusion kb automatically regularization setting display outperform simplicity advantage something lead improvement kb information encode complementary except wide roughly counterpart hard soft performance perform share confirm pre different embedding essential properly collect embedding constrain pre encode complementary performance performance soft model type classify relationship cardinality tail argument variety head vice versa pair classify respectively result constructive predict tail remarkably relationship filter model cccc c predict head predict soft ex previous regularization similar soft term bad confirm embedding actually different via e good relationship seem happen bad model around twice h insight relationship behavior detail notice simplicity predict triple use counterpart present triple triple triple subset contain triple learn one triple rank decompose overall adequate particular original paper counterpart triple use expect well counterpart train soft counterpart instead account prediction entity relation test triple slot answer triple prediction display row want among find make team topic answer type country movie may operate relationship could actually attribute relationship entity head website website entity little among relationship leave hand nearly impossible cm project embedding project use usa cluster separate except correspond thompson cluster appear triple heterogeneous category illustrate however look neighbor embedding entity entity like work movie tail together triple form act object triple predict fit expect answer triple top enter make take enter leave move join lead join conduct carry convert release produce include base become establish dominate name enter ex like good third instance show heterogeneous relationship explain good fit express channel lead rank target list pair similar rank high triple unbalanced frequency rare rank much bad appearance tend rank sometimes match due influence frobenius norm relation impose norm norm impact importance frequency tuning enforce example answer triple provide carry use save visit enter come know include join bring reach become join say help leave make involve support take carry carry move leave release produce become include take include leave run name take call move form establish dominate break relationship translate entity embedding explain argument unbalanced factorization combine good pattern embed phase pre benchmark different strength actually conclusion hyperparameter soft soft soft hard hard configuration soft c hard soft soft c hard hard hard configuration soft com facebook research de paris france universit universit cs france problem basis entity previous attempt complex connectivity pattern overfitte rare relationship capacity frequent capacity simple train variant kind regularization combination strategy show result benchmark tool rise retrieve digital kind area domain biological purpose kb google knowledge kind provide capability knowledge engine language internal kb answer language processing task translation formalize relational entity encode various kind
accumulate big annotation enable outli contrary majority voting outli detection vote green correct annotation red outlier number vote receive majority voting remove bad save receive one vote voting detect examine outli example comparison label validate closely huber robust contrast rank statistical ranking train datum huber ranking rank score store rank instance huber otherwise regard outli loss huber robust outli outlier ranking design huber huber huber difference robust cost low instance main objective rank thing common ability detect framework could remove introduce low huber differ consider low critical huber instance denote decompose svd complement svd project column space compute outlier solve sparse approximation dimension huber lasso huber lasso cyclic ranking perform operation rewrite eq e analysis huber effective detect outlier exploit low projection able well identify outlier especially pairwise annotation training huber always pca sure huber outlier validate dimensionality expect ridge level pair dim net age original model pca outlier five benchmark dataset fall category visual video estimating attribute recognition human age face set human however model comparative deviation plot qualitative box annotate interesting success box detect box failure box agree annotation later annotate image consist scene comparison annotation human annotation reasonable pairwise rank ranking mean order predict method voting outli pruning remove outlier vote score regression learn image dataset enough robust outli conventional huber follow estimate use outli annotation clearly significantly outperform global local outli superior joint outlier enable result outli order reliable hundred comparison per met suggest weak global majority interestingly comparable annotation discrimination majority affected see improve pruning rate outlier big showing stop see box bottom ground indicate nice predict odd hold camera visually unlikely capture camera aspect video digital product video give complete interesting video annotation noisy invariant feature sift coefficient incorrectly annotate attribute answer failure case cause unique building colour consider age key evaluate truth person enable perform depth significance alternative factor annotation outli accuracy measure directly age individual label truth range compose generate error accord pilot pairwise collect age fitting error age human age error introduce bad worker provide label human crowdsource mixture thus setting error error result add around unless ground truth give compare method experiment four show similar correlation comparison show global robust rate feature dimension chance identify outlier peak importantly stay annotation compare majority voting compare majority voting prediction accuracy pruning rate pass aggregate pair outlier aggregate voting effect ratio roc measure relationship pruning age vary comparison amount error train prune fix show true deal non effectiveness employ outli examine decrease rank list comparison accord outli relationship pruning ground outlier large age tend first conservative pruning obvious reliably framework visual advantage voting detect outli detection ranking prediction formulation conventional outli comparison effectiveness alternative validate effectiveness outli also human going include extend application denoise iterative field economics fu degree university degree china currently post video understanding degree currently university research interest model machine vision mining interactive major international member degree computer national reader associate school engineering computer science interest include vision machine international co behaviour semantic currently degree china research interest topological high datum electrical receive college university research interest include computer vision wang wang china vice institute digital medium medium video technology receive electrical engineering ph california work ph dr wang research interest computational vision digital visual institute technology mathematics ph mathematics california berkeley stanford sciences china interest topological geometric vision member american mathematical statistic apply area mathematics neural ac school sciences university china email edu cn correspond wang school china email wang cn estimating attract increase visual image intermediate visual recognition challenging make recent crowdsourcing tool video interesting give separately introduce outlier rely majority voting annotation require amount pairwise collect detection cause principled way annotation visual property problem outli jointly pairwise label together rank lead well annotation outlier benchmark alternative property detection ranking path computer image video scene indicate scene category image object object interest e represent bounding box face one age gender person property little ambiguity estimate less variety example estimating improves automatically predict people video start prediction world increasingly rely retrieve increase video application useful visual recognition people face like meaningful refer visual prediction challenge primarily difficulty obtain annotated score range cast problem low level annotate value annotation people example especially note human pair visual easier exist comparative predict pairwise amount annotation instead compare study thus resort crowdsource amazon economic conventional laboratory bring crowd affected worker provide wrong annotation cause nature regardless worker rank familiar figure face number comparison big instance crowdsource tool annotation remain e pair compare deal outli majority voting annotate allocate annotation pair vote limit infeasible cause error effectively majority voting base inconsistent pairwise ranking pair ranking eliminate ranking ranking cause locally consistent vote outlier focus outli method property collected crowdsource first outlier majority follow formulate unified robust framework jointly voting operate integrate local together correspond receive vote thus comparison operate sparsity optimisation formulation statistical make suitable unseen video formulation video dataset relative attribute demonstrate method state art effort aspect include relate correlation people al systematic contribute preference refer certain type natural input video receive less attention perhaps hard understand mean liu frames essentially treat work benchmark video video early cast problem absolute social collect pairwise comparison crowdsource majority voting remove outlier comparison employ learn rank unseen video compare experiment unified robust majority vote formulation broad attribute base gain popularity recently intermediate attribute use include shot shoot previous binary attribute relative predict semantic focus due intra interactive address annotation outlier majority voting necessity heuristic primarily annotation sparsity majority voting respect global voting theory study computer aggregate huber lasso potential robust local unseen learn addition order address critical problem theoretically experimentally solve outli ranking prediction novel visual pairwise comparison rank first detect outlier theoretically experimentally superior exist majority voting ranking early version work focus image video model noisy pairwise video training instance represent instance comparison tool direct comparison label give comparison strong save save aggregate cast vote ij indicate similarly e edge word ij indicate instance vote carry node consist task remove outli problem prediction consider feature predict coefficient level formulation introduce vector edge notation convenience vertex ideal vote outlier vote majority outli crowd propose globally jointly end variable outli coefficient unified edge model magnitude edge outli expect discrepancy annotation nonzero whole ec e e annotation keep sparse note vote many discrepancy need vote represent edge sparsity outli unify robust identify outlier globally integrate ideally sparsity
adopt representation reflect group aim representation tool classical invariant reader invariance haar kernel haar integration binary classification highlight conceptual advantage perspective explicit subset radial act haar unitary invariant integration see z haar group action framework since turn classification give label class minimize empirical lipschitz belong hypothesis kernel call space optimal nx ix f gx nx group invariant translate assume follow x x x identity function rkh posterior since unchanged endowed group eq misclassification dx preserve core imagine cardinality cardinality review haar setup reduce core kernel haar integration test time computationally virtual transform make usefulness contribution fold feature derive theory introduce haar integration around kernel large unseen approximate rkh assess empirical minimization mnist random theory haar integration start cumulative variable draw accord haar latter define truncate cdf gaussian vector rejection template behind control concentration theorem unitary concentrate rejection hold sphere sx sampling control dot pt xt pt uniformly group cdf template independently gaussian ready section study geometric state explicitly advantage map outline store invariant category plausible feature invariant proof supplementary material haar compute distance expectation invariant hold sx gx gx hold restrict sampling group asymptotically constraint relaxed r invariant result dot product around invariant large bin template element n n put corollary capture distance distance x c universal constant assume template unitary draw equivalent indeed gaussian rejection template proportional template assess template kernel interesting template achieve minimum point future set unseen architecture aim risk cdf sx combination x dense constant approximate via empirical restrict sampling restrict infinity relaxed practice n template feature arbitrarily function core unseen hence integration achieve invariant random space rkh govern template summarize datum perform f f cdf onto complete toolbox explore sequence element alphabet character assign character regardless invariant permutation version template sequence translation rotation template translate pixel degree child digit speak template play speed detail template template outperform bag invariant invariant good template match bar remove clarity supplement z g unitary virtue unitary compact particular template eq g gaussian variable rotation invariance note write gaussian analytically chi freedom bound upper tail chi chi freedom upper together equation put equation z k sx e noting unitary gx z eq product sx z x symmetric proof two template ss sx turn cdf cdf cdf sx z n nx nz z sx j nz cardinality note dense sx dt pt ff prove need preliminary assess approximation certain function function f jx jx tf fm lemma x j f j equation follow function translate exist proof dx dx risk sample fix ready f f let approximate know template element optimality union template cumulative empirical let lipschitz respect rademacher iid symmetric bernoulli take create dataset exploit permutation invariance provide group access take alphabet give total character choose target sequence position character likewise character binary positive sequence preserve permutation sequence belong version another character vector every position form represent character invariant representation standard pool cdf baseline dimensional split positive remain template encode datum template fix number template improve accuracy bin
domain guarantee existence local minimizer conversely constraint restrict consequence descent local never elastic mkl lagrangian identical lagrange correspond elastic optimizer include minimum show necessarily manuscript consist step alternate problem composite kx k kx efficiently solver attack sum elastic net constraint manuscript novel simple solution sub remain section solution optimization abstract original mkl please vector like problem change implicitly account elastic scaling find practice minima corollary c gx sx sx gx sg cone show simple calculus give efficiently iterate iterated stop meet later provide behind ease hereafter next interpret gradient offset constant p sx substituting follow p gx hyperplane level find tangent guarantee decrease monotonically fact point sx make start descent algorithm give ref ref outside limit convergent subsequence continuously function mapping continuity condition simple easy sx sequence exception condition satisfy provide name n follow q first unfortunately origin inequality violate consider readily q importantly equality start show imply convex equality define point condition sequence converge new iterate lie positive cx x hx hx g suboptimal know exact condition predefine q terminate iterate satisfied hx hx focus bind elastic algorithm detail kernel coordinate classifier optimization compare exist external library wide applicability readily open source library report assertion follow state name series show open suffice remain evaluate x sx gx sx sx cone hz sx x convexity sx sx sx substituting rearrange gx gx gx condition gx x sx gx gx statement proof sx gx hx sx gx function global unique reasoning strict convexity sx gx sx statement side r sx sx substitute rearrange n z concavity square
order tight leverage recently svms helps sample risk ridge regression feature design future direction feature provable nsf problem square analogue power selection leverage score randomize spectral leverage sample risk perform synthetic world indicate popular machine penalize simple square long space require technique deterministic provable numerous empirically randomize like provable empirically failure accurate feature algorithm feature irrespective feature class label provably provable non bad feature fix design subset deterministic spectral provably feature unsupervised setting score selection error failure feature complexity pick training provide approximation guarantee guarantee training score unsupervise ridge design comparable risk full feature guarantee feature selection set single set information gain qr report run time offline experimental indicate spectral perform dataset observe small number require deterministic achieve good whose dimensionality identity singular orthogonal diagonal singular value singular q nr indicator one row respectively sample matrix rescale dimension hilbert rkhs square class consist throughout eqn eq function generalize classification goal perform algorithm train proportional rank transform dimension giving subsequent portion similar second lie outside span consider real ridge norm coefficient towards fix ridge rr dual tn n dataset study svd full closely circumstance selection base sampling dominate svd single set randomize setting however focused reduction combination hadamard lower subsequently solve regression dimensional spectral technique one vector symmetric definite low respectively potential measure low lt dominate satisfy q construct lemma rescale combine sampling score sampling vector train choose probability norm leave select trial scale row parameter describe bound ridge bss invertible definite let theory show guarantee bind bss depend accuracy feature bss leverage technique definite bss ram provide I suggest pick break tie previous column never inner pick column use single matlab vector compare bss numerical algebra community rank qr slightly abuse matrix thin polynomial preserve rank uniformly replacement serve get sample five score randomness pick presence absence strategy whereas unsupervised bss ig bss ig bss feature bss experiment synthetic relevant working randomly probability choose relevant among power follow run ht value bss repeat ten compare bss ig leverage sampling table across set top pick bss good pick ten frequently select relevant able bss document matrix project comprehensive web maintain contain pair category binary collect bag document systematic use bss remove group category whose appear perform fold cross validation repeat ten time dataset regularization parameter offline music us uk product us bss chemical laboratory school music business south north leverage score analysis analytical library service service south bss score regularization show bss ig bss average document fig bss well leverage score achieve bss bss bss outperform sample bss due worse supervise metric list word bss seven validation experiment name document matrix experiment
come improve query j return strategy query query group rotation iteration suffer acquisition easy regret factor regret section surprise move coordinate update k perform bayesian update axis maximum produce run pt optimisation dimensional expensive scale tackle challenge additive expressive additive function dimension scientific naive additive application optimisation evaluate example tune machine strategy scientific interact bandit reinforcement either optimum exploitation bayesian refer tackle challenge unknown pair optimisation successfully tune hyperparameter high field design knowledge date identify challenge scale dimension lower often exponential sample reflect regret optimisation heuristic attempt high effectively concern work challenge treat mutually exclusive acquisition high regret dimension additive experiment simulator detection matlab implementation online next set acquisition include improvement thompson upper confidence interest literature study variant acquisition batch vary work carefully restrictive experimental method meet expressive contain along entire kernel dd ex additive additive additive assume additive even statistic optimisation force regime function develop additive additive naive query dominate application precision monte exponential bottleneck paradigm evaluate expensive result complexity believe restrictive cost online learn act second real time smoothness tractable bayesian paradigm sample kernel exponential respectively write convenience gp analytically keep sample gp independent covariance imply gp formally mean need define call act act variable kernel look natural run kernel true still alternatively approach fraction query group easy approach budget group proceed approach place hyperplane entire suffer high elaborate sequential additive first gp pair time gp next since dimension z bayesian condition aspect specify tend require tradeoff exploitation note reality rarely gp base tuning recommendation point always treat marginal likelihood decomposition infeasible randomly select decomposition choose exhaustive decomposition random part rich risk kernel bias low fairly budget hope recommendation practical observe specification original uniformly random hyper
difference parameter generate obtain assess trace check meaning spread beta noise uniformly hyperparameter simulate dataset sign noise simulate dataset error simulation display th exploratory patient model covariate trajectory cross patient cancer collect treatment month period survey send respective treatment response cancer index answer treatment age cancer condition filter author training filter goal opt constitute remove curve patient filter fail criterion remove patient secondly whose representative constrain treatment datum patient patient report row trust report patient patient note fail criterion sum criterion filter patient tb identify potential recovery curve fitting post shape scatter plot correlate identify treatment patient correlate scatter relationship likely decide categorical whether patient age categorical level author clinical experience pre variance patient covariate bin age thus patient belong depend four interval pre treatment lie add visualize patient patient category general model patient pre value justify show superior analogue consider paper first possess value patient average average scale value patient average curve figure analogous separate patient feature word patient scale regression prediction scale treatment inverse assume see predict scale superior sample tb difference time population parametric shape curve unimodal cluster important visually extract likely give along cluster say clear result certainly make closed predictive boundary peak tb shape treatment patient series respective recovery level dependence show fit categorical see recovery separately figure examine patient curve treatment control patient year level see patient age large old value however level pre asymptotic patient range distinct small proportional drop initial pre treatment appear depend age age patient drop function treatment scale patient treatment value high patient year age level treatment level monotonic unlike time entire post trajectory regardless still compare finding model function treatment low age link binary satisfactory age past analysis pre measure link post function agreement finding unlike previous dependence longitudinal function treatment patient depend emphasize prediction shape however strength visually recovery goal facilitate flow statistically interpretability believe medical practitioner easily particular predict recovery curve domain furthermore visualization encourage recovery curve clear used analyze patient post patient age produce agree supplement past finding medical quantified model produce believe produce benefit context acknowledgement support national foundation grant discussion beta show beta gamma vary tb figure value parameter hyperparameter tie remain tie hyperparameter aforementione style gray sep pt style rectangle draw dash inner sep fit author technology curve curve interest cancer patient utility aid produce interpretable relationship supplement medical event disease level extent perturb recovery many follow stroke exhibit recovery curve initial instantaneous drop towards function predict patient available aid treatment would particular function patient post prediction decision close interval treatment medical widely adopt aid merely patient readily particularly curve restrict curve thus event trajectory curve level drop smoothly lie treatment furthermore encourage predictive trajectory curve plot visually interpret posteriori apply express measure index evaluate convert answer cancer affect stage usually treatment effect localized affect able crucially past study illustrate average patient function time select patient post treatment patient method post prediction score obvious naturally cutoff serious logistic longitudinal outcome whereas post treatment trajectory longitudinal relate treatment growth rich past exist possesse curve monotonically pre recovery growth growth trying predict trajectory include place enforce regardless applicability contexts shape medical growth technique sum score basis incorporation patient specialized stroke time specific contribution expect possess ensure predict trajectory actually shape clinical curve function value satisfy q parameterization thus post event trajectory post event trajectory event parameterization refer scale post denote patient normalize treatment shape scale value parameter asymptotic drop drop scale function drop recovery mean recovery curve block parameterization pre patient appropriate throughout introduction tb observe patient post treatment towards adopt hierarchical bayesian observation patient accord function shrinkage accomplish let covariate model generalize analogous section post treatment curve patient support recovery profile support patient pre patient outcome pre constraint model beta respectively distribution unimodal example constrain analogous constraint ensure beta gamma interval come elaborate patient shape happen shape curve model center section curve parameterization section generalize vector specialize beta spread property pt sep latent pt ib ic iy east xshift yshift patient draw sep pt fit white sep south east xshift yshift b f curve support beta gamma ex specialized parameterization beta detailed mode
superiority propose visual dimensional propose achieve new art wikipedia basic program china cb fundamental project address institute university china advanced technology china email com edu cn com key laboratory vision china university china department electrical computer sciences berkeley electrical engineering university technology datum audio widely internet example text web page conduct recent decade retrieval across modality attract modal span different space heterogeneous characteristic challenge media retrieval address media semantic propose modality observe focus couple mapping project modality modality maximize subspace closeness sufficient media retrieval since modal semantic unite common subspace although modal may text streaming wikipedia http paper cross media retrieval different method retrieval method semantic modal semantic unite common latent fig wise closeness modal learn projection couple projection reason accurate image semantic query hard retrieve image label correlation term optimize main contribution dependent media retrieval datum different modality media retrieval validate effectiveness compare state powerful feature far evaluation feature publicly available remainder organize briefly work media retrieval cross media retrieval experimental past numerous medium retrieval try subspace project representation modality directly modal popular cca pls find couple mapping variable cross media retrieval investigate media retrieval abstraction hypothesis obtain generate medium media retrieval correlation generic multi analysis work view introduce separation modal address nearest hashing approach large search medium propose cross hash hash hash code maximize sparse multi modal obtain code modality modal cross modal development medium deep semantic identify visual label obtain document mapping mechanism modal inter relationship source stack auto beyond problem bi media bi directional ranking embed medium inter media heterogeneous cross medium visual medium retrieval retrieval dependent retrieval text e space cluster instance I original assume ij th media problem map dataset consist pair wise closeness modal semantic mapping subsection framework address media retrieval image retrieve respectively denote frobenius norm matrix paper define media retrieval retrieve regression semantic framework present optimization unconstraine convex problem local solution design stationary fix fix specifically minimization q partial updating summarize procedure I easily semantic matrix n cv size convergence v w evaluate systematically wikipedia totally text utilize available sift lda besides media cnn latent feature base token use description firstly base token stop utilize lda text annotation remove pair belong category treat utilize firstly feature token compute topic semantic text experiment euclidean retrieval precision map retrieval specifically precision ap query item rank pls cca sm cca mainly cca sm semantic three cca cca discriminant wikipedia firstly publicly I sift lda experimentally optimization map score method see effective compare learn validate necessity medium term eq could compare dependent htbp wikipedia unified scheme wikipedia cross media retrieval division average image text map
network exploit recall variable moment draw approximation bind approximation distance scale nonlinear line target x frequency magnitude spectrum linearly overcomplete linear independence frequency lift suppose learn lift satisfy theorem arbitrary neural magnitude fourier formal bind frequency make intuitive fluctuation second lift factor sample method network term neuron generalization impose vanish idea complete proof appendix lift part fourier technique estimate label neural architecture label score tensor decomposition network specify proved function yield operator complete magnitude phase lem manifold fourier exploit argue bound impose set say hilbert satisfying term multilinear multilinear form bind matrix technique vanish assumption sample lift know see tensor exact prove sample propose follow magnitude relate sample lem fourier fx fourier transform entry noise equality equality zero final equality manifold compute uniform ss lf integral simplify I proof property delta q use equality introduce change q sake implicitly second delta property repeat finally eq argument impose l thus integral limit magnitude weight concentration desire nn lift satisfy prove concentration use label denote final use apply inequality concentration h v proof phase actually generalize norm rgb derivation claim observation example time bold bold ff non optimization backpropagation descent novel neural efficient input tensor provably set mild degeneracy standard descent sgd neural moment tensor decomposition network vision recognition understand neural paper training guarantee error ability unseen analyze training overfitte poor new classical bound extensive guarantee neuron np local per optima example analysis guarantee lead relevance network wide guarantee hardness refer bad condition input tractable tensor formulate tensor finding sum fit achieve computationally technique mild degeneracy model address trivial question work use tensor linear activation adapt set behave perturbation establish address question efficient term tensor lift sgd training scale train layer feedforward layer start neural high lift decomposition use estimate bias lift operation fourier dataset complexity comparable lift transformation transformation depend learn manner without many theoretical distribution fitting error nn lift polynomially etc guarantee lift well approximated architecture natural expect guarantee meet continuous input discrete reduce function collapse single precisely characterize redundancy neuron network weight distribution target thus achieve generalization matrix etc moment correlation lift tensor yield third identifiable tensor realistic assume orthogonal require vector tensor network exceed input overcomplete exceed incorporate tensor recent perturbation despite convexity decomposition provable lift nn lift layer recursively layer principle analysis control estimation establish formulation column refer operator th rao member product euclidean space second refer rd order outer unit say cp transform call frequency neural know continuous feedforward neural nonlinear fit target neural estimation network finite ability hide label weight overall estimation establish combine generalization detailed minimum pt sep draw fill cm mm circle cm dash mm f name name name name observe width line width width red width red line line red mm line width mm width mm red blue mm blue width network name lift use component denote second estimating bias unknown compare part explain third pdf mild regularity probability derivative differential operator discuss next see various score score addition learn auto find encoding decode reconstruction denoise unsupervise unlabeled argue approximately first estimate recursive estimate high score score representation extract use training neural ia yield show eq th refer notion reason behind score recover tensor tensor appendix power power rd tensor ti l multilinear guarantee iteration orthogonal tensor develop literature whitening apply tensor iteration perform start mini small initialization overall complexity nn lift parallel iteration first auto approximate dominant computational complexity tensor comparable computational backpropagation processor estimate weight bias decomposition fourier frequency fouri entry dimensional manifold intersection sphere actually spherical draw spherical coordinate angle directly cone angle draw pseudo spherical estimate density function l v need function network function second case high cross network overcomplete product full overcomplete similarly extend degeneracy vector full singular target overcomplete product smooth
matrix g projection useful important reader understand embed intuitively follow vector x speedup solve problem frobenius error hold strong speak rank approximation almost column solve svd svd expensive line matlab reader try near randomize hadamard sketch projection combine know seminal gaussian implement line code c guarantee advantage implement line matlab quality sketch even sparse hadamard transform hadamard fashion uniform nine line matlab notice product perform low complexity n subspace embed count sketch property sketch apply uniformly reader notice sketch fact entry line code true j sketch memory keep pass theoretical rank disadvantage sketch attain accuracy improvement sketch matrix cs product subspace hold nearly efficiently small prove sketch count projection cs satisfy q property property subsection sampling leverage random projection column selection visit entry preserve negativity sketch whole avoid every entry matrix leverage define equivalently coherence great leverage score small good study define column leverage column score roughly leverage score k theoretical leverage sampling satisfy sampling accord score leverage expensive svd leverage score practical way score little uniform leverage effective heuristic finding representative zhang centroids class centroid sketch heuristic little make simply solve centroid unnecessary centroid run centroid suffice local dataset supervise associate centroid sketch row correspond datum contain label regression computer science economics inverse svd solve cg machine cg cg attain x roughly time cg heavily slow ill condition long subspace embed solve gaussian count sketch inexact implement matlab matlab sketch sketch thing inexact sketch logarithm thus high subspace indicate z sketch score find standard discuss cg efficient matrix cg let triangular matrix qr compute one easily thus factor probability qr r cg initial subsection extension svd efficient extension describe solve section particularly cs complicated solve implement matlab function section uniform nearly sampling complexitie logarithm iteration weakly gap block implement u c end attain machine pass infeasible memory trade cost go pass store volume disk ram keep sketch iteration svd nk kk kn describe frobenius norm target matrix qr n low gaussian matrix sketch property hold minimization problem matrix find orthonormal matrix randomize depend implement line code empirically discard line algorithm keep prototype solving randomize even fast reader inexact present trick cs embed solve frobenius randomize svd approximate decompose sm cs cs p r p svd k sketch sketch sketch sketch rd clear thing go pass cost k line remove l program fortunately pass ram store disk block consider kernel social matrix fisher find low symmetric show sketch row rbf kernel compute matlab sigma minus form rbf matrix computed function sigma presence million kernel fortunately sketch efficiently matlab code sigma sigma require solve exact solution time approximately need identity expand gradient naive low rank inversion possible spectral fix approximation besides diagonal even approximately discard way low approach svd show chosen error apply follow prototype count line code go pass fit store disk enough despite several drawback cost visit entry serious apply kernel point time avoid every entry reader notice column approximately tb integer column qr try leverage proportional norm row high probability sampling overall matrix sketch contains empirically enforce improve empirically large sufficient line k true z kp give give row rbf sigma sigma true unique sigma computing propose w become approximation call nystr om nystr om use literature figure illustration nystr method thing nystr nystr om rough moderately accuracy thus inverse numerically inverse drop bottom many discussion nystr om implement matlab code nystr om w w sigma sigma rbf tune enhance notice small affected approximation nystr om kernel efficient nystr om inexact mean top efficiently use thing speedup square svm speedup eigenvalue significantly unstable reader well implementation extension nystr om selection c much nystr reason speedup subsection rectangular matrix matrix form multiply large generalization kernel fortunately matrix merely give cr prototype prototype every compute let solve kernel avoid whole leverage column quality approximation speak uniform sampling well suffice quality pc c c pr pr pc pr unique u pr x rbf procedure c sigma r pr c sc sr sigma pc pr pr sr pr sc sigma pr generalization spectral transpose kernel rbf goal extract feature step datum th kernel th entry feature eigenvalue empirically accurate nystr om cost thus fast matlab lambda sigma k sigma clear u lambda lambda lambda k u lambda sigma end row output perform use test user decomposition normalize uk fourth scalability spectral al nystr om make spectral cluster scalable form accurate nystr om line matlab sigma n sigma times replicate fast instead efficient replace sigma gaussian bayesian hyper parameter transpose rbf form tune label compute inversion empirically apply speedup discuss similar nystr nystr matlab sigma alpha l sigma alpha end intensity train matrix entry compute predictive four sigma sigma apply generalization straightforwardly speedup sigma max r sigma c orthonormal column b b solution qr decomposition full row
broad art model special significantly remarkably neural performance comparable vocabulary problem language application speech parse task generality translation paper neural translation fall category encoder decoder sentence automatic represent sentence bi directional alignment network target sentence rnn alignment decoder achieve less instance superiority efficiency mainly mechanism avoid vector dynamic mechanism external inspire novel architecture name carry task series input different eventually intermediate stacked layer stacking generalize network introduce tailor sequence special case architecture importantly deep capability superior cc start discuss read operation memory generalize form illustrate transformation memory read head controller controller operate read memory modify value location memory write basic machine architecture implementation infinite size determine memory implementation always instance determine controller controller operate read write head discussion put simplicity read layer high another reading convention get controller influence controller read address head controller location core controller long memory rnn lstm state controller read write return read turn update reading allow one example suppose omit notational simplicity read memory unit main respectively operator read writing relatively effective reading write address address read address run determine time important suggest go forward read parameterize differently address address f implement dnn un memory mechanism advantage address way therefore introduce flexibility hybrid address address read read address controller worth note address address read head address allow read different base address address writing simple keep unchanged write determined network tn normalize weight dnn clearly transform memory embed shape specify parameter argue offer complicated introduce flexibility address read get unit rnn stack right read invoke dependency spatial order recover deep layer read operation address read writing offer major address add layer design especially couple next layer deep later learn base representation need read transformation induce read address strategy list since combine address read read amount specify omit due design different way address writing stack together stack architecture analogous representation suited layer transformation stack greatly efficiency language chinese english figure leave stack apply transformation low layer base layer stack manner entire diagram right start layer reach layer read operation output rely lstm generating memory read follow symbol guess state generate lstm read flow layer target pure address reading stop generate token different read learn different homogeneous mostly transform activation sensible transformation greatly performance little future read write reading allow read specifically follow cc memory require strict alignment together create structure section read layer potentially inner read right address read address read head write reading memory accordingly flow correction scale signal start output layer signal back control machine lstm read write location address optimization practice descent sgd control discuss four representative show proposal c address intermediate layer diagram right address address read read layer put together layer address sequence read different memory use memory layer differ address address form together predict target deep architecture address strategy generate layer address write address address read head bundle layer layer put intermediate among special efficacy address write form address read memory write address read address address layer transformation read address cc interestingly seminal translation automatic right employ address read address write address target intermediate nontrivial write operation empirical english art machine training sentence corpora million chinese million choose mt mt mt mt number sentence mt mt mt mt insensitive evaluation significance segmentation chinese stanford nlp english frequent chinese map token translation corpora direction grow diag adopt model gram
additional latent transformation volume transformation operator mcmc langevin elegant approach iteration true disadvantage langevin hamiltonian one gradient throughout effect flow inference training monte estimate version parameter sample version result schedule go variable deterministic maxout linearity windows maxout window take mini batch collect time average score estimate insight normalizing flow set unnormalized list e w w characteristic normalizing length transformation diagonal volume preserve flow nice achieve performance grow flow flow far parameter flow nice initialize matrix nice figure height fig posterior posterior kl divergence digit contain image ten handwritten digit train latent flow nf approximation volume preserve nice nice different summarize systematically kl approximate approach normalize nice wide result specification nf nf nice k nice nice c consist rgb size extract patch convert logit x summarize increase length systematically improve log posterior ccccc develop density transformation complex normalize flow inference clear improvement view normalize flow unified perspective closely flexible point conclusion flow rich approximation normalize flow convergence class see able competitive default make rigorous research normalizing flow simply flow base alternative transform design g lie transformation allow thank radial flow always invertible linearity choose condition solve parallel expand take dot product yield invertible h enforce modify produce compactly write always invertible splitting uniquely scalar take q suffice impose constraint impose choice inference application employ family inference approximation impact specify flexible scalable construct normalize whereby invertible transformation view develop category flow view theoretical true combine variational interest increasingly complex problem increasingly large core large model default chemical despite advance limit power wide default limitation choice address know approximation approximation imply solution method inferential regime unable recover posterior rich sigmoid belief field posterior auto clear evidence limit posterior provide exposition widely choose posterior result map rich typically mean incorporate dependency within powerful evaluation present approach specify variational base inference carlo section propose flow distribution transform probability invertible mapping sect normalize sect show normalizing flow admit us regime variational present unified view normalize flow sect normalize flow systematically compete sufficient marginalization variable integration marginal likelihood latent variable principle jensen prior latent often refer approximate prior act parameter variational use mini batch descent scale address variational log approximate tool way log expectation approximation analytically mini carlo compute center approximation affine backpropagation involve latent know location transformation backpropagation carlo monte carlo variate exist alternative backpropagation variable continuous backpropagation model variable among compete use map compute variational time cost generalizing deep deep deep latent gaussian deep direct hierarchy latent variable transform composition form successive distribution transformation law know expectation write jacobian depend invertible flow reduce point towards interior reduce outside formalism normalizing flow give variational appropriate transformation factorize length increasingly modal normalize term transformation partial differential evolve time langevin sde wiener process vector diffusion transform langevin flow transformation kolmogorov transform evolve often langevin importantly density evolve sample langevin sde accord density term result machine make hamiltonian allow scalable inference normalizing flow jacobian straightforward equation g invertible approach jacobian determinant dimension furthermore gradient jacobian determinant several involve numerically unstable normalize flow allow jacobian determinant transform define transformation series expansion hence refer map modify density around parameter radial density flow visualization show spherical successive form invertible discuss satisfy appendix posterior flow free write normalize flow free variational generalize variational construct
security additive interactive database original statistical query statistical decrease perturbation interactive differential result interactive correlate perturb get get survey present berkeley edu present private aim adversary sensitive subject intend possible perfect privacy regardless preserve sufficient health sharing grow volume phenomenon storage social law public patient lack etc even whole use area interest collect health technology health patient monitor status intervention overall scenario collect privacy preserve patient privacy life cycle processing finding focus phase sensitive private thus aim prevent adversary intend linkage maintain privacy private subject might consider become argue private private circumstance reveal little adversary private piece privacy converse also complementary follow survey relate motivation discuss experimental discuss conclusion future research technique security database intersection randomization survey analysis privacy field conduct brevity brief study survey area recently attention health spread health record medical concern regard health research medical therefore domain reader privacy database statistical database review lattice limiting risk lattice control security privacy security rigorous survey datum privacy statistical database privacy grow privacy differential privacy quasi subject anonymous every record anonymous individual individual extension include closeness differential single record differentially private sd show attribute differentially diverse review privacy preserve piece private exchange use shorthand capital realization respectively variable marginal function instead write mass primarily monitor health share nature infer piece like health patient weight generally want piece infer private information guarantee information potentially use inference passive circumstance ability infer auto thick circle fill draw font edge edge leave node concrete follow health index already know status category consider however category infer want ensure privacy create encode public status group encode privacy preserving sense status category security objective adversarial infer private text perform know complement private ability adversary infer towards consider send send information passive information exploit know decode send e ability inference send minimize private similarly define piece encode send call encoding output since use message describe treat continuous reader substituting function cover case nature use appropriate function need adversary different subject model recall information intend know intend send message requirement finally like minimize carry sake infer adversary present reader correspond concern discrete mixture discrete continuous yy intuitively bit mutual independence conditionally objective class information hold privacy communication transaction piece information belong transaction piece information send belong apply function address question formulate condition space pc pz pz dx pc privacy adversary conditionally prior adversary already possess need ask privacy ever privacy message extra adversary surprising utility usually utility kullback define know get pc pz mapping fact pz r valuable privacy mapping observation case membership function adversary regardless auxiliary serve privacy verify origin zero mean similar omit every distribution shape cc describe publicly toolbox question privacy category hand order maintain weight status base scenario monitoring fit assumption motivation early matlab toolbox affine extra degree problem c yield affine class ground depict class note calculate know train classifier procedure encode datum confusion information drop highly indistinguishable since weight category class trivial predict datum bias category category class histogram category preserve piece without look category decode weight decoding point also infer derive preserve private theoretical perfect privacy preserve utility show achievable provide close perfect privacy adversary knowledge show perfect adversary auxiliary knowledge information subsequently discuss demonstrate control weight status set drop bind guarantee achieve approach private information infer propose serve alternative say adversary get access message suffer curse dimensionality number framework estimating appeal option model mixture computationally mutual extend scenario clearly general wide applicability present indeed message sense well belief adversary private research include mapping necessary condition change approach solution analogous minimal oppose privacy acknowledgment discussion significantly improve team research science foundation toolbox toolbox toolbox similar code keyword subsection reader familiar helpful toolbox organize short manual toolbox describe engine privacy structure parametrize definition dimension hand two keyword begin code subsection worth note nested block allow skeleton skeleton begin variable atom user search parameter fx variable implement toolbox parameterization convenience var shorthand var variable either expression convex constraint var expression expression object mathematical involve repeatedly equal code engine create constraint constraint later engine feasible problem treat general object pass separately learn subject mark begin constraint nothing line keyword toolbox provide keyword toolbox current implementation datum histogram allow bin example weight another private class class name string call one try variable useful example class provide store variable come convention list keyword toolbox ht variable cn expression parametrize end end definition block program rule satisfied program start end maintain consistency assume toolbox therefore toolbox assume dimension toolbox computation follow assume variable constraint latter create instance map inequality v symbolic constraint implication later relaxed fix correspond entry exist
relation relation yes counting list set simple knowledge conjunction compound reasoning induction reasoning task language present learner learner ai hope loop solve believe language solve ai fail task go ai beyond variant also highlight extension hope task fail motivate loop develop task believe language understand training task ai learner ai present memory interesting beyond highlight propose extension still several supervision support typically require task human couple additional supervision signal hope develop solve lead research produce applicable reasoning language building agent goal argue usefulness set proxy evaluate read answer simple many design aim many set researcher identify recently introduce able motivate motivate semi supervised equation uci continue component dataset latter relevant work synthetic amount datum researcher elaborate base datum example gram compete far researcher synthetic try break work develop automatically ai response task answer question think cast propose capability common build unify classic whereby actor object interact kind hope task current help loop break task sec benchmark result analyse failure development propose memory show unable solve open project recently unlike task like especially scenario appeal research question human child require ai organize collection question intend machine reader aspect remain complicated indeed able task answer etc capability improvement modification project come scale feature acquire corpus argue actually understand extraction representation remain highly rely lot choose collection failure success feedback capability schema challenge thank help receive help result challenge mostly center around system background diverse self train need setup amount test relate one lambda compositional semantic provide task software computer ideally test simple aspect subsequent build publicly successful perform supervision answer statement answer may task correct else cut noiseless human potentially try choose human reader formal semantic logic representation simulation character object move around interact location generate many giving include task support provide person thus office already simplest hard answer support question office pick support difficulty letter alphabet dataset ip I ip vb ip language english produce task task le support fact pick office drop office statement answer answer recognize subject consider extreme sentence bag office north north office word different answer separate argument task give receive give question two actor mention test support ability answer question pick yes perform count operation hold design answer ability produce answer list entity pick pick hold database operation union apart one type support fact imply office office office office yes yes statement describe possibility office test simplest detect office typically label study sophisticated phenomena address multiple subject task refer actor office far implicitly understand expression school school go world evaluate via property induction color white induction scope produce induction test spatial component red sphere right sphere square yes red triangle yes task yes question inspire reasoning schema box fit yes three yes task north east west north ask certain address actor behave game generating location object hope well model within environment real complement version task class apply memory index learnable execute feature convert internal representation memory I output component use parse simple incoming example leave responsible reading g calculate produce actual module produce feature support scoring support memory support square output module produce recurrent rnn limit response ranking dictionary function feature role map text every representation depend support model gram ng multilinear ml indicate task extension last analysis column amount achievable training c c strong supervision c fact support fact relation yes question fail set fail conjunction compound time reasoning induction fail reasoning extension model former order need know sentence directly sentence old triple pair question carry gradient answer support try task fail bag sec max involve support fact unless rnn provide set required finding improvement variable number support fact dependent ask support x support fact stop embedding learn hard loop computation stop module word iteration conditional I I rx I way model bag variant gram bag disadvantage dictionary rapidly neural multilinear map position position employ whereby word position follow nonlinearity mapping tag rather consider nonlinear embed network side bag gram follow nonlinearity compare I long recurrent network describe gram baseline produce bag gram share least answer use gram use filter method similar answer support fact disadvantage testing hyperparameter give result experimental separately outperform consistent still task build failure expect fact answer bag failure yes linear scoring query yes answer interaction response sec approach gram ng multilinear ml plus combination give straight forward support multi output remain difficult gram modeling clear improvement task gram seem substitute embed outperform gram especially yes fail g combine ng improved task multilinear useful gram ng example perform example quickly task require require latter solve pick subtract drop task finding solve even advanced build
order random tuple bundle ii kn label enforce pseudo label pseudo random tuple pseudo say q exist efficient input variable pseudo random q conclude return almost term kk arbitrarily case choose c kn google fellowship research author thank discussion many I work terminology counter hypothesis theorem conjecture learner access use prove complexity formula show guarantee even favorable strong version logarithmic arbitrarily low case proper access hx learn parameter learner run emphasize general improper good bound efficient poor assumption exact algorithm theory establish hardness gap problem unclear belong learn recently bound possible certain hardness certain indicate low recent hardness learn hardness average problem yet natural rather hardness recognize direction overcome proved hardness spirit work hardness formula framework strong allow hardness e proper hardness currently low base concern study assumption hardness approximation public tuple coordinate collection tuple denote collection tuple distinguish formula hard gets gets solve seem light problem evidence addition performance hardness instance approximately interpret hard semi formula relaxation relaxation sum problem relaxation hierarchy low bound another analyze bound statistical solve formula extensively formula efficiently make cn algorithm trivial distribution output underlie belong family time imply constant section outline implication hardness upper efficiently linear much hard currently guarantee achieve marginal hardness assumption security hardness derive formula show assume marginal nothing hardness concrete algorithm bound hardness approximation restrict algorithms classifier computational already soon theorem later consider author assume efficient distinguish instance instance even rather methodology hardness basic certain hard learn restrict boolean cube sample fair denote distinguish easy output efficient contain toward return distribution maximal random example lie bit describe bit efficient use l generate choose random example l use description oracle n weak end explain course reduce problem conceptual problem correspond constraint namely problem formulate minimal problem distinguish n distinguish point sense yet measure next address easy reduction strongly hard error map reduction address replace independent second show produce property independently uniformly new indeed w either produce note together reduction reduce specify w fail reduction reduction n reduction randomness h test give next deal ks r describe pseudo property convenient whose un concatenation indicator product remain nk class consist therefore show ks ds ks ds j indeed ds j c strong pc strong theorem unfortunately explain pseudo set close check fraction tuple polynomial h nk c cp k eq polynomial z kp prove completeness assumption arbitrarily close conclusion aspect theorem restrictive give majority completeness toward realize define eq nh w w n show efficient distribution distribution analyze formula assignment variable every mapping partial output different remain size leave remain unchanged partial tuple hoeffding tuple pseudo tuple formula every z fix z j note u j
error assess finish small scheme associate may fail probability inequality sample probability imply finish task interval inequality depict hold short upper furthermore present transition behavior confidence critical exponentially quantity performance reveal provide extreme possess generalization regression kernel localization domain control essentially linear thus automatically square expectation differently present exponential purpose term ill employ guarantee number bind deduce attain minimal guarantee suitable generalization capability study capability kernel whose lead specific regularizer smoothly toward exactly vary form choice application purpose show utilize impact capability range real may bring suitable notice assertion conclusion heavily behave square distinguish regularization range regularization attain highlighted knowledge among encourage reader lasso regularization see accord lasso generalization capability choice capability determine take non generalization consideration smoothness explain domain near construct arbitrary sample estimate section respectively deduce x rewrite sample subsection type describe rkh norm define addition formula since v fx u k p dt u vx dx dt nn u dt imply k n proof complete lemma deduce subsection standard typical one bernstein inequality probability space deduce q q imply eq q desire difficult empirical bind main norm denote cover bound properly inequality deal class bernstein single almost everywhere everywhere everywhere everywhere apply eq hold provide hold rate ridge regression proposition ef ef confidence cm r actually chapter constant complete simple traditionally space error regularization regard nature data dependence attribute essential characteristic scheme excellent localization frequency reproduce guarantee present formula spherical polynomial help deduce approximation deduce probabilistic inequality spherical polynomial definition space u u generate operator interior help deduce real without small convention set space q follow lemma note polynomial taylor formula ep qx p np cn since polynomial e deduce eq confidence hold np np prove functional operator restriction hold old introduce decomposition ef ef e ef e ef ef z q ef ef ef since inequality ef ef qp ep present error deduce old implie represent deduce regularization cn ef confidence set tackle spherical due perfect localization frequency paper suggest usage spherical contribution firstly selection kernel totally require computation set excellent localization truncation add sense optimality mean discrete parameter reduce computation secondly bridge bring utilize kernel capability ridge excellent property domain recognize tool tackle spherical datum domain nonparametric due localization regularization ridge arbitrarily consequence excellent localization property far associate include bridge possess almost reveal utilize choice strong capability modeling arbitrarily smoothness computational complexity nonparametric spherical scientific explore predictor variable phenomenon euclidean neural support machine appropriate exclusive useful recent focus considerable spherical spherical poor spherical localize spread sphere formulate theoretical alternate sphere reproduce hilbert utilize spherical popular polynomial similar drawback method remain exclusive spherical sphere localization requirement develop technique cope nonparametric regression exclusive localization paper organize follow introduce generalization capability ridge capability scheme regression main result useful sphere integer homogeneous harmonic sphere spherical class spherical spherical degree course comprise restriction polynomial
forward fairly smooth parameter mse linearization approximation enter approximation fix smooth linearization quality estimate wide comparison map ml kalman filter kalman kf kalman filter state lag pf backward maximization link e mail liu se division mail electrical engineering mail newton gradient hessian compute gradient identity explore linearization linearization computationally model cost ml control overview focus denote time function dynamic denote solve hessian quantity smoother explore recursively sensitivity derivative analytically intractable result solve end linearization likelihood extend smoothing gauss particle filtering smoothing method asymptotically estimate computational beneficial linearization approximation small linearization typically vary evaluation focus approximate likelihood subsequently newton approximate finite difference approach ascent particle challenge skewed lastly method gaussian optimization method advantage estimate product intractable distribution section use linearization return solve ml hessian schedule newton gradient hessian pt obtain label use use gauss newton similar particle height pt run label sample multinomial iw first give section locate hessian equal counterpart consist compare propose initialize estimate set present parameter positive plane low alg far alg alg alg term bias argue accurate alg r alg alg alg scale dynamic apply necessity would expect compare
cm fix design device implementation dct architecture version expansion correspond expansion design design list input dct array device via interface hardware processing toolbox define within bit typical ai architecture throughout ai encode ensure offer accuracy architecture resource device bring term slice look table resource though design expansion possess accuracy compare design hardware resource architecture require considerable amount ai propose real bivariate ai encode ai encode dct hardware completely ai domain completely quantization free final ai tb dct operation entirely ai intermediate ai dct quantization noise final location quantization channel dct noise level remain hardware test ai encode bit relevance bit realization operational frequency ghz propose dct embed resolution word dct publish architecture arithmetic row column dct transform precision affect dct acknowledgment proposition definition cr tag cr dct architecture integer exact mail algebraic integer base architecture propose ai encode discrete cosine dct free dct without dct ai user dct multiplier expansion architecture high digital video computation propose validate hardware implementation realize bit size simulate among bit input design imply ghz embed image device dct digital video demand video image automatic surveillance traffic security wireless video system operate associate hardware throughput complexity efficient circuit capable operation numerical need video resolution minimal noise consume possible two discrete cosine transform dct system circuit dct relate noise circuit consumption video device dct successive call apply dft area point dct require multiplication computational rational dct implementation employ operation introduce address employ algebraic integer ai ai encode possibly integer exact dct multiplier computation ai back usual besides quantization dct dct coefficient correlation noise video signal noise concern dct ai dct architecture form ai complexity multiplication eight thus naturally low foundation propose optimize architecture require reconstructed format column mean reconstruction code enter dct ideal propagation intermediate propose digital dct throughput quantization ai concept reconstruction truly occur structure prevent computation result correlation final coefficient totally doubly ai dct precision dct speed section dct base fully architecture fundamental difference intermediate doubly ai scheme architecture characteristic dct absence operation quantization architecture aim bi encode ai basis optimize multiplier realize gate array review exist ai bring description hardware architecture detail propose test measurement report conclude ai digital dct bivariate encode bivariate dct dct area processing circuit dct core conventional arithmetic architecture architecture dct buffer implementation dct block size suitable application available linear array dct hardware realization array architecture forward dct report dct core cyclic performance transform employ scheduling dct array algorithm base dct realization dct block cycle ip core dct synthesis arithmetic dct processor dct architecture unique describe ai dct computation wise application dct core ai architecture realization also application dct dct refer encoding ai dct quantization report implementation synthesis ai encoding make realization prevent adopt ai mapping link integer array major classic exposition widely clarity depth bring explanation emphasis integer follow focus practical ai useful number call algebraic root coefficient algebraic mathematical form multiplication ai encode follow format integer array integer always integer arbitrary decode operation array constitute ai basis hardware ai basis require ai ai ta integer represent decode principle limit employ integer encode exact multiplier dct encode dct algebraic ai sequence represent ai interpret modular multiplication polynomial ai particular illustrative multiplication fast multiplication consideration ai possess constant represent error ii integer element representation small facilitate encoding decode ai constant yield transform dct cosine arcs adopt dct element particular dct encode adopt array encoding encode scale encoding specific ai cosine possess free utilize integer arithmetic moreover employ hardware shift ai encode dct arbitrary real usage essentially exhaustive search however unnecessary encode express term usually hereafter identify bivariate ai coefficient representation indicate encode integer element emphasize ai encode algebraic multiplication modular hold tailor technique handle ai dct mathematically express usual dct notice correspond column application dct dct result dct use ai encode decode section place column dct operation intermediate introduction quantization noise component contrast employ bivariate ai encode maintain computation ai arithmetic avoid arithmetic error ai dct ai dct column tb ai dct block wise architecture five input circuit ai encode dct block fig column ai buffer connection obtain iv ai dct computation eight circuit ai encode complement format via connection input format already block block stack pixel dct modular refer modular input bit serial fed rate aside stream optical transmission throughput drive sequence row mean eight operate eight consist ai code computation ai dct wise dct core architecture employ arithmetic entirely ai transformation ai encode channel index modular hardware ai parallel channel ai encode integer cf four connection channel ai transpose buffer ai show pre wise dct partially transform wise dct represent encode ai tb fig operation eight cycle ai ai column wise eight therefore output buffer require transpose dct connection cross brevity ai tb period master subsequently ai element ai dct core operate parallel continuously partially dct component bit channel row ai dct provide input rate dct input dct every eight cycle signal perform row dct order complete dct doubly ai representation output channel ai number complement format architecture architecture differ circuit compute indeed circuit employ dct prevent dct channel quantization uncorrelate doubly encode element final dct summation return term rational number two remain binary close sign list number consequently respectively cl input fix
smooth distribution achieve study learner decision expert learner set follow loss also multiplicative weight bind generate base entropy think loss return learner generate assumption eq q f te sf nd nx q fact lemma contradiction mm find center minimum denote entity w entity remove large remove small element mm word correctness see overall reduce half stop round except need two round overall communication institute technology agnostic setting noise general boost concept space noise computationally communication prohibitive demonstrate scalability increase amount attention datum common fit want process utilize one entity evenly inherently entitie example scenario scientific customer deal datum partition traditional algorithm care bottleneck communication baseline entity center vc communication advanced communication complexity recent work distribute boost logarithmic standard set noiseless impossible communication inefficient boost agnostic much enjoy communication baseline efficient concept finite dimension insight example learn weak hypothesis boost result challenge design agnostic set agnostic boost weak list open work rate algorithm communication distribute identify boost agnostic adapt agnostic require call learner previous learner set class agnostic learn flexible weak learner learner centralize distribute make easy confirm theoretical empirically synthetic promise introduce agnostic problem access low often within denote common function agnostic entity entity act learn much convenient count problem paper bind communication complexity boost vc boost hypothesis assumption agnostic setting even set poorly set access learner agnostic rate discussion existence centralize reason boost agnostic tend weight end put noisy overcome smoothed boost technique enjoy nice shown originally analyze hard first weight additional current weight bregman projection technique find distance bregman boost always generate verify convex hypothesis center call weak entity sum center across sum vc rate underlie sufficient find error good weak entity summation center communication index find sort coordinate inefficient fortunately advanced find median median potential large em subset mm mm em mm mm mm mm entity entity w w project set complexity direct adaptation centralize proof correctness search run find respectively must remove median center half stop except update round find easy base find candidate communication agnostic algorithm theoretical rate use round involve per boost draw ks centralized theorem thus achieve communication round communication weak bound note vc generalization weak learner tn iv center agnostic weak learner union return update normalize project algorithm mm empirical boost algorithm synthetic dataset adaboost logistic implementation three amazon ec trial still run synthetic dataset interesting potential boost choose odd sample coordinate equal set example machine approach poorly achieve adaboost experiment real million datum repository yahoo yahoo positive example
incorrect right sentence action formulate problem action world encoder decoder infer action lstm encode alignment lstm decoder sequence neural encode encoder state encode version rnn use arrive decoder hide maximize determine sequence approach decoder employ function term dependencie vanish gradient alignment salient abstraction detail illustrate encoder take natural sentence treat word vocabulary rnn summarize relationship word annotation annotation sequence architecture affine sigmoid forget lstm cell activation cell summarize forget lstm gate affect hidden encoder encode backward direction recognition machine hide annotation encoder annotation include word improve decoder level context also word permit match salient sentence g weight extent position around match model perceptron architecture lstm decoder take current previous matrix learn encoder decoder action give draw corpus demonstrate loss train sequence find posteriori action learn search sequence search initialize sentence list step action world agent line domain include define bag word world publicly contain six virtual pair sequence deviation ht sentence train fold retain latter tune repeat fold weight fold later refer training procedure adopt whereby train decide empirically effective decay decay find increase epoch converge within epoch regularization use early metric end strict evaluation metric position exactly orientation goal still challenge error overall sentence benchmark present study first investigate ability intend language report overall accuracy sentence directly use linguistic resource sentence really amount paragraph art employ specialized linguistic semantic multi strategy sentence additional enhance reinforcement path correction chen consequence accuracy vary different aforementione average five previous art stable randomly use evaluation ensemble improve ensemble single multi sentence table ht sentence reach exact model reach fraction reach reach encoder encoder understand encoder experiment directly randomly initialize embedding decoder rely alignment table present demonstrate encode sentence information sequentially help resolve turn encoder utilize focus salient effect training vector unweighted eqn encoder rnn evaluate single sentence execution competitive multi despite work train linguistic resource study currently extension embedding reinforcement add acknowledgment david chen prop pt edu sequence language natural recurrent rnn sentence environment propose salient region alignment help resource e seed benchmark sentence dataset give competitive sentence able understand execute people environment ambiguity amount detail require specialize annotation goal end linguistic knowledge I compositional semantic raw pair able language propose neural long memory encode action suit task temporal machine action use base approach representation achieve sentence prior seed use exhibit stable paragraph perform specialized series study primary conclude direction form calculus weak supervision learn multi inference generative structure include model spatial discriminative compositional factor express correspondence linguistic object location action alternative formulation treat end neural
map subsequently compactly ready next contain advantage apply arbitrary nonempty define expansion sample expansion cf assume coefficient sample sufficient converge coefficient likewise make ensure expansion assertion converge proposition mutually constant assertion converge conclude prove consistent sum turn expansion necessarily formal qualitative end approximation unknown along construct kernel close point appear lead effectively enough work definite uniquely substitute arbitrarily rkh consistent least since state estimator technical assumption e dominate sufficient dependency issue consistency asymptotically arithmetic virtue order estimate parametric gaussians z point variable jointly need value characteristic general expansion think propose applicable dependent interesting field consider jointly causal direct acyclic dag statistic causal direction suppose eq simplicity eq independent copy basic expression multiplication division scalar rkhs independent proxy lead sufficiently estimator range approximation kernel reduce size optimize coefficient rkh analogously I evaluate kernel gaussian rbf kernel bandwidth choose distance distinct depict operation three see sample increase bivariate inference interested identifying cause cause observational datum cause benchmark x square fit degree illustrate next direction score specifically decide rbf heuristic speed adopt map see value rate force range scatter benchmark depict pair benchmark method achieve force developed base approach value propose rkh cause encouraging material remain unify describe hope future le song comment eps stroke acknowledgement mail statistical mathematic mail ac mail reproduce applicable respective distribution probabilistic programming structural derive structure crucial computation carry string determine well operation permit typically composite simple one operation applicable propose build applicable approach nonparametric categorical general pay either input require characteristic mapping associate represent distribution hilbert space generate kernel reduce weighted operation expansion remainder article organize describe analysis conclude limited represent reproduce hilbert attract rkhs briefly start live nonempty experiment kernel equality strictly kernel pca hilbert map positive allow canonical whenever write satisfying reproduce space g string possibility helpful map order one generalize point eq density provide nonnegative map apply kernel take negative normalize moreover would kernel map map symbol borel hold support distribution sometimes distinguish map special sometimes instead retain represent object map summarize result table moment see conclude characteristic universal include moment rv equal equality homogeneity independence latter estimate aside stein construct cf probability requirement use rkh sufficiently bound approximate kernel substitute alternatively method match way road linear algebra question interest far rather conditional update connect lead realization rv induce probability instead distribution value random define operation density belong elementary form spectrum resort operation real world system uncertainty arithmetic measurement connected serial establish arithmetic independent suggest fourier transform people propose cumulative become loose repeat use computation consider numerical arithmetic represent piecewise chebyshev long behave approach scientific g goal another generalize propagation help inference conditional variety probabilistic programming propose map expression operation apply expression show desire property operation rv resort exist complexity result expansion limit benefit three fold first data domain kernel finally density intermediate dimension function take since speak leave I measurable consistent moreover convergence
value positive let cm relaxation supplementary lp relaxation tight df satisfy condition solution lp rr energy bind follow lp responsible marginalization w paragraph relaxation constraint redundant potential order potential equal first potential labeling two mrfs potential reduce marginalization pairwise lp extra lagrangian potential method discuss sec approach dd perform mrf create mrfs semantic mrfs mrfs connectivity group grid value lower bind bundle segmentation dataset reasoning bfgs aggregated bundle remain bfgs author implementation implement dynamic max implementation programming iterate normalize energy energy expansion unary potential code dd subgradient bundle material detailed plot outperform optimization outperform solution energie energy expansion robust semantic segmentation segmentation construct energy unary unary potential piecewise train unary potential unary potential boost grid potential shift segmentation produce segmentation bandwidth colour domain pair colour set model choose optimizer decomposition split horizontal diagonal order potential form subproblem unary potential evenly horizontal within variable potential individual variable bfgs step routine apply image call bound energy bound energy label subproblem aggregate converge obtain sophisticated could potentially intersect intersect equivalent report expansion energy optimality instance find energy model exclude energy global equal energy table report energy median gap analogous running second robust potential cc exp indicator interactive segmentation segmentation task synthetic neighborhood grid unary potential pairwise potential significantly different unconstrained energy linear constraints loop loop solve lp outer lagrangian low lb lp unary potential solve lb dd solve simplex energy constraint conclude converge fast primal comparison energie energy global percentage pixel consistent curve propose mrf comparison exist hypergraph tree require art method potentially high theoretical provide equivalence relaxation acknowledgment kolmogorov valuable discussion anonymous great support microsoft technology additional plot paper segmentation main iteration version minimize pairwise main potential positive elaborate true sec pairwise affect potential retain long submodular minimize cube solve standard lp lagrangian constraint describe follow pseudo apply equal standard lp concave piecewise relaxation minimization specifically assign unary equal set explore experimentally sec low pairwise equal lp minimize set unary correspondingly lp relaxation look lp lp target set feasibility negativity feasibility complete standard express lagrangian consistency constraint add corollary together trick lp versa define set comparison gap dd set sec specifically energie label unary potential potential weight note report table tight trick cccc percentile pairwise energy pairwise potential maximum prove min energy unary potential eq z ff z equality finish fact case potential function sum pd hold sum new dual function contrary point exist label l converge consequently mean number belong segment restrict generality analogously show opposite equal contradict opposite possible contradict feasible contradict form specify statement ph work bayesian supervision currently team sup research interest receive sc ph degrees state university head science school economic method research include computer researcher sup paris mail fr science school economics university e mail address mrf motivate numerous approximate propose submodular unlike energy minimize mrfs take account global property experimentally field combinatorial cut field mrf many apply computer paper one important inference posteriori refer mrf energy inference type combinatorial mrf minimization potential two unary potential set energy np exactly polynomial define energy submodular one pairwise energy potential image representation potential preferred energy potential lagrangian consistency constraint pseudo boolean binary submodular efficiently max min relaxation expression experimentally applicability world rest paper follow sec well present sec analysis sec sec way analogously call cut lagrangian relaxation encoding run expansion generalization minimize nevertheless pairwise w submodular label submodular possible applicable cut rely cut multiple time oppose minimize energy popular bind graph approach problem try agreement solution provide give obtain energy ensure drawback method two type subproblem aware way enforce agreement message pass message pass energy flexibility relaxation lagrangian relaxation define acyclic graph tree low point option submodular problem variable submodular subproblem flow submodular generalize energy potential put subproblem several apply task without create take subproblem relaxation reach agreement advantage depend result take joint denote mapping ci ai number cardinality correspondingly hypergraph accord hypergraph mapping potential convenient notational ci ic unary potential energy notation take indicator minimize number indicate popular approach perform solve formally get eq continuous strict convex relaxation paper dual primal energy r unary polytope number intractable polytope marginalization constraint constraint local polytope call g recent paper mrfs sec potential global version indicator rewrite energy constrain constraint label assign describe way approach potential simultaneously either compact representation linear indicator important special class intensity image pixel separately number node take g lagrangian inequality submodular indicator thus dual piecewise concave thus convex maximize construct approach ascent maximize bundle smooth proximal directly applicable oracle evaluate function compute subgradient flow request computation proximal prox log solve sum explicit conjugate lp min cut aware possibility prove tradeoff optimize face subgradient short bundle bfgs compute value subgradient subgradient optimum solution overall tp ip primal stepsize mrf energy bundle collection compute parameter intend keep current step bundle perform bundle replace suggest choose adaptively serious bundle another size choose step work try default size typically line search bfgs implement hessian maximum mm l select potential min old lagrangian min label old rule potential intuition mm coordinate fig order analytical variable numerically give dual practical issue relaxation framework primal solution subgradient maximize dual aspect heuristic
number simulator deterministic simulation deterministic case worth improve dual abc carlo close small possible tell result possible optimization apart execute communication equally solid blue grey jacobian describe weight stem change proportional length line segment indicate contour statistic help illustrate one stand idea delta peak replace average evaluating arbitrarily inside ball derive posterior small enough jacobian end optimization since assume define remain restrict inside next expansion volume last compute normalization accurate optimization solution distance observation assume reject reject reject difficult mix situation depict trace form surface may intersect manifold intersect q pseudo around ball define manifold volume I crucially optimization sort particle smc use sequential round smc simulator newton smooth random compute sided expense place max simulation round note time lack error bar deviation break notational convention exactly translate pseudo pseudo statistic explain unknown smc use ss ss ss value ess normal ss ess decrease ess remain result ss experiment ss ess dropping drop smc ess ess significantly remain whereas allows effectively switch run expensive fine grain still effective v ie bottom queue plot std sort plot ess ess discrepancy ess work explore control simulator transform procedure simulator parallel quality optimization apply problem allow jacobian computation note high expensive infeasible include ad library incorporate incorporate add expensive method start view pseudo simulator number outcome know simulator prior jacobian ensemble monte parallel handling resource sample procedure validate demand spectrum whether biology tumor cancer research galaxy weather forecast science movement economic physics aim likelihood simulator base model target correct distribution simple rejection process synchronization e inefficient benefit title considerable aim sequential smc particle cascade introduce stream minimal management processor independently trick generation simulator simulator piece simulator inverse jacobian result get value optimization core mcmc free model note simulator reach use alternative randomness relate estimation indirect abc make create development indirect independently manuscript jacobian dividing restrict introduce abc novel evidence correctness primary simulator simulator pseudo treat auxiliary
boundary discuss recall order q indicate ucb average budget rank context time rank event e lemma error detail proof supplementary give ucb agent heterogeneous contextual bandit horizon identify algorithm near optimal multi method ucb ucb regret boundary achieve regret insight design contextual future study system general contextual bandit constraint context exploitation bandit context identical approximation combine ucb expect unknown ucb ucb certain system contextual armed mab exploitation tradeoff contextual bandit observe receive function agent historical potentially motivate example crowdsource sublinear regret contextual bandit context recent computationally efficient however traditional bandit capture characteristic constraint resource action horizon crowdsource pay worker consider work mab arm budget constrain mab observable contextual bandit time agent horizon incur context bandit process total reward case setting possibly shown achieve regret benchmark computationally inefficient resource bandit focus static action unclear extend arrival context address challenge practical cost constraint agent achieve consider contextual bandit still bandit system current context remain theoretically scenario computationally curse regret implement manner total third start unit cost context normalize oracle approximation context expect context capture reward well bad context less agent unless remain budget remain context incur make decision medium context balance expect reward resort specifically bind lp budget constraint static lp suboptimal propose adaptive replace performance use budget near within boundary insight reward note algorithm order reward estimation method expect short systems ucb systems two system set ucb relax assumption system heterogeneous agent statistic coupling achieve unit ucb identical summary regret contextual bandit round accord identical generate condition cost incur action context insight constrain contextual bandit cost cost take observable round reward begin round observe agent round round otherwise agent action neither reward paper horizon end agent run contextual bandit observation reward context reward budget greater compare know include let interested infinity point unit system capture expect reward expect good action context reward suboptimal agent oracle context present recall cost capture reward know k j oracle action context skip depend budget unless verify context arrival general computationally horizon resort approximation constrain linear lp propose bind denote time budget lp threshold verify value view reward consider entire horizon reward hard oracle hard budget horizon budget upper reward oracle later upper reward time propose programming randomization probability instantaneous remain budget replace follow round remain therefore reward take total verify remain remain nothing budget consider replacement draw take action draw white ball budget follow budget symmetry evolution budget property distribution ready investigate bind regret budget unchanged thus stay static change budget stay average budget budget boundary critical threshold achieve good certain cumulative optimality possible decay please refer consider similarly supplementary contextual bandit agent reward still focus know order reward probability short combine ucb ucb contextual bandit take reward reward pair traditional ucb well ucb states action reward long execute property widely armed lemma minor well constrain input horizon remain remain j kt j jj j tb kt kt kt maintain ucb implement define next seem regret
overview run server challenge recent key processing analyze meanwhile problem framework example thousand problem interest become ever organization wikipedia example automate document document classify rare remain despite available available class add imbalance level hierarchy statistical pose challenge new specific major challenge number web repository million account complex relationship wikipedia category different scientific event include semantic indexing answer international challenge series imagenet challenge http www net challenge extreme research microsoft com en people classify web series challenge aim assess hundred thousand hundred thousand multi setting track main corpora wikipedia www www datasets http gr may run dataset get performance rank source http know http www consist indexing indexing removal create indexing description page manually format file example instance sparse format category comma category correspond feature feature internal indexing ignore token map unique year track track mapping use track instance validation train datum file participant free create use file belong category meaning keep participant track file file child file challenge file hierarchy parent depth deep depth parent visit root parent omit allow leaf artificial child track label challenge split track vector type less hierarchy track cycle track medium number stem category track third addition medium text instance process track b track flat prediction include gold wrong gold hand account predict gold way wrong various measure gold flat first negative tn positive tp count many label truly gold label gold fp prediction accuracy precision divide category macro version multi flat implement version micro calculate micro micro follow false positive false negative significance evaluation test p two flat challenge account thorough evaluation challenge run attract world challenge subsequent challenge conference briefly present regard track participant track track flat description polynomial svms online approach centroid description svms online centroid knn bm similarity meta feature prune multinomial naive centroid track track win couple post score one knn top system close participant first track track hierarchical competitive system multi class meta usual hierarchical construct meta extract tree meta thresholding classify top learn hierarchy pruning improve multi class flat competitive multinomial optimization strategy
query produce text wikipedia training feature text author text lda apply assign view feature text inference overcome unbalanced dimensionality image feature text representation quadratic variational text placing without account learn text optimize search variational distance latent produce ranking evaluate curve l query avg sm cm art algorithm match sm correlation fisher paper query drop compare report image query lack input capability introduce make latent scaling introduce switch explicitly provide integrate spike acceleration dimension model spike state modal retrieval structural become important spike ibp prior enable composite process fig process choice provide principled dimension way scale effectiveness real view process share latent manifold determination spike view art apply cross modal task reduce dimensionality dimensional gp nature use enable compact gp various domain gp cell gene gp verification human dimension large overfitte choose dimensionality dimension large covariance negligible therefore dimension length relate automatic determination ard ard limitation threshold hand non kernel like whether slowly decide really drive spike latent allow discard use principled however intractable form marginal close form likelihood efficient switching spike problematic variational switch closely datum simultaneously determine active literature spike monte switch enable representation regression input select nature spike unsupervise code truncate covariance extend learn explicit view determination space amongst different view particular formulation inter dimension assigning decide parameter must ignore space switch unnecessary principled spike introduce counterpart spike gp extension effectiveness dimension new aim latent simplicity th maximize fitting therefore introduce derive variational gp formulation rely induce becomes induce log x marginal jensen w integrated lead lower marginal latent rely call scale e covariance typical scale principle latent usage variable determine latent switch variable control usage dimension wise binary distribution prior marginal eq tractable inference introduce approximation spike variable define variational posterior dimension view posterior representation view consistent operation posterior view view latent posterior fall integrate latent lower correspondingly exactly divide evaluate distribute way demonstrate signal multi dimensional switch latent dataset correspond curve colored variance posterior learn different color reconstruct artificial source evenly interval signal recover st rd transform generate way combine nd spike offer signal latent st nd variance two dimension first explain big difference use match gp perfectly match apply spike generate share recover assignment dimension signal latent recover variance dimension private recover nd signal infer signal significantly true signal infer view parameter st nd switch nd th scale answer quantify digit take mnist digit take image training spike dimensional optimization purely label use latent dimension scale
different reformulate positive semi compressive sense demonstrate choose relaxation provably recover semidefinite quite notably consider choose probability assume random subsequently initialization provably vector number measurement logarithmic modify remove consider restrict demonstrate sub vector recover follow provably show appear ball center explicitly compute minimizer fall global minimizer method global strong actually result sub measurement one semidefinite empirically reformulate manifold exact present broad problem rank generalize matrix relaxation technique derive remain restrict noiseless sub satisfy eigenvalue th moment calculation random vector fu positive aim sign measurement high minimizer finding condition normalize eigenvector eigenvalue iteratively via output rule around global minimizer convexity convexity result sub require namely moment include first main convexity hold sub gaussian refer reader broad measurement thereby establish quantitative strong guarantee state I gaussian measurement fix sample vector covariance probability initialization lie around satisfie remark one entirely tolerance fix accurate want unstable saddle point rest paper convexity result concentration initialization prove produce region finally numerical robustness equivalently semi neighborhood draw initialization assume moment convexity coherence proof quadratic polynomial insight bind region loose provide enough information guarantee stochastic via concentration uniform ensure convexity section lemma refer reader make geometric exist depend norm bernstein improve state indicate unique minimum region however overview begin know consequently would normalize norm produce regime conditioning hold broad sub measurement thereby quantitative convexity parameter fourth parameter x eigenvalue quantify strong guarantee incoherent eigenvalue remain initialize result initialization sake completeness state gaussian eigenvector main extend wide study initialize noise numerical present follow measurement ensemble experiment unit noiseless meta matlab build quasi newton measurement ensemble problem global numerically meta noisy mean zero noise ratio meta relative reconstruction signal random ensemble additive rgb
version domain suppose ideal verify distribution firstly domain marginal secondly prove kind result theorem domain marginal hypothesis defer theorem open pac adaptation one domain learn imply trade direction domain bind improve distribution proved rely triangle inequality detail therefore seem good distribution improvement rely contrary domain imply bind close equation stand disagreement consistency except instead abstract pair obtain finish equation probability negative mr apply inequality find bind leave measure exact r th td proof follow guarantee abstract distribution every proof process theorem appendix rescale result bound theorem apply j sg nh abuse notation classifier j correspond r define mm kl j least choice substitution et de universit st fr universit universit st universit france contribution well hand propose previous average tight adaptation bind classifier show sentiment annotation task generalize adaptation allow domain adaptation make adaptation pac human think education student course make acquire previous learning learning draw strong real task adapt spam filter system poorly another another need tackle framework arise generate differ generate situation learn come refer domain approach weight covariate shift direct exploit labeling transfer source unlabele common unlabele execute source g present source target hypothesis consider behind look preserve good measure easier much related issue loss deduce bind distance divergence value loss generalization kernel situation domain adaptation view multiple trade adaptation divergence marginal exploit different propose take prior majority prefer construct model contribution explore domain adaptation situation sometimes domain family pac focus distribution evaluate divergence disagreement many last estimate disagreement derive domain adaptation averaging provide improve easy thank independent three contrast majority tailor multiple imply quantity pac divergence risk correspond structural domain deal seminal work pac completeness time deviation tailor adaptation pac adaptation section section derive comparison provide review seminal measure domain consider adaptation space sp tx sd tackle challenge target identically b ss objective learn target lead low expect agree respective empirical source disagreement target depend source I view disagreement assign label respect adaptation target impossible solve derive domain adaptation source domain easier differ marginal different function happen take account labeling hypothesis situation target representation marginal close work adaptation divergence hypothesis perform source h distance marginal bind depend disagreement hypothesis quantify marginal last domain act quality adapt vc bind trade complexity symmetric td divergence bind differ disagreement tight equation trade disagreement rademacher hypothesis detect difference perform divergence disagreement essence pac pac introduce succeed tight vote rely set various derive new machine pac orient create adaptive traditionally pac vote hypothesis distribution learner aim find distribution vote classifier risk classifier gibbs draw error disagreement suggest fix numerator denominator great joint pac one sg kl main bayes kl divergence bernoulli handle least every low sg sg interpret risk give close task pac bayes theorem first propose straightforwardly least suggest minimize perform trade minimization complexity nature control distribution theorem become grow describe closely relate isotropic us trade classifier dot restrict gaussian specialized pac classifier vector property prediction gauss finally base theory bayesian completeness conference algorithm supervise work hyperparameter explore choose thank descent vector similarly express classifier regularizer trick substitute recover kernel many extensive achieve accuracy time replace derivative note figure contribution domain theoretically bayesian domain pac adaptation adaptation present classifier disagreement domain adaptation pac derivation relaxed generalization guarantee behind bind equation h disagreement triangle jointly error disagreement easy need disagreement maximize minimize instance family modification contrary pac quantity disagreement divergence posterior simplicity type disagreement prior choice defer straightforwardly set pac bind derive adaptation deferred theorem domain disagreement put indeed grow choice differ term logarithm pac domain disagreement notice marginal p mm g pg therefore sg sg e sg general trade term risk domain marginal expect joint target source accord good adaptation possible deviation target labeling come discussion provide next pac pac source disagreement bayesian domain hypothesis pac domain marginal prior number upper theorem respectively q choice theorem agreement distribution imply negligible pac theorem notice adaptation bind risk divergence justification section design inspire adopt pac theory linear spherical classifier sample negligible minimize minimize recover disagreement sg section equal minimize function descent even task empirical convex name algorithm source hyperparameter algorithm gauss function derivative function evaluate trick allow dual augment space kernel term vector toy sentiment minimize function implement source domain adaptation light library source domain adaptation algorithm try iteratively self library co adaptation look train show fold cross via fold reverse circular source target point logarithm crucial rely validation approach tune similar circular fold fold target parametrize tuple follow firstly label example set unlabele secondly use algorithm reverse finally reverse summarize repeat reverse cross across fold reverse circular domain classical inter accord seven angle angle difficult evaluate algorithm make kernel problem repeat ten table co nice adaptation probably seem situation appear illustrate source maintain source focus behavior domain c svm mm rotation angle green target grey plot correspond source tune highlight adaptation black eq q q popular amazon review review amazon book review follow set equal star keep appear ten time task remain process standard tf weighting correspond task book label evaluate competitive costly increase run clear advantage jointly step tackle adaptation ask whether combine stack denoise autoencoder new denoise autoencoder representation reconstruct reconstruct execute source input value execute hyper select representation representation using execute amazon slightly tf source execute representation sentiment cccc db kb select hyperparameter validate svm result hyperparameter achievable advantageous mix label cross strategy exploratory perform reverse interestingly section analysis consider source
learn mapping space word vocabulary learn share contiguous sentence book try reconstruct sentence I home color indicate share token book use near sentence run inside still copy I sure say I party say turn although start become pressure vision could ram behind far chance stroke head towards house probably answer sharp I say come place piece break reach framework encoder decoder gain lot encoder english translation english encoder decoder pair convnet lstm lstm dynamically attention translation activation decoder identical encoder translation show well lstm conceptually simple rnns model decoder tuple embed part encoder encoder word produce state sentence encode iterate drop subscript gate gate decoder language condition computation update gate gate hide decoder second decoder separate decoder exception vocabulary weight connect computing decoder next analogous computation hide decoder iterate drop subscript denote tuple optimize probability forward backward sentence condition encoder tuple c amplitude neuron encoder vocabulary word rnn vocabulary large w space un regularize matrix map query vocabulary initialize rnn word softmax vocabulary decoder hundred avoid train character capability training learn encoder sentence involve computing compute describe detail train linear extract fine backpropagation skip restrict gain throughout experiment non scope strength representation become induce skip subsequently skip sentence sentence dimensional skip training recurrent initialization recurrent weight initialize batch use roughly also report concatenation vector skip result vector since extent gain trivially skip skip vocabulary rnn encoder purpose skip thought vocabulary vocabulary though skip train skip pre sentence nlp mean rnn lstm skip bi skip combine acc dp skip bi skip skip combine skip semantic metric pearson second microsoft corpus metric autoencoder sentence score related sentence average relate dataset come predefine derive image metric pearson difficulty employ engineering heavily lstm task take completely sentence skip component wise together score setup regression compute derive train obtain exist table result previous remarkable simplicity approach lack highlight skip think suited semantic task dependency lstm dependency expensive collect embed perform lstm table challenging drastically sentence person little look little little look little look drive car car drive drive car stream stream stream person person remove task microsoft sentence predict training consist positive pair sentence component whether sentence semantic two sentence result baseline well publish right skip alone dynamic pooling use recursive pooling skip combine basic competitive incorporate much promising pair fine grain detail signal cccc search ranking gmm skip bi skip skip rank sentence publicly available description annotate consider task annotation search annotation sentence rank reverse retrieve good query development split set image k rank within retrieve vice versa close ground result rank well rnn encoding sentence sentence jointly represent strong rnn baseline compare experiment use skip pairwise input loss skip sentence incorrect image sentence similarity margin model performance development using skip thought sentence get image skip thought representative combine also perform well high image available quantitative commonly evaluate sentence dataset movie mr customer cr opinion dataset skip train classifier top pre define train split tune l mr cr svm paragraph skip bi skip combine skip combine skip nb group properly bag nb skip thought give alternative easy bayes nb improve performance present skip bag baseline sentence sentiment learn unsupervised big skip nb particularly mr new baseline text skip bag train skip remarkably sentence skip property language also generate generation sentence generate previous book read albeit
computational algorithm risk erm overcome acceptable achieve robustness acceptable computation unlabele poor short robust solution construct minimize inside good never maintain parsimonious specify cover practical label analysis present demonstrate effectively maintain label tight label disagreement algorithm substantially superior tractable show substantially well theory key aspect technical analysis appendix address empirically reveal degree effectively erm extensive simulate active streaming call superior array summary figure show fraction dataset test sub different query rate detail regime classifier simplicity relax use respect label example w hx h regret take empirical receive label hide unless query goal label decision pick whenever inverse weight specifically unbiased h importance radius schedule ss unlabele I add increment update epoch epoch long technical always query label epoch compute maintain note consist label radius level consist accord notion measure empirical epoch determine constant state epoch schedule difference come solved consideration epoch query erm time obtain compute essence problem encode generalization maintain specific optimization objective encourage might odd objective encourage query barrier algorithmic constraint importance key ensure later bernstein style constraint estimate measure weight apply example region label importance weight rhs ensure feasibility always regret make satisfy crucial complexity benefit see might force disagreement region one hypothesis include slack implementation alone adequate ensure concentration impose region predict bound albeit biased meaning describe efficiently indeed feasible see consequently make suffer choice compute provide next counterpart crucially rely disagreement erm capture inherent problem recall overall describe section epoch maintain identical amongst critical proof exploit fact label introduce bias favor thereby drop classifier since erm always classifier additionally set h actual imply however corollary start set suppose unlabele set good control disagreement region constant bad disagreement classifier follow epoch defer appendix worth note rate value epoch drop leave reader algorithm query thereby guarantee passive generalization guarantee label complexity favorable begin agnostic quantify extent passive disagreement define intuitively learn good label term set aspect attain difference label close comparison dependence corollary also epoch defer label indeed example match theoretic corollary query recall corollary highlight improvement attain result completely method entirely query disagreement result refined disagreement however completely query oppose use region query rarely even regard illustrate quantify complexity virtue disagreement region classifier predict differently gain single classifier query disagreement everywhere imply finite distinguish let uniform hx rx h h h hx h h problem ideally uninformative however determine query uninformative different see uninformative consider focus query region fix constraint mx region pick rhs consequently satisfie find label sum thing check query baseline algorithm erm operation passive testing schedule still follow efficiently apply adequate solve hx big challenge every constraint bind infinitely primitive available true expectation access true expectation challenge variable difficulty lagrange call erm become clear classifier violate level level barrier notice objective barrier parameter x solve large variable dual ascent access erm lagrange multipli hx algorithm approximately present degree rescaling solve violate reduce call erm constraint q hand risk sample may scale last approximate violate constraint erm appropriate detail primal approximately execute appropriately erm primal epoch p constraint solution iteration vary solve unlabele substitute solve sample original follow expectation replace expectation slack every solve draw streaming collect sample size additive slack intuitively solution since large concentration argument guarantee proof statement finite boundedness replace example query solution initialize importance classifier stream estimate I point p ix iy iy w erm store query may need demand discuss test setup epoch schedule assign new epoch explicit dependence current entry correspond elsewhere explain connection start erm threshold instead erm erm stream without store oracle logistic computing maintain intend weighted update coordinate ascent derive use stream numerator denominator enforce negativity point via online detailed appendix algorithm slight modification maintain estimate query threshold quantity decrease variant current disagreement maintain label batch erm style predict label sub draw exist dataset feature characteristic dataset appendix run evaluate testing goal trade query algorithm select available look set thereby individual detail parameter appendix query performance label pass achievable query rate dominate agnostic except strong reveal difference hardness dataset scale error dataset hyper minimum minimum error quantile dataset par query query high specific dataset relatively example possess level advantage ap right horizontal reveal well vertical reveal typical h setting select optimize query hyperparameter dataset figure optimize cumulative achieve mean rate test competitive extreme much superior rate algorithm vary marker vertical bar th quantile relative error achieve across error comparable query w ap w independently much improvement hyper examine different hyper possibility hyper setting may dominate achieve generalization baseline diverse present broken regret appropriately control constraint term epochs corollary set appropriate inductive claim intuition precise notation introduce prove technical notation z importance define sequence population restrict region expectation importance epoch center around error concern example also term entire biased define regret simplify sometimes shorthand play bias early notation h h adopt convention zero epoch notation biased introduce favor evaluate favorable h key ingredient appropriately control term hold lemma intuitively importance disagreement disagreement well behave highlight natural keep handle lemma hold handle event proposition concentration regret analogous erm epoch concentration epoch epoch proposition prove general version corollary prove give proof theorem corollary follow clearly statement establish case hypothesis establish epoch conclusion inductive event hold epoch lemma intuitively h appendix lemma algebra finally epoch directly second observe empirical furthermore yield manner indeed hold epoch invoke lemma substitute lemma eq substitute yield substitute obtain complete inductive claim almost establish proposition condition complete simply use pick epoch rearrange desire trivially h c last rearrange agnostic active streaming guarantee good generalization label favorable show disagreement complexity special condition additionally interesting highlight structural complexity improvement entire limitation achieve careful defining probability refine datum estimate also extensive online well baseline indeed comprehensive diverse agnostic active knowledge believe reveal characterization disagreement likely fine grain easy active disagreement development number example need ideally solve epoch perhaps important future attractive implementation impractical obtaining close would close acknowledgement author thank helpful initial adaptation martingale adapt exist quantity depend define direct present several threshold define satisfy inequality empirical h imply finally prove schedule summation second inequality second inequality provide proposition prove lemma pick h lemma properly difference epoch bind desire clutter round pair instantaneous ix ix h I associate measurable form martingale adapt accord identify inequality use independent past consider concentration empirical random martingale difference furthermore event choose desire proof proposition inequality control q note q rewrite disagreement simply mh substituting obtain far substitute back cauchy schwarz inequality use assumption eq rewrite bound application cauchy inequality complete proposition lemma
singular value svd singular leave orthonormal leave freedom unit norm freedom constraint unit total degree need completion space unknown completion uniquely pick space minimize affine propose nuclear singular via semi thus become nuclear affine strict minima probabilistic natural element construct sample want entry matrix subject research year application collaborative dimensionality reconstruct correspond low matrix heuristic solve replace nuclear objective observe see th entry entry freedom via uniform sampling contain entry unless almost observe impossible zero space extract value product vector therefore row restrict recover observe eliminate exactly requirement rank incoherent inner recover probability sum leverage observe element recover require incorporate column leverage reconstruct probability entry sum observe exactly nuclear norm also obtain rank incoherent row arbitrarily coherent provable leverage score relax leverage leverage improvement recover finally achieve additive improvement size even sample incoherent briefly notation natural bold scalar entry th component clear transpose respectively trace square product act letter singular op I unique sampling independently define leverage singular denote negative column space orthonormal state main n towards score reconstruct leverage observation exact completion notice subtract relaxation element recover matrix regardless accord match relaxed leverage score observe incoherence two completion well know bind achieve sample dependence show observe recover exactly case consequently entry discuss completion incoherent domain datum adapt incoherent column high observe step nuclear minimization problem recover correctness leverage score algorithm total sample couple exist theorem recover exactly via exactly comparable failure theorem arbitrary recovered leverage reality phase completion knowledge budget entry uniformly score second estimate I heuristic synthetic zero poorly relaxed leverage score replacement probability nuclear relaxed leverage similar rest organize size proof condition intermediate lemma experimental matrix software write sample observe constant entry leverage optimization recover solution say success successful behave close suggest seed collaborative web edu rating movie user rate movie low perform truncation create choose repository text categorization th entry th appear choose remove unit observe spectrum rank figure score close incoherent nature reasonably coherent dataset high power law relax leverage sampling singular value leverage leverage score constant c use relaxed score leverage overall result use relaxed leverage rank via outline unique optimal give closely need notation th span complement span operator similarly onto indicator sum ij road map optimal solution hold op subspace lemma optimality universal prove proposition discuss scheme satisfie proposition recall I e eqn bernoulli I lemma sample hold claim hold follow hold control norm set hold least eq hold construct op apply fail failure write apply hold derive fail recursively lemma failure sum total failure exceed failure lemma call tight depend related failure leverage relax set sampling need leverage score entry independently
remain tree small different reduce poorly poor subset matter trial try simply try forest also instead well reduce performance adaboost adaboost adaboost want create grow forest find adjust tw happen adaboost fail otherwise constant grow update repeat use intuitively weight grow next tree force concentrate case random tree make aggregate unlike aggregate prediction weight median algorithm read document score word score score relative document score multiply word score document divide square statistically different scale test score text frequent text score scale dispersion metric document scale raw transformation raw score deviation scaling show sample total batch score cover country I country corpus remain corpus extraction lda extraction asymmetric tree forest correlation table correlation com ccccc forest rand topics lda poorly correlation high ratio every lda specification coherent influential drive drop score weakly good call automate ad available com ad ad ht range limit english world low middle east ad follow ad summary ht mean std ad house etc generating behind score ad bias raise conservative bias investigate possibility mean test country country dataset political orientation ad two ad difference country std n ad country seem tend check biased toward economic country economic median score ad std error mean reward free market perhaps right positively ad less capture economic state resource country sure whether ad inefficient ad economic policy bias favor market policy circular partly merely ad country ad ad less country get case unite indistinguishable ad united states statistically indistinguishable country country indistinguishable bad ad reason ad one million article word total word go denominator word ad something believe model categorical like analysis extract latent behind house index categorical conclusion house etc fine grain regime surprising grained difference subtle ad ad address ad narrow ad effective already country expert collect create training replicate exist ad ad could incorporate regression variable daily index country text help news day would enough produce pick month period precede date want score nlp de n proceed n allocation state comment g processing code score ad ad article period indice ad enough ad produce semantic analysis combination regression one economic political economic affect country go political concern question researcher assign score house index mixture competition yet none adequate uncertainty rely directly country expert checking box box check score odd increasingly country boost adopt policy rule much minor political operate code must moderate political challenge regime always observe moderate bias test association regard measure house house instead proper house give prevent know say statistically indistinguishable regressor whenever almost conclusion create al ordinal latent measure among house come confidence interval quantile big improvement indistinguishable year year moment write pair diverse regime new actually use create idea article say north contain news article article relate create ad I try work try method regression outperform I I news repository content list american york usa post daily france english internal identify select contain regime particular choose article one tag international security system news database period actual coverage vary source ad cover provide reliably retrieve news union east country think regime news million article total country spurious association proper help prevent high news country occurrence proper noun remove country year news article document matrix try news proper frequent language corpus corpus million adopt case learn word score frequent knowledge period country unified sample sample select extract text tell b appear frequency column entry article country instance france transformation entry multiply number word appear transformation word appear appear whole corpus increase appear lot detail step document rarely news united appear news unique large normalize column transform ready decompose weight step extract principled choose large something topic svd decompose follow singular vector value whose vector conjugate transpose create decompose keeping row call truncate truncate matrix map topic run collection medical result look topic topic mutation heart heart large absolute text diabetes contain weight corpus word also etc topic topic cancer real usually large common top across two topic real corpus extract topic three turn onto document collection article product look ht document document diabetes disease diabetes diabete importantly extract order first column variation third topic topic decompose topic weight expect generate create topic row topic influential improve clear topic change corpus try memory limitation word core truncate work onto topic interpretability get know represent document say motivation allocation lda ng whereas free text gain interpretability word weight topic clear mean unclear tend result topic extract try lda transform text lda value assume generate draw heavily document draw poisson topic continue topic diabetes heart disease cancer ready draw word may diabetes document specific word variable everything else randomness datum generating observe inference algorithm create suitable also though topic document draw topic corpus motivate explain topic score principle ols explain year score respective score predict sample ol run topic may need recursively allocate leave split leaf follow branch branch parameter try point homogeneous use gr non need specify interact like large
nuclear incoherence probability exact sufficiently observe although problem formulate completion rank want upper bind enough challenge practitioner prefer nuclear regularize efficient problem due hand rate linear certain unclear correspondence unknown happen compressive solid theoretical efficient bridge gap theory investigate full like kind compressive classical completion relative summarize result general study good error tighter develop compare small notice never vanish denote nuclear absolute respectively brief mathematical prove obey incoherence convex great highly elegant analyzing give bound bind require assumption simplification completion alternative study matrix also however mention completion completion side completion coherent universal completion name amongst many study completion characterize behavior investigation rank completion follow upper restrictive view least theoretical generic generalization additive ignore investigate contain propose condition relative work high completion subspace subspace multi step algorithm recover matrix unclear formulation follow norm constraint thus restrict convexity incoherence derive upper contrast contain study define two operator eq replacement simplify guarantee nm n eq simplicity factor comparable rely incoherence lemma simplify result bound relative tight tight additive derive interpret describe arbitrarily make necessary utilize theoretical bind relative bind please although require easily completion correspond bind theorem bind become work analysis upon guarantee rank q q optimality base convex analysis main bit two intermediate nm go lemma throughout care except last provide construction partition least subgradient subgradient convexity get form verify complete continue proof three utilize conclusion put get basic mn constant instead ensure later due ta plug substitute eq plug inequality
policy horizon main penalize uniformly respect policy framework penalize completely case detail markov process two armed sharp devoted regret analysis weak limit rescale arm proof go n multi armed arm first independent success occur value sequel generality real define sequence probability ni na distribution otherwise behaviour arm fail keep case success represents spread arm weight decrease case study recall fast success exist wrong ns bad remark competitive game prediction step forecaster choose receive use next arm step seminal reward assess compute action paragraph refer minimax strategy supremum possible general overview minimax several kind replace statistical analysis pseudo quantity bernoulli reward uniformly integer refer first uniform order focus pseudo order lead bind good accord lead conclusion growth absolutely competitive constant take last ns convenient drive ns well ns rely failure use probability arm modify opposite penalize ns carry failure note procedure probability select ns decrease failure decrease whereas probability possess become theoretically uniform pseudo capacity control uniformity define random q introduce play precisely modification increment weak mention role exploration efficient completion penalize procedure ni probability arm fairly arm alternative draw arm penalize state main understand ns regret arm ns page armed stationary generator act stress constant may minimize seem potential minimize r h make comment armed order competitive bind replace theorem regret competitive viewpoint sequel penalize penalize algorithm completely recursive horizon trick section dependent horizon bandit simple handle numerical view arm state result n armed penalize lack generate mean sequence carefully explain penalization illustrate let armed furthermore choice make term obtain figure remark upper sharp penalize satisfy uniform red leading converge result boundedness numerical bandit ucb therein well kl worth phenomenon evolution p penalize color dash colored penalize algorithm multi armed situation describe pointwise penalize armed bandit provide sharp weak establishe toward unique act compactly limit markov normalization study argument spirit existence thesis case may write one resp exponentially jump equal positivity start well jump could easily represent h exact trajectory drive right pointed depend relation jump unique invariant existence uniqueness wasserstein uniqueness show strongly ergodic left limit ergodicity positivity distance wasserstein variation state wasserstein distance initial law bandit drive p recursion upper exponent open appear drive distribution almost generator start one dimensional deduce build wasserstein couple bring path sufficiently wasserstein stick really intrinsic explicit exponent particular property different integer binomial lead hx formulation exhibit contraction x h contraction useful soon sequel remark recursion r previous n aim apply lemma defer section argument deduce key need section bind application careful inspection increment decomposition satisfy thus competitive regret choice careful inspection lead deduce remark multiplication expectation sum x p n also r r r controls eq since reach maximal function ip z sharp reader keep mind penalize need third polynomial way careful coefficient lead fulfil remark soon possess consequently unique simplify negative computation fulfil idea use sharp tt p c integral conclude consider power increment z plugging series yield eq lead regret sketch variant remain rr adaptation want argument first satisfied remark q soon rough estimation yield previous fulfil exist resp deduce result proposition penalization bandit multi permit ix main increment v n n x j c ode method ode possesse number equilibrium identify equation equation straightforward recursion equilibrium discriminate decide stability lyapunov close vx j x soon unstable arm increment consequence true toward start martingale increment event toeplitz deduce put together last conclusion boundedness ii argument line prove generator martingale invariant detail rest let differentiable generator sake clarity iy p rewrite p iy I fy fy n fy iy f iy f iy iy n f approximation iy iy fy ix ix ix us decompose equation part f iy iy iy behaviour iy iy iy ig c iy g c n g iy I iy iy end trajectory bandit armed ergodicity generator particular us convention relation control exploit obtain suitable possesse position jump follow jump jump procedure generator symmetric invariant drive exchange act immediate check use imply integration consequence inequality result argument base path close wasserstein try jump path couple establish couple bx xt yx xx aim build sharp triple bt x st naturally law independent pt deduce consequence coupling denote te bt deduce use decrease conclude plug ergodicity w distance start idea wasserstein couple stick alternative b x wasserstein preserve every b bx try deduce moment small optimization e check author numerous motivating n introduce bandit competitive point result competitive bound precisely penalization modification penalize make explicit exist penalize convergence process suitable finally multi process ns bandit type algorithms recursive markovian seminal bandit necessary study slot arm play arm none aware want design strategy linear define represent select
remainder section mlp write r b rl l n k activation similarity form lin fix particular radial rbf amount associate r mlp unit space similarity fix space select equivalent elaborate suffice mlp mlp constant r z x z conv sim implement layer process processing color mlp describe extension locality sharing focus layer enhance context convolution field extension processing accordingly refer across incoming successively stack coherent bank summarize follow principle mlp incoming map summarize via fig location similarity template rl classify rule ij z patch sec relate underlie base patch word patch extension maintain patch kernel addition operator density sec capture desirable coordinate seek linearly transform independent refer literature independent would input multiply measure similarity weight template rise dimensionality reduction matrix low component produce dimension whitening follow conv sim figure dimensional filter match template produce similarity output one similarity conv sim structure similarity sec conv particular filter perform whiten intend construct sec whiten conv sim arbitrarily deep start account sim depth network general sim similarity income weighted template similarity map max use classify final final local classification class conv sim template conv conv sim follow briefly describe pre layer conv sim filter template property scheme unsupervise ii rise channel conv sim layer forward define suffice consider conv recall conv sim linear transformation reduce input measurement accordingly filter turn initialization weight input template weight define define gaussian prior coordinate nonetheless regularization art recognition challenge hand enable evaluate architecture color partition category hold learn implementation toolbox code near future softmax nesterov acceleration momentum weight rate least choice consistently momentum epoch mostly initialize naturally estimation experiment similarity conv convnet choose design convnet accordance layer convolutional relu max pooling align spatial size relatively whiten compare vary run convnet validation accuracies convnet similarity plot operation classify computational budget comparable convnet fall behind publicly compact convnet recent deal none choose cifar convnet network follow relu pooling dense relu layer score ccccc convnet layer convnet cifar accuracy classify learn compare convnet general outline maximize alignment convnet channel pooling summarize convnet cut see accurate play contrast network specify much problem architecture expressive burden explore reach ccccc maxout c supervise net art cifar augmentation exclude comparison channel layer field average case pool window augmentation nature input conv sim datum augmentation rescaling improve orthogonal convnet distinction simple architecture comparison reach augmentation art check extremely compact three dropout multiplicative leave parameter classify inherent outperform compare call generalize architecture drive operator product non capability generalization interesting architecture operator neuron incorporate locality sharing realize type simple feature exponential gaussian include special dynamically generalize multiple equip argue abstraction trait mobile application cifar validate concern thus abstraction advantage architecture endow scheme unlabele besides aid determine channel hide pattern determine unlabeled capability probabilistic unsupervised also plan evaluate intel grant definition lemma proposition university university university deep architecture generalize convolutional architecture call drive similarity inner mean exp neuron space realize simple setting space powerful include case even dynamically learn learn contain abstraction convnet enhance impose mobile empirical resource also concern vision speech domain train end rely manually feature introduce preserve effectiveness inner lies convnet inner control conventional argue design abstraction mobile approximation high level abstraction generalize neural role activation capability beyond conduct cifar accuracy perform complexity limit feed connect comprise operator layer neuron activation mlp forward parameter bias neuron two function template r map linear similarity note unlike mlp
penalize neighbor voting proof need essentially continue job establish follow consider end event op cd lp op b op op op op op theorem term right side lemma appropriate exercise corollary proposition false chapter claim corollary lemma exercise proposition section analysis guarantee procedure theoretic limit space computationally proportion model regularity two stage penalize consistent apply assignment achieve competitive numerical cluster analysis spectral clustering become topic computer science observe among subject computer people instance underlying process belong algorithmic great advance make threshold among effort state art solution yet reach comparable problem know limit computationally major present network propose stochastic provable statistical optimality describe sbm adjacency matrix sbm community zeros bernoulli label node respectively connect community refer proportion wrong permutation shall break regime proportion justified physics recent possibly size established ensure misclassification strong equal later arguably intermediate vanishing grow usually call consistency literature weak strong consistency network go among various way study vanish refinement guarantee important devote investigation later tackle em another likelihood mle relaxation indeed achieve definite recently zhang establish misclassification sbm weak form size minimum enyi order precise statement achieve computationally none tractable likelihood base error match exponent lie computationally feasible provably misclassification proportion establish adaptively weak cover size regime addition compute even node bind match misclassification exist boundary weakly condition strongly necessary sufficient could even polynomial word enjoy statistical core refinement detection estimation initial certain refinement able improve high separately optimization step completely drive hence penalty play ensure community probability cluster normalize variant satisfy need subsequent refinement scheme consider fashion essence shall local stage desire also localization idea play lead problem example phase retrieval high dimensional closely relate linear instance therein refinement method literature provably achieve optimal wide configuration organize present method demonstrate simulate section discussion investigation defer notation frobenius usual probability determine context independent notice may change line problem two refinement shall node wise neighbor voting discuss several initialization cluster tailor current stochastic completely symmetric label sbm community nearly equal size assume paper grow assume parameter community connection bound connection proposition throughout rest induce community structure label permutation symbol therefore error misclassification define stand permutation main refinement community penalize combinatorial intractable wise know reduce quantity neighbor first node connection advance label apply exclude submatrix column remove htb nk neighbor ij j define consensus submatrix row able cluster node category present description refinement step first basis obtain give assignment assign node except penalty add connectivity equal way connectivity assignment voting rhs count penalty vector basic step community assignment determine aim consensus look consensus possibly assignment truth algorithms note apply cluster speak spectral clustering call unnormalized spectral normalize introduce study bound sparse adjacency node replace argue remove particular denote unnormalized clustering normalize spectral cluster another important popular choice mean establish spectral community close look small proportional setting lead inferior address inspire center population kt u last algorithm spectral consistency state property govern critical quantity enyi bernoulli probability throughout p hellinger distance distribution space tend parameter essentially community community stage misclassification proportion essence lie long refinement constant suppose sequence addition conclusion continue achieved reduce simply consistent rate misclassification converge give wrong community misclassification proportion least space extra estimating connectivity adaptive estimation without directly check initialization step either conclusion improve require different refinement state suppose compare condition condition sufficient weak consistency follow characterize spectral sufficiently sufficiently q conclusion assume regularization dense dense regime moreover due conjecture bind theorem quantity step full three different dense community dense community cluster lead report set base draw achieve simplified algorithm obtain different refine simplified initialization thus simulation similar consider precise refer reader generate node community consist four spectral achieve refinement regardless reduce misclassifie around use remove discussion sensitive recall average mis standard respectively hence community great misclassifie different initialization reduce ht version much stochastic block simulate around misclassifie initialization either initialization reduce misclassifie simulation four initialization consider agreement theoretical property political connect contain large component pre conservative naturally panel figure likely node group panel node group political summarize simplify dataset average removal node initialization misclassifie misclassifie except initialization refinement na na misclassifie stand application initialization method application simplify refinement multiple misclassifie node keep misclassification misclassifie refinement misclassifie depend initialization three misclassifie node converge within include due inferior iteration art method score achieve comparable worth note correct fit sbm presence spectral design semi definite method lead political competitive well important issue relate theory misclassification bound recall suppose addition hold vanish exponent replace drive arbitrarily define neighbor voting need truncate theorem obtain initialize hold parameter replace achieve mild misclassification weak consistency long regardless behavior ensure regardless behavior strong comparable consistency algorithm case fix without regime misclassification paper multipli exponent comparison result much broad last provably strong consistency grow algorithm initialize slightly use initialization initialize sufficiently hold parameter space point key large eq space replace instead large replace condition refinement version description similar simulate iterative simplify keep drive misclassifie great interest establish simplify iterative certain think interesting knowledge date drive research propose block validation information ratio paper
mutual fix become variable update conditional small making average actually marginal average similar backward note solve empirically convexity calculate triplet look bind mutual observable pair apply selection mnist noisy mnist empirically backward improve convert thresholding pick digit datum conduct initialization em close drop poor component mixed digit mix proportion digit training set digit refine prevent able regard regularization improvement active digit upper part look pixel pixel contrary pixel bottom strong two firstly digit style bottom style want hard get stagewise marginal datum model marginal calculate mutual add active component mix proportion component eq one step usual diagonal whose th active decide add need stagewise em stop criterion q mutual q mention stagewise big forward decrease empirically find empirical mutual start split e active local convexity empirically converge local digit result stagewise job poor job stagewise local experiment dataset stagewise perfectly mutual drop zero sufficient stagewise em mnist mnist bottom mnist stagewise em learn digit lot digit original em experimentally converge local maximum maximum mask small stagewise em dramatically converge quickly stagewise em open component experiment split encourage science new develop bernoulli information theoretic propose backward active guide em stagewise em analogous stagewise irrelevant mnist approach diverse biology tool generalized categorical theoretically identifiable condition learn mixture develop em continuous gmm well identify parameter information problem backward strong interaction show robustness importantly tn behave tend weak much pattern contribute eliminate question order eliminate selection forward
belief sample belief guide reward convergence estimation observable probabilistic infinite first chapter represent mix problem simulate anneal schedule metropolis comparison figure evaluation annealing despite ability tune anneal schedule finite infinite continuous case may boundary randomize span beyond estimation base powerful general u fa david frank ac uk search posteriori program ascent probabilistic mutually dependent countable random map search compare map artificial intelligence planning reasoning utility recommendation map problem wide artificial intelligence representation graphical typically represent powerful model plan program represent expressive probabilistic separate allow inference restriction sampling finding map would scheme posterior single joint multi contribute optimize say optimize simple setting bayesian monte countable dependency program program construct draw value define probability program probabilistic property initially argument return argument return without upon return terminate repeatedly implicitly denote deterministic trace proportional discover implementation programming usually algorithm valid program probability finding maximize inference paper map propose extend marginal exactly advanced probabilistic anneal simulated annealing sa constitute approach anneal gradually change analogy physical anneal acceptance course sa fail annealing sa program bayesian monte monte unlike information assignment know planning game playing planning root certain number must determine simultaneously program often finite countable variable variant mix type open use introduce way independently search estimate previous go quality solution algorithm compute weight trace trace previous line belief log log ig k weight discover improve select domain random high randomize thompson many context scheme maintain belief reward select belief belief sample extend randomized domain maximize know type choice unified maintain reward guess reward belief choice choice add choice base
relate form control issue method assessment parsimonious logistic introduce hasting compare discuss limitation report database individual logistic article binary specifically suffer explanatory indicate presence absence drug drug consumption define influence drug coefficient zero coefficient group induce belong observation write q obviously log event profile appendix solve likelihood mle also mle eq absence presence drug consumption suggest account compete set model high uniformity hold distribution laplace logarithm integrate compete model huge exhaustive compute bic therefore hasting perform unique stationary propose move neighbourhood copy iteration current element sample mix candidate accept pr return maximize different algorithm visit r present compare comment method evaluate performance contain four event study nine case control across interest support clinical report ccccc positive extract consumption mention drug among control negative status four study htp choose method report odd ratio reporting statistic drug pair cccc ref mid result apply regression obtain ten fold misclassification permit signal intercept zero difficulty roughly weak event event bic compete event dimension ccccc negative detect compete ht ns cv good control find lasso explain misclassification constrain obtain detect control worse negative probably relate detect could cpu number times nb signal control control minute require realization profile ccccc signal control strongly reduce appendix allow list detect obtain poor penalty determine misclassification event moreover deduce control black relate criterion indicate dot hard well present penalty result optimize figure reality nature seem calibration practice individual reporting database avoid drawback co effect lead throughout parsimonious regression lasso challenge calibration reference signal detect method event show present evolution event exhaustive compute compete selection whole database several day propose investigate drug reducing grateful il support drug obviously coordinate belong many profile profile eq drug ccccc event ab positive bb positive kx l bc ac positive ac ac aa aa bb aa positive aa ad positive ab c ab ba ax ec bc ac ac j ac aa positive ab unknown ba unknown ba aa c db bf n ba aa ab aa unknown em phone event detect drug use onto contingency association method regression penalty approach limit drawback influence logistic regression selection metropolis hasting bic penalty threshold database approach reference drug association propose
neighbor f offline help offline fast process occur batch ms completion time adapt rate cause benefit particularly evolve mention initial gibbs converge allow impose computational load propose online sequential hmm capable batch incremental main unsupervised adaptation batch accomplish dynamically balance batch memory far sequential adaptation hdp hmm include evolutionary thereby test segmentation accuracy improvement thank hdp solution attention literature intervention candidate streaming application stationary load significantly balance effect posterior inference parameter observation accumulate summary online parameter q accumulate however control versus impact accumulate conjugacy conjugate prior derive posterior investigate gamma conjugate iw rate inverse wishart derive hyper canonical parameter simplify try thank remove ideally affect hyper initial dependent conjugate scale parameter conjugate derive posterior general conjugate conjugacy expand derive hyper parameter presence single extend case observation proportional appendix explore mean inverse wishart unchanged scale draw scale whereas former tending nee sequential context daily surveillance stock flow address focus pre principled capable class delay streaming context far enhance responsible balancing extent streaming observation evolve remarkable evolutionary sequence unseen segmentation attract domain segment classify finance understanding annotation human computer interaction date main slide window hmm structural covering spectrum discriminative margin increasingly dataset challenge adaptation dynamic remain address limitation accommodate model hierarchical prior hide hmm exploit adaptation adaptive joint incremental set hdp hmm sequential ii buffer tune class unseen continue entire iii bootstrap supervised manner operation process stream obviously life problem learn rate biased adapt pattern evolve rest present hdp expand compare benchmark amongst hierarchical hdp principled nonparametric typically inference variety hmm data state decode dynamically find domain varied problem approach entire learning system obviously stream response demand dedicate processing mini batch inspire recursive study online refer repeatedly online optimisation formal seminal work monitor stream class comprehensive time appear adaptation adaptation drift knowledge periodic ad hoc costly absence expert balance problem assign likelihood complex choice dependent dynamic exponential decay recently step adaptation introduce novel statistic supervision slightly evolve significantly life continuous follow time tackle thereby dynamically dirichlet think distribution infinite control base measurable location repeatedly establish name hdp process similar top level dirichlet process various element application continuous take parameter wishart yet hierarchical hdp property diverse collection book genetic marker population hdp switch markov hdp interpretation step current property explain hdp observation sequential hdp pz group coincide add hdp worth hdp tendency segment unbounded add towards change yet brevity hdp extension estimate distribution hyper derive extensive mainly gibbs sample variational simple significant slowly remain local minima mixing variational usually fast derivation analytical suffer low approximation initial rapid accurate indicator meaningful correspondence ground truth label classification obvious hdp correspondence hamming ground truth frame adaptive online inference extent annotation comprehensive costly annotation brief variable truth reach conclusion process batch batch alternative batch pass stream fy figure hdp emission transition mean posterior next adaptation imply accumulate non nature carry buffer unbounded latent dirichlet unbounded buffer extend memory requirement process respective batch propose system learn note responsible set prior parameter current accumulated adapting likelihood worth note batch compare play relative accordingly pseudo respective distribution belong exponential bayes property canonical accordingly rate bold font notation convert canonical canonical prior ultimately please proportional purpose thank coefficient merged exponent scale canonical parameter sample hmm hdp dirichlet member unify form section learn text mention early exponent merge hyper convert impact ultimately convert transformation learn conjugate derive prior sufficient gamma however wishart gamma parameter derive posteriori hyper parameter video datum importantly category hdp context introduce segmentation recall detect boundary segment regard correctly detect interval frame ground truth percent segment additional boundary positive detect action ideal report table colour instance estimate plot label provide plot view colour univariate around dirichlet matrix generative hdp hmm replicate absence please configuration run sequence train leave batch size time unit batch propose online hdp segment percent probe far add increase considerable overlap despite noise significantly accurate percent level repeat datum percentage undesirable row accuracy figure term precision infer l cardinality ex stationary noisy evolutionary evolutionary evolutionary combine evolutionary bottom half adapt truth class learn colour combine challenge slight decrease extra nevertheless considerable performance visible synthetic evolve distribution involve shift unseen class deviation examine appearing shift generation demonstrate distributional comparison drop undesirable class shift yet around class colour learn consistently batch class percent mostly thank adjust adaptation rate highly percentage combination new need distinguished fold mode misclassifie ii merge shifted experiment close challenging distribution give hdp prove highly percent cardinality thank mechanism hdp perturb observe evolutionary increase keep prevent drift evolve around zero absence hdp overall undesirable drift exist considerable allow evolve ultimately merge neighbor single section video action video actor action sequence segmentation way action frame feature centroid centroid actor contour c cardinality hmm online offline hdp hmm ex offline avg c hdp remarkable qualitative segmentation offline variant represent batch run comparison offline study yet operate class fix trend due stationary test addition accuracy offline consist activity contain annotate separately actor leave relevant reach something reach subtle even action boundary even annotation segmentation sequence provide leave validation test run typical one order run actor frequency occurrence compare show frame similar sequence vs minor frame difference infer visual colour segment distinction leave subtle back blue figure freedom number explain sequence emission change transition adapt due remarkable mainly one phenomena object show sequence hdp consistent future model inherent
problem work direction become compute processing consensus consensus admm additionally fitting machine domain variation admm variant decentralize inexact subproblem update solve ax enable begin lagrange multiplier generate ax k ax ax stepsize ix global solves update f share global finally lagrange multiplier force iteration regularize problem jx dx b scalar lasso consensus solver ix tn consensus form compute admm single entire equivalent td tb solve server store single aggregate server compute cloud use admm forward split latter transpose exploit complex dataset consensus subset aim solve remove yield fy dx step solution represent proximal decomposable minimization line simple solve square exploit transpose reduction decompose iy server place datum gram even far converge central iy ix ix ix classifier mapping kx admm proximal optimization solution compute form solution solve hinge give proximal simply note svm form support solver require variable fitting reduced setting compute happen column row fortunately handle dual f ball otherwise formation server rather admm general result guarantee rate rate iterate admm multiplier thus iterate primal dual feasibility decrease formulation iterate satisfy optimal feasibility large ask optimality go lipschitz constant global constant spectral begin write optimality ty optimality equivalently fy dx fy k know result note satisfie rate convex accelerate convergence possible reduction consensus synthetic transpose consensus optimization resource center range extremely implement transpose consensus consensus routine author use stepsize parameter substantially tune tune scale stepsize make solver regression svm warm start subproblem limit bfgs method warm accelerate transpose reduce require square accomplish backward splitting solve svm solver well warm solve solver use feature core b datum transpose admm minute band geometric every star feature interaction require space run decrease show transpose converge consensus experiment experiment store different core node little transpose method far efficient load report core advantage transpose confirm transpose powerful realistic oppose used core computing core core classify consensus admm terminate core transpose reduction consensus transpose consensus nearly amount problem transpose grow consensus transpose whereas require transpose gram send consensus local particularly overall solve short note however total computation time shorter transpose subproblem entire corpus consensus portion distribute apparent heterogeneous across datum different homogeneous consensus method datum across problem differ tendency solution consensus consensus take heterogeneous transpose transpose solve entire insensitive heterogeneity transpose reduction figure datum admm tradeoff node solve problem consensus admm require despite transpose highly dramatically server send consensus consensus admm solve node regression svm consensus contrast admm close transpose reduction stay consensus admm communication consensus terminate time especially across node inner overhead admm cause call algorithm allow stay naturally transpose model problem global square distribute consensus distribute across transpose particularly advantageous consensus solve consensus apparent original put form svm solve approach attack dual advantageous act coordinate efficiently approach popular powerful consensus solve method inspire implementation iteration solver bias consensus central server treat differently warm accelerate outperform problem core processor solve feature exclude oppose sub svm dimension processor average second solver core per core total corpus truncate transpose second computation transpose second second total consensus core day day day day day day day day day day day day day day day day day day day day day day f day day
connect represent connect distribute engine computation iterative adapt factorize force graph oppose automate partitioning factorize improve balance assign also minimize communication edge partition undesirable cut apply vertex span multiple partition minimize copy vertex definition directly cause inter detailed method uniformly onto compute vertex add node master vertex add vertex induce computing vertex respect master node master vertex receive vertex update master master integrate implement computation graph model edge cut node master vertex overhead incur message master replica reduce reduce communication overhead total hold since replica provide efficient balanced complicated evaluate diverse include light field hyper face illumination study behavior consist select atom light field array collect light patch ii consist atom light light consist produce band video dictionary patch image produce image illumination addition decompose level decomposition light cpu processor ram light field large node machine amazon ec core two intel ram per synthetic evaluate computing core processor ram per level abstraction enable implement architecture map efficiently accelerate use framework matrix library compressed format use system distribute factorize core cluster dotted scale behavior almost linearly minimization utility denoise employ vector approximation signal zero select make signal lie certain classification show fista full gram fista square vary fista remove decomposition obtain full gram zero zero light light field ii capture slightly viewpoint combine enable view represent observer position device trade capture light result image resolution resolution complete light collect dictionary field different fista decompose tailor dataset regular dense peak ratio ratio maximum recommend db fista norm fista fista runtime achieve order magnitude fast compare decompose reach comparison achieve run fista decompose db eigenvalue relative runtime decompose normalize evaluate power light various power error expect accumulate decompose versus significant improvement power finally propose synthetic decompose advantage regular format efficiency sparsity model zero would overhead base vary density decrease degradation worse overhead represent number scale processor processor single processor inter purpose scale processor gap large processor baseline depend specification select systematic future b processor introduce distribute apply iterative framework dataset dependency component underlie platform load balance significantly communication method demonstrate significant offline decompose subsequent computation overhead example light section core complete less overhead justify consider patch use light matrix ask operate zero store advantage communication approximately diagonal reduce communication communication proportional difference large general specification objective complexity massive dataset decomposition scalable subset svd require costly create scalable decomposition introduce successfully implement create decomposition execution densely lemma conjecture platform aware framework dataset introduce aware end execution massive iterative platform scalable mapping enable arithmetic platform message pass incur update available resource subspace flow iterative trade level base facilitate automate perform optimize power dataset amazon ec usage execution compare prior many modern prominent algorithm application belief propagation achieve matrix multiplication involve gram single update become highly challenging communication number distribute iterative adopt parallel run parallel gain communication partition effective movement parallelism accelerate machine readily exhibit non format store addition infeasible exist datum dense dependency wide field medical image problem find densely dependent execution broad apparent low lie union property overhead impractical dataset transformation analysis critical system execution memory usage overhead accelerate matrix multiplication require dense structured rewrite far data automate method partition decompose factor within bind computational depending decompose iterative model base pass vertex write programming develop computation decompose write amazon service utilize available explicit contribution domain specific knowledge extraction dependency structure hardware resource scalable onto contain zero result dependency systematic way tune desire application efficient model partition decompose datum aware demonstrate magnitude domain specific use provide well rank extracting improve learn setting span powerful approximation kk kk singular truncate analysis seek subspace approximate least leave provide find good rank sparse dictionary dl apply large column independently batch omp zero independently store small column write property dataset predict accuracy factor decomposition impact predict decomposition exhibit lie exactly linearly low characterize decrease factorization select column difference good rank decrease exponentially another gain low illumination structure signal low subspace union effectively bound sparsity sparsity column independent number zero increasing control algorithm guarantee select span ambient exactly subspace approximately low introduce number far increase discuss associate introduce naturally control also achieve question specific application connection error gram accuracy include exploit aim generic approach specify introduce iteratively compact decomposition already establish decomposition framework learn tune map result decomposition decomposition resource large value resource
distribution response regression see random cdf without z diag diag case special design subset due another kind rank sampling perfect use used error subset design matrix stochastic f likelihood give r n rx follow sampling design condition rx ne g n r r rx rx rx content counterpart analytical numerical tables rx rx negative proof theorem effect member counterpart value report table comprise replication table effect unknown content size content content compare counterpart first calculate content sample fix size perfect different ranking scenario value order match scale provide value element size efficiency simulate replication apparent key parameter observe design moderately design parameter increase rank effect model propose design design clutter covariate compute probability estimate compute design carlo comprising table explore error content sample mixture exponential handle calculate counterpart ex ex logistic uncertainty structure aspect include shannon nevertheless worth play role inferential design likelihood ml cs ex ex ex exponential ex ex ex ex ex content ex ex ex ex exponential ex ex ex content ex ex information propose aspect concept engineer shannon entropy quantify quantify uncertainty inherent technology censor datum testing therein shannon entropy design ranking perfect associate integral shannon quantitative information technology computer practice shannon sx dx complete enyi data r enyi entropy enyi entropy include enyi engineering etc enyi investigation enyi eq pdf sx sx dx u sx complete kullback leibler measure quantify kl measure kf dt quantify instead rank design underlie design denote compare sampling design one interpret hypothesis within dy fy dy l sample content perfect informative observation population simple set sampling extra information measure enyi perfect interest suggest test seem goodness investigation criterion appeal university fellowship unbalance unbalanced unbalanced size sub group cycle th cycle say unit end observation unbalance unbalanced cycle subset second cycle respectively unit subset ccccc present unbalanced continuous pdf q pdf th statistics latent iv iv sum I iy unbalanced design u subset fy obtain unbalanced counterpart ex ex ex let distribution unbalanced n ie I unbalanced I unbalance counterpart size follow example field research university department statistics mb abstract different environmental study superior fisher compare rank counterpart uncertainty enyi kullback leibler kl discrimination several subject phrase shannon kullback leibler rank sample underlie sampling fairly accurately without actual measurement little measurement costly unit rank environmental estimating stream area management association exposure cancer stock abundance previous rank initial sample take unit call perfect small unit rank small ask rank difficult unit particularly rank consequently partially rank order aim burden require flexibility unit subset rank subset observation collect mean propose statistical selection measurement basic post information formal perfect give joint l u vector rr furthermore marginal easily give first analytic size give modelling involve play theory behaviour maximum likelihood rao regularity calculate derivative function matrix size indicate negative
measurement contain allow compare classification bic compare body diabetes follow name component clearly perform compare mixture family body fit heavy eight gender heavy tail component mixture diabetes model classification select close also imply shape mixture four freedom four select model freedom despite differ greatly respectively ask comment parameter fit restrict three ari ari model overall real note bic pick datum set commonly bank fit ari mixture expand model improve procedure propose previously tail tail suffer datum often show gaussian mixture student handle heavy light tailed component allow thin flat well purpose range eight eigen outlier restrict exponential become enable fit estimation mixture skew future suit asymmetric lastly mixture power may high outlier work grant engineering award research density scale valid see appendix different present comparison estimation estimate parameter hold initialize log heavily determination convergence point axis procedure plot dimensional conduct extensive equivalent scale concavity exponential log q derivative fix update get jacobian p j analogous ig context ig ig log write calculate iteration log update form newton update g ig alternatively implement function obtain closed update utilize increase accelerate search orthogonal manifold construct surrogate employ maximized step list initialize depend check go back constrain equal group alternatively implement involve update base ig refer construct use support hyperplane maximize lead g construct simplify ig ig ig ig ig ig ig ig take eq q ig ig ig ig ig accelerate orthogonal orthonormal tangent objective reasonably decrease ig q unconstrained descent tangent hence smooth curve move qr decomposition herein denote ig ig ig g ig g surrogate q maximize obtain update identity g g ig ig two depend use ig ig ig q ig ig ig derivative obtain q g ig ig ig ig ig accelerate orthogonal manifold function minimize unconstrained gradient show space refer isotropic derivative hence recall scale refer yield detail group ig ig ig update decomposition ig ig ig weight less follow proceed similar ig ig ig ig ig ig minimize matrix eigenvalue idea ig ideas ig ig ig ig schwarz commonly development extensively bic maximize likelihood size put observation otherwise iterative initial need em poor anneal pick run multiple start initialize random mean cluster constrain degree freedom mixture sometimes constrain acceleration commonly progress criterion stop result acceleration iteration herein adjust rand determine classification group label ari rand chance calculate ari perfect agreement ari random thorough diabetes class name mm mm multivariate fit skewness receive vary tail parsimonious eigen maximization lastly model benchmark popular investigate heterogeneity classification refer advance mixture popular presence become tackle weight deal skewness lin mode herein utilize base generalized parameter kind characterize peak heavy tail peak thin tail quite flexible furthermore difficulty estimate shape yet distribution parameter none previously geodesic unconstrained newton focus special impose model term decomposition previously family five see log negative infinite herein guarantee monotonicity make accelerate line manifold estimation wide family weight mixture toy suggest random definite identifiability issue give determine special distribution covariance multidimensional denote distribution scale th mixture previously identifiable practice component geometrically diagonal matrix proportional orthogonal eigenvector order eigenvalue equal variable parsimonious eight parsimonious option constrain structure natural extension structure structure group denote eigen spherical align axis align equal pp g p g algorithm complete likelihood maximize conditional replace em maximization computationally expensive step rather maximize expect give numerical schwarz criterion acceleration adjust rand assessment detail appendix compare simulate modify package use program utilize work dimension metropolis rule easily refer bic family bic select ari family median similarly ari even select ari select range value scatter mean give ht deviation parameter family success overall distribution dimensional common scale generate dimensional generating follow report frobenius norm bias clearly generate simulation investigate parameter estimation perfect ht lrr family component compare family mixture sample binomial zero component generate scale perform run five select component low size seem tailed component cluster unique group similarly time generate family heavy due fact clearly separate mixture model mean ari similarly
track display improvement hold network hold task similarly fix consistently suggest amount task add hold apply number type white area bar smoothed represent useful hold datum growth curve learn growth curve hold fold validation note many bad baseline initialize dataset never positive negative transfer stronger train large well average auc absence include training curve experimental across hold run randomly result section benefit explain consider question active biological target context imply contain inactive inactive inactive reason active physical mechanism hence similarity plot call occurrence compound dataset compound coordinate odds auc eq dataset single odd reduce auc baseline discuss exclude moderate portion effect determine likely benefit many collection exclude correlation improvement improvement exclude qualitatively gain suggest framework datum overlap odd improvement vs sided unique improvement sign target give confidence suggest unlikely affected target collection investigate virtual screening collection significant explore aspect performance still task introduce additional contain large amount observe effect strong investigate possible active moderately correlate improvement biological accurately model target efficacy availability amount critical possess private measurement argument increase sharing benefit maximize achievable architecture algorithmic publish deep virtual aware comparable metric field direction use realize target another improve unsupervised explore chemical deep offer drug discovery process remain deep couple field field optimistic acknowledgment support foundation fellowship acknowledge support nsf gm nsf american recovery act architecture drug discovery architecture public source dataset measurement across target aspect study network accuracy significantly method improve task add contribute significantly improvement sharing innovation drug disease challenge traditionally drug year move start rate suitable identify first target million drug like attractive automate virtual screening attempt replace throughput screen virtual machine learning apply virtual supervise target virtual screening active special handling care must inactive artificial virtual screening impact learn drug great predictive learning combine multiple flexible predictive facilitate share limit particular aspect virtual screen collection million train achieve improvement machine learning method add task yield collection significant feature extract contain presence moderately class rich discovery predict drug activity relevance voting combine belief networks retrieval virtual relate deep recursive neural predict extract feature small discovery notably competition molecular experimentally predict activity hold team model set virtual choice hyperparameter forest drug major concern size well occur gain justify virtual work target train far million nearly million highlight gain network focused improvement collaborative virtual screening network network outperform great language unify recognition pattern introduce go winner publicly divide four e database wang virtual maximum group interaction protein target drug test construct contain dataset detail fold classifier train fold recommendation literature comparison network perform transformation respectively layer nonlinearity feed softmax predict learn backpropagation softmax dataset number capability provide boost architecture network chemical benefit subsection series table highlight architecture outperform include whose performance lr extremely dataset fold average comparison uninformative exclude subsequent affect performance dataset show basic include single lr cross give lr rf task net hide net neural consistent network overfitte dataset positive hundred overfitte issue strong motivation wide narrow layer implementation lower expressive narrow specific train task good understand design work datum indicate substitute alternate present combination layer size architecture shift show sensitive datum demonstrate section understand increase growth curve visually performance initially improve fall initially performance improve construct dataset ten hold target hold precede
ft proof equation ft ft norm ft ft term term minimum eigenvalue large result ft side respect know exist q figure title show title first show line real dash residual display plot variance realize ba realize realize covariance theorem forecasting asset crucial finance availability frequency suffer curse model factor realize extensive parameter significantly maintain autoregressive dimension forecasting covariance volatility asset crucial financial field portfolio asset pricing asset lead realize covariance major arise realize covariance transaction different frequency secondly observe frequency price price think noisy version several way tackle overlap moreover realize realize covariance fit covariance natural value automatically generate impose wishart put wishart autoregressive wishart centrality fix autoregressive distribution dependent wishart wishart instance issue realize dimensionality covariance entry realize matrix grows need quite challenging model practice probably reason study literature realize limited say realize counterpart covariance build improve realize average volatility method technique construct volatility matrix inspire realize volatility estimator realize length day infinity parametric realize vector autoregressive var covariance significantly need covariance var factor matrix fit need factor article approach realize matrix overcome var fit extract several advantage matrix impose additional secondly excellent approach indeed less combination extraction modeling study also propose thorough theoretical setup extract section middle asset conclusion proof theory price process continuous diffusion standard motion matrix integrate volatility th define inherent trading trade asset day allow several arise asynchronous number adopt threshold volatility denote attractive dimensional integrated use construct raw realize covariance kind estimator thresholde dimension definite normalize hand let next obtain corresponding eigenvalue finally calculate fitting ft ft ft ft ft latter fs tt central wishart freedom order coefficient still grow factor practically efficiency coefficient support tend covariance history variance covariance study find achieve performance parsimonious require however b q carry likelihood bfgs optimization positivity maximization empirical analysis root large eigenvalue frobenius theory use follow model satisfy slowly constant factor xt xt ergodic matrix condition consistency paper use nd te addition go go infinity assume value eigenvector large go maximized datum dimensional finite fourth least model adequate series solid fit residual plot residual day ahead var day realize day ahead plug forecast model get frobenius first day day parameter take average period error except far good need need imagine become need prediction day ahead forecast performance check matrix actually positive covariance stock trade research start end totally day firstly outside pm exchange open pm eliminate open transaction price multiple transaction use price entry price outlier treat price sample mean around price enough price entry realize covariance realize variance realize variance covariance skew skewness realize variance covariance big tail variance year graph subsection show diagonal day treat evaluate choose factor drop less let corresponding factor volatility matrix fit diagonal series order namely every log parameter fit var r criterion ft ft dimensional square white process moment absolute ensure stationarity var check almost forecast realize ahead factor expectation forecast covariance norm day estimate upon inverse contain follow frobenius except parameterization order parameter var need day ahead addition day forecast find deal covariance become dimension model perform require model frobenius spectral parameter latter realize covariance mean entry stock sd skewness realize variance ba cat realize ba cat ba cat cat ba cat fit var aic sc final prediction report model show frobenius sn spectral sn sn inverse var sn sn var realize variance covariance dataset skewness
stick optimize choice strong baseline trial experiment accuracy good accuracy generally search resource bayesian hyperparameter easy realistic environment certainly impractical increasingly machine trial separate learn e course trial evaluate network take sophisticated simple classification logistic choice example include character gram word different language representation document serve initialization nonconvex work approach text representation categorization sequential identifies sophisticated sentiment relatively link linguistic big effect see black nlp raw manual tuning acknowledgement support project fa computing amazon em institute school computer science pa usa edu apply nlp choice make text big researcher module sequential sophisticated sentiment towards nlp system manual tune nlp amount compare machine learn differ way text bag weighting lead big little consistency across task language learner suppose decision automate way hyperparameter selection strength lasso space choice interact decision hyperparameter argue high gram need training iteration decision human al work hyperparameter popular little nlp range task consistently perform baseline linear train network overall goal likelihood etc hold proceed map representation transform n c composition learn concatenation datum simplicity rest clear context perhaps focus select wish carry select optimization family select et iteratively make evaluate trial search option choice algorithm th select function surrogate assess hold probabilistic nonparametric trial initialize ty describe acquisition surrogate use acquisition return either predict uncertainty high balance classic tradeoff choice exceed perform discover acquisition combination ei widely acquisition show et estimate expensive exactly density previous trial less use quantile prefer draw note give explicit need joint depend compute trial hyperparameter th range reweighte count occurrence place set great distance neighbor multiply probability every version multiply probability relevant path exclude gaussian like preliminary configuration advantageous exploit allow research implementation publicly available library function treat box newton hyperparameter experiment consider choice categorization representation type gram scheme removal gram length minimum term include side least predict vote speech segment topic task first classify topic classify vs realistic remove information article often stanford amazon review science graphic benchmark result eight four table dataset l r dataset acc weight strength conv sentiment tf amazon review n graphic classification accuracy acc accuracy regression max correspond gram gram removal regularization strength conv tolerance strength round baseline case publish svm expert method overall always evaluate testing split case development set datum set summarize hyperparameter baseline use weight baseline art recursive vector show logistic comparable recursive acc na I paragraph stanford sentiment base convolutional neural feed zhang use varied log frequency vector weight finding outperform achieve weighting consider ht acc svm gram nn gram lr sequential amazon score amazon come feed restrict acc gram gram rbm gram nn gram lr grams lr bag words cnn sequential cnn comparison rbm nn gram outperform base weak baseline well acc svm link u svm svm method include apply learn author elaborate logistic baseline use normalization net binary weighting
parameterize significant redundancy deep remove integer activation exploit linear appropriate rank et compress quantization storage one memory reference codebook quantization orthogonal pruning attempt replace global network art benchmark adopt learn new problem et al motivate enable pruning complexity fit optimal brain brain hessian loss pruning prune weight decay scale activation testing activation thus layer remain layer randomly hash connection hash value pruning point et sparsity minimize hash hashing pruning give pruning employ via unlike conventional final connection remain connection significantly choose performance prune non result pruning dropout prevent adjust account dropout regard drop drop regarded hard dropout pruning chance informative fitting pruning capacity original work dropout pruning follow retain pruning cnn co layer layer network layer retain less back propagate entire suffer vanish deeply make pruning error hard recover prevent co un connection pruning follow find accuracy method boost pruning compare single pruning iteration connection prune pruning connection connection pruning connection connection zero contribution loss lead connection neuron automatically prune modify add mask tensor network pruning choose quality deviation carry prune l parameter l l ref top parameter l ad ad naive cut parameter inferior work cut layer much reduce single svd neurons achieve rate mnist convolutional convolutional layer mnist prune pruning achieve number activation prune percentage operation prune sparse act weight fc fc weight k conv k fc total pruning region pattern layer colored correspond digit network center performance prune dataset example million parameter top accuracy take train prune whole rate require layer original size reduce pruning connect act weight conv conv conv conv conv fc fc fc iterative pruning give show green pruning still much drop much connection reduce accuracy regularization accuracy pruning dot close green towards extension regularization pruning phase mode adapt prune solid red solid line circle dot curve green pruning drop achieve pruning believe reduce layer convolution pruning layer prune memory operation appropriate hardware activation column multiply ratio layer layer convolutional intensive convolutional figure pruning layer sensitive pruning convolutional sensitive pruning due redundancy layer small layer etc histogram fully panel weight tail drop quickly almost prune center remove parameter adjust spread also convolutional pruning pre prune pruning conv fc fc fc x promise whole size yet chance layer gain prune connect reduction layer image process reduce requirement prohibitive right part brain pruning highlight imagenet show convolutional reducing without accuracy network capacity
sir state sis eq sir constant mild assumption cf ess adaptive particle particle ess technical rigorous justification speak allow fall parameter behave regularity condition ensure effective particle sir measure effective ess bound whereas ess measure ess ess ess condition pg geometrically specifically regularity pg geometrically soon gibbs sampler geometrically condition particle exist pg exist furthermore condition exist maintain ingredient smc pg provide concern smc primary expect smc measurable knowledge investigation propagation kl sense kl investigate encode information lose beyond justification study appear importance estimate particle informally region low probability small measure produce sample
statistic use procedure reduce support assertion attain approach readily gain interest graphical multivariate aid multiple connection vertex lack connection vertex edge vertex connect distinct edge undirected series encode direction distinct degree moderate practically challenge assume model value time direct undirected subsequently direct absence partial frequency involve hold partial coherence jk terminology assessment indirect partial exactly test every suggest partial frequency nan approximation see considerable kullback leibler stationary determine know graphical interaction autoregressive var ic select appropriate exhaustive exhaustive search topology selection penalize term reflect pair partial coherence determine everywhere nan determine missing edge constrain order ic case ic select possibility misspecification arise address world identification leibler kl divergence test simple nan allow test particular subgraph determine edge pose real constraint iterative less computationally costly especially decomposable statistic employ impose select much identifying model instead iterative exactly miss hypothesis hypothesis method error test nan exceed specify obviously decrease fitting simulation original review time series model correct graph summarize work computationally employ statistic two approach empirically statistical power section conclude provide refer otherwise state take edge connection uncorrelated precise remove residual u prediction residual partial jk partial correlation graph gaussian nan partial correlation independence conditional spectral jk jk f assume exist denote element coherence jk jx partially uncorrelated corresponding concept use graphical correct impose way fig cover cover impose edge true correct brief relevant var process jointly stress applicable autoregressive vector value white process vector covariance pp fu I uncorrelated purpose name series therefore satisfy constraint correct result determine correct calculate observe difference suggest correct uncorrelated uncorrelated spectral pf pf ft nj average derivation sufficient e estimator expect window cross recursion purpose along estimate leibler divergence fully miss graphical construct form hypothesis thus concern exist correct incorrect miss correct true incorrect hence obvious v v e h v procedure z accept accept z l l h c test iterative consequently easily family number conservative common statistic distribution function critical formula level z c c h mean estimate edge compare difference miss classify miss reject hypothesis reject find calculate proof edge total number statistic sparsity asymptotically regardless length statistic miss edge exclude increase remove time time test make direct comparison calculate reasonable compare second fig plot expect rapid algorithm generate large gradient fig generate completion different specify two ii var case base replication outside enough simply comparison purpose ignore take infinity miss true true false fig construct multiple varied replication step record proportion replication recorded replication essentially miss hypothesis create varied form concentrate carry replication fig effective hypothesis state procedure fig turn miss boundary c replication effective state miss perform relatively dimension dimension think moderately fig algorithm take day type encounter averaging model repeat column percentage ratio accept type ii ratio graph number present value varied derive rise connection satisfactory behave expect contrast dependency statistic cpu upon calculation assign due compute simply core calculate scale high processor inverse processor eeg control rare clinical patient discuss detail interest detect brain
nmf thresholded plus standard entry entry quickly apparent structured introduce significant whereas compression heavily seem mixed character book pt active link character book exhibit characteristic real life network character appear jointly book column th zero coefficient identify character column nmf correctly identify thing human character ff nmf correctly character recover use sc compression compression summarize bring nmf introduce compression seem cost reconstruction consistently compression library perform loading main regular resort first publicly regular use column algorithm structure compression adjust r produce matrix vary fast explain qr always investigate fast fix varie qr approach propose reflect observe compression explain compressed curve towards generate error conclusion explain analyze qr approach shape compress order qr decay reconstruction column increase row extraction extremely efficient compression nmf last representative frame video examine frame movie comparison least magnitude video truly rank matrix yield qr compression impose comparison extract element fact reflect example compare matrix compression greatly enforce help find b qr qr core core pt core control comparable x second sample frame per channel compression representative normalize coefficient take compute respectively test complete source movie movie minute long process two matrix file gb fitting compressed extract extreme frame second process gb structure nmf formulation technique namely nonnegative algorithm compress random projection compute compute extremely matrix useful decomposition currently investigate replace norm fast cauchy suitable structured rescale sparse author thank scientific help library direct simple direct stack factorize component qr stacking factor compute multiply matrix eq orthonormal matrix multiplication orthonormal matrix qr matrix decompose matrix block extract store need indexing hold augment ij use method multiplier successively respect fix closed set nr r nk k ij k preliminary propose follow closely tucker kkt hadamard accumulation kkt plug clearly limit point leave non combined get ij ij identical argument apply prove accumulation kkt kkt kkt ideally kkt provide fig rgb edu nmf establish numerous usage situation challenge year increasingly grow science exploit nmf separable nmf subset formulation representative show result technique shape limit presentation numerous structured projection analysis rapidly economic collect speed database transaction social rapidly raw insight guide management whereas usefulness theoretical challenge information science aspect become algorithm cope present tool power rich create increase big communication algorithm broad include secondary pass even substantially technique lastly massive parallelism model numerical adapt environment benefit boost recent nonnegative nmf frequently since good way model recommender system audio nmf seek I matrix popularity combination factorization interpretable nonnegative formally nonnegative entry appeal advantage present np pose matrix exhibit efficiently exist stack separable present nmf denote choice frobenius easy improve year popularity partial decomposition partial decomposition key observation identify project matrix desire low robustness thorough propose algorithmic computing compute decomposition beyond focus compression loading memory need use projection increase boost multiplicative active nonnegative admm structured projection reach nmf case rank random arbitrarily available organized provide propose diverse technique medium scale finally conclude remark describe projection guarantee limit deal way overcome limitation desire factorization follow become define whose entry realization independent identically follow step compute approximate range orthonormal factor factor exploit try technique compression let h nr r l decomposition rank execute iteration note hypothesis assume failure beyond prove theorems justification grant freedom grow execute r extra factor note analogous open gaussian fourier significantly use give automatic speedup running times structure leave multiply preserve reference therein however structured compression achieve compression agnostic whereas structured theoretical research justify gap reduce necessary computation arise happen number even store secondary I line qr completeness design parallel computation cost perfectly use implement algorithm involve matrix compression introduce scalable many matrix decomposition e nmf work nuclear norm already decomposition commonly norm become increasingly popular recent year widely audio frobenius matrix contaminate noise adapt right investigate fast alternative goal large aim new nmf make try decomposition lose might already conceptually compute structured compression difference qr let r r storing magnitude become huge gain speed fast easy understand compute qr decomposition ratio decrease fast trivially detailed qr order nmf similar model negativity row similarity problem replace pseudo norm possibly solve contain reason present alternative present numerous example support structured random projection nmf speed compression allow core algorithm disk fair structured compression slow approximately small matrix overhead matrix per block evident well impact algorithm performance summary great compression respect core compression present slow exhibit computation come slow compression representative use multiplicative update variant admm structure matlab far show nmf variant although structured end reflect variant converge observation nmf variant counterpart admm gain structured compression lastly compression come indicate case pt sparsity generate uniformly zero stand structured mean sc generally original match sc nmf hyperspectral emission visually nmf compression error seem several statistic reflect visual computing time statistic south east north thick thick thick multiplicative admm sc south thick rectangle rectangle south east north west thick thick image south north west thick thick image north thick thick thick rectangle east image north thick rectangle thick thick sc image west thick thick south rectangle thick thick south east north west thick
signature expert algorithm component couple describe combine resample svms report svm bag ad hc boost hc discriminative svm code patch quantify deep extract either filter convolutional layer study pre stack autoencoder tune convolution operation contrary p ad hc vs hc ad patch ad hc vs hc ad ad hc hc way ad hc ad third hc vs hc patient baseline hc longitudinal subject baseline vs hc hc hc vs vs set vs hc hc data ad hc hc hc ad hc hc hc ad hc vs convolutional validation evaluate performance unseen give architecture superior comparison well hc comparison superiority interpretation deep architecture class fourth autoencoder l ad hc vs design pattern autoencoder network primarily assess relatively patient experiment local boost classification albeit small future hyper system layer pre autoencoder improve complexity stage would share pattern recent autoencoder convolutional predict disease status scan historical network outperform report result refer disease unite million mild cognitive mild change without field create try ad human machine predict ad great develop image able discriminate hc artificial autoencoder convolutional image yield performance slice experiment report obtain hc ad hc ad vs hc describe approach offer brief review literature discussion part disease clinical genetic early dataset originally ad hc international brain template template weight normalise divide deviation result voxel slice scan whereby initially autoencoder filter neural whose use autoencoder interested compare present convolutional detailed autoencoder neural network extract feature autoencoder layer several input map input decoder reconstruct bias function analogously estimate identity decoder real sigmoid would tie intensity error autoencoder hide decide try autoencoder overcomplete autoencoder hide autoencoder overcomplete layer autoencoder minimize reconstruction potentially experiment autoencoder investigate autoencoder autoencoder enforce operation advantageous context encourage may underlie controlling variability image mean activation hide average try unit hide unit try extract spatially digit artificial neural convolutional connectivity hide sharing describe hide spatially beneficial number one number architecture modelling detect part unit hide whole reduce number within position input array let filter connect let bias feature add filter array convolution map give output obtain convolutional layer basis sparse autoencoder previously learn convolutional applying basis convolutional patch basis term apply sigmoid activation likely discover topology convolutional follow pooling feature adjacent every neighbourhood retain pooling approach apply max operation pool feature ignore output pool stack map input e output hide unit sigmoid unit softmax activation represent class ad hc network number summing function label decay early network train batch randomly layer momentum
frequently mobile device read book user use old mobile device percentage medium video making indicate rather technology dynamic seem finance across ad category finance continue finance video respectively video seem group finance ad completion figure mobile device percentage finance fitness interact ad product appeal user aware user mobile device profile correspond music video social medium preference probably phone playing game rule day profile target profile ad video rule video lift video leave support lift ad rule example class play confidence health aware video lift video lift mobile user ad simple suitable numerous rule lift rule small result rule average lift could accomplished algorithm future work describe cluster analyse medium find profile ad future expand study investigate ad investigate association sufficient ac uk mobile rapidly success user interact application mining mobile whole interaction ten consider base percentage validate investigate difference way ten interact various association perform find user certain interact difference interact interact rapidly expand mobile ad mobile uk digital ad growth international overall operate focus mobile ad network mobile package operate side side work medium globally customer ad month interest receive interaction click identify association age interact ad display refer attribute cluster mobile user determine interact mobile ad researcher medium mobile ad phone email web history good knowledge investigate cluster cluster ten investigate difference interact finance ad profile association profile day ad interaction demonstrate type user time follow section analyse profile ad million user period interact ad ad ad user click ad video either finish ad play ad view video video general record stage load video video video video video however point ad play video unique user anonymous use multiple ad list mobile device date ad play etc name site site ad user nd nd file consist approximately million million people ten type ad ad ad ad finance ad device list date site finance finance pm site finance finance pm site u pm site finance finance load finance finance pm finance finance pm finance finance pm site finance finance site example collect site record cause whether video looking association interaction unlikely user datum mean preliminary investigate distance sum distance performance increase however undesirable perspective represent vector represent user binary likely issue curse computationally expensive consider map user category normalise user correspond feature space category category category dot category vector user divide ij table finance finance category finance rather one interaction feature map category f normalise treat squared point cluster ten interact ad profile calculate mean k specify first cluster cluster denote receive user user play video l specify interact investigate association profile day total rule mining relationship consequence commonly restrict support consequence occur item database rule consequence support confidence extremely efficiently issue overcome left support instead contain lift value cluster ten classified group many classification total total day occur find create profile time u cluster assume profile record low user minimum support interest filter rule lift rule rule consequence play video video store access r library mine ten present score profile percentage category profile low total consist category interaction ten profile finance ad ad ad identify user percentage game less average percentage user numerous centre profile mobile device work user index interaction low video consider finance due financial playing finance video getting decrease profile finance ad ad investigate suitable health aware medical weather making slightly health fitness average health aware tend mobile device guide health weather may often aware profile finance ad respectively user video finance ad aware user average continue suggest design ad modify encourage ad profile suggest finance user tend mobile fact video likely people user alone index finance ad finance ad value ad drop play video interested ad ad index dynamic ad increase video complete click ad interested majority search percentage business consider business looking high finance suggest life indicate mobile device profile mobile device interaction category video present play finance index finance decrease index completion less ad value interact video expect finance explain interest finance ad lack ad
widely availability include parametric find build work also consistency nonparametric density build consistency condition set completeness organize model place bayesian slice nonparametric combination weak true provide study weather predictive daily return daily wind speed paper statistical expert tuple cdf let pool combination cdf beta interpret calibration act calibration function linear pool pool admit lebesgue probability serve achieve seek combination weight assign importance pool flexible certain calibration mixture beta k interval beta mixture illustrate flexibility mixture raise general mixture bm flexible transform flexibility unknown treat unbounde mixture bm calibration aggregate alternative interpretation combination major whereas imply continuous model express beta pdf symbol denote correspond cdf horizon predictive time respective realization calibration map cdf use calibration combination aggregate predictive subsequent leave ease burden aggregate cdf eq bm w positive parameter approach beta density proportional gamma proportional uninformative prior parameter respectively adopt datum augmentation introduce allocation likelihood bm draw ii number desire ahead distribution beta transform propose gibbs account namely approximate advantage credible interval calibrate easily output mixture number beta procedure choose study series instability thus pool dramatically converge select subset properly instability scheme select give finite beta increase one benefit infinite answer finite mixture propose estimating include uncertainty infinite calibration assume bm dirichlet standard result dirichlet eq breaking stick base base stick break calibration ty gd ty ty first sign driving introduce new dispersion depend crucially infinite usually assume provide allow dirichlet method dirichlet truncation mixture propose use observation complete finite conditioning slice introduction auxiliary variable introduce allocation finite observation complete k dimensional breaking dispersion complete assumption distribution ty joint adapting describe collapse gibbs sequentially full ii ahead cumulative weak mixture density speak put near general cover simulation forecast non heavily ergodic markov spectral posterior calibration density space say converge weak neighbourhood prior assign satisfied formally kullback leibler neighbourhood size kullback leibler hold short kullback via parameter density see joint calibration kernel dirichlet sake pooling parameter case dp state recall f proof assume strictly turn check similarly assumption check satisfied gaussian student consider cumulative need check interior g end mixture variance c yy use ii check analogous consideration easy satisfied calibration base concentration turn analogous corollary give necessarily spirit w dp p kl combine normal location equal hyper prior posterior affect calibration less secondly improper distribution model possible still improper lp pool estimate recursive score equal bm beta pool bm component bm define bc mixture bm mcmc iteration burn purpose arbitrarily bc combination respectively arbitrarily bc cc bm bm bm cc bm bm bm bm table empirical transform standard black dataset bc cdf green uniformity nc difficulty combination part ccc flexible cdf linear calibrate cdf blue line close show decompose mixture consider solid left figure contribution mainly part component result weight bc calibration assign weight solid line component component pool calibration mainly positive part thank assign weight model quantity choice p cc graph nc gray black blue random figure infinite beta burn calibrate line belong interval cdf calibrate account uncertainty component chart dispersion model see always row infinite particularly accurate also wide gray line concentrated suggest informative second distribution degree freedom predictive combination nc bc result calibrate calibrate cdf evidence bc tails bc calibrate tail assume unknown infinite mixture lead calibration calibrate middle nc gray prior black posterior number component panel panel bc contrary calibrate tail see distribution mixture change dispersion parameter substantial hyper investigate period kullback leibler utilize forecast density compute variable density though compete sufficient average rank cdf pdf consider daily version ml trading year ahead first ahead form combine density score non property split sample period calibration investigate sample therefore period relate period times window day day ahead forecast confirm cc individual measure period degree ideal confidence nc calibrate tail nc differ substantially simulation number beta concentrated attention average forecasting provide score perform approach model combination normal period nc begin version apply improve predictive consider ten ground prediction european centre weather forecast restrict prediction speed ensemble bilinear interpolation initialize
long period temperature rapidly increase dimensionality constant add per month science advantageous computational large first vs consist pixel use pre convolutional extract train remain size gb non normalize mse also attain varie need case reflected training process difference error difference minor slightly datum worker error speedup speedup achieve correspond speedup average run obtain duality gap vary small iteration iteration achieve speedup attain decrease run suffer split number observation remain dimensionality decrease together shorter plausible illustrate splitting row present generalize wide variety worker result bound notably worker task achieve convenient smooth loss however demonstrate function communication deterministic ahead beneficial additional block feature physical location clearly theoretical improve exponentially research amount communication minimax supplementary lemma random value column raw random worker loss problem raw proceed result relate quantity ease notation subscript follow definition define cauchy schwarz write combine definition contain contain row sum simply obtain sampling orthonormal balanced partition quantity subset row coordinate without replacement denote row express q uniformly replacement chernoff b subsampling fact eq take block define plug low upper chernoff ease paper chernoff semidefinite dimension replacement low different bound compare optimization variety estimation form depend covariate furthermore convex lipschitz ridge long even deal distribute datum many core cluster common portion incremental update number point sparse setting slowly fundamentally approach worker access portion preferable reason well scale challenge separable across high encountered bioinformatics science furthermore beneficial vector deep privacy block feature medical record include solve ridge locally number wider intuitive tight regression dependence rather yet good guarantee experimental dimensional world vision art ascent show optimal implementation environment asynchronous propose solve memory environment large possible asynchronous datum dense slow distribute impractical cost al coordinate resp worker make update allow trading communication speed notably show worker thus consider assumption hand relatively assume recently ridge distribute across round optimize sub portion master structure able setting indeed split able entire extensively use sample domain factorization square context randomize hadamard consist subsample independently normalize hadamard key sum operation impractical applicable distribute sum random dimensionality worker random combine aspect single worker worker compute solution sdca worker way final primal discard feature consequently especially consider matrix subsampling return bind final solution obtain worker union mainly determine relie rank often fulfil high arrive application recovery estimate primary distribute implementation detail implement increasingly community benefit many easily library diverse task greatly facilitate application worker worker require feature locally sum scheme illustrate increase number worker simply layer load benefit increase relatively lead demonstrate practice datum well compete hinge therefore suggest
acyclic panel node neighbor node arbitrarily panel edge restriction absence imply showing may constant normal may edge db use show edge differ negative candidate fit select direction include graphical model appear literature divergence structure stand information complement row second similarly column equal obtain inequality second inequality mutual result immediate consequence schwarz rearrange yield distribution let information respect nonnegative term note expression accord integral nonnegative conclude conditional kl equal uniquely determine l ex ex theorem kl set corresponding model bound sample order increase popularity scientific network vision molecular biology year substantial advance broadly estimate edge structure encode independence relationship goal treat separately reduce dimensionality subsequent advantageous graph observation scenario propagate misspecification may g estimation procedure edge albeit fairly need consistency explore incorrectly close fit respect graphical restrict constant true close single conjunction true accuracy order relate kl gaussian mutual discrepancy mutual procedure propose al focus kl recent ise unclear whether kl ise paper organize relevant statement kl term parameter accord kl extension kl example separation grow manuscript write similarly write constant use determinant mean covariance p undirected miss cx j exposition cite therein distribution quantify distance via divergence fix distribution wish infimum range distribution subgraph infimum hence least impose impose frobenius diagonal analyze ise heuristic identifiability follow signal covariance furthermore fact semidefinite quantify conditional equality impose side possible express conditional mutual information next accordingly quantity submatrix index definite quantity stem relationship necessary validity equal graphical message true graph constant result al upper hamming distance model estimate whenever hamming divergence close already lie whereas inside conclusion may interpret divergence kl divergence inverse matrix make edge introduce corollary intrinsic equality explore separation example distance affect purely appear account suppose set graph model analyze distribution let let respect minimum discuss frobenius remark corollary objective agnostic minimizer draw multivariate pm require size graphical size somewhat impose regression incoherence take min assume previous discuss parameter frobenius inverse matrix class somewhat undesirable frobenius norm g jointly create even requirement impose
skewness gm furthermore gm tail near five four suffice model location spread skewness filter pf cope skewed computational increase state dimension smoother retain computational kf introduce flexibility skew heavy model skew bayes vb simulation compare pf approximation skew et series unimodal introduction skew univariate skew spread shape density pdf gamma student freedom pdf six shape skew skew introduction multivariate version skew univariate cdf skew multivariate factor letter general assume pdfs covariance index read conditionally skew parameter whose shape mutually entry letter derive filter bayesian smooth vb hierarchical diagonal denote squared denote gamma pdf analytically tractable approximation vb kullback factorize expect side constant recursion convergent integrate hand discard normal filter posterior derivation expectation kk cp xy ax kp k ax kp kp q xx cx kk u k kk c cp c xy cx xx cx k k u r ii ax kp ap qx kk simulation skew skew bayes smoother compare particle pf kf kf kf component innovation quantile kf covariance times covariance computation walk vb change process low square error rmse kf kf kf slowly discount kf large error ht rmse skewness kf kf th position bias vary linearize linearization negligible process international service average carlo replication computed study vb speed slow increase fast outperform reduction negligible slow iteration iteration pf numerical acc acc delta show rmse rmse levels box quantile replication measure correlate vb posterior work dominate variance snr rmse rmse histogram distribution set histogram rmse replication indicate robust real compare algorithm skew percentile difference small filter smooth smooth bayes simulation outperform symmetric skewness present burden depend simulation cost lemma
similar computational complex summation ensure suffice proof theorem define chebyshev converse fix arbitrary present theorem measurement require vanish concentration inequality proceed step separately converse start former observe condition mean ni si chebyshev q finding constant function define precede remain unchanged leave implicit throughout suffice generally within set vanish overall case logarithm therein vanish impose condition dominant heavily concentration use theorem combine probability conditioning low recall function ensure achieve restrict similarly always put everything replace pair weak lead significantly nearly proof key difference step theorem obtain sufficient condition sequence sx characterize use probability make dependent vanish ensure negligible vanish contribution negligible remainder vanish cf remark construct deduce step deduce experience procedure come less concentration chebyshev converse obtain converse however provide tight concentration bernstein analysis generalize rather repeat attention q analog theorem often powerful small one theoretic limit recovery use end combine refined argument analyze section incorporate argument discuss advantage precise mutual require turn require straightforward discuss main difficulty concentration add converse add difficulty derive converse exist technique powerful comparable recover observation focus throughout example use mention vanish inequality chebyshev moreover lie follow hold chebyshev fact moment general bernstein observation cs section group general converse explicitly concentration inequality consider form generality consider fix vector noise information procedure trivially contain support concentration accordance kk yet fashion readily kk hold term behave rearrange suffice behave suffice imply condition imply converse analogous left hand combinatorial reveal combine precede apply precede setup provide conversely converse hold equality decoder additional bad therein behave numerator behave final numerator arise permutation low term behave immediately dominate objective behave iii condition dominate behave kb numerator scale b readily verify maximizer numerator factor two identical threshold coincide advantage handling strong converse converse part notable however maximize easily evaluate corollary dominate term law would allow coincide ideally factor former characterize avoid brevity analogous law comparison discuss constant snr logarithmic iii corollary state regardless achievable corollary reveal enough coefficient yield constant provably suboptimal optimize constant strong converse contain simplicity accept suffice intuitively enough handle large thus sharp converse setup two change distribution recovery recovery eq auxiliary see proceed follow characterize magnitude define variable increase magnitude one empirical k surely immediately small converge integral write verify typical set distribution entry consequence proposition behave ensure contain focus also focus realization simplifie give vanishe imply converse weak combine previous setup whenever obtain numerator coincide remainder factor k op behave factor vanishing without generality readily obtain turn proposition ok dominate hold numerator follow continuity handle focus dd k desire converse law maxima achieve hence coincide numerical counterpart uniformly absolute one quantization multiplicative mutual precede mutual mutual quantity iv variance correspond give simultaneously present step setting trivial chebyshev proposition choose therein behave second vanishe provide set describe part handle theorem thus substitute combine apply asymptotic corollary precede conversely whenever part identity remain term factor equality numerator hold equality evaluate achieve set identity precisely case increase factor describe counterpart necessary behave bit require super ambient dimension counterpart seek setup appendix along obtain counterpart focus limitation handling avoid bound function set gs minimize amount fix converge converge accordance make concentration proposition set ensure term control typical see small mutual information observation behave hence provide imply converse part analogously weak suffice previous obtain conversely whenever usual moreover identical vanishing remainder factor right simplify whenever similarly converse behave tend bit unbounded corollary bit set snr high similarly present represent db asymptotic corollary replace arbitrarily normalize number divide linear setting necessary interesting bound converse measurement multiplicative great generality behavior grow discussion moreover model kk bernoulli measurement depend one depends readily grow well understand noiseless testing q proceed trivial singleton inequalitie accordance write precise small sequence appendix converse immediately effort summarize finding choice kk idea dominate provide well behavior corollary combine previous precede setup noiseless conversely consider part maximum substitute slowly q identity p objective growth eq whenever bound behave thus maximum remainder condition maximum arbitrarily value behave achieve arbitrarily one converse part maximize satisfied equality readily verify coincide yield threshold test limit counterpart denote fall see attempt provide follow accordingly proposition proof conceptual analog let optimization follow precede setup noisy conversely numerical matching converse noiseless precisely characterize tail term dominant sufficiently term tend imply assume achieve logarithmic hold limit oppose see partial recovery cf fact testing follow proof corollary concentration converse multiplicative corollary e recovery reduction multiplicative exist switch see bound bound combinatorial matching pursuit dd algorithm assumption c asymptotic converse adaptive key implication asymptotic measurement important decode scheme unclear dd noisy e narrow converse adaptive adaptive yield asymptotic optimal limit density output alphabet notable example group generally strong variant probability vanish argument replace eq side right vanish coincide recover recent combinatorial additive behave whereas dominant wide section provide proof theorem change result mention proof mix channel due part discussion code density search exist setup decoder choose exhibit symmetry say subsequently remain bound intersection event one mu mu ss mu mu mu indicator substitute count simple I fix apply writing term upper bound union term equal ss write eq logarithm appear density bound recall theoretic converse bound reveal important entry prove symmetry pair permutation index claim fact among function maximize low realization appear respect permutation vector generate uniformly subset reveal vector estimate occur modified setup converse converse first condition indicator follow shorthand recover write joint distribution namely ss section coincide realization permutation immediately already amount support search oppose converse section immediate recall without generality cardinality almost surely decoder term remove restrict change disjoint count fall set dp channel appear develop bit test exact threshold number converse direction focus interest consider converse constraint move model structure sparsity probabilistic guarantee minimax poisson interesting similar analysis argument triangle subsequently give form proceed value variable distribute direct substituting apply hand side definition statement define probability follow moment combine lemma obtain result identify e substitute identity calculation conclude fact p substitution random freedom definite identity b relevant value x similarly partition write minus amount multiply substitute derivative part eq use difference part one upper substituting note fourth argument therein decay term part iii assumed may also loss imply factor simplify assumption easily taylor convenience obtain taylor expand middle follow qx qx qx e qx qx follow substitute complete show behave follow follow integral half verify handle similarly briefly integral thus decay part assume x iw function vector imply derivative writing logarithm difference lie hence yield effort accord w respectively combination upper f w case universal case two account I combine recall throughout oppose bayes independent write mutual information negative p write upper therein follow take expand hold mention split expand integration bit quantization mutual latter recall without loss generality precede use continuity binary analogous limit use begin brevity sp ok consider k identity assumption expansion measurement respectively imply one combine statement right use fact chernoff tail binomial write first case logarithm summation note follow summation provide substitute analogous along law readily suffice fact reduce group attention differ noiseless appendix begin later substitute proceeding follow precede probability information behave example follow expand logarithm analog noisy testing sequence level index obtain recall average apply expansion x h section index depend hold identity bernstein base expression common numerator denominator let right identical proposition keep rather towards choice remark consider slowly maximize yield upon write kk numerator form keep clearly set write translate verify decrease obtain set compare hand write verify occur symmetric remark support arise setting compressive sense unify sparse channel converse characterize error measurement variant law necessary threshold matching advantage broad parameter converse fail vanish tend support recovery limit compressive sense compressive sense recovery determining arise sense cs vary significantly considerable interest unify sparse observation
initial fitness greedy fitness action bt composition explain bt degenerate bt change change bt composition encode decrease fitness bt perform continue success agent run fitness fitness stop bt learn bt bt goal anti possibly inefficient node due window due act tt simulated environment fitness value fitness fitness accordance complete fitness point acquire game character agent spend game bt tree reach genetic operator size fair fair selection gp generate bt gp restriction bt reduce size bt property bt unnecessary enumeration run bt fitness latter tree create bt new procedure stop find present code procedure tree experimentally ai super game initially ai control character namely level view walk state act jump one onto collect score collect presence life currently small big benchmark walk walk field choose pass left end around agent move end show phase illustrate bt learn move phase show bt action fig illustrate bt version benchmark learn bt www bt node bt conventional scalability free ap involve greedy goal whereas relevant base framework detail address subject ap learn experience exploitation knowledge simulate benchmark work illustration available result encourage art work examine ai extensive dynamic accordingly inspire work plan possibility supervise bt regard develop model bt strength lie http supervise implement example example play remark em height se definition automate ap impractical fail drawback conventional system finite machine describe plan ap represent valid alternative present term modularity framework programming gp bt observable environment illustrate source character include evolutionary genetic planning branch artificial ai concern realization action typically unlike must multidimensional concern automate ap example pattern finally ap word environment evaluate case environment online extend toy planning present come despite suffer planning description environment goal exact planning ap recent tackle control planning well plan fitness execution composition computer game namely bt modular robot task artificial intelligence meet player flexibility ease human popular grow attention compare transition switch leave actually many language transition govern call return pass flexible call modern language exhibit many gain call add remove reveal connect state avoid redundant transition bt relation define child relation gp entire yield optimize plan base generate bt agent achieve goal lie modular dividing goal successfully evolve behavior methodology mobile apply generate simulation environment benchmark evolutionary outperform reinforcement learn possess ambiguity player work bt evolutionary create ai controller agent game use genetic learning micro air application evolve bt conduct control robot demonstrate bt compare bt take bt go free work framework robust observable make work behavior graphical modeling execution become popular intelligence computer hierarchical network bt direct tree check return accordingly node running generate label biological bt use gp optimize randomly bt determine bt ability tool genetic mutation part fitness function satisfy reach final bt use exist bt fitness generating bt necessary control optimize bt gp child child mutation three parent mutation replace bt function perform bt sub bt cross use avoid unary mutation mutation replace node versa gp call mutation several population bt gradually start avoid fitness diversity goal population population select proportional sub goal fitness naive fitness divided fitness individual ensure rank high sort population fitness fitness define follow fitness individual survival lie fully environment initial final take fitness derive fitness bt fitness value move second bt execute continuously fitness fitness assess determine course use gp provide increase significantly pure achieving bt satisfy goal bt consist execute find execute second fitness keep action add bt find bt assignment consist action
fix size name encode size rely constant factor word mechanism use word provide unique code word long factor properly select language fed enable dependency corpus compression outperform lstm implementation assume vocabulary adopt word vocabulary representation independent representation use history encoding symbol namely symbol represent representation first history eq control influence position symbol vocabulary obviously discrete recursive context power property language gradually nearby role code vocabulary symbol always perfectly recover simple value without ambiguity since many far context close countable choice theorem almost countable chance choose isolate extremely almost quantization verify run element wise code less figure case show chance allow appear time obviously normally run corpora nine show except feed lm scale base figure encode neural encoding direct platform computation multiplication particularly mini run code code compute multiplication code word triangular code position mini consist mini compose n sequence code mini batch eq code show compute activation project look matrix embed vector calculate gradient propagation bp vocabulary limit k preprocesse validation compression benchmark article part validation testing set vocabulary replace vocabulary token reader reproduce c evaluate word gram neural projection hide layer per net initialize initialization sgd mini keep fix set continue six epoch training epoch net architecture mini mini compose several word truncate sentence investigate factor order nd experimental figure factor lm rest paper summarize test outperform lstm indicate long without feedback nd knowledge kn gram st order gram lm rnn lm lm far examine much large text corpus article wikipedia gram ii traditional rnn lm sentence speed examine base input two st output initial batch sentence rnn experimental outperform popular lm yield paper almost uniquely work apply neural network nlp sentence machine translation part technology development china fundamental central china discovery microsoft constructive suggestion everywhere countable choice code ambiguity decode multiple happen happen equation either word ahead root eq total total root fraction root except choice
goal easy satisfied arbitrarily either induction satisfied focus choice eq complementary satisfie set restrict formalize separate mass deduce couple useful side go induction arbitrary result outcome couple marginally couple measure simply shorthand couple exactly interest couple miss distinguish couple confusion couple force identical since depend probability sample differ symbol miss claim probability positive absolute explicitly fact coverage couple identical allow event divide localization range probability coupling equation family see probability coupling compute meanwhile conditionally simply complement least value appear k k choice verify event always away zero step combine complete establishing begin proof validity couple equation coupling recall sake similarly hold event distribution violated formalize q hand recall combine choice occur occur occur satisfied event contradict establish closely relate simple task tail exceed miss problem concern probability completely positive subsequence pac tail sketch theorem force tail unchanged adjacent concern sample coarse instead concrete justification tail theory concern even clean characterization sufficient learnable family cover let family distribution pac miss mass cite technical rely chebyshev readily albeit much demand surprising dirichlet process produce tailed distribution mass paper unseen symbol discrete contribution learn completely well failure light tail place discussion continuous tail estimation show light learnable assume nothing familiar notion kind failure study landscape reveal open concern establish mass paper family heavy tail learnable smooth parametric learnable plug question concern establish rate fact established slow convergence lack uniformity lastly learn make fail accounting critical geometric plug symbol mass estimator miss geometric coverage parameter eq let chebyshev inequality indeed eq q bound convergence care symbol true portion convenient segment formalize segment segment contiguous coverage coverage segment localization segment give integer appear induction turn geometric give nothing thus integer event complement event n claim union analyze hold neither hold put show pac mass equivalent fix great satisfy large intersection give see equality numerator denominator sx claim proceed step yet prove basic positive number base fraction z induction continue previous large z z effectively bound converge roughly instead regardless probability satisfy desire school california berkeley california rare structural obtain outcome coverage event miss distribution learnable relative proof constructive rely couple rare one heavy event light tail give datum traditionally event rare rare symbol answer mass arbitrary distribution precisely sequence miss say consist mass relative empirical trivial answer mass mass good interpretation derivation cross contribute fundamentally derive form framework later refinement also focus shift relative pac property good power geometric tight concentration interpret learn light tail include addition power leave exist pac learn miss fashion insight justify assumption event failure good light tailed distribution barrier success law
margin demonstrate jointly video number truth performance step illustrate car lift car one three frame language appearance video despite large variability text manual supervision please project website assess cluster vision order cluster allow relation align even similarity car car however want able refine group direct car experiment vision improve similarity recover recall alignment step represent object relation obtain truth manually give similarity precision curve similarity average nlp nlp vision object relation average varie list nlp use step previous remove object relation signal case overlap direct consider evaluate perfect stream encode alignment video text constraint order video encode assign video type iii let feasible still optimize relaxed frank give detail step frank solve separately linear quadratic video follow frank wolfe oracle q program entry equal recalling mean exactly type ii th column go jump time subsequent possibility continuous path column cost value every end size illustrate transpose red gray cost gray entry page interested additional constraint alignment region wolfe stack definition look among initial relax several option turn use z stack automatically change car video contribution develop natural take two internet video contain car experimentally automatically within video demonstrate joint people change flat machine video cognitive ability would virtual work video sequence video demonstrate car consecutive car remove addition learn visual linguistic video discover step video task expression variability video little car start appearance vary video people viewpoint perform vary finally variability step slightly challenge joint appearance benefit modality assume sequence step call nlp input video individual learn modeling video advance video task link joint output discover step temporal video validate compose video supervise relate like natural description discover scale purpose scenario differently video internet event description help description lift car car video work rely scenario differently aim video consider latent structured perceptron align video laboratory protocol whereas form speech video computer recognition video weakly action video address explore training annotated discover event video description discover structure video action category unlike video exploit visual modality approach exploit improve nlp linguistic expression appearance people video visual video recent video model cluster fold address automatically discover video unsupervise improve video frame stream task raw text process token main step give input input link achieve stream video stream sequence latent variable indicate step interval token problem video stream cluster raw text video convert direct illustration video sequence align direct align together sense g right main description step roughly preserve overcome challenge complete involve interaction representation specifically direct pair compose complement dependency object extract correspond object video signal sequence token length key convert raw easy direct formulate relation together occur frequently keep maximize pairwise similarity key match together direct object measure object particular sense find constrained change sequence threshold step common align object step exceed case less output extract formulate video stream vector extra feature regularize video stack design classifier among video video stack matrix similarly concatenation function measure target class assignment square simplifie respect additional optimize matrix encode incorporate obtain constraint link section cluster video section use cluster task people perform approximately strongly together cluster second constraint encode sequence order assignment link single clustering detail constraint encode mention video alignment action alignment action video issue first global action happen interval encode direct assign video note encode video interval object match constraint identity video strong text encode alignment maintain order constraint similar particular video step see appendix previously summarize without constraint fix text cluster subject frank wolfe give satisfactory video far similarity text cluster ground top head lift lift continue continue annotate video feature sec localization video visual description task search keyword video speech manually correct obtain video length video frame second video manually annotate video ground step task video truth represent sequence relation dependency object object represent give object respectively frame represent appearance histogram optical descriptor aggregate frame vocabulary appearance obtain descriptor video output descriptor aggregate aggregation descriptor dimension descriptor normalize single represent discover report result alignment step step step mostly recover sequence repetition car fine lift include quantify precision proportion step recover step bottom partly cause coarse
meet forward pass scoring configuration normalize score soft computation probability iii back gradient iv state complex severe output prevent line amount configuration score efficiently representation normalize chen discuss extend scoring scoring score small restriction function local require marginal restriction pass width approximation reweighte alternative iterative method summarize forward pass compute local marginal approximate backward pass utilize repeat propagation message pass backward pass parameter fail decomposition assume computationally require restriction densely pixel possibly pixel densely yield segmentation densely consider context log set aim fully convolutional discuss within probabilistic model get approach mixture detailed simplicity assume function high two scoring generalization begin computationally efficient approximation n leibler assume field require valid due update iterate convergence update marginal point neighbor densely bottleneck arise involve densely complexity marginal marginal require importantly dimensional marginal achievable mixture formally label compatibility feature ensure convergence cost restriction pairwise formally compatibility semi definite readily convex term convex proceed iteratively result linearize program linearization solve filter linear system entropy relate filter marginal cost observe address restriction attention finding parameter exact expensive field marginal surrogate loss truth labeling perform parameter loss w marginal perform w carefully investigate field update recursive derivative x depend early iteration hence desired successively track fortunately back substitution gradient require back refer reader regard assume give logistic crf generalizing gradient arbitrary gradient marginal truth predict unary scoring connect repeat via backward pass update compatibility kernel summarize functional network subsequently via efficient tracking evaluate approach summarize challenge segmentation background training addition report intersection metric validation training fine convert aim large probability mask skip take employ fact two adapt object assume dimension size spatial however unary crf parameter size intermediate probability image bi interpolation scoring variable image perform update original loss track propagate r unary term bi interpolation network shape compatibility crf detailed update many successively due stage connection present tune imagenet dataset due gb memory gpu mini data car cat c tv valid convolutional compatibility shape crf arise pairwise employ use contain position contain dimensional pixel channel nine compatibility parameter shape ccc mention part neither tune unary approach unary second training crf compatibility shape fig indicate unary visualize performance number peak largely accuracy peak report chen visual approach segment image variation pose challenge present also boundary jointly train convolutional fully generalize make publicly dense fully long employ modify architecture present joint incorporate unary potential tractable even li investigate objective linear remain regressor computationally pose heuristic normalization convex obtain ccc ccc convolutional neural neural approximation
way correct sentence describe artificial model model explicit et find tree generalization decaying gradually find decay generalize structure find recognize syntactic natural syntactic task simple encode sentence linearize lstm syntactic also develop encode structure massive token leave learn unseen object show learn understand obstacle kind structure define logical consequence hold draw suited sentence model develop sentence task force allow representation effectively mutually exclusive logical distinguish equivalence semantic independence tp p p brevity show language et highlight complexity straightforwardly interpret statement logic label interpret logic crucially conjunction new sentence allow arbitrary conjunction arbitrary sentence datum come use model word tree sequence model embedding structure sentence corpus use multiplication rank third tensor find help rather sentence encoding transform jointly embedding input word eqn activation activation rnn sequence include parameter minibatch descent negative regularization tune test train epoch largely converge peak figure reach accuracy slightly generalize structure familiar tree example size reach test sentence smoothly quickly lstm fall lstm considerably well set baseline frequent short bin unlikely level lstm lag far behind four exploit sentence complex unseen bias interpret architecture role sentence recursive syntactic explicit make small set architecture constrain suggest linguistic natural language sequence exploit structure acknowledgment acknowledge google advanced project filter air force laboratory contract fa national grant department office research grant finding conclusion author reflect google nsf ex citation stanford stanford nlp stanford computer science network geometry sentence network outperform model neural like fact discover compositional structure possibility recursive compositional learn tree well able informative sentence real value vector array nlp include parse analysis base build representation base linguistic phrase model principle align linguistic
desire subsample variability benchmark except state benchmark sample candidate normal draw replacement early methodology replicate detection detect anomaly logistic discriminate anomaly normal al parameter choose validation point normal response anomaly provide quantitative level difficulty score score difficulty score difficulty score create benchmark bin dataset pd normal class decide impact practical consuming partition equal apply separately difficulty anomaly influence comes create compose entirely anomalous control contaminate benchmark draw candidate anomaly benchmark anomaly draw candidate anomaly benchmark candidate anomaly benchmark draw anomaly candidate anomaly configuration level yield anomaly reach rf normal target normal achieve candidate normal anomaly benchmark normal state impact anomaly detection relative anomaly normal desired dataset anomalous normalize sample variance normal point anomaly normalize anomaly exhibit semantic candidate anomaly algorithm set cluster measure euclidean distance location point cluster choose anomaly difficulty candidate anomaly cl level cl benchmark anomaly cl cl cl normalize normalize cl normalize benchmark anomaly score mathematically purpose score densely anomaly anomaly pose anomaly unlike many benchmark create instead run anomaly datum set ran choose clustered anomaly measure bin benchmark contain discuss early detect outlier assume offer set purpose irrelevant introduce add pairwise increase level average ratio ratio point feature sampling replacement original point marginal status point preserve real determine irrelevant feature expect distance vector want need extent set set anomaly four pd pd pd pd difficulty pd summarize anomaly c pd pd pd note anomalie pd even benchmark wish ignore point entirely anomaly exhibit undesirable multiclass majority candidate anomaly candidate pd pd many pd suggest offer great flexibility control difficulty benchmark benchmark entirely point difficulty vary anomaly point difficulty induce neighbor induce success normalize nc success nc great summarize near binary neighbor near benchmark anomaly especially binary multiclass anomaly usually create anomaly draw binary less anomaly requirement anomaly create benchmark anomaly great flexibility difficulty level truly flexible methodology unclear setting note cluster anomaly create match application domain anomalie binary difficulty effort seek create target level pairwise ratio level report add irrelevant assess effectiveness conduct describe tune employ cross validation maximize make effort maximize conservative implementation one shift datum away search boundary separate employ available radial basis execute reliably distance determine point support vector find available radial outside surface distance determine outli algorithm nearest compute average near neighbor roughly point anomalous significantly neighbor package choose small reliably detection probabilistic point anomaly model em numerical analysis retain single gmm robust select combine generate member ensemble varied value bootstrap replicate randomly replicate bag discard less log point compute assign gmm work fit know complicated robust et employ radial basis optimize forest liu anomalous isolate parallel forest tree subsample data select split uniformly observe datum totally isolated point leaf contain depth anomaly score depth intuition outlier easily isolate anomaly subsample anomalous anomaly address liu et criterion forest employ axis internal forest implementation subsample entire benchmark benchmark micro auc roc achieve micro probability anomalous analyze four anomaly bernoulli random begin robust package pd relative frequency anomalie pd rf model believe portion benchmark optimize apply auc expect particular benchmark logit prevent auc assume uniform beta much correction parameter place prior anomalous rank fit model level impact predict intercept remain fit tell change odd benchmark baseline baseline configuration pd pd rf rf cl perform factor factor produces yield large examine value auc figure quite anomaly benchmark construct learn show assessment classic excellent surprising research recommend design improvement worse recommend although control variation disadvantage make hard interpret intuitive display median average away similar drop third contain show relative pd notion difficulty intuition difficulty result easy pd pd hard pd anomaly confirm auc show intuition supervise anomaly easy intuition anomaly become imagine case additional anomaly become distant away anomalous span tend general become match plot advantage analysis steady plot performance mix capable produce cl cl impact relative easily methodology create benchmark show serious maintain find surprising euclidean poorly seek uncorrelated hence begin micro discuss sensor drift produce consequently factor present analysis point impact estimate good tune hyper maximize still tune write assessment propose publicly significantly bad matter concern argue discuss employ liu contrast publicly implementation liu examine code liu differ suggestion make forest sample algorithms parameter notably prove impractical level projection outperform easy recommend implementation real rarely know anomaly encourage perform irrelevant irrelevant understand add performance anomaly implication confirms extremely solve relative suggest practitioner g anomaly use domain obtain anomaly anomaly improve reduce anomaly eliminate e pose important challenge research anomaly handle clustered well suggest practitioner reason believe supervise method anomaly feature well recommend use setting may aspect produce big try forest recommend use start methodology detection control important benchmark offer thousand demonstrate four strongly influence anomaly benchmark accurate robust kernel estimation problem dimension great influence systematic anomaly benchmark anomaly suffer lack realistic construction limit ability understand determine anomaly anomaly thousand supervise benchmark difficulty anomaly anomaly superior anomaly variation four dimension choice advanced project contract nf finding recommendation material view research office anomaly task application security novel phenomena broken failure cancer cell field anomaly detection evaluate study hoc dataset publicly dataset consequence first compare progress understand dimension anomaly problem influence anomaly guide development standardized methodology statistical detection anomaly create realistic dataset systematically demonstrate experimentally exist assess anomaly detection method set requirement procedure construct realistic benchmark validate evaluate lead systematic anomaly detection clearly inferior third anomaly significantly understand lie choose anomaly anomaly detection value normal anomalous detection cite anomalous point anomalous natural alarm recall fraction anomalous composite area roc curve focus auc point understand sufficient training anomaly understand anomalous computer attack supervise study base discover machine failure law fail cancer disease rather whole understand mechanism anomaly anomaly rely target benchmark anomaly detection heavy combine assess anomaly kind anomaly synthetic dataset dataset construct supervise treat anomaly dataset help publicly security dataset anomaly anomaly decade experience complex validity finally retain e treat dataset without exception anomaly another et anomalie dataset treat anomaly idea systematically property exist anomaly benchmark systematically feature power irrelevant methodology achieve list first transform dataset repository classification anomalous distinct process rather distribution requirement anomalous criterion feature define dataset ensure construction benchmark work uci repository uci begin match follow multi time categorical ignore value exception anomaly cover detection focus common identically iid explore nominal ordinal network yield collection segmentation array segmentation letter optical recognition handwritten digits page concrete compressive year
analyze way interest principal dimensional work differ classical encourage factor author exhibit favorable array exceed make practitioner exist decomposition encourage outperform pca pattern interpretable penalty encourage gap propose tensor margin array present trivial intersection detail method recover factor exhibit interesting sparse temporal design smooth filter structure recover accurately exist array statistical ease array three observation hide however vary smoothly locally indice situation array spatial main generalized array penalize decomposition main challenge face result however unlike sparse unconstraine contribution exploit find penalize research case penalize work successful protein mass measure microarray use interpretable fused encourage neighboring solution filtering trend area need comparative genomic genetic close along different different column reference component array dimensional interpretability feasibility propose successfully apply area tensor decomposition describe sparse point sparse outperform decomposition directly penalize generalization tensor incorporate solution include relate follow notation definition throughout mathematical formulation brief state use highlight literature component discuss orthogonal decomposition address tucker experiment extension connect datum introduce material capital bar thus th formally likewise follow outer outer nj nx case three yield third mode tensor j matrix successively nn nn n wise scalar frobenius generalize scalar product tensor frobenius eq conclude scatter tensor scatter u ny ng nu u nj j nj n j ng j nj ng j ng j q lagrange eq q lagrangian observe separate special eq focus q dual eq path therefore
hessian adjust apply determine length matrix covariance kalman calculate sensitivity derivative likelihood use obtain evaluation derivative direction nontrivial posterior difficulty factor properly normalise analytically integration amount intractable intractable distribution monte method use approximately particularly integrate latent stochastic process tackle shall simulate markov construct call interest mean spatial average coincide average chain sure note commonly systematic way construct metropolis reject iteration distribution design newly probability newly add establish nice algorithm mh random walk proposal analogue present result run discard burn think way link amount treat together use augmentation terminology sequence naturally simple contrary via suggest make address bayesian identification identification procedure relate complete log likelihood iteratively step monotonic increase likelihood intermediate require quantity q directly intermediate tp tp covariance kalman reader explicit implement blue em conditional augmentation complete firstly provide secondly many sample trajectory follow generation variable form mutual represent method call admit stationary ergodic markov use expectation interested store discard worth pointing method instance sample exactly valid replace mh similarly nonlinear available trajectory gibbs apply implement trajectory simulation rely filter denote model density sample posterior gamma part sampler mh note augmentation implement approximate sequential associate non gaussian particle smooth approximate smooth marginal intuitively kalman nonlinear approximation distribution often spread particle represent system possible probable strategy arise identify derivative order find search likelihood come augmentation strategy quantity order particle principled filtering specific result kalman filter general particle application use expression generate approximation importance particle proposal account discrepancy particle assign proposal normalise q approximate furthermore result inspire proposal mixture choice part proposal mixture step use randomly component particle component particle time component ix refer particle conditionally also refer particle propagate selection replacement among weight account discrepancy proposal weight particle empirical complete particle freedom jx particle particle simulate time bring complexity sample work particle early influential arguably simple nevertheless propagation distribution particle appeal due choice unfortunately suboptimal account simulate particle proposal well strategy reduce computational weight computation instead explicitly introduce importance sampler mixture computational refer particle simply refer random simulating advance around refer particle generate generate variable depend identify interest x pf compute investigate convergence bound exist book give limit theorem weak regularity pf state reveal rate carlo ask variance recall approximation interpretation exponentially fortunately scenario ensure pf analogy pf identification pf approximation compute importance predictive proposal sharp first tn expectation randomness pf sequel estimator convergent normalise hence albeit however selecting derive approximate q approximate result index trace particle get full serious limitation smoothing know due resample particle weight resample degeneracy propagate degeneracy degeneracy important direction make simulate complete approximately smoothing particle promise smoothing introduction particle filter distribution position outline standard optimisation method possibly via identity intermediate log approximation smoother discuss hessian example identity note gradient way step method gradient asymptotic identification via algorithm intuitive idea simply replace intractable sketch auxiliary quantity auxiliary marginal distribution use unbiased dependence operate despite employ likelihood eq fact q recover extended target despite explain interpretation likelihood obtain current sample sample practice track likelihood estimate store development generate particle constitute one member seminal extend sampler use standard target proposal parameter together target discuss posterior pilot upper present result indicate dotted augmentation strategy treat integrate algorithm introduce closely link discuss correspond smooth natural particle smoother discuss solve subproblem idea monte approximate new simulated approximation current discard entirely result work inefficient particle construction serve subsequent member sampling intractable situation exact introduce systematic particle aforementione state retain particle interested connect compute x jx ix input trajectory return trajectory view trajectory refer usefulness come mcmc ergodic generate particle intermediate quantity variable employ intermediate quantity later mcmc nonlinear run accord base outline particle particle contribute factor efficient iteration fact even particle sample use parameter kernel complete state block invariance ergodicity property sampler arbitrarily conditionally set draw make parameter need available generate rejection instrumental dotted point direction challenge believe start provide inherent much applicable continue possibly entirely new concrete also call imagine weather example spatio promising tackle high dimensional know duality learning problem couple various fundamentally learn policy control work concrete trend couple various powerful continue evolve believe play role contract contract system mail se united mail
interaction video human recognize activity paper hierarchical video cut hmms semi graphical model divide category generative discriminative assumption concern reflect attribute probability regardless environment combination correlate temporal sensor scenario recognition crf popular linear chain inference limit capture within extra therefore names crf crf hide spatial interaction type action object object node within segment fully connect modeling interaction outperform solve less exact linear convergence study latent latent layer et al solve inference inference solve latent variable approach utilize explicitly independence transform wherein efficient human feature learn directly drive latent outperform graphical fig temporal video kk source pose accelerate correlated nature model implicitly learn consider semantic clarity imagine type action greatly video however capture formulate total activity recognize variable five potential score observation feature concatenation connectivity avoid conditional conditionally case consider aforementioned example dependent potential score coupling either represent potential potential characterize compare potential model rich contextual contain also latent fourth potential model compatibility among consecutive activity scalar compatibility activity compatibility activity global interpret potential sequence potential potential evaluate equal joint space objective therefore explicitly rather latent learn however initialization therefore great graphical loop general make semi maximize generally np hard applied efficiently acyclic loop transform chain tractable become result chain crf cardinality efficiently perform programming chain compute maximal evaluate record contribute max segment optimal segment know assignment track good assignment solve cost activity show max margin graphical example nn na activity unobserve automatically train goal objective avoid normalization provide balance fitting make incorrect activity compute loss return zero truth indicator loss view form previous recognize leave graphical unchanged framework track incorrectly predict action regardless directly involve substitute surrogate factor add exact inference compute solve concave tangent hyperplane serve term transform constraint add slack infer datum constraint solve cut learn em variable state learn decrease objective global avoid local minimum presented extensively transform action activity activity model insight show outperform contain sequence collect sensor skeleton person quite dataset label action subject room office include contrast video activity include show activity detect object illustrate challenge perform actor actor differently term view front b large label video partial also dataset video actor completely difficult body directly compare ensure feature consider action three baseline detail introduction evaluate dataset contain label environment label unchanged environment activity predict video sequence level zero focus action label structure svm margin rescale slack adopt initialization initialization state apply set well minimal object upon label categorical encoding transform therefore drive single activity learn structured upon baseline infer action classify activity multi augment activity refer successively hierarchical joint activity video manually annotate segmentation motion together segmentation enable fair two generalize performance evaluate fold fold choose hyper segmentation subject training testing validation training video perform generalization across data result average fold recall enable dataset consider remain imbalance report report score precision single al model latent precision recall f precision layer et dataset score report performance different action ground testing result pose wherein video activity label achieve average segmentation standard percentage point use truth term activitie information transition aspect activity table similar prediction percentage hierarchical approach exhibit improvement percentage reduce importance variable recognition activity notably recall percentage ground truth base performance percentage recall significant latent case illustrate contrast table start good vary level complexity adjust successively recognize show segmentation notably activity increase point percentage recall gain recall layer label prediction provide environment learn experiment outperform art precision percentage achieve precision score percentage group base location train test different term room al et std compares dataset al activity estimate action activity activity notably ground truth activity improve percentage row show confusion task overall recognition stacking value simultaneously activity exploit joint activity traditional focus use result jointly benchmark ph institute research interest robot vision human activity sc intelligence software china research ph computer university focus automate university computer track human across camera machine network sensor research interaction human sc china currently ph research interest include computer ambient university digital life research focus interactive device apply service ensure health security user artificial intelligence system paper scientific book issue conference surveillance association ai activity recognition contrast approach successively recognize activity action unify embed capture rich contextual although loop overall chain tractable therefore learn learn structured structured drive therefore manual result two model human activity people daily currently numerous provide physical fundamental recognize activity decide physical robot recognize person people continue need recognize activity robot interaction people propose hierarchical activity care water detect water rgb recognize activity build recognition object skeleton skeleton human activity type sensor task adopt
line number vary pdfs discuss experiment skewed density multimodal decrease htbp ccc integrate htbp pdf mse pdf type require kde multivariate multivariate multivariate bandwidth kde article notation report section multivariate require multivariate multivariate dimensional spread multidimensional kernel vector bandwidth determinant matrix symbol bound symmetric matrix give simplification diagonal precisely direction definition product derivative taylor series notation expand origin let dimensional vector differentiable df natural expanded taylor expand taylor relationship permutation put element keep unchanged eq derivation multivariate find derivation generalize gram series characteristic multivariate vector follow fourier take inverse transform bring convolution derive equality q pdf identify series satisfy multivariate reason notation use kronecker product differential variance estimation equation q kde assume near pdf required bandwidth multivariate use kernel product whiten direction derivative equation q need kernel q kernel bandwidth derivative visualization article address bandwidth kde multivariate series assumption estimate gaussian density unimodal case skew well multimodal density skewed asymmetric multivariate kde estimation generalize rule various encouraging realization parameter definite bound mostly quantify measure kullback divergence smoothing bandwidth obtain base criteria square expand taylor expand pdf sample obtain equation small whereas bias depend preliminary kronecker product repeat operator product dimension size general np nm f f jacobian match calculus operator also apply article f q matrix stack column one operator derivative differential obtain jacobian list scaling simplification simplification constant estimation article derive novel extended kde exist bandwidth minimization mean error pdf integration pdf well rule series expansion verification derive article extend gaussian gaussian univariate kde kde multivariate elementary calculus calculus derivation differential notation derivative kde gram derivative estimation widely processing upon select decide bandwidth limited pdf avoid either variety rule vary drive bandwidth asymptotic mean estimate actual criterion require square function estimate satisfy selector reasonably good estimator integrate square functional computation use fast order kde accuracy article pdf restrictive infinite selection extend gram empirical selection unimodal include skewed outlier univariate similar kde multivariate derivative estimation kde multivariate taylor expansion derive gradient vector calculus derivative involve polynomial use elementary calculus repeat product differential expression also elementary comprehensive counterpart use derive notation overall multivariate taylor series polynomial multivariate rule derive univariate exist datum bandwidth derive kronecker taylor series others derivation multivariate derive vector kronecker product series derive kde derive realizations pdf estimate bandwidth mostly follow property accuracy pdf quantify measure pdfs norm integrate error divergence optimal smoothing bandwidth measure integrate appendix identify I slow increase large optimal minimize total give derivative vary criterion vary criterion various rule bandwidth differ way name rough estimate gaussian accordingly deviation parametric family rule base computation plug rule derivative functional actual require order pilot density pilot bandwidth rule assume optimize put performance rule selection cross bandwidth criterion list rule low bandwidth selector restrictive estimation derive pdfs expansion verification concept pdf gx gx order approximate quantity obtain gram series bandwidth give separate selector performance test density
sufficiently additionally reconstruct word far reconstruct describe representation tf tf weight document document task word train language along regularization encourage align parallel corpus look representation phrase mention linear mapping separately train skip gram learn language align also useful representation phrase extract neural architecture training discuss tree autoencoder technique enable capture language language closely publicly language classify document achieve leverage corpora corpora language language directly language english en de english en en es embedding roughly million language pre processing remove remove label experiment document top category pre corpus mini batch speed merge adjacent pair pair hyperparameter tune performance portion available language compare use language regularization encourage aligned word embedding document describe mt document translate training phrase mt default embedding document report embedding mt baseline summarize report vice versa good outperform last table network document autoencoder aim make align close align sentence train embedding publicly en sentence accuracy respectively en one train de plus document cost en de en improve model still en en good en en report last de en en es mt majority supervise brevity observe importantly remarkably learn meaningful en cc en de en pr say pr office shall microsoft en de en microsoft microsoft markets competition competitive business also capture across language en english close word distance word language excellent merging bag word still sentence level essential merge bag merge several sentence word embedding exact essential reach good h l l en de en en fr fr en en label example application setup addition study consider determine wherein whether cca english language pair equivalence common approximately pair character represent english serve test standard news title original task give english word title pair obtain dimensional correlation representation threshold tune news seen clearly share learn cca mae common representation view propose capability one ensures learn two correlate scalable benefit world learn language representation learn believe view application amount example parallel english language common english act provide code valuable correspond author deep neural network autoencoder common abstract common representation wherein embed attention analysis approach cca joint representation maximize subspace learn representation reconstruct approach cca outperform task scalable approach neural explicitly mention approach correlate representation far employ cross language representation learn state datum view modality video movie audio video may available lot common representation view view application motivate importance view transfer item view learn reconstruct autoencoder reconstruct view reconstruct audio video available detector movie common representation view detector compute view common representation fed finally name write one language language view subspace item view correlate common formally view concatenation respectively highly correlate possible reconstruct vice versa canonical cca correlate representation suffer drawback scalable course try make cca scalable come capability reconstruct view cca put severe disadvantage view one view multimodal autoencoder common modality idea kind view notice mae explicit encouraging capacity view word develop mae view subspace verify report cca mae variant produce representation lack capability mae aim self guarantee representation representation combine approach main propose method allow cross reconstruction unlike cca mae capability useful view reconstruct view unlike mae ensure particularly item view descent mini batch modify world view three use mnist digit reconstruct ability common representation usefulness setup digit view language project parallel two language subspace employ representation task perform finally aim view correlate representation well cca mae organize describe architecture representation present characteristic cca mae language conclude remark highlight describe early common two reconstruct cca two view correlate autoencoder together autoencoder two propose variant autoencoder view goal section describe training propose layer single input layer consider discussion denote compute projection activation function layer try output activation architecture parameter sub outline goal formally view would self reconstruct minimize reconstruct view w b objective add common take create step deep layer common make hard describe specifically training minimize error training cross error contrast stacking outline objective pre align describe neural explain differ three feedforward neural network view single maximally train maximize learn covariate reconstruction clear neural cca single cca objective correlation maximization self self multimodal autoencoder mae neural though mae mae three error reconstruct ii reconstructing reconstruct fourth mae force network correlate secondly manner accordingly try minimize procedure three separately cca employ maximally maximize correlation experiment cca cca mae ability reconstruct learn representation mae handwritten digits dataset represent image use image set tuning list mae view half suggest reconstruction mae mae correlation view obviously learn view well correlate self reconstruction reconstruction half equally reconstruction emphasize show follow capture dimension four mae check plot correlation tune hyper dimension embedding aim transfer task digits image learn representation dimensional representation consider use provide classifier fold accuracy two leave right right cca mae decrease image less perform mae term analyze function term consider term capture reconstruct learn repeat function performance term row immediately wise row immediately
normalize statistic appear throughout regime illustrate efficacy real bi occur point work mix yield improve estimate property learn extensively symmetric whose entropy distribution distance algorithmic result establish thorough development area specific closeness identity versus main know summarize previous thought relatively essentially intuition element et optimal matching low bind distribution pp rd small mass remove tight sufficient introduce community refer version sample low factor al closeness slightly closeness analysis two set sized asymmetric closeness determine require versus distinguish large et al imply distinguish two distribution bind gap result testing mix chain go back expansion pick testing graph independently provably rapid test computation relate symmetric nonzero task achieve use test markov often comment testing hope additive strictly difficult distinguish result test distance estimate sample unknown theoretically wants distinguish test distinguish exact bad sample distinguishing understand though case logarithmic linearly exponent go closeness constant factor bad sample theoretically distinguish lower uniformity step propose two versus regime algorithm incorporate appear appendix robustness hypothese complexity application extreme mixing factor oracle mix finite chain query versus time obtain mix improved theorem testing chain closeness run begin state throughout portion assume sample distribution number occurrence expectation complexity set obtain complete calculation employ defer testing testing finally section empirical suggest statistic illustration gram contain word testing give extreme regime extreme modification basic wish test partition sample denote respectively occur draw I check accept reject intuition probability capture empirical frequency frequency second set element use modification size similar instead numerator statistic possibly note seem poor performance additionally unweighte easier heavy light ease empirical suggest want certainly perform heavy light independent copy tractable need extreme component qp n I I appropriately choose accept asymmetric assumption algorithm appendix variance draw variance threshold tailor design distinguish hypothesis unbalanced modification closeness case follow algorithm work whenever suppose pt b I reject otherwise reject summarize probability worth robustness extreme involve modify present take probability versus proposition closeness size markov matrix stationary start step step product zero everywhere chain xy markov al closeness test closeness improvement markov mix test mixing involve every running start state runtime markov chain sample average state whether accept characterize markov sample versus least stationary apply geometrically repeat one obtain formal involve likely provide statistic core small primitive natural language distinguish surprisingly small select occurrence random occurrence books corpus google books dataset henceforth refer set bi gram involve convenience whose illustrate statistic range rather essentially identical pair grey differ variation preference bi contain first word second vary provide reference value correspond e depict red variation empirical distribution word versus different size subtle
sub exponential loss long sense characterize thresholded descent size match retrieval explain unchanged change banach simplicity detail thresholded apply thresholded numerical illustrate relative estimation size sparsity signal thresholded proof give defer thresholded solution due avoid away initialization crucial provide justify assume give magnitude statistically heavy deviation square recent progress modern preferable due square empirical update step reach appropriate condition gradient flow global complex direct ideal phase retrieval utilize signal contamination noise incorporate priori end guess update thresholde tn compute update recover avoid greatly result reality knowledge additional descent intend behavior restrict update thresholding thresholded flow noiseless retrieval thresholde flow drive motivated independent act combination thresholding literature although choice justified notice validate later ht j select crucially first minimizer thresholded propose yield diagonal thresholding collection estimator focus marginal coordinate select coordinate least focus treat constant choice later thresholded flow j independent sub maximum fundamental sub absolute efficacy lemma explain thresholded defer interpret follow noiseless probability signal noisy mt upper intractable contribution fast ignore thresholde guarantee condition sensing previously principal hardness plant paper size computable discretized establish bound gaussian design future word thresholding idea interesting converge rate thresholde iterative project optimality satisfie property convexity smoothness thresholded aim dimensional apply risk satisfy rsc matter sample show optimal precision recovery rsc thresholded regularize widely dimensional thresholding e alternative sparse risk non interesting guarantee precision local optima long condition possibly satisfy strong convexity appear empirical eigenvalue se however phase rsc optima consistent penalize version strongly large global lie sample minimizer optimal question matrix respectively cl ki si consequence moreover eq let exist satisfy c l k j notice first know sub bernstein see absolute chebyshev bind eq q l x l eq next probability k mp ss moreover probability least eq eigenvector unit e c support upper separately lemma inequality provided least provide inequality summary guarantee absolute describe lemma argument thresholded iteratively n se ne soft eq q know support moreover support assertion establish show q eq p theorem imply condition straight constant take satisfy proposition due lipschitz function whereas imply satisfie proof independent exponential constant probability least sub eq chebyshev inequality let absolute sphere exponential bernstein probability probability rgb chapter remark exercise conjecture subsection height consider phase recover measurement goal computationally optimal rate retrieval thresholded show adaptively achieve minimax sparsity provide retrieval recovery thresholde researcher recover signal contaminate rise terminology extensive theory ray array quantum information impossible observe one treat multi refer interested scientific engineering background relate sparse deterministic linear transformation without rest paper case setting may
would streaming scenario unique near neighbor find offline forest need sub complicated test answer decision tree question r tree end search portion see slow metric informative online scaling learn bf projection speedup real tracking pass raw information boundary learn dna come dna protein dna letter mnist protein number recommend dataset repository data dna letter mnist letter dna letter protein dna forest bf supervise forest store see datum online bf crucial application need input flexible learn speed setting art learn often practical situation seek learn ii fast iii iv satisfy particularly importance problem stream forest bf boundary bt store bt structure quickly modify relate near boundary different class arbitrarily shape fix bf retrieval include geometric neighbor nearest build structure need access entire bf tree ball costly require volume dimensionality bad case calculate computation example tree bf latter decision forest find method preferable decision make bf projection speedup maintain create tree fact tree well tree online latter library naive near neighbor extensive try reduce store near neighbor add compression enough build forest root consist represent tree start show example independently trivially associate retrieval associate specify output real need child tree move compare child query unless close child case none child locally close child could potentially get child add show finite low negligible label compare vector position boundary node consist child tree child smaller add close break position call take set vector training call min bt min x position label vector iy process leave locally close happen task retrieval close close approximate give position locally vector weight answer use would bf locally close tree locally query example label vector tree whether new position label edge decision classification label need add bf follow root bf practice root training emphasize training strictly find power assess training example accord ask fall say good fraction fig within law time bf example note bt bf node bt comparison bottleneck bf algorithm observe time sublinear switch happen bt appear child understand consider artificial situation point remove go node child recursively stop connect node stop traversal child stop probability eventually logarithmic coincide root scale bt train dimensional hypercube bt behave like bt plus correction node child child grow metric root child child argument set tree behave root child query balanced metric power child must root query root increase inner bias bt train hypercube increase phenomenon cover ct handle neighbor train online use ct approximate nearest ann ct study previously ann ct point ct little speed important base original suggest publicly default change increase scaling amount bf rescale bf thing ct ct triangle obvious bf ct draw hypercube example bf see bf neighbor computational maintaining since traversal bf informative bf perform implementation multiple give well good tree visit number lead give repository use metric intensity even though give generalization bf ghz intel cpu
ii iii fixing vi moderate size correlation function increment simulate set material ridge probability size increase importance tend sample size rough recognize quite make concern screening screen use computation efficiency sis fix record fix sub vary sis compute inefficient comparison provide sis efficient sis forward sis incur approximately active time preferred computational complexity step refine evaluate screening selection stage method six choose scad use extend determine minimize model stage sub use extend model compare scad scad sis fr fr scad lasso scad apply sis measurement negative wrong wrong true select true false negative second simulate sis sis calculate discuss cpu responsible probe logarithm normalize link gene high nine study examine fold cross validation prediction report error final choose nan ccc final scad sis scad fr scad sis scad cross might interesting gene full detailed supplementary material choose report scad marginal efficiently compute successful computation extensive among good screening circumstance resource dimensional ridge go zero concern close degeneracy matrix dominant magnitude may dominate illustrate phenomenon ii screen propose issue theorem ridge xx middle employ combine ensure subset divide close great deal regression paper study nonlinearity compress sensing sense satisfie fourth graphical elsewhere thank constructive comment wang es national institute environmental health sciences definition lemma far exceed independence screen sis reduce sure screening rarely reality overcome simple screening technique ordinary possesse screen consistent correlation ridge simulation correlation disease illustrate inverse ordinary square sure rapid advance technology complex exceed ordinary square estimate ol long lack sufficient recent develop handling set assumption affect response loss sparsity lasso group selector elastic accurate estimation discrete although dimensional case computation concern desirable reduce refined analysis concern dimensionality quickly sure preserve property sure play role sis screening hazard response extension correlation deal marginal correlation aspect screen design operator consideration computational screening estimator possess sure assumption otherwise sis operate estimator efficiently scale sure important away refer hereafter set often correlate highly hand important correlate marginally reason iterative sis repeatedly apply sis finite classical name projection motivate sis efficient sure screening restrictive correlation discuss version theoretically possess sure interestingly ridge retain true tend extensive elaborate motivate compare analysis confirm conclude proof material familiar error alternatively realization distribution assume invertible true nonzero cardinality look map sis emphasis screen important maintain nonzero relatively q combination preserve satisfy part would part argument least square degenerate identity motivate kind material particular long term dominate quickly iii iv propose diagonal scenario pattern singular decomposition orthogonal diagonal belong see supplementary material intuitively impact random prove supplementary dominate dimensional least square ordinary square follow select retained screening projection ol project onto use capture forward regression another screening give dominant likely contrast forward goodness whenever mark analysis motivate ridge sis ridge inverse formula supplementary material gives see extreme opposed let real often ridge degenerate implement sis computational sis scale sis invariant affected predictor response denote respectively standard easy identity tail behavior different family tail zero independent follow classical admit characterization analog show include random exponential moment integer various place denote constant symmetric large small assume error tail sis need easily violate correlate weak concentration directly screen fact noiseless sparsity establish present see interest strong consistency far specify simply thresholde state choose guarantee select seem surprising however consistency condition parameter pre consistency screening assume standardized depend become recall control similar property satisfie hold recommend suggest screening procedure practical preserve choose extended simplicity mainly provide screen robust fr numerically assess various successful screening screen procedure screen sis forward evaluate report motivate sis investigate iterative variable entry iterative slightly much consideration decide simulation adjust theoretical high covariate six simulation report include true selecting due cost predictor coefficient structure use distribution coefficient factor reduction independent distribution choose specify correlation pattern independent control comprehensive generate motivate example response predictor predictor predictor extremely important predictor l sis ii iii autoregressive group vi symmetry iii autoregressive extreme supplementary material summarize noise
elimination state indicate gain assumption first result large difference result difference consider set r n reduction difficult recommend obtain use effect compare activate closely successive elimination benchmark two world begin multiply loose recommend theorem ever value suffice size preference simulate respective moreover theorem state log web set microsoft rank approximately query contains label microsoft data label ranking feature whenever ranking feature rank relevance document document result flip find winner alternate hypothesis differ index score select arm arm since assume pa j b j iterate arm prove technical bernoulli ji bernstein bound repeat triangle ti j jt ready probability hoeffding begin prescribe arm definition never side greater equal n tt rt arm sparse discard arbitrary I theorem analogous hand equal inequality step recall remark electrical computer engineering armed bandit pair arm new arm gap suboptimal arm particular new sparsity winner experimentally synthetic result sparsity improvement classic armed action rather arm represent bit response people ask paper exploit different notion primarily concerned winner arm probably every drawback winner unless underlying comparison show winner winner exist make impractical drawback prefer choose winner always assume paper winner winner armed score order oppose quadratic winner winner consider motivated existence multi armed bandit exploit structure explore capture behavior candidate compete winner mostly irrelevant predict winner concern successive arm well experimental standard approach dependent low winner show multi armed essentially logarithmic structural structural assumption top arm distinguished arm bandit show complexity improvement naive multi armed experimental demonstrate modification arm bit indicate well prefer constant th entry exist focus exploit furthermore make existence winner winner arm winner arm say high winner attention winner winner reduce bandit arm simulated arm arm however far winner winner web winner matrix allow comparison violate practice secondly winner winner winner would prefer case arm winner winner arm marginally arm prefer marginally preferred winner design robust estimation matrix vote winner winner finally define win choose uniformly unknown become bound depend preference preference score hard another course apply algorithm winner collect matrix exhibit kind difference motivate preference matrix index number arm gap like approximately argue certain unknown permutation winner recall ignore argue permutation winner winner probability arm uniformly random hoeffde reduce arm identify arm consider index winner easier reduce top two guarantee winner scale argue gap winner ask know e index learn exploit answer argue dependent complexity arm general algorithm tight reduction bandit pac select winner inspire bind j pac bandit algorithm winner choose imply preference extreme real dataset aspect winner distinguished subset find winner sort structure look two dataset consider microsoft web list suffice arm arm cardinality contain great indicate reach plot winner order small away message unnecessary estimate score arm gap
user request receive uniform width know give sample guess sequentially randomness query return return passive strategy budget avoid outside let use passive estimate noise margin characterize around threshold give constant boundary half boundary together threshold getting seems substantially easy threshold capture brevity denote passive analyse minimax equivalently mean mean follow label risk nk n mx threshold interpret quantity exponentially e rate rate paper prediction noise true change noise noiseless passive learner notice remain noise noiseless active learner rate passive vary smoothly coincide coincide well furthermore might inequality intuition help noiseless active learner add however subtle claim rate get small large even behave quite information idea quite flat convolution noise less behave linearly region nearly regardless drop assumption threshold shift quite trivial ensure behaviour flat linear shift intuitively help contradiction control passive describe handle making cross see lead leave first technique bound passive active eq lower follow passive intuition generalize passive passive approach bound feature suffice hypothesis let measure kullback leibler hold minimax loss metric threshold two threshold least use noise distribution risk learning determine model follow past passive iterate need passive reveal convolution noise regression grow region differ iii region vary active learning choose passive apply since differ interval rise passive similar passive minimax calculate notational convenience shift q explicitly understand e implicitly threshold respectively point proposition verify bind length difference point na n verify level large point equally stay epoch end phase induction always start epoch epoch phase nk setting detect noiseless argue work noiseless suffice unchanged proof bin except bin leave claim I r get point two phase phase large second verify must design intuitively query geometrically query successive epoch noiseless error shrink claim w final propose threshold setup analyse learn already feature passive level minimax noiseless continue large level achieve passive passive seem powerful carry beyond get emphasize possibly due denominator class constant observation function convolution uniform seem figure without quite task base tight acknowledgement thank support nsf big x cx cx calculation satisfy whenever c prove detailed calculation check prove cx cx cx w w cx dx w cx boundary cx dx cx k f w cx k w k k k cx dx f f w k k k proposition allow w dx cx dx boundary w w cx cx k dx w cx cx cx w w k w cx w cx cx k k f give k cx dx cx w cx w w k respect k cx cx hence k f verify complete ease presentation assume expansion look like particular let break form claim immediately see outside b small enough make set epoch feature noise epoch induction high trivially epoch epoch epoch threshold point stay epoch epoch imply final n active notice choose er ec r ne epoch since become theorem theorem definition department department university user sequentially transmission oracle returns label feature noise error variable study extensively study additive uniform set one
tackle jointly exhibit profile survival particular offer direct natural extension risk order survival survival event condition common fitting lasso prevent enforcing efficiently descent report breast cancer experiment outperform cox logistic predict task consider survival analysis make associate censor tuple censor whenever class patient expect survival patient risk model consider supervision viewpoint logistic specific patient interpret score correspond either respect look survival event compute patient patient still since patient risk see patient low also express aggregated partial event event cox model survival except censor computation restrict define event time conditionally compute naturally formulate cox enforce net solve gradually decrease model include zero extent feature covariate normalize partition group first group feature predictive survival group represent combination parametrize feature run model cox hazard select column illustrate feature survival report term correct harmonic outperform approach task predictive cox breast study repository sample distant common patient feature large variance objective tumor low versus equal survival report replacement various feature predictive run harmonic mean aggregate representative performance overall benefit model original mm generalize predict survival jointly classify induce offer embed discover informative specific generalization multi continuous
arm fix eq identification sample population solve confidence draw reward obviously computation induce solve ergodicity readily available finite regular reward one regardless gap complexity bound reward index arm correspond sample reward th variance estimate arm I drop notational simplicity context optimal identification adjust finite set algorithm maintain confidence law iterate logarithm high update algorithm terminate reader detail ucb achieve complexity reward arm could ucb terminate criterion arm sec ucb terminate reward replacement convergence take maintain mini sub stop remain equal schedule leave whenever alg stop within iteration computational former price specific alg arm adapt draw tight positive typical inference size mini uncertainty index find good eliminate sub gd I I gd c inequality replacement improve later subsample mh miss valid uncertain empirical bernstein bound concentration make assumption provide prop let replacement match mean covariance p central joint assumption uncertainty intuitively cdf standard choose normal equation union account notice runtime mini batch provide arm reward nd depend gap signal technique reduce compute efficiently reward adopt expansion reference point gradient analytically moment typical choice mostly depend reward conservative trick exploit close trick extend rejection optimal fix tailor discrete problem ucb latter assume distribution reward hold algorithm subsampling mh respectively evenly free take schedule replace original binary sampling also alg family subsampling sampler discuss sampling adapt ucb tb interval proportion desirable adapt ucb reward close heavy tight pairwise estimate draw loose ucb sharp large direct uncertainty joint author mention author name parameterize take empty case name sample subsample important mini union label label f use varie vary score almost identical negligible relative evaluate initial dependency algorithms framework guarantee improve original also evaluation speed inferior arm future relax assumption efficiently distribution condition prop almost I reward chernoff population without replacement adapt second valid tight theorem adapt never big prove output adapt ucb update change arm eventually original arm return mean alg iteration arm become require g sub eliminate last correct prove arm alg marginal union none happen eliminate use x high estimate alg marginal estimate pe define alg rhs plug stop upper bound alg pe follow proposition vary iteration visualization release code numerically e e e perform plot pairwise also variance perform empirical never exceed adaptive exceed heuristic theoretical ucb perform significantly ucb thm perform significantly bad ucb term scale log scale confidence show sample plot log scale sample overlap easy log scale estimate sample normal stochastic volatility let logarithm return asset remove problem auto treat inference mcmc introduce complete model assign regular without reversible mcmc ideally model conditional except mh normal control likelihood could order magnitude sure sample often wang approximately fix compare separately stochastic langevin bins quantile notice subsampling size approximately thousand already mini batch abuse use empirical error mis independent exploit modification pairwise strictly score communication discrete algorithm sampling suffer high burden typical scale efficient approximate solution connection arm bandit finite population provide empirical robustness efficiency algorithm synthetic real conditional core operation necessary component filtering apply challenge burden type large large random describe chain monte
model trajectory system besides fail long term mark separately trajectory cost c e next control system line start angle balance reconstruction face difficulty angular velocity image discretization error pixel exact markov property stack inner sep draw west center cm edge west showing position topology model trajectory space almost perfect reconstruction learn meaningful position remarkable table well stable dynamic balance explain non magnitude unstable real globally high six six input reconstruction visual classical history raw robot pixel image latent dimensional cart control arm dimension control use previous space depict image execute slightly obtain operate real start material experiment representation deep autoencoder ignore transition g stream employ learn raw non autoencoder forward train neither prediction learn dynamic control approach recent variant deep observe desire transformation transform learn receive considerable year refer recent overview implement multiplicative interaction find model filter recent discussion system stochastic stream state benchmark reveal embedding ease system acknowledgment thank provide link would discussion partly foundation grant program parameter propagate start approximate identity without network truly sample method main simulate dynamic simulator produce input robot arm link link compare always report use real cost goal state execution achievable control model cost dynamic create training one subset prediction autoencoder extract combine dynamic method fundamentally achieve reconstruction hyperparameter minimum make phase unfold trivial minimization architecture relu conv dimensionality encoder relu relu relu relu relu sigmoid relu relu c dimension dimensionality relu decoder relu relu relu relu except c dimension encoder relu relu relu relu relu relu relu relu conv relu relu relu input action relu relu max pooling conv relu relu relu relu relu relu relu conv relu relu relu qualitatively predictive accuracy latent latent position multi start angle velocity predict force apply angle force prediction cart long cart unstable fix dynamic cart move change angular velocity move right c prediction bias consistently cart attribute angle make accurately image input left depict case trajectory omit image figure system one ahead prediction combine model predictive control supplementary deal space cart robot arm compare locally model always velocity always robot angle slight advantage h rgb de uk non system raw deep belong autoencoder learn locally linear latent support prediction exhibit variety dynamical system broad reinforcement prominent aim algorithms linearization combine problem relatively ultimately need capable complex difficult apply usually thousand content typically advanced algorithm raw turning locally high identifying locally learn latent class autoencoder derive formulation probabilistic trajectory plan train fully compare aside briefly review dynamical introduce locally finish derivation dynamical step system smooth use notation restriction require control perform learn map dimensional solve account noise equivalently approximated transition tf system analogue assume control instantaneous cost tf minimize control optimal locally time reference trajectory linearize offset efficient control weight c formulation control result trajectory optimization locally trajectory formulation appropriate pz property information enable reconstruction prediction latent next observation require highly non crucially hard subsequently transition linearize directly impose desire prediction next property representation formalize pz z state generate access follow bayes resort outer sep draw corner height outer sep line pt thick angle black fill solid north mlp south leave leave sigma sigma gauss east edge gauss east gauss center box z z north east south north tr west box right tr z north east edge tr north edge west tr west z east fill white xshift white north north yshift fill white leave close sigma sigma gauss inter gauss sigma close gauss center north gauss mlp west east south east north shift north z east west north west south mm cm none sep encode width east draw none width mm cm draw inner shift th diagonal z x n compute network activation hide layer learnable encode weight bias variance natural expressive come gradient enforce yield generative generative operate aim dynamic reconstruction image decode network mean white include make learn linearization offset covariance transformation base network offset tn z prediction require distribution planning enforce latent lead possible valid action since come prediction fail model markov chain form model datum tuple obtain interaction dynamical true log loss per inference generative isotropic mean unit additional contraction weight agreement chain transition kl expectation via take give sample minimize descent highlight layer evaluate visual visual balance cart control link arm detail type connect moderately encoder accordance adequate throughout state action cart list parameter hyperparameter choice open
decision treatment decision sensitive receive consider use target ordinary learner aim find minimize give learning viewpoint individual aim little viewpoint company want decision job include include gender viewpoint wish base job supervise say aware attempt avoid minimizing minimization misclassification therefore need trade misclassification dependency resolve accomplished term add minimization lead give dependency predictor empirical low sensitive generalization job predictor exist might fair decision job empirical decision dependency except theoretical provide dependency unify aware universal divergence exist variational divergence aware erm aware predictor upper bound generalization dependency extra restrict low upper bound estimating divergence aware state constrain divergence guarantee tight divergence achieve generalization divergence provide expand divergence second bind loose purpose tight divergence introduce discrepancy formulate erm aware employ generalization generalization rademacher complexity regular erm thank compare estimation divergence divergence constant aware measure represent readily solve solver setup describe elimination viewpoint insufficient achieving viewpoint indirect job train exclude hypothesis may address may correlate indirect exist work construct naive employ discrimination difference aware erm total preserve propose ff learn design couple dependency measure unseen dependency derive error term dependency iid extensively distance divergence loss divergence moment match property conjugate derive estimator divergence analysis derive upper dependency exist obtain measurable addition seek measurable rate viewpoint misclassification generalization learner directly empirical erm find risk relatively number theoretically dependency viewpoint general measure dependency difference divergence suppose compact absolutely continuous divergence divergence measure generality subdifferential confirm change include divergence kl attempt rf f trade parameterize eq generalization learner empirical evaluate underlie procedure divergence upper empirical objective aware maximum mmd estimating estimation estimate evaluate estimate expect mmd mmd fr function universal discrepancy dependent choice u statistic give regularizer mmd ensure empirically addition regularizer estimation surprisingly rf rv addition true state mmd contain subdifferential almost q divergence divergence mmd divergence divergence minimize mmd order divergence large lead rate empirical cover divergence formulate procedure aware define indicate nf mmd us effect estimation depend small hellinger divergence various convexity respect assumption convex simple hypothesis form rkh give appear appear problem rewrite relax prove claim convex problem error bind algorithm theorem x n obtain error aware learn hypothesis appear generalization fx learn divergence upper divergence compare erm accordingly hypothese divergence reduction cause restrict divergence rademacher x fx appear aware misclassification viewpoint contribution estimate aware estimation introduce erm aware bind generalization mmd considerable product include
head associate intrinsic intrinsic set consider always appear appear simply include conversely satisfy condition head tail h ji ta vertex edge remove cause subgraph standard run nest markov parameterize parameter index see detail consequence algebraic define closure irreducible strictly simplex semi irreducible closure dimension prove section fairly technical particular cone marginal nest space prove py ax py vx keeping span space uniform totally cone around collection empty subset direction vector contain tangent cone idea nested graph constraint lie show model perturb rv thick minimum mm sep control control control u u achieve random function formalize parent parent assume value variable parent label vertex value fact fixing initially independently sample lead completely uniform observe clearly generate control distribution us direction tangent know argument direction x x direction give additional since parameterization direction cone tangent locally restrict loss concern marginal satisfy run intersection order edge share contain order intersection call choose rather idea terminology dag edge may decomposable simple cycle run intersection edge share different call addition vertex order q ordering since take remain latent attack result paper contain exactly variable residual behave visible parent vx vx value take say decomposable fix dag replace structure b give variable dag contain parent rv minimum inner sep distance u control rv thick sep mm parent take vertex parent set construction direct remainder induce conditional variable represent dag satisfy parent three component additionally parent parent lastly formulation tell rather parent must remainder set lemma vertex exist proving within remainder I exist vertice root degenerate q lie span connect I change nothing connect rv draw thick mm mm node nest non empty consider cc apply around uniform nest marginal associate induce inequality joint nested marginal identical theorem far fix space without tangent compose subgraph circle mm sep mm xshift xshift subgraph suitable tc span tangent subgraph variable choose mechanism share parent result distribution satisfy markov distribution uniform perturbation generate follow choice tc proof cycle nest graph subgraph however ii graph q obey nest use subgraph apply see span contain tangent nest model tangent together everywhere manifold boundary describe algebraic nest closure irreducible variety furthermore property set interior marginal irreducible variety closure subset exponential nice statistical asymptotic boundary active distribution generally complicated effort computational generalize graphical criterion obtaining np algebraic elimination latent variable discrete without generality marginal define inequality probability conjecture however graph marginal state space potentially cause latent model nest maximum ml latent mle claim loss denote px ax q lastly count dimension give whole build degenerate sum degenerate degenerate degenerate matrix clearly degenerate degenerate replace degenerate degenerate degenerate sum degenerate disjoint particular degenerate degenerate j db particular degenerate last general method proof though observable f vx v c I w w aa w suppose degenerate degenerate letting combination follow linearity part summation degenerate sum identical summation first construct definition impose maximal claim begin inductive imply root subgraph lem lem proposition lem lem conjecture lem remark lem mit imply latent nest nested well possible variable avoid inequality extremely complicated asymptotic identifiable represent exponential easily fit parameterization acyclic dag widely machine causal discrete jointly greatly flexibility unobserved cost create consider latent identifiable regular asymptotic specify parametric latent variable additional assumption generally difficult implicitly avoid characterization rv thick sep node mm rv color five show graph represent treat write form margin dag deduce word satisfie four distribution constraint subset immediately model might restriction constraint sufficient describe model inequality equivalent markov property consider conditional presence refined account model know large marginal always sense model yshift bend bend black black circle n sep node xshift xshift size mm tc dag situation lie within nested sharing tangent full algebraic characterization model represent sensible complicated nested work easily fit addition regular suitable propertie discovery currently causal structure without make assumption hyper associate dag model margin carefully nest outline main show marginal manifold begin elementary collection distinct parent similarly denote vertex dag visually dag vertex edge generalization dag introduce dag vertex restriction vertex disjoint rv circle inner rv yshift reduce dag denote square one round identification associate probability determine density obey dag density reduce familiar extra represent also factorization dag exist random noise variable criterion equivalent always property model structural obeys satisfy obeys dag possibility integrate latent marginal random vertex margin principle however existence dags dag instead additional figure latent rv circle mm treat random hyper element inclusion fix aspect change theory necessary nest dag node circle rv thick inner sep node u u yshift control represent dag replace edge child vertex canonical dag dag spirit let marginal model distribution construct margin distributional marginal define dag represent appear restrict variable parent vertex edge edge b rv draw inner xshift cm together parent direct ignore subgraph
recurrence relation appear mainly additional epoch define I recurrence n simultaneously difficulty infinitely overcome bound net infinitely still need probability enough compare already bound epoch large small period epoch precisely epoch bound epoch phase end epoch denote note end epoch bind finally event key appendix mention small enough lemma failure probability complete determine certainly check consider recall qr decomposition r h qr I turn block power refer block sample allow different precisely step block block later basically grow call space easy result focus dependence simplify analysis bind n recall would like suffice start fact next I I q r ii rely suppose eq q fact estimate begin save sample keep lemma appendix enough sample algorithm remain bind achieve rely put complexity represent bound omit experiment thousand match normalize generally report optimistic algorithm fix follow block l propose block insufficient ratio run dataset evaluation value lead visualize clearly figure check algorithm pt choice competitive figure small block achieve error sometimes block slow keep block ensure update result less block begin far reduce error error match reduction choice easy tune favorable rather figure similar affect update less slow result tune deep proper show support converge however immediate block stream family fairly theoretical empirical side sgd principal family dynamic enjoy fast original study demonstrate dynamically justify family block new research competitive result substantial hide conjecture hand suggest hence worth study could improve use omit proof well appendix remark therefore assume induction accord recurrence appendix instead look net happen eq therefore assumption I lemma satisfy recurrence relation q finally choose n give u u matrix mean chernoff bind eq first second fact lin national chi study recover stream family previously set exist easy analyze representative family moreover set dynamically empirical family real advantage datum goal component principal component covariance store moderate algorithm run batch streaming recent streaming store feed solver streaming huge arise modern pca main restrict usage amount measurement goodness streaming project point low spectral span solution span component th eigenvalue th reach somewhat hard spectral algorithm meet consider pca along regret guarantee reconstruction order convergence choose proper size real contribution perspective block easy guarantee well conduct experiment real provide concrete recommendation notation enough let integer spectral streaming point assumption sample
simultaneously contribute result conclusion plausible consider product product force inference jointly plausible individually noise configuration conclusion efficient inference variable useful sum variable want conclusion max specialized transition probability hmm state forward backward viterbi product hmms step process convolution viterbi currently perform max convolution valid sum utilize max convolution speed convolution yet max convolution x n case prefer sort guarantee useful decrease amount preserve unitary sorting explore event adapt convolution replace pair max product require fast try derive equivalent max challenging convolution use operation value number convolution apply probability operation convolution convolution regardless convolution addition two l operator prevent exploitation lagrange convolution exclude highly specialized achieve contain value convolution applicable runtime bad certain expect proceed sorting list head update yet still index overall runtime become despite pose suggest due overhead sophisticated algorithm neither sort value suggest result theoretical suggest subsequently extend min alignment wherein collection circular align sophisticated highly complicated short path max runtime improve pair short solve previously mention solve practically moderately sized max easily library transform input output turn max chebyshev shift rewrite consist rewrite chebyshev ignore expand back original place appear r yield strategy introduce result library nonnegative use numerical max convolution r v r v often loss precision input late way unnecessary recognize since normalize estimate computation final divide dominant parameter max l r l r l convolution implement automatically naive numerical naive overhead roughly expect running reduce briefly numerical compare implement package convolution implement fast k pair vector uniform element draw result ok convolution fast demonstrate substantial speedup curve speedup axis dominate empirical demonstrate max computed value respectively demonstrate replicate numerical naive different value low high value scale manner large satisfy compete large significant maximal term contribute although sophisticated yield exploit relative fairly intuitive formula relative consideration normalize maximum convergent substantially zero high numerical yield improvement call note piecewise extend increase runtime decrease routine terminate index adequate estimate convolution nonnegative return max l optimize specific application p f enough issue avoid lastly way piecewise convolution code code solve simulated people person price person knowledge cost bin fuzzy amount try amount spend gaussian likelihood variance plus problem instance likelihood appear convolution several case second distribution naive fast plotted demonstrate product sum close ultimately setting mention sum far computationally occur numerical many problem practical high quality probabilistic subset problem solve convolution mode indicate bar large sum operator less discriminative curve connection max pairs short path fast method science numerical theoretical solution depth would decrease one transform convolution argument series perform suited transform represent locally choose value uniform max close zero thus high chance high index indice maximum toward take element current dividing may number large could accurate property solve max bound subproblems subproblem numerical initial subsequent call qr way use furthermore obtain always interest improvement ever method traditionally empirically max inference perspective rather perform product inference view hyperparameter position event end product value sense convolution already moderate max convolution piecewise available acknowledgement edu perform transform convolution min convolution estimate max nonnegative two mass length numerical demonstrate fast fast inference arbitrary contiguous convolution specialized runtime viterbi path problem mass similar mass occur small nuclear mass read rna location genome sum many copy read protein source responsible discover protein knowledge individual information inference particularly increase field sum biology effectively utilize sum ability meet collective turn convert several back address peak multiple read discard limit direction might substantial effort distinguish result peak binary discover arbitrary discrete decompose information
machine aspect share preliminary compose expect insight boltzmann brief boltzmann machine truly boltzmann element note english slight article originally recently boltzmann machines boltzmann without number pseudo boltzmann machine use successfully boltzmann investigate system boltzmann truly boltzmann machine compose element although machine two give dynamic large boltzmann since variable boltzmann machines spin physics boltzmann probabilistic addition internal state boltzmann internal namely initial r eqs boltzmann machine variable map boltzmann machine map rectangular boltzmann follow state space differential rectangular move space change rotation rotation ergodic uniform distribution move velocity brief note boltzmann quasi periodic probabilistic rotation number rational lebesgue carlo zero
length tree help sequence true would expect great long sentence fig relationship measure specific bar omit clarity observe significantly sequential short encode structural information applicability variety nlp task substantial learning long build rnn avoid confusion neural tree rnn associate child composition numerous variant basic tree rnn phrase representation word classify sentiment sentence generalization topology arbitrary branching factor lstm two task control dimensionality lstm outperform sequential suggest role sentence thank anonymous valuable stanford university natural project exploration filter program air force laboratory contract finding conclusion recommendation express view recurrent modeling lstm far exhibit syntactic combine lstm structure lstm baseline sentiment stanford representation phrase sentence model value represent mean fall model bag word representation example representation contrast token lastly phrase sentence syntactic insensitive insufficient mean syntactic tree model relation syntactic interpretation sentence natural extent tree oppose address art nlp structured generalization x x chain structure lstm structure branching due arbitrary length recurrent rnn modeling recently rnn architecture effectiveness capture successfully variety prediction notably speech recognition program execution standard lstm topology superiority represent sentence lstm current lstm hidden arbitrarily unit special case lstm internal child evaluation demonstrate sentence evaluate architecture pair sentiment sentence movie review experiment tree outperform available rnn input receive previous token commonly transition affine nonlinearity problem rnn decay exponentially learn address introduce able preserve numerous lstm version define lstm state lstm lstm multiplication intuitively forget gate control extent previous gate control update gate hide lstm unit therefore internal memory vary element basic consist step lstm concatenation forward backward setup allow capture future hide lstm lstm long dependencie input sequence propagation extension lstm variant rich topology able incorporate tree lstm unit index input dependent unit additionally single forget gate lstm contain forget gate lstm unit incorporate child tree preserve rich lstm lstm lstm head take word node right mm child node sum tree follow component unit hide unit child dependency application gate open input word since child lstm unit well suit tree branch child head highly variable child dependency lstm use branching factor order index node child follow eqs eqs reduce transition eqs introduction grain regularization hyperparameter value integer sequence ordinal scale indicate ground human sentence pair tree lstm model sentence representation consider distance empirically outperform multiplicative interpret sign expect rating predict rating function kl divergence pair th sentence lstm sentiment sample movie review sequential dimensionality state summarize predict sentiment review stanford fine grain five neutral classification grain classification neutral exclude annotated sequential baseline predict sentiment phrase representation lstm lstm train tree sec tree sentence sentiment span match sentence semantic involve compositional sentence score completely rating assign eqs layer produce dependency lstm lstm grain rnn paragraph static lstm lstm tree initialize sentiment nlp rnn lstm lstm layer lstm lstm lstm baseline structure development initialize available sentiment update task hold improvement tune train minibatch regularize minibatch sentiment classifier regularize dropout gain dropout semantic system fine grain state tree least dependency dependency corresponding match span find fine word boost fine minor gain gain originally train capture sentiment summarize pearson square metric correlation evaluation compare baseline rnn vector dependency sum transform child follow nonlinearity dt transformation baseline lstm perform system semantic share mean nlp generally use combination lexical lstm outperform without achieve dependency lstm model receive supervision contrast sentiment supervision
bound le easier relate performance risk define detailed treatment variational trade measure proximity hard distinguish proximity distinguish action must corrupt wish decrease experiment advantageous work variational invoke generalized common theoretical hellinger alpha divergence divergence follow collection ii function lemma define yield bound fashion many function divergence often apply sake conceptual proceed corrupted experiment processing divergence le rank corruption invertible meaningful allow processing divergence variational divergence seek prove occur kernel tp possible matter require kernel k mx markov apply tx ki provide processing theorem generic compositional kernel hence occur call ergodicity stationary loss simple define ever le repetition clean experiment repetition matter suggest measure amount summarize e ft one clean many machine corrupt problem label le kt suggest cost total problem admit greedy choose high pick high previous upper bound r hence r greatly occur increase particular interest get compositional reconstruction furthermore statement first use follow norm implication optimistic bad arrive bound apart know consider present rank proxy theorem thing theorem case loss theorem combine worst insensitive choosing corrupt clean slow rate reader preliminary corrupt fast minimizer surrogate link ultimately work v convex notice predict expect learn proceed proper attempt careful leave corrupt corruption rank corrupt make informed decision introduce corrupted bound tight facilitate ranking corrupt bad future refine proportion corrupt problem corruption directly losse particular classification case original even loss rescale shifted loss ie take margin develop result noisy directly symmetric l cc tend marginally less easy semi r confirm unbiased one simply leave behind label average l cc omit rational present three class variant noisy ccc r follow partial label spurious label add ccc case give complicate available closed good bound pac assess appear pac prior furthermore combine generating lead appendix corrupt first generating yield max risk reduce consider supremum inequality relate find quantity mean act simultaneously classifier infimum take firstly definition complete follow convexity finally fourth forward reverse implication must positive summing e k proceed follow inequality definition definition definition focus bernstein loss ex finite theorem pac bernstein least draw choose position bernstein erm fast erm erm bernstein eq define relative utilize yield erm minimize side meaning generalize secondly bernstein chose always high version question ask bernstein condition rule erm learn quickly slowly converse true compatible bernstein compatibility condition final corrupted one symmetry right condition need q easy confirm take useful pair bernstein separable classifier class noisy pt identify pattern joint label pair provide low learner many real corrupt many type corruption yet mean ease corruption develop introduce risk corruption inform economic corruption process sense coefficient ergodicity calculate proceed appear early goal relationship function learner comprise iid empirical risk minimization problem label learner observe label variant multiple usefulness type theory inform economic decision place abstract decision develop risk corruption problem abstract main unbiased early theorem corrupt learn mean bound corrupt theorem answer corrupt contribution progress toward final goal inform regard main deal start actually action decision maker rather observe corrupt different corruption triple convenience ideally compare le form l le year label term term tr transformation form learn include label
average encode observation fitting training unbiased cross evaluate mle construct penalty promise adopt convention understand true powerful compare difference parameterization ic approximations compute first originally rise criterion aic great distribution unknown freedom criterion aic encoding parameterize mle aic selection many problem fail context reason failure aic mle normally especially precision determine eigenvalue poorly failure laplace information criterion analogy aic aic approximation consider true complexity write predictive call analogy evaluate mle mle value large analytic elsewhere difference nest represent general increment specify level direct aic expression exploit approximation let information implicitly dependent eq harmonic procedure chi square random freedom harmonic freedom parameter discuss derive rigorously information criterion bic mathematical statistic argument statistic bic approximation bic result bic large bic mathematical prior ignore principle aim I aic ii dependence encode measurement super sense simplify particular bin bin instead smoothly periodic equal bin clear mathematical necessity list know intensity daily parameterization therefore experimental discrete represent finite datum bin simulate fit expand discuss mle encode sequential underlie coefficient vector select respective mle identically execute add fourier integer except cutoff index determine aic aic perspective complexity count complexity ambiguity mle predict invoke argument q clearly low towards fourier greedy fourier follow fouri row correspond fouri initialize encode parameter execute sequential procedure choose magnitude already cutoff complexity aic parameter one add expect expect initialize large coefficient mean complexity identifiable ambiguity recover aic fouri coefficient chi correspond piecewise respect panel argument aic contribution fouri coefficient integer sensible uninformative bin assume source visualization true simulate sequential greedy qualitatively red green fourier frequency model dot cutoff correspond coefficient red correctly identify red function optimum cross entropy happen cutoff cutoff index sequential encode fouri coefficient per greedy slope dash complexity cutoff sequential prefer compare algorithm distinction bin panel index since poor true slope independent observation greedy complexity like proportional note number observation greedy like regime see index predict eqn complexity capture vs aic bic fail correct scale aic predict correct bic cutoff large bic extremely strict reverse greedy aic weak lead whereas correct scaling even coefficient incorrect accurately complexity scenario determine encode complexity sequential representation unlike formalism depend encode oppose formalism key presence greedy mle small consequence equivalent encoding need resolve ambiguity arise consequence zero fisher result non fisher lead identifiable complexity general system give sequential clearly scale scaling like counter example model therefore scale like elsewhere conclusion information tractable parameter generally must unlike applicable approximate hoc prior need implicitly understand frequentist specify confidence typically inference therefore generally applicable plot simulated encode fit panel coefficient magnitude plot index cutoff dot qualitative agreement fit selection identify illustrate true encoding information information information blue dash index solid represent equivalent criterion match simulate sequential dot algorithm dot slope slope cutoff case true predict solid application analyze degree intensity gaussian variance intensity derivation intensity model relevant discuss encoding
stream rate wherein hence uniform cdf distribution cdf clarity presentation discovery choice goal namely ed denote cdf marginal apply eq arbitrary exponent rate denote apply surely exist sequence q finding condition corollary discovery define condition discovery mixture gaussian get alternative side follow classical performance alpha power consider setup concern normal statistic random lead power compute significance choose way apply decision rule alpha denote discovery propose context incorporate substantial hypothesis reject truly appear exploit scenario scenario mean stream explain power underlie domain research hypothese reject stream early stress relative alpha show procedure scenario total figure several almost identical procedure reject hypothesis via note alpha drop substantially acceptance henceforth metric five procedure scenario drop procedure support discussion compute ii several figure discovery rate scenario relative scenario relative evaluate adjustment dependency trial statistic diagonal uniformly cope dependency control interestingly adjust show rule presence dependency microarray wherein gene arrive stream would one horizon cancer expression control cancer patient compare tumor tumor cancer patient test hypothesis gene cumulative degree since expression adjusted describe one subset return fdr online manner recover state fact deterministic get first proceed proof clear sum result next discovery false union fact nan moreover due rule adopt q true false nan note complete apply I e occur clearly eq rearrange recall rule arrive false occur write rhs false time equivalently write use th discovery mixture c I decrease q straightforward concave arrive bind relax eq eq apply inequality plug suppose induction clearly claim use prove rhs equivalently use rhs eq inequality successive inter discovery elementary pt minus pt minus conjecture consequence replica test core inference proportion false nan control pre procedure multiple testing possibly hypothesis must whether reject access well first control whose alpha rule manner develop lower truly nan independent accord nan hypothesis adjustment address arbitrary procedure synthetic datum compare alpha scientific discovery typically hypothese significance family fdr microarray expression level thousand gene cancer patient gene association cancer hypothesis say nan expect procedure large false false particular truly get finding time stress challenge increase especially case line numerous hypothesis time researcher central carry account generate cancer environmental stream false association previous could obtain research run cancer raw issue e carry instance hypothesis control decentralized nature prevent decide basis evidence information previous remark motivate formal hypothesis test need one implement discovery step unlikely seminal widely serve acceptable reduce spurious false strong interest hypothesis instead predictor sequential hereafter fdr testing briefly significance level define reject every test significance address describe control hypothesis arrive value aim ensure remain pre assign rule function one testing allow flexible use collection hypothesis reject hypothesis increase reject gain power stein control false alpha spend hypothesis occur alpha toward refer alpha fdr online fashion work online discovery adjustment cope choose hence test truly fix arbitrary independently accord non nan hypothesis validate management concern desire limitation currently database assume quality underlie motivated concern practical insight query paper fdr building upon alpha procedure computationally search pool alpha avoid leverage control incorporate procedure feature selection provably year hypothesis conjunction coordinate selective control testing context regressor regressor fall short address nan past discovery current score accept alone achieve nan indicate asymptotically throughout online level stand number second discovery occur stand level choose give equation less control simple condition alpha exploit control rule adapt discovery discovery sequence next discovery let recent eq show control every control control follow control rule procedure introduction control capture among test relaxed assumption subset nan control conduct prove control basic significance budget spend rule increment hypothesis alpha rule test outcome alpha stay alpha proceed sequentially might
applicable symmetric approach crf validate apply previous future speed method quasi eigen vector product arbitrary property l l write u equivalently reformulate binary problem correspondence k x sdp relaxation drop solve u denote variable w u u n l use requirement far bf pt bf claim conjecture token conditional crf widely conventional neighboring pixel long contextual approximate sensitive initialization make develop yet fully sdp rank algorithm tailor quasi specialized sdp dual apply fully connect solve pixel level co image segmentation vision category clear satisfactory contextual image successful semantic pixel solve posteriori contain unary potential typically feature texture potential consist term disagreement pixel contextual relationship crf encouraging contextual fully challenge stem crf pixel million usually infeasible case approximation inference fully base accelerate filter crf relationship depend relative spatially inference kernel semidefinite programming sdp relaxation accurate solution relaxation interior sdp constraint method map alternate multiplier admm estimation wang present still work sdp map large significant improvement present integrated sdp accelerate expensive part term mixture filter method field much applicable achieve superior knowledge first level co super make tractable sdp relaxation generally project quasi newton semidefinite notation list rl bold letter bold low case letter cone semidefinite identity scalar wise diagonal whose indicator otherwise product hadamard two kronecker product derivative order factorial crf energy conditioning drop notational assume unary map inference follow I I unary term represent compatibility l l l pixel compatibility base accelerate two briefly respective limitation approximation marginals crf suppose kullback leibler kl divergence recall complete factorization kl divergence view keep marginal fix form solution equation iteratively monotonically decrease limitation converge one problem optimize convex consequence non sensitive define bottleneck update equation express matrix naive need time next speed pairwise gaussian convolution r viewpoint signal processing convolution proportional standard filter filter convolution complexity filter approach limitation general euclidean feature dimension filter complexity time lattice work accumulate semidefinite programming convex minimize semidefinite q b integer denote number relaxation develop optimize typically solve follow lift relaxed round original solve method scale poorly associate requirement solve approximately solve solution respectively accurate close advantage much firstly intersection contain column rr prove several strategy sample representative mean method column entire nystr positive semidefinite approximated th summation sdp paper compatibility arbitrary label compatibility function discuss define j f quadratic x encode constraint drop relaxation constraint follow major improvement scalable key fast interior issue address show drop several efficiently several spend eigen decomposition variable spend prohibitive bfgs convergence condition continuously necessarily differentiable algorithm map nr k h next improvement scalability initialization n traditionally round round carry quasi converge round quasi dual value increase dramatically also drop quasi newton adopt round semidefinite round scheme express step standard variance discretization bottleneck eigen require iteratively accelerate utilize lr specific structure w constraint l make u j accordingly lr descent eigen decomposition computational crf approximation superiority segmentation image co experiment iteration lr set work pairwise image color pixel respectively similarly appearance adjacent compatibility function product c k f p k k operation bring memory image resolution need perform position nystr om low representative original image c unary mf mf c l c unary mf lr times na truth provide respectively field cpu memory vector product nystr om evaluate refer pixel around qualitative result segmentation mean field quantitative demonstrate complexity optimize filter limited gaussian achieve significantly energy viewpoint unfortunately superiority performance actually evaluate similar lr mf scale level perform sometimes converge undesirable see detail lose c mf co
requirement qualitative level statement concrete formulation course practical instance satisfy non trivial requirement however consideration serve filter restrictive generative formalize requirement instance satisfy give requirement significance run instance input guarantee satisfie terminate however problem far bad desirable ability namely advantage requirement allow direct checking one collection representative practical cluster input extent requirement somewhat relate namely requirement explanation success cluster lead main question follow rather metric euclidean euclidean explicitly instance find instance set cost depend course objective simplify cluster dx k kx give cluster denote format objective ig objective sum distance objective hard np hard approximate datum furthermore otherwise discuss algorithm difference approximate probably objective clustering require arise different type specific clustering define center clustering say c solution require measure implication notion particular mean c roughly discuss imply cluster year line clustering appropriately exhaustive major notion perturbation general sake similarity notion namely formulate center optimal however scale add diameter stability multiply multiplicative perturbation robustness optimal every namely cluster remain small multiplicative relaxation optimal introduce discuss objective definition initially respect cost factor easily hold relate optimum significantly optimal respect objective cluster every instance satisfy condition q optimal cluster center obtain center instance list almost except perturbation yield imply vast separate provide version characteristic notion showing carry efficiently sound plausible concrete support plausibility paper present quantitative assumption essential evaluate plausibility currently desire expect satisfy keep focused relatively level major notion determine concrete need evaluate gap optimistic thesis distinction context concern mean hardness cluster determine clearly input exhaustive search cluster feasible refer task space cluster g take runtime dependence feasible requirement demand relevant target cluster find instance diameter input recall np consider get runtime allow depend runtime respectively obtain similar optimal perturbation max cut consider cluster show existence cut focus note cluster task small find clustering mean propose variant solution arbitrary cluster clustering clustering distance optimal clustering r objective output cluster cluster pruning examine list value suffice efficiency corresponding requirement think average point input setup imply hardness obtain probably somewhat relevant exceed center somewhat trivial cluster allow find cluster viewpoint relatively gap np almost stability become claim extremely strong formula average optimal exponent minimum distance runtime claim every grow pick one cluster point follow implication condition bound cluster center distance cluster bound bind average distance read every outside consider aim overcome vast distant outside cluster comes test key clustering np hard whether conjecture np property clustering say clustering imply evenly spread singleton cluster datum show algorithm follow stable every efficiently find pruning concern cost pruning define datum weakly median clear I arbitrarily stable come notion test guarantee aware currently variant popular come show application yield quality clustering relax recent view ask convex recover address generative balanced notion range requirement range rather trivially suffice currently rather requirement currently thesis thesis well case many yield clustering way know clustering sense explanation result may proof come notion list notion close match hardness imply np vs notion almost variant rely implication prove efficiency condition believe yet light notion way separation demonstrate assumption restrictive yield finally paragraph remark argue really wish well current practice varied detect record basis patient advance aim extent input like transmission objective usefulness optimize compression distortion furthermore restriction clustering cluster make cluster hard imagine realistic situation number focus analyze case meaningful currently satisfactory conclusion intend intuition stem cluster obvious open status thesis challenge section technical whose answer meaningful complexity condition stability notion arbitrary euclidean though relatively significance optimize objective stable become hard significant open question carry contract whose leave singleton pick dissimilarity node already create agglomerative tree stable input datum pruning notion linkage cluster output tree properly nice pruning notion cluster mean nice median cluster notion nice exist nice programming relaxation come naturally approximate approximation guarantee hardness grateful david discussion concern input instance attention lemma theorem example theorem cluster hard optimize practice cluster provide discrepancy notion distinguish hope provably hope matter believe extent line critical conclusion thesis formally requirement meet validity thesis list imply requirement examine exist requirement outline open two fold I biased overview research concern assumption work quite community arise motivation technical aim attention gap encourage theory quantification resource worst understand approach hardness hard exist infinitely many compare experience np solve instance actual hardness give approach notion well expect come behave behaved area paper
quite complicated derive implement common ep traditional mcmc poorly dataset gradient langevin draw estimate include monte carlo improve fisher langevin dynamics infer simplex etc mcmc whether sample ep vb thing plug need memory big since store experiment dnn store paper parametric carlo teacher student student online give detail past parametric student teacher mixture student online large dataset use deep neural also train student teacher extend approximated single training student fit net teacher level teacher generate improve classification reliable prediction datum dark knowledge represent hidden teacher student bayesian dark knowledge first combine mcmc network kind prediction lead score compare recently ep vb train minimize kl teacher student form teacher monte approximation copy network single furthermore architecture student deep net fewer wide problem uncertainty capture softmax neural multimodal unimodal py fx gx fx gx variance fix independent kind multimodal dimensional dnn output dimension train student predictive teacher teacher network expressive effect nn I test want prediction teacher network integrate train student denote point ground label teacher choice control make accurate prediction uniformly eq quite integral tractable however sample put together take show low hyper control teacher student spherical precision strength datum teacher two reason teacher single whereas student predict argue second teacher pass input minibatch iteration schedule teacher teacher minibatch j teacher softmax approximate use also softmax output eqn function cross loss output compute propagate py py fx w train student network twice teacher predict avoid deal positivity train back propagate section approximate sgd ep vb hamiltonian carlo hmc nets sgd library hmc perform ideally apply enable open source code ep support vb number third small toy compare dataset vb hmc mnist start toy illustrate performance dimension per class fit perceptron mlp layer relu softmax output result decision boundary high bottom corner figure hmc true wish discard keep every th see monte student mlp train random encourage student predict accurately location include student teacher capture student get well kl hmc pointwise grid qualitative cccc c k consider mnist digit problem example preprocesse tune strictly comparable whole relu activation minibatch final hyper minibatch rate dropout believe perform averaging use unweighted teacher mc generate student gaussian sophisticated datum two teacher use rate reduce every network furthermore make small teacher make sgd sgd sgd report run unit sgd report sgd start toy regression order visually illustrate mlp unit activation student see density vb ep finally incur lot computationally avg ep sgd regression dataset training repeat hidden relu remain hyper minibatch noise table fit iteration interval well teacher initial rate reduce every student use precision every log bad ep method vb show kind seem
amenable normalize bethe amp bethe likelihood normalization evolution behavior via two parameter express variate bethe appear definition evolution simplify physics intuition behind bayes signal describe chosen overlap randomly propagation amp evolution statistical physics widely weak notably mmse compute state evolution log likelihood amp state fix trivial mse sign completely noise whenever evolution mse sense problem hard study fix expand equation trivial matter linearization away linearization contraction mean transition behavior translate iterative amp small quite remarkable universal detail agree phase remarkably spectral transition canonical transition mean variate gauss dirac criterion amp matrix fix point give informative preserved identity play assume note invariant always rank line result evolution instance mse reach uninformative se amp reach red middle mmse several investigation amp evolution likelihood different compare amp algorithm excellent agreement amp able find transition mmse green nd informative transition gauss trivial stable case mean single mmse previous limit r depict result e zero trivial translate fact amp theoretically blue amp amp mmse red mmse amp suboptimal region blue red nd informative mean phase transition remarkably density depict density amp transition mmse support size analyze probabilistic evolution rely statistical physics state fix amp large important topic future signal regime region sub picture signal asymptotic mmse mmse false negative mmse amp unless hard region stay barrier algorithm part european union analysis zero employ pass algorithm state theoretically minimal amp size prove phase transition suggest amp fail matrix consist gaussian noise stem constrained underlie analyze minimize large theoretically minimal square mmse achieve approximate amp estimate marginal se rely exactly amp experience reach principal technique describe small component pca search facilitate describe variability constraint pca simplicity report model straightforward comparable abundance pca algorithmic development theoretical study g many concerned recovery possible number zero zero probabilistic amp bernoulli state evolution describe remarkably achieve amp theoretically transition mmse rank derive asymptotic amp least reason question possible
cumulative usual learner q decision environment depend decision deterministic achieve sublinear assumption decision learner set round choose accord learner suffer observe framework accommodate plan minimum span cut set accordingly considerable attention differ make round regardless mean information scheme learner observe associate ii seminal learner minimize fix choose concern exploitation mix result distribution modification concerned variant exponentially average forecaster match normalize forecaster refine popular scheme sp information come combinatorial optimization overview consider paper problem name full set online mirror method use prove expect semi coincide completeness forecaster know attain regret semi case outline work full scheme picture weighting efficiently hull describe constraint work prohibitive list efficiently efficient semi bandit perturb later offer efficient idea draw perturbation minimize perturb loss conceptual simplicity efficiency due reason good scale bandit efficient regret straightforward obstacle expression importance issue efficient guarantee online contribute wave concern besides mention show implement form hull decision perturbation regard recently show intuitive excellent information scheme call recurrence efficiently principle besides full concern variant regret information access increase result close gap performance semi section main recurrence weight geometric change name concept broadly use statistic specific armed bandit access basis decision decision vector sigma round notation estimate prediction estimate otherwise I operate utilize importance operate many algorithm fall arguably online operate recurrence weight even computable variable repeat execute round geometrically distribute random construct estimate whenever notice surely case time might offer problem combinatorial offline combinatorial optimization feedback critical recurrence combinatorial action mapping let algebra history algorithm pick action problem importance close recurrence weighting manner follow learner draw well everything ready follow perturb recurrence define st draw component exponential implicitly cumulative equivalent perturbation emphasize additional construct bandit compute static overhead loss computation sample take sample control high observe I recurrence weight statement notice prove martingale follow suggest running probability proof statement concern present recurrence present tool analyze put theorem idea recurrence weight replace importance amount distribution yield unbiased want sampling termination introduce bias concern matter first expression estimate generate estimate fix satisfy simplicity write combine important recurrence estimate ensure rely sense second property ensure learner loss estimate statement q hold control calculation take last concern loss estimate well current satisfy infeasible somewhat surprisingly fix simplify copy geometric law analyze component synthesis style idea nevertheless combinatorial know tool develop gap semi bandit statement know work perturbation step let virtual pick eq crucially conditionally exploit use numerous first virtual sequence proof completeness also slightly improve replace usual fix refer side follow md prove next relate actual rely loss trick relate rule fix arbitrary kp lemma notice proof sum everything ready expect fix put regret prove central done remain arise begin use consistently simple inequality key trivially fix multiply side sum relate start increment mb obtain q holds imply together increment least prove lemma order arise fix hold martingale increment statement theorem also enable particular exponential full define satisfie become combine online semi tune properly study weighted forecaster whether remain gap
uniformly draw validate follow introduce benchmark dataset handwritten digits interesting classifier generally level label facilitate new classification create employ artificial building image principal contribute flat pyramid great challenge significant occur capture second cover various kind air water partially cast tree great dataset extreme pyramid unbalanced classification edge employ extract image prototype characterize primary ridge type synthetic fig synthetic validate handwritten digits uci repository totally handwritten people set generate dataset interpolation configuration rx process channel experimental benefit utilize sec scenario encode input rbf employ classification image reconstruct fed convolutional cnn create digit recognition summarize comparison cc visualization dot synthetic synthetic distribution synthetic bridge synthetic autoencoder autoencoder reconstruct encode layer comparison handwritten digit get c cnn corresponding autoencoder c thing synthetic achieve real synthetic result synthetic reconstruct encode encode much appearance reconstruct correlation reconstruct synthetic shown intuitively help datum ht identify synthetic solving problem could learn synthetic novel multiple standard synthetic gap jointly robust facilitate image introduce method validate supplementary material branch target balance balance branch optimize turn cause two branch avoid branch minimization quasi regularization add difference oppose parameter exact gradient vary control source one red accurately control roughly later difference synthetic real note directly autoencoder autoencoder purely place synthetic output gap reconstruction contrary pattern validate extend bridge gap reconstruct divergence cc sf opt sf edu fu research fu propose synthetic classifier normally bridge jointly propose show possible learn experiment type two validate efficiency model methodology large normally crucial world adequate even crowdsource amazon necessary classifier per object mean object extensive cloud label classifier label consume expert labeling effort practically point solve problem sample transfer hold instance nevertheless attribute nontrivial learn ability angle utilize synthetic g learn well b synthetic synthetic development cognitive artificial intelligence vision learn parent example svm work create sense train extremely challenge firstly generate shift illustrate obstacle synthetic potential useful knowledge address literature practically label available automatically synthetic novel sparse autoencoder synthetic real datum try enforce transfer synthetic generate applied facilitate image dataset need challenge instance demonstrate handwritten uci learning repository datum generate basic result approach highlight contribution knowledge synthetic gap propose gap vision community annotation several image classifier synthetic create mesh simplification visual quality recently build point cloud indicate semantic cloud normalization employ one space degradation help line handwritten apply moderate training boost enhance document degradation degradation degradation degradation texture degradation handwritten recognition success limit methodology handwritten digits aim applie find helpful sentiment page zero shot image video unsupervise transfer fall transfer nonetheless previous domain task synthetic gap cause shift feature real idea autoencoder vector input autoencoder follow pre train deep autoencoder different activation layer sparsity layer sparse autoencoder train autoencoder purely place synthetic bridge synthetic problem reconstruction complement synthetic real real synthetic synthetic identical datum leverage rx new autoencoder channel encode enforce reconstruct common task divide autoencoder channel task two channel decode flexible autoencoder knowledge reconstruction together balance channel speed optimize minimize require cause fast situation channel channel propagation newton balance task compute please material autoencoder learn pattern simultaneously capture another input sx r rx autoencoder autoencoder autoencoder learn configuration unbalanced optimization biased autoencoder topic similarity difference augment datum aim classifier nevertheless bring preserving highlight stage synthetic well could generate stage interpolation set respectively propose synthetic represent model simulate real appearance point prototype location iteratively minimize associate connect get match synthetic htb real point position prototype image initial converge generate generate prototype manually learn section generation prototype control propose zhang pre knowledge prototype design
ng default induce may recommend next covariance value mix try trajectory age ng vary summarize number bayesian criterion bic describe posterior class class include subject subject classify classify classified posterior goodness fit assess l age false panel residual prediction weight mean observation accord age option prediction contain soon covariate specify include associate standard tool seq length age age col age year normalize mmse r type c add na predict trajectory model implement intercept random age link beta age link data spline default spline knot mixed default difference parameterization intercept residual rescaling ordinal might probit mix rarely complexity integration log likelihood assume estimation recommend satisfy inclusion show data threshold age data threshold take hour depend object mix fit age subject link node criteria aic bic discrete log aic aic per per per subject likelihood longitudinal se intercept age covariance intercept intercept link se spline spline spline spline spline intercept constrain involve link give involve approximated spline knot provide probit guide evaluate nonlinearity relationship longitudinal marker normal estimate link provide confidence band col col add col col col col add col legend linear spline quantile col q legend na col n latent process option beta knot quantile confidence band spline knot quantile plot observation provide break outcome compute accord covariate default draw line code computing seq age var draw trajectory display col age legend n col gender confidence band class class use section mixed multivariate implement trajectory cognitive since entry entry far investigate effect gender marker correlate marker process mix beta cdf function summary center age time link beta latent maximum mmse time subject link beta observation function cdf mmse cdf criterion iteration derivative goodness maximum aic longitudinal se intercept age center mmse coefficient sum effect intercept intercept se bm mmse error beta mmse beta mmse beta mmse beta beta beta beta beta fix summary object common marker global test covariance marker specific along standard finally marker provide link plot band col c col mmse mmse mmse mmse mmse mmse draw asymptotic distribution vary depend seed percentage compute percentage measurement error call explain datum process input apply object joint mix implement study trajectory risk indeed cognitive closely study natural cognitive dynamic risk cognitive simplicity illustration account compete risk death incidence trajectory model assume delay give latent one latent class default survival ng age mixture age survival hazard ng age survival hazard ng function table bic proportion latent compare select latent specific effect intercept bic proportion automatic choice maxima criterion reach class solution default systematically try different example illustrate use initial similarly class mixture age survival hazard ng mixture random survival hazard ng class value beyond example table b g latent provide note latent bic avoid computation latent outcome risk maximum age age ng model event event baseline risk criterion iteration e derivative fit ci p maximum reference class se intercept class event se intercept intercept intercept age class age class effect intercept residual similar summary depend whether conditional longitudinal survival although provide longitudinal option predict longitudinal option age decade weight subject model predict longitudinal use covariate profile plot risk baseline option predict baseline survival class seq r b c decade year year ht marginal survival class summarize longitudinal time class longitudinal classification provide classification objective dynamic function use plot visualize var age age r age main validate estimate col col true col main c add col legend col c c observe model surprisingly change incidence give predictive low simple class although joint perform age finally indicate computation individual include illustration purpose class good candidate highlight year c horizon draw col age age year main old subject would thank implementation ne implemented share subsample de grant extend latent mixed theory include mix longitudinal ordinal longitudinal multivariate longitudinal outcome latent gaussian outcome censor compete setting modify strict criterion base second derivative likelihood provide various fit goodness trajectory predictive constitute introduce give analyze longitudinal outcome assess longitudinal study enter outcome gaussian ordinal e absence presence longitudinal especially biological measure life longitudinal process disease observe may exist unknown disease gene cognitive complexity longitudinal directly one gaussian variable asymmetric distribution death jointly finally population subject group toward estimation function heterogeneity latent trajectory model model theory powerful iterative goodness compute v organized implement analysis function conclude package subsection type longitudinal subsection dedicate longitudinal marker subject subject vector asset linear mix measurement subject individual visit greatly follow mixed vector respective vector shape trajectory design fit spline measurement brownian stationary w parameter involve model random effect cholesky longitudinal marker measurement covariate effect entire marker longitudinal outcome scale extend longitudinal outcome longitudinal marker define mixed latent process model separate structural interest link observation continuous longitudinal flexible observe measurement parameterize latent quantitative monotonic linear mixed rescale cumulative canonical reason follow ij basis spline knot splines ordinal marker probit cumulative ij latent mixed intercept process parameter separation longitudinal observe longitudinal marker unique marker multivariate mixed covariate measurement extend set take specific longitudinal marker measurement differ subject marker flexibility account aspect intermediate marker effect marker induce take intercept capture would capture marker model marker cdf univariate constraint require identify set latent constrain intercept location intercept intercept allow structural model tb assume population mixed consist population heterogeneous subject profile subject membership equal latent describe multinomial intercept covariate identifiability scalar covariate predict class probability profile covariate latent mixed model standard fix effect gaussian outcome previously still call distribution proportional identifiability error apply latent mixed replace structural constraint intercept constraint remain remain assume heterogeneity affect underlie interest include marker longitudinal process simultaneously longitudinal survival death disease capture correlation family ensure positivity piecewise tn knot spline specify tn lt cubic spline three family baseline parameter restrict square transformation exponential paragraph event multiple cause censor nature occur censor hazard covariate cause baseline cause baseline function proportional class model p g mixed vector involve individual likelihood contribution likelihood matrix link process function jacobian transformation rescale ordinal link function level define conditionally moment q ij variable presence effect integral random gauss quadrature unique gauss quadrature currently continuous function process individual determinant link covariance definition matrix block ik nn ni identity contribution linear variance row mix longitudinal marker individual specific cause censor q cause longitudinal outcome class delay entry contribution maximize type model choose speed find extend update default knot time first knot knot regular event time knot manually risk function transformation square imply specification hazard note unconstrained range event specification suited number baseline function cumulative output iterative initialize generate package one default yy respectively couple survival risk z n presence least crucial program put model log multiple maxima might converge maxima algorithm ensure convergence recommend initial manually aware begin grid work discovery program automatic ensure automatic deriving initial assumption value g assumption automatic initial actually estimate estimation analysis symbol subsection apply likelihood directly give matrix maximum likelihood latter triangular effect directly summarie cholesky estimation error variance compute function test calculation class goodness model class collect longitudinal latent complete class membership subject membership ig provide correspond probability base longitudinal fit selection discrimination derive two classified subject table compute belong class perfect would elsewhere indicate belong n ig n g ig g ig g ig mix empirical longitudinal four linear empirical li equation univariate assume predict transform marker kk ki involve ig z ig g mix z ig process mix involve empirical bayes g ig residual estimation residual ss ij mr specific ij ij ig marginal prediction ig ss g ig ij mr ij ss ij ss prediction residual ij mr ij ss mr ss transformed provide marginal specific graph membership link compute marker compute object multivariate latent threshold longitudinal marker standard cumulative function predict trajectory marker profile compute computation longitudinal computed refer marker class posterior approximate monte large marker value maximum link function option compute inverse carlo use estimate value ordinal link constitute cumulative mixed effect acceptable aic function risk option survival
list seven intervention absolute gram diagonal violate mechanism fit well mechanism intervention activity abundance activity change mechanism abundance abundance global seven intervention approximately mechanism experimental intervention case violate concern relation connectivity correspond activity target point success occur american intervention variance peak american value contradict cycle exist row write generality fix definition j l equation furthermore vector analogously replace reasoning invertible furthermore diagonal element path cycle hence cycle show cast complexity let write analogously row diag eq p define loss minimize exist theorem cyclic cyclic cyclic record strength necessary fulfil almost three distinct pure observational demonstrate simulated series discover causal effect fundamentally challenge various public study economic application life acyclic include observational alone assumption interest characterize relation self loop semi matrix existence equilibrium term govern equilibrium invertible also converge iterate e condition large eigenvalue assumption strength feedback cycle eq product clearly graph cycle strictly identifiability see strictly small identical cycle intersect cycle cycle intersect eigenvalue strictly situation cycle cycle solution iteration stable either still theory arguably little interest summary interesting observational alone cyclic show parent contribution effectively intervention occur income location node neither limit exact strength often different environment see financial time series call uncertain bayesian intervention simply give variable determine intervention assume demand variable discuss section detect location leverage environment matrix sufficient identifiability simulate flow let two external input member flow assume external explicitly consecutive observation section furthermore except except environment j let mean center version n state matrix invertible thus strictly latent connectivity identical intervention uncorrelated c matrix imply characterize aim reconstruct observation environment unknown intervention strength environment additionally detect assumption reconstruct main transform setting change matrix hand stem shift define setting imply intervention shift side let restricted space one product assumption l j alternative replace throughout counterpart subsequently enforce important step detail compute minimize counterpart constraint invertible scaling row diagonal element result challenge product follow lead cycle seem variant last cycle product one satisfy return met op problem compute cycle product problem exploit difference observation environment shift intervention specifically equal variable shift environment another gram read strong proceed wider adapt replace gram practice unclear approximately weak exploit location intervention namely environment difference intervention convention minimal intervention alternatively observational serve intervention variance intervention environment identifiability provide appendix solution intervention intervention variance variable must environment variance intervention shift across identifiability environment identifiability intervention environment absolutely lebesgue relaxed achieve identifiability generic environment present synthetic set various property besides assess stability retrieve code compare cyclic case observational data cyclic generate specifically environment intervention draw intervention act observe strength intervention sample present sample intervention tp minimum inner mm font circle circle draw blue blue circle blue dash dash dash dash blue w metric precision coincide point exclude close return achieve absolute value illustrate relative magnitude hamming show hidden hide intensity illustrate estimate width edge coefficient absolute retain setting assume cope present pool interpret come variable follow pose cover satisfied five obtain recall increase adjacency improve causal require worse somewhat well positive identical converge value precision accurate setting set increase intervention strength false positive return stability selection
size profile contain give profile assign assign class standard inner train model label nb distribution popular include discriminative aim class otherwise label helpful prevent overfitte svm hinge loss meet world compositional square lead technology numerous computational challenge implementation large read model approximately base profile distinct genome reach million train thousand reference may choice redundancy still useful properly intra inter specie explore large interesting dimensional real life actual hundred massive multiclass scenario reach efficiently dedicated exploit approximate sgd require fast scalable exploit train lead storage refer interested relevance sgd long disk count map hash impact reference database refer database genome cover list different generate one specie one database represent situation reference validation complete filter sequence accord keep less filter genome short specie represent specie pick sample genomic sequence remain sequence reference database adding describe represent genome refer therefore involve solely database database use base database represent gradually increase batch cover nucleotide coverage maximal value length complete train performance computing specie proportion correctly median multiclass bias performance axis color start respectively sufficient still increase beyond systematically increase steady length increase coverage marginally dataset involve size drawing length consider increase bring improvement vector hashing feature hash multiclass hash divide consider store hash observe increase actually decrease great middle mean specie micro compare comparative compositional set profile abundance use affect genome return maximal pick specie correspond repetition discriminative never outperform alignment nb performance shorter obtain outperform nb bp show level cover genome gold performance report dataset right bp species predictor train cover coverage equal solid compositional naive green dotted alignment grey dot experiment relevance feasibility performance establish learn database specie large database learn configuration database allow respectively species hash database database compatible evaluate read base genome around sequence approach previous concept nb reference specie median number little performance vs compositional nb dropping ability reference grey compositional performance report reference database median accuracy bayes nb reference compositional approach performance performance perform error sequence challenging sequencing read sequence read simulation sequence error commonly g able read evaluation systematically length evaluate impact error kind drop small reference impact compositional drop less case nb drop use database severe impact drop around nb consider reference almost nb respectively compare alignment approach around profile read mutation mean half read show impact see relatively severe compositional mutation mutation error implement empirically mutation current probably calibrate short median mutation read agreement publication impact length modify model increase mutation default configuration type alignment mutation median mutation performance drop large hand obtain compositional mutation database drop mutation drop even severe remain around mutation great nb keeps reach high mutation outperform nb configuration current experiment significant drop compositional moderate mutation rate especially mutation performance nb respectively impact alignment realistic alignment show high performance median reach database figure grey approach evaluate accuracy obtain nb grey solid line rate turn comparative compositional aspect indeed large volume generation sequence constitute motivation base measure time take base involve experiment read read mutation database allow investigate involved reference sequencing read compositional computing specie dot classify obtain dot product efficiently procedure procedure nucleotide encode convert memory lie define cpu gb memory summarize show variation across reach classify around read read increase sequence impact need database compositional systematically offer prediction time read top bottom horizontal line require represent ratio take modern scale datum extensively performance scale regard iii specie involve reference detail robustness simulate read baseline comparative compositional generative demonstrate impact estimate model highlight configuration reach svm compositional offer high nb classifiers competitive alignment tool involve sequence error result however compositional still limit hundred specie compositional exhibit alignment approach sequence error confirm compositional systematically list compositional approach species level emphasize provide memory scale linearly database could compositional alignment fast memory sequence improve learn simulate allow tune sequence technology producing read provide model properly characterize reduce memory could straightforwardly learn store prediction multiclass suggest address issue art compositional broad spectrum emphasize remain amount sequence error specie
censor enyi node noiseless span graph impossible happen recover turn limit average isolate average grow average remain fix ask question infer assignment plant positively quantity strictly overlap vanish guess unity recovery task positively assignment task knowledge belief bp conjecture part prove practical bp describe spectral show rigorously sense additionally without knowledge gap method large fast trivial interpretation connect try community membership censor observation relationship graph cluster discuss contribution development detect recovery spectral operator traditional adjacency interestingly statistical spin plant spin line notation know spin backtrack sec threshold non backtrack bethe property backtrack operator relevant bethe non backtrack bethe backtracking act graph motivation uninformative sec entry neighbor favor graph edge ensure positively plant backtrack call bethe bethe otherwise relation second lead stability bethe hessian arbitrary turn algorithm belief locally optimal overlap achieve bp strictly small avoid propagation optimal detecting observe overlap bethe always superior backtrack bethe h noise vary instance positively correlate soon overlap transition require concern assignment positively plant notice unweighted backtrack spectrum uninformative contain disk plane informative disk follow theorem generalize main enyi graph average vertex uniformly random backtracking decrease magnitude tend positively plant illustrate fig straightforward assignment positively correlate plant sketch proof proof orient transpose start ef ef eigenvector easier ef p problem case know contiguous trace allow graph neighborhood radius cycle moment e symmetry symmetry prove small eigenvector adapt bound h allow eigenvalue compute quantity explain ball large neighborhood branching process natural branching generation path yu natural martingale reasoning couple branch martingale translate backtrack operator eigenvalue spectrum circle eigenvalue plant relate spectra generalize define ki v site convenience define note bethe value thus must zero turn eigenvector eigenvector eigenvalue need limitation
v nm nm sn sn mn contradiction clearly x rr sample implicit possibly dependent construct sequence min tm tm limit constant recognize convergent since follow xt contradiction form accumulation statement development stochastic involve mild function set requirement map conclusion upper semi pointwise analyze stochastic generalization asynchronous couple field general classical approach I algorithm differential system approach stochastic track inclusion follow map reader step exposition book heuristic reinforcement execute couple iterate size satisfy martingale lipschitz function couple iterate could project onto compact ensure euclidean tackle problem generalize word follow couple recursion h create single asynchronous notation paper I al present reference di km dx mm xx da dx compact invariant call subset follow convex map radius close represent couple recursion eq h k scalar bn bn square sequence k generality k upper closure contain globally lyapunov standard essentially wise map course clear key requirement link slow iterate mild show enable describe start analyze exist say convenience claim f proceed gx sequence convergent sub n nk w nk gx yx remain bx requirement next martingale proof sake similarly prove enough q follow prove technical exist convergent sequence nk n gx n nk gx contradiction statement gx nk n convergent nm nm proceed trajectory let construct trajectory bin sn sn sn ns sn sn corresponding ty write I surely recursion asymptotic reader refer trajectory evolution iterate surely satisfy assumption I trajectory track follow trivially
insight many theoretical recent paper stochastic mechanism exploit motivate recent optimization present threshold minimax rate adapt unknown noisy sign along gradient line solve provably achieve rate gradient part stochastic convex noisy repeatedly perform optimal adaptive smoothness convex active seem inherent nature field role feedback action condition bound technique however unclear idea common field aforementione new inspire adaptive uniform design parameter simple pool active access learner subroutine randomize procedure uniformly simple return noisy sign coordinate full value gradient result adaptive two preliminary insight describe minimize function query diameter convex convexity arise dual strongly equivalently deal parameter estimate internal randomness algorithm query return optimum alternatively error oracle sign internal randomness paper motivated application computing gradient huge amount compute coordinate compute expensive multiply however require expense vector keep track coordinate proportional expense sign weak actually obtain noise zero next return sign derivative easier small circumstance calculate value could much easy require expect spirit power crucially sign round error precision get round precision flip assume length draw side half hence allow sequentially dependent label guess close formal cf common minimax exponent version condition classifier strategy measure excess threshold minimax notation clearly idea subroutine optimally unknown prove active subroutine convergence rate adaptive argue access noisy switch sign sign exponentially deterministic fx fx fx fx uniformly dimension uniformly jx mathematically use bounding bound error setting exponent exponent mention require condition growth tight similar around directional minima growth strong smoothness strongly strongly function strong relate unbiased around uniform reproduce clarity sketch sign convex jx jx boundary switch erm vc argument ignore passive learning procedure ball know contain threshold close threshold constant kk k within argue risk point stay sized region bind high ht diameter budget passive choose r generalize epoch repeatedly passive epoch en epoch radius epoch epoch otherwise threshold passive least treat clarity exposition factor diameter least limitation algebra appropriately epoch analysis sufficiently eq lie range equation issue completion though start might far away radius round round imply secondly round may geometrically epoch far like epoch close summing word hold epoch radius actually mathematically assume epoch notice previous completion would er e something strong epoch e epoch least epoch upper lemma justify result conclude dimension stochastic subroutine gradient sign optimally knowledge simply coordinate approximate search subroutine active algorithm call descent coordinate vector choose due approximate accomplish optimal fix time search set diameter budget stochastic sign oracle return let number epoch approximate use subroutine allow q subtract denote take cauchy k c lr epoch subroutine subroutine also adaptive appropriately calculate summary give information unknown convexity smoothness minimize concern store limit affect gradient remain unbiased might first surprising reveal rounding flip sign drop possible return
equation information sparsity specify fix specify spatio kronecker van covariance ik qr contextual kronecker kronecker rank temporal kronecker matrix widely physical output passive sense non move array target completely contextual target case specify kronecker kronecker unknown kronecker theory recently discuss estimator structure incorporate entry mean expectation mse relative decrease piece information complexity covariance form map log mse directly table map uninformative contextual information specify glasso add l penalty glasso precision determine algorithm contextual kronecker kronecker kronecker kronecker sparse kronecker factor laplacian type factor add contextual assess study sample complexity asymptotic dimension maintain go spatio mse take rd row contextual information contextual delta c kronecker sparse represent prior contextual nd rd information specify spatial coordinate norm type contextual row column correspond gauss markov field sufficiently large kronecker corresponding row rank kronecker factor kronecker complexity regime th row various quantifie value quantify change mse contextual kronecker kronecker additional determine show e fix plot set constant mse variable knowledge kronecker valuable illustrate right structure kronecker information integer curve contour row plane equal curve reduction require contextual inverse alone kronecker alone curve case inverse label dominate covariance maximal free value per one primary support correlation correlation measurement model consider estimating model year learn sparse inverse entry model broadly method bayesian space sparse mining gaussian penalize method pseudo base base entail sparse develop maximize penalize likelihood inverse quantify line problem complexity estimation tend infinity grow dimension via complexity recently graphical seek maximize eq element matrix denote th iterative along cover property state detail covariance vary accurate exist hold large denote diagonal entry n jj furthermore ii iii selection establish iii hold minimizer ij establishe precise spirit literature remark consistency guarantee size consistency tail heavy size grow polynomial correlation seek discover topological characteristic precision treat screen presence connect node high correlation topology easy covariance table high specifie structure inverse block screen edge perform apply estimate partial correlation place exceed plug inverse correlation develop study discovery local node variety include gaussian regression testing sparsity pattern screen illustrate complexity determine also vector bound block go go one relation pp ne ga nn problem screen occur false constrain rate type I control remarkably true attain give number rate zero correlation decrease critical direct thm great positive following quantify intrinsic variable large size require reliably detect great needed ten screen higher quantify value curve surface similar positive detect curve panel phase differently reveal detect reliably increase small correlation desire often mine possibly existence highly value specification confidence reliable quantification require critical task inference science recall require go infinity summarize regime rd regime increase screening correlation detect mean correlation partial false correlation screen false specifie increase infinity rate converge satisfy support covariance support include priori apply union subset cardinality bind sum pp p infinity thm regime regime critical report table recovery derive particular tend detail variable impose coincide convergence relax determine frobenius norm square limit mse set estimation specify example critical region optimally anomaly estimate nan outlier square function empirical density minimax risk asymptotic critical screening pe pn n covariance st specify rd sample row increasingly size require limit detection existence asymptotic positive one give existence magnitude mixed limit false critical mixed asymptotic misclassification correlation asymptotic square frobenius error covariance finally performance bind high asymptotic mse borel constant conclude unlike screen scalability glasso reduce computational contrast screen non due building thresholded ball ann ann datasets million hundred issue appropriate inferential classification decision inference lack account credible inference completely dataset population variable focus mining infer population reliability inference limit mathematically associate specify ensure complexity regime infinity purely dimensional go comparative sampling correlation mine different regime screen govern purely rate quantification require screen require acquire adapt inference estimation uncertainty quantification acquire strategy prediction regression acknowledgement partially air office scientific grant award office grant nf w nf foundation award national health grant technology us energy national nuclear security award support air office scientific award fa national foundation dms dms dms research project smc corollary conjecture ann usa stanford usa reliable draw context answer implication scale like dataset rich acquire replicate far neuron recent focus understanding especially dimension grow gap unified quantifie sample various inferential task divide category size go comparable purely go regime scale problem correlation regime mine dimensional covariance task keyword big mining correlation correlation screen graphical increase availability drive science big phrase business scientific media concentrate issue research community issue statistical largely recognize stand success scientific consequence insufficient especially inference big big column row index statistical theory develop big correlation discover correlation limit mathematical reconstruct population covariance sample underlie mining significant challenge term use specify requirement latter challenge covariance include finance communication sense science differently depend entire covariance far explore thresholding correlation covariance relate error support zero vector interest context special reliably mining presence correlate might accurately estimate entry matrix densely population structure emphasis set correlation network annotate human thousand human subject correlation level sometimes significant spatio temporal clutter full spatio clutter hundred thousand bin discovery profile thousand ip address point profile recommender preference category music fmri brain activation brain currently researcher hundred pattern brain practitioner face spurious correlation essential understand intrinsic requirement study requirement fall control asymptotic theory inference small regime cover go regime lebesgue plug covariance sparse say say function additive natural addition correspond specify pairwise global graphical also node zero support zero indicate correspond correspond covariance reason scale version give predictor inverse covariance depend inverse covariance many classical discriminant analysis variance fourth entry th coefficient prediction residual physical sparse last inverse example physics poisson integrable laplacian differential operator solution poisson heat transfer extract graphical discretized convert smooth discretized equation diagonal spatio drive gauss structure diagonal full parsimonious visualization realization simulation discretize support partial user adjacency
hx hx hx il contribution lp il h sum give b r r h thus ij ij nk nk n nk nk nk set indeed proof observe nk novel constructive kolmogorov importantly subsection state basic intuition support dimensional metric turn intuition formalize I distribution cdf let namely map invertible distinct uniquely distinct cdf make non exist neighborhood continuously jacobian point define distinct statement continuously argument jacobian entry lemma parameter matrix matrix calculate jx ix ii quantity respect equal multiply take particular q ij p jk p contain complete ready interior inverse mapping specifically define n k completes take neighborhood ii lemma argument sd ns bs intersection hand disjoint intersection jacobian open open proof structural constructive cover kolmogorov metric must identical packing subset denote cdf immediately suppose contradiction lie point inequality cdf kolmogorov cover union volume volume ns proof complexity prove prove theoretic argument subsection tv main theoretic absolute complete define I apply root coefficient ix ix ix qx assume proceed far simple give two z z z c z x another application triangle x complete characterize statement kk x variation equal k must mx ok ok enough lemma c bc x x contribution small discrete x prove x I must mx ok eps expectation large enough lemma x achievable obtain cover easy need deal single cover indeed straightforward discrete appear produce approximation fortunately require negligible large element mean couple minimum c increase expectation assume component leave unchanged k cx nk nc nk pair symmetry may integer constant long c let complete approximate element efficiently give appropriately variance construct x theorem universal support deviation exist integer mode satisfy generality mode know eq similarly term recall basic fact concern variation start processing domain function draw next total use variance tv pt claim observation theorem observation ac uk california edu university com sum independent near sample variation use nearly cover admit ok cover transform structural transform namely analytic argument concern integer variable order distinction distinction sum independent arise special poisson trivial binomial know survey fundamental form special chernoff hoeffding long research random decade near efficient tight upper constructive explain cover elaborate context motivation work variable definition learn natural analogue well pac boolean unsupervise set topic rich extensive literature context year body study perspective computationally gold setting theoretically ideally near main run output description hypothesis learn variation require sample sample provably logarithmic case previously near run polynomial high sample algorithm understand computational consideration theorem give tight problem would conjecture complexity case distance distinguish fair biased coin perhaps surprisingly learn arbitrary distance separation conjecture learn theoretic rely understand space elaborate cover say cover cover cover exploit variety cover kolmogorov role theory book statistic book upper bound distance upper constructive cover prove construct polynomial size least comparison non cover quasi cover theorem consequence learn game theory combine algorithm imply output run sort enumeration cover equilibrium show constructive upper size standard along additive nash equilibria correspond hence constructive cover approach lead implication hardness computing equilibrium anonymous support denote cumulative cdf variation distance variation kolmogorov f give overview ingredient cover moment matching agree tight proposition explicit agree variation unfortunately moment moment periodic structure distinguish odd integer proposition explicit agree work limit regularity variation support integer close force proceed hypothesis bottleneck arbitrary exploit beyond aforementioned upper transform tool give fail type fourier fouri random product fourier transform similar starting essential new small assume effective know extremely simple point transform everywhere exploit sparsity fourier transform complicated transform precisely cover show transform necessarily transform logarithm approximate interval actual root fourier circle therefore provide coefficient logarithm description geometric defining probability mass distribution function fact interior understanding allow expectation roughly change region distinct effect word effectively size remark structured family polynomial approximation provably apply sense lead piecewise necessarily incur structure bind sample additionally exploit dependence idea I detail completeness tv discrete transform dft function integer dft dft dft onto give intuitive explanation fourier transform apply fouri good likely error bind believe may interest fourier effective standard fourier big effective effective support could dft idea hard compute b otherwise proceed sm depend fourier transform appropriately small effective proof efficiently ii constructive beyond remark upper analysis since copy note consider dft write ij take relate claim integer exist integer application claim proof ii follow rhs integer uniformly note claim q ij integer ready theorem run dft calculate ok ok correctness e henceforth give tv size learn total henceforth assume indeed application kt come consider high automatically absolute mean chernoff nr union nr bind get nr ok contribution case error ii eq note lemma compute expect q like outside use show I x real complete upper size proceed construction minimum size upper proceed start size case desire polynomially cover base point order close discretize random constructing cover belong theorem translation subsection moreover ii cover claim sub variation cover support discretized variance large variance want discrete interval discretization geometric grid proposition cover size last inequality reduction note proof cover proof notation identity pdf view function circle plane entire complex agree e conceptually logarithm additive taylor polynomial assume desire true reason logarithm arc base arc lemma magnitude divide root arc aforementione cover transform logarithm appropriate nearby relate variation transform equality analyze polynomial defer fix root qx qx suppose root list qx x jj triangle lemma claim root standard proceed prove difference first jx nk nk eq multiplicative next replace q assume seek hx h j mx rhs satisfie assumption f hx hence complete induction require proposition size z fact tv nk ok associate follow lemma root consist part real I show cover claim ok dd first relatively arcs number take possible arc note strong arc I z z z part w taylor give near cover time integer exist run ok kn k ok kk build establish subsection cover case close exploited cover spurious point large efficiently construct proper cover spurious careful argument defer possibility constant possibility observation coordinate variation suffice find probability henceforth main possible taylor fourier transform additive dynamic program problem let sufficiently divide unit circle arcs associate root mc ic ic b c ic ii concatenation near exception less algorithm
cnn latent crf recover object condition compute feed forward energy yield microsoft capability mit recover coherent category appear room together measure improve classifier take category entire gain object gain combine pre classification performance unsupervised node image neighborhood latent activation node potential instance scene image variable scene diverse traffic another represent various kind framework capturing unsupervise scene mit scene probability match use misclassification baseline neural capture scene engineering combine strength learn expensive dataset employ advance technique instead small finally pass category detect contextual co occur tree graphical dependency incorporate category detector probabilistic use simple detector degradation contrast pre train incorporate context many contextual addition thus framework object classification imagenet vision task popular number train cnn object framework plan future independent incorporate probabilistic localization object bayesian optimization grain believe setting learn cnns scene classification scene label available training training automatically label scene localization segmentation task perform multiple coherent account spatial location interest body work contextual future plan scene expect recent attempt probabilistic combine crf joint joint framework deep learning feature account dependency variable train network mrf network train latent lead finally work multi technique text rest overview fc train model dataset compute likelihood eqn image train imagenet consider extracted effectively multiple image predict achieve dependency structure relate dependency object label condition input allow tractable structure leverage extract moment rkhs distance recovery kernel conditional conditional give setting modal component transformation x rkh empirical distribution reproduce hilbert rkhs yx xx iy given employ rbf estimate tree gram parameter l l work among available provable guarantee statistical k tn employ cl learn many structured probabilistic graphical structured absence design machine view special energy energy particular output compatible potential graph use configuration eqn define find perform parameterized net gradient loss classifying multiple category b ms training image label object independent classifier recall tree network correspond avoid potential covariate use along map compute back use dropout viterbi message latent l l l l l l l l l l ht measure ex classifier classifier layer learn recover structure relate appendix tree role divide scene node object room cluster around car instance object precision layer conditional train layer feed network classifier label decision category neural gain significant object like percent percent percent percent percent b contain test different marginal precision comparison investigate activation potential image result high activation effectively capture contain result image appear belong scene relevant use scene scene capability mit room optimally misclassification never scene use marginal probability hide neural table show input hide result misclassification ex place co appearance train set gain neural network capture semantic distinguish level information image information manner co knowledge apply like california art classification imagenet unify strength multiple deep microsoft image incorporate contextual latent co condition extract fc train imagenet pairwise object occurrence take fc object learn conditional significant gain measure ms especially object capture scene infer alone scene mit use present scene deep performance computer task scene parse pose focus train imagenet consist object far challenge currently framework use simple approach predict category mutually exclusive classification decision however natural label mutually exclusive ignore label share knowledge prediction explore expensive
case significantly hope classifier accurate perturbation theorem vertical horizontal independently risk similarly two robustness adversarial straightforward calculation unlike use perturbation switch illustrate result practical classification confirm identify linear quadratic large adversarial robustness suggest linear svm svm rbf width classifier validation close perform find satisfie procedure obtained define I point robustness follow perturbation opposite baseline adversarial say uniform line find large condition estimate svm svm adversarial perturb f switch cubic rbf classifier g original perturb g ht vs first mnist handwritten digits digit task train small translation image unit euclidean report adversarial perturbation despite perform fairly well small adversarial perturbation visually translate perturbation instability adversarial perturbation surprising table addition improve classifier important implication establish limit interest hence classifier though random robustness classifier hope adversarial perturbation get close classifier design specifically account robustness identify limit perturbation theorem would identify towards understand deep net human successively event observe moreover cauchy schwarz inequality fx fx rf schwarz together adversarial conclude sphere spherical show follow sample sphere prove bound note armed concentration n deduce result n f fc decrease pd negativity take side obtain result prove conclude inequality first exist generality lemma successively inequality get negative z similar fx follow use get norm rf p last use note hold get conclude solve perturbation label thank point reference arbitrary small possibly robustness perturbation adversarial perturbation express result task involve adversarial robustness random perturbation perturbation knowledge theoretical address phenomenon instability recently limited give explanation adversarial instability proportion misclassifie evaluate robustness perturbation highly desirable particular change paper robustness classifier classifier perturbation differently average lack robustness perturbation data adversarial vs car task small car plane therefore seek understand perturbation formally study robustness perturbation set robustness linear fundamental robustness adversarial perturbation express specifically classifier robustness classifier implication involve small classifier misclassification compare notion noise robustness much former showing notion classification task value illustrate newly concept theoretical run practical task surprisingly unstable adversarial receive instability adversarial perturbation raise challenge generalize unable correctly paper show perturbation flip classifier theoretically adversarial instability network several attempt network adversarial perturbation relate explore argue nature high dimension contrary network go general trend problem involve flexible adversarial even low risk security adversarial attack work e decision counter attack robustness adversarial extensively differ classifier paper sec introduce problem introduce throughout sec quadratic classifier conclusion adversarial conclusion leave future associate simply take misclassification focus classifier perturbation ambient perturbation noise flip nature perturbation perturb point outside support robustness adversarial perturbation minimal perturbation flip estimate label note independent adversarial perturbation perturbation robustness definition assume observe region sample label robustness radius center sample classified illustration give fig perturbation outside quantity quantity risk robustness adversarial perturbation uniform introduce run robustness risk consider binary vertical resp horizontal constant class image background resp illustrate permit separate line vs vertical valid task separate despite detect visually image orientation classifier exploit achieve indeed resp risk achieve capture fail orientation separable adversarial minor computation robustness satisfy maximize adversarial perturbation unlike orientation direction robustness classifier fig great extent classifier image difference robust adversarial noise example fact perturbation orientation unlike classifier f say capture essence equal evaluate similarly perturbation world partial enough concept essence robustness class adversarial perturbation classifier adversarial random perturbation equal distance hyperplane classifier assume q intercept eq robustness adversarial constant represent vs diagram diagram region attain importantly interesting quantity intra intra class geometric transformation vision even task adversarial average class robustness perturbation linear illustrative diagram achievable robustness classifier adversarial perturbation bound behave
true separate consider confirm converge htbp simulate inversion procedure determine run linear cost stop iteration multiplication multiplication power multiplication first risk reduce cost nr fast nr hundreds great low agnostic exception smoothly low explain iteration slight improvement eigenvector guarantee interpretable tradeoff risk computational cost estimating setting suggest single update allow risk computational illustrate iterative simplifying various allow subsequent choice estimator tool measure aspect together employ whenever study throughput read development methodology spectral numerical preserve edu foundation mathematical research department university partially science foundation grant dms office nf office grant research institute advance grateful constructive greatly research recent need trade tractable practitioner computational theoretical give problem tradeoff computation analytic risk computationally constrained estimator conclude risk termination iterative computation family massive field curse improvement cost largely field gradient descent relaxation conversely introduce describe procedure bag little implementation bootstrap massive classical problem constraint chain carlo consistent mix order detection detection compute algorithmic complexity likely never bad aspect understand fine specifically framework address cost risk understand gain assess degradation risk exponential outline basic section illustrate normal idea general robust extend idea problem estimate value random value denote estimate seek r optimality principle maintain add formally compute statistical estimate equip algorithm compute hence together runtime straightforward example manuscript property compute outside set collection storage keep much processing exploit put among feasible estimator must know estimator plot illustrate achieve risk balance examine investigate computation identically variable explore index generalize allow allow linear compute operation look compute store datum perform operation sufficient paradigm look perform mle omit algorithms q consider streaming near zero extremely select intuitively say computing streaming setting possibly use collection estimate index index impact computational cost assign indicate b unique change blue cost illustrate row row risk fisher overall risk signal present note optimal great regime figure time family density q parameter model ie n n p maximum convenient index subset estimate fisher pairwise intersection kk kl analog parameter covariance sample verify well frequently compute runtime compute sum risk relative importance component able estimation general inverse nontrivial task generally especially graphical finding compute np parameter nonetheless runtime tradeoff less investigation possible consider estimate define arise contaminate central degree scale percent contaminated proportion datum decompose time pairwise sum comparison data simulation computation sample approximately describe compare replacement pair replacement median estimate three estimate cost simulate contamination
far pf exhibit centralize pf ii require communication neighboring sensor iii consensus average implementation sensor communication distribute achieve account decentralize pf multi network involve motivate broadly mean parsimonious representation term low dimensional facilitate interpretability enhance predictive deal decentralize algorithm rank argue internet traffic anomaly measurement moreover rf subsequently development layer wireless cr decentralize linear algorithm outline aforementioned ip abstraction node operational origin traffic flow denote traffic link interval count across single adopt mean traffic link source accordingly horizon count flow traffic relate term carry flow traffic matrix temporal traffic periodic traffic typically intuitive validate term failure attack attack service let traffic flow explicitly flow traffic carry error anomaly traffic flow effect link anomaly anomalous spike interference flow stem miss link measurement operational reality rely indirect measurement traffic link measurement tuple available introduce l traffic compact operator entry keep unchanged flow traffic matrix link traffic rate short relative flow suppose anomalous instant row flow put plus decomposition albeit natural criterion np optimize surrogate accordingly sparsity control optimization appeal accelerate complexity develop network interestingly link subsequently exploit spatio temporal link anomaly shoot estimation turn outperform latter value leverage sparsity jointly red anomaly continuously traffic monitoring aggregation anomaly operational adopt network associate minimize reduce translate miss raise central represent isolated point traffic anomaly anomaly carry locally rely count cardinality block likewise edge terminate term oriented incidence incidence orient denote small nonzero eigenvalue algebraic theory establish define np lp notational convenience use rewrite form lp lagrange constraint lp collect multiplier associate multiplier augment back dual make local cost local cost gradient constant hold function point I f b aggregate aggregate gradient guarantee constant hence aggregate cost differentiable sufficient convergence decentralize admm far sequel well respective consensus attained form stack comprise stack copy optimal may multiple strongly exist lagrange converge one establish dual lie space convergence prove analysis define convergence several contraction distance euclidean context ergodic rate establish prove refer speed successive primal dual vanish optima convergence next four monotonic namely hold show monotonically ergodic primal solution prove straightforward kkt condition main establish decentralize multipli primal solution initialization admm local cost close specify dual solution indeed primal solution ultimately ergodic iterate difference prove recent decentralized contraction inequality linearly convergent convergent meaning convergence decentralize admm algorithm initial multipli c iterate lie convex lipschitz multipli column guarantee converge lie converge contraction indeed lipschitz continuity eigenvalue orient eigenvalue laplacian admm penalty arbitrary insight influence find value right aggregate condition note graph condition dominate imply contraction cost dominate acknowledgement friend author extract j wu support grant gm wireless communication science technology china china ny tx put framework decentralize acquire importance communication central cost privacy reason term network paradigm decentralize maintain broad decentralized comprise wireless medium internet use refine local hierarchy local estimator fully exploit maximize suffice accurate task decentralize favorable structure direction multiplier admm iterative method back process decentralize price wide encourage single inter communication th single neighborhood inter symmetric undirected edge represent communication clear domain wireless wireless electrical cognitive example across scheme decentralize local processor refer fc internet collaborative agent perform centralized raise concern processor represent isolated failure objective develop decentralized setup exhibit coincide correspond keep overhead communication neighborhood argue admm wireless decentralized solver classify operating handling constrain domain subgradient variant incremental gradient average subgradient inexact achieve price stepsize order primal form local applicable depend local iterate without admm numerical convergence chapter solve subproblem demand fortunately subproblem solve run iteration burden remainder describe admm heart algorithms chapter network section focus estimation unsupervised inference deal estimation collect sequentially internet spectrum wireless cr network motivate fundamental admm state straightforward decompose overcome idea local represent local per network formulate equality constraint coincide neighborhood extend turn amenable decentralized leverage favorable alternate direction method multiplier g employ minimize fashion whereby converge centralized facilitate application one variable eventually eliminate multiplier j lagrangian coefficient entail comprise decomposable separability come primal turn lead step admm decentralize algorithm jk admm decentralize redundant eliminate say track redundant end store node iteration local cost attain consensus likelihood mle square posteriori formulate minimization centralize fall short capability spatially sense outline decentralize far outline depend technique accordingly centralize mle capturing pdf weighted vector map yield decentralize decentralize example decentralize recent advance nonlinear least monitor grid arise ac global interestingly estimation decentralized programming leveraging base centralized sdp sensor employ yield wish blue fashion local I unconstraine admit close decentralize case fitting offer decentralize wide rule local suitably average originally allow decentralized exhibit inter quadratic centralized mle problem tackle task reformulate outer linear know structure formulate ensure matrix outer drop non semidefinite solve decentralize decentralize complex node system magnitude newton tool iterative linearization therein suffer variability grid capability decentralize multiple control attract grow three area centralize htb decentralize neighbor error converge decentralized sdp successfully address solver overview along decentralized estimation variety task rely sense resource constrain message wireless ap limited capability desirable decentralize sensor attain another environmental monitoring local sense decentralize framework sensor network available everywhere fc diverse challenge inter node exchange allow overhead decentralized detection framework decode task wireless communication scenario know codebook belong assume symbol symmetric channel conditionally sensor know characterize pdf noise unable reliably message information sensor global centralized centralized ml decoder likely multiple propagation approach centralize even cardinality exponentially introduce burden sensor centralize decode become il py likelihood ignore decode objective give n clearly statistic equivalently bit interestingly discuss admm decentralized framework section allow attain sufficient length complexity decentralize decoding since posteriori rely average sufficient extension alphabet consider tb bit versus snr demonstrate performance decode code numerical test involve ap scheme sensor curve mark initialize decentralized iteration correspond iii decoder consensus iv admm decentralize decoder exhibit convergence consensus average counterpart iteration suffice bring decentralized relate common ap message map entry finite per sensor admit form sensor channel additive assume uncorrelated ap neighbor ml covariance suffice wide constitute argument decentralize decode lead attain locally constitute decentralized order centralize demonstrate task develop decentralize minimal centralized significant reduction communication svms centralize setting task surveillance often limited acquire central processing costly scalability communication overhead environmental structural monitoring diagnosis medical condition record available seek follow slack scalar allow discriminant mapping possibly centralized decentralized reach decentralized svm take pair slack decomposable structure decentralize identify b I decentralized iteration algorithmic decentralize admm svm incorporate consider node draw gaussian matrix respectively optimal depict global training set centralized iteration admm rule centralize counterpart local unsupervise exploratory infer structure collect set design decentralize capable joint processing various centralized centroid denote prototype element amount specify centroid error minimize convex coefficient program consequently admit solution rise term cluster optimal membership suboptimal proceed r fix least nonetheless require availability information per challenge reason topology offer address yet decentralize leverage neighbor albeit decentralized extension decentralize method environmental typical monitoring option less motivate decentralized processing decentralized scheme sensor identify temperature measurement group connectivity average centralize test include note converge iteration whereby datum exchange reach available description interest varying motivate decentralized scheme node collect datum recursively refine develop tracking approach kalman particle scope facilitate processing network decentralize adaptive possibly nod collaborative fashion communication linear criterion per instant node without generality estimator interest jointly well wiener tt develop decentralized building form amenable via stochastic handle variation statistical step instant apparent obtain root equation cost solve available statistical acquire algorithm find expected size local constitute counterpart section decentralize first stochastic approximation step parallel admm constant tracking capability operate presence node see expression mean relate slowly vary vector specifically link variance depict local evolution mean noisy ideal link closely follow theoretical trajectory steady accurate penalty link fig adaptation affect adaptation respective adaptation closely track variation fail square well online estimation signal track especially attractive rate offer valuable admm decentralized scheme distribute spectrum claim set minimize form history past discard enable tracking decompose global estimate utilize follow decentralized iteration q involve node recommend recursion estimate converge invariant bound mse along comparison diffusion regression arise spectrum monitoring suppose sensor comprise interest spectral peak reveal heat source channel contaminate additive sensor source frequency band operate lead decentralized estimation evolution signal
completion mc preference incomplete feedback ill fortunately structure texture motion lie subspace dimensionality recent convex remarkable fact complete rank select well suffer robustness even minor sensor failure environment mc far ground resolve issue effort devote mc solid theoretical chen al truth subspace detect column corrupt expansion please advance physics quantum apply exist try resolve show extend robust mc coefficient basis column intrinsic corrupt filtering able solve numerically polynomial expansion however perturbation might fitting far original reduce sometimes expansion coefficient possibility fouri due failure signal carry outlier success mc quantum r experimental overcome basis severe commonly justify exact robust dataset corrupt try segmentation face observation miss resolve issue relate subspace model call robust low lrr thus mc remove outlier suppose whose range probably mc formulate recent sense objective nuclear envelope ball eq worth note standard propose general mc ground cardinality traditional issue sensitive minor occur failure environment mc mr range space probably principal component analysis pca outlier even corruption resolve successful pca via convex xu work truth sample outli pursuit apply subspace alignment texture etc unfortunately value work worth note mention mc outlier mutually limitation recent mc complete detect simultaneously correspondingly relaxed chen range sufficient column input exactly corrupted sample report mc recommendation research basis limit challenging task extend robust mc general demonstrate extend robust mc succeed traditional basis ambiguity fraction comparison significantly robust regularization parameter choose reduce algorithm incoherence relate extend lrr algorithm immediately subspace fine structure follow describe setup present detailed proof establish theoretical application cluster validity theory section conclude column general consider exact corrupted recover element hope recover column mostly rank cover situation problem mc low original normal unfortunately replace relaxed brevity rewrite project onto rank total word approximate successfully low several example sparse identify expect space low svd problem condition adopt incoherence condition please table explanation incoherence incoherence suffice column sample bernoulli entry non select corrupt incoherent identify clean guarantee analogously issue column sparse isotropic ambiguity assumption ambiguity number comparable ambiguity condition geometrically zero scatter matter main entry measure assumption position specifically position determine event guarantee exact noiseless l severe work summarize use notation support matrix grow product capital zero column whose sum column truth optimal solution h xx result surprisingly column close measure w fraction column exact robust mc recover probability subject model parameter automatically imply distribute traditional low rank mc seek rank bind consideration arbitrary partially recover rank matrix fraction rank miss recover corrupt robust al incoherence ambiguity extend recover extended result extend mc elimination extend mc recover succeed solution mc ks km succeed lt lt argument bernoulli high c probability nc fix small proof I j assume provide eq nc p extend mc accord feasible note h prove brevity construct inductive eq obeys lemma q three inequality first eq dual exact h norm l hence f first inequality mp remainder construct q triangle dual suffice assumption obey dual condition check net unit cardinality show adjoint accord automatically bernoulli variable hoeffding inequality eq accord stand I ie unitary second inequality hold fact proof complete mc efficiently alternate multiplier admm nuclear minimization problem fast term algorithm speak recover ground truth select norm least speed scale subproblem recover suppose I theorem form submatrix brevity submatrix solve scale mapping restrict section bernoulli column rl ks r regression admm nearly fortunately separability norm decompose subproblem equivalently matrix problem admit l l r filter subspace randomly sample recover seed solve conduct line remain recover low dimensional subspace filter range column end step probabilitie recovery seed column line span range range recover justify recover ground truth subspace seed though row check range examine column outlier informative select line bind parameter intuitively property element element zero high suffice guarantee two measurement connect incoherence define eqn space see condition definition corollary incoherence ready follow illustrate result suppose bernoulli sample well condition hold large happen coherent enough exact rank low select roughly typically probability suppose provide appropriate chernoff obtain theorem lemma ground subspace seed fulfil incoherence succeed numerical constant filtering justify outli outli matrix e l condition incoherence identify even complexity filter case bad algorithm require recover seed factorization multiplication r mn r subspace svd multiplication converge significantly discuss missing demonstrate validity application subspace clustering aim lie e face motion etc probably effective lrr clean representation low mathematically solution cluster cut lie robust lrr widely handle situation commonly failure extend robust modify lrr np incur great difficulty efficient show mutually form solution mc conversely lrr find approximate original obtain extend lrr far conduct validity algorithm I matrix column sample I optimal compare range hamming run illustrate regularization enable recover range show independent magnitude record truth vary succeed matter magnitude range simulation plot region succeed comparable working ht speed admm list cpu hamming distance significantly comparable c admm filter admm filter admm filter fraction
universe physics feasible thank development extract everything feasible range growth universe analysis invoke transformation estimate act prominent example appear instance value quantity routine invoke fouri combination non computable calculation give remainder formalism implicit provide perspective science summarize sec implicitly deal set storage computer computer routine implement mapping science technique determinant covariance require constrain denote adjoint exploiting estimate wise vector analogously operator trace method spirit require purely stochastic cost phrase trace subsequently linear split case implicit operator physics approximation violate introduce q time sufficiently together represent determinant operator evaluate time integration order taylor expand determinant dropping pseudo case deal dominant determinant coarse numerical correction computational cost partition ref community signal stochastic novel implicitly previously impossible see address bayesian determinant homogeneity physical field parametrize power assumption detail become space respective spectrum position kk x determinant form q set diagonal dominant whereas diagonal fig ht matrix refer explicit implicit explicit well determinant regard separation apply well perform integration realize furthermore study value dependency interval fig precise matrix eqs applicability discretization interval chosen see particular illustrate numerical determinant fully minimizer step calculate selection vast topic henceforth present example find describe measurement signal independent gaussian e represent operator variable relate covariance q signal respectively denote phase bayesian evidence might deal implicit perform integral last method nest wants infer switching exchange affect determinant calculation field reconstruct ip explicit dependency follow calibration integration perform produce general contain instead routine implicit matrix probe variety scientific affected extraction background calibration realistic unknown calibration amplitude parametrize specific assume gaussian gets affect mask cut still uncorrelated measurement equation device calibration calibration regard external calibration sufficiently strong infer amplitude simultaneously approach prior sec calibration noise generate realization part numerically calibration give data efficiency use eight trace peak eq determine determinant determinant integral involve discretization integration operator obey fine discretization necessary fact might keep cost universal inference comparison deal instance calculation determinant acknowledge realize determinant operator expand method expansion approximate theoretically single pseudo step enable derivative integrate pseudo integral representation determinant integral representation ref
wavelet wavelet efficient compactly wavelet useful wavelet introduce soft family wavelet look sort haar wavelet beta wavelet fine tune keep loose orthogonality property haar address wavelet remain wavelet approximate wavelet advantage I cycle iv central signal nature compose cycle signal successive probably wavelet cope another detect analyze power system investigation dr lot regard limit theorem continuous wavelet beta distribution limit theory compactly wavelet recently insight wavelet present reading wave wavelet link wavelet infinitely wavelet wavelet transform wavelet unbounded support wavelet wavelet order interpretation average smooth derivative derivative derivative wavelet transform low pass version kind central limit unbounde compactly variable chi square play central wavelet unbounded wavelet concept impose beta efficient concept entropy recently reveal wavelet one link beta valuable practical role unbounded support ip random ji lattice dirac accord kolmogorov density q wavelet well know application discover wavelet compactly couple factor generalize factorial function beta function easily variable transform unity extreme ia guarantee wavelet cycle wavelet smooth wavelet spectrum function q wavelet prove spectrum carry wavelet sense spectrum symmetric haar wavelet support check reliability computation wavelet etc occur expected wavelet half cycle first spectral due unimodal feature wavelet henceforth refer wavelet algebraic handling expression order play beta relate h couple beta wavelet aim investigation potential wavelet write happen compact beta wavelet ability provide wavelet efficiently concern focused location main drawback haar wavelet contrast
risk suggest genomic promise spurious association identify variant disease high throughput popular important genomic level achieve due restriction share level summary datum score frequently share significant shared commonly value normal transformation generalize weight combine issue dominate setting irrelevant high cause wrong inference correlation lose disease associate genetic identify study conduct summary variant simultaneously genetic comprehensive review perform genomic genomic genetic novel drawback genomic genetic complex besides genetic disease trait genetic new genetic genomic possesse work admit single genetic study second two share genetic signal solid support third produce unique minimizer solution method conduct comparison experiment method formulation selection discussion mathematically multiple relate express entry transform score study goal detect genetic variant sparsity indicate please example study identify type genetic recover genetic genomic variant irrelevant noise rank correspond causal snps disease trait correspond causal disease trait component measurement zero count intractable effectively nuclear singular surrogate rank low prove powerful prove solve alternatively theoretical reduce regularize least problem follow value singular singular value rewrite close ij refer optimize summarize global variable thresholde soft input two control probability recover state value adjust specific matrix number snps share snps rarely use step expect snps sparse absolute deviation four method method search association distinguish result meet predefine cutoff default setting use decomposition try result method precision suitable irrelevant method trait relate trait material study snps disease trait disease trait convert pc investigate work apply share snps present rank component individual snps sparse recover disease trait edu college pressure pressure density tc disease trait causal snps finding disease trait cluster together relate detect snp snp snp map gene supplementary material besides snps value moderate snp whose college detect snp method snp report value map gene severe include fold pressure pressure publish additional low snp respectively gene pressure identify causal trait snps match confirm disease trait supplementary material clearly show recognize detect snps moderate disease task disease trait discover year hundred carry systematically investigate comprehensive genetic complex disease trait disease snps divide share trait snps individual recover noise formulate optimization demonstrate method conduct several different setting method outperform many study successively datum discover share propose easily well understand disease mainly development technology annotation structural acknowledgment support grant university grant explore genome data table table simulation table four four simulation description disease trait body index height ratio adjust five spectrum tag total density high diabetes pressure pressure social college trait disease disease diabetes body disease http www pressure pressure cm attention major diabetes type diabetes htbp c c snr snr low recall htbp c snr precision f cm true genome thousand individual widely identify disease increment explain genetic variation disease miss disease disease common genetic variant explore correlation promise remove spurious identify genetic complex disease identify genetic scale genomic dataset formulate aim multiple disease trait trait convex solve dataset experimental show reconstruct wide scenario matlab code human disease diabetes cancer influence environmental health great interest find advance insight complex substantial support
example reaction increase course depend incorrect branching splitting step strongly brownian motion start brownian motion euler method time choose inverse order magnitude run highlight proper splitting version happen yield consider exactly namely maximum small reaction precisely version biased modify section work newly replica explain remark remain may pick work maximum level equal end probability eq consistent updating formula exactly resample step work small initial current level equal accordingly strict version enter increase modification increase iteration explain result estimation give stable value check standard direct carlo table e estimation version negligible probability version incorrect implementation order easily extract modification size go variant maximum see go recall variant present htbp c e e c e choice reaction efficiency dimensional langevin dynamic wiener euler denote simulation numerical scheme g condition give plot figure potential connect channel minimum saddle open around trajectory temperature lot interested time reaction associate perform run empirical confidence solid line blue circle represent low bound line evolution htp small bias phenomenon sufficiently large agreement fact unbiased reaction coordinate fluctuation lot reaction seem computed consider empirical dramatically phenomenon apparent large get reaction computation plot equivalent kind evolution confidence jump already use large reaction relate pathway relate paper one potential reaction value situation reaction pathway estimation reaction coordinate carlo sizes dynamic langevin euler depend double minima saddle whereas latter saddle decrease x reaction coordinate figure reaction green red circle reaction criterion q always average realization take test behavior go saddle evolution realization overlap reaction comparison standard monte able reliable direct reasonable realization saddle figure evolution interval realization realization realization temperature fluctuation reaction realization interval section pathway rather pathway symmetric role reaction apparent split htp summarize finding always simulation accordance result lot reaction coordinate poor average limit remark reaction thus tail interval reaction coordinate trajectory go unlikely reaction coordinate particular reaction multiple reach certain level relative channel reach maximum example reaction adaptive reaction update get close algorithm direction reliable branching resample implementation build section property check minimum reaction coordinate particular recommend simulation reaction minimal realization regime scale instance parallelization trivial namely thus acknowledgement grateful position grant grateful conduct european european agreement grateful project would many proposition heuristic method example discrete dynamic path chain estimator rare event choice practical illustrate experiment efficient reliability molecular us molecular let discretization langevin dynamic q giving position position energy configuration variance remain long call locate reality molecular event denote disjoint reach molecular path path assumed simplicity deterministic trajectory start chain reach smaller naive carlo reliable estimate example molecular technique quantity sample splitting splitting ingredient q use advance towards reaction molecular call useful requirement existence path markov call system stop remove path keep discuss generalization sup soon fit path remove replace fit remove path sample go computation determine remove remain fit path iii resample path iterate one obtains successively stop feature level remove maximum threshold iteratively fix priori deterministic adaptive splitting generally standard sequential carlo chain stochastic reason mainly discretize numerical discrete monte carlo markov context raise question context resample whether time remove exactly implementation section main appropriate implementation splitting yield estimator rare event splitting enter call framework various classical remove reduce sort procedure moreover chain numerically toy example implementation splitting numerical experiment recommendation reliable see property possible reaction quality concern number minimum reaction coordinate interpretation reaction spirit mutation resample analogy precise see reaction know resampling accord condition reach remove case practical interest condition meet monte statistical error crucial show concern choice discussion reaction large relate apparent stress resample definition suit trajectory another terminology mind static consider describe actually splitting detail write variant yield highlight property produce unbiased quantity theoretical illustrate efficiency rare event go denote recall standard disjoint borel give trajectory probability distribution endow test probability measurable associate probability transition test notation dx respectively value rigorously variable measurable element endow x n endow denote write ensemble subset replica label n replace space goal define markov unbiased treat time treat many continuous chain transition generality deterministic random condition introduce sup corresponding path endow measurable disjoint borel mainly occur region neighborhood resp resp close stop probability see interest rewrite generally allow stop time stop endowed ingredient importance reaction coordinate molecular value q aim choice unbiased estimator impose define stop reaction coordinate q stop emphasize strict time equivalent stop average memory splitting advance contribute copy fit accord resample kernel denote define law q stop identically generate probability stop reach branching resample resample modify dirac follow specific trajectory unbiased generalized splitting reaction two minimum iteration replica decrease iteration order work maximum equal keep resample procedure replica use resample kernel complete trajectory time label end th iteration construction level denote subscript retain terminology refer stop generate iteratively resample estimator observable consider set namely maximum equal union refer working label estimator position detail algorithm define initial n th order remark remark satisfied case step work equal label denote branch notice I criterion fulfil parent replica replica procedure replica branching replica old one construction observe weight replica soon replica branching replica q q eq set th statistic x time loop consist none first replace step imply stop three maximum level split resample occur representation replica small replica beyond replica reach maximum procedure replica refer replica maximum level replica square replica action replica replica iteration level work iterate one obtain even replica create level next especially discretization time process observe carefully definition test phenomenon describe illustrate splitting resample unbiased estimator bound observable realization highlight choice bx obtains contribute one retain specific observable bx give namely work investigate introduce contain refer framework splitting framework fit highlight essential mathematical produce prove framework estimator propose variant organize main framework introduce path markov analogy branching resample mutation precise variant illustrate flexibility order introduce section consistent context space assume section path chain q aim estimate main introduce two section framework structure index therefore application measurable construct consider subset z back e z convention consequence introduce stop level stop level level stop field characterize particular interest application z x ingredient field consistent resample resample probability index resample use resample continuity mapping right introduce dy consistency introduce assume consistency relation distribute eq assumption law replica necessary finally mention assumption view implicitly accord measure ii initialization step resample framework aim introduce general refer adaptive sequel section introduction successive step splitting satisfied random iteration n x become precise iii level correspond item many variant section framework adaptive splitting index measure initialize z stop perform branching satisfy replica replica old new child replica thus q q q label update child parent replica label parent replica resample procedure replica child parent branching replica set q field field sample next level assume level increment go follow iterative procedure labeling way framework field index z endow field necessary three random property branching assumed sample conditionally estimator claim emphasize requirement level instrumental optimal easily get property emphasize measurable stop section moreover replace algorithm section martingale algorithm thus go describe prove section endow topology explain define pz assumption consequence continuity property crucially open implie kernel precisely resample piecewise x ax xx x explain practical splitting criterion branching indeed branch splitting iv check satisfy requirement notice branching positive particular strict weight formula ng n nn consistent g p framework n let requirement satisfy computation actually result highlight variable subset measurable consequence fact last result assumption convention q partition I thank sure mass indeed weight induction satisfy estimator define notice actually one obtain explicit algorithm sequential monte smc sampler familiar smc method reaction value understand sequential importance introduction algorithm highlight reaction coordinate label successive rather iteration iteration system index level check standard sequential iterate step reach set accord split total level path stop define smc crucial smc resample parent section precisely resample one unchanged check discussion method comprehensive mathematical particular dynamical interpret discrete path stop iii hard obstacle reach differ presentation use change picture construction unbiased form standard smc language normalize ratio average rely smc reaction extend reaction consider discrete section variant generalize improve section particular three illustrate setting enter dynamical setting path markov design possibility level lead exactly require resample simulation variant resample define modify follow sample kernel chain random variable distribute use stop reach order markov chain stop time z additional markov chain euler langevin dynamic see initialization since algorithm sort entire idea level th flexibility parallelization notice modify branching branching number affect important spirit sequential importance bi enforce probability branching visit channel sufficiently implement branching present stop reaction enter path dependent reaction coordinate duration continuous process jump branch homogeneous stochastic etc bridge brownian interpret dx x mx distribution lemma fact natural deterministic build gaussian empirical prove content state section test function
instead arise anomaly unknown proportion mixture mixture proportion mixture set address development sec jointly optimize scale many discussion datum contamination let empirical population consider membership property convex q simultaneously observe contaminate parameter search category line distribution category dash show divided point mass numerical conduct deterministic input run show mixture conduct solve average total second implement broadly classify fit testing recent bernoulli quantify contamination address probability fisher exact solution contamination pearson limitation approximation category employ optimization category ingredient readily thus eq complete combine confirm particular assume contaminate dx b three step first write second property regard separation property close solution valid lastly separation equivalently form unique large factor let order show kkt eq simply suffice proof confirm complementary verify primal feasible lastly check primal trivially kkt satisfied range strictly approach infinity approach examine behavior constraint allow minimize increase monotone existence uniqueness fix decrease statement require arbitrarily objective realize bind close divergence ready difference must q result thm remark thm prop university identify anomaly variety estimate contamination contribution contamination control appeal contamination series program contamination goodness testing detect wide environmental science motivate anomaly contamination communication computer system application management internet broadly distinguish include dimensionality anomaly base g distributional threshold identify anomaly establish norm threshold anomaly false alarm see section base estimate anomaly level consider contamination free specify comprise distributional standard method compare contamination testing base base answer question consist empirical member distribution subset attribute inequality problem whose geometric application model number model finite lastly show category denote index empirical occurrence leibler jointly lastly probability simplex distribution category entropy mixture distribution correspond concerned specify significance sample random model contaminate quantify define statistical significance contribution herein order sample q xx iff order definition typical empirical interpret become require increase sequence typical original insight continuous create discard contamination contaminate contaminate must attribute contamination agnostic limitation case contaminate full empty significance level time consistent report zero contamination reporting proportion minimum know important grow result grow theorem exception involve category directly check contaminate alternatively deviation particular contaminate kl provide way contaminate level numerical efficiently check contaminate empirical size single check contaminate contaminate follows exclude long answer question discard kl still remove empirical create discard possibility provide twice discard violate exclude interpret number time appear discard kl empirical sample remove check contaminate note meet convention imply empty contaminate
mind expectation randomness learner literature find slowly regret environment feedback achieve regret bound least logarithmic factor bandit notable exception achieve set bandit guarantee extension set also propose version match though substantially realization sequence improvement specific arguably improvement replace round loss action standard bound take exist superior cognitive channel service answer question full aware bound show prove previous armed discussion scale size simple implement combinatorial computationally efficient similar combinatorial bandit approach perturb show appropriately tune largely minimax whenever become notice inferior bound know tune adaptive sense way bad guarantee armed bandit action e tend large e online good ad satisfactory bound worse worse increase problem consider much type provide sequence stay condition variation obviously easy construct linearly summary conclude order depend quadratic variation capture loss discussion full reference comment bandit set vector every combinatorial bandit name bandit give bit confusion combinatorial bandit learner observe tt line prove highlight algorithm distribution dependent note compare regret bound expect regret rather argument regret actually optimal refined assume expect regret argument variation explain key underlie many know bandit algorithm entropy regularization tuning depend loss prove may easy give reasoning replace keep close bind course bias challenge rate schedule information perturbation perturbation know achieve guarantee bandit satisfied action perturbation specifically perturbation variant exponential truncate tuning ix algorithm hold truncated perturbation implicit exploration parameter draw technical going proceed comment probability computable close efficient equivalent expectation resort presentation otherwise implement access efficient introduce answer deeply reader might reader answer without price however relate truncated perturbation select use exponentially perturbation vector play round particular establish total relate quantity ease upper similar important highlight generate entail hold algorithm follow integrate side md md concern state hold proof defer armed ready prove thus dl statement expectation substitute achieve bound requires trick overcome difficulty issue modify tune solely observation tune corollary notation notation ensure hold tune random known analysis adaptive deterministic largely simplify see treat performance guarantee result regret simultaneously nonnegative together hand ready theorem plug equation expectation jensen prove bit care rule bound eq solving auxiliary equation truncate perturbation allow step forecaster lemma result replace provide bind term assume prove final lemma quantify proof observe discuss implication extension really hold truncated perturbation perturbation bound along line arise perturbation additional become note induce paper essential result high suggest would handle corollary proving confidence variant leave future acknowledgment high education research author thank france optimization repeatedly combinatorial loss associate learner action propose improve scale loss action feedback combinatorial combinatorial feedback
intercept constant f trial count choice iterative initialize reconstruction implement range grid reconstruction fig reconstruction realization regularization optimize reconstruction visually several weak may attribute element average realization poisson total count minimize total increase fig reach continue improve thank sparsity constraint employ improve reconstruction objective fast time reconstruction measurement identity adopt link intercept solve criterion performance integer reconstruct substitution toeplitz element constant select element gate average cpu show normalize measurement yield reconstruction unknown small compete almost second well similarly large performance unknown time slow known omit density simulate orthonormal haar level circular mask sense transpose fan platform circular mask ray rotation center platform image pixel choose space vary correspond collect detector ray detector maximum element number projection c reconstruction projection take show reconstruction projection reconstruction visually reconstruction list reconstruction signal truncation truncate thank projection well bring take minute converge reconstruct nonnegative transform proximal scheme nesterov acceleration propose iteration decrease computationally state reconstruction signal discover crucial size handle fidelity constant avoid convergence develop sparsity poisson remarkably focus incorporate sparse ray maintain publicly package unknown substituting q upon ignore maximize ignore establish convexity remark concentrate hessian prove positive apply cauchy schwarz eq q r label step invertible except subgradient minimize solve proximal objective construct ordinary optimum optimum dual minimize algorithm j proximal close use get direction since make large step take scale relax orthonormal linearize linearization regardless indicator quadratic denote instead linearize ordinary condition english english proposition remark edu accelerate proximal reconstruct nonnegative motivated signal nonnegative represent material hyperspectral band adopt data fidelity indicator accelerate account vary provide numerical accelerate construct sense reconstruction gaussian generalize wavelet achieve method reconstruction compressed compressive paradigm transform domain number much appropriate signal vector negligible magnitude idea compress value p noiseless measurement compress focus nonnegative encounter hyperspectral dna monitor hide see transform nonnegative practical application ray ray correspond material map activity pixel concentration region interest nonnegative transform domain recently consider linearly toolbox adopt difference onto computationally impose hessian norm poisson link advantage paper also adopt unconstrained minimize scalar constant quantifying term impose orthonormal c transpose zero identity logarithm soft I follow poisson often adopt optical hyperspectral count particle detector poisson c mean identity ignore term linear summarize function hessian identity identity count optical emission deep light physics identity scan adopt identity link simplify intercept link function account imaging refer treat disease mapping combine norm intercept unknown replace ignore concentrate appendix regularization constant define certain intercept thought nonnegative constrain problem integrate toolbox substituting linear propose nesterov numerical conclude whose u proximal discuss nesterov acceleration achieve iteration second momentum accelerate iterative denoise perform impose size satisfy acceleration iterate monotonically place monotonic carry accelerate discuss j j usually set obtain sum exact orthonormal appendix linearize see onto nonnegative yield objective wish outer consecutive return index respectively within criterion I I j sufficiently improvement loop achieve compare loop return sense outer size backtrack inner splitting part threshold identity interest identity second derivative decrease global constant indicate coefficient minimize large step towards noticed analytical size design seek conservative iteration size consecutive iteration otherwise multiply keep subject signal reach type without increase step would fail constant algorithm illustrate advantage adaptive backtrack signal sparsity employ equivalent approach employ scheme modify monotonicity condition accelerate scheme regularization desire return initialize especially parameter usually unlike decrease convergence initial final indeed range parameter keep close reasonable scenario identity repeat u map intermediate threshold c guarantee ensure decrease together reduce intermediate general inspire adapt intermediate example follow explain regularization hold ref regularization constraint converse hold minimize appendix discuss implication motivate serve scheme thank use focus gradually decrease convergence threshold among realization fig reconstruction sense optimal average imposing improve greatly metric impose sparsity final minor reconstruction reduce fig omit report time partly matlab measurement realization sense matrix measurement group traditional signal sparsity separate achieve thus incorporate via active signal possible implement line truncation group bring method regard fail reach reconstruction much employ explain valuable small converge achieve convergence start fail black color solve fast within competitive term attribute step explain similar unstable behavior objective function benefit convergence scheme thank adaptive run run convergence occur complete reach phenomenon trial completion explain threshold section consider ray conventional image generate matlab orthonormal construct haar decomposition full circular mask transpose operator platform circular mask detector array set initialize reconstruction store allow storage method implementation accept vary achieve fig show average compete fig two method group ii reach upon fig projection fix choose equally vary coincide projection projection fig time coincide benefit signal fix projection achieve projection achieve accounting bring significant benefit time projection time perform poorly projection consider converge gap employ show reach low reconstruction similarly reach reconstruction time fast reconstruction group consume gain reconstruction increase fig converge slowly perfect fail achieve objective compare realization projection beyond point proximity minima convergence threshold reduce far result decrease convergence signal separate method reconstruction reconstruction space realization show gray start reconstruction true subsequent plot suffer inferior nonnegative group signal sparsity reconstruction impose sparsity achieve quality pt mm adopt sense stability generalize divergence small cause nesterov acceleration make infeasible would address logarithm q observe continuous step size parameter minimize regularize model applicable package option fitting
fdr aggregation conditional uniformly symmetry interpret shall expect likely original odd enter early also hypothese hypothesis likely early improve sequential decentralize center normal variance randomness filter pilot order value aggregate hypothesis let summary statistic measurable evidence correspond particularly interested function decrease condition include x mx x iw one size order place aggregated simply number behind feature nonzero binomial translate refined far summarize filter get shoot communication aggregated statistic fdr call false rate fdr special accept reject rule introduce randomized assign hypothesis close fdr motivation fdr close whereas randomization undesirable practical fdr potential find internet randomize decision occur frequently randomization testing g mention refined statistic motivate nan matter away false nan nan jointly binomial stochastically generalization incorporate obey interested reader nominal eq convention reject hypothesis present control set summary function q lemma parallel proof lemma let martingale run backward know variable proof theorem idea rank argument prove without get eq stop time stop theorem stochastically would control interpretation favorable reader refer attractive translate control information improve still control aggregation communication decentralize send sign piece encode bit require logarithmic respectively original median low control fdr maintain big instead decentralized simple achieve vanishing start column far extremely augment design orthogonal take cost otherwise uniformly cube denote reject last tend aggregation slowly nominal obey hence aggregation capable distinguish move send model center hypothese piece message non interactive bit budget interactive hold tend exponent constant coin hamming hence signal design strength row draw length randomly level save hold take scenario universal detection access decentralize allow summary simulation ip achieve level vary potential case meanwhile aggregation least square ol procedure estimator obey pm ignore value hereafter choose send center majority vote share signal illustration show sd ols lasso validate e fdr despite validation still term satisfactory aggregation ol nominal though aggregation around nominal ol spread sometimes information contrast ol undesirable aggregate run decentralize aggregation enjoy fdr provide exhibit address framework cover broad link room investigation fashion last interesting incorporate lead much strong manuscript false hypothesis similarly hypothesis conditional concatenation proof respectively binomial extensively use know hold without hypothesis agree exchangeability proceed jensen reveal obey right rhs attain obeys give jensen give rhs shall decrease consequence remain hence tend permutation rejection rule take almost vanish generality replace make extensive eqn sufficiently eqn give shannon consequence move since exponent obey next proceed finish nothing v pt theorem proposition theorem definition lemma remark section stanford usa control false spurious target possibly manner start filter shoot fdr asymptotically signal setting scientific significance summarize explore big number difficulty hypothesis simultaneously sophisticated tuning arise chance care address community decade fdr elegant fdr
furthermore provide hill simple problem asymmetric insight experimentally asymmetric likelihood markov stock index flexibility introduce also insight entropy organize constrain asymmetric normal condition compare asymmetric normal one real world advantage finding characterize existence underlie split segment must base provide segment continuous weight describe simplify associate create laplace constrain x pdf argument partition describe continuity pdf mixture guarantee preserve build force pdf place side map constraint underlie pdf redundant use volume unitary place part describe non pdf split base exponential parameter behave therefore produce stack sufficient produce stacking hope without fix partitioned separate parameter mixture asymmetric introduce laplace normal version optimize appendix separation place mode follow avoid mixture laplace prove generalize asymmetric laplace constraint give respectively arrive laplace define family equation match laplace get close negative less exhibit describe asymmetric let q satisfie appendix arrive normal asymmetric define exponential show asymmetric case adjust model involve indicator belong current markov asymmetric identify parameter specifie give make simultaneously optimize either formulation likelihood optima closed optimize numerically describe alternatively define exponential show sound new intractable approximate conjugate therefore optimize important highlight particular likelihood give equation entropy divergence therefore third likelihood depend move implicit come define increase fitting loss become asymmetric asymmetric optimality pdf optimum weighted median optimal look check partition optima median create value outside require alternatively consider show conjugate give equation equation asymmetric conjugate density give gamma gamma distribution laplace format write distribution parameter bernoulli beta gamma parameter hyperparameter link way partition optimization asymmetric pdf normal give optimum similarly asymmetric look partition valid similarly laplace instead asymmetric define equation asymmetric conjugate asymmetric give equation prior eq beta distribution agreement prior variance prior section show maximize likelihood hill initial tolerance adjustment hill algorithm go algorithm keep compare move direction maximize likelihood repeat asymmetric example choose fix behave hill allow change able avoid first solve minima flexibility asymmetric new application version understand explore deeply normal frequently standard therefore asymmetric able adapt gamma regression may user may people familiar asymmetric normal since use emission flexibility linear parameter compute offset asymmetric asymmetric weighting c figure asymmetric straight relationship dot since standard fit point concentrate simulation value uniformly set symmetric fit dash line since describe symmetric symmetric case likelihood asymmetric asymmetric higher perform however symmetric asymmetric likelihood asymmetric fitting decrease asymmetric explain distant incorrect noise model equation prevent error weight equation prevent inherent equation impose much line confidence clearly far parameter sample like instance draw estimate specify noise outperform motivate asymmetric able interpretability insight use state hmm asymmetric improve initial estimate hmm emission iteration every emission asymmetric hmm method describe last price consider distribution economic compose logarithm consecutive day sample th day happen sample miss day symmetry normal may reflect reality market return market expect introduce emission asymmetric state asymmetric flexibility state increase subsequent symmetric asymmetric b emission subtle component large preserve considerably certain none symmetric indicate likelihood appear b evaluate become fourth transition entropy clear transition c miss entropy reduce entropy occur state histogram normalize entropy divide hmms considerably state version version less spike emphasize figure quantile represent identity high first reach latter almost always equivalent quantile low additionally curve consider indicate asymmetric lack introduce use normal create new asymmetric underlie keeping prior moreover distribution inherent term regularization directly likelihood impose avoid symmetric underlie compare asymmetric understanding operate symmetric version asymmetric must asymmetric normal hmm stock flexibility allow distribution
encourage powerful et al et purpose logic logic al coding kind knowledge subsection arrange representation spread representation representation min percentage representation et b et et hyper sphere representation et hull logic fuzzy fuzzy fuzzy et et al al representation stack b genetic programming x fuzzy representation group category present rule schema format correspond representation partition space result paper apply value part define interval handle range interval represent tuple value match classifier argue many vary interval phenotype phenotype mapping perform mutation operator truncation must keep bias frequency appear condition representation name value match dimension truncation comparison frequency interval issue order operator might produce infeasible order previous name phenotype one mutation provide pressure unlike mutation pressure raise inconsistent block architecture ga run al present name maintain encode et problem equivalent unlikely change htb difficulty boundary long successful extension classifier hyper condition axis advantageous new condition base sphere shape sphere common condition deviation axis parallel condition hyper axis name transformation indicate fully ellipsoid represent ellipsoid center ellipsoid transformation ellipsoid diagonal initialize zero hyper angular represent ellipsoid encode mutation operator act al decrease successful evolutionary investigate condition rotation hyper ellipsoid three angle hyper ellipsoid highlight mention htb activation center current activation use match parallel ellipsoid hyper respectively promise approach report al continue general condition dependency representation note suited boundary parallel effectively another represent value concept hull depict lie inside form representation convex hull fine complex region asymmetric htb represent hull hull present angle hull hull variable sized condition hull base representation fast condition classifier consist rule base system interpretability must considerable besides fuzzy logic mechanism fuzzy goal behind effort capability fine fuzzy online briefly comprehensive notable approach fuzzy logic integrate fuzzy fuzzy message list researcher fuzzy fuzzy fuzzy task al many reinforcement reinforcement et reinforcement relation membership especially control et al address classic competition versus fuzzy style name divided produce classifier base apply reinforcement agent et learning framework call analyze model classify literature al al fuzzy logic rule format issue system successful ability proposal et name fuzzy mining comprehensive description fuzzy fuzzy fuzzy represent label fuzzy evolve consistent fuzzy rule mapping real nominal common rule consist fuzzy input linguistic operator linguistic meanwhile represent linguistic classifier code schema integer code schema action propose bit linguistic dimension dimension linguistic part classifier bit linguistic bit appearance linguistic utilize linguistic show portion input cover linguistic rectangle htb example fuzzy visualization area fuzzy classifier white part fuzzy feature offer miss fuzzy support absence consist promising robot simulation online learning increase adapt fuzzy well issue modify predefine fuzzy linguistic employ produce fuzzy modify fuzzy manner original fuzzy linguistic knowledge well linguistic fuzzy technique modify inclusion cause improvement expression environmental well understand condition classifier value exceed add feature translation ga representation phenomenon happen occur important evolution ability illustrate like partition expression tree form accord different namely encode phenotype classifier way classifier form seven encode gene initialize terminal arbitrary visualization cover area phenotype advantage fine environmental linear tree also verify validation operator attractive make gp like representation report show environment slow compact dynamical representation term dynamical graph condition wherein boolean present node input node correspond input connection common benchmark computational rl scheme name classifier like perceptron mlp condition ga mechanism besides number node rule see extra whether member member correspond high test single step name show addition examine root fuzzy radial component evolve scheme namely ability problem continuous action promise include also applicable single hybrid propose ga self adaptation explore system well adaptive improvement optimally complex one al replacement make advantage ability issue name drive mining maintain predictive part encode predicate et well structure space htb space visualization highlight et define produce cover hyper rectangle extend code code common successful tackle learn space map overlap partition input hyper prediction resolution ga replace match specify input partitioning suggest extend test take car evolve reach converge technique strength effectively system suited identify problem area proper whole complexity effectively problem setting relative handle problem category worth category boundary aim kind solve environment word researcher propertie knowledge string interval good boundary htb positive instance white negative instance boundary representation technique kind simple popular technique might classifier space ability parallel boundary continuous action handle round interval independent method use produce recent htb region white problem properly consider boundary contrary hull representation decision due shape flexibility cover model obviously ability cover unknown dimensional p applicable environment implement well contain boundary limit value integer value mining al medical mining al data set et al suit wherein dependency dimensional shape applicable real value environment length complex generalization interval convex ga fuzzy fuzzy applicable capability maximal set interpretability fuzzy support action handle mixed attribute miss produce limited freedom flexible adapt mechanism et reinforcement al present among ga operator relational well suit nominal able offer ignore input add relevant applicable value offer great ignore add step datum free able produce interpretability every accept necessary multiple et hard number problem problem string one attribute real subproblem value value attribute subproblem solve mixed attribute representation attribute gp like fuzzy technique mention representation remarkable provide table representation main advantageous domain test technique exist far improvement knowledge seek prominent capacity efficacy decade interest propose representation grouping incorporate category representation schema representation technique partition extensive knowledge illumination domain comparative analysis hope stream research usage since representation like properly survey interest researcher practitioner choose knowledge stream research key rule include insight partition turn prominent generalization whole attention mining community efficiency find comprehensive yet elaborate knowledge group incorporate schema format support precise technique partition extensive experimental provide view technique comparative interest researcher practitioner research topic framework cognitive year originally general principle evolution cognitive framework ensure reward evolutionary population production genetic reinforcement technique production system population methodology address classifier system inspire university de classifiers system usually university member ga complete population form mechanism reward tool lose rise reinforcement agent simplify use reinforcement accuracy system classifier system wherein fitness rule use effectiveness paradigm extend wide computational economic mining light control al al diverse ever valuable domain author try research indeed aim development et give past try look ahead system various domain focus sequential introduce historical major algorithmic difference principle interested develop specific good exist try common consider provide research area might efficient enough progress regard handle dedicate survey individually focus modify give consequently survey attempt description knowledge attention mining term efficacy identify environmental support system attempt elaborate exist knowledge incorporate schema precise additionally comparative exist conventional current providing choose problem address none trivial issue hope survey understand stream research choose representation interest researcher practitioner stream provide description describe incorporate subsection schema format explanation comparative conventional general include conclusion remark system belong combine genetic ga paradigm solve specific representation component provide environment reinforcement responsible incoming discovery create mechanism evolve ga idea keep notable solve compatibility develop style environmental detector environment eventually reward select action efficacy reinforcement try receive produce current reinforcement subproblem subproblem usually use evolve review mark appearance successful popular recent et show online mechanism continuous classifier call empty beginning randomly different condition action integer payoff apply fitness etc receive environmental match set environmental classifier match operator predefine current
design small advantage increase design reduce optimize advantageous large fill dimension situation preferable remain computationally expensive variability specifically application random compute eq nd measure variability n take lebesgue functional distance distance variability variance cause uncertain quantification simulation fine grid estimate method generate fashion quality grid thus possible two select possible precision simulation simulation simulation grid distance distance transform transform variability equation benchmark realization realization simulation conditional field simulate time design use obtain three grid field krige algebraic mark benchmark grid obtain interpolation simulation obtain simulation approximate enhance example show substantial cost optimize b well point optimize benchmark behave measure reconstruct point optimize criteria b time optimize interpolation total approximation benchmark could point results optimize show value reconstructed obtain simulation point blue dotted level denote represent variable kolmogorov kolmogorov statistics test distribution optimize point rejection distribution optimize point estimate volume reconstruct computational reduce cpu simulate design total second intel ghz gb ram simulating realization however random fine impractical fine approximation realization conditional literature define distance variability variability appeal grid show cost uncertainty quantification could improve volume monte carlo simulation fine design attain volume six indistinguishable attention regularity predict design predict realization correct center issue need bias field approach extension optimization generic box analytical dx dominate prove pick almost everywhere conversely zero word everywhere almost density standard thank helpful insight remark lemma quantify uncertainty evaluation budget adopt objective function realization rise approach analytically tractable expectation variability carlo rely random choose realization fine design predictor computational cost simulation enable prediction dimension fine grid uncertainty volume six process number application behavior practitioner get input forward prescribe inverse problem scalar quantifying equivalently pay reliability engineering describe configuration lead nuclear science phenomenon costly evaluate typically consequence systematic space fine reach reconstruction interest evaluation rely net mainly focus modeling become evaluate rely drastically reference therein approximation enable quantify uncertainty conditional idea appeal context estimate like reference therein realization simulate field location question arise notion scatter variable context quantification present base field carlo often obtain simulate fine design need especially choose approximate predictor obtain set unbiased predictor point introduce way specific reconstruct set possible divide section introduce formula present explain introduce limit advantage present optimize simulation application show g approach allow transform uncertainty quantification transform heavily quantification dimensional volume monte method unknown reduce computational time definition come theory close main continuous objective df paths kernel range borel tt simplicity generalization interval pre image concept theory notion role approach algebra cs one expect distance borel close particular notion measurable review another expectation notion variability empty set define appropriate space close respect variability let infimum say addition define notion heart discrete simulate e moderate dr become impractical involve rapidly consist fine simulation rely basis quantify uncertainty actually end trace fine simulate propose replace remain construct way expect evaluate essence affine predictor deterministic krige predictor case review simulation krige context notably uncertainty complex address purpose criterion measure borel eq mean bivariate krige ne k ne ne eq random conditionally prove conditional moment tm ne ne e ne conclusion field suffice nk ms simulation consistent necessary proposition established select ordinary predictor algorithm find assume simulation fix package characterization field several technique already design simulate approach lead rely analytical gradient slow follow heuristic optimize previous current add reach bivariate time optimizer rely krige krige model sequential krige formula new approximate bivariate bivariate package fast standard bivariate cdf
possibility one employ hardware device restriction small sized baseline ann describe neural wireless able prediction come layer output activation ann output hide step wise step activation available weight bias equation vector previous consequently validate database real accuracy demand simple architecture hardware wireless network promising advance wireless physical wireless capability provide application monitor health physical environmental ambient network sensor home environment house develop home paper temperature home resource study european primary energy demand consumption home half consumption air conditioning thus major overall necessary develop home demand efficiently consider plausible artificial intelligence soft compute artificial learn apply wide devote system problem normally historical traditional back propagation could minute hour number stopping criterion receive datum consume lot mention nevertheless could trained totally learn successfully regard bp application line arrive soon observation discard without necessity store historical imply necessity storage prior many generalization learn computing resource idea integrate technology cost concern idea resource variable consumption sequential objective hardware device intelligence forecasting environment cost far monitor ann ann require storage whether feasible ann resolution estimation short period innovation bp time income feasible device hardware resource hardware describe work wireless topology slightly forecast back bp neural depict implement resource experimental conclusion explain draw future power environment devote monitor current activity control actually embed capability distance wireless medium thousand sensor typically part internal circuit embed device sensor node also sensor sensor correspond memory communication advance mesh propagation technique network besides unique characteristic constraint state dense sensor sensor impossible change severe constraint resource focus sensor node application sensor physical failure topology change topology node channel identification address overhead identification traffic pattern multiple certain requirement different objective capability typically influential design sensor node cost low consumption self reliability tolerance security wireless acquire usually devote collect come computer store persistent device want acquire pc network node wireless network add figure display pc validation purpose four sensor capture room allow power extend four sensor technology tx cc ghz system wireless combine excellent rf enhanced memory kb ram powerful limitation cc suit consumption several advanced operate systems wireless configuration also power within temperature security transmission exchangeable four sensor room temperature ambient calibrate digital point temperature two general output expansion sensor addition reach depend connect hardware sensor wireless sensor sl monitor control security alarm sensor implement learn advanced architecture nevertheless highly accept lot area market control moreover excellent application device competitive really device implementation consumption record forecast future energy resource record repeatedly record interesting formalize scalar process appropriate model internal consider forecasting method smooth fit autoregressive move restrict storage much possible forecasting linear perceptron perceptron mlp purpose comparison perceptron estimate result standard storage complex computational expect however comparison assess base device propose summary difference vector measure mae mae several mae sample paradigm frame deal frame modify time failure raw incoming frame hardware resource buffer incoming time variant back propagation probably noisy computation high model compare batch bp random traversal well series statement implementation device algorithm skip allow ignore incoming depend skip additionally powerful device buffer sample also skip rule behavior take consideration least training scale tackle adopt belong input unique capture solve incorporate regression response predictor concern predictor non complete one condition variable could additional column consequently coefficient write represent identity predict receive value series must element simple forecast consequence build need start represent assumption auto correlate use incorporate feedback violate introduce lead biased consider skip primary building estimation prediction acceptable storage perform scientific absence available assume bayesian could incorporate parameter context estimation informative prediction predictor first demonstrate solve inverse employ term storage remain expensive must future way introduce estimation context informative datum utilize treat additional last assessment previous follow read represent thus predictive assessment high process storage cost high hardware concern standard model simple computational resource requirement model ann consider device resource train ann state propose sequential bp compare baseline perceptron mlp matrix bias input ann describe bp ann layer moreover influence bp kind need algebra operation dot product logistic activation implement library requirement implement ann element bp real perceptron real mlp hidden output correspondingly bp number resource bp mlp implement ann memory data temperature experimental node room house continuously hour pc mainly configuration place room home central control energy purpose competition loop show start initialization minimal rf interface issue wireless protocol rf aim rf aspect correspond ann establish random ann system start decide core main receive sensor sensor place room temperature among message come wireless subroutine receive temperature average min ann average one ready main loop negligible requirement use pass consumption communication depend frame implementation goal wireless h subroutine display subroutine receive iteration responsible second happen call compute temperature nature consider implement aggregation temperature aggregation consecutive integrate datum pair second use store call time miss bt store return case consecutive consecutive different non lose case temperature slope compute intercept begin line segment second increment temperature complete mean value subroutine frame system negligible use aggregate lose temperature subroutine compute consecutive auxiliary buffer length buffer control forecast buffer counter equal buffer counter use static call execute compute increase unit success circular equation condition check buffer forecast follow cumulative add input call static circular buffer number memory consumption whole bp control add current counter need buffer buffer size assume position buffer least matrix initialize start use different model iteration forecast add algorithm forecast state train aggregate second system plot mid window receive delayed value forecast condition air temperature relate receive pass directly study complete first generate large baseline real application uci repository dataset house world competition energy structure ready explore simulate consecutive randomly modify
mt mt sentence mt sentence limit corpora direction use diag final balance gram portion million chinese stanford trees vocabulary frequent chinese english approximately map special token descent train minibatch word gram lm cnn convolution layer dnn multi perceptron softmax insensitive gram baseline decode string configuration cnn target target side hereafter test hierarchical phrase base dependency integrate table proceed give report clearly cnn cnn baseline average mt indicate decode worth informative fact avoid propagation alignment cnn cnn win table signal complementary cnn cnn cnn head cnn pooling cnn pooling cnn strategy max extent max pooling max pooling replace local pooling layer pooling layer guarantee clearly see pool pool well conjecture relevant translation mainly source sentence pool seminal text recently context word model clearly relate work instead ad hoc window cover model effectively leverage form sentence cnn rnn decoder representation source therefore inferior directly integrate apply decode signal weighted sum proceed importantly nonlinear retrieve summarize source propose devise consider linguistic enhance cm lemma proposition conjecture conjecture thm remark institute technology chinese com adapt centre school city recently neural gram systematic treatment convolutional decode design architecture part target source form representation language word fed network dnn strong experiment english task achieve improvement space source language attract much statistical translation model neural sentence encoder decoder encode part decode process notably model gram achieve architecture dynamically cover entire sentence effectively part information language decode convolution architecture sentence unified representation together fed dnn purely decoder cnn signal decode joint art chinese english translation show able improvement outperform cnn b cnn start overview convolutional key decode experiment report cnn cnn predict language probability target source p stand cnn index source stand cnn figure translate target proceeding word gram lm source word source cnn generate cnn g generic architecture encoder basic generic cnn encoder six form length shorter put begin convolution layer simply sum feature window size global final convolution operate slide window carry high sentence ff ff eq previous cnns nlp take convolution fusion select value map soft template keep convolution separate release score convolution composition overlap window map convolutional layer segment strategy merge embed logistic model windows layer parameterized assign weight representation train along embed proceeding target dnn soft word procedure corpus seek maximize one parallel optimization perform back propagation batch cnn describe layer cnn extra embedding indicate word treat regular embed tag parameterize supervised predict propagation learn put make stand adjust cnn predict learn alignment unlike tell location cnn proceeding encoder retrieve essentially attention alignment attention generative network rnn decoder basically proceed window specifically window index convolution dnn sigmoid retrieve sentence transform retrieved segment representation word target language cnn proceeding provide complementary verify decode improvement purely integrate decoder adopt integrate gram hierarchical extend include index word align align source word integrate joint dependency translation decoder efficacy describe dependency mt art dependency string dependency string employ rule head string head tree top
empirical max convolution nearly identical max method even fairly rough fast second second naive approach speedup increase dramatically term essentially viterbi claim state track train vertical viterbi right track viterbi fast affine nearly exact substantial particularly multipli particle observable universe slow speedup basic approximate chebyshev accurately suggest possible compute contour value real real magnitude numerically approximate convolution history long would interesting rational rather highly increase precision arithmetic operation small increase optimize error runtime base contour search contour base optimize trade runtime max convolution affine modification matrix vector tensor likewise element multidimensional norm norm enough piecewise multidimensional convolution tensor likewise via row transform demonstrate fast numerical convolution method run tensor without speedup tensor convolution considerably tensor reason speedup become even increase dimension width cost convolution mean graph adjacency use distance node concrete example max convolution naive compute tensor replace argument size convolution likewise additive transition multidimensional allow dimensional american deconvolution computing already stable denominator large piecewise method absolute piecewise piecewise bad contour correction negligible must nearly vertical scatter plot exist index create scatter correct conversely entirely contour zero achieve requirement simultaneously point absolute individual scatter bad case affine absolute absolute code convolution numerical would thank suggestion max closely convolution convolution occur field approximate chebyshev derive error propose bound viterbi markov approximate viterbi image calculus equilibrium generate wherein probable identifying protein convolution operate semi mean identically convolution operation convolution min convolution operate semi convolution effort quadratic max vector function analogous nonnegative max convolution find l max convolution equivalent find probability event version subsequently numerous small drawback category accurate method worst conversely type bind category solve convolution either complicated sorting create numerical solve convolution equivalence process index vector chebyshev norm step choose yield fourier algorithm date demonstrate good task particular convolution efficiently probable discrete probabilistic generalization sum hard evenly spaced possible value analysis formalize closely note value multiply back conversely suffer less perform numerical bound worst demonstrate begin compare analyzing method improve variant use max summarize introduction max give numerical convolution two parameter scale maximal standard still numerical error maximal chebyshev exact note converse numerical achieving performance significant convolution high method cutoff previously method increase empirical numerical stability boundary formalize convolution argument fast convolution approximation stable boundary replicate max perform source see depict left mode mode large index regardless occur go match tolerance piecewise implementation make accuracy result dominant convolution employ large choice characterize well approximate unstable unstable pose improve give number runtime respect term maximum introduce compare high piecewise method convolution vector value estimate r I result l derive scale mention refer scale demonstrate problem stable fast convolution easily element chebyshev norm p u p denote one element simplify therefore reason bad contour bind piecewise step achieve desire practically particle observable universe full approximation stable many method stability vs index tight elliptical contour slope contour lower depict scatter plot exact piecewise every exact correlate contour slope approximate contour slope contour bound constrain scatter plot point inside envelope figure previous ideal affine contour exploit correct bias small within contour f u max single index result via naive quadratic long index already cost small contour affine compute correct contour specific trend approximate exact numerical estimate error use possible return convolution slope I I contour contour I I I r combination absolute affine piecewise also qualitatively point affine choose manner value value propagate affine thereby avoid error affine absolute affine affine transformation dramatically error fast negative viterbi additive transition arbitrary property compute variable viterbi algorithm exploit self small specialize hmm leave max vector multiplication backtrack thus enable left pass note modification perform complex pass vector pmf max matrix use describe return viterbi valid j additive transition economic country price predict figure
communication allow error begin input additive implie perform subset matrix argue construct define associated row let construct verify eigenvalue canonical consequently u I suppose randomize introduce shorthand thus compute solve study show bit scaling suggest establish scale complexity achieve rank tight question special compute easy algorithm communication rate prove tight low special many precisely rank compute large give rough connection observe rank matrix determine sum computing reduction series binary search whether turn goal testing psd testing reduction machine inclusion coin complexity moderate amount definite vector norm decide note minimize solve particular ix strictly solver allow importance rank estimation characterize communication acknowledgement partially grant office partially laboratory research office contract grant nf fellowship monotonically refine eq summary chebyshev guarantee yield th row subspace orthogonal th great suffice satisfy sample dimension loss dimension expand projection expand reach projection note project sphere subspace q define since combine claim completeness degree variable replace compare conclusion parameter exponential apply last exponential notice occur since plug proposition definition ccccc zhang berkeley electrical university california berkeley hold separate machine deterministic solving must randomized bit match demonstrate semidefinite eigenvalue generalize large analysis expensive order determine determine robust pca collaborative filter algebra scale divide approximate number locate determine motivate set decompose matrix store estimation formulation suppose want determine aggregated root determine rank paper computing exploit law decomposition entry exact limitation delay bottleneck efficiency reduce communication moderate eigenvalue chebyshev polynomial pass cost degree chebyshev polynomial algorithm establish derive efficient result deterministic communication deterministic approximate algorithm able corresponding algorithm randomize approximate bit relative bit establish communication low randomized eigenvalue randomize easier long history seminal characterize communication algebraic question low task non arbitrarily requirement practical applicable allow inexact practice li test inverse solving linear well know whether related streaming distinguish model generalize rank well semidefinite denote give generalize usual rank motivation terminology assume machine sum store distribute protocol machine exchange arrive close sense bind strictly eigenvalue close distinguish basic communication complexity see book detail standard party communication etc input string string communication scheme player common read protocol consists construct early player information communication protocol bit deterministic deterministic broad class protocol allow public randomness access string message randomize protocol correctly least probability framework minor define correctness public set master communication notion correctness model protocol error rank protocol satisfie definition deterministic study quantity allow contrast communication input substantially hard round discuss sequel discretization little communication complexity devote consequence trivial essentially assume bind deterministic surprisingly large scale matrix analyze encode receive order bind party holding suppose substantially slow composite function stage recurrence prove numerically stage substitute evaluate overall qx ia chebyshev expansion polynomial generate compute fa repeat overall combination generalize q logarithmic pre factor experiment algorithm practically suppose receive point uniformly random datum ccc eigenvalue b version estimate matrix nr ji sample sum generate choose choice motivate degree repetition let output evaluate square run experiment distribution eigenvalue eigenvalue generate show centralized set panel achieve communication efficient distribute case approximation chebyshev composite replace pass x chebyshev expansion method substantially communication bind spectral interval rank satisfy section true communication match choose bind communication achieve length answer code length open interesting natural machine deep investigation party communication say obvious currently upper bound lower defer low state hold sum operator rank addition mutually exclusive alternative hold use particular reduce two machine hold achieve rank otherwise conjunction test bound say orthogonal orthogonal appendix shorthand n conjunction lemma side inequality orthogonal use stre binary string communication perform reduction string encode communication complete proof psd find ib bit entry define final inequality yield recall q
replace represent entity abstraction abstraction abstraction word belong abstraction name type abstraction ability next abuse x abstraction part contribute sub describe match weather paragraph meaningful pattern mining graph discriminate one roughly evidence efficient success fortunately abstraction uniquely minimal direct mining reduce jointly side mine avoid find match remainder however mining abstraction grow graph discriminative ability start simple tweet recursively grow size side remove pair threshold grow efficient pattern give algorithm tweet extend nm nm nn nm mine abstraction abstraction name entity resolution grow counting replace entity appear side group pattern x stand merge give match abstraction stand similar abstraction x note tree correspondence sentence occurrence explore superiority tweet fig assign high spurious york occurrence mining pattern relationship text match text contrast text translation translation dependency deep incorporate determine text diagram text obtain look dependency convert binary text match learn neural network raw layer suit dense continuous building sparse demand take refer connect approximately underlie much go representation architecture sigmoid hide significant triple objective e margin control margin measure match basically tweet calculate candidate select score one original good chance average tweet translation tweet variant model calculate text representation since tend tweet text word text perceptron mlp concatenation input topic network layer neural logistic pattern input view roughly pattern embed base make trained descent insensitive mini present report large remove architecture performance setting detail performance architecture match regularization dropout prevent influence salient hidden generally architecture deep bring improvement slow margin one pattern l baseline test vast gap pattern embed fairly one one performance drop dramatically versus nine maintain drop contrast suggest space certain match case reliable fold hyper greatly improve suitable pool accuracy represent show vs observation partially deep cc response determine role abstraction improve p vs real cc want response candidate abstraction stand name entity role assign specific filter mining processing use matching extend notion match subgraphs domain common subgraph capture hierarchical relation pattern simply type string generate pattern learn difference consider tree discover vast match dnn task tweet margins work china national cb liu partially support ce adapt centre city university institute computing centre next city university translation answer matching sentence text propose approach consist mining discover matching text define product dependency text build matching tweet social medium hard match outperform margin central importance problem language processing formalize match two text match information retrieval matching modeling translation sentence meaning need appropriate response neural suited processing embed building embed answer short text match text represent relation sophisticated text correspondence hard capture embed study short text matching name discover subtle corpus pair short text network dnn decision text pattern task tweet chinese
north anchor dense north south white softmax softmax north south white north anchor south dense mm north anchor softmax softmax mm north south cnn north anchor south softmax north south font sep height text fill draw inner sep minimum height center fill outer pt sep minimum cm cm text outer sep inner width height text center bend right cnn cnn cnn xshift cnn right cnn xshift cm cnn east pool pool cnn north west south pool mm pool anchor dense dense north anchor south softmax softmax north anchor south font rgb rectangle inner sep center text height height cm sep fill rectangle thick bend bend cnn cnn cnn xshift cnn cnn cnn cnn xshift cm xshift cm yshift anchor yshift height minimum anchor south outer xshift yshift cm xshift yshift rgb pt width text center outer sep height text angle sep true height inner black inner width cm height cm text black fill inner sep black rectangle inner sep text center fill thick bend bend conv right xshift west west minimum conv north west anchor west conv right xshift xshift west east north south width conv north anchor west dm dm xshift right xshift cm mp dm north south xshift densely thick fit mp cm south north softmax north anchor south south anchor anchor right east inner sep west anchor cm base frame architecture network span video recurrence pool mp temporal briefly recognition overview depict figure pay fully connect number cell base individually mention optimize architecture model cause difference hyper preprocesse single architecture work video nevertheless much image contribute convolution layer stack overlap shorthand architecture denote map layer layer show promise second exploit strategy suggest ng et connect across video network window event lose across frame core rnns create memory temporal conventional recurrent build frame wise recurrence direction formally microsoft depth sequence user record front camera performing stand pose stream microsoft sign mean contain several performance pose vocabulary dataset kind video minute sample hz include varying position imbalance frame annotate contain skeleton show achieve good depth end architecture mini work mini exponential rate initialization describe file consist recurrent produce summarize channel shorthand optimize cell model base rnn generally locate frame correctly forward backward feed frame frame cnn frame recurrent optimize architecture frame score test frame well outcome cnn fine temporal max pooling give slightly observe temporal dimensional pooling show consider frame target frame video slide single many deep train pixel vertical direction horizontal rotation factor temporal factor interval value video online furthermore conventional spatial temporal overfitte recall pooling conv rnn lstm conv lstm follow challenge score dataset competition score category binary rate among category sequence architecture prediction baseline pooling vs max pooling last network surprisingly act cnn feature cell small temporal long combine temporal convolution architecture rnn cell improve score network multi table work outperform et al rgb remove depth preprocesse rescaling achieve need depth pose vs b l c depth knn et multi dnn conv lstm video information usage stream image prediction architecture sequence frame classify accurate difficulty boundary prediction recurrence temporal user feature map inactive without move activation movement suggest motion feature tb inner pt outer thick bend right bend outer anchor west bend bend layer map architecture without extract strong activation move learn paper recurrence rnn pooling architecture need account aspect add architecture notable impact able motion cell equally rnn recurrent able beginning frame great model future build subtle part write language annotation translation simultaneously channel sign translate audio acknowledgment gpu lead innovation science reference van van recent machine video however question temporal aspect architecture neural incorporate temporal recurrence crucial approach dataset art core human interaction become increasingly enable device towards subtle due vary performance camera hand motion target cnns de computer vision cnns abstraction impact like pose video classify video frame aggregate cnn pooling apart collection frames rnns either short memory lstm cells temporal dependency allow researcher achieve recurrent extraction motivate cnns rnns cnns spatial add capability recurrence video recognition almost play motion particular category beneficial spatial explore end apply wise challenge
leave invert thresholding class separate distance think moment anomaly technique information compare window simplification detect anomaly regular approach identify anomalous state estimation class classification color etc disadvantage encounter know available representative normal behavior probability moment normal compute observation anomalous threshold challenge infinite sequel optimization find financial economic wide application hold back intractable advance problem formally borel sign borel integrable stand sdp solve efficiently thereby tool attempt solve medium state solver relaxation optimality optimistic moment aforementione demonstrate anomalous real mean semidefinite column polynomial dimension denote polynomial coefficient variable th give understand column contain element illustration eq check positive reader detail refer polynomial matrix describe subsection discuss variable express indicator sdp relaxation moment tell optimality reach include completeness generally semidefinite program attain optimal bound approach work small density incoming anomalous classify compare threshold receiver operate roc curve metric standard way assess obtain matlab kde toolbox matlab automatic tune standardized option reader simple decide anomalous anomaly select incoming neighborhood sphere box form select fact set like account measurement equation moment present power scale quickly direction fewer available sdp solver whiten technique discuss unit may possible moment size resource whiten subtract univariate process store transformation obtain density anomaly detector moment contain two especially point increase three recover contour binary pose tool window mode estimate test consist outlier experiment moment c c complexity pdfs greatly multi distribution portion svm point outlier table moment include another trend continue h roll equation create consist h familiar
etc non mathematically respect dominate I possibly share refer set independent prominent stochastic covariate parameter life application covariate researcher rather fix situation include clinical pre treatment level etc I nh residual residual variance covariate robustness general nh case nh minimize average originally density excellent property approach also model alternative tail suitable property nh boundedness applicability test generalize glm covariate present supplement comparative remark end proof supplement refer condition ensure normality nh supplement make section nh take correctly specify density measure correspond datum density f satisfie coincide distribution I observation different nh statistic remark present define pm satisfie approximation testing significance nan significance test consistent pre solution require least robustness nh observation nh multipli nh functional sample size depend size contaminate contamination contamination direction contamination derive f l eq r p w derive condition respect supplement true parameter define supplement case always independent normal regression covariate invertible limit test glm directly set glm nan z ti base power direct develop report brevity robustness hypothesis denote first reduce significance statistic z contamination restrict fix order form nuisance glm like need row column result asymptotic q similarly element along perform brevity present online supplement popular originally discuss test hypothesis distinct robust robust unknown specify consider value parameter unlike case incorrect estimate robust h theoretical contiguous pure decrease robustness term power increase increase choose suitably robustness property test mostly extent compatible proper nh set proposal desirable yield inference hypothesis contiguous alternative hold present supplement
connection biological target feedback unit feedback put burden propagate message separate channel allow layer target direct connection output backpropagation obviously case implement feedback feedback slow distinct recurrent rapid dynamic tune rapid g combine generative recognition circuit feedback connection serve note weight typically associated neuron problem transmission neuron income output reach neuron principle close substantial degree organization neuron spatially feedforward neuron require transmission spatially close nature depend notion target channel geometry regardless channel weight target feed ultimately change taylor remainder system full hessian interest expansion unitary approximation interpret term approximate precision level bit bit turn bit real direction correspond component gradient thus main question bit budget per gradient box around cone unitary expectation approximated good channel include instance curvature case like approximation significant practically useful improvement learn compare target weight transmission backward channel interested estimating limit precise primarily interested computed example epoch want scale term elementary operation elementary include computing transfer function transfer backward essence implementation computer assumption computation forward pass backpropagation network scale send back target information send back hide derive double represent instance backpropagation bit bit backward divide information correspond least operation weight consider ultimately improvement dot descent g perturbation perturbation stochastically produce notion large magnitude various direction produce g u unnecessary generation uniformly sphere approximate calculation equivalently norm tend normally simple tend normally calculation normally tends implement descent descent associate identify w perturbation target weight either binary perturbation indicate improvement presence repeat brevity local global offer additional insight perturb amount perturbation since decrease however detail stochastic algorithm amount real represent change computation weight different deep target binary thus essence sign component locate random unitary direction target unit perturb provide derivative use compute perturbation produce case binary feedback perturbation lead except perturbation real feed dot small whether deep perturbation propagation lead w first small forward measure e ij back computational scale propagation step bit final thus bit unit well information improvement perturbation hyperplane correspond good refined version worth use perturbation unitary normally distribute mean direction scale deviation multiplicative constant provide feedback per direction produce dot high direction orthogonal select unitary take backpropagation furthermore unlikely backpropagation maximal improvement conclusion backpropagation algorithm maximum improvement h concept play role network learn six concept systematic beneficial mathematical expression adjustment describe physical study neural capable associate stack feedforward input even available learn feedback capable information deep deep implementation channel reverse connection carry feedback interpret bit capacity channel gradient divide require per capacity calculated backpropagation capacity remarkable optimality necessity biological neural must biological relevance carry simplified match biological biological question extend reinforcement analysis carry feedforward recurrent carry question whether capture biological obviously biological neuron machine seem favor descent biological neural obtain biological deep must locality principle provide uniqueness network view network connect connection asynchronous minima vector function store minima energy induce acyclic orientation dimensional hypercube isometry polynomial hypercube need case construct force rule derive spin spin appendix rule logistic goal factor actually convergent rule simple supervise supervised decay decay version eq q decay depend alternative bound range weight version gradient appendix deep target target activity follow activity forward may consideration isolated target instance may minimize respect procedure generalization use autoencoder schedule inner loop well provide include layer alternate pass along architecture variation backpropagation convolutional architecture momentum dropout adjustment phase backpropagation target rather layer apply focus derive leverage contain monotonic figure jump beneficial avoid poor hidden activity correspond entire guarantee stay hence case perfect exhaustive convergent convergent stochastic deep target target target hide successive refine target part award physical neural processing rule adjust available post systematic framework studying must nature functional tie capability network discovery rule stack deep feedforward deep output target layer input backward nature target information provide divide capacity backpropagation outperform theory concept learnable explain sparsity learn discover network unsupervise backpropagation problem view away try important backpropagation backpropagation vision energy physics attempt biological unsupervise within general precise capability limitation backpropagation core idea could require state case adjust activity appropriately input suffice deep think plausible book organization rather cell b repeatedly cell fire b neuron book appear thousand cite lack become obvious soon raise simple rule molecular biology opinion progress address basic happen feedforward network capability partly observation far rule backpropagation familiar perceptron delta variation create situation newton raise broader particular discover delta origin distinct idea associate neuron learn depend spectrum possibility narrow end spectrum proportional activity pre neuron activity pre sense ultimately govern environment possibility organize study systematic implementation adjust clarity must give backpropagation error variable backpropagation local g fire concept decide local model learn type rule also transfer function g threshold topology autoencoder feedforward line batch identity bias input value unit assume corresponding formalism recurrent primarily feedforward issue important within formalism activity activity formalism also instance consider component perceptron assume issue section share possible local look coordinate treatment beyond scope change bring narrow apply sense formalism problematic always transformation apply use transformation transform rule quadratic homogeneous system multiplicative sensitive behavior sensitive generally change slowly average epoch however neuron connection different different fourth consider consist connect well characterize quadratic quadratic orientation hypercube edge neighbor orient store set acyclic orientation hypercube hamming see kind elementary sign happen simple lead acyclic orientation hypercube show ultimately acyclic orientation hypercube dynamic towards isometry hypercube yield new vector hence rule new acyclic orientation see cubic although form rational rule although system invariant neuron apparent effective depend degree effective shall classify expect word recommendation local assume ij local rule could denote quadratic would include correlation require average form possible also rather cubic complexity concept degree input line stochastic fluctuation presentation term behavior average epoch assume rapidly datum compare assume epoch analysis weight constant change training epoch instantaneous govern write epoch analysis must recurrence restrict unsupervise et I supervise epoch architecture primarily feedforward architecture feedforward close feedforward layer input essence problem expectation case case input output consider case linear purely limitation backward channel section time move away learn replace definition goal systematically study local rule feedforward network reduce behavior local expectation let correspond training polynomial solve standard great general precisely less covariance require compute systematically transpose vector diagonal square component component represent elsewhere order moment vector moment ei ii notation compute thus expectation list expectation cubic moment datum term diag n diag diag eliminate recurrence expectation precisely compute epoch iterate furthermore invertible write symmetric matrix power cd c diag equation become kb c I iw diag w iw w www diag w diag iw w diag w w diag w iw h var diag diag w n diag ei tn vector epoch independent write give second example version get magnitude linearly epoch remain case quick approximation grow direction center rule lead notable learn anti rule bias include vector properly descent converge linear rule solve covariance effective rule recurrence relation must e reason let drop equation sign origin sign either asymptotically decrease try boolean fan architecture train learn boolean initialization learnable learn least one single train layer adaptive adaptive target learn decay linearly boolean rule learn converse deep rule learn demonstrate complex function layer top see boolean total learnable learnable boolean function total circuit comprise recursive method boolean function instance boolean function input learnable single fashion learnable layer network combination question learnable hope threshold gate perceptron solved perceptron gate implement perceptron state linearly separable local converge separate gradient separable perceptron behave sense relatively small compact supervise target independently update training linearly separable without separate simplify throughout separable learnable separate hyperplane every condition put ensure target pair learnable every learnable learnable first square cosine uv cv supervise canonical bias necessary hyperplane learnable start vector equivalently orthogonal learnable sum row column since canonical epoch learn gate epoch sufficient effect alternatively decrease rate condition thus cause term epoch sum allow simply take epoch preserve sign rotation special note fair coin learnable length binary equation become obvious true vector bias necessarily start component finally canonical previous check rule vector lead w I ei ei learnable separable go sufficient learnable angle lie equivalently learnable start length learnable sum cosine multiply target weight initialize update gate logarithm variable logarithm boolean polynomially therein monotone boolean learn single polynomial bound argument short bound function learn fraction boolean learn network significant possibility iterate deep architecture simulation learnable learnable year attempt seek plausible alternative local simple autoencoder local first broadly try learn purely local rate hyperparameter attempt fail feedforward layer layer layer layer activity process fairly arbitrary differentiable input extend take differentiable supervise consider local feedforward deep critical locally globally weight deep show target weight likewise depend layer layer point input target strictly learn scheme deep weight input thus reason stack autoencoder backpropagation course local stack technique globally would completely phrase exclude difficult capture precision entirely datum input simple feedforward architecture physical reach locally architecture physical target back deep raise question regard nature channel channel target see implementation target remain target weight deep previous local target become local incorporated rule I local deep deep target algorithm depend thus deep q see target available adapt work practice deep solve good deep deep target target unit backpropagation view
centroid independence analogous neither depend operational scheme generalize cope simplicity exposition counterpart alg alg although line alg notable difference validation estimate augment dimension vector n confusion pdf zero alg overall application phase r around r core validate involve test full ii rp scheme rp score iv algorithm association randomly datum attractive attribute draw initialization usage initialization keep finally association result distance centroid cluster percentage point assign cluster cluster pc core gb ram test run server eight bridge processor ghz gb memory matlab inherent capability core server plot curve average carlo realizations model per integer centroid cluster cluster randomly select demonstrate alg approach execution separable map prescribe linearly separable kernel end handwritten digit b accordance sigmoid store accuracy time second accuracy require scalability draw execute parallel moreover exploit capability multiplication rp fig parallelization beneficial compete idea novel algorithmic propose member tailor streaming mode cluster third member family trick separable fourth intermediate complexity synthetic art projection research rigorous implementation appendix vector product limited look r span notational convenience express moreover evaluation let th entry n follow linearity r term show solve task r r r alg efficiently term evaluation follow edu response tune introduce efficient huge possibly build consensus context robust operate batch fashion streaming operation separable family offer member user select trade family minimal subset mean iteration extensive algorithm competitive huge datum collect image mobile device medical big big volume impossible traditional stand alone processor e examine face comprise refer group numerous prominent thank point via hyperplane term probabilistic tool cope key question contain huge informative efficient computation retain albeit distinct introduce task angle combinatorial scheme well big require solve latent sparse efficient dimensional subspace sample randomized scheme non term score unfortunately leverage computation svd impossible computationally rely rp leave multiply agnostic rp reduce employ flexible attribute rank datum offer trading develop include step efficiency streaming mode operation member big datum even fourth extensive numerical highlight massive population art rp alternative letter indicate matrix vector letter stand respectively denote massive centroid accordingly model comprise centroid say association euclidean centroid per iteratively association assignment initialize iterative solve success denote square hard outlier k metric centroid term need carry qualitative long become otherwise g various distance centroid generalize potentially dimensional transform cf association probabilistic iii incorporate regularizer generalization unify replace per map linearly whose term knowledge association confirm whereas canonical k readily yield centroid recognize also pdf parameterized mean k multiple mixture pdf kk iteration maximizer likelihood unknown although module sec algorithms novel scheme unified remain follow trial mean dimension upon validate phase repeat final starting per realization dimension uniformly nk move phase procedure assess draw select draw extra dimension cf eq n centroid measuring association cluster space per draw trial validation phase prescribe realization last r k alg phase draw row obtain run centroid k close identify validation cf straightforward vs f unbiased large separable cluster light calculated validation burden numerator fdr choice avoid concentrated incur plus alg available resource computation alg dimensional probabilistic argument determine practice draw along denote realization meaning mean association mean characteristic draw repetition probability quantify carry leverage informative di I capture locate confidence centroid pdf per draw informative independence cluster interesting percentage validation ranking realization r randomly centroid initialize obtain sample k alg associate identify validation drawing validation phase may computationally especially prohibitive motivate feature per augmentation add flexibility meet consideration development alg phase alg remain alg phase one cf likewise small current memory draw test reject possibly bad clustering perform satisfactory augmentation difference across augment drop alg smaller equivalently detailed next cluster full prominent separable define distance induce pre kernel simplicity novel big hard extension similar norm form mean store step implicit association substitute eq distance list alg comprise realization trial line phase initialize comprise column distance centroid involve step evaluation cf operate dimension line major tailor deal across huge phase specify alg centroid centroid map cluster group distance generate summarize I randomly centroid draw versa common change validation q realization trial identify alg alg store alg incur data centroid randomly associate close centroid centroid cf associate centroid nonetheless practical term select run investigate single limitation require random draw sample section introduce number assess complexity perform examine draw center assess draw end follow pdf stand define parameterize pdf kernel link well population translate actual select clearly representative estimate fig pdf whole population
along theorem case regularization base start support employ example gradient evident pointwise distance euclidean readily minimax theorem condition vector exist last scalar projection reward vector mix compute build generate vector next satisfie observe consider apply discussion vector satisfied repeat satisfied choose w rx tr rx eq actions eq restrict require reduce restricted set handle concrete apply gradient specify evaluate since set modify q one set examine relation coincide use equivalence convergence convex algorithm recall choose action via need show give substitute turn attain scaling accord support positive induce sublinear summarize compact regret consequently proof recall tw sf fr eq bound integral emphasize simple lead rate recursively rather view rely logarithmic acknowledgement author helpful preliminary foundation follow outline proof lemma accommodate prove induction therefore eq strongly maximizer generally observe follow proposition hold trivially proceed logarithmic lemma point coincide outer unit normal due shrink property projection tr eq unit hold particular obtain sum yield regret bind small affect sum need establish proceeding bind obtain conjecture remark notion repeat game payoff introduce along geometric condition rely average payoff set regret set embed high original convergence regret learn decision presence adversary address feasibility repeat payoff payoff nature geometric extensive implication play game agent obtain action nature action strategy regret sub regret adopt last decade community online extend offer overview online recent survey may know already explicit show optimization online target convex carry original high present direct mentioned support along relation euclidean algorithms may require sequence vector concern present recover via propose bit logarithmic general observe algorithm still meta algorithm rely generic discuss demonstrate obtain outline conclude remark standard product norm euclidean diameter set maximal programming focus player extend mixed action bilinear stage stage action vector pure history action pure nature strategy restrict attention agent independent across stage furthermore smoothed reward average rather reward exist strategy exist exist strategy arbitrarily strategy recent dual avoid propose strategy elaborate smoothed reward useful benefit probabilistic nature online may meaningful martingale smooth reward agent nature mean reward extra pure action affect reward mix particular restrict assign action past mix
low formulation short moreover cost prove cost take full polytope subset denote index component index basis cardinality optimal solution low spirit tool cost find failure shape obtain linear program necessary basis use follow linear basis optimal restrict preserve satisfactory basis expression easily summarize independence component formula turn short path represent linear instance shortest equivalent whose component incidence orient degradation start end incidence row column index edge extra column edge incidence program encode path correspond nonzero totally every submatrix drawback incidence rank recall rank introduce extended incidence incidence graph path solution tucker equation cost mc nice result contribution vector feasible surprisingly bad possible remarkable show happen replace bind trivial cost assume independent exponentially bind minimize one hand index thus intuitive prescribe program replace cost far safe providing bound address inspection explain main random application failure complex degradation state residual incomplete gamma function define distribute moreover omit integrate obtain readily see positive crucial commonly encounter satisfy direct explicit distribution consider component program let satisfying theorem variable thus subsequent eq easier involve obtain trick triangle function belong simple moreover e desire path optimal path programming extend mc nature application subsequent property section inspection time reasonable shortest direct degradation sometimes class linear program cost upper programming short may describe failure degradation scheme study path problem degradation identify system degradation node system suppose evolve degradation neighbor assume distribution expert number merge start system start policy reach degradation
model single pf improve informed way consider subset assign close fig row single dataset htb initialization efficient mixture ise explain beneficial method field well ise overhead future selection centre coin introduce problem physics ise learn propose ise synthetic physic challenge intractable normalizing inference forward parameter form markov chain approach widely another successfully apply e go recover high amount currently paper parameter learn model overhead ise model ise superposition ise mix ise external field ise intractable ise learn bn n k q maximize side sample represent course due constant mcmc accomplish build iterative estimate mention field propose e instead efficient optimization ir ising show good single mixture coupling surface row curve agree generate obtain curve dataset bottom neither contrary surface bottom see coincide peak htb dataset ise leave mixture basically ise accomplish collection pl
framework impose extra formally background use solve difference provide notation denote mapping empty denote absolutely almost unique every inclusion induce ordinary differential equation format pose every real number great ii chain aa invariant neighbourhood topology dynamical apply result application trajectory neighbourhood neighbourhood call lyapunov converge consider e lipschitz continuous globally stable trajectory converge lyapunov continuity initial fundamental neighbourhood reader lemma couple iteration jointly latter q lipschitz uniformly latter condition depend martingale increase field scalar individual specify respectively state kernel need space countable ball equivalent uniformly continuous iff recall relatively tight definition assume follow eq invariant prescribed prove map close close let I f z w w compact follow close compact upper w ng nz ng z I pointwise uniform w g w w I nz nz w w similar continuity require analysis inclusion singleton inclusion h exact correspond first single restriction space use reference single scale eq time piecewise linear e dirac solution family lemma sn sn sn sn sn sn un un un n tn km limit point meaningful measurable note member fix e ease understanding point limit problematic far explore auxiliary tracking lemma lemma one trajectory track hence iteration control drive difference require modify version precisely tn tn tn rest tracking lemma topology surely limit countable zero martingale martingale bound quadratic martingale converge eq fact eventually claim fact compact use inside fix one c ns z sg sg sn sn sn sn sn sn due uniform continuity un compact union continuous argument lebesgue distribution control noise iterate see unchanged solution suppose every absolutely continuous moreover hence w z bn converge almost differential w converge track solution let construction tn tn tn te k note correspond continuous jointly integral pointwise convergence fact eq show every satisfie integrable martingale process convergent theorem choice eventually increase latter property follow due z continuous uniformly compact w proof ns f un z z sg un help ga compact union lemma thereby lebesgue e absolutely satisfy inclusion main almost set differential state specify transition behaviour sub approach transition keep discard triplet next policy increase weighting trajectory use allow unlike introduce currently target policy generate sampling sample policy td policy introduce importance weight policy gain reason policy trajectory policy analyze previous behavior represent action reward find value discount factor x px ps rs temporal reward nx hence project bellman mean bellman error descent contain expectation model iterate expectation correction importance weight iteration weight irreducible behavior px n ne tx td e w iii due homogeneity section value finite transition mdp iii q state argument third uniformly w third iv vi martingale increase field w condition iii fast clearly globally tx slow nx xx assumption equality assumption statement assume iterate condition stability markov subsequently put stability satisfied theorem boundedness iterate operator projection trace call irreducible e depend iterate
detailed e multiplicative namely four go purely appear replace transform arithmetic multiplication diagram depict transform matrix hadamard transform addition hadamard hadamard matrix govern arcs ba aa consequently therefore absolute combine hadamard arise matrix point algorithm aside element new step represent agree next make observe post complete multiplication depict derive transform extend derive figure diagram fast transform general proposition general decomposition achieve multiplicative value additive digital processor acknowledgment partially lemma conjecture cr tag cr tag email de mail r de discrete transform discrete fourier dft transform transform fast derive low dft achieve approach transform field role application area integral year dft existence ft computing connect promise transform field transform paper minimal dft signal pair cc dft vice besides real self exploit ft minimal point multiplication transform correspond give follow matrix value
increment keep walk price define drift suggest return term one write student generic explicitly integrable become moderate resort increment computation still impossible financial market recursive part nk k computation nn correction student bias walk keep expansion sum ns become become expansion contribute negligible accordingly relevant fact student distribution converge record first term walk find q formula confirm limit reason number record first increment see correction heavy tail important record mean drift make possible derive insight derive state distribution tail large discuss come different student tail keep law intuitive argument student part law tail start e student approximate derivative sharp transition line student z find numerically approximate record increase linearly ce record increment walk increment one show surprisingly increment record previous price record ensemble stock daily price record day result global negligible may back stock likely back compute record record probability uncorrelated case ratio directly expansion equivalence price warm statistic stock price least coincide unbiased exponent really package implement daily yield confidence standard determine allow keep mind valid record upper record average I measure discrepancy approximation reasonably accurate determination record problematic record number stock asset log price path log return permutation record interestingly two version version uniformly powerful exponent focus straightforward permutation distribute infer er record ratio infer dr spline straightforward deviation various completeness record price increment estimator efficiency interestingly close single sum estimator statistic upper price record estimate ratio return regard outlier contribute lack estimator record ratio numerically hour computer conceptual limitation record statistic uncorrelated variable main computation ar alternatively permutation estimator fa suggestion code available com gray rgb much use finance yet record interval increment variance record uncorrelated attractive new expect record increment tail remarkably expect numerically asymptotic record analysis nothing else hence finance live world arise bias correction serial correlation block actual price return price return fundamental walk depend precise walk asset price approximation connection price context study distribution low walk quite remarkably universal depend increment latter occurrence record record robustness r non parametric ratio trend price passing reason unbiased generic review classify treatment record number time importantly increment focus student upper asset walk increment first time increment notably zero inverting estimate
plug resources university liu national speedup platform processor ghz study core gb speedup memory mix lda sampler compare fully sequential collapse standard include start seed iteration collapse sampler gold topic indicator collapse pc lda run collapse sampler collapse mcmc topic spectral package table gave report conclude efficiency collapse sampler pc sampler collapse gold two sampler quite situation extreme dominate propose sampling contribute systematic token investigate effect time seed sampling threshold corpus small runtime run clear figure short sampler get seem independent popular use ad topic indicator effect initial core partition effect mode converge joint study core sure due unlikely randomly study effect seed seed remove h exception tendency converge core insight convergence ad lda display sparsity decrease increase core interpret posterior towards end collapse collapse end see end partially collapse sampler run example sparse mode core indicator detail decrease progress gibbs topic indicator ad lda token core number mode partition different probably find word result especially situation large token prior influence speed sparse lda different sampler clear benchmark actually pc relative switch alternative see total medium sized sample corpus determine runtime parallel scaling characteristics setup measurement aspect runtime want fast burn convergence occur sparse likelihood definition collapse sampler recognize gold previously assess aspect time sampler burn ideally initialize seed sampler core prior sparse ad speedup good program take lda fast sampler configuration good program pc lda real speedup eight speedup time eight core roughly core dataset offset gain parallelism compare core matter core relatively get corpora sampler go topic offset indicator thus gain parallelization speedup characteristic sampler ad eight probably cache ad describe never reach collapse configuration leave figure core total speedup penalize get speedup core core topic dataset roughly five eight characteristic sampler increase increase core corpus core real compare pc sampler relative execution core sampler seed topic topic table figure cm speedup core pc sparse ad lda hour ad hour pc slow pc lda make large conclusion speedup characteristic sparse ad term speedup parallelization overhead notable relatively several characteristic large topic affect therefore heavily sampler choice speedup generally lda scale eight core dataset handle weak scaling core ad small medium lda number pc lda already variable force strong become much quite introduce situation quite towards reduce spike prior small perform count draw model speed lda pc lda enhanced partially collapse lda art important indicate parallelization efficient contrary commonly collapse sampler nearly mcmc gold sequential collapse sampler enjoy parallelization moderately corpora core corpus important conjugate regularize topic pc spike give increase thereby solution interpretable collapse lda algorithmic improvement pc fast model collapse ad acknowledgment part systems united ns university david allocation model text collapse indicator sample draw popularity sampler stem balanced combination efficiency inherently implementation growing size complexity lda model computationally infeasible sampling therefore indicator basic exploit far collapse contrary parallel implementation collapse sampler well know corpus partial speed parallelization corpora keyword topic parallelism latent dirichlet probability distribution word indicator th denote document lda develop trend supervision inferential markov carlo collapse block advanced topic suffer sequential nature practically impossible way still generate sample serious number computational since approximate bayes solution use distribute collapse algorithm approximate bound error check inference sampler topic document remain sample iterate topic conditionally document row regard regard collapse sampler partially collapse note quickly mcmc collapse generally efficiency increase must benefit parallelization show collapse lda compare collapse actually small setting theoretically complexity token nonzero document basic pc version table search scan less frequently partial elaborate fully collapse use gibbs model study together collapse indicator collapse gibbs topic indicator word corpus scalar hyperparameter type sequential nature conditionally dependent indicator whole corpus efficient collapse approximate lda lda idea processor work parallel count processor collapse processor sample processor guarantee converge find ad sampling indicator number processor total parallelization suggest improve speed collapse sampler lda attempt sampler introduce document evenly document heuristic load document collapse sampling since fully collapse sampler job allocate core load sample job performance overhead large job introduce synchronization job decrease topic way topic indicator typically store topic indicator cache sampler collapse sampler add cost number token basic collapse sampler token complexity reduce collapse sampler grow determine overall pc language quite law relationship token often topic dirichlet pc lda sampler nk di nk eq exist extreme pc topic moderate section topic start costly scan type corpus typically rare scan frequently common type reduce compute much scan gibbs probability iteration start sampling theorem scan ergodic
construct equivalent bn subsection sampler sampler ce markov exact model reference therein build probabilistic constraint surprising equilibria game continuous known presentation technique book ai well specific book subject linearity simplify ce need clique consistent full joint following adapt tell sufficient condition pairwise clique let decomposable clique locally exist joint distribution unique possible value random index large sx sx state bn direct decomposable well appendix alternative brief process page decomposable necessary force marginal clique clique hypergraph decomposable systematically extra edge become decomposable compute decomposable graph whose linear large therein alternative throughout remainder decomposable construction feasibility ce make clique every constraint ce pairwise exist need discussion clique clique variables system constraint w constraint l lemma paragraph pairwise marginal clique decomposable decomposable game game equilibria marginal clique decomposable system feasibility linear clique decomposable submodular discussion game hypergraph game equilibrium polynomial game well bn ce corollary correlated equilibria action clique large take thus mostly grow true yet algorithm still practice open guarantee adapt present online ce ellipsoid attempt reasonably sized fail consistency marginal unfortunately guarantee problematic unclear test consistency whether convergence sampler reasonably keep add increasingly consistent np estimation mrfs instance ce idea leave future process compute joint tree primal linear system join joint assign join constraint local make sure truly consistent clique ce normalization join join player intersection edge local ce player express marginal clique ii ci must join neighborhood originally hypergraph clearly ni ni ci ni ce pair ia ni ia ni new extended player player node appear easily local player ci ni ia pair ci ia ci ia ci join tree time join thus local number ce variate marginals ce clique join uniquely decomposable graphical size variable represent see model believe process ph page page pdf thus simple game assume game provide mrfs lot work equilibrium researcher economic bring view current go algorithmic advance mrfs immediately advance graphical model ce games result yet width find practically e intuitive perspective property ce ce etc unclear I kind property ce circle motivation concern could result original apply avoid truly idea field simple computing equilibria game graphical equilibria bayesian implementation equilibrium distribution bn respect direct acyclic finite factor table bn hypergraph primal undirected parent arc implicit variable conditionally parent px bn arbitrary mrfs permit stochastic bn simple cumulative px I tf apply parent node value parent uniquely determine space connection undirecte graphical direct still graphical bn mrf bn assume course mrf x likely mrf cx normalizing function concern outcome evidence index evidence variable connect remove turn mrf clique c c posteriori posteriori map likely mrf computing normalizing equivalence presentation problem belief inference mrfs general np hard although reference therein usually characterize graph deterministic exist result equilibrium game summary book introduction one equivalent open problem ne ne games pls common statement ne think bad computing game essentially mirror mrfs constraint polynomial intractable probabilistic ai therefore share characteristic heuristic graph ce games feasibility problem include graphical game joint strategy individual player action I ix notation clique except player clique strategy respect play mixed call equilibrium ne payoff accord nash equilibrium ne ne ip ix p ix possible mixed strategy player differ formally I ix ip corollary remark game originally early economic existence equilibria graphical potential game applicability area engineering beyond economic intelligence study resource g public segmentation graphical potential back several game consider literature local game party game game leverage know probabilistic model particularly work establish playing imply game originally economic important class game equilibrium pure originally inference equilibrium game note version game potential inherently nash strategy game fundamental broad introduce potential back special potential engineering economic area artificial intelligence machine network social dynamical resource allocation public image segmentation selection spread social please describe paragraph implicitly special graphical potential specifically relaxation labeling ai vision see connection recognize dynamical ne mrf introduce goal propose metropolis minima annealing base exploration also work explore ne new class interaction game lattice game party instance graphical game graphical game payoff player neighbor game player potential game game leverage literature major contribution present establish playing rule imply player graphical delay economic recently science end large game sum clique neighbor node graph game symmetric previous important impose imply must game playing games mrf structure game run consistently converge graphical game play play rule steady distribution neighborhood game game special trees cycle grid terminology theory finally certain playing establish via connection monte preliminary terminology concept graphical game graphical role establish connection mrfs equilibrium game note characterize game certain kind establish mcmc gibbs sampler random mrfs graphical shift reduction belief mrfs equilibria potential call game introduce basic notation model ia undirected set include clique mutually connected useful concept generalization hypergraph think clique graph acyclic e direct length denote node e direct start elegant graph theory impact modern effective language system correspond miss mrf respect I variable clique joint mrf function mrfs familiar mrf px class suit fine structural graph extremely fine graphical wide variety probabilistic game theoretic offer respective hypergraph structure gibbs hypergraph px gibbs mrf primal conditional mrf hypergraph maximal behavior outcome rational individual paper individual maximize utility act want good central equilibrium equilibrium set player let denote action pure play joint action action player compact representation game graphical model ai inspire mrfs payoff node player game payoff player hypergraph payoff cx ci actions player ni unclear game polynomial contrast player clique fact several player exactly game j player player graphical addition game graphical pairwise multi define direct player arc hypergraph clique payoff player ix ni cx ix ni multi player clique clique implicitly cx ni respectively clique singleton obtain game clique define neighborhood undirected version game game game property unclear I game exactly polynomial correlated equilibrium contrast game player clique local appear summation game game set clique pair involve every player subset game game player particular game game dominate representation payoff emphasis representation remainder considerably small game neighborhood symmetric size dominated clique matrix exponential clique size game size game matrix equilibria game equilibria equilibrium x player improve payoff prescribe stick like problematic sensible come strategy try way randomization yes player distribution capture joint play component player play player except joint mixed play ne strategy play every ne approximation version equilibrium except gain ne one interesting contrast ne ce players conceptual equilibrium external player implement draw thing infer player mechanism encode conditional qx qx qx I action receive switch payoff would player switch formally equilibrium ce qx qx x marginal play qx qx qx ix ne q ne relaxed deviation gain approximate ce replace qx qx ce conceptual ne payoff achievable guarantee consistent something ne response player player play role mrfs game game game game player payoff matrix satisfy q weight last replace weight game call introduce probabilistic model facilitate derivation graph function strictly transform context play transform preference transform transform payoff ix transform transform games player potential ordinal graphical neighbor satisfy ix potential exact potential weighted potential characterize transform potential local potential function clique clique game potential graphical transform game graphical transform potential potential derivation mrf game gibbs graph clique local potential normalize respectively define strong transform potential potential ordinal potential potential ordinal graphical corollary dimensional game scale ix ix iw simply say payoff totally open neighborhood node subgraph connect totally neighborhood tree cycle grids potential scale payoff maximal corresponding addition neighborhood symmetric respective neighborhood potential potential implication reason expect potential say difference payoff would tell payoff simple dimensional matrix generalization dimension game local use potential hypergraph undirected graph node clique player play stochastic adjustment literature learning game sequel let sequence let playing plan select play formally player player observe action graphical game say neighbor graphical type conditional give every compose consecutive joint outcome joint play empirical round round undirected graph vertex set clique potential define play transform mrf consistent conditional scheme regardless condition play always weight player scheme correspond leverage game theory graphical model establish strong connection answer question computing equilibrium last discover develop early computational game equilibrium may relate area plan strong belief artificial intelligence equilibrium largely development back advance understanding belief single mrfs computing ce ne graphical game follow section mrf graph gibbs potential node clique neighborhood player hypergraph game order mrf individual payoff player game graphical game mrf game exact ix x ix remark converge finite regardless play refer player observe maximize dynamic implement assignment mrf guarantee maxima mrf mrf game characterize maxima ordinal potential game game induce mrf potential whose maxima solve local mrfs pls mrf ising model symmetric similarly network game party game arbitrary reference support notion strong player payoff map game induce mrf short heuristic mrfs simplify expression definition mrf maximum one whether mrf induce game reduction implication reduce likely perform mrf would theory characterize normal game therein consideration game additional insight mrfs graph expansion property limit poisson expect game enyi sufficiently average high mrf games low games something maxima suggest max maxima critical mrf state induce rich game go begin establish strong connection equilibria player joint condition equilibria mrf express simplify equivalent simplification highlight need induce ce thus maintain size ce linear programs ce alternative mrf algebraic get remark useful denote marginal play player condition ce mrf game kind mrf entropy summarize mrf equilibria game player ix ix px leibler qx qx player player equivalent hence ce mrf game kind optimum critical kind summarize remark mrf equilibrium induce satisfie local ce ne ne strategy marginal joint action play player except probability player entropy variable imply hold turn express ne locally field except ne mrf nash game induce note property imply field view compute mix formally player ix I I call game player strategy nash assumption mrfs extra game continuous set strategy equilibria reasonable action nonempty euclidean quasi concave infinite want mixed strategy mixed strategy game infinite gibbs potential product mix derive individual mixed pure equilibrium game local payoff also connection game learn repeat game existence equilibria difference involve mixed regularization factor strict player consistency infinite payoff overall play behave depend indeed good player mrf induce variational recursive condition equivalent minimize function parallel strategy monotonically optimum equilibrium nash equilibria game surprising property broad game interest modify ne go opposite also suggest treat mrf play game converge ne explore connection learning propose play estimation game mrf maxima potential game everywhere maximize support measure pure ne game might equilibria optimum critical point connection nash equilibria support limit ce mrf heuristic high fact argue desirable lead well capture aspect would approximations multi modal approximation mrf induce game polynomial ce polynomially sized distribution case ce action play correlate product follow concept correlate equilibria ce potentially ce iff player qx derivation seem novel improved mixture polynomial ce algorithm correlate equilibria mixture regardless one ce mixture ellipsoid ellipsoid practical algorithm interior simple arbitrary polynomial guarantee guarantee ce describe section concept introduce way spirit see g reference therein connection game equilibria probabilistic inference future end recent work equilibria comprehensive evaluation relaxation labeling ai vision connection game yet recognize connection game reduction pure concentrate
ba double double double double cp p space double cp double string true string scenario describe generic scalar intercept double lambda cp cp space cp rand double cp block wise cp cp cp cp true mr job mr double double double mr double double mr matrix mr double double output label cp matrix double double cp cp double double double double cost scalar scalar example plan runtime plan remove remove directly relate execution code variable specific describe program dag small translation see first operator transpose self matrix multiply exploit unary computation transform prevent transpose construction runtime available meta scenario memory execution mr runtime accordingly job make generate hybrid runtime plan mr mr aggregation aggregation select call multiplication small cache rewrite execute transpose mr transpose exceed local budget mr job fourth mr mr single mr job share prevent decide cp read partition w task input demand prevent repeat read optimizer runtime column prevent optimizer select map operator mr small parallelism transpose prevent intermediate scenario already given generate similar mr configuration budget block constraint third combine operator mr summarize major plan characteristic decision runtime bottom runtime computation important runtime dag rather size runtime reflect give plan use box compute cost model job entire program runtime aware resource see memory constraint explicitly parallelism estimator allow size disk parallelism available virtual core resource execution runtime plan skeleton cost track memory pass runtime program compute estimate per aggregate accordingly program track fundamental individual dense format weight read bandwidth compute maximum multiplier operation introduce early requirement convert execution assume example correction ghz processor cost white mr job consist task time reduce read compute computation degree parallelism reduce mr job account without aggregation job job read cost result write cost effective degree parallelism available parallelism block take mr degree parallelism put runtime discuss main program cp lambda cp cp generic cp cp x cp cp rand cp cp cp e cp cp cp cp cp cp write plan simplify annotate program total plan plain cost show couple cost e operation dominate execution cost generic c cp cp cp line cp cp rand cp cp c mr job input map mr mr mr mr ix cp c plan scenario cost mr job pure simplified scenario annotate comparison adapt increase operator read second generate mr job factor contribute include job reduce parallelism reflect read degree intensive job include cache partition time time actual read third despite remain memory hybrid exchange account accuracy example estimate cost within actual simplify fundamental limitation give general reasonable complex ml program run however conservative scalable ensure plan validity however mr job infer case lead commonly make optimizer pruning partially buffer cost address box buffer live sake buffer box acceptable buffer pool usually small fraction total many ml flow loop branch recursive especially number heuristic predefine reflect loop body execute repeatedly allow code motion iteration future summarize allow runtime program reflect decision importantly analytical without relevant aware runtime execution learn true false red black pt ml aim specification ml algorithm level language automatic range memory computation mr framework exhibit advanced cost technique model decision share runtime generating runtime automatically successive phase runtime loop branch cost time advanced resource global optimization optimization state system aim ml language ml construct ml algorithm underlie runtime cluster program automatically memory runtime level full physical independence representation efficiency scalability multiplication decision distribute operator quantify several optimize potentially program characteristic runtime potentially rely run orthogonal cost cost intermediate program available parallelism aware cluster resource ml program loop branch call complex program simple robust runtime several world example algorithm feasible rarely w ordinary problem read intercept lambda intercept one tx beta ask compute intercept program construct write discuss input cluster characteristic generate select optimizer task leverage runtime create x ram intel ghz ram disk storage gb map reduce used default memory budget ratio max size overview scenario size c size ds ds scenario input plan generation give range use case detail show dense follow discuss select dag program runtime scenario well memory estimate selection multiple transform prevent unnecessary intermediate memory intermediate accordingly challenge
contradict part shorthand choose prove n v reward prove follow case less classifier know guarantee since maximization two apply following make maximization invoke note step valid measure benefit value value lemma follow conv measure reward point hoeffde online batch guarantee length stage challenge procedure respectively es union ds require accurate frequently mild severe label imbalance requirement classification measure non decomposable measure pose challenge application family possible implement family pseudo linear measure core contribution adaptive scheme technique truly base update concave dual pseudo alternate method demonstrate significant similar datum severe label imbalance requirement negative include spam classification anomaly medical imbalance classification suited situation trivial optimize predict class instead measure case entire decomposable include consistent effort optimize year result broad approach convex indirect include cost approach solve plug approach rely fairly scale large memory style approach cut plane well plug solve class approach prevent moreover take large streaming take preferable however decomposable recently surrogate optimization maintain prohibitive state develop novel two broad family decomposable truly wise buffer pass intuitive level linearization amenable sgd feed see write include mean etc exploit variable parallel linearization maximize stochastic mirror fractional need outline structure develop optimize combination via alternate strategy converge optima strategy batch validation experiment class fast plug style measure surrogate base indirect applicable measure learn compute exist dedicated optimize pseudo linearity maximization cross validation considerably improve multi label decomposable plug style challenging role designing solver non decomposable define challenge generic therein style buffer maintain guarantee special application care denote instance denote positive sake calculate r averages reward concave lipschitz value range publicly available benchmark dataset mnist severe imbalance min also compare specialized sake unweighted compare stochastic method implement rest testing plug cross validate split hinge execute level actual solver implement since rapidly pass datum epoch every allow runtime solver allow q accuracy greatly accelerate fairly find accelerate fail slow least find classification due imbalance report similar competitive accuracy f accuracy similar else slow reason confirm buffer method cause rate secondly style poor accuracy compare plug work measure plug explain acknowledgement fellowship lem write iff region us primal dual exclude prove direction conclude vector great dual inside prove direction region radius eq define f result stream execute feasible shall notice update write interpret respect involve constant note monotone conjugacy far tr tr functions individually p monotonicity second stability inequality concavity jensen write eq projection concave involve show ascent observe linearity step concavity third conjugacy measure reward satisfy probability eq
proceed previous thank notation ratio hull result optimize intersection radius complement ball radius x converge zero j proposition use obtain eq dot previously propose straightforwardly modification adapt elastic net allow early discard derivative propose version duality consideration create safe region zero screen wide convergence support cope solver descent significant safe mid attract attention explanatory context square refer lasso pursuit processing enjoy theoretical name many issue accelerate dimension indeed method rely hundred non scad mcp lasso often mention homotopy lar homotopy choice particularly method problem seminal exploit discard guarantee allow reduce burden introduction safe potentially variable post operate choose solver safe safe static orient towards commonly well drive manner consecutive know call strategy road think warm start screen warm sequential safe rule keep improve screening efficient proceed argument leverage safe safe safe contribution introduction safe look associated safe concept screen rule converge rule rule equivalently inactive also rule build dual converge safe safe region gap safe sequential descent solver propose standard report safe denote observation approximate norm norm control fidelity estimator solution primal eq denote feasible formulation read see refer onto rule center radius dynamic dynamic safe lasso primal link eq tucker kkt kkt primal screening consider safe rule exploit equation soon challenge unknown safe construct set contain safe helpful benefit construct region denote define cast differently safe primal explanatory thank close convex hull restrict safe safe explanatory safe safe safe safe region possible safe safe set contain support lasso solution safe whose rely safe sphere center radius simplicity safe commonly safe review safe strategy consist safe region however evaluate priori also experiment gap safe safe sphere convex dark blue provide safe build converge let duality provide light dual feasible thank see tangent let read late insight one recover safe primal dual objective resp gap solver next establish oracle radius quantity pick dynamically available safe ensure converge eq converge primal sequence define safe safe safe safe sphere safe include gap radius safe rule respectively property satisfied radius warm start safe make safe inherently sequential approximately handle approximate produce safe screening safe sequence next equivalently safe evolve follow sequentially safe gap safe replace definition one access primal solution safe still approximation screening choose well suited especially coordinate require process commonly operator wavelet thank x implement rule coordinate code level necessary dense array sparse pseudo present pass stop gap involve know safe norm evaluation gap stop strong rule safe need processing also sequential require exact solution lasso prevent converge phenomenon present safe proportion safe screening depend much gap safe especially tuning scale safe rule especially test bring improvement cost gain hence prescribe safe dataset present feature graphic
rich assignment heuristic cope make outcome parse dependency parsing may similarly complex combination simplicity removal concern employ cost multiclass algorithm ensure error policy oracle competition advance overhead neural remove essentially write decoder oracle english nine language competitive recent publish label strong parse complex broad learning approach include perceptron production search policy take action search word n word n output build framework write decoder parse loss state consider middle deviation complete loss write speech annotation aside library computation reference trivial hamming prediction tag previous machine arise answer try one yield efficiently epoch execute trajectory loss alternative execute act loss context learn policy able initial step deviation trajectory vary policy manner regression instance mixture epoch subsequent epoch reference reference l stack buffer arc root root root root root cc thick node style base b bend leave bend bend bend leave bend cm every anchor base bend leave bend bend leave derive gold framework implement stack maintain buffer keep arc triple ease buffer stack dependency arc direct word terminate arc parent derive parent word parse assign tag arc dependency head unlabeled extension label top arc hybrid transition system arc buffer contain stack arcs root parsing take buffer stack e terminate take one move word add arc arc valid show execution parse parse dependency parsing associate derive gold transition action move root ref mention framework decoder reference oracle policy optimal derive annotate pseudo decoder dependency discuss take stack buffer speech tag list template generate configuration change parse oracle lead minimal library learn system policy automatically therefore action implement arc predict action configuration predict annotation wrong loss effect decoder implement unlabele label loss tree gold assign arc feature feature library fold decode system second framework library ease quadratic cubic mechanism provide unified allow base learner argument modify mention framework reduce sensitive reduce framework employ study base analyze learner baseline recent baseline greedy stanford wide different language show achieves language transition perceptron dynamic avg assumption language hence obtain substantially bad language root exclude conduct language chinese convert head split testing pos tag evaluation stanford accuracy split last datum development need gold pos tag learn policy policy decrease round round experiment reference preferable roll reference roll dynamic tag embedding regularize particularly neural hyperparameter transition stanford network setting arc hybrid system exploration thus set development seed test external resource embedding randomly suggest setting fair exclude run unlabeled exclude evaluation compare tune cc leverage well learner update state base learner stochastic gradient improve metric importance single hide neural hide nn regularize multiclass multiclass nn rule gold label oracle learn detail l bi gram template comprehensive transition average parse algorithm combine learn explore principled way search sensitive classification evaluate end treat bad approach labeling resolution graph unified first interface parse broadly structure probabilistic programming language however rely language describe implement furthermore provide wide advanced optimization language extend room line stanford support code parse stanford implementation usually code comment gd sensitive h offset initialize finish label valid valid action gold stack arc right arcs arc arc arc ec variable task action option add option value ensure sentence label value arc label index ex index get bc bb dd ff ef db df triple vector triple triple condition feature return l cost task task label gold tag stack v cs ex child ns size mask f ex ex sum ns ex begin ns index ns ex arc hybrid datum task array stack stack gold gold tag gold tag tag tag v array child child stack return last stack child stack stack child stack stack child stack stack size stack child tag stack stack stack gold tag stack stack return stack child child stack child tag stack last stack stack stack return extract search ec task mask mask multipli shift stack stack tag datum ec ec ex ex mask multipli ec ec stack ec ec ec ec ec buffer ec ec ec ec ec feature stack sl ec sl ec ec sr stack empty last stack ec child stack ec bl ec bl ec child ec ec stack stack child end ec stack I fs ec begin fs ec fs ignore continue offset ti offset ec k ec fs k fs quadratic offset mask multipli ec fs ec multipli ex offset mask multipli stack empty stack stack stack stack child stack last min stack empty stack tag child additional offset offset offset mask index datum ex data ns string begin end count ex vector
require solution datum distance except privacy differentially q private naturally prevent attack power differential interpret theory reader several privacy firstly dp dp privacy automatically allow dp dp advance make simple boundedness single posterior denote preserve free classic consistent differentially preserve alternatively domain g preserve privacy result thing boundedness l lr familiar notice mechanism preserve output exactly posterior exponential simply notation specify thing effort posterior privacy b b x boundedness usually small decrease super exponentially paper convenience practice predefine threshold release perform great generality consistent briefly consistency bayesian sense great consistency apply consistency frequentist sense prior posterior consistency hard consistency prior promise bayesian find distribution either equivalence weakly suitably bernstein von posterior distribution von theorem hold obey normality independent interesting class near similar class leave work proposition asymptotic relative function key idea scale different fitting include mild converge mass mle remain bayesian whenever hold bernstein von asymptotic near optimality mle rescale posterior obeys likelihood correct equality likelihood close generate since difference scale minimum regularity invoke modified bernstein theorem say converge nj nj note interesting remark proposition log sharp previous intrinsic eigenvalue depend implication essentially generalize classic result confidence interval test generalize ratio private use trade powerful easy extend handle agnostic leave claim privacy sampler rare complicated often option sampler never something privacy sampling preserve differential privacy sampling procedure preserve dp commonly proposition clean interface need privacy bad news g approximation easy sampling lda suggest arbitrarily near log concavity distribution imply confirm differential privacy constraint nice thing modify go dp whenever tractable provide insight bind seem barrier privacy rather hold achieve differentially private erm objective perturbation work function prior differentiable strongly threshold restriction hinge huber privacy hard work view stem intrinsic privacy require implementation application convexity need add additional may give sample privacy many section answer look technique year show differentially free simply release differentially private sgd advanced composition allow privacy composition advanced mechanism dp fold adaptive eq constant simplify expression see apply taylor addition dp subsampling randomly evenly random sampling sensitivity gaussian differentially minibatch regularizer empirical gradient tool would avoid iteration update parameter mini ordinary stepsize strongly minimax optimal convergent later prove choose iterate stepsize stochastic minibatch converge gaussian stepsize slow discretization approximation obeys due translate assume mean estimator study minor modification burn minibatch number pass initial minibatch coordinate define burn period lipschitz collect carlo differentially private minibatch also smooth preserve differential privacy every iteration access l technique use tt advanced failure accordingly proof choose level big reduce failure converge alternatively suggest pass variance langevin use iteration collect stepsize need overcome estimator calibrate initial already posterior long stepsize valid claim different ensure privacy internal bad however run strong privacy modify balance get privacy practical drawback mixing describe attempt resolve use auxiliary variable counter use stepsize gradient therefore briefly langevin hamiltonian ignore hmc proposal distant enable rapid hmc author restrict arbitrarily long simulate correct noise dominate get quickly become discuss still exactly true correct trivially variable serve similar appropriately describe interpret momentum sgd get flexibility range gradient chain posterior parametric von idea use inverse fisher key stepsize speed stepsize far differentially learn near release bar true true go noise benefit principle many collect stepsize collect able adaptively adjust temperature unbiased sense equation unstable train pass sufficiently large involve fine direct way privacy constant matter use decide perturbation degree often conservative bad stochastic differentially private sampler logarithmic thousand well illustration sampler linear illustrate converge like become able produce unbiased level evaluate page uci repository logistic hybrid risk minimization privacy result figure improve classification privacy use laplace mechanism perturbation solve bfgs numerical long confident minibatch pass choose plain initialization perform equally slightly especially curse constant early first aware develop focus mostly conjugate point boundedness differential computational use provide efficiency aware scheme normal strong result require unbounded semantic point develop tool perform privacy completely different post processing procedure aim denoise integrate far boost investigate effect beyond scope differentially stochastic private party modification gaussian match logarithmic confident contribution extension preserve disjoint require setting applicable pass well replicate find objective perturbation originally version compare sample solution differentially private sample intermediate iteration conceptually inherently differentially get exponential estimator parametric algorithmic langevin dynamic variant preliminary practice theoretically practically meaningful provide intermediate think case exploit randomize hash dropout thing hope differentially private movie recommendation goal
challenge near prediction quickly curse overfitte limited mostly univariate strategy ideally power grow combination prohibitive limitation drastically reduce possible predictive causal make tractable criterion demonstrate nonlinear delay even forward selection suggest fit improve prediction index ni predict goal traditionally since neighbor neural ahead nearby mostly state reconstruct take nature multivariate information dimensionality nearest impossible predictor redundant information perspective variable mutual curse predictor mutual searching subset lag exponentially search strategy prohibitive due therefore propose demonstrate approach predictor theoretically recently serie much allow globally search strategy case additionally criterion selecting subset predictor compare even cross much run forward suggest framework also problem series underlying mechanism understand firstly also drive fit free relevant efficiently improve understand mechanism index ni causal sect information selection prediction sect explain sect causal criterion discuss computational sect sect sect analyze prediction apply sect evolution function drive possibly time lag represent drive quantify shannon entropy latter conditional level predict past entropy uncertainty maximally perfectly truncate lag dimensionality many actually carry merely poorly goal thus carry still new possible avoid combinatorial search predictor include selection iteratively conditional lead computational sect globally strategy might drive fail sect key use satisfying state remain term theoretically parent add variable increase parent parent describe sect drive drive series drive selection parent causal sect p start iterate combination node hx hx h stop combination test cardinality else one iterate initial previously need underlie independence order guarantee graph entail conditional relation violate certain assumption fulfil algorithm imply predictor inference yield analyze scheme far backward step algorithm ix propose ix n mix value sort numerical positive come surrogate drawback adapt predictor fix threshold level low causal predictor complexity optimization illustrate mi fail prediction select causal parameter predictive mutual subset analyze mi certain combination large causal maximal mi
omit take w matrix identity diagonal normalization element rotation rotation rotation step k compute h estimate source comment insight account rotation numerical bss moreover show bss whiten numerical cost increase linearly life environment vary adaptive estimation utilize slide sample show matrix encounter parameter complex rotation follow resp resp avoid non one typically need sample convergence method algorithm present mm criterion show comparison bss performance signal interference ratio filter matrix channel source system symbol pass generate mean specify signal snr channel simulation htb db htb different examine number figure compare draw unchanged perform compare db figure propose snr expect significant compare notice perform pre whiten g low obtain algorithm suitable notice db number nearly db well snr respectively figure sample notice nearly however high figure iterative batch name use pre whiten operation reduce recursive unitary unitary rotation modulus instead maintain mainly deconvolution signal favorable bss notice number case criteria together alphabet eq double angle identity write bottom write element subsection order generalized write order scale scale al laboratory france mail fr also department electrical engineering technology mail address multiple deconvolution bss algorithm modulus criterion show design quite real maintain modulus whitening rotation improve size rotation whitening occur bss interference symbol bss blind modulus rotation source bss implement pilot symbol utilize reduce bandwidth efficiency g pilot symbol valuable tool meet demand rate wireless system pilot contamination output system bss signal estimate source unknown bss source signals bss find criterion multi criterion attract interest cm criterion mm utilize separately several present mm numerous cm algebraic name analytical constant modulus capable separate mode overcome drawback numerical batch bss similarly mm modulus outperform mm minimize name firstly bss implement manner two batch bss communication unitary unitary rotation utilize present name algorithm pass whiten operation unitary unitary filter filter signal convert rotation iteratively find algorithm slow convergence fast developed show propose case manner provide comparison perform much bss organize bss brief bss rotation real design algorithm rotation detail propose section present notation along th transpose complex conjugate transpose pre filter modulus part matrix real element symbol pass channel model symbol instant independent source white noise signal utilize prior source inherent bss bss channel mean value signal add r n h bss receive matrix w receiver vector global system receiver rotation large whiten efficient unitary decompose rotation lead size whiten inefficient rotation unitary brief review unitary rotation identity two diagonal angle unitary rotation except h pp qp like algorithm decompose rotation decompose product rotation denote order rotation compute desire unitary transformation accord write seven involve simplification thus complicated rotation motivate come deal mention challenge work previously difficultie version use receive convert contain maintain rotation rotation necessary preserve sequence rotation show rotation apply successively parameter rotation rotation shift diagonal criterion explain iterative transformation accord q rotation unchanged modify rotation angle assume express identity express similarly replace last I irrelevant determining solution minimize eigenvector correspond eigenvalue rotation similarly apply successively function norm eigenvector initialize summarize table whiten construct use rotation separation matrix small whiten effective channel unitary unitary real rotation rotation overcome limitation product elementary rotation rotation refer transformation rotation
derive mutual group computer science biology algorithm ref different evaluate find important perform model plant partition political annotate expert partition community reference partition similarity easily label maximize range permutation see overlap partition roughly partition label refine normalize overlap however problem group modularity ill another accept well study randomly select mutual information leibler kl detect nothing detect ground practice joint approximate group shannon joint gain know gain knowledge nothing obviously similar evaluate one way bound identical become popular consistent fig partition plant sbm call plant plant generate independently commonly size distinct un phase network plant modularity three give plant configuration come modularity report guess group bb line bar top bottom use evaluate similarity partition give systematic statistical value compare configuration detect could configuration large significance compare nan use science structure modularity compares expect edge graph configuration compute average usually already less plot algorithm find plant sbm plant detect generate stochastic block benchmark happen plant partition plant overlap consistent un perfectly phase maximize permutation un similarity partition fig algorithm modularity benchmark measure work benchmark right tell use community detection mutual network fig average realization network modularity bp benchmark different size exponent distribution exponent size show numerically
hardware round operate quantization precision asynchronous nonzero entry high sgd algorithm precision increase result corollary negligible unfortunately rate sgd expect particular martingale arise track principle unit eigenvector base simplicity focus outline entry condition update randomly index uniformly origin equivalently recover show require incoherent unit ease run bound times martingale save initial value expression problem horizon parameter cb determine initial appropriately run failure bound analyze convex illustrate analyze asynchronous precision sgd complex include validate matrix implementation precision like run update bit input limit decrease modern row bite k gb gb conjunction terminal explanation color package graphic terminal graphic ltb lt lt lt ltb lt lt lt bp r logistic regression bit ltb ltb conjunction explanation color package graphic explanation graphic macro ltb lt lt lt ltb lt lt r sequential ltb version claim discuss application show precision change ran analyze report forest music music glm report logistic display speedup sgd axis six core ram low arithmetic algorithm sgd combine data table update compare convergence ten eigenvalue differ somewhat randomness asynchronous version behave qualitatively dataset take run take second speedup unified produce rate asynchronous precision random stochastic martingale base sequential easily give asynchronous modern hardware resource algorithm acknowledgment thank helpful author acknowledge contract air heterogeneous graph stream mid da library language high dna sequencing specific national energy system stanford parallelism fa program simplex national science foundation nsf award office research national image big http www american view conclusion herein policy either express imply nsf detail body long tx w r x k next g x g indexing apply h k apply continuity v gx h r side produce h h r k h r tx apply update distance r h r tx x r x e use k entry substituting conclude algorithm success occur actual take apply hardware law expectation state convex sgd horizon section result except quantization set prove lemma purpose piecewise logarithm lemma order concave armed prove lemma optimum tx tx x f bound fx assign fx x side jensen rate occur x success occur occur negativity rate verify lemma statement occur lemma tx definition tx tx clearly expression maximize tx x lipschitz mean value theorem next lipschitz index fy apply round apply lemma lemma appear state first version specialized update include use another combination lemma proof x define stop stop stop b tx x occur stop yet stop negativity tx rate next bind time first give x therefore x x n assumption give bind x j u apply incoherence j agree assignment produce proof substitute c f produce desire result simplify martingale consider elementary tx x noise due delay update x I e e k c x entry value substitute therefore prove secondary literature stanford electrical engineering computer stanford stanford machine researcher technique runtime asynchronous execution capture rich specifically use new way relaxed sparsity asynchronous sgd completion design analyze asynchronous low arithmetic experimentally algorithm efficiently variety modern problem eq sgd wide range application machine widely learn poorly success practitioner asynchronous asynchronous include deep recommender system practitioner way also asynchronous version stochastic stochastic proximal producing propose asynchronous unfortunately sgd approach entirely precision ideally could martingale enable extension unify technique relax assumption asynchronous sgd matrix asynchronous sgd quantization fix point validate experimentally algorithm theoretically describe analyze asynchronous challenging copy asynchronous core separate copy cache core write handle possible central store atomic solely dependent function write denote write think independent reasonable though since delay delay occur equip convergence asynchronous continuity collect
efficiently reconstruct input usually sparse classifier risk framework drive truth mapping space kernel carry representation solve neighboring code reconstruct neighboring pixel sparse code pixel encourage neighboring pixel sparse pattern extend drive joint pixel extend general drive dictionary use section generality input consist pixel label parameter jointly minimizer choose quadratic joint input difficulty function define row active perturbation locally active gradient bit involved omit limitation optimal kernel theory observe yield satisfactory nonconvex properly university spread pixel range band water remove processing randomly available training range spread pixel split training datum rest pixel choose per set parameter outline drive propose purpose name enforce neighboring set neighboring jointly evaluate propose kernel svm l latter construct prior accordingly university hyperspectral show table formulation achieve enforce among pixel propose performance competitive compare dictionary construct dictionary propose compact dictionary translate computationally formulation enjoy among pixel show equip university hyperspectral image formulation dictionary readily task research topic sparsity test corollary example laboratory md successfully discriminative dictionary constrain prior advantage domain hyperspectral classification formulation dictionary jointly optimal performance supervise hyperspectral suit prior enforce neighbor illustrate hyperspectral hyperspectral increasingly become target shown achieve construct collect sample pixel lie form generate unstable enforce code neighbor pixel joint neighboring lie stable improve base learn generally dictionary aim find yield task supervise drive art task jointly machine high dimensional counterpart rational class space class sample typically subspace discriminative code
dft number input compute multiplication complex coefficient implement compute polynomial polynomial real number polynomial division division multiplication two multiplication necessary multiplication compute dft polynomial division autoregressive fed circuit dft component dft fast attractive procedure dft need compute introduce root inversion factorization euler dft component polynomial combine produce multiplicative single dft iv section dft consider zero write autoregressive q polynomial division multiplication component multiplication due multiplicative multiplication assume polynomial multiplication indicate hardware q compute hardware implementation symmetry numerator polynomial therefore form algorithm multiplication multiplication attractive fed shift circuit dft element namely lead dft obtain output correspond hardware multiplication multiplication require j algorithms component transform well complexity far compute respect fix
combination specifically assumption dispersion around spherical contain compactly ps ps ps laplacian identifiable structure carry necessary ica natural seem ica however properly progress work unconstrained model source visible size difficulty deal limit attention image reveal basic block multi architecture propagation handle transform reference therein compose block block handle diagram amenable implementation rapid source row matrix q kronecker identity probability contribute space represent distribution flow network branch forward message usually numerical stability version backward combine propagation direction multiplication loop branch compute normalize element reader rigorous translation marginalization flexibility bi directional message generation delta three propagation distribution variable version display result follow delta backward propagation distribution factorial code decoding pattern subset available bottom delta backward miss step propagation collect product observation forward reduce elsewhere omit reason therein report mnist binary pixel architecture figure delta maximum block prior generation figure increase forward message picture number increasingly accurate pattern character shape build representation definition factorial code learn marginal less kronecker present set delta posterior source act soft configuration factorial code present note code sharp column decode graph decoder sharp figure network backward pixel forward posterior miss ht ht question mnist ica matlab retain density confirm ica source try look patch natural ica sources generative preserve structured composition set figure mnist image unsupervise addition information backward block learn bar represent encoding row naturally experiment architecture backward show could consider bar simultaneous ht forward posterior encoding propagation unified framework image datum code correct flexible alphabet greater report elsewhere currently universit pe te le belief bipartite factor graph inference full image mnist dataset show factorial code implement source contribute build generative ica information propagation become aim capture visible popular map source source visible variable signal ica filter seem converge pattern visual explore possibility constrain factorial code feed difficulty naturally product limit attention perhaps
ram hour train show imbalance varied address remain operational search fine grid could operational baseline choose satisfied drop operational ii instance mean free violate operational meet cv test cv final produce constraint favor mixed ensure operational constraint htbp ptc instance free cart c ridge elastic choose lin svm interpretability train operational constraint satisfy lr ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc elastic ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc lin ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc none ptc ptc ptc ptc ptc ptc difficulty operational among elastic net produce operational constraint cart produce regularization ridge lin rbf unable require level operational include emphasize point pt operational crucial implementation adjust model sparsity incorporate suitable operational correctly handle operational high extensive never operational r choose parameter predictive several fold free maximize predictive operational free unfortunately operational htbp lr lasso elastic lin svm rbf cart acceptable produce acceptable scoring produce significantly expect minimize elastic surrogate operational sparsity max sign net need least sensitivity plot sign score net roc evident sensitivity specificity lin sign coefficient vs real interpretability acceptable head head operational lasso elastic htbp point align domain knowledge sign large coefficient model screen tool poor sensitivity elastic high sensitivity ability provide relationship response score suited provide kind qualitative understanding quick computer help user work example human cognitive entity association may also help influence input sparsity require score help follow simple rule interpretation sparsity system popular train sized minute ran learn summarize choose explore varied nature process categorical value process resource htbp breast breast cancer high risk heart disease detect breast cancer predict mail spam baseline publicly available package subset accuracy validation sparsity interpretability run baseline time grid free training run per allocate minute ip ghz machine gb ram hour train scoring method dataset function htbp cart cart default rule penalty lasso value figure report represent coefficient model lasso ridge elastic lin leave cart rule base c box svm dataset bar cv addition regularization path error varie level ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc range ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc ptc error error ptc ptc ptc ptc ptc ptc ptc ptc scoring unable show figure sparsity evidence minute sparsity trade directly optimize necessarily accuracy discrete lasso baseline restriction arguably mainly suggest accuracy dataset find restriction relax ridge lin interpretability interpretability focus nice comparison predictive figure model omit attain perfect cart far lin integer express line mutually exclusive prediction hand help find use hand hierarchical make assess input interpretability benefit interpretability notion depend light benefit practitioner directly encode interpretability requirement operational point htbp mml htbp text center height font draw gray near fill text center font green font none em yes leave safe root root end yes narrow right safe leave narrow safe block right yes leave safe yes yes line yes yes node end yes node yes cv none narrow none narrow mean c lr lr lr lr specialized problem interpretability set interpretability interpretable heavily exclusive interpretable set require scoring system train accuracy train scoring ip l p j x j r j identical binary ensure interpretability interpretability ensure interpretability specialize base real convert convert feature value threshold use discretization form unchanged categorical yield binary jt threshold tt benefit compute table originally model net stand fully form coefficient n gain require c tumor rule cv solve r j j ip thresholded version suit value intensive use threshold feature optimize real exhaustive value classifier threshold interpretability small rule per include constraint agree ensure maintain monotonically system ip p j max variable p big parameter identical interpretability penalty interpretability least one rule count value variable limit binary rule constraint constraint encode create drive scoring show scoring system optimize operational constraint come computation approximation surrogate practitioner need integer programming software real integer practitioner way allow choose acknowledgment thank comment addition dr dr general acknowledge version vector loss using ensure example define I statement fact cauchy show ensure classify three margin margin lie margin I I definition minimum expression whenever put yield nz c minimizer theorem set q remove remove sign mean statement sufficient assume satisfie use would contradiction iii iv look rhs inequalities incorrect management institute add multiply assess risk serious medical quick extensive computer american medical currently whereby system create scoring difficult create integer explicit operational number approximation round surrogate round address operational impose hard false calculate impossible mean loss operational constraint free parameter model operational alone score integer optimize current classification produce scoring optimize sparsity wide complicated operational without contribution pt approach learn score system operational advantage derive particular present discretization bound sufficiently addition relate novel reduction portion reduction laboratory tailor significant million people state alone eight classification publicly accurate sparse matter paper rest special accommodate operational constraint medical system discrete present reduction laboratory create tailor screening report experimental specialized stream medical scoring integer classification medical medical systems pt iii ii iii patient death detect criterion score sparse model small scoring system use medical scoring case heuristic ii construct multiply integer approach odd fact solution programming system hand suggest history point factor increase rr subsequent year dm stroke easily create medical scoring cognitive poor trade loss attain good theoretic accuracy similarly restrictive rarely recover lin linear balance rely greedy part stream create linear discrete minimize small coefficient formulation optimize hinge loss discretization novel model discrete reproduce create converse necessarily method oppose sparsity difference may eliminate produce operational train integer ip goal restrict integer tackle albeit formulation minimize refined formulation integer comparison since goal simultaneous tuning encode important operational constraint restrict whose lp significantly tighter design norm minimize discriminant overview mix lee early misclassification feasible dataset mathematical body focus improve scalability procedure cut specialized branch remove redundant dataset x ty label represent coefficient intercept value datum may great encode absolutely control balance soft set interpretability separable directly optimize restrict finite operational purpose restrict objective coefficient add dropping adjust trade additional adjustment factor choose drop entirely remark feature penalize training scoring finite coefficient e bad real minimum large magnitude training classifier less round resolution set discrete coefficient attain classifier baseline attain directly margin resolution margin resolution denote coefficient train coefficient set parameter discretize construct discretize discretize py follow provide uniform important bind q classifier obeys hypothesis space motivation indicate include increase exclude provably suboptimal scoring linear every classifier obeys appendix parameter model theorem exploit system coefficient express discrete classifier linear classifier integer obey counting use plot improvement show reduce classifier improvement associate model discard redundant technique classification carry well suited optimization reduction general computationally represent feasible represent function aim discard change solution require specify easy initial surrogate identify redundant training solve provide reduction work problem q objective surrogate denote level original
compare stein shrinkage en estimator penalty scad iii stein estimator restrict preliminary shrinkage ridge rr en en en represent estimate en en elastic net rather elastic estimator ridge estimation retain notation ridge lasso lasso en en portion define see elastic easy among estimator rr en facilitate indicate superiority estimator efficiency finding summarize ridge efficiency estimator dominate figure shrinkage positive always well find zero estimator term relative stein dominate simulation scad estimator result table begin various configuration increase observe relative may order stein dominate neither scad stein dominate outperform covariate correlate elastic ridge en decrease value en dominate type stein estimator en dominate significant simultaneous whereas stein methodology focus en en en inf rr en inf inf en en inf inf al scad al scad l scad l scad l c scad scad scad scad scad al scad al scad scad scad scad scad scad al inf inf en en en inf inf l en en en inf inf l al en en c inf inf en en cm lemma section development predictive key formulate search popular adaptive scad elastic net analytically characteristic ridge rr elastic net en versus restrict preliminary stein space rr dominate scad en dominate neither stein dominate lasso scad en significant uniformly dominate stein dimension efficiency en penalty depend en dominate stein rr estimator analysis error keyword phrase stein shrinkage uncertain classical normal admissible quadratic give birth partial document preliminary stein stein type estimator expand include stein paper appear estimator shrinkage rr minimization thus least criterion penalty bridge generalize criterion ridge become smoothly net devoted characteristic like elastic net stein ridge stein type simultaneous exceed dimension ridge estimator limitation conclusion base efficiency organization follow discuss penalty preliminary test estimator table graph provide unbiased penalty simple example namely coefficient estimator ls criterion sphere minimize yield ridge parameter regression problem linear minimize q pn bridge reduce ridge estimator variable popularity later glm course estimator good penalty unbiased unnecessary modeling coefficient estimator avoid instability well possess property estimation call smoothly absolute deviation continuous define q tuning expression scad adaptive consistent vector equation minimizer computational methodology propose elastic net ridge component ridge estimate mix cross validation elastic advantageous highly predictor elastic net lasso shrinkage elastic oracle property elastic property group variable traditional group inefficient inconsistent overcome extended group tool oracle regression vs chi freedom df may optimality assess lose continuous stein type inherent problem due happen interpretation become another namely stein five group improve preliminary estimator asymptotic uncertain hypothesis consistent function unity p distributional cdf chi square give difference give perform whenever hence consider optimum
addressing I example denote behaviour noise variance quantify selection eq rv provide magnitude show well define combine pure ranking dependency examine tend prove important relate behaviour variance since become noise infinity free consistently prefer consistently outperform rs whereas consistently prefer bad consistently select example alone impossible directly term possibility leave e rewrite give magnitude proof namely unbiased quantify combination selection determine entirely behaviour random exceed likely argument pool comparison argument serve illustrate rs guarantee receive existence unbiased estimator algorithm capture ideal unbiased ignore central estimation expect new include component estimate label multiple quantity dataset raise statistical choice term bootstrapping I testing three three I bootstrappe replacement give two clarity variation estimate estimation classifier estimate estimate classifier close bayes loss increase small precisely estimate near optimisation reasoning suggest practical application algorithm simple component broadly reduce component estimate second classifier base compute loss immediately estimate classifier also loss produce suffer use lead argument require component two estimate na I motivate development term computational evaluate pool popular random seek way provide reasonably estimator bootstrapping class classifier third together estimate estimate sampling pool j k j e median seek show define c statistical generating independent component component unbiased ideal dataset unbiased completely unbiased application neither classifier large test available open research suffer development algorithm ideal estimation estimator asymptotic rate result suggest reasonably class raise whether estimation merely reasonably bias selection size prediction cost estimation differ explore describe varied al substantially classifier capability nn na I appendix explore group experimental use another conclusion experimental explore variation evaluate al compare classification fall natural benchmark se third define method logistic forest classifier diverse open research weighting sometimes recommend literature however weight theoretically weight primary competitive method weighting defer exploration performance continue entire label label final loss much metric describe classification monte replicate use experimental experimental al evaluate performance iterate produce profile loss label label class learn experiment nn al metric assess improvement al rs quantity method literature performance evaluate function label metric create single curve iterate rank employ al total overall rank overall avoid metric experimental assess evaluate seven method tie brevity overall rank table reasonably insensitive al address variability label pool test draw source namely initially label c relative al discover perform whole end loss average al imply ranking finally classifier c se rs classifier replicate al al method mean ranking average ranking clarity method classifier ranking se se rs se rs calculations ten monte replicate loss ranking group average overall ranking three average appendix e overall rs classifier ranking rs twice replicate problem table central study ranking method rank rank rank rank se al conclusion different suggests consistently outperform se outperform rs experimental argument algorithm consistently outperform section examine method literature consistently entropy leibler somewhat way pool classifier rs clear benefit exception rs individual classifier estimate experimental despite difficulty performance achieve statistical behaviour via classifier practical insight al competitive statistical begin target al individual batch iterate definition examine know exactly suboptimal unbiased motivate construction algorithm comprehensive experimental study several result competitive recommend choice estimation algorithm various bootstrap e resample sophisticated estimator research motivate superior subject work al heuristic experimental effectiveness univariate gaussian rv sized cdf dt rv b rv giving chebyshev hence apply rv strictly since six six discriminant na I bayes logistic analysis lda classifier discuss na I independence logistic svm popular classifier r classifier detail lda use implementation r implementation covariate covariate na I r implementation e assume regression svm svm radial score mle computing optimisation diverse al fall set problem datum split group another large problem error uci provide term property covariate b dim class dim illustrate probability balanced uniform prior problem multipli multipli mixture boundary appendix nn I logistic show table six problem monte rank mean tie se rs se se rs se na I rs se b c c rs rs se rs se h se se rs se logistic se se rs se rs section ranking rs quantify rs regret naturally difference give actual benefit rs behaviour treat reduction estimation novel provide framework framework central motivate allow al behaviour heuristic make abstract al outperform reveal issue turn motivate new al experimental al competitive effort bias improvement central certain case learn al seek select base classifier example include medical diagnosis approach review method performance assess experimental al consist classifier improve systematically formulation raise classification loss suggest optimality suggest reduction quantity basis improvement novel statistical framework strong advantage theoretical practical describe framework formally define al behaviour abstraction behaviour heuristic optimal context ideal unbiased crucially motivate algorithm strongly compare estimation type different explore source variation classifier abstract result perform background classification define illustrate abstract estimation scale experimental conclude background context al brief method later model covariate prior denote produce probable allocate objective somewhat example index indexing dataset may division subset discriminant regression regard fix give fitting notation extend become contain store near classifier role produces use denote prediction assess performance assume assess example error quantify focus allocate denote empirical performance reason generalise loss log denote hereafter refer example al abundance label expensive al select obtain oracle expert provide label classifier improvement pool usually small label denote consider scenario al repeat al time generate define al exploration amount label grow al contrast iterate selection occur iterate step critical iterate label covariate create example pool turn rs preference receive label example thus provide reasonable benchmark al classifier even benchmark receive training explore assessment rs hence address relative rather loss label blue horizontal uci nn shannon al reduction number classifier form fundamental al seek metric comparison common significant comparison goal label need single pool random expect define label form denote denote loss one give denote loss j kt future enhanced difference define loss pool example novel abstract problem indeed iterate generate covariate extend al smoothed classifier batch pool batch consist example expect improvement examine label actual reduction example select pool expect example denote batch analog incur major huge candidate consider pool batch candidate size candidate jump batch al generate selection candidate present major calculation selection require require calculation greatly severe make estimation extremely challenging batch al option recommend estimate individual al target iterate rest foundation framework abstract classification illustrate character calculation reason pool explore al examine denote make shannon imagine binary balanced c boundary rate split equally pure subset sampling hold prior classifier classifier denote calculate
follow resp provide resp range accordance integer fix say corollary quantify probability sparsity relate independence suffice comment length corollary equal q complete statement thing remain many way deal satisfy decrease eq integer side give sign edge correspond strategy probable growth result positivity part assumption separate kl divergence continue assume still appear straightforward counterpart satisfy begin recall notation collection definition least possibly define pa ps correct fix satisfie argument long throughout expression maximize subset cardinality pg pg version appropriate hand differ second belong account account place criterion complete simultaneously factor appear proposition proposition claim general three reason number third parameter choice condition equivalent interval nonempty take eq final task accomplish way namely sum condition thereby parameter quantity least derivation explanation second fourth obtain final eq change relation part also take bounded condition quantity form state throughout probability emission necessarily marginal list elementary distribution q q lemma taylor write taylor test taylor expand base base eq use hypothesis define cm monotonicity prove compute zero critical point inequality uniform expansion number contingency least upper bind bound bind difference define together find amount quadratic analytically fix let assume formula purpose proof substitute accord quadratic interested nonnegative must positivity proposition return inside numerator numerator q carry outlined beginning eq minimum purpose mutual information set say follow give claim probability completion depend make statement yet expression responsible asymptotic maximum one involve know exponent denominator make ignore asymptotic asymptotic see case use b asymptotic theorem refine throughout proposition ultimately chernoff ultimately draw lead sparsity boost specify growth sequence consist draw great statement apply statement integer least consequently let least observation apply union multiplying form complexity hypothesis precede proposition replace second statement estimate dominate large usual statement state involve asymptotic explicit expression make arbitrary adequate term first order proposition achieve leave subtract decrease coefficient result precede term obtain third accordance first quantity theorem large aid elementary asymptotic behavior log sided eq apply inequality right recursive identity yield claim asymptotic recursive familiar single say apply finite complexity behavior non positive integer well let follow immediately fix difficult formulate q obtain last divide n appropriate union multiplicative second q multiplicative second explain part computer describe concrete implement code package package follow theory type alphabet whose integer outcome element sum terminology frequency equivalence belong class associate define namely characteristic indicator aa ap expand emission probability reduce notational clutter reader marginal carry compute offline reasonably pair file table produce interpolation raise offline computation main issue interpolation main issue precisely carry statistic derive answer question computation carry store slow desire consideration balance another step obtain want table ram possible find explore first computation method exact computation second combinatorial integral approximate replacement integration scheme determine iteration concern construct selection point accord pick weight scheme heavily second covering weighting develop method exactly feasible cdf random mutual step finite store structure keep track point cdf hash structure part practical type length type possible advantage type amount account separate various list arrange rooted tree root locate parent leaf root level pass tree iterate give method tree traversal reason traversal recursive dft def datum process def boolean generator return else return generator def child else process leave tree element proper processing node input must emission type calculate constant finitely function implementation depend used list time certainly pr emission order compute make call n parallelization e plot dependence calculation algorithm experiment possible improvement far serial simple parallelization computation break branch subtree branch separate return object accounting type branch return merge accounting although parallelization agree closely paradigm carry parallelization auxiliary accounting list cdf generalization internal object carry parallelization modulus collection cdf library one available core processor entire eight processor hour processor call constitute color branch language marginal marginal p follow explain family define finitely increase ratio finitely many nf pdf gaussian variable unit standard term paragraph finitely approximate shape path explain pick maximum path still consume though marginal path statistic namely scale calculation marginal define manner segment contain center ram reality sort speed linear thought suffice fortunately theory analysis discover follow dependence likewise relationship emphasis statement care moderately small moderately large explain demonstrate claim affect store statistic linear marginal adopt statistic ultimately illustrate give intuitive make relationship conjecture great projection onto form relationship htb plot statistic present case small present demonstrate fix care strong thousand figure linear already except consider boost least safe interpolation range relationship trend difficult estimate become range end unnecessary algorithm whose intermediate procedure close nearby record table enough reflect edge left reason come last decision table inform consideration section cover statistic precise meaning meaningful determine illustrate reliable grow practice seem experience adopt table experiment dependence recall denote positive real distribution statistic note order direction obtain parameterization good think unit care getting become concentrated towards produce scheme object invoke generate list store member invoke signature stepsize rational separated unit rule bt def return bt interpolation actually separate associate contain soon encounter function convert kl send already explain choose crucial sparsity boost function simply evenly side statistic replace practice boost reason example could rise sample possible boost difference matter find reasonable return belong small path marginal path pass reason factor drop location length whether contribution arithmetic portion boost huge prevent sparsity show put threshold empirical mutual information computation clear power observe make entirely alternative interpolation language def gamma small near gamma gamma gamma interpolation linearly simplex interpolation neighboring point rigorous manner close reader exponent exponent believe exponent exponent result boost finite boost term boost function rule skeleton boost structure finally polynomially restrict consider approach attempt usually independence lead greedy relaxation max running constraint orientation produce initial relaxation relaxed contrast identification undirecte skeleton final network edge greedy scoring hybrid still separate constraint approach distinct step rather independence term directly score remove need heuristic one believe could valuable outside implicit experiment research compare publish result bic score quickly every boost boost evidence test preliminary experiment maximization suggest local able scoring sample issue find prune parent parent parent exclude consideration variable significantly section criterion parent whose contribution like cut regard parent view parent set independence line investigation choice overall ii error currently implement marginal restrictive alternatively fix way insensitive marginal somewhat adopt strong marginal must marginal every approach approximate without concrete future explore incorporate order boost effective certain lead point discrete structure throughout field paradigm hypothesis learn researcher paragraph characterize precisely onto least onto projection onto illustrate need text understand projection considerably simplify interest close marginal entirely constant serve justify interest uniform component conjecture decrease conjecture consider product symmetry distribution distribution identical far product write let fix correspond partial derivative say variable derivative q principal real unique branch map interval fashion namely decrease interval graph critical increase value follow order lie uniquely solution one symmetry point claim come function trivial interval principal branch htb conclude give conjecture marginal marginal point divergence nearly case p quickly move away enough lead long conjecture small conjecture empirical show heuristic chance appearing state minima exist state recall notion relate likelihood field probably exception characterization kl recall notation conditional joint use entropy conditional parent dag use term nb arbitrary relationship express dag collection bn know appropriately count extend naturally relationship kl divergence since involve entropy characterize accord side factor marginalization minimize kl simultaneously set matter resp mostly probability underlie bayesian scoring assign among closely though content empirical elementary state directly prove eq estimate eliminate absolute sign use condition small inside cite union theorem conjecture incorporate score mm com david cs edu scoring bayesian traditional dependent work property become prove polynomial distribution generating whenever exist perfect generating distribution although new score together explanation relation hide conceptual automated reasoning prediction concern system concern discrete write factorization number basic estimation perhaps fundamentally give structural factor world artificial intelligence speech biological medical even factor factor system sparse preferable representation task wish tractable proportion appear formally pair acyclic dag follow condition node edge correlation influence relation give explanation simple connect dag term every independent see follow equivalent represent rewrite conditioning except parent parent dag imply call independence one bayesian framework overview attempt comment literature simple case bn call bn notion bn change statement connect devise observation distribution distribution avoid possibility error seek identical study classic discuss evident different situation differently relationship imply terminology learn strictly consistent network distribution true clear consideration probability unobserved perfect map thus generate network framework count costly map two avoid complexity bn bayesian entirely obtain dag undirecte long share skeleton cause situation cause common effect cause dependent cause oracle conditional distinguish break problem follow structure reason appearance v responsible orientation learn structure vertex sift thing many recurrence relation bn complete even restrict bn independence constrain bn naive iterate structure take advantage achieve much relaxed study construct scoring assign score fit penalty bias structure fewer simple principle bic score score know justification make maximize np hard hill program seek conditional observe one approach commonly refer constraint key parent run approach drawback propagation difficult together knowledge combine main approach statistical testing view true keep function incorporate hypothesis add bic skeleton bic experimental log score act prevent boost close sensitive dag parameterized depend observe first score weighting penalty contribution perhaps go parent node paper parent bound iterate possibility second potentially produce sparsity boost exist hand conditional fix minimum sparsity intuitively boost calculate two really dependence truly exist want ensure independent zero mutual go empirical reference dependence independent inside logarithm event strength remain remain test pearson nan independence observe cdf justification namely pearson mutual information really reason threshold decision pearson small denote type relationship powerful explanation pearson classify evidence minus complement sparsity ii pearson test instead produce associate powerful henceforth theorem refer case equivalent elementary correct correct correct sense sense second learn network skeleton dependent ultimately learn network mention depend contingency represent conditional occur set express contingency stay contingency less integer contingency restrict set size subset contain pair substantially tolerance parameter appear variable role function sake formula main readily deduce n n would relatively remove however network basis access table finitely software look accurate chapter approximation publicly point generating contrast divergence generating benefit familiar relate representative put achieve polynomial number may exponential compete rather recover skeleton make discussion hybrid approach author acknowledge useful discussion definition need sample course follow invoke detailed classic theory deviation introduction topic refer notion concern chapter section chapter composite problem structure examine contingency table conditional variable denote convention cardinality denote summing quantity eq often restrict various proof random product xx l simplex identify contingency table sum element standard parameterization contingency table number consider denote identifiable equal sum row column say contingency contingency table denote atomic event atomic contingency sum second distribution notation represent certain product first distribution marginal case denote special distribution fundamental hypothesis mutual kullback leibler divergence minimize kl constraint onto share marginal complementary edge network denote term set consist one family distribution share parameterization distribution binary contingency range range share say binary definition course interval fix kl unique positive sharing marginal reference clear tool quantify strength quantitative strength parameter important quantity sake idea probability consider frequently bn bayesian acyclic dag without tuple obtain joint assignment variable write distribution write level sequence n p g bayesian network distribution competing obtain normalize produce conditioning order objective minus place element convenient shorthand understand informally mapping subset family bn collection wish learn consist vertex bs ga gb b ga behind separate collection think distinguish dropping edge another reciprocal structure strength notion separate make polynomial line bic boost mainly place state order finite sample complexity elementary need make involve define thing understand statement need lemma complexity allow learn return quantify bn state divergence divergence arbitrary notion familiar accord onto factor solution map namely distance simply word map denote bayesian satisfying map cm namely cm network whose distance quantity though content independently deviation specific need quantitative concern probable estimate various entropy multinomial mutual entropy mutual combine standard theory deviation probability estimate concern continuous function primarily case effort lead variable let frequency exceed fall theorem easy side chernoff achieve large deviation chernoff lemma theorem need complicated appendix simple one side inequality multiplicative elementary analytic sequel draw bernoulli second breaking argument derivative interval term approach infinity derivative value entire unit occur absolute value divide thus find denominator namely elementary monotonicity zero differential calculus increase interval clear slope possible interval value attain value bind denominator turn attention chernoff exponent use claim interval attain decrease keep letting interval actually suppose contradiction side consequently assume proof claim possibility mutually exclusive possibility exhaustive proposition able derive concern involve note state quantity apply know entropy minus entropy far side observation define principal variable follow sign desire estimate factor outside bound term mutual proposition collect fact function simplify formula eq monotonic minimum reason decrease two though still decrease monotonic decrease q decrease fact three decrease eq q drop differentiable say eq define point derive decrease similarly eq one inside less quantity multiply side decrease function impose eq statement relate statement prove accord unlike application probability generate learner control marginal go naturally large must condition hold calculate inequality proposition implicit circumstance preferable condition conjunction q imply follow form let upper eq express finite error local distribution theorem parameter relevant state come first complexity concentrate case refer initial boost consider straightforward complexity sparsity boost weight complete analogous formula technical future choice hyper affect asymptotic growth choose quantity q probability examine result may arise free particular obvious close optimal appear denominator comment hyperparameter carry however merely nod eq asymptotic dependence appear determine dependence replace say element cause dependence namely instead chernoff cm great define sample probability function combine chernoff bound asymptotic quantity finite complexity node mutual minimum edge strength perfect separate define let free pa pa ss quantities follow let q reader theorem edge graph refinement asymptotic strength free large q asymptotic nm part theorem asymptotic theorems theorem begin collect factor asymptotic asymptotic actually assume elementary list last logarithm suitably difficult go define dominate completes verify dominate asymptotic function define n asymptotic dominate asymptotic statement properly comment fewer technical essential corollary reason boost boost objective extra edge false quickly term network second presence boost false miss false proposition key quickly network skeleton distinct structure possible distribution fully boost mapping objective reflect composition first test generate marginal derivation case point quantify converge enough moderately multipli time proposition penalty grow moderately proposition finite multipli order growth start choose conclusion upper readily probable bound boost let satisfying condition assumption q reason large new convenient corollary first derive
generate monte actual cost hierarchical substantial sample consider procedure previously moreover certain generate augmentation emphasize location apparent assume parametric parameter choose scale location write proposal exactly independent pdf possible prior simplicity draw draw proposal pdfs drawing distribute accord closely resemble target interpret estimation x clearly infeasible main sampling obtain layer instance base combine mis role devote level tune mechanism know scheme walk mh monte method implicitly subsection notable case independent walk mh collapse technique layer although measure previously target proposal chain matrix mh summarize pdf x strategy well target parameter properly many mode mean proposal choice tuning provide depict proposal proposal walk mh burn iteration already reach stationary burn proposal walk method burn period imply walk generate follow hierarchical interpretation implication draw target walk well independent proposal roughly tune non generating procedure certain denote write pdf density q estimation clearly walk generation build figure importance sampler population step initial draw x accord tt generating cast nn advantage play approximate mc technique adapt cloud mis population proposal alternative literature static mis mis sampler highlight challenge consist population pdfs refer mis target assume exactly exactly ensure robustness scenario mis mis mis mis pdfs sample distribute mis yield statistically dm mis pdfs evaluation proposal evaluation target weight accord load instance mixture dm mis p mis divide proposal disjoint form set case definition capture proposal mis weight jj weight whereas mis htb mis c mis mis dm mis sample set integral measure proposal pdfs monte past scenario pdfs subscript proposal framework adaptive multiple include eq characterization show adaptation many procedure instance build mis discuss figure representation show spatial proposal pdfs htb xt proposal dm mis case partial mis appear weight single iteration form use scheme sample employ htb mis mis dm mis suggest mis dm mis summarize different grouping proposal pdfs divide proposal index pr cost increase total grow indeed proposal evaluate hence number perform scheme build use remark pdfs generate advance usual adaptation algorithm convert mis normalize simple iterative adapt proposal mode iteration version th previously need show c express indeed final estimation eqs express recursively estimation state start n ms see appendix estimator simply proposal pdfs weight ratio finally consistency version choose proposal pdfs specific generation draw weighting given normalize return pair tn adapt pdfs scheme markov adaptation location mis proposals mis section adaptation procedure estimation argument observe adaptation underlie markov depend update parameter technique type adaptation variant like procedure walk interact pi doubly interact importance detail algorithm cost simple specifically mcmc technique coincide proposal pdf motivation one hierarchical mh underlie approximation interact markov base markov markov markov location mh draw nm normalize easily pdfs number iteration step specify mcmc adaptation first one parallel chain interact adaptive pi pi version doubly interact case initialization choose pdfs mcmc technique step draw mn normalize eq return simple applying mcmc consider mh pdf n n scenario location pi layer partial dm mis manner explain build pdfs incorporate adjust langevin mala sharing introduce hence pi detailed mh mh working location sequential different adaptation rest discuss extend coincides pdf subsection interact pdf fig simple technique type transition form draw pi accept differ describe probability dramatically suitable purpose q idea potentially burn period one iteration consist candidate choose correspond accept difference one iteration need already compute step mh sequentially mh draw pdf total employ pdfs mis final population change diversity preserve unlike resample update section alternative describe update involved total big specifically target pi mh gibbs generation acceptance choose pi mh uniform extended multinomial evaluation require recall base evaluation general monte bad easily rest observe jointly different n robustness result suggest approach sake covariance covariance strategy consider proposal suggest performance benchmark tackle issue carlo consider parameter wireless propose multimodal specifically multimodal gaussians I eq matrix approximate monte normalize adequate approximation ability square mse three describe partial dm mis parallel independent pi method moreover static mis mis dm final fair implement total evaluation pdfs specifically order proposal approach importance level pdfs different matrix initialization denote initialization region mode thus improve specifically select show initialization initialization mode eq result experiment respectively table highlight face computation per sake good among simulation pi choices initializations pdfs initially localize pi improve adaptation pi depict circle configuration proposal density pi pi htb literature nature mathematically carlo approximations true exhaustive deterministic thin order mse pi proposal randomly n case mixture algorithm keep number mixture component different st sn order pi adaptation pi pdfs pi see simulation small highlight outperform example also display circle proposal run represent approximately mass output triangle diversity pi ensure well cover htb adapt covariance mass localization wireless sensor plane realization range sensor locate identical pdfs also p prior receive observation sensor fixing compute expect order deviation pi three scheme description proposal initialize randomly pdf consider proposal also isotropic diagonal fair pi choose average pi outperform robustness pi algorithm walk proposal mh furthermore use adaptive importance scheme class employ reduce dm strategy include different differ extent term scheme consider partial dm computational confirm benefit sampling project foundation de grant european network grant mathematics university es circuit de represent approximate complicated multidimensional target simple proposal draw
abstraction principle bellman mdp regardless ai reward element ai aggregate mdp aggregate mdp algorithm vi mdp us option consist termination tell primitive expect reward primitive analogous discount give option execute state pi state pi matrix matrix option format introduce format function value option evaluate brief survey hierarchical stress reason macro run date except option vi generalizations bellman require mdp macro operator largely focus hierarchy discovery option discuss hierarchy also controller temporal abstraction differently abstraction option construct hierarchy policy hierarchy mdp hierarchy describe mdp termination start plain vi proceed notation state eq select action rewrite correspond come execute note multiplication vi next reward define associate pick improve update execute policy policy stop value reach possibility termination stage termination therefore terminate terminate conceptually think specification termination diagonal however summarize termination behave terminate identity actual update iterate tend go state state exceed induce policy introduction action give formally boolean action allow follow q benefit irrelevant whole become macro give rise state solve immediate availability use macro operator hierarchy contrast convert mdp aggregate state stress valid describe reach macro repeat fast aggregate example vi eqs state use help mdp model convert new transformation aggregate compute option state model option termination termination policy eq state follow state q aggregate terminate towards evaluate row option valid primitive accord follow contain row option terminate row option primitive add action action original time g mdp macro action macro original supplement observation algorithm mdp bad thing happen macro iteration take look eqn may domain run four computing vi complexity take version aggregation get iteration complexity algorithm need jump due converge version see aggregate state map original position gets map aggregate proceed stage vi obtain aggregate value time less original combine stage compressed action get iteration complexity fast add action original action move sensible movement converge run plain fig vi make comparison fair speed vi version summarize construct stochastic move qualitatively combine option stress deterministic follow iteration many follow cm aggregation option cm aggregation option try different aggregation speed compressed extract use aggregation happen leave apply never intend aspect system eliminate vi tuple disk take denote action disk disk vi iteration speed abstraction solve e abstraction ignore place sub ignore proceed solve move linear mean speed vi plain vi whole state times vi plain computing options vi plain vi plan denote allow place map onto mark belong use alone trial able reach time plain intuition behind already time amount speed figure bellman equation sound solution medium sized mdps combine option abstraction notable problem finally experimentally option realize apply appendix background information concern aggregation mdps adapt entirely due concern ai I vector reward ai aggregate define architecture value conversely define state aggregate matrix represent aggregate state state bellman mdp call exactly q aggregate equation operate short course since present contain operate expand follow aggregate lead row state column operate state introduce namely exact single aggregate equation aggregate state transition mdp solve equation mdp action
direction constraint w partial derivative lagrangian tangent strict saddle regularity smoothness problem strict saddle project descent algorithm output defer apply stochastic give strict saddle ica decomposition goal find symmetry valid symmetry tensor saddle property solve problem warm saddle apply prove stochastic descent iteratively careful try single component misspecification straight view satisfy strict saddle stochastic gradient unstable expand form scalar rewrite tu l li tu permutation sign global program strict saddle minima defer appendix strict minima permutation base objective oracle application tensor multilinear operation tensor multilinear gx u j u u orthonormal transformation transformation orthonormal observe use technique simplicity follow multilinear z ta ia lemma verify closely relate u function therefore construct order gradient one sample straight share inner take apply result predict reconstruction formulation orthogonal carefully measure f way sample compute gradient sample set easy ica introduce compute stochastic variance use mini size simple generating reconstruction converge exhibit cause saddle significant negative new ica stay decrease rate htb bc bc htb bc bc paper saddle descent converge decomposition step gradient saddle property handle symmetry give detailed strict randomness step assumption add simplicity gradient extend di descent saddle bound smooth exist minimum factor dependent proof analyze behavior three assumption iteration choose equation initial inequality w specify correctness never always make saddle know locally neighborhood smoothness w know imply initialization fw tw tt max relax max max sketch sequence update next lemma approximation around sgd simplicity analytically follow simultaneously q substitute q hoeffding summing union directly eq finish prove generate sgd substitute sgd w carefully event enough fourth know contribution come product hoeffding equivalent finish ready proof denote fw fw fw fw carefully bind choice max finish finally fw know definition locally close minimum bind probability know p I sum since enter hold repeat lemma discuss equality constraint mild point w project argument unconstraine could slightly convert standard unconstraine interested satisfy easily di introduce constrain optimization material technical deal modify constrained want condition quantification common problem say independence constraint gradient linearly constrain unconstrained introduce properly regularity well define easy check curvature everywhere quantitative bound case exponentially close partial lagrangian tangent iw dm know interpretation tangent normal complement vector tangent space assume w w fw give know q necessary due fact optimality please suppose continuously multipli tucker equality lagrange multiplier sufficient equality constraint lagrange multipli kkt know multiplier satisfy therefore strong implication thing unconstrained constraint effectively consider feasible point technical relate equality space much constraint w w w q give give see serve constraint serve curvature next tangent nearby point eq calculation conclude constraint c projection iw since close hand know proof use lemma add project back feasible smooth p first inside ball w w constrain problem continuously tucker know eq ready saddle constrain follow next smooth run direct manifold bound curvature local dynamic locally similarity unconstraine point saddle function smooth lipschitz lipschitz proof condition exist lipschitz bound lipschitz thus eventually need smooth feasible follow without ambiguity calculation linear derivative inverse every transpose rd finally lipschitz bounded lipschitz list essential require modification eq small theorem notation follow calculus know saddle neighborhood everything else proof notation tangent previous subsection define characterize couple sequence lemma notation lemma around tangent project e tw satisfy hold q notation w update gradient remainder project space denote p tw immediately tw later prevent ambiguity event carefully know w recursive easy case hand also choose combine w w fact us saddle point combine prove proof lemma optimization problem strict saddle trying first specify dynamic equivalent section compute gradient lagrangian multipli second partial function constraint therefore strict dependency far strict saddle respectively intuition choice parameter I eq symmetry pick pick line pass neighborhood diagonal mean local finally proof immediately thing local minima transformed problem investigate lemma satisfy know close local must lemma minimum symmetry change coordinate perform change effect I fu support constraint give express coordinate coordinate q derivative lagrangian index indice satisfy strict try dependency exactly strict saddle follow version around saddle relative choice parameter I ik eq suffice j diagonal swap coordinate argument would j know quadratic swap either suppose hand since therefore know negative entry clearly coordinate u du u u concatenation know unit tangent finally c eq proof finally ready theorem saddle lemma thing local permutation sign argument proof chi optimize convex reasonable concern update saddle point strict saddle convex use polynomial knowledge give stochastic gradient descent convex function saddle decomposition rich class optimization formulation tensor saddle property tensor basic solve problem pair convergence gradient descent understand convex stochastic backpropagation success transfer optimize np hard come non convex many minima among even minimum saddle minima point local discrete analog pls deep network main bottleneck minima saddle gradient particular saddle saddle rely hessian usually computation time gradient empirically saddle answer give saddle efficiently call saddle intuitively local progress first efficiently give framework tensor core latent see saddle issue different permutation valid symmetry creates exponentially optimization analysis give online scalable twice call stationary stationary saddle identify convex strict hessian saddle negative seem counter intuitive stochastic descent show stochastic help saddle strict output step saddle may application orthogonal discussion strict saddle give orthogonal correspond valid analyze stochastic setting get first decomposition guarantee economic work property twice function call point stationary could minima maxima saddle hessian minimum criterion positive semidefinite equal point saddle degenerate say strict saddle twice local minima stationary intuitively saddle point taylor
histogram next form protein final element experiment outlier small remove outlier protein cluster row product structure element cluster shape transformation one see inside similar detect protein get rand evident good shape protein class class present assume flexible configuration dirichlet realize chinese restaurant distribution reasonable automate partitioning visualization cluster curve shape broad scientific technique shape assume cluster elastic metric joint wishart assign carefully automatic markov carlo chinese cluster protein shape chinese restaurant wishart automate object area large choose homogeneity minimize homogeneity cluster address researcher g component clustering assume e maximize useful quantification probability population population population introduce comes assume upper assumed convenience parametrize multivariate alternative appeal infer cluster almost focused primarily availability center cluster g shape unlike cluster center quantitie quantification homogeneity obvious important use cluster object preserve scale parametrization cluster however choose exist shape shape project preserve cluster euclidean avoid mean cluster base cluster datum summary encode cluster salient elastic preserve transformation inner product wishart configuration induce chinese restaurant avoid idea configuration model organize follow start study mathematical metric product specification present cell shape protein conclusion study involve protein biology discover structure state protein term protein structure evolutionary origin protein manual classification protein snapshot protein trace evolutionary structural automate shape diagnosis genetic research extract contour shape extract contour medical diagnosis database describe b cell contour paper article shape shape cell visually denote appearance method object broadly base riemannian pairwise develop wishart cluster instead method computationally product specific square transformation drawback nonparametric elastic introduce inner square velocity let parameterized curve domain attention absolutely open curve close curve p purpose study shape square integrable translation elastic curve nice interpretation simplification term care variability protein biological camera distance rescale curve unit c follow rotation element preserve composition parameterization join accord riemannian rotation analyze equivalent translation scale product shape give curve inner well perform optimal optimal rotation whether product u n article distinction specify irrespective illustrated pose curve inner wishart generalize product non sum easily parameter shape inner respectively denote partition represent membership come sub population shape population place inner product panel three block define enable identity encode cluster information introduce conjugate prior intuitively control similarity inner indicate association goal develop infer membership prior let place constitute prior induce partition chinese restaurant crp induce bi n ji th calculating calculate euclidean membership matrix onto threshold cluster thresholding calculate mean euclidean posterior observation cluster find threshold thresholded rare posterior b wishart crp w crp section euclidean discard number trace plot main mode euclidean model px gx gx w control within crp produce keep fig three cluster class contain histogram upper panel right appear compare lead tight intuition right upper right cluster small big experiment present investigate relation fig upper correspond observation infer exceed certain infer increase reasonable crp prior induce typically situation strong sensitivity relation specify fig heat range fix tend tend role probability small recommend choose preliminary association strong tend tight class cc dataset gaussians lead inconsistent finite mixture prior mixture assume assign panel panel cc comparison gaussians unlike case mean euclidean result panel show gaussians gaussian crp confirm convergence slow although theoretically cc c gaussians data shape shape shape shape shape observation definition cluster final result number cluster shape class I histogram number freedom wishart euclidean cluster robust compare shape method class shape result shape assign recognize misclassifie dominant shape class number shape sensitive rand quality similarity ground ground truth quantity belong different assign class rand compare rand index method fourier descriptor vector markov weight elastic shape pairwise model wishart crp elastic inner crp rand I include cost generating mcmc calculate elastic inner since know hmm neighbor result c hmm classification rand index section shape crucial datum hard rough estimate visually grind available provide
importance sec reduction learn sec call adaptive sgd factorization modify lead support generalization first useful thing view one learn use sampling training minimize reduction addition aforementione like estimate conjugate potential straightforwardly properly second method slow however estimate quasi limited memory bfgs sgd tool motivate sgd keep family absolutely include via omit clarity estimator estimator use variance denote variance exist almost nan reduction minimize respect step average equation f sgd lead machine loss function reinforcement scenario follow sequential change variance speedup sgd sample uniformly importance evolve goal expectation accord base iteration instead gradient properly I would depend reasonable rate decrease schedule build function solve sub gradient gradient variance equation e approximately stochastic involve sample direction simplify standard since remain remain valid strongly convex upper weighted standard sgd benefit define minimize moment sgd moment simplicity differentiable strictly minimum assume convex exact appendix strongly smooth step exist inspire reference adaptation rate practice constant optimization imbalance hyper parameter standard mining positive negative importance positive important negative cross good imbalance expensive instead sgd biased sampling mm acceleration convergence benchmark rest feature pre image classification provide strong consider image negative image odd sgd initialize sgd regard parameter sgd sampling sgd fast acceleration sgd parameter gradually learn hard also depend decomposition observe loss softmax square sgd currently one integral policy algorithm expect reward reward predefine trajectory action p grid environment four grid change distribution trajectory canonical squared consider discount terminal state locate successful terminal start optimize correspond sample variance tune learn grid close improvement success sgd sgd benefit sgd problem three connection reinforcement application reduction strong inference integral sgd integral also intractable machine potential importance without technique around gradient compute h minimal strongly convexity minimize inequality tt argument tt smoothness assume objective know proposition www w w line large reason property prove suppose respect condition satisfy definition evolution class negative negative correspond object varied class exception positive intra note dynamic stochastic process value self tune exploit extra cost memory access ns inspire know multiply factor access time weight divide sample achieve sgd plot algorithm time time account sgd speedup compute epoch divide slow access big overhead read memory local seek sgd sgd overhead choice epoch sgd speedup slow actually correctly convergence try use notice sgd improvement factorization column parametrization embedding non slightly fast behave index index prove reduction choose integral efficiently minimize derivative derivative equation main decrease zero unbounded ax qx ax qx minimize equation non sampling optimizing need store implement store denominator axiom conclusion theorem criterion theorem theorem problem theorem summary sgd online machine accelerate adaptively example first estimator sgd optimize show convergence
symmetry get desire sx differentiable l q use mean hoeffding union bernstein maximize exponent though low long original final sx sx dr ff contain q assume hoeffding bernstein bernstein exponent middle exponent final statement solve suppose lipschitz tool slight special collection although result thus pick mean via hoeffding generate entropy except distribute except additive condition event error shift however q obvious achieve convenient mean cast q absolute value l q agree define hold argument get guarantee sign extremely error shift invariant phase thing university approach lead scalability dataset handle approximation error bind give use learn surprisingly two elegant traditional contain attractive approximate shift exploit embedding scalable kernel date embedding fourier transform eq continuous transform let eq b b obvious twice add non formulation use remainder embed count publication practically aware implementation learn library popular use kernel previous analysis increase nystr om fouri embedding problem study complementary view embedding preferable popular prove bad constant expectation concentration discuss effect evaluate two analysis kx sx bt green let approximation error claim diameter embed bind definite diameter fx achieve probability long place net centers net hoeffding replace inequality originally low exist moment finite first moment finite moment material bt gaussian place increase constant though many fx kx full note unfortunately integrate minimum use kl appear unbounded bind truncate relate gaussian kx increase otherwise note somewhat let kernel bernstein style f f claim lower let f tight concentration possible concentrate high stochastically less finite justified kernel likewise p x thus exact result decision use training value eq k kx ii say sx k z hx hx require loose achieve induce p unbiased strong sample serve compare microarray different situation merge database approximate embed bias unbiased estimator simplify noticed subsampling pair reduce favor approximate solver approximation test slow far block quality fx f advantage bind apply uniformly set single two tight consider x evaluate change tell exponential unbiased easily combine unbounded extend uniform evenly spaced matrix embedding curve predict z expected numerically integrate loose first depend tight constant loose old loose predict empirically mm survival mean former low slope black color mean mean squared uniform natural substantially constraint turn distinguish I
learn principle optima many binary hash nonconvex nonsmooth much issue image hash optimize assume continuous code learn code affinity way round optimally continuous code thresholding introduce bit result classifier hash various technique several approach produce function retrieval precision need code binary affinity since million optimize one competitive precision art slow improve provide option three code learn hash join learn code learn hash code element problem code hash incorporate optimize lower learn hash function iterate correct version code iterate emphasis optimize mac cause stage review function mac base propose evaluate use hash focus hash perform emphasis attempt preserve attempt preserve similarity achieve affinity affinity subset optimize hash embedding hash supervise hash sequentially element affinity although approach find affinity usually approximately code alternate hash number embedding free parameter encode base unsupervise representative l nm try project close projection laplacian locally nonlinear embedding objective exist separate point optimize embedding code use two hashing long embedding though focus difficult thresholded hash recently meta construct nest proceed stage transform recognize constrain deterministic hash second augment lagrangian former simplicity unconstraine dependent penalty third apply hash iterating later optimize code projection close try modification optimize target except latter would nothing correspond optimize practice start initialize mac result slowly optimize nn hence unlike method affect optimization optima give overall mac hash optimize affinity ex pca approximate minimizer cycle stop appear term simply hash minimize hence hash classification problem even enforce eventually increase svm optimize slack penalty simplify code complete fortunately alternate would correspond hash surrogate start alternate bit remain fix next start describe modification regularize binary base hamming distance binary consider well mi replace help rewrite define n optimization q tn quadratic qp definite qp qp qp binary use spectral relaxation constraint relax eigenvector small truncate eigenvector qp solve bfgs np complete expect skip update avoid minimum bit bit np bit form variable propose proceed possibly group function code binary code etc alternate bit ignore rewrite rewrite equation block submodular proof use alternate block submodular objective combination hash dataset art hashing focus elastic linear svm hash train radial center svm use gram use svm give use feature sift image image sift image cifar extract feature image test digit feature original near unsupervised training label supervise dataset retrieve nearest hamming hamming inside indicate loss schedule mac hash initialize mac increase code stop stop find consistently low objective unsupervise state art algorithm method preferable mac optima introduce framework effectiveness retrieve neighbor use bit hash compare way optimize step hash mac use subset hash mac svms mac linear hash top iteration mac show mac operate augment trading decrease enforce step typically bottom show reduce sift sift sift retrieved hamming hashing supervise hash sift bit nearest retrieve near image query search mac quadratic step thresholde two hash iterative quantization locality sensitive hashing hash spherical hashing mac create affinity neighbor mac subset affinity less achieve mac take wide training point method mac mac truth near retrieve neighbor practice one would increase retrieve use sift mac training neighbor sift retrieve near set plot mac outperform precision bit wide truth retrieve hamming hamming ex sift retrieve ground truth ground nearest training near hamming precision bit precision retrieve neighbor ham hamming hamming digit retrieve hamming hamming ex cifar neighbor hamming distance ex retrieve cifar ground retrieve image search binary code digits precision bit use bit neighbor search binary code mac use step hash hash iterative quantization supervise mac create mac point show achieve mac overall winner consider retrieve neighbor hamming distance bit change retrieve almost mac achieve well show result cifar bit hash distribution equal code extent hash discussion precision depend user ground compare single mac mac indicate mac achieve well well note retrieve drop large mac ham find number expect precision sift k cifar algorithm bit sift cifar plot code bottom row hashing code dash line horizontal dotted algorithm set mac mac cifar dataset hash hashing proceed greedy find code fit hash final randomly cifar dataset create label iteration mac training bit row test hash first iteration mac function almost case precision bad happen many point neighbor ham precision retrieve indicate avoid available last systematically try number retrieve precision cifar linear svms cifar runtime case mac well precision subset noted work quadratic try nonzero limit hash subset relation well enough amount entry use problem subset precision bit retrieve neighbor ham hamming hamming retrieve cifar mac top mac precision hash bit cifar image ground retrieve image search retrieve bit kernel hash entire originally propose method mac hash mac hashing hamming hamming b precision retrieve neighbor cifar retrieve hamming hamming c bit truth point query image retrieve near image hamming search binary code bit range neighbor code learn ignore interaction hash nonlinear learn tradeoff gradually term find make hash gain function know situation arise selection classifier classifier accord filter hash act produce map input neighborhood role hash pattern pattern preserve relation regardless easy mapping code function hash act objective nature attempt hash optimize fit strict suboptimal separation hash mac still objective involve code hash iterate penalty code close hash e achieve hash quality retrieve code dimensional reason hash retrieval ease hardware software library etc optimally hash simply involve change runtime mac approach iterate besides iteration except warm
important bayes shannon sensitive loss produce function information bottleneck bregman attempt extract solve e regularize mutual information derivation follow concave bregman firstly secondly insensitive seek assume column two surrogate learning assess function page distortion curve firstly loss plot one see mutual greatly mistake classify class error way fp recover mutual information processing inequality eq case mutual algorithm optimization department science theorem proposition definition conjecture feature aim automate fashion drive behind current trend method characterize generalization lead learn linear code independence test unsupervise course old fashion learn familiar flow chart method exist measure performance sake seek overall end feature scheme present transfer result distortion understand well unsupervised possible framework characterization lose map understand surrogate distortion throughout denote instance action space arbitrary include amongst denote norm denote denote divergence shorthand measurable markov give x space learn draw observe learner incur part place distribution act act loss p suboptimal distribution action r xy xy function ie markov xy purpose risk randomization help bayes respectively standard p p apply rule normally restrict sample focus information computation curse wish possibly randomize learn protocol draw observe incur feature move map loss gap feature versus raw non closely notion independent stein theorem label zero contain predict average x posterior sufficient prior assign zero well lead class pick iff lp xy surrogate surrogate log lead bottleneck bregman divergence losse another surrogate minimization include alternate rate distortion theory first something involve third restrict practice know care surrogate greatly contain difference loss varied study field know comparison experiment focus eq sense noise notion appear stein theorem randomization lp suggest calculate material fast alternate behave toy drawback interest relative predict many compact might seem one could use many case much well structure model automate search structure behind deep restriction relationship marginal learn xy l p minimize lost need reconstruct deep belief network highlight reconstruct find structure interest regret choose nearby equip metric wish map equivalent ex dx x xy material require high good feature surrogate entropy prior small reconstruct map view principle reconstruction many surrogate reconstruct divergence restrict normal standard autoencoder autoencoder justify autoencoder terminology involve end performance complete capture distortion rank map loss mutual map obtain form surrogate ideally calculate distortion question surrogate provide least answer information distortion fp xy tight information algorithmic implication ultimately restrict lie know optimize general rather deep hierarchical rather map learn final map n composition map chain final scheme layer fashion analyse system invoke union reconstruction material belief surrogate semi learn classifier comprise normally via map label allow analyse generalization joint something know sample complexity combine complexity work square allow supervised give scheme operate give example particular sufficient greatly contain particularly hence bayes show classifying however jump say gap sensitive generic interest insensitive weighted bottleneck toward class map feature map ht loss determine conditional bottleneck separate function versus experiment fashion example sort concentrate one cost map row stochastic prior plot
hyperplane dt candidate generalize nonlinear probabilistic use probably bind ensure learn safe inductive invariant program abstract hyperplane form generation benchmark literature benchmark tool dt simple ml fundamental verification loop construct program hard approach come boolean inequality reason abstraction learn label task learn unseen ml formalize probably pac one sample diverse bioinformatics finance vision artificial intelligence program consist loop verify value reach program record execution loop head program execution terminate fail loop maintain loop begin maintain existence prove nothing binary classification two good bad loop ml algorithm condition worth note classifier counter point classifier good linear arithmetic program must invariant must ml describe set simple elegant method approach discuss illustrative end inductive invariant restrict program control program begin execution state reach start consistent good start loop head loop assertion fail directly go assertion fail assertion side state safe inductive indicate hyperplane bad automatically assume else em overview overview bad domain default hyperplane translation domain learn dt correctness invariant pick satisfying collect state point good reach bad state good choose invariant benchmark domain inequality hyperplane domain transform bad sample dt process dt rule binary point path equal threshold child leave specify list easily split last bad split state right bad state state child good child pattern pick bad fall computed dt dt follow path root lead good conjunction path simplified annotate invariant pass verify invariant prove program think transition transition safe state want program remain respect program least similarly complement show imply separate good invariant generation space label hypothesis approximate often term think common instance view compute safe set program label characteristic safe safe inductive program assume partition consider represent domain represented dt binary inner node child inner leaf evaluate trace tree inner node hyperplane many dt np separate well follow leave separation normally entropy commonly low reach homogeneous co formally look reach define splitting point reach condition split perfectly separate bad heuristic pick entropy measure use dt dt get abstract program return matrix program hyperplane describe procedure surprisingly get sample function transform dt learning transform tree correctly formula simple procedure program annotate verify box necessary safe inductive encoding formula tt generation dt convert formula reach predicate thus classify leave path leave conjunction predicate formula recursively learn sample output boolean combination inequality return moreover assume generation sound terminate return inductive hard generation dt learner could refinement loop make role less incorrect could potentially refinement large justify probably note key sample justify include pac empirically successful formally guarantee et hypothesis output true vc might class bind finite boolean hyperplane large label unfortunately point arbitrarily point leave end leaf labeling learn arbitrarily nod basic dimension polynomial probabilistic invariant time measure however run routine hyperplane point set sample compare cover hyperplane consider take candidate ease generalize nonlinear add column final occurrence easy correctly require nonlinear benchmark require see algorithm library dt cart learn greedy simple consider variable state run loop state reach state state margin run program fail assertion state program program benchmark domain program class appear learn work state lower dimensional would reduce run algorithm good pca respect basis point form program allow verify static mainly focused consider tool abstract ice ice version use pick similar run version well benchmark base learn meaningful sampler implementation default abstraction interpolation software interpolation inference tool linear combine solve testing benchmark various benchmark choose benchmark among solve reflect tool instead valuable technique dt second sc total follow tool second safe invariant find program arithmetic tool safe repetition case ghz benchmark minute table experiment depth detail ice program solver run ice stop limit ice solve predicate constant ice similarly difficulty sc higher also run easily handle benchmark mainly specialized dealing boolean weak slow handle spend dt learn benefit approximation guide ad hoc benchmark implement verify modulus certain finally one specifically guide overhead algorithms generation discuss sc learn tree benchmark well learner hyperplane hand time hyperplane hyperplane candidate slope sample point yield generation view binary loop learn inductive unclear good ice implication refinement handle however able infer nevertheless consider ice become complex paper infer ice learning refer
theory chapter first real connected follow riemannian th point connect send send point independent rank connect row connect send manifold nearly identical operator depend particular bandwidth ad else unknown graph connect bandwidth compactly support directly connect choose article choice use use principal ideally l denote construct final modify elimination eigenvector evaluate define q elimination denote sample column divide make note accord close complete generate eigenvector create map x eq ball center radius lebesgue ambient small amount embed horizontal circle radius horizontal noise show show eigenfunction pass algorithm color green blue h thm thm example thm lem prop major computer science geometric topological geometry survey space topological begin support lack topological concentrated approximate statistical laplacian manifold related use eigenvector laplacian datum orient persistent homology use directly heat topological community well drive exploit cluster required also non find outside usual proof long apply rigorous justification validation study automatically choose kernel
second equality chebyshev equivalently dt z e protocol give compute normal probability otherwise x bit protocol x stability normal cumulative var var
cg dual erm variant note dual dual runtime sdca could dual repeatedly dual produce run un minimize erm randomized way desire running yield run achieve fast dual primal mapping map subsequent mapping viewpoint warm begin bound center change error dual center sub measurement strongly convex invoke obtain q strongly convex eq recall lemma establish primal let iteration notational convenience fx f g dual oracle q fact combine induction case invariant since hold combine result subsection evaluation rate grain match terminology spend stage pass gradient analysis iterative meanwhile stage dual current coordinate negligible overhead valid time primal dual xy sx nice property solve error rapidly switch empirically size appear inner take amenable mid tuning dual sdca convenient derive update single sdca square locally efficiently decrease algorithmic require tuning end optimizer sdca sgd demonstrate tradeoff approximate proximal advantage shrinkage yet improve stability stage desire bias mnist cifar protein mnist classify vs rest classify vs mnist cifar transformation significantly normalize row take k meanwhile protein pre train randomly hold sdca decay sdca minimizer notion inner period exact simplicity advantage count stage report erm introduce vanish bias towards point investigate minimize erm regularize erm run optimizer extent center help sdca regularization erm heavily meanwhile convexity place cifar recall convexity add sdca least sdca low final line erm sdca legend l erm lastly demonstrate extent statistically desirable cifar take sdca achieve protein effectively erm sdca plot incur sharp meanwhile sdca converge smoothly exhibit degradation stage take research institute berkeley nsf research fellowship several stand alone lemma regard smooth convex combination quadratic strongly smoothness convexity quadratic function convex consequently erm objective error erm notation convex duality optimization turn saddle lagrangian mapping imply duality equality attain primal primal substitute recall primal dual error furthermore f recalling yield international machine france email accelerate fast minimization erm linear wide setting classical provide strongly accelerate box exhibit stability advantageous strongly term correspond erm predictor minimize sample focus linear logistic regressor z capture problem five year increase mild despite solve dependence impact running application erm notion arise correlation solve erm solve erm know scale solve erm option bridge black box erm approximate erm problem regularize erm accelerate erm inner precise approximation operate ascent run erm summarize improvement accelerate proximal convex suppose possibly convergent minimizing order multiplicative previous art well multiclass structured variety erm instance space smoothness exposition clear capture argument illustrate several machine proximal accelerate proximal enable acceleration impose believe presentation simplify broad loop employ accelerate proximal sdca lead erm regularity smooth refer case erm subject comparison algorithm consider interaction primitive refer denote context dual convex denote certain present hold make erm operate namely decrease dual primal risk gd gd sag sdca ap sdca l naive r acc mark regularize mark algorithm system mark minimize square denote run least square briefly explain run context erm gd refer canonical gd nesterov accelerate reduce sag stochastic gradient ap sdca accelerate sdca accelerate latter restrictive solve proportional time general guarantee time minimizer number use run ignore problem variety three slight simplest apply strongly complicated apply prove require erm natural choice conjunction minimizer operate popular desirable analyze proximal concern involve general use derivation technical lemma simple introduce sequel abstraction minimizer finally design quantify minimizer accounting ensure abstraction inner give oracle property typical erm erm accelerate oracle primal complexity erm state one time duality gap error primal induce apply perform dual mapping yield proximal section geometric contraction possibly immediately time un give erm within immediately oracle take yield aside effect namely prove relate minimum convexity immediately contraction every multiplicative prove oracle accelerate minimization factor compare yield accelerate run run part oracle require fix constant primal py x central regard accelerate erm p primal erm accelerate section erm remark similar result
hide update parallel ps visible accord visible hidden versa step update full gibbs long step collect calculate average square gs bethe gs indicate accurate instance grow strength reasonable operating cavity remove node function gibbs reliable replace mean gradient likelihood numerical inference locally visible reach expect bring insight understand rbm role work science institute study brain education technology restrict boltzmann function restrict boltzmann advanced bethe theory evaluate free gradient compare expensive boltzmann rbm deep belief able learn internal representation structured rbm biology activity rbm hard log every accomplish tradeoff require careful way evaluate log likelihood cavity bethe efficient distribute rbm remarkable efficiency confirm likelihood visible layer layer layer visible external connect visible hide visible assume matrix identically layer distribution mean denote ratio hide visible node fig complexity become advanced certain simulation propose bethe original fig boltzmann cavity visible contribution follow consistent equation normalization neighbor factor denote neighbor except visible auxiliary quantity represent product correlation bethe stability due summation iw nearly imply iw b I recursive message eq dependency drop cavity understand node cavity pass pass approximation factorize near neighbor bethe approximation bethe free express e I pass rbm basically densely typical precisely initialize cavity factor iterate converge prescribe time amenable cm free apply confirm result gradient log analysis also equation rbms strength vary display size theoretical enumeration external step single energy bethe fast reach also apart cavity evolution iw instability spin bethe inconsistent single fig step maximal become
computation poisson intensity hyperparameter iteration coverage correct compare interval credible hierarchical model uninformative interval percentile bias correction correction zero bootstrap quantile bootstrap bias obtain resample iteratively correct medium replication correct close nominal coverage minor coverage near behavior bad interval vary bootstrap coverage basic correct percentile fail near coverage compare correct correction iteration credible percentile basic interval number coverage horizontal gain correction simply percentile interval bias iteration improve increase figure prefer produce interval correct suffer bootstrap without resample play role uncertainty coverage pattern situation effectively attain nominal average interval peak difficult correct close nominal coverage part spectrum estimate bias interval somewhat online supplement small number illustrate unfold real large invariant publish decay show histogram count unfold mass intensity tail bias cover peak whole spectrum particularly somewhat physics unfold involve quantification correction technique way hyperparameter performance appeal choose strength cross discrepancy well require specification method beyond unfold frequentist quantification correction correction crucial frequentist base resample credible raise interesting correct govern iteration appear able coverage expense increase interval iteration might optimize within nominal supplement evident interval near boundary space improve quality confidence usually systematic fact parametric unknown generally however may need uncertainty incorporate loop confidence finally point find situation unfold reconstruction smoothness appropriate instance true intensity contain sharp rapid biased correction sufficiently penalty vary second consideration highlight inference family interpret mind acknowledgement thank discussion ed physics unfold estimate elementary resolution consist observe unfold proceed one intensity frequentist solution propose form regularization strength marginal observe credible interval bootstrap achieve frequentist problem inherent introduce iteratively correct confidence enable achieve frequentist coverage methodology apply experiment regularization em call unfold arise european organization research world powerful property produce energy detector vast produce order law physics pose challenge unfold detector detector production angle induce detector version observe produce physics unfold study momentum production challenge unfold ill sense space trivial exhibit solution two denote unfold intensity relate via detector hand inference intensity technique valid unfold account rarely close challenge unfold use maximization smoothed consecutive em smoothed stop analysis variant regularization terminology somewhat svd call account effect incorrectly multiplicative correction recently main iteration physical interpretation regularization stop svd unfold nature enforce positivity solution strength quantification handle standard heuristic worst simply quantify uncertainty confidence propagation little know coverage aim satisfactory regularization frequentist quantification correction bootstrap interval unfold properly take positivity spectrum impose curvature physical helpful unfold separate subproblem point quantification construct frequentist pose discrepancy technique context alternative bayes advantage literature statistical inference process empirical follow use form point like variability energy frequentist statement preferred interval good frequentist coverage challenge otherwise ill pose simulation credible propose employ variability correct estimate interval remarkably length explain unfold unfold process explain detail simulation study world analysis consist unfold invariant close conclude reader supplement technical quantity detector corrupt detector section supplement observe energy follow always additive furthermore sophisticated one usually close analysis detector simulation measurement detector point unfold measurement analysis analyse know phenomenon particle obtain unfold discovery analysis unfold play indirect attempt discover need arise purpose direct compare measurement distribution monte generators spectra spectra published directly compare might measurement sometimes since alternatively detector prediction could reveal exist document server four paper make unfold unfold use physics bin bin em form penalization expect similar mechanism model e compact interval random process intensity fs mb nb poisson set detector spectrum compact negative intensity two relate bounded kernel reality associate unfold process problem pose case operator na I unstable fluctuation understand unfold denote dirac identically sx detector thing happen limit efficiency mathematically indicator z pz intensity point constitute poisson whose identify notice unfold deconvolution unfold formalize methodology discretize histogram intensity model spline b function sampler carlo quantification percentile correct pointwise enable principled unfold uncertainty quantification detail argue technique natural inverse observable discretize first observable histogram application discrete detector reason analyse million observe treating would partition denote see histogram employ follow function basis denote unfold reduce spectra high energy physics function spline attractive whose restriction interval degree interior interior knot freedom order cubic spline consist polynomial continuously order give spline spline conceptual simplicity unfold literature go toolbox spline recursive order detail negativity spline negative negativity note condition positivity spline impose restrict subset spline spline bayesian regression scale decide bayesian reason mass plausible way empirical bayes explain truncate smoothness I interpretation curvature hyperparameter smoother enforce boundary result improper depend orientation posterior undesirable bayes require furthermore boundary near condition smoothness penalty introduce matrix hyperparameter plug obtain bayes posterior point j available intractable hence resort markov sample computed mean monte unfortunately elementary sampler unfold full conditional belong family proposal different scale able efficiently hasting denote sampler posterior conditional sampler tractable density full truncation negative negative replace tail detail supplement provide mix inverse attractive admit strength marginal introduction bayes appear bayes hyperparameter hyperparameter maximizer respect trivial close use integration monte one question rough issue use maximization poisson inverse originally image reconstruction later receive little unfold read complete log step unknown spline conditional hyperparameter depend hyperparameter step guarantee coincide em enable hyperparameter integral monte expectation monte hasting sampler em call monte monotonicity property iteration reach maximizer clear iteration summarize find hyperparameter iterate compute intuitive summarize understand tune vary match sample become closed take normalization hence q constant plug iteration size mcmc hasting result summarize start chain step devise rule mcmc large extent fully bayes unknown allow joint mean compare hierarchical bayes gamma parameter convenient conjugate full single component metropolis loop metropolis correct acceptance attractive unfold base analyst belief typical physics consensus unlikely especially quantity regularization uninformative specification uninformative unfortunately vary limited amount highlight major issue pose frequentist quantification nonparametric inference generally problem see chapter approach bootstrappe various construction bayesian interval argue coverage provide guarantee coverage interval significant demonstrate building intensity estimator problem major bootstrap overcome issue directly variability construct confidence first correct interval similarity quantification de problem methodology seem reduce bias confidence ill pose iterative interval improve procedure indicate stop correction attain nominal coverage interval probability close nominal residual length phenomenon bias bias bootstrap point context model previously employ ill pose nonparametric bootstrap g replace word bias version obtain use estimate correct estimator ignore reason replace less bias naturally I replace version positivity constraint reason value resample hyperparameter correction bias single metropolis compute element correct spline coefficient correct intensity variability basic bootstrap probe band fs property fs fs e pointwise inference point multiple issue elsewhere generate maximum observation r reason obtain bias correct intensity form interval deviation form standard suffer due skewness induce positivity intensity interval formal justification approximately sampling section first demonstrate unfold use intensity intensity fs refer sample interval noise unit discard far deconvolution error discretize bin uniformly place ill pose boundary computer setup core ghz intel processor outer core exception iteration parameter sample sampler start negative least square
logic interpret logic infinite logic truth extend truth enable concept completely false proposition sensor high protein capture represent naturally scale actual sensor level continuous valuable many entirely extension require conjunction operator correspond boolean logic operators max eq mrf variable logical map identical relaxed logic subsection relaxation convex program reason generalize mrfs kind probabilistic mrfs preserve scalable modeling additionally rich dependency mrfs variable semantic variable generalize interpretation explore round represent naturally degree ranking describe mrfs unify objective several semantic eq examine least occur also observe explicitly unweighted far relax easily logical clause linear distance define clause discuss domain satisfy higher penalize relaxed relaxed constraint satisfied tool enable exclusive possibility background restrict feasible aggregate far include value equality introduce mrfs useful treat relaxed constraint mrfs component region linearly hyperplane loss useful section thing piecewise make winner preferable highly reduce weight example follow optimization optimizer term ambiguity objective potential probabilistic preferable reflect smoothly objective term hinge loss optimizer influence function intuitively presence require exclusive strength two exclusive possibility could many prediction specialize similarity optimizer include evidence support square relative optimizer informative complete inference either hinge loss mrfs choice potential map subsection state convenience definition fully empty domain vector constraint index denote constraint give nonnegative free loss energy mrfs place input constrain energy condition probability explore mrfs wide range structure problem mrf rich purpose programming language soft mrfs easily define datum repeat include strength tie people network closure predict exactly define across repeat dependency template template define abstract constraint single dependency template model template random template introduce model template program mrfs parameterize provide interface hinge template logical rule arithmetic mapping clause hinge provide additional wide hinge potential constraint essential support many mrfs setting development kind mrfs convenient definition potential many name letter zero digit program universe must string universe program element universe constant denote person person person program string double character encode within constant constant represent constant logical predicate refer either predicate name every name example friend relate string entity attribute subject predicate combine create atom atom call ground atom reasoning unknown interest take atom whether friend ground atom stand substitute type state template hinge potential induce mrf predicate predicate atom observe predicate unobserve atom atom maps either unobserve valid map redundant sense ultimately specific different scenario agnostic define specify base universe universe eq universe six four type predicate include type close predicate relationship open predicate final observation value atom could value default value remain text formal annotate tag atom type example atom list map map specific rule mrfs induce mrf atom include variable define hinge mrf rest rule logical atom atom use refers minus atom value logical rule logical unweighted logical annotated express logical boolean logic operator de law implication rewrite body head therefore implication body head valid kind logical weighted rule template potential satisfy logical rule end logical induce unweighted logical template logical induce enforce note emphasize weighted rule atom agree atom map produce contain rule interpret induce mrf without loss generality contain replace ground unify objective ground set ground likewise map logical rule annotate potential mrf annotate add mrf unweighted constraint hard consider q atom interpret hinge logical efficiently mrfs inference objective logical basis objective mrfs potential rule arithmetic template logical unweighted arithmetic arithmetic logical ground arithmetic define atom example substitution define arithmetic flexible atom sum atom augment term augment substituting constant atom augment dependency arithmetic possible variable select logical arithmetic predicate constant sum statement restrict corresponding statement evaluate statement affect precede clause ground treat imagine restrict summation arithmetic constant satisfy property select constant arithmetic coefficient piece count without arithmetic atom enable rule depend piece coefficient build coefficient implementation include distinguish take scalar rule maximum far define template loss arithmetic template weight arithmetic instead hinge arithmetic define potential completeness definition equality relate combination augment arithmetic annotated arithmetic rule unweighted atom rule replace agree atom consistently map appropriate possibly statement coefficient arithmetic input arithmetic unweighted ground add instead arithmetic rule unweighted add set index add arithmetic include rule weight equality potential include arithmetic rule annotate induce flexible language usage come many predict among predicate background entity exactly arithmetic express say functional alternatively imagine task predict relationship student one write finally imagine aligned person person align rule say predicate additional argument range incorporate predicate predicate multiplier copy potential mrf potential convex hinge could solve interior expensive admm update find quickly optimizer region region optimize optimizer correct region check replace optimizer optimize case optimizer gradient vector optimizer solution trivially solve inspection definite via cholesky potential often share structure perhaps cholesky among potential optimizer modify fact modify modified objective problem reduce projection operation subproblem easy equality inequality feasible define constraint optimizer optimizer local copy lagrange multiplier local correspond lagrange q likewise mrf em initialize copy appear initialize converge em lagrange multipli copy iteratively update lagrange multiplier local copy subproblem local depend iteration update back subproblem parallelization fast scalable one interesting mrfs map subset must potential map state amount time identify map perform potential repeat section method mrfs maximize find discriminate truth shared potential template mrfs mrf template template associate weight partition template potential template shorthand potential template mrf th hinge loss weight template energy learn mrfs maximize derivative log perceptron direction average point outside region project back smooth ascent divide th template intractable current variant likelihood intractable condition I occur derivative integral admit expectation interval accurate linear group e set block sample uniformly represent mutual label drop interpretation view mrf prediction large shift produce accurate model produce accurate map task structure describe large margin cutting intuition behind truth alternate continuous margin disagreement expect separable relax slack large user rescale infinite follow subject grow iteratively add update subject plane full objective infinite inactive bad separation oracle augment inference potential mrf truth loss simply augment mrf violate standard augment mrf ground interior base loss oracle non interior ground local optimum since concave portion loss objective ground current great round flip variable state correspond svm objective iteratively invoke separation oracle find violate violate solution add repeat one margin slack mrfs square potential potential always distance ground case slack quadratic margin intelligence interest predict inductive programming structural logic several many dependency correlation implication compactly specify proposition adapt domain inference proposition consistent limited uncertainty approach hold dependency rare broad research probabilistic model distribution relationship explicitly allow compactly parameterize represent assignment random describe compact benefit probabilistic operate conditionally piece wide mrfs bayesian construct design usually researcher approach first relational relational schema schema dependency template use server query schema logic boolean mrfs logical template set proposition whether potential ground satisfied proposition value energy probable clause potential unweighted way similar boolean mrfs discrete information relationship mrfs mrfs basis field structure generalize multiclass task appropriate score true structure structure energy model mrfs connect structure show mrfs objective train train structure incorporate regularize broad class structure admit cut plane learn terminate size large margin violate accomplished function often equally challenging structure distance view mrf search structure np focus approximation tractable technique inference view optimization potential indicator certain polytope marginal polytope local consistency relaxation relaxed marginal potential state sense sum quickly predictor research highly message approach dd solve descent dd solve pass solve optimization iteratively nearby relaxed solution allow fast another primal objective optimize binary mrfs support example analog ad use admm optimize objective approximate programming relaxation enforce individual order consistency technique relaxation mrf configuration particular explore relaxation previous relaxation relax graphical bipartite potential condition nevertheless case useful criterion admit researcher relaxation local code also convex relaxation programming therein search discrete dd formulate recent programming discrete mixed programming relaxation inspire relaxation guarantee solution mrfs already effectively domain include vision drug natural language traffic model user prediction easily mrfs discover medium student massive open communication researcher mrfs develop implementation mrfs graphical model boolean logic fuzzy logic capture relax boolean fuzzy model mrfs allow programming mrfs mrfs refine algorithm set tool enable practitioner large accurate model relational acknowledgement would people suggestion development mrfs grant contract pc reproduce purpose annotation view conclusion contain herein necessarily policy imply appendix objective equivalent unique begin hierarchical polytope inner program mrf fix program objective mrfs clause potential constraint redundant simplify analysis make give deriving reasoning maximizer guarantee parameter fully sense potential simplex show fully constraint summing component bound imply complete parameterize fully via tucker kkt maximization kkt necessary optimum write kkt reason variable relevant kkt value constraint simplex also kkt constraint resp fully lemma reason prove lemma imply since constraint exclude convenience fully value trivially yield case definition multiply complete equivalence logical clause nonnegative max vice versa optimization max relaxation definition ex mm hinge markov mrfs suited prediction derive mrfs generalize three scalable consistency field reason fuzzy logic mrfs logic language mrfs pass exact mrfs well algorithm domain well model jointly example biological web vision relational inductive logic seek ever grow capture rich rely combinatorial mean approach scale graphical scalable rich call mrfs mrfs mrfs proportional functions discrete mrfs mrfs continuous variable lose mrfs useful discrete class trade complex connectivity dependency computationally challenging learn mrfs address crucial hinge admit highly scalable without restriction connectivity wide range useful relationship expressive technique structure three approach scalable inference randomize markov field reason continuous mrfs generalize unified inference hinge feature mrfs generalize reasoning relational logical retain language mrfs relational explore class markov discrete mrfs dependency dependency model build hinge logical clause probabilistic enable tool aggregate mrfs applicable probable assignment unobserved variable mrfs leverage map mrfs show decompose method multiplier subproblem loss potential mrfs easily million potential novel run mrfs overlap constraint show mrfs margin margin rely inference estimation mrfs excellent mrfs core collective mrfs offer accuracy comparable dramatically organize follow structured logical clause approach mrfs precise extremely scalable pass inference discuss wide range relate useful tool dependencie domain usually noisy address logical incorporate task one logic probabilistic mrfs popular class graphical rich structured informally mrf logical clause potential mrfs assign configuration mrf behaves assign configuration potential prediction logic excellent formalism define relational logic clause variable index logical clause form expressive equivalently structured clause express dependency condition imply
runtime modify stop probability still maintain true finding contain number operation event tree algorithm different precisely runtime implement seem transform signal advance translate estimation sparse covariance typically whereas nonetheless r sub sparse corresponding sparse albeit say gaussian every sufficient condition assume ct entry significant vice versa probabilistic strong large replace tending find operation leave number sparse entry nn direct runtime lsh library use select diagonal definite increase chose space complexity algorithm store exceed memory simultaneously process coarse modification theoretical particularly reduce space probability dimension increase use small large entry value lead discovery large lsh hash pick number whereas table large runtime various surprisingly opposite sign runtime factor tree slow runtime search number lsh design cope lsh set small thus increase low dimension direct clearly preferable slope direct clearly large slope application artificial decide previous examine ignore effort spend process use average query runtime false choose way inner runtime simulation success single lsh lsh projections lsh reduce unchanged query lsh lsh nn direct scenario near query unlike tune lsh preferable direct dimension importantly runtime query operation sub calculation lsh operation constant logarithmic aspect detect give either requirement underlie demonstrate one raise question future oracle give precise low raise theoretical consider g could lead algorithm require efficiently parameter datum definition q output satisfy union bind conclude proof detect q thresholde entry fail contradict step calculate operation algorithm step exceed query multiply number conclude alternative eq respectively concentration derive alarm divide node disjoint set say visit execution algorithm tree visit fast avoid process tree thus suffice w visit th since sparse therefore suffice occur enter skip denote complementary readily obtain prove absolute consider variable non positive absolute reduce center sum follow proof coefficient sub gaussian let matrix eq since entry absolute probability show sparsity sr ts every either condition gap union bind argument instead imply acknowledgment discussion ram gray reference intel intelligence ci computer science matrix fundamental analysis estimate computation slow population approximately raise question assume approximate detect much theoretically large entry sample operation sufficient approximate real value random covariance denote operation computation storage modern computer several however approximately whereby small even precisely array interested possibly detect location compute randomize sub quadratic reduction fast fourier task multiple call recently simpler prove sparsity furthermore p operation suitable normalization datum sample whose section assumption absolute value sample thresholded statistic study statistical main motivation estimate assume threshold slowly various concerned effort thresholde mainly approximately reality may population approximately provide p approximate empirically entry addition include near potential corpus represent fast multiplication currently fast multiplication square complexity hence expand matrix good knowledge nonetheless work relate problem goal inner product retrieve generalize provide runtime dataset recently simple reduction exact search study decade reference example exact algorithm lsh suitable dimension lsh algorithm approximate approximate covariance perhaps matrix slow runtime guarantee empirically sub tree relate goal recover form partial closely task exact suitable mainly hash algorithm rapidly pair similarity uniformly distribute apart correlation detect correlate vector method sub exponent fast offer op tree weak correlation value entry entry read coordinate priori demand entry simultaneously succeed suffice invoke output row union index run row corollary run suffice inequality omit multiple ij potentially compactly prove appendix runtime guarantee accuracy row compact entry p li r runtime operation entry method detect entry assume dimensional level tree coarse fine subset simultaneously whether contain query way resolve query compute calculation require sample q estimate variance calculation operation construction subset form describe pre process construction suffice lead large row apply start check divide disjoint continue reduce lead efficiently construct full binary height start th vector tree require future denote node level consider
tree recursively bottom work use matrix child parent bottom compute reach root node training minimize propagation structure rnn employ parse parse dependency parse semantic role network direct field neural extension tackle relate translation right role history activation condition back propagation time vanish quickly propagation addition dependency input long enhance lstm idea maintain back vanish receive preserved compute output cell memory gate forget gate gate sigmoid output activation correspond element bias sigmoid activation see intuitively network gate decide output gate access forget gate rnn figure combine child parent child option store information later simplicity lstm two forget child lstm activation gate parent decide gate moreover forget gate child gate forget gate child store compute composition instance beneficial information pass higher gate forget gate cover main interestingly child contribute correspond gate forget gate happen rnns sentiment lstm rnn rnn composition lstm cover word neutral softmax see equation assign embedding leaf similarly leaf leaf node inner let dimension word matrix leaf inner size inner regularization sentiment representation training batch thank propagation method analyse rnn forward phase classification complexity backward gradient multiplication carry sentiment multiplication inner sentence assume unary branch node node inner approximately eq lstm rnn approximately lstm rnn times high difference take core modern sentence second evaluate sentence stanford sentiment sentiment label negative neutral phrase sentence standard splitting also sentence development testing support sentiment remove neutral lead sentence development give initialize train word symmetric n unit lstm test function softmax choose representation epoch network achieve high accuracy accuracy final grain binary task lstm rnn outperform rnn activation function activation seem rnn lstm rnn fine classification rnn fine grained often show work experiment lstm work sigmoid agree common research lstm rnn tensor convolutional neural deep neural rnn keep rnn multiplication composition purpose make deep rnns cnn cnn convolutional pooling handle length hierarchical convolutional lstm rnn achieve performance lstm rnn clearly bad task bad grain cnn lstm rnn focus rnn four model cnn word whereas word fact grain task binary task perform conclude embedding use dropout neuron dropout strong regularization co adapt share neural thank fine grain experiment boost inspire try embedding dropout work lstm might dropout corrupted training difficult pay embedding train hyper parameter experiment select get high development dash lstm perform par grain task taking outperform propose extend long memory lstm recurrent lstm rnn rnn lstm rnns lstm recurrent explain lstm behave pass filter focus lstm keep information region region composition could compression auto boost lstm work compare rnn give lstm embedding gain significant thank dropout lstm boost topic future acknowledgment thank comment propose neural memory allow tree store memory much vanishing allow
possibly estimate suffer high guarantee estimate exactly obtain nonlinearity approximation aforementioned end structure parameter call context address e maximize ml square property oppose decompose nonlinearity coefficient impulse make throughout connection propose give identification toolbox identification formulate modeling give experiment conclusion end output time relation static nonlinear measurable signal strictly stable describe impulse response output corrupt combination input convenience impulse large static know scale fact input output equally suggest introduce impulse gain e symbol indicate product bilinear property dimension denote give introduce result throughout paper lemma extend toeplitz construct dynamic mean vector contain constitute parameter kronecker formalize kronecker let uniquely vector column complete constitute regularize detail frobenius assume drawback square possibly despite suffer large see square error estimate equivalently gaussian vector property subsection design system incorporate first zero smooth typical spline multiply impulse response hyperparameter redundant working case kernel kronecker kernel rank gaussian highlight estimate eq determine step solve establish decompose estimate next become scheme vector review system base measurement relation spline description parameter estimate via maximization solve square estimate next strong produce kernel decompose kronecker product dimensional thus vector equivalence recall bilinear kronecker prove measurement marginal lemma ml prove ml equivalently eq recall bilinear product recall q since statement follow estimate detailed impulse response nonlinearity procedure ht identification solve decompose consist carlo run accord follow pick uniform nonlinearity coefficient depend different snr c cccc snr impulse follow paper problem initialize sample variance implement least review working condition square linear detail well know accuracy index impulse generate experiment static nonlinear op ratio snr estimator identify snr estimate version rank l op need estimate method nonparametric order system particular start spline kernel novel regularize impulse
inform support expensive base incoming message consider message ep infer variable px mf I parametric g family approximate ep f ix x xx I know cavity projection satisfy factor message typically factor complicated integral defining become intractable quadrature numerical integration technique learn message signature v f numerator inference cast message parameter moment importance send offline inference need firstly assume factor conditional sample incoming message rough incoming message ep need sufficiently cover relevant incoming message distributional input assumption message characterize dimensional hence simplify regression contain characterize input treat vector regression allow parametrization operator message analytically opt simplicity online learning inference rich support consider guarantee computing expectation output uv seek regression square loss I define enter straightforwardly message appeal set choose make expensive unseen point eliminate fourier v back nothing generate incoming message study distribution rkh incoming message lr ls random expectation cr dl define distribution lr expect product preliminary experiment logistic use regression incoming message fx operator choose truth message operator output expectation moment kl kl number choose see improvement ridge incoming message convert compute multiplying virtue derive active inference variance similar incoming message importance otherwise message efficiently
entropy fit author average entropy plot see slight benchmark evaluation identical solely ph well link lda cluster remain lda tie mention marginally representation turn surprising cluster cluster cluster leverage leverage learn author show truly representative community understand examine recall reader set agnostic intersection hold user benchmark select another document great us document ph co similarity select another document possess tuple summarize lda low ph co th extreme high high optimizing within cluster explain cluster document rather document belong author prevent community present augment blockmodel item community community membership content fit real accuracy distinguish world community give nsf nsf fa provide helpful david liu community infer community latent item ignore item item tendency cluster share content augment datum enhance community item state arise interact article highly respect metric representative pattern motivate scientific document generative document associate scalar indicate membership stochastic blockmodel interaction document application motivate develop fine grain categorization paper currently fine grained daily newly paper within preference visit website date physics researcher discover people entity interact interact document video form bipartite kind interaction document ignore community tend differently community tend interact document community interact frequently bipartite alone add document cluster observable occur argue community argue hold paper interact tend article preferred community community community interact model co provide document distinguish interaction observable model distinguished comment co advantage access match community distinguish co cluster document refine co advantage cluster address grow interest item content recommender community discovery user detail deal user potentially encode feedback define relevant item belong community user belong membership cluster membership paper completely find explicitly model assume encode sample community follow thing assume blockmodel without notion content item represent trait occur force item vector item choose place thick black gamma gamma right beta w edge connect connect edge connect connect w edge connect c expectation j xy xy describe stochastic blockmodel follow variational dependency grow cluster community document link consider author jointly text allocation mixed stochastic relatively response link draw content article powerful attribute study node node attribute vertex closeness distance introduce initially popularity assume describe node community membership depend popularity membership belong specific node technique variable associate parameterized kl topic choose co category high energy physics user visit website date category vast representative frequent primary validate benchmark lda model allocation lda mixed blockmodel learns satisfying structure lda benchmark treat user cluster al generative preference learn preference vector document recommendation rating document draw eq offset recommendation fit go benchmark et propose link formation depend extend incorporate content require assign content model benchmark modify without another article train vector text pdfs article character appear vocabulary cluster article train ph th article choose benchmark proper increase qualitative evaluation conference proceeding conference conference practitioner research paper present paper author link dataset study conference represent field cluster evaluate form cloud frequently occur form scalar proportion paper belong cloud scheme popular across cluster frequently occur th limit appear display clearly mathematical community separation relax real website complete well purely retrieve research community evaluation ph ga create category go ph ga ph ga paper discuss galaxy paper discuss galaxy ph enforce interpretation galaxy ph ph ga communities interested differ paper separate old paper ph ph ph ga item content paper ph co compare clustering truth ph co ga truth manually naive
essentially require monte carlo rapidly range technical review appear comprehensive survey currently journal area sufficiently finance integrated practitioner reason carlo option sde section motivation performance build conclude brief sde drift motion euler compute approximation stepsize increment sde roughly convergence mean result impose sde standard satisfy constant eq euclidean appropriate euler order absolute euler understood borel small argue relevant sde important make convergence remark sde complete sde financial growth infinity also unbounded occur interest positive constant break nonlinear sde euler requirement fix concrete wish position independently answer would euler path monte average overall error independent depend accurately method approximate sde path discretization bias arise sde remain solution decrease reduce stepsize describe weak behave width arrive like word scale random generator either per proportional hence argue euler like replace argument scale like establish extra place achieve euler carlo computational suppose exact expression sde apply numerical interval evaluation overall quickly sde curve infinite truncate affect later fine detail build resolution fine impact picture wiener six show monte require box sde approximately box euler comes turn sample long exploit obtain quickly large path low frequency path might idea next focus wish mind represent asset risk neutral european style option european call exercise simplicity argument payoff satisfie lipschitz cover different discretization level level stepsize precise simplicity think limit index large stepsize covering interval one refined euler apply sde variable linearity operator carlo expansion hand side indirect hand think widely usual path q illustrate general brownian suitably construct variable knowledge nearby tractable nature sde knowledge explain author sde similarity geometrically refined grid grid resolve important mind distinct notion pass refinement cycle early monte sample via discretization type analysis summarize constant p estimator bind estimator replace euler european style option make achievable formalize confirm euler fail weak asymptotic closely relate direct whether combination euler carlo sure euler event euler rare extremely unlikely gaussian sde euler convergence analysis indicate improvement confirm potential practice matlab code http people ac uk asset geometric european digital price payoff digital option event code monte carlo request picture path level decrease added right indicate term precisely dash line cost predict picture equivalent computation large grow give digital version see monte carlo htb output carlo code site show number target scale call option digital summarize advance relevant option comprehensive maintain http people uk source date coarse refined payoff logic behind euler path close refined payoff lipschitz european put option must refine european option option value depend option problematic sde path payoff close asset change barrier option sensitive barrier logical must able path exception case european call put accept slight digital option consider barrier digital option far option option option barrier option american options monte common variance integration conditional produce estimator quasi carlo low replace outperform finally methodology extend asset brownian motion aim explain manner idea behind base financial monte widely heart rely tight sufficiently implement straightforwardly gain circumstance approach specific developing scenario carlo
review publish gaussian noise represent error summarize satisfy function follow spectral eq j modify expansion formulae k integer z logarithm derivative eq q euler therefore g g l polynomial constitute orthogonal hilbert either admits decrease open condition unknown belong sense say nd satisfied parametric eq weakly limit dependent error asymptotic obtain certain range define rewrite include consideration condition formulate spectral measure gaussian singular spectrum asymptotic vector weakly multidimensional b ccc kb kb b assumption limit obtain class q nonlinear regression q g behavior play crucial role distribution properly coincide b capacity polynomial case random stationary process seem weakly consistent sense seem concern asymptotic normality observation normality simulate value generation quantile chi degree number square mahalanobis plot graph distribution multivariate apply simulated set compose replication note rate need increase verify suggest estimator discretization step numerically matlab function base graph contour form equation ie constant contour study inside plot represent red dot ellipsoid tt right bottom case tt bottom h right discretization b leave top bottom h case h subsection aim normality perform section test plot combination contour plot subsection figure consider b h h tt discretization case b top case bottom tt discretization bottom tt discretization size case h bottom right gaussian harmonic constitute parameter check limit result prove confirm simulation validity spectra regression convergence differs include partially european ga research grant dp corollary theorem theorem european ga partially grant dp harmonic address property numerical prove consistency asymptotic parameter prove two long science harmonic technique formal treatment matrix formulate assume zero count continuous nonlinear regression independent weakly therein nonlinear slowly dependence obtain volume present asymptotic distribution class estimate nonlinear estimation study consistency zero transformation process process complete follow
reconstruction application denoise noisy diffusion recover smooth relate diffusion relate retain embed noisy clean coordinate mapping encoder decoder recover variation recover low new autoencoder clean version point provide denoise assumption addition add clean clean smooth representation implementation net autoencoder procedure neural train unsupervised input previous initialize yield poor deep quantity little employ gain large beneficial encourage unit hide corrupted set autoencoder average divergence mass impose unit specific limited bfgs bfgs line matlab package deep bias layer normal zero mean regularization experimental examine new constraint mini typically dataset show gradient entail multiply problematic mini point local influence eigenvector extract subset mini batch output walk manifold constraint unnecessary mini batch differ deep imagenet cifar etc perform offline typically nystr om nm retain embed train matrix necessary thus train nystr om geometric save embedding training matrix additional application memory approximate laplacian derivation equip bi mean smooth compact equip locally laplacian extension combination eigenfunction exist real demonstrate autoencoder constraint especially noisy finally outli demonstrate verify agree training examine add eigenvector encoder toy effect network layer point diffusion point th great train datum average encoder mse single calculate mse encoder hide row output add know compactly represent layer number unit hide reconstruction embed add hidden improved serve noise decrease encoder trivial minimizing impose constraint yield twice high noisy run vary train layer unit average mse realization several consistently noise include proper estimating diffusion result perform dependence unit give clean I autoencoder denoise capability decrease order fit encoder autoencoder previously diffusion train autoencoder encoder layer decoder fig denoise capability diffusion std example fit scenario autoencoder outlier state necessarily apply distinguish display image acquire sized center image separately patch image sized training dimensionality training set autoencoder image easily capture main structure datum differ diffusion autoencoder reconstruct mapping autoencoder embed properly b point extension point embed due true embed decompose geometric eigenvector similar rotation experiment sec heat equation thus solution pde eigenvector eigenfunction eigenvector eigenvalue take curve diffusion embed rotation embedding share calculate rotation calculate error realization value embed encoder solid encoder much new deep manifold propose designing encoder vice propose training encoder preserve locality embed demonstrate encoder enable efficient linear embedding decoder enable stack together deep autoencoder enable denoise autoencoder properly represents present noisy scenario demonstrate focus perform training processing manifold embed dictionary lead good datum evolve develop instead regular add layer tune datum autoencoder remain recover embed work affect harmonic constraint enable minimal surface encoder decoder decoder decoder provide theoretical decoder expand net average texture synthesis examine affect decoder research examine determine number maximal need system addition explore deep neural implicitly incorporate manifold embed support foundation support award dms author thank suggestion rely bound n let suffice moment fourier rely proposition let limited band bx dx need intermediate address claim fx dx n let transform clearly band mean bx l ball show kx bx bx dx f notation manifold learning enable sample datum embed decoder stacking encoder autoencoder net extension detection net constraint preserve prove encoder also storage space world yet embed ambient reveal year learn develop geometry within neighborhood include kernel laplacian maps affinity capture representation noise outlier capture structure unlike reduction world lie hyperplane preserve apart mining processing compute entire complexity eigen embed extend new representative common approach processing application deep learn gain popularity achieving handle deep net increasingly abstract perturbation representation globally without incorporate geometry laplacian autoencoder locality preserve affinity pre autoencoder formulation reconstruction regularizer embed ensure representation paper approach apply incorporate manifold embed address embed image datum outli representation insufficient three goal encoder layer approximate map perform train inverse decoder recover point stack two term diffusion diffusion outli encoder due outli diffusion denoise reconstruct clean memory unnecessary retain efficiency enable quantity organize provide manifold learn deep neural propose learning enable pre proof encoder present discuss map technique various signal review diffusion see graph point connect measure similarity negative radial use neighbor point neighbor point create normalize set view transition eigen eigenvector eigenvalue calculate diffusion two stationary depend spectrum approximated equation set set general diffusion propose extend new point example create harmonic eigenfunction analytically nystr give extend upon nystr om treat instability due extend eigenvector significant scale dependent eigenvector adapt complexity eigenvector separately implicitly cross validation avoid minimize distance embed embed respect covariance incorporate property datum determined perform necessary memory affinity euclidean complexity add enable network typically layer layer feed cycle loop layer net layer densely computed mapping layer term linear element denote net successfully etc achieving result task minimize supervised predict consist weight prevent layer weight regression multi square q penalty net train variant stochastic minimize loss weight compute backpropagation start output learn use manner autoencoder autoencoder encoder decoder stacking encoder autoencoder train minimize reconstruction try unit dimension tune output autoencoder autoencoder autoencoder classification denoise image lie smooth compact embed calculate euclidean structure address problem detection purpose extension extension pre calculate back calculation datum aim new newly good nystr om scale kernel x near prediction affinity evaluate zero work however uninformative lead mistake indicate outli net encoder decoder deep autoencoder instead autoencoder middle see datum embed output layer layer map manifold output impose constraint preserve decoder inverse back pre stack decoder obtain compute diffusion autoencoder denoise present ht diffusion initialize encoder pre stack parameter tune new initialize decoder stack optimize network tuning cost calculate image point decoder ht stack decoder alg encoder alg autoencoder reconstruction autoencoder outli score learn encoder
fmri particularly create contrast pearson correlation adapt brain theory mutual information pairwise information measure benefit stable test cluster stability cluster area prior prior covariance choice one rule first priori sort covariance allow consistency covariance hierarchy wishart derive prior integrate result turn inverse wishart last simplicity family prior advantage marginalization wishart freedom strategy freedom still correlation identity since correspond uniform correlation admit content inverse prior simultaneously seem variance ideally separate variance likelihood could issue contrast algorithm perspective influence behavior outperform exist method variable mutually independent block one solve difference stem inverse wishart wishart difference relate covariance expect base confirm behave automate stopping yield arguably intuitively structure normal exist start consider rigorously normal grid support de support study case normal mutual usual use matrix e suffer systematic bias additive chi freedom kullback besides I eigenvalue mean determinant mutual lead mutual see mutual information value mutual favor merge marginal correlation systematically give likelihood optimization contain equation equivalent e fmri liu publicly project fmri tr analysis motion estimate time dataset inter median fmri dl transform template body fmri functional nuisance parameter voxel basis hz pass white well principal six body parameter fmri volume spatially mm isotropic condition time spatially brain include cluster apply subject exclude enough volume subject exclude fmri visual thus actually degree scale equation manuscript translate wishart z lead derive wishart matrix j proportional freedom paris universit paris france paris france paris department center op centre de universit mail fr measure agglomerative hierarchical clustering raise correction merging multivariate procedure naturally empirical priori g automated scale measure dimensionality log asymptotically additive dimensionality criterion mutual encouraging derive outperform well toy lead advantage automate fmri dataset identify establish mutual systematically hierarchical keyword cluster model mutual normalize mutual analysis vast variety agglomerative hierarchical sequentially cluster measure shape critical use distance popularity notably information shannon kullback cover tm feature interest univariate apply arbitrary dimensionality suffer dependent mutual normalize version mutual dimension normalize however correction paper normal measure compare covariance element admissible log bayes log marginal sum restriction express matrix automate cluster remain term local global asymptotically I sample enough explicit variable mutual dataset aim asymptotic dataset toy imaging detailed introduce framework namely likelihood assumption respectively section examine compare union decide whether compete model marginal restriction block respectively identically log ratio integral j dependence quantity parameter q calculation j tw lead normalization scale b yield n j wishart degree scale j hypothesis block block submatrix introduction j j I far sake I turn marginalization I inverse incorporate equation wishart degree incorporate equation equation yield quantify amount support multivariate distribution start define covariance right hand expand k k term sum use fact k n taylor expansion give plug multivariate see independence development aim e step read quantify marginal likelihood quantity similarity submatrix previously l expand turn degree diagonal compatible incorporate equation calculation correspond cluster unchanged consequence th obtain merge bayes c quantity nothing use successive degree low optimize cluster yield covariance freedom matrix yield distribution note equation involve advantage bic automatic fact belong consequence therefore likely pair mean probably stop correspond apply cluster automated stop behavior examine synthetic bayes method automatic comparison implement pearson correlation linkage method hierarchical mutual algorithms purpose block precision maximum penalization penalization sm version criterion transformation unconstraine base von either original partial correlation approach repetition mean cluster measure define namely sign absolute time perform code implementation mean simulation perform equal occurrence take normal sample wishart freedom rescale correlation matrix correlation j student equation length vary increment assess various thresholded quantify use proportion classification adjust rand fraction cluster correct rand pool number length multivariate bad index adjust rand percentile adjust rand percentile adjust rand smallest rand proportion classification definite consequence run algorithm operational simulation htbp c rand classifications htbp computational time adjust rand right summarize figure globally affect classify well never outperform method already publish automatic compare particular outperform variant prove simulation analyze conditional study investigate early diagnosis child various component g count statistic give spectral repetition result discard htbp summary correlation partial correlation clustering give table hierarchical started confirm behavior classify accordingly g cluster create variable partition variable cluster partitioning compose g yield cluster represent note identical bayes merge successive strongly bayes analysis lead follow favor independent belong g cluster step bottom row cluster part grey cluster favor hierarchical tool organization brain state fmri identifying brain region highly good simulation see brain aim establish cluster algorithm yield solution examine several literature similarity cluster subject rand rand index subject result capture similarity visually identify cluster rand large include generate rand index close large rand cluster split cluster correlation mutual similar method know plausible sp analogy variant mutual test generate solution decide cluster rand subject panel associate consensus brain panel weight visual brain order hierarchical examine cluster variant yield solution consensus ensemble individual consensus accumulation represent adjacency area across subject method element code brain cluster use criterion one consensus brain region cluster probability subject figure stability method output
early geometry processing shape albeit heuristic interpretation insight many improve compose map cycle attempt method patch manifold likely inaccurate due diffusion visualization score compute global shape bundle mean correspondence manifold analyze discretized bundle point goal global make analogy terminology flat induce flat bundle geometry bundle triplet datum union shall denote value product neighbor edge neighbor moreover also terminology pair block edge symmetric matrix since triplet graph also construction eq set distance decomposition eigen decomposition since eigenvector define length h eigenvector graph theory vector spectral block frobenius norm eq inner produce base data base affinity wise non closely map adjacency laplacian connection g negativity negativity eigenvalue allow power power theorem view addition base capable entry th segment could euclidean preserve automatically similarity simplicity call component bring common template close embed reconstruct interpolation map build correspondence map implicit sometimes version unit sphere laplacian define multiplicative map goal relate differential importance tangent riemannian extend build adopt notation sample bundle review support manifold accord shall equip riemannian tensor inner tangent q unless adopt summation product shall denote q connect curve compact positive geodesic define orientation preserve isometry tangent parameter w symmetric isometry definition standard rotation borel riemannian call define respect normalize asymptotic relative tangent riemannian tm tm tm constant bundle unit tangent proof sufficiently whereas theorem volume see modify geodesic induce metric still symmetric notation shall tangent contexts notation eq unit manifold constant depend appendix laplace project well define proposition tangent completely counterpart practice much bundle base make finite unit tangent sampling tangent tangent map acquire proof appendix strategy technical assumption dimensional riemannian two compactly derivative therefore automatically compactly derivative inverse polynomial demonstrate compactly datum satisfy step point projection respect recall difference euclidean use define probability practice show procedure good basis tangent suffice repeatedly point truly space manifold tangent characterize coordinate basis express approximate map pca coordinate change choice describe detail simultaneously throughout summarize near eq carry svd tangent plane singular singular arrange q decay take decomposition neighboring point schmidt norm minimization efficient namely eq basis coordinate explain parallel compose basis expansion summarize definition collection uniformly unit sd tangent unit column space q plane isometry uniformly projection isometry j therefore parallel q assumption suppose b realize sphere compare near cloud sample unit length unit circle collecting denote neighbor among neighbor choice explain sphere explicitly rotation finally normalize entry diagonal choose various observe experiment investigate influence ratio eigenvalue size laplacian approximate laplacian approximate manifold limit moreover coincide eigenvalue laplacian spectrum similar sampling sphere cloud discretization sampling tangent space tangent k bn construction noiseless non generalize obtain sampling b embedding diffusion similar specify pairwise point template euclidean embed take place illustrate unit bundle stand unit black choose unit tangent give rise discretization tangent bundle interpolation canonical connection vector close discrete extend interpolation neighbor geodesic segment construct close possible topology truly manifold interpret generate connection connectivity store mesh use fundamentally cloud manifold oppose structured triangular mesh formulae provide interpretation experiment follow coordinate eq index consequently definition depend differential operator coordinate jacobian invariance reason define volume respect volume element geodesic flow bundle tangent bundle manifold base manifold see tangent induce volume tangent shall work compact manifold constraint prefer compact non compact motivate unit tangent bundle compact notation riemannian manifold inner metric induce volume know geodesic flow arise horizontal extension extension flow start start write coordinate eq equation define parallel word isometry similarly construct convention everywhere otherwise compactly compact manifold radius parameter sufficiently operator eq moment operator scalar curvature put geodesic geodesic coordinate orthonormal basis eq recall normal tensor meanwhile read q thus symmetry domain kernel vanish argue explicitly characterize coordinate conclusion follow study riemannian geodesic coordinate parallel denote geodesic connect normal coordinate chart center field equivalently geodesic iteratively equality geodesic normal coordinate derivative expression q jt lemma high expansion geodesic hand meanwhile geodesic parametrization combine crucial computation expansion homogeneous coordinate drop armed ready start investigation incorporate parallel deal soon denote horizontal define consider geodesic around sufficiently geodesic radius center contain neighborhood geodesic support geodesic put th lead q geodesic taylor expand coordinate rest simply substitute taylor expansion integrate symmetry computation drop thank equality use geodesic coordinate bundle laplacian define note compatibility isometry naturally isometry carry prove assume dimensional horizontal spherical operator scalar curvature integral ball since orthonormal direct give eq expand numerator expansion denominator numerator conclude compose computation bundle version instead proof proof recall drop higher argue prove path lemma bridge manifold ambient close yx reason diffusion geodesic euclidean ambient diffusion quantity replace euclidean since construct euclidean exact parallel estimate parallel establish asymptotic version lemma compactly assume close manifold embed radius kernel euclidean integral operator associate constant moment laplace curvature fundamental form expand put geodesic coordinate neighborhood geodesic support accord high order geodesic normal coordinate taylor expand around eq note symmetry proof adopt volume denote fundamental sphere integrated place go affect conclusion specifically numerator still replace fact apply expansion pick I key establish last piece large step strategy point I generally directly tangent bundle number stand component explicitly next note manifold variable observation iterate limit leave generally iterate expectation expectation respectively density real value density respect probability denote purpose due bernstein individual summation stem independent come fp bn bn therefore last replace fix trivial notation like manner easily bound reduce compute various uniformly explicitly recall remark interested small note suffice moment moment create notation coordinate lemma side apply determined obtain remain plug expansion back scenario thus direct computation yield short bind constant interestingly term sense control bandwidth find positive q lead error determine notation shall merely dependency bound root q later rewrite q second noise result order accordance reflect accumulate grow linearly increase make unless one appropriately term equivalently q prevent instance pointwise case law shall estimate case replace would since like union interested long asymptotically heat laplacian eq constant small bound high specifically eq eq depend bound ensure probability establishe provide estimate adapt notation compact dimensional subspace minimizer orthonormal frobenius geodesic eq taylor notation large I fact f assumption theorem theorem introduce generalize map massive affinity diffusion vector geometry likewise consider scenario possess structure interest investigate tool study augment goal obtain nearby analyze tangent connection sub riemannian geometry family operator massive science challenging analyze understand datum wide inference mention interest direction laplacian base hessian alignment map diffusion image text etc abstract vertex similarity connect pair build dm interpretation appropriately smooth eigenvector precisely eigenvector preserve appropriate lead euclidean manifold estimate original opposed diffusion wide task precision robustness early instance construct random manifold translate reveal orientation associate algorithm eigen name analogously finite although embed much benefit incorporate tangent sign successful geometry dimensionality tangent utilize bundle notice origin eq isotropic embed blue beyond contrast incorporate tangent yield see sense algebraic geometry curve x dissimilarity score intersection belong distinct parametrization use methodology broad context geometric indeed structural typically encode circumstance detail vertex face mesh text collection shape desirable variation across collection shape score simplify similarity always clear similarity practical heuristic addition situation shape surface set persistent diagram direction laplacian analyzing carry huge degree freedom contain feature oppose characteristic g triangular persistent case admissible surface correspondence surface substantial miss mining generalize dm take different path scenario individual consideration manifold dm denote dimension augmentation around augment look like universal template manifold intuitively parametrization template sense compatible appropriate restriction compatibility datum picture family play role development geometry shall underlie adopt terminology geometry universal template manifold step transition occur either adjacent within bundle refer look application augment object also incorporate bundle formulation transition distinct certain directional impose analogy counterpart manifold lift walk base mild manifold name eigenvector differential new turn couple parameter informative partial bundle geometry experiment eigenvector though focus tangent study tangent bundle differ terminology describe characterize graph tangent shown conclude propose include geometry tangent implicit freedom reasonable ambient semi supervise supervised build label training high reduce simplify data manifold differential bundle manifold manifold parametrize manifold bundle consist class distinction bundle coordinate use act open neighborhood component element correspondence neighborhood base manifold state bundle approximately bundle object single assumption reduce manifold bundle especially bundle manifold piece intersect manifold bundle get trivial bundle restriction understand equally understanding become insufficient datum interest collection surface biological shape govern freedom learn manifold apply diffusion shape distance yet infer variation geometry individual shape away point add interpretability coordinate keep similar coordinate belong
normalize resort rejection sampling ram proposal arise sample proposal perhaps type reasonably behave target example unimodal chain long logarithm dependence proposal manifold extension cost provide nontrivial proposal rw proposal algorithm explicit discretization certain diffusion variance take scale translate inverse number require almost impractical limit target dimensional effective target design problem discretized refinement mesh towards independent introduce weighting proposal discretization sde go use adaptively posterior covariance space adaptively discretization langevin sde general amount elaborate world adjoint possibility avoid code valuable motivation algorithm present attempt combine well without arise discretization arise euler discretization diffusion infinity versus viewpoint former even herein preserve empirical past yield adaptive behave gaussian nonetheless gain quantity limited dimension much large become factor operation traditionally multiplication operation prevent high dimension algorithmic advance outline cholesky inversion immediately cost assume evaluate logarithm unnormalize feasible use low update acceleration algebra operation fundamental level benefit hardware stress significant hierarchy attain hardware operation limit fast negligible memory accelerate hardware gpu memory device thin express magnitude bandwidth gpu illustrated across gpu reduce slow operation increase memory bandwidth gpu processor logic nearly physical due frequency forward intel etc future value force parallelization traditional inherently nature nonetheless justify merge parallel chain within potential scale reduction diagnostic parallelization work work explore partition forward herein parallelization efficiency chain periodic slight strong black principle apply note elaborate parallelization recently tackle consume core available gpu operation provide optimize e hardware thank advanced sampler multi way operation propose great impact black rest precisely finally present diagnostic justification experiment acceleration extend multiple highlight probability indicates give number assumption correlation assume readily evaluate present notation algebra cause confusion know observation observation vary inverse problem formulate discretization limit measure mind big scenario statistic need imply explain dimensional space increase potential increase inverse observation parameter intrinsic property full informative space effective bayesian problem albeit connect expect markov transition dx qx follow notation qx dx analogy quantitie stochastic measure ergodicity tv x rate approximately function calculation identity time limit effective metropolis refine version perhaps amongst essentially arbitrarily compose accept follow satisfy give define many behavior kernel one subsequent turn large size metropolis hasting right computation mention subsection basic hasting indeed advanced black box mention sec finally present work begin increment motion euler discretization give walk rw n identity although bayesian maximizer turn symmetric distribution use start point posterior furthermore multiplication q notice preserve imply acceptance nothing discretization one extend extend introduce proposal reversible general proposal name derive incorporation inform proposal allow effective play role inform direction gradient present work upon proposal plug identifie propose step acceptance consider presentation substitute account eq I else may approximation case kullback independence target acceptance point non proposal coincide turn proposal trace proposal par target necessarily target additive performance although dimension exist investigation parallel serial proof rely nonetheless diagnostic chain covariance diagnostic justify chain chain batch interval follow within merged moment mention potential scale diagnostic chain merging start quantitie chain chain variance indicator systematic begin normal generate eigenvalue standard log fix eigenvalue dense matrix multiplication pde forward solver realistic simple shaped target exactly computable case consider diag I order correspond jacobian determinant change maximizer furthermore course two distribution spread target practice reduce highly covariance big target reduce clear target problem posterior example pde forward decay spectrum parameter big hard sample gaussian acceptance initially four autocorrelation measure rw panel autocorrelation function projection eigenvector eigenvalue subtle competitive although suffer target whose condition increase show algorithm perform eigen bottom perform worse expect give multiplication involve fast fourier argument indicate roughly eigenvalue inverse eigenvalue prior pre least turn outperform outperform outcome simulation aside specific length burn parameter affect proposal case run separately various varied reach convergence fig necessary show due operation case increase curve due effect average simulation converge multiple value target suffice experiment total effect cholesky update large operation consequence large memory discuss section illustrate convergence reduction factor describe require right hand stop chain relative covariance fall convergence latter euclidean gpu accelerate library hardware landscape x call cpu gpu thousand core computing capability magnitude peak compare standard speed gpu memory cpu bandwidth cpu specification challenge gpu order magnitude memory application technology e parallelism move communication reduce communication software stack hardware kernel available particular dense linear operation thank regularity three basic algebra e multiplication multiplication mostly bandwidth kernel thank kernel bottom software chain critical parallel architecture library library mkl library implement last cpu high inversion flow target x x ref x ref x x accept accept n compute batch update lag target evaluation weight every cholesky matrix inversion bottleneck increase operation therefore basically compose generation triangular operation perform general perform general matrix operation perform mostly compute symmetric factorization mostly compose gpu library implementation determine kernel need platform operation well symmetric dense multiplication highlight cholesky inversion compute intensive overall parallel frequently hand memory exhibit arithmetic lag investigation exist implementation operation library operation occur intel mkl library data movement cpu gpu slow try operate persistent gpu memory motion asynchronous overhead bridge library ensure parallelism thank parallel rely join parallel programming parallelism batch batch processing share facilitate safe synchronization load imbalance sophisticated parallelism another cpu intel library chain implement mkl mkl running chain core therefore critical various define cpu specification bandwidth stream benchmark
error statistic look small minimize realistic parameter skew high estimate distinct fold mid however optimistic frequency compare versus stream observe limited agree stream estimate platform counting frequency distribution cv frequency rather rely large body toolbox deterministic work quantile vector linearity imply processing frequency statistic design distribution support query effective skewed overhead relevance monotone statistic encoding update outperform particular sampling use sketch sample roughly obtain replacement sample replacement skewed repetition overhead sample query weight linear presence conclusion statistic analysis accurately frequency estimate compute bring distinct frequency look ahead framework extend boundarie attention use start partial function weight start get particular function claim sampling randomization use randomization assign element adjust observe randomization density threshold correctness change remain value determined value depend randomization preserve claim claim respect initial case density u du maintain count condition notion dominate distribution threshold dominate dominate fy dy ready upper cv build function distribution seed dominate th small exponential exponentially distribute condition small difference order seed exponential combination equal order parameter sample weight segment cv extend take seed whereas since consider weight population key perspective small seed dominate suffice upper bound expectation different square variance inverse estimate minimize unbiased estimator sum per surprisingly cv segment cv condition contribution dy inequality relation seed take zero analysis work seed exponential condition seed density seed key xy dy e decrease hold xy dy dy eq consider since substitute dominate small seed draw dominate latter ready inclusion key eq inequality relation substituting remain treat establish maximize size e recall already assumption denominator minimize substitute establish tw last ready conclude dominate obtain key showing segment proportion cv variance outline pass conditional tw tw coefficient bind tw eq theorem w inequality use dx tt subject w dx w e increase e te w maximize useful work set think cache drop threshold back queue depend rand return rand kx xx x threshold x design compute segment estimate stream construction store need key counter stream cv logarithmic present logarithmic apply stream string return score counter stream key stream stream key string string key thus counter would rough inherent counter weighted question extend objective sample feed program optimize inverse eq relation obtain substitute last var tw py exp title exp exp title x title thm thm example thm thm google ca usa pt pt hash seed diverse source web service ip traffic utilize express segment apply frequency key segment parameter statistic computation instead would include easily aggregate costly state active ideally without aggregation present pass stream two pass design classic provide decrease gold tight single stream utilize segment special define frequency active practice sample sketch approximate exist design stream general frequency estimate monotone benefit unified single service interaction service ip traffic key universe stream distribute storage aggregate consist active occur total uniform frequency key proportional frequency frequency query nonnegative segment population typically carry contribution less prominent moment frequency eq special moment sum element segment mid typically frequent mid limit typically target segment say place pose facilitate interactive exact frequency aggregate representation aggregated produce distinct resource process stream need element maintain translate communication pass discard ip statistic live address retain sampling include replacement poisson weight approximate segment frequency understand tradeoff segment value cv normalize root square segment fraction interval actual segment cv gold design estimator aggregate pass maintain query hold family another sum query suit distinct query distinct support unbiased frequency meet cv bind contribution sampling specify scoring element tailor want cast stream cv gold frequency offer arbitrary spectrum algorithm parametrize exceed maximum derive admissible statistic specify integer continuous parameter derive continuous everywhere statistic differentiable surprisingly perhaps spectrum elegant simple estimate statistic upper cv estimate unbiased monotone decrease nonnegative make transform frequency invert transform inverse transform minimum variance unbiased nonnegative meaning optimally inversion estimate flow sample spectrum estimator address application estimate multiple close propose tradeoff spectrum aggregated notion simple technical resemble application include demonstrate accuracy suit universe set appear key key pass maintain size execute sequentially location quick scheme dataset specify frequency interpret key estimator key key guarantee estimate segment normalize well bad scheme statistical fix size quality correlation scheme draw seed scheme treatment successively key seed threshold take seed take weight sampling inverse depend available seed interpret condition randomization covariance query sample pre turn statistic cv respect aggregate generally moment cast distinct sampling cache sample scheme distribution seed key minimum sample include small seed value seed apply exact datum detail pure streaming setting platform scheme fully pass stream flexible execute process provide process first pass identify seed score fix summary small summary summary summary merge union seed summary seed summary exceed compute summarie weight merge two summary union pass stream sampling distribute parallel rand hash kx w fix threshold sample initialize discrete initialization continuous rand hash return xx cache stream element key xx algorithm discrete algorithm maintain keep count seed processing element increment result key seed fully reflect seed currently repeat key seed set iterate count either become latter rand xx supremum range work stream per maintain queue active cache distinct final count place low value count element queue decrease get queue probability number key seed attempt key seed way seed roughly whereas roughly regardless illustrate key select terminal font line set left title lc red title lc blue title lc green title lc title form express parameter th contribution sample express f non element express element otherwise grow suffice entry use write note suffice entry pass inverse invert streaming estimator random express relation triangular iy q computing sampling entry substitute total unique admissible need computed sketch show monotone claim positive induction show rearrange nonnegative follow hypothesis key nonnegative nonnegative nonnegative monotonicity sum present continuous scheme update continuous offer fix explicitly maintain value implicitly derivative spectrum multi stream base hash draw return minimum key detail minimum exponentially variable happen seed satisfie qualitatively exponentially result property roughly pass scoring need key weight sample statistic proof appendix cv smoothly cv inclusion gap aggregate relative gold large ratio conjunction terminal option explanation load color graphic graphic macro ltb lt lt lt lt lt lt ltb lt lt lt r p ltb ltb r ltb maximum normalize streaming pass perform compute break otherwise break initialize assign mass return xx cache stream size algorithm maintain threshold key working begin randomization key randomization compute randomization threshold maximum remain elaborate key computed cache score randomization necessary process element key enter simply entry rule key enter cache enter cache rand hash xx initialize cache w z xy h adjust b verify key depend cache statement condition randomization provide key comment cache expect however reduce modification cache full th present next rand hash x initialize cache least xx size transform state obtain relation coefficient seek unbiased nonnegative continuous differentiable unbiased treat first coefficient count count consider expect fw w fw fw fw unbiased dx dx continuous monotone requirement cv appendix cv bound cv cv establish estimate statistical query statistic suffice improve basic instead set improvement ensure sample proportional coordinate randomization randomization constitute variable element score express seed include value fewer surprisingly perhaps fix na
polynomial like arise newton like method walk adjacency diagonal weighted degree walk induction undirected graph walk consequently walk polynomial size dense laplacian algorithmic spectral recent technique degree paper nearly spectral walk observation design utilize mathematical critical walk weight negative construct spectral zero walk constant even construct eq approximate handle degree directly invoke even degree degree om build overcome matrix appear would expensive due multiplication power rely clique specialized polynomials newton like find equation cubic polynomial root factor one challenge slow complex new series careful adaptation error spectral condition approximate simple mathematical algorithmic advance middle turn matrix induce walk offer enough loose similarity algorithm together preprocesse query regard effective logarithmic due connection widely nearly useful variety undirected vertex transition walk analysis make spectral semidefinite positive semi definite usual approximation positive scalar situation lower relate electrical flow view recall potential current else equal obeys add graph suffice guarantee weakly critical upper bind apply standard number upper efficiently two support integer fraction draw draw fix u preprocesse draw edge two random walk length time combine algorithm efficiently low g run even effective edge first path accord integer subsection put give spectral eq lemma r om complement vertice regard complement semi definite rearrange add eq extend routine walk walk side composition use expand since q us write support edge laplacian om nonzero support first q u describe walk replace uniform proportional additional occur chain approximation however remain spectral combine degree invoke lemma reach approximation invoke approximation final dd rd g dd term effect sampling splitting split diagonal laplacian upper difference need extra lemma analyze error matrix definite nonnegative nonzero nonnegative r n om decompose sum column incidence odd correspond weight u upper walk exact lemma exist q via multiplication nearly motivated equation high degree routine call range mathematical future nearly routine widely well explain effectiveness symmetric prove laplacian sum follow induction ii unitary scalar claim graph na entry everywhere theorem algorithmic question theory random undirected matrix recall walk graph polynomial become hence enjoy nearly challenge path precise calculation would expensive multiplication power nearly edge laplacian polynomial well numerical equation motivated problem classic sample transition reversible application random fast challenge polynomial slow efficient structure multi reversible markov task field mathematic encode various physical taylor fast numerical responsible engineering range weather forecasting galaxy polynomial arise mathematical dynamical matrix equation adjacency undirected graph use transition matrix walk power step walk graph prohibitive memory walk motivate walk close representation classical newton reduce radius sampling gaussian graphical model random field specify gaussian distribution representation convert vector
article informative feature yield give block theorem embed human md usa center md usa mathematics statistics md university fidelity manifold dissimilarity multiple dissimilarity optimize fidelity modality stress transform special inherent efficiently compute transform greatly measured modality yield object dissimilarity fidelity object modality common dissimilarity one optimize fidelity e preserve within dissimilarity across modality preserve raw multidimensional exploit employ dramatically procedure see synthetic matching common dimensional wherein joint investigate application computer machine example survey manifold match broad entire correspondence proceed raw multidimensional represent inter treat missing represent procedure embed miss embed scaling dissimilarity potentially embed see cross dissimilarity routine attempt minimize raw criterion row euclidean fidelity versus embed optimal preserve dissimilarity expense correspondence cross correspondence expense dissimilarity optimal subject see detail raw stress optimize fidelity regard cca optimize embed regard fidelity see pairwise distance fidelity pair distance point modality parallel combinatorial define scaling write iterative generate transform derive hence algorithm closely metric multidimensional stress multidimensional scale weight embed initialization I value expensive fortunately application sometimes simplification familiar unit permit much simplified calculation algorithm first notation mn hermitian follow block dissimilarity weight dimension configuration orthogonal onto x I remark output iteration diagonal block denote qr additionally iteration algorithmic give algorithm algorithmic parallel iteration algorithmic complexity serial parallel potential dramatically increase achievable nm nm plot replicate increase versus variety ghz processor gb let set represent measure modality nm nm identical iteration average monte replicate average average mc replicate relatively increase dramatically figure serial ratio time suggest contrast fix vary average mc replicate ratio versus nearly factor utility multimodal wikipedia article english wikipedia geometry exist article link consider undirected correspondence article wikipedia associate short path graph cosine semantic article across modality serial factor embed embed wikipedia article article preserve pairwise result dendrogram two cluster dissimilarity preserve two call dendrogram merge histogram article article article less dissimilarity preserve across label dendrogram height rand ari clustering ground
sum regular radius provide minimax estimator prediction appendix base standard pack critical fundamental quantity appropriate benchmark estimate consequence notion sketch achieve achievable corollary type randomize choose illustrate experimental simulation tu u column correspond eigenvector lead whereas remain sketch matrix whereas formalize sketch follow noise upper sample sketch great matrix sketch apart design randomness randomness proof source project noiseless define whereas fit space elementary sketch matrix induce range estimation error lemma approximation prediction sketch however computational conjecture dimension compute nonetheless various randomized construction sketch various form randomize proportional dimension previously throughout require sketch equation constant purpose state high function guarantee gaussian I problem satisfy universal three overhead sample statistical sketch scale statistical scales cube x illustrate perform begin u unknown spaced theory kernel ridge hadamard note predict prediction confirm plot prediction error three curve sketch sketch tend prediction versus decay simulation sketch dimension panel rescale remain x versus rate complexity nystr approximation om sketch row error nystr column match number substantially dimension performance nystr om poor nystr om sketch empirical diagonal k kk sketch take degenerate depend lead substantial approximation ccc approximation panel panel rescale top covariate arrange unit om compare nystr sketch return kernel h interval type nystr om show regular function namely perturb case original closely contrast nystr om approximation behave intuition precede nystr om recent work sampling involve extra leverage obtain kernel matrix minimax regularization scale put piece sampling must requirement large prohibitive statistical contrast whether approximate leverage score statistically optimal theorem definition remainder technical lemma theorem matrix lemma conjunction accordingly prove two simplify proof rescale statement program must feasibility applying obtain basic least appendix auxiliary analysis basic inequality imply sketch randomness imply inequality rearrange universal exhibit recall u u constructive transform block assumption recall triangle j put piece sketch value turn remaining previously state combine two randomize specialized broadly analyze save computation hope result support office grant dms air grant microsoft fellowship show end nystr om kernel constrain program qp estimate solution via I turn write constrain nystr om formulation base om apply formulation begin g j consider x version subject draw index condition I remains pack mutual ball packing norm q take packing kl regular yield prove w standard jj variate tail union underlie noise vector consequently note union combine last display z violate bind k standard consequently concentration gaussians eq moreover inequality ii radius recall rescale set tail completes prove schwarz inequality suffice show choice side family since increase setting eq equip bind violate ba b violate vector second lower bind early claim theorem state projection guarantee formal split straightforward discretization argument j consequently union part v measure gaussian turning inequality follow piece mt taking complete argument analogous letting sketch c eq euclidean thus result lipschitz eq row c claim follow section assumption definition em university california berkeley electrical computer science department kernel ridge reproduce respectively prohibitive dimension preserve optimality prove sketch proportional regression goal parametric regression make prediction covariate vector say covariate covariate standard assume function regularity enforce reproduce kernel rkhs short assumption estimate lead estimator process appropriately yield minimax attractive property complexity take different machine base partition combine average divide approach zhang give split guarantee forming rank nystr complexity exclude estimator section classical widely algorithmic context project choose subspace involve complexity approximate processing complexity suitably projection tt cluster pose connect definition projection constrain project dimension minimax optimality result estimator contribution several projection organize devote background regression reproduce dimension statement sketch consequence class confirm nystr om devote proof matrix advantage orthonormal dft hadamard say detail focus orthonormal bound meaning entry dft identity p identity understand sketch identity orthonormal use randomization nystr om characterize definite u correspond rescale sum truncate function g uniqueness radius radius bound
dependent word embedding consistent model produce discriminative counterpart word embedding candidate select candidate mean contextual word across language similarity principle phrase recurrent rnn recursive autoencoder long convolutional semantic sentence accurately architecture task largely synthesis sentence one hypothesis distinguish candidate contribute procedure turn reason set work context phrase context improve performance obtain score integrate deep architecture dependent translation way exploit contextual information target partial novel translation statistical translation convolutional language specifically design encode semantic translation pair context phrase similarity translation pair adopt classify medium gradually ability represent phrase context use example difficult experimental significantly outperform conventional translation translation construct word word align corpus phrase however utilize translation surface form capture translation pair utilize phrase distance phrase incorporate method prove decode pair treat context accordingly fail local context context capture propose neural semantic similarity phrase pair language phrase pair sentence phrase source phrase layer perceptron classify triple phrase negative distinguish phrase negative train similarity phrase dependent semantic convolutional matching phrase system outperform proving incorporate local context context similarity phrase section learn strategy section research build phrase employ word phrase employ sentence context word guide treat distinct exploit contextual part tag word pair similarity sum matching score phrase directly similarity convolutional strength network focus capture context instance match occur sentence different precise sentence sentence moreover phrase derive build train large phrase match representation phrase similar language translation semantic mean continuous project source continuous language phrase exploit two semantic capture similarity pair context context useful match dependent sentence convolutional sentence capture dependent translation phrase compare perceptron symbol indicate turn model consist convolutional sentence mean phrase compare perceptron sentence phrase project feature compute matching take train elsewhere summarize mean layer reach convolution take slide window involve composition slide convolution gate function whether activation relu convolution unit I word vector slide term distinguish pair additional embedding phrase transform embedding convolutional take composition slide slide phrase exploit contextual source phrase zero dependent phrase vary embedding language embedding capture across language capture level could embedding language similar language representation utilize mt encourage similar word embedding embedding contextual inspire study word embed exploit word align nearby window word word representation return embedding matching score accord word train eq either machine refer sequence strategy learn easy gradually increase learn benefit give rise randomly present organize gradually negative difficulty distinguish phrase target phrase sentence target phrase want semantic target phrase medium semantic phrase vary context difficult mix mix reach local minima random mix example alg different training consist example easy increase line use sgd meanwhile capture language reach minima reach terminate propose baseline outperform context counterpart embedding counterpart translation chinese english score local translation initialization conventional embedding embedding counterpart indicate embedding well experiment chinese english contain come dataset portion language portion corpus use mt mt minimum insensitive translation neural pooling layer map slide window development produce high phrase decode collect phrase obtain phrase pair phrase context phrase least corresponding phrase phrase remove undesirable mt mt two baseline system system translation phrase convolutional previous degree phrase contextual information serve phrase phrase baseline matter significantly translation gain
hold add equation acknowledgment discussion lemma let increase calculus increase j eq use k k substitution hellinger simplicity depend j kp pp c constant depend decrease substitution ratio multiply therefore adapt constant square substitution remain let k x substitution I c constant sum get hellinger distance bind q combine k kp lemma part constant expression regime chernoff depend appropriately substitution compute hellinger get sum hellinger distance use let whereas computation previous similarly let chebyshev enough tight relie relax collection gene section split equal sized disjoint sub convenience proceed infer compare quantile bin algorithm succeed probability whenever sum repeat lie monotonicity p c come q vice versa follow together consider three specie triplet molecular rgb rgb berkeley reconstruction establish seek connection trade reconstruction gene branch corrupt typically application establish error explicitly boundary paper establish signal reconstruction multiple gene population genetic evolutionary subject study survey particular grow body understand research history theoretical connection derive boundary trade need accurately reconstruct signal extract gene method e evolutionary evolutionary relationship species leaf problem common gene seek reconstruct principle time difference formally estimation root leaf leave root one simple markovian model molecular evolution mutation mutation derive continuous markov mutation associate root u u two adjacent root combine leave also identifiable long evolutionary biology survey learning problem theoretical sequence theoretic upper general molecular evolution context g access gene abundance new survey evolutionary gene deep horizontal loss sorting phenomenon paper accounting leaf label binary specie past divergence share consideration associate gene mutation pick specie mutation parameter inter time gene conditionally specie sort give combine genetic explicit perfect base method distance distance sufficient accuracy reconstruct detection lower label leave early distance estimate differently require leaf limit distribution respectively notation gene sparse positive total variation distinguish bound spectrum imply need bind recently spectrum regime regime gene tree gene reconstruction decrease analysis whenever test distinguish reconstruction molecular trial equation recent formula boundary bind instead hellinger q prove proof regime sample test test gene specie gene event different exponential depend effective theory get specie success property give leaf specie specie gene conditioning leave look branch exponentially independently branch merge population proceed
integer sum partition subset number instance consider inequality evenly partition verify lemma assign sum equal lemma continuity decreasing must satisfy meanwhile minimizer minimizer interval discussion interval overlap since overlap element thus must global lie proof due lemma case strictly need equality become fan comment definition thm wang keyword nonconvex concave penalty penalize statistic past decade high review recent advance refer reader two smoothly scad concave mcp concave penalization enjoy computational concave penalize due intrinsic structure many local quadratic linear however regularize remain author penalty np hard assume accordance assume main monotonicity decrease concavity concave hard literature fall category global strongly arise naturally select generalization penalization penalization correspond bridge obtain thresholding penalty satisfy widely scad mcp penalty study function also prove result follow decrease follow concave lipschitz np strictly np hardness among example work concave penalty addition proof considerably proof end property penalty furthermore
autoencoder look quantitative aware imagenet generative convolutional imagenet lead finding include layer preserve color show color present retrieve object convolutional pixel layer precisely layer information non zero activation precise value object small supporting allow generate somewhat look feature representation principle visual representation could modality acknowledgement acknowledge start grateful sharing thank comment supplementary material figure similar one main reconstruction reconstruction reconstruction reconstruction autoencoder figure contain reconstruction autoencoder reconstruction vector two autoencoder example generate feature standard shift leave percentile normalize feature multiply average generate significantly complicate th main feature autoencoder less realistic image image way every say neuron look class reconstruction somewhat interpretable much image cm cm autoencoder cm cm cm cm cm cm cm cccc cm rgb particular good learning representation extract new inverting convolutional imagenet numerous insight representation color rough contour reconstruct activation even class convolutional network cnns large impact technique vision nonetheless typically towards understanding cnns allow image perturb vector insight task invert trivial usually pose train noise illumination mapping many indistinguishable interested inverting n inversion impose manually implicitly natural convolutional allow supervise squared reconstruction natural cnn imagenet insight probability sufficient reconstruct input image activation detail propagate activation identify responsible activation activation make extra maxima information crucial inverting also feature seek minimize loss regularizer enforce optimize solve feature produce feature approach map try hard distinguish care moreover feature representation relatively gpu costly inversion forward per image gpu representation example reconstruct base invert traditional computer invert restrict invert neural work make backpropagation advance architecture allow modern convolutional architecture generative study pre website layer process please h c c conv conv conv conv conv fc fc drop fc relu norm relu relu relu relu relu relu channel cm c layer conv conv conv channel c fc fc channel train set convolutional minimize image speed computation image slightly improve convolutional map value original corner comparable varied architecture generative depend layer architecture reconstruct convolutional layer reconstruct connected architecture reconstruct fully connect three five layer convolutional layer depend layer reconstruct slope imagenet use optimizer mini gradually learn towards design low square interpretable favor suppose visually even reconstruction cm cm net decoder representation encode autoencoder stay fix decoder autoencoder reconstruction net autoencoder invert high layer neither much error reconstruct qualitative reconstruction autoencoder even reconstruct autoencoder training estimate lose autoencoder reconstruction reconstruction actually convolution max pooling much beneficial slightly reconstruct deep deep reconstruction even deep layer error reconstruction autoencoder low cccc cm cm cm check color preserve high network green image layer softmax give large pass inversion zero lead classification color precisely reconstruct probability depend maximal top small probability probability predict carry test high preserve reconstruction preserve motion interestingly high image indicate horizontal image cm bin level preserve rich represent gain insight representation perturb change reconstruction carry reconstruction dropout sign vector unchanged try except feature negative set non dropout normalize unchanged normalization qualitative perturbation feature layer quantitative surprisingly dropout reconstruction quality although know convnet well reconstruct important code activation binary training autoencoder sensitive perturbation code dropping dropping
rapidly network efficient way analyse modern genomic network analyse genomic association genomic year dna breast cancer platform cancer genome pre process mapping snps remove removal gene annotation remove probe replace knn imputation annotation information platform package check effect component phenotype batch significant would individual patient survival age individual potential dna clinical validate expression node gene rna protein domain alpha I domain contain fusion protein activate ii protein box sort protein activate protein alpha gamma beta family member neural box family signal protein relate early cell contain cd cd adjacent ap bind protein family member family interactive rich rbm rna bind active reading member nucleotide interact protein beta protein protein neutral neutral heat bind protein contain contain node box alpha protein member homology c open member contain member breast associate domain contain repeat protein like bind dominant negative protein alpha activate death nr member alpha neutral ab cdr protein cell sec h alpha gap like bind node l protein alpha four domain alpha b nk bind sub member sort six six open reading repeat protein code rna protein protein di alpha protein open read rna cd b body protein activate member protein bb bb dna domain member alpha similarity member g ph domain transfer protein growth b bp ph domain protein l open read activate protein domain like alpha sec member associate dna sorting b member reading frame dependent member protein st alpha alpha channel repeat factor group box repeat domain long non protein code rna g domain domain contain member activation half bind protein gamma alpha interact protein family protein activate alpha like iii domain contain contain open reading protein h couple interact loop translation gamma cd cd protein induce protein bind protein pt process dna increasingly disease cancer change reflect environmental amongst pre change dna promise develop dna base measure pattern assess genomic analyse group interaction quantity happen genome dna measure genomic infer genomic community science system model know include social science network much shift gene pathway gene cell examine gene successful principle group gene statistical possible gene mean expression particular gene amongst store modification chemical dna di information acquire phenotype interface genome also environmental dna change human offer identify develop early dna think promise development wide play major reflect change dna highly change dna individual dna whereas lead datum noisy dna gene analyse show activity genomic region associate way represent analyse large quantity produce balance fidelity efficiency widely study blockmodel observe interaction block modularity quantify observe particular community present certain fitting blockmodel modularity spectral network genomic decompose module functional biological community detect genomic sbm think module recently reasonable sbm mechanism optimum blockmodel representation mean blockmodel biological genomic law mean visible use module identify isolated biological fitting cancer network arrange phenotype advantageous cancer patient outcome breast could gene interaction gene dna dna measure infer genomic node indicate module within network gene interactive term dna behaviour link relevance network patient dna represent new methodology dna dna numerous form dna analysis cca novel interaction gene single patient dna dna quantify extent dna profile gene dna profile gene act surrogate extent pair may include co type interaction gene presence product amongst cca cca discover combination variable explain way combine location particularly particular deviation profile probably gene along correlate cca across cca seek dimensional space xx profile patient measurement gene gene patient dna interaction profile analogy equation eq cancer gene reflect well correlate behaviour patient measure dna typical explain profile c gene vary typical equivalent varie vary interaction number measurement ordering identify interaction gene represent adjacency node accord measure problematic carry level edge pair network cox proportional model quantify dna patient cox hence adjust clinical covariate dna normally variance mixture gaussian infer lead mixture equation interpret function take likelihood observe standard mixture gaussian prior tail use practical implementation slight mis specification gene find pairwise comparison mostly detect correspond generally detect high gene adaptation degree obtain interact social functional biological detection allow constructed interact differently cancer predictive advanced term detect interact relative serious disease interact serious community fit correct blockmodel community divide network histogram community module identify way represent dna network community summary accord gene whereas gene correspond increasingly positive care network community magnitude network interaction pair gene pair community multiply cox cox fit predictor hazard factor event g death occur hazard coefficient issue fit dna interaction measure generate follow network infer cox dna disease green increase score dna interaction decrease poor interact examine dna amongst genomic interaction calculate gene expression gene patient intuition non zero measure interactive statistic tool breast cancer cancer genome initial batch dna train together clinical disease disease dna dna profile potential community datum set clinical validate training gene infer adjacency presence dna interaction associate edge extract connect community range size relate ht adjacency blue detect outline table indicate label survival divide median score initial hazard value calculate cox community survival patient group community divide training display value cox validate note identify potential network carry infer cox calculate community unseen test association survival unseen test sample five potential validate way outline cox network community datum clinical covariate test l ci age disease network r l ci network ci disease ci
provide human fm translation visual help ambiguity sentence modify component multimodal show failure case mistake reason incorrect method say actually try focus small look similar fourth answering output sign e try issue incorporate visual linguistic information question university california present question short lstm convolutional network extract lstm store linguistic answer component question evaluate chinese english answer human mix answer provide human human score indicate quality human rapid progress deep deep cnn recurrent memory lstm annotation play discuss description image interaction preference paper answering need answer content address four figure lstm encode sentence dense second convolutional extract representation train imagenet fix third component lstm encode previous dense fourth predict next jointly train fourth answer weight sharing also allow layer fully layer answer fm detail ms chinese english allow content annotation contain ai position object e base visual column encourage study area variability answer accurately evaluate visual human mix answer generate label human determine human addition also ask e pass treat human average failure ask answer tb recent significant progress neural natural computer vision cnn classification rnn lstm widely speech recognition inspire task deep cnn vision rnn extend ask question image use question pre template contain kind language translation set requirement complementary lead interesting topic visual answer lstm question answer word prefer single answer feed lstm answer fm dataset annotation dataset four pre color location dataset weight share manner different different component please see short extract semantic convolutional cnn extract lstm current answer linguistic incorporate part generate next jointly four word answer e non zero index sign begin answer use generate answer stage stage input image start start search keep probability accord softmax repeat question extract embed memory cell function layer map semantic feed dense neural design vanish layer serve bridge e lstm gate calculate stand activation word denote matrix multimodal word softmax layer share word softmax detail similar use sigmoid activation relu non cell non activation intermediate answer question mean answer embed third component weight softmax layer word embed function pseudo word reduce concept task cnn imagenet adopt word answer function answer fourth decrease epoch stop epoch chinese answer treat equivalently english train question answering describe annotation publication start image newly ms image annotation annotation amazon type long get question beneficial question annotation monitoring set quality labeling monitoring question answer per satisfactory correct interesting question require reasoning give rest set good bad example answer reference annotation image far refine portion question answer tb answer annotation length question answer chinese answer question ai capability contain simple understanding question g object person among computer object e color contain question vision language answer hold tool problem answer part answer phrase e give answer automatic metric word similarity wu answer dataset complete sentence metric image metric answer critical tend give keyword evaluation suffer severe answer conduct generate well grain evaluation ccc rate rate score visual answer determine base answer pass
side sup sup f contraction last sup sup sup sup h old jensen previous chain combine give part inequality hoeffding n sup I sup use combine bind well bind complicated simple clarity occur definition f nm f iid value hoeffding remain thm bind among task representation task highly beneficial study statistical general rigorous justification benefit illustrate linear specification iteratively linear transformation sigmoid representation learn analysis case specialized consist function different choice note feature function hilbert method apply large representation would hour height ne ne pt skip ne compatibility abstract chapter head part head head head head head head compatibility conjecture proposition compatibility compatibility true true false mu false size ex h mu mu th h mu mu align align end end end end end end align style style use style array type allow true tag tag true hour depth ne f ne pt abstract head head head head compatibility corollary proposition ex em compatibility compatibility compatibility true em em mu mu mu mu mu false mu mu h mu mu align align end end end end environment environment style allow use allow false false tag true uk science college uk ac uk uk method justification setting method advantage task focus derive regime beneficial independent sample number intrinsic feature learn multiple jointly become increasingly range preference object mention similarity study mid connection neural structured perform task arguably intelligence experience task influential research transfer mean jointly task ai hierarchical representation multiple empirical remarkable increase interest component work remain discuss advantage domain derive representation enough reliably model precise half propose vast paper incomplete considerable representation learn inductive learn analysis perform paper main goes consider base cover reproducing avoid factor dependent subspace good main specificity beneficial worth effort half noiseless isolate task method match performance know advantage good agreement theory organize problem learn far half rigorously error suggest interpret member interpret learn model pair output predict constant predictor handle simple scale task simplify task composition map define specialized sequel class refer representation specialized require multi sequel exponent iid representation solve concerned rather propertie hand important assume probabilistic law keep parametrization learn guarantee correspond decrease typically contribution first play observe average independent tn many interest include kernel machines gaussian rbf discussion learning vanish term govern equivalent map specific special phenomenon considerable sample subspace learn show operator size understand support task like dependent eq q play basis quantity assume subset hope obtain suffice hypothesis albeit unknown subspace isometry bound radius specifically k k appear dictionary call certain allow nonlinear activation atom drop allow trade class specialized identity express simply theorem let g theorem least excess eq compete bind bad whenever approach slow course break distribution obtain excess risk occurrence norm covariance imply large number term trace covariance easily marginal concentrated eigenvalue c term appropriate available highlight potential order computational task superior really dimension regularizer large proportional empirical burden dimension vanish section explain case noiseless classification half performance superior marginal unit sphere classify loss function let define without partial isometry mapping onto unit sup da f excess get probability expect use estimation vanish limit roughly share bind class transform orthogonal algorithm produce correspondingly algorithms rotation misclassification unit least every u bind bind high advantage learn purpose setting beneficial binary task namely generate ground vector orthonormal haar select sampling build hinge optimize weight assess consider report repeat report average input task average suggest despite find axis instance experiment experiment help diagram accord experiment scheme previously error similar average test trial diagram generate dark compute bottom column reader parameter difference parameter partly accumulation loose derivation another apply agnostic nature bottom agreement dictionary truth regime could permutation change overcome similarity tr permutation ds tr vertical represent horizontal task theorem establish right excess
flexible whose tucker decomposition depend denote column usually need advance however due apply significantly less stability seek elegant multilinear avoid employ sparsity prior latent straightforwardly place independent prior interaction inaccurate multilinear account n precision place propose student let core gaussian integrate term tucker l laplace prior simplicity thus estimate logarithm however map aim infer distribution treatment perform inference scalable sampling inference technique conjugate address inference provide appendix vb aim seek approximate posterior p factorize optimize expectation distribution variable follow variational posterior q linearly relate evaluate show expectation computational polynomially complexity scale introduce theorem scalar eq detailed scalar spectral factorize product lead operation individual eigenvalue decomposition matrix operation diagonal matrix significantly memory significantly reduce format product posterior multilinear operation evaluate memory multilinear kronecker avoid explicitly kronecker note factorize operation cost gamma rigorous approximation play automatic determination multilinear rank distribution conjugate accord expectation square account automatic model f n factor unnecessary slice machine precision expectation laplace prior lead difficulty solve problem inverse distribution inverse gamma inverse hyperparameter l mode hyper variational represented posterior straightforwardly mode avoid computational modify essential difference student one n lie student laplace manually hyper straightforwardly derive mode modify function variational hyperparameter update expectation residual alternatively multilinear hence computational reduce denote iteration bayesian tucker tensor completion position tuple index entry tucker multilinear incomplete tucker consider new tucker represent model tucker incomplete ill role successful determination multilinear significantly affect strategy cross multilinear varie dramatically ratio multilinear occur core minimum multilinear automatically efficient elegant repeat many memory overall complexity n nr memory scale polynomially suitable multilinear automatic reduce rapidly iteration computational decrease automatic determination multilinear enable low tucker secondly uncertainty problem uncertainty deterministic scalable convenient related ard tucker carefully performance ard tucker automatically tucker fit vary show discrepancy baseline three automatically complete vary noise level range db range f theoretically rapid surprisingly improvement snr ard tucker automatically infer however db fig exactly except snr ard tucker infer accurately runtime second ard tucker ratio range snr mr rapid increase able optimize base tensor rapidly infer tensor condition snr mr runtime summary mr completion evaluate recover miss test different water pe b sample represent ideally three original corrupted snr db tucker need predefine tensor infer tensor rank underlie capture stable estimation datum performance case ard tucker always well noise computation ard tucker also randomly datum repeat present tensor infer sensitive five htb noisy original noisy noisy original ard r n runtime runtime flow global generally prediction become challenge globally rank small block tensor overlap tensor completion tensor handle performance adapt vary noise condition outperform require medical test evaluation rank specify infer ratio performance rank global yield completion visual quality method c tucker structural introduce group prior hierarchical inference especially multilinear observe significant advantage completion propose theorem multilinear operation scalability empirical validate superiority ph degree china laboratory advanced signal interest vision brain computer interface paper international zhang ph degree china department engineering china interests cover theory visual publish paper receive ph dr electrical technology team laboratory advanced processing scientific paper translate proposition zhang bayesian tucker completion modern attract extraction compressive sense challenging determination multilinear especially probabilistic tucker decomposition structural multilinear fully inference inference model efficient multilinear automatically complexity multilinear principle comparison synthetic remarkable recover ground multilinear entry tensor tucker multilinear structural completion aim seek multilinear array technique modern mining factorization structural model multilinear lead dimension compressive tucker decomposition attract tucker multilinear operation factor core basis core tensor multilinear whole tensor basis dimensional coefficient profile datum tucker decomposition contrast strict limited become low compact representation one fundamental problem determination tensor matrix numerous study either tensor always manually general need tucker possibility exponentially tensor automatic determination exploit fail true tensor case highly decomposition value decomposition focus tensor focus semi aim predict element tuple index important issue completion relate miss underlie tensor key appropriate obtain whereas tensor specifically applicable correlation observe decomposition develop tensor rank determination cp contrast multilinear degree framework assumption minimize relaxation minimization avoid specify manually selection problem nuclear tensor completion attractive year impose exploited type obtain carefully implicit issue nuclear correspond weighted rank denote dimension core model previously another nuclear tucker tensor multilinear rank solely datum automatic student facilitate enforce group sparsity framework non miss parameter addition multilinear rest multilinear modeling induce present bayesian tucker tensor tucker completion issue discuss experimental tensor matrix ni n standard tucker nn
order detect signal pressure patient might event offer attractive unknown base particle mix slowly base maximization size nonparametric furthermore counterpart degenerate maximum directly observe case inferential challenge jump predict patient show degeneracy take develop trajectory parametric variance asymptotic proven infer extend connection gaussian variance gaussian approach idea obtain motivated objective scalable challenging dirichlet hmms approach parametric function suffer solution degeneracy lead robust inference case construct exponential nonparametric jump parametric jump attractive improvement case reduction mean error prior provide degenerate solution probable comparable outperform reconstruction finite space probability state exponentially leave transition kkt ts kt k states trajectory state state time inference expectation maximization state suitable finding carry efficiently degenerate spent visit far thus slow often carlo propose mcmc address issue probable model contain pz pd pz approach limit map variance simplify likely time time simplicity interval let place gamma prior detailed rely asymptotic stable maximize limit variance exponential two shape hence mean scale ft gamma scale writing drop modify long markov stability maximum ht kt kt grow penalize expect penalize manner however remain long interpretation trajectorie usually trivial system almost mle mode superior valid difference case observation case consider observe thus objective trivial know jump combinatorial sequence continuous complexity resort alternate optimize spirit modify viterbi jump optimally distinct jump jump point modify viterbi keep dependent upon state infer eq restrictive jump jump eliminate find optimize jump mean similar place indirect viterbi step initialize converge poor bl jump nonparametric ms pressure run em number hold state synthetic synthetic ms model exponential process construct gamma generate state parametric replace scale hierarchical gamma h component mt error jump uniform scan hyperparameter respectively mcmc jump em par error jump em run obtain significant state generate probability em likelihood use uniform initialization amount jump jump mean hide state jump perform slow trajectory slow likelihood disease cast patient state represent disease trajectory aid disease care real world phase clinical trial drug dataset track different randomly evaluate value record initialization tb infer jump mean patient ms dataset jump significantly achieve reconstruction fig provide trajectory jump maximum include reflect realistic patient amount trajectory produce take account stage picture pressure collect hour period observation length patient keep test initialize uniform matrix hyperparameter particle state inference particle category pressure run em report good em bt evaluation category jump significantly reduction error iteration cpu need example infer color assign pressure degenerate trajectory tune show histogram run fraction run setting lead large hence study robustness jump choice hyperparameter runtime jump increasingly handle scale linearly decrease increase bt jump inference asymptotic algorithm experiment obtain degenerate offer parametric art problem acknowledgment comment air office scientific research national fellowship begin scale natural uniquely integral ft shape let ft incomplete gamma state give bregman divergence multinomial multinomial writing trajectory family apply expansion fact process base partition define gamma key ep conjugacy conjugate family proof analogous proposition key additional insight denote b suffice xt exchangeable times exchangeable integrate generative manner restaurant obtain dirichlet omit transition reach likelihood k logarithm expansion term mt retain asymptotic asymptotic rewrite integrated ignore yield
error plot cost probability observable n exist cover nonetheless implement numerical enkf probability near slightly slow achieve approximate rmse sde analytically transform q noisy introduce eq upon define noisy sde artificial ensemble numerically integrate ensemble mean ensemble provide return integration solved numerically validate exist integration euler scheme mesh parameter set gold rmse slightly enkf measure rmse mean plot measure attempt monte equivalently optimality kalman however limit view linear filter convergence limiting model sequel publication support rt member quantification sampling carlo filter enkf yield kalman provably superior asymptotic carlo filtering kalman sequential parameter system sequential incorporation complete estimation probability conditional solution formulae kalman filter close form resort probabilistic leverage kalman enkf approach converge kalman filter solution incorporate evolution give may computable integral approximate grid dimensional even solve enkf intractable close less accelerate idea iterative solution pde early iterative pde pre become random context sde pde beyond discretization proportional cost computation sample monte context markov chain knowledge author yet extension methodology explore enkf implementation enkf limit distribution bayesian something else rest organize review enkf first sub favorable field equation direction filter gaussian state section implementation kalman filter ensemble enkf time sigma generate positive definite track aim variable variable seek admits evaluate exactly rather associate confusion concern herein follow sde w satisfy follow condition fit framework wiener principle come unknown must lead hierarchy approximation notational time refer non easily extend merely notational convenience approximation solution follow depend solver verify cf arise coefficient dependence map simplify uniquely iterative computing covariance give derive use one may derive verify formula derive verify fix side quadratic obtain shorthand summarize classical kalman formula describe equation nonetheless responsible enkf appear particle enkf propagation kalman filter n consist nonetheless step one interval realization sample random variable realization realization confusion latter become apparent suffice assume map numerical satisfactory complete path mean ad hoc indeed implementation formulation filter know perturb nonetheless functional converge lipschitz limit particle increase discretization let denote simulation numerical increment consist update pairwise couple realization level level unlike return similarly enkf ensemble I enkf predict level compute level convenience convention introduction second argument sde initial hierarchy section use constant contain implication scheme possible sufficient regularity sde euler norm assimilation vs monte cf repetition notice realization give boundedness tolerance large step grow polynomial exist scalar locally polynomial growth infinity cf sample update formulae enkf distribution sequence line high practice introduce limit l evolve covariance gain limit formulae intra level member identically I independent solve limit system approximate replace last sample perturb limit independent come correlation crucial proximity two particle require great mean lemma bound boundedness predict ensemble ultimately ensemble covariance virtue convergence consider omit avoid unnecessary gain follow micro control let notice normalize notice eq continuity depend notice ml q cost denote predict final system forward let triangle turn lemma term respectively discretization boundedness contain second inequality comes triangle complete observing imply difference hold define bind surely expectation quantity triangle avoid similarly aa ba ba cauchy arise sure depend cf use hand boundedness eq next covariance see continuity covariance similar term inequality term old proof show predict ensemble error carry make rigorous ensemble begin use old plug hand summing recall induction remains induction actually able definition denote treat separately relate
quadratic perform allow stochastic gpu implementation show converge correct low structure goal structure input explain want input avoid serious utilize hide cost spurious spurious signal must subsequent supervise additional expert generative pp median regularization approximate constrain normalize model posterior maximize leibler achieve extract desire structure impose generative pt factor loading component unit correlate fig center consist stack analysis posterior maximize give expectation em minimize first constrain I component wise hide solve project feasible start newton try reduce fail ensure alternatively method project fast projected newton require define gradient projection step iteration program benchmark mnist speedup project solver ratio mnist cifar speedup update essential computer restriction quadratic reconstruction eq step gradient allow gpu dropout gradient using require euclidean posterior euclidean euclidean projection positive non positive otherwise supplementary project project scale active column posterior alternate dropout zero predefine dropout long hold cm project gradient project n cm kk complexity step project project step gradient alg alg converge maximize sketch alg ensure minimum alg decrease gradient projection requirement fulfil generalize thus update view gpu implementation covariance capture good ridge sample noise strength randomly noise deviation large evaluate method code generative percentage value small sum reconstruction supplement detail hyperparameter average overcomplete unit instance dataset supplement confirm level yield since variational low yield ex performance factor network stack obtain pass architecture layer machine stack denoise autoencoder stack autoencoder iv restrict machine rbm stack boltzmann machine report bold overlap validation select bad nine experiment perform significantly p ex mnist basic rand ex ex ex benchmark dataset mnist iii random iv mnist discrimination discrimination rectangular image discrimination generic category validation center selection set default fine fine stop rate learn bold significantly good nine perform project lead company aim pde project project expression identify event expression gene event stem formation panel rare small negative feedback pathway event detect relevant project example design previous code row gene red green inactive gene formation confirm analysis green panel feedback pathway cancer factor construct code normalize alternate minimization prove correct code low yield sp improve deep network drug detect rare module unsupervise relevant study large code factor network efficiently sparse dimensional rare interference unit explain covariance generalize derive posterior unsupervised autoencoder rbms ica sparse code reconstruction test deep vision superior rbms autoencoder expression drug discovery detect module highly insight learn advanced representation relu advantage representation code importantly much use representation break bioinformatic dna human representation code event vast majority suppose would fluctuation
answer answer set output hide call map feature map follow output hide sentence word slide windows input convolution traditional combine sentence joint pair formalize input sentence perform sequence key cell time cell gate gate gate lstm study discuss time memory gate forget gate equation lstm keep context discard forget gate add compute update cell unit matrix accord conditional answer answer sequence softmax training test conduct dataset challenge training contain answer answer good bad accounting answer question bag model dirichlet crf crf approach word apply belief predict answer multimodal learn cnns representation question answer answer score development hyperparameter joint modeling question sentence pattern question notable powerful svm crf answer potential svm crf answer tendency good answer reason distribute deep architecture capture semantic crf suffer noisy question answer f crf cnn cnn cnn superiority pair sentence cnn convolution operators r cnn tensor sure representation semantic feature bag improve answer sequence answer complement learn context previous answer modify valuable pass rnn main improvement cnn potential answer much cnn potential answer answer intermediate r cnn score multi ok please response indicate question easy performance cnn correlation answer bad cnn sequence model integrate lstm unit cnn successive relevance answer explore improve overall label world support part national china foundation china anonymous comments intelligence institute technology chen cn answer question answer regard sequence task novel approach apply cnns representation question pair firstly use joint long short lstm learn answer question answer conduct answer valuable knowledge information retrieval matching answer typical question explore study exploit syntactic measure semantic answer require external resource directly disadvantage semantic question answer selection figure answer intuitively answer answer recently especially short memory superiority long term short work use convolutional cnns sentence sentiment labeling identify answer cnn answer pair use lstm bad answer study answer generally treat problem explore pair structural represent machine classify train lr classifier answers fields crf answer match additionally language translation suffer perform applicability symbolic belief net semantic learn
triplet visualization generalization independent triplet gain joint task measure difference triplet triplet consistency average triplet learn embedding large triplet triplet consistency synthetic pose significant gain approach triplet high consistency gain significant tb cm cluster triplet consistency jointly measure triplet view specify specific mahalanobi demonstrate conventional although hinge easily generalize trade view real application triplet expensive jointly metric preferable empirically triplet consistency view view great gain show multiple error future study similarity classification label annotate top leave illustrate five triplet translation square match work categorie attribute p attribute shape cs similarity multi similarity specific view perspective jointly exist view achieve triplet generalization grouping learn independently improvement large triplet role application content recommendation speech similarity object abstract represent explicit parameterization inner product matrix gram implicit representation operation complex demonstrate embedding object triplet supervision embedding comparison proximity use word similar back head human embedding reflect similarity comparison easy absolute scale ambiguity head head back ambiguity annotation result poor desire perspective measure view enable human loop interactive fine grain thereby main drawback view comparison undesirable triplet jointly exploit view training car take angle model onto play view view iterative dataset realistic domain namely pose dataset crowd similarity collect datum per view low triplet naive independent approach propose learn well cluster leave joint embedding base triplet wise ordering dissimilarity set kernel crowd alone triplet relation van triplet similarity embed cluster study four focused supervision aim learn object cluster extend multiple task label different fold triplet comparison learn instead multiple aim embedding instead triplet capture complementary user effort collect triplet triplet collection embedding triplet dd distance agree j set study van propose loss hinge non crowd stochastic triplet distance jj minimization q trace gram convex embed embed however optimization true similarity case object multiple measure obtain aspect comparison triplet corresponding notion embedding view end hybrid approach combine global gram base view correspond global object gram define mahalanobi ti formulate learning generalize literature regularization term add trace produce ambiguity scale scaling hinge respect since descent via iteratively take project result cone summarize term scale carefully trace geometric reach value depend product hyperparameter view argument tell view dimension much enable well leave whose throughout classifier van triplet lead error triplet necessarily low metric aspect conduct triplet task adopt package author cross sample hypercube center randomly hypercube six project datum five triplet possible triplet pose construct image base pose translation associate range vary view additionally leave point evaluate produce unbalanced belong class similarity color public figure public face create consist attribute real value appearance person aspect ten attribute attribute group image label specie collect show user various contain triplet collect crowd manner ask nine image interface specie display partition nine set triplet specie l sim triplet constraint acquire region various fig procedure triplet cast similarity whole cast localize region breast triplet test balanced manually super class challenge sense situation triplet relation nature crowdsource triplet incorporate human feedback recognition embedding data conduct learn triplet generalization leave error plot achieve triplet generalization cluster dataset triplet learn triplet error error pose embed triplet object lie leave orientation triplet embedding show triplet generalization learn triplet become indistinguishable see joint independent pose public embed triplet set triplet generalization error triplet error reduce triplet increase dimension understand bias triple learn understand term leave continue well triplet embed dimensional visualization bottom jointly show triplet embedding dimensional
impose draw vector selection existence meaningful approach learn bayesian class sample base factorization single vector aforementioned control class discriminative infer use desire dictionary restrictive overcome c weight standard precision distinguish parameter r arrive representation conjugate place mention latent dictionary atom datum dictionary notation assume place hyper normal distribution f gamma representation model variational et effective easier relate sampling process conventional sparse svd expression conjugacy analytically start analytical expression posterior isotropic simplification significantly approach atom dictionary atom atom dictionary codes dictionary atom drop update contribution atom write aforementioned eq expression probability concern therefore ik eq sample normalize bernoulli simplify arrive expression light expression sample conjugacy sampling eq sampling give weight assume express isotropic conjugacy distribution must sample write arrive set vector k discussion desire draw estimating dictionary mean size present regard closely take discrimination appear model simplify c dictionary require vector non later happen ok atom simultaneously accord drop atom bring remove redundant arrive probability probability c atom commonly represent class word arrange large learn appear location infer clustering discriminative dictionary character six probability different training extend scene respectively plot represent vector plot distinct query follow methodology encodes contain query assign component technique joint optimize separately appear class computing denote modeling matrix write h framework gibbs use instead infer new infer code query learn discriminative dictionary underlie learn classify regard exist coupling probabilistic exact hence jointly coupling keep term match omp query greedy pursuit efficiently search omp code infer select omp support initial initial equally effective atom initially getting select finally serve similarly vector compute vector initialize computation complete drop category categorization scene category evaluation representation consistent svd learn separate datum dl comparison unsupervise use dictionary acquire public code implement toolbox public perform intel cpu ghz ram mention carefully difference illumination ar illustration follow subject test pixel image projection ten lc lc whereas result expense computation time reasonably lc lc original distinguished section result lc dl residual tolerance small give classify denote work l accuracy time lc lc svd lc lc term accuracie fairly I reduction rate require propose exist l c c zhang al et wang dl lc comprise object class tree sign number vary sift descriptor extract patch densely pixel extract spatial pyramid extract grid codebook pyramid train pyramid protocol select experiment repeat accuracy experiment number lc lc result sparsity et suggest give result select well lc dl clear consistently compete approach case lc propose increase favor sample result precise posterior distribution setting inherently technique inference lc verify testing batch class contribute efficiency table include propose sec lc scene category category kind country etc pyramid descriptor vector feature consider propose value set suggest al lc dl original database lc dl approach propose lc pyramid lc lc l lc dl comprise channel video include dataset evaluation protocol performed fold one five summarize lc lc parameter along accuracy action also dl take literature optimize report outperform validation database propose well art claim insensitive large precision clean noisy give less training class increase availability clean therefore precision large dataset among similar without easily verify mention desire atom plot training represent complete training correct mention convergence initialization work code learn dictionary sample svd lc fair parametric employ infer bernoulli atom say atom specific datum also correct hierarchical exploit classifier code instance use evaluate scene comparison art discriminative representation outperform prove advantage exist base us dictionary secondly principled dictionary atom manner make inherently online specific prior knowledge principled signature hyperspectral spectral signature adapt hyperspectral image classification future arc grant dp lemma edu discriminative dictionaries dictionary atoms association dictionary atom class parametric infer dictionary exploit separately classification instance encode learn fed face scene public state discriminative representation experiment propose consistently discriminative redundant root human technique digit well instance dictionary redundant atom effective g wavelet domain decade favor unsupervised learn signal supervise dictionary discriminative representation discriminative dictionary use training dictionary atom query assign associate maximally representation query result achieve become considerable dictionary allow computational learn force dictionary specific associate constrain learn exclusive class separate atom exist strategy assign atom adjust accuracy principle perspective representation beta adaptively build association atom character fig bayesian discriminative learn atom bernoulli code later learn atom wise code datum classify infer classify code exploit bernoulli distribution test database action database scene art approach efficiency exist paper follow review explain propose propose experimental setting conclusion main approach learn dictionary al recognition atom dictionary et term texture segmentation sparse code compute specific dictionary action atom coefficient negativity training apply detection recognition use encouraging incoherence among specific dictionary allowed represent incoherence mention mainly associate atom directly single minimum query stage representation dictionary approach single force encourage discriminative objective already author coefficient common learn classifier joint dictionary learn zhang li enhance along coefficient minimize task classify sparse code learn dictionary classification stage dictionary also fall category take hybrid discriminative representation dictionary et
equivalent kernel quadrature rule definite measurable lead eigenvalue integral logarithmic particular quadrature general beyond match special preserve cm topological equip borel integrable family matrix reproducing also reproduce element respect weak rkhs kx dy kx adjoint semi definite trace df h extension sequence eigenvalue eigenvalue dense element orthonormal ne covariance rkhs dx operator eigenvalue cm generic make equipped probability measure consider additional give term show x equal always attain cm usual decomposition kernel fouri kernel form periodic definite negative kx expectation usual uniform traditional space kx geometric decay eigenvalue kx tx tx fourier split decay study decay decay decay dd extension tensor decay integrable derivative multi integrable derivative last decay avoid linear x strong expectation x nx surely kernel uniform subset function rkh key difficulty general include characterize approximation measure look possible possible definition n e n scaling choose well respect n v cm square integrable integral combination allow depend fashion cauchy equal quadrature formulate quantity possible mean standard weight note correspond respect require fix well integrated accommodate respect respect robust cm mx many always x gx quadrature expansion note form approximation approximation qx gx many constant set compact line interpolation decay uniformly fourth quadrature one integral basis orthogonal quadrature polynomial derivative good quadrature orthogonal quadrature rule gauss quadrature lebesgue measure sequence point univariate adaptation smooth generalize interval typically quadrature essentially quadrature paper quadrature weight positivity property integral preserve constant exactly constraint require kernel novel conditional gradient improve several setting comprehensive example space good quadrature integrable property eigenvalue integral thus allow extension manifold quadrature rule sequentially improve quadrature error characterize space go recover partially perspective optimize adaptively interest outside scope minimum noiseless problem guarantee bandit bound outline section quadrature section eigenvalue operator quadrature expansion proposition I rely f properly bernstein concentration column sample namely small explain may quantity refer open order relate directly state eigenvalue also allow degree freedom decay max q prop thus need logarithmic sample decay eigenvalue get geometric decay cm cm em surprisingly tool constrain norm interpretation tolerance quadrature prop end quadrature namely equivalently computation density quadrature approximation norm strict qx e converge tend decay recover upper quadrature gx dx gx dx several kernel space cm output fx minimizer sampling feature lead prop base require worst n however regardless cm build evaluation amount noise decrease happen smooth rkh h l op rkhs number quadrature max rkhs compute quadrature bit get note estimation quadrature regularize consider characterize norm I op decay notice eigenfunction constant periodic simplicity dr prop degradation plus show quadrature rule expansion positive approximation quadrature applicable improve within work variety quadrature framework kernel parametric improve consequence stochastic support centre european author write paper three eigenvalue eigenvalue easily eigenvalue regular number multi easily time beta uniform optimize report space function integrate parameterized quadrature parameterize average convergence matching integrate less quadrature potentially bad compare quadrature gauss quadrature take uniformly spread compute kx large computed point happen dl norm eq introduce equal minimize define heuristic equal ax ax v dx adjoint overall f f f f performance goal v x eigenvalue less
display display fig reality collect generate cm generate period retain obtain histogram marginal long histogram together design observe posterior display appear decentralize towards posterior retain shape seem gain correspond layer gain characteristic design posterior posteriori show agreement slight overall red black incorporate bayes discretized x j kernel variance reference kx top fig posterior burn period retain obtain distribution display significant divergence shape seem information gain present location collect low informative something assumption reality prior present maximize information additional address issue well computational optimization apply problem identification contaminate system medium flow location measure set adopt code simulator derivative optimization process address accelerate surrogate crucial otherwise validate inference show location informative posterior true limited resource performance acquisition highlight monitoring environmental risk economic I I precisely proportional clearly control control bayesian two phase crucial restriction informative bayesian translate update expect design task alternative design criterion addition burden address concern contaminate area validity simulator approximation evaluation methodology demonstrate set field increasingly need area accelerate expansion range health consequence monitor never key credible procedure characterization across ultimately physics assessment flow functional assessment issue resource vast challenge year decision work worth result uncertainty reduction next address analogous recently formalism experiment characterize flow media work worth phase phase make utility quantify worth collect evaluate general design latter usually add economic concern role uncertainty design attention present arise objective explore criterion candidate worth without concerned economic providing decision criterion functional fisher counterpart design criterion nonlinear choose utility work maximize gain high model intensive surrogate monte involve evaluate criterion optimal design rational basis decision mention fig paper actual site collect site locate figure site dot site specific investigate step storage result initial flow medium correspond specific observable distribution observable concentration kullback kl fix optimal location determine distance location provide update improve prediction concentration criterion bayesian elsewhere scope address algorithmic implementation monitoring strategy organize optimal inference stochastic approximation toy apply site subsection accelerate computationally intensive experimental validate generate scenario conclusion summarize fix resource reduce uncertainty sense design fix thus instance smoothing kernel alternatively present vector coordinate plane also consist nonlinear functional denote prior attain restrictive attain reduction uncertainty numerically conceptual difficulty infer parameter posterior posterior update rule integral pd shannon kullback leibler kl divergence quantify gain spirit define expect output set gain experimental trivial unknown evaluate approximated replace estimate evidence prior carlo identify exhaustive grid search limitation computational expense report easily become infeasible design involve expensive forward incorporate adapt monte carlo instead adapt gain maximize arise application direct entail appear proportional sample require also furthermore variance loop sample application include hundred prohibitive contrary bind see unbiased loop something achieve accuracy much low number sample denote measurement normally relate typically incorporate purpose rather assumption among measurement finally use jensen namely q low equation eventually need term datum new linear regression problem minimize expand eq see carlo derive difficult design solver evaluation become prohibitive maximized turn root noisy root optimize iterative constant respect explicit attractive versus update step appropriate random zero correspond success probability goal explore bind substitute gain numerical two respect uncertain quantity prior initially explore carry second location observe case direct monte posterior sparse quadrature infer reproduce gain distribute sake illustration display perform carry present real show use finite simulator module simulate volume energy consist estimate increment step several density etc various r module intend water change decay phase model law decay decay interpret decay explain dependent characterize r water air parent phase simulate locate approximately deep discretized cell dimension paragraph investigate uncertainty site take domain although mainly effect assign half molecular describing detected name molecular gram molecular weight gram simulation c formation heat heat j factor initial pressure pa initial assign pressure close state ground initial mass zero pressure initial simulation assign initial condition location approximately area storage see purpose inactive volume cell assign value per randomly simulation minute finish implement thousand impractical necessary create would evaluation run unknown material choose make cover cm si semi material find solely material independent uniform admit q modulus variable independent gaussian gamma basis multidimensional polynomial paragraph polynomial version write number expansion coefficient fashion need general done implement convenient simulation see coefficient calculate multidimensional method calculate evaluate root least linear regression take tn form select square give exist demand simulator impractical processor one week allow n use hypercube expansion include coefficient close choice concern convergence polynomial fall scope purpose coefficient sample first know statistic test truncation estimate statistic expansion leave regard particularly expansion thorough include fit along east boundary truncation error expansion median upper quantile good bottom paragraph expansion model output bottom perform bayesian enhance potentially gain analysis observe depth generality good location moderate appear satisfactory validate beyond x x subject additional indicate involved information gain bind derivation solver sophisticated deviation proportional quantity factor contradict derivation red location source plot maximize performance case monte vary variance unbiased maintain three evaluate result run objective evaluation iterate much maintain low fig idea
perform convolutional datum independently first see fully convolutional hand pathway demonstrate depth intensity significant quality alone describe modal pre share one correlated taking suboptimal resolve architecture require test blind fusion fundamentally correlation see capture layer hyper modal initially separate intensity video channel effective stage nature hand modality correlate rarely beneficial channel hand fuse cross complementary skeleton motion audio initialization specific pre fusion pre relate network train network effective fusion quick degradation share fusion strategy gradually powerful strategy among fusion strategy mean classifier complex gradient descent implement early non straightforward dropout activation geometric output arithmetic well quality consistency geometric fusion output layer initialize architecture diagram indicate matrix conventional visualize interpret structure clarity vertical size modality hide specific hide target share layer size output weight think matrix unit column share specific block meaning initialization phase force procedure modality train cross modality capture impose evolve eq initially relate notation stand share output channel relate number first contain weight responsible inter correlation force comprises unit modality layer block softmax activate else initialization force output fusion mean modality initial force stage relax later fusion multimodal concept number weakly separate modality shared avoid modality handle channel key share would meaningful modality expect signal formally consider model represent q output ground regularization weight initialize diagonal relaxed objective formulate indicate modality formulate fine path modal drop certain accordingly non zero minimize correspond advance follow network activation input denote come output weight unit l ns come unit minimize cross target drop eq pt consider corresponding modality modality drop preserve formulate activation output relate involve bernoulli selector activate channel activation concern output unit gradient correspond q drop bernoulli e get expectation expression approximate exception selector derivative calculate derivative sum weight minus modality need stress correlation involve multiplication channel cross product analyse case channel unit come modality uncorrelated network expectation eq q pt network lyapunov lyapunov central product input tend result centralized magnitude input assume vanish number regularization prevent interesting belong modality positively product grow logic apply input correlate enforce correlation accordingly correlate modality act cross regularizer discover signal dropout multiplier proportional sigmoid activation magnitude weight mid range play less role weight introduce adaptive regularization input unit vote strategy fusion single introduce weight meta classifier quickly per output prediction obtain rate increase wider slide window stroke post stroke overlap pose class stage appearance vast temporal employ simple address additional period activity precisely point fully pose descriptor frame example frame right consider negative thus motion module frame output end typically noisy boundary close switching point detect boundary people rgb stream vocabulary category recognize rest participant explore dynamic modal version annotation distance phrase due alone surprisingly challenge dataset augment take neural table architecture identical temporal unit module tangent optimize early prevent overfitte additional fusion temporal scale section deep implement library operate frame gpu c filter unit share pt evaluation challenge adopt quantify sequence prediction frame mark rest addition ensemble iterative descriptor purpose explore relative beneficial combine depth intensity extract hand pose plane histogram pose third dimension third comprise depth reflect temporal hand extremely randomized fusion iterative architecture baseline recognition pre order word phrase period activity class list treat c team team chen et wu author modality cm cm pose video video audio localization challenge win hybrid combination deep baseline second note multi achieve one work percentage optimize architecture video skeleton path employ advanced fusion procedure challenge neural architecture modal per test useful capacity typically temporal pose correspond temporal prediction refine localization module stream contain also insensitive spatio temporal nevertheless duration roughly length cover participant channel propose alternative pose video subset entry al competition wu validation test test competition video index localization virtual visual modality hybrid visual modality isolate provide architecture mostly alone gain obtain alone localization couple experiment localization module contribute significantly c precision recognition comparison recognition audio extend introduce speech dataset actor result gain performance modality audio alone next quality audio temporal performance dynamic pose duration overall speech alone perform partly result audio localization annotate phrase moreover style either delay ahead alone predict poor compare representation baseline involve recognition report accurate localization audio possible recall case detect temporal truth context employ drastically improve recognition different start audio fusion multi modal classic deep consist handwritten digits augmentation hide convolutional digit formulation obtain dynamic modal optimize architecture modality channel aspect c fully training dropout dropout visible segment segment clean segment corrupt corrupted segment segment currently fully activation mnist exploit strategy redundancy switch structured network separate layer connect capacity case uniformly modal optimize unit channel turn due drop drop separate capacity restriction place row error table mnist sensitive dropout pt modal optimize balance operate hard constraint real capacity experiment insufficient modelling specific lead degradation whole typically thorough hyper fusion initialize fine layer share layer block section speed observe bias critical mnist optimize validation drop degradation architecture comparative analysis report provide per indicator path layer pre block case block effectiveness observe positive interestingly dropout network noisy channel regularization result respect dropout signal audio channel hand method modality information spatial body whole operate temporal extend augment channel depend sensor pathway without structure video scale integrate explore aspect multi modal term complete modality drop channel wise obtain stable input corrupt partly g student france work action recognition modal aspect taylor university google interested computer vision motion science technology national institute science university machine inspire vision emphasis university thesis co lead different project dedicate human robot de france work computer team human rgb multi spatial modal base scale multi modal capture motion body operate temporal strategy initialization modality fusion dropping channel cross preserve representation recognition track modality allow classifier well noise channel ensure robustness miss channel produce available modality demonstrate applicability fusion modality nature augment audio neural learning deep rapidly grow human interaction effective variable take typical scenario infinitely kind motion real constraint computer demonstrate previously object localization recognition galaxy claim reach face extend explore partially explain orient version competition core aspect approach employ network call dynamic scale visual modality integrate intensity depth pose make decompose multiple spatially grain pay special develop label challenge strategy network robust corrupted channel scheme augment channel arbitrary audio classification major present develop modal detection localization augment channel arbitrary nature inclusion fusion multiple target co ensure missing audio enhance immediate recognition address raw multimodal fusion action distant recognition video extraction spatio descriptor follow classification near accurate reconstruction dedicate infer multi resolution spatial pyramid frame path pose output per score modality depth video
rate far warm next initialize latent price entail w yield department engineering david computer university price develop probabilistic learn strategy set historical fit price new modeling decision estimation solve variety mechanism variable network scalability space price company minimal high transaction company bid bid price maximize company want price future bid imagine company advantageous exactly pay company price run million item learn historical word item advantage paper predict might potential average predictor price maximize second price bid dash smoothed approximate actual typical value function asymmetric high bid bid bid formally fig put bid predict bid bad price regressor fail reflect price advance yield probabilistic difficult seek specifically formulate price study predictor historical maximize problem turn posteriori new objective variable price parameterize draw objective note objective model imagine parameter prefer parameter find spirit technique decision decision find help decision generalize neural price price ref demonstrate optimize price quantify yahoo previous optimizing build idea research demonstrate mechanism nonlinear sec relate recent idea reinforcement markov process amount maximize binary reward likelihood solve similar addition learn simple policy set high bid bid various characteristic date time day date average price open market execute price determine receive price illustrate historical price feature feature regularize regularization regularizer optimization price set price mapping bid consider highest high high second b much predict price much account directly high bid difficult convex address iteratively dc solve result dc expectation price center around become probit principle parameter however latent em update e replace nonlinear predictor parameter specifically high bid next bid interpret relate pz b ir around observed outcome maximize center linear equal smoothed plus involve thus smoothed tp line distinguish set mm distance mm thick connect connect edge connect latent price red proportional time historical attribute imagine variable regularizer compute given take respect previously model price ascent bind optimize price standard appendix integrate real posterior expectation function complete model predictor step amount ridge initialize price update step integrate price terminate change threshold least square advantage change parameterized prediction technique nonlinear predictor outperform linear algorithm nonlinear unchanged e change feature price become kernel gram product degree gram without evaluate operate space technical demonstrate replace lead work computational nonlinearity network layer h analytic instead neural dc term compute oracle know bid advance report ten train split exist
criterion program propose tracking technique ucb armed propose preserve reasonable compare original probabilistic computation argument value value return draw argument call upon return argument upon return program run termination produce sequence induce pair distribution program trace choice simulate invariant mh drawing sample distribution reject next mh offline adjust mcmc criterion proposal target dimensional either systematic step modification sample metropolis hasting select subset proposal point vary probability provide target scheme program course via joint trace differ algorithm random select execution resample start initialization provide lie support random precede choice sample choice equal form trace let trace define modification accept reject accepted output description algorithm indeed essential aspect one influence output quantify influence must translate trace part computation extensive literature probabilistic user output program variable mcmc objective accept indistinguishable reject acceptance parameter proposal propose variable new accepted change bernoulli modification change choice change consideration program probabilistic produce type choose identical output quantify introduce quantify fraction output choice define total hamming adjust compute reward generative modify program remain trace update value accept early reflect program section probability variable variable due update scheme modify delay variable variable maintain component variable show line maintain history ensure cause get modification ensure ergodicity condition degenerate equilibrium selection weight accept change trace accept analyse sequence shall subsequent arrival change occurrence reward count begin sequence add history sequence reward count end probability geometric unit match proportional shall analyse summation substituting appendix note program dimensional distribution program unit reward variable zero ensure unit reward family bandit ucb compute factor ucb lower preferable different bandit arm adaptive expect equilibrium proportional arbitrarily history mcmc ergodic fundamental algorithm adaptation informally must decrease zero technical program broad specification density many adaptive invoke concept versus choice crucially preserve restrict admit program choice reduce adaptive mh algorithm ergodic suitable necessary satisfied ensure across positive language restriction induce regularity leave precise program adaptive scheme adaptation ergodic next demonstrate program evaluate many observe verify number adaptive program sample effort engine kullback kl time number difference scale plot run reward hmm transition trace predict divergence bar plot reward choice adaptive exhibit fast whole many approximation median quantile bar provide insight bar bar reward bar right bar plot height bar unit exploration low unit reward final immediately select often unit converge study define form mx ax bx kx x program value hyperparameter observation predict infer distribution run take bar reward sample count adaptive range bar choice choice predict require low acceptance bar green bar unit reward converge dynamic choice selection involve large amount classify specie dataset observation indicator variable fit split leave dataset belong specie run exhibit fast half many classification kalman previously describe dimensional impose additional assume simple velocity predict prior posterior condition simulated matrix qualitative consecutive chain
would aid start knowledge rate another trade payment help otherwise pay truth scheme payment tackle challenge digital scheme truth identify probably side summary complete prototype impact prototype remainder organize prototype experimental work section adversary payment exhibit behave digital independent e group view omit view due besides act indeed want server specification different assumption different scheme simplify assumption purely digital zero weight happen weight scenario view truth view suppose view separately adopt weight weight prototype server start want specify payment server server apply payment finally trade confirm payment define view introduce three prototype three stage detail st payment confidence pay whereas pay specification weight server server nd mode view server total view assignment server evaluate calculate rd truth payment else function payment e payment trading rely prototype assign view calculate introduce prototype assign view implement server start several truth reliability derive mean error view view start server eq adjust view divide group adversary average equation calculate view restrict suppose maximize great value maximize small combination gaussian unfortunately verify consequence unable derive likely namely weight assignment normalize view probability prototype usually statistic source say influence external limited china statistic approach relationship provide indicator ground deviation view ne bb statistic growth year weight prototype view assign median value calculate result source implement k vote spirit relate category vote let assign view view weight equal difference growth voting source initially year statistic full name error capital formation production worker fix balance secondary ti cccc source payment function different payment sort view v mc v unit ccccc ccccc method vote voting source source method vote voting source ccccc ccccc voting source payment th payment use ground trading growth rate say confidence level confirm payment list payment voting see among weight improve three factor prevent find improvement per payment truth payment fast payment follow view see payment grow increase payment consider variance factor factor randomly see grow rate change payment decrease receive payment least level small similar design scheme crowd relate focus issue firstly blind digital signature trading consideration signature construct signature publicly fair recently add truth suffer quality view find heuristic knowledge fact sensitivity specificity use reliability source calculate crowd
take related gradient even represent however helpful naive implementation hmc update iii term order result sec dynamic desire system framework consider correction distribution outline momentum r scenario explore complex dirichlet large wikipedia find rapid trait sampler kl divergence right bar aim correctness assess high full choose sampler naive implementation converge correct addition efficiently show pre hamiltonian help element explore contour plot versus number wikipedia include fisher membership corpus wikipedia three run lowest report expand parametrization sampling distribution discuss incorporate riemannian riemannian sampler benefit gain hmc pt present sampler markov construct sde two matrix skew devise continuous process prove cast particularly stochastic sampler propose scalability method streaming wikipedia fa mr wu complete write far decompose term q equality compact constructive theorem existence notice matrix hand fourier multiplying arrive write side substitute nice eq n variable clear skew arrive inverse fourier sl process turn new skew ik convolution real dimensional q differential ingredient motion position surface r discretized hmc practice continuous careful show naive hmc interestingly author prove correct eq stochastic noise r interestingly physical interpretation interpretation term langevin dynamic framework hmc rely additionally detailed momentum langevin correspond take variance finite stepsize simulation accurate lead score information metric sampler dynamic q fall correction term take correction lebesgue determined framework provide correct method incorporate idea far auxiliary algorithm take r ccc dynamic framework university edu markov adaptation define transition explore mcmc via subsampling gradient require physical modify account gradient general sampler include gradient trivially previously stochastic propose adaptive streaming sampler become mix model scale poorly decade rise provide efficient hamiltonian monte carlo define explore landscape enable proposal gain burden large quite langevin minibatch mcmc notion langevin dynamic adding amount iterate posterior hamiltonian monte build incorporate provide hmc momentum term naive efficiency mcmc leverage hmc show complex desire stationary novel dynamic ensure challenging require physical natural gradient minibatch method target quite importantly jump hmc variant development stochastic positive diffusion matrix skew symmetric matrix stochastic dynamic explicitly vary explore mcmc maintain stationary completeness although provide take avoid significant specific modification leave question define explore direction choice framework new building sampler synthetic streaming mcmc drawing distribution like hmc auxiliary desired represent augment state discard perform desire marginalization hmc translate simulate sde discuss characterize stochastic simulate stochastic sample mh straightforwardly meet step costly entire mh correction short period stochastic dynamic correction sampling write sde deterministic relate dimensional wiener clearly devise red continuous represent continuous define choice theorem stationary stationary blue corresponding method discretization sde lead rule calculate computationally intensive eq potential form unbiased full gradient key gradient distribution analyze impact make result hamiltonian update variance satisfy rule term stochastic gradient get bias distribution design meet maintain target avoid need however small practice bias tradeoff sampler addition pt mcmc choice sampler mistake implementation ingredient hamiltonian simulate motion object position simulation special u stochastic discretize hmc update arise careful hmc interestingly author naive indeed compare see physical interpretation term langevin fit hmc sampler correct mistake rely intuition readily sampler sampler propose momentum
instead difficulty know polynomial even norm tensor approximate nevertheless approximate performance theoretic completion get good connect appeal atomic observation agree hierarchy universal broad polynomial end natural upper sort tradeoff present previous inverse considerable recent interest understanding area possess inherent run efficiently problem information widely however assumption prove problem assumption preserve input sense step powerful think explore machine algorithm design something provably well reach sum hierarchy sharp phase impose phase transition section decomposition numerous learn hmms modeling detection perhaps random contrast tensor observe tensor tensor highly tensor factor constrain orthogonal enable decomposition tensor tensor tensor whose work elaborate tensor useful keep view offer satisfie make precise think standard transformation clause observation informally maximal agreement readily upper clause refer clause hierarchy bind fraction clause improve work formula clause go clause algorithmic machine also connection make powerful tool complexity need round straightforward hierarchy originally natural hierarchy upper extend tensor follow unfold author complete picture work view pseudo see eq observe entry j triangle inequality chernoff feasible good true type optimization design atomic balanced atomic even approximately well computationally hard balanced require dual hard third instead induce square atomic operator pseudo polynomial suppose say every behind dd converse true nonetheless upper pseudo satisfy kx throughout definition size set exist call sum atomic bind resolution norm completeness see tensor independent bad give j j concavity rademacher random k k j complexity generalization bound replacement set move model invariant think triple index choose tensor tensor convention section sum order triple triple inner remark straightforward rademacher atomic discretization considerably let convenient suppose arbitrary expand moreover chernoff bind q set soon nearly good hope algorithm bound norm albeit dependent resolution strongly formula six repeatedly degree expectation pd bound bound respectively eq think map case multiply maximum care remark index index come clause decompose matrix separately matrix pseudo hence hand matrix separately part claim expectation proceed pseudo nonnegative claim easy section follow eq row still index clause instead triple use otherwise vice versa eq event triple contribute contribute high probability moreover contribute ensemble ensemble q sign independent expand trace k u v indicator cover early ignore variable odd time encode encoding appear distinct distinct give encode convenient encoding question appearance appear appear step value visit visit visit already answer easy answer arise work remove encode compactly term bind encoding uv occur exactly identical encoding step new visit current similarly expectation exactly distinct distinct set positively mutually variable return task triple q conclude equality recall arbitrary element rademacher q readily make plug square follow convex sized plug concentration feasible return error rademacher theorem translate imply refer satisfie fraction clause satisfied whenever recall constraint balanced fraction clause desire hold moreover rademacher immediately low absolute typical value think invoke prove os rr fraction let k k invoke solve satisfie noiseless noise fr definition atomic show rademacher relaxation resolution norm follow introduce hierarchy boolean f v f f kf f f satisfy feasible program solution relaxation think convenient correspond character round hierarchy u ks switch feasible clause constraint feasible solution function complement moreover feasible consider identical since thus complete particular multilinear define multilinear replace multilinear z construction repeat multilinear need c verify accordance complete random formula permit immediate rademacher tensor resolution norm even relaxation round square optimize trivial hold tensor atomic th norm immediate general formula completion ignore entirely almost conclusion predict truly possess measurement need inefficient conjecture formula clause condition fraction clause imply clause thank note give show upper fraction clause direction implication like term computational semidefinite interesting explore several notable possible speed semidefinite programming e round even hope speed application find speed prediction provable minimization work orthonormal guarantee like many helpful discussion copy rgb lemma theorem restriction corollary conjecture definition rgb rgb support mit google award accurately tensor sum hierarchy work attempt hierarchy moderately theoretically suffice broad square hierarchy linear natural characterize rademacher connection formula advance broad solving problem recover unknown object possess special structure perhaps compressed sensing show incoherent observe general inverse nonzero low challenge turn obtain relaxation q solve efficiently interest succeed compressed px nuclear result computationally theoretic significantly prediction define many phase retrieval principal resolution
overcomplete accuracy compete recovery vector removal impulse handwritten remark future direction selector incorporate overcomplete dictionary signal reliably suppose sensor noise overcomplete spike concatenation transform concatenation orthonormal basis component admit selector sense dictionary express use overcomplete two representation bernoulli element fix largely incoherent employ sample bernoulli overcomplete basis frame give isometry successful compressive sense selector fix equation involve proximity equation finding let convex indicator exist therefore also proximity selector incorporate overcomplete proximity proximity guess generate criterion meet construct signal iteration ideally terminate reach meet b selector tend support let define whose element selector solve least square begin equation stage compute multiplication stopping criterion meet contribute length vector iteration complete separation composite signal demonstrate code implement intel ram observation sense sample unit noise respectively algorithm cpu run recovery fourth world united service handwritten digits composite signal overcomplete individual composite composite accuracy however significantly complex dictionary problem table follow second recover value separate wavelet cosine transform composite wavelet length signal select signal observe overcomplete form level haar discrete cosine transform cpu component simulation c std std std std std mean std std std noise composite one dirac spike signal location coefficient experiment vector support entry illustrate numerically value simulation standard simulation std mean std std level std std std level composite signal select set overcomplete concatenation fouri specify signal observe accord plot signal htbp separate std std signal handwritten handwritten digit class nine collection vector form form abuse whose vector select test random overcomplete set principal two digit composition recover digits integer column overcomplete dictionary apply component yield small dictionary coefficient vector explain digit composite appropriate overcomplete decomposition leave vector recover composite space principal identify reduce determine residual generate match accuracy separation image algorithm overcomplete training experiment read composite recovered strength noisy composite signal experiment table composite separate use demonstrate distinguish range figure separate smooth impulse demonstrate separate overcomplete component fairly increase system practice demonstrate noisy underlie overcomplete element yet fast moreover readily involve dictionary use real introduce selector dictionary separate composite signal additionally iterative experiment support cpu applicable wide problem foundation separation signal include image moreover advance compression communication composition reliably component another add compressive composite selector incorporate overcomplete collection composite selector noisy minimize
distance dx kf f dx dx essentially quantify close disjoint band present km introduce km replace define set step psd bt accord window identify entry step construct z normalize km observation carry psd bt x compare distance meaningful comparable average henceforth adjacency determine form component assign come generative hence eigenvalue laplacian bt psd compute contiguous fw white variance across autocorrelation function f moment l correspond length result false connection connection although property alone guarantee sensible appear difficult finite length accord describe km guarantee thank km easy proof theorem km overlap observation contaminate provide sufficiently window bt psd quantify tradeoff overlap k rhs vanish particular increase large km observation length unknown quantity large like come close spirit show come sense via psd distance positive measure finally process cluster cast vector demonstrate clearly thank exploit available synthetic performance solid index table mark dot x index plot anchor west xlabel xlabel xlabel height entry km km font style font legend font none x mark none dash black index mark none dot black mark none black index human activity experiment motion contain sequence activity marker body record optical sequence respectively cluster marker subject difference length length normalize bt psd power ce confusion matrix value ce report subject perform subject outperform algorithm km c c c ce ce ce human proof prove condition theorem say condition observation model close measure follow property imply km select observation contain generative noting suppose km l g generative underlie run guarantee iterate contain model f triangle lead similarly rhs imply upper bounding u g I e sequence e r r last remain bind accomplished tail obey u r toeplitz covariance consecutive element tail concentration namely toeplitz f b red green rgb definition cm cluster finite ergodic nonparametric knowledge generative algorithm dissimilarity term near knowledge rely via simply km initialization consider literature albeit dissimilarity km provide length tradeoff noise synthetic stationary want knowledge number generative divided meaningful processing example audio video sequence production dissimilarity euclidean divergence g divergence distributional fed cluster infer posteriori effective analytical result mostly
put subsection theorem sequence f induction lemma generate f dx estimate boundedness learn q exist desire firstly yield increase must hence part eq complete know rademacher average class function rademacher random rademacher average define g independent state principle useful gaussian gaussian process countable average let j index parameter let variable process follow denote lemma x k hold q jj gx desire f critical part let jk combine complete analogy easily counterpart guarantee give tx jk gradient z partly derive sufficient state hold e x induction theorem notice apply put third certainly verify eq recall eq b bt r uniformly recursive q analogy argument fourth classification learn guarantee establish explicit rate decay size contrast mainly focus online novel refine property average direction firstly instance least loss achieve rate remain secondly kf remove loss clearly future remove loss function popular hinge loss remain algorithm hinge lastly would interesting convergence last iterate acknowledgement thank comment suggestion grateful lemma paper grant proposition corollary explicit extend refine continuous establish guarantee iterate establish first polynomially decay tool refine reproduce hilbert bipartite ranking complete consider learn identically classification univariate predict label predictor generalization classification notable bipartite auc aim pairwise true pairwise online pairwise hilbert rkhs semi rkh linear satisfy reproduce learn throughout specific define follow concept introduce loss typical loss loss logistic purpose online usually draw tx square vary deriving refine square loss gradient contrast size form step iterate proof soon new simple powerful handle learn rkhs eq pairwise involve g turn rate explicit uniformly furth f maximal loss gradient maximal theorem immediately since bound f derive role h old l l definition desire role part eq b generalize general case complete end comment main algorithm inequality old part part take
subsection utilize min band threshold data vary size available lie outside explain optimal day slot detail supervise unsupervised algorithm size accuracy svm attain htbp comparison vector machine naive tree hide member paper technique technique naive vector unsupervise study highest detailed perform classified status utilize secondary spectrum share numerical svm unsupervise propose new support vector cognitive compose type secondary core behind cr access band interference understand spectrum towards realistic spectrum usage various spectrum cover wide study lead researcher spectrum characteristic depth exploit spectrum many statistic spectral frequency autoregressive availability achieve datum secondary utilize transmission purpose less switch control cognitive series evaluate tool limited assumption require derive whether tool tool machine assumption conventional ml spectrum cr aim comprehensive investigation analyze motivation often good ml list propose study use advantageous capable environment region space sense ml spectrum cr management sensing discuss analyze collect walk aggregate cell band dt svm unsupervise hidden classify status status far utilize evaluate status supervise modify hmm lr investigate spectrum approach outperform well technique outperform unsupervise organize ii detailed explanation contain eight band eight band bin band example band bin band bin arrange row frequency represent four constitute minute column vary number bin band user band interference let slot frequency bin total frequency bin energy time slot bin q represent zero computation explain status bin three minute frequency bin decide bin minute quantify chance slot decide rule minimum consecutive bin represents consider vary band frequency day evaluate order guarantee transmission lie apply apply evaluate slot free bin slot vice versa b ml construct feature train train n fed classifier successfully ready test tp I n sequence assume divide reference status slot class therefore correctly give alarm occur evaluate slot allow utilize slot free transmission occur length consecutive present start index evaluated utilize predict status supervise dt lr unsupervised motivation five characteristic naive bayesian call dependency feature account slot represent status response status evaluate bayes classify classified find rule explain iv b decision build tree leaf dt divide subset node label case tree regression tree label label represent record belong iv fraction affect classification unlike dt prevent svm separable datum separable classifier vector define division represent divide divide give q separation margin box I try light intensity evaluate find another intensity determined represent position flexibility motivation slot criterion select square criterion aic square increase unsupervised need sequence recover observation state state value define array produce array main emission probability utilize sequence produce viterbi likely generate viterbi match accuracy forward matrix emission maximum estimate evaluate viterbi eight statistic
edge edge edge edge edge edge edge edge edge nonparametric hand document q equal convenient slice beta constrain property propose gibbs distribution truncation big technique difficult truncation form distribution conditional proposal eq finally w w summarize implement parallel initial k update eq truncate commonly accept maintain consume elegant call resolve adaptively version slice decrease construction gamma posterior sampling distribution update update update eq whole slice summarize effectiveness number topic usefulness generate explore hide choose set ground generate global dirichlet distribution parameterized interest topic parameterize interest follow number word firstly draw document word finally row word relation document inner product relationship retain one adjust dataset topic count bar rough gibbs mix citation link publication dataset consist unique citation publication absence unique cross performance split five stage four scalable toolbox document link evaluation design link link link document interest topic document word link vector word evaluate topic topic interest probability link influence model e topic link document word accord interest fold show fig clarity denote slice algorithm slice version slice keep initial guess normally mixed left setting fit word outperform every notice less accurate see link come topic tend reach link possible number within topic around knowledge topic argue absence accurate domain well discover topic document world predefine relax distribution relational necessity time introduce difficulty therefore present truncated slice experiment dataset real world dataset ability future interested scalable network mrfs mrf constraint support research arc grant dp china foundation china receive china currently technology research interest machine network web associate engineering technology interest support publish book five research discovery grant grant excellent serve international intelligence special issue international six international zhang engineering technology mathematics university mathematics university china associate interest decision fuzzy fuzzy four book conference four grant xu receive engineering south ph sciences university technology school compute communication computer vision computer china currently receive master technology china grid technology chinese science main interest cognitive co appear science computation experience computing etc transaction system technology also number include program co member zhang da xu traditional way discover hide document prediction benefit reveal hide advance impractical relax relational topic probability generative elegant bring spatially document tend document resolve assign global subsampling design discover simultaneously capability importantly nonparametric markov corpus instance paper person understand corpus discover corpus topic service paper organization resolve concern help understand interest person provide accurate service citation example link citation link document nature apparent discovery study network successfully develop mining topic make link consider topic drawback dimensional multinomial gamma require hidden advance normally domain difficult fail document drawback topic remove necessity fix simply dimensional express infinite gamma process two tendency feature find requirement formally relational network document database document retain design superior performance hide topic note use example contribution nonparametric relax assumption use truncate version summarize model derivation synthetic conclude study discussion briefly review aim network link generation link trivial call mixture suppose infer traditional dirichlet normal avoid process replace former distribution limit satisfie dirichlet dirichlet good alternative dirichlet three method schema chinese restaurant dirichlet schema cluster chinese restaurant process constructive process infinite mixture infinite finite gaussian dirichlet dirichlet topic allocation chinese restaurant number dirichlet infinite properly successful mcmc inference successfully many many relational topic gamma extend finite relational propose nonparametric detail handle need gamma document topic process
frame tight frame ds invertible frame shift invariant frame leave discrete index label shift instance illustrate base discrete directional wavelet atom label collection invariant countable q cm circle cm circle tensor discrete discrete right fix translation thank key let frame upper l minor thank ingredient argument band function yield operator l da integral kx du l dl k kx end xx kx inequality complete q remark convolutional specifically call identical frame prove translation invariance purpose importantly frame frames wavelet layer class transform invariant result wavelet base feature particular algebraic frame may want detect handwritten digit temporal location handwritten digits robust linear success practical theory modulus wavelet translation stable linear moreover effective deal signal dominate natural transformation audio transformation contribution goal theory cope transformation wavelet translation major contribution stability wide structural transform simplify short proof invariance wavelet behind theory continuous notation material dx j rx df f fx operator dl dl derivative rapidly decay df dm fx operator denote gradient jacobian jacobian vx vx filtering technique modulus extract pass filter function label wavelet filter satisfy reader short discrete frame wavelet apply element wise prove prove state q distance mm child node f fill child child fill none child fill none child parent none child fill parent draw child child child atom reader might want frame frame applicable g frame course wavelet introduce network generalize allow addition layer require modulus convolution index frame I l f operator note f inequality gene network frame put piece nn atom semi define mm mm circle child parent none none parent child none parent none parent child node pt child circle fill circle fill circle child pt fill child child fill none parent child fill parent none child circle child circle atom discrete frame associate result feature translation time wide one pass band stable depend appendix result retain feature derive condition easily normalize frame accordingly neither translation see technique algebraic frame accomplish state employ argument
quantification order assertion due apply corollary duality assertion follow apply calculated conjugate programming duality safe empty assertion substitute expression open variable value belong uncertainty optimal set e ks ib ns coincide sample outside analogous reduce inside polytope major challenge implicitly define program linearly uncertain problem observable loss generality dependence stage two stage programming uncertainty polytope corollaries feasible non empty compact give parametric empty compact vertex eq assertion infimum indeed follow classical applie assertion follow well assertion rely equality hold due strong apply set exploit elementary observation express pointwise linear assertion ii expect ambiguity free variable case reduce computational wasserstein reduce program piece underlie except uncertainty program scale polynomially description tractable number vertex polytope program hx kx reformulate program hx q hx hx contrast examine wasserstein rich first problem view separable study uncertain maxima concave assumption bad uncertain stochastic assume jt appear instance open loop induced wasserstein reduce norm summation express combination result would solution may overcome wasserstein define process small satisfy every bad case summation separability auxiliary hold provide inf note tractable case wasserstein ball satisfy irrespective sequence decision whose supremum wasserstein attain convex program omit brevity exposition emphasize theorem discretization unless affine pointwise concave function may loose upper expectation wasserstein loss equal bad expectation coincide exactly radius small ball contain effective conjugate term arithmetic interpret bi domain conjugate conjunction minimax maximization obtain denote seem simple arbitrary uncertainty maximization lead corollary employ function conservative substituting trivially coincide extend sample loss adjust proposition highlight theorem mm lipschitz e directly conjugacy equality explicitly consequently ball imply portfolio problem simulation provide optimization capital market return capture short range simplex mx iii aim single q portfolio quantify average high portfolio replace definition solve counterpart wasserstein set proceed ambiguity optimal portfolio wasserstein ambiguity set portfolio portfolio ambiguity analytical consequence computational non x x n nx dominate equality substitution hold minimizer portfolio cone readily recognize cone portfolio wasserstein uncertainty k n x uncertainty nonnegative shift asset definition unique portfolio assertion observation positive price far return decomposable systematic asset constitute high exponent say distance whereby reduce portfolio asset bottom dark asset dark red solve figure average independent run numerical confirm insight weighted wasserstein portfolio constitute average run line critical wasserstein fact across provide adopt robust figure portfolio solid represent event jx likely wasserstein improve consistently unable validate theoretically visual inspection suggest wasserstein respectively indicate empirical consistent indeed exponent wasserstein radius wish portfolio outperform various benchmark quantification section quantify portfolio distribution unknown dataset bind compute moreover replace interior another weak linear line line dataset visualize green line solid wasserstein respectively figure empirical emphasize basis dataset constitute rare bound coincide contain drop wasserstein reach imply almost fall proportional agreement radius magnitude small first large wasserstein curse whereby portfolio portfolio keep optimizer curse grateful valuable grant cm thm thm corollary thm definition thm robust wasserstein guarantee program uncertain finite training wasserstein ball seek view wasserstein state quickly mild wasserstein ball program leverage concentration solution risk portfolio uncertainty quantification powerful paradigm uncertainty generic stochastic find uncertain problem encounter increasingly world never must infer dataset two dataset term optimizer overfitte integral constitute affine distribute hypercube robust paradigm expect ambiguity characterize seminal gain modern optimization decade robust benefit adopt optimizer curse characteristic tractable even though stochastic surprisingly may offer decision counterpart easy ambiguity set ingredient ambiguity set rich confidence ambiguity exclude would conservative ambiguity easy ideally tractable structured program solve robust ambiguity therein metric leibler divergence wasserstein metric etc set distribution close nominal prescribed metric adjust radius ambiguity underlie radius drop ambiguity singleton nominal ambiguity stochastic wasserstein ambiguity identically sample wasserstein ambiguity empirical wasserstein modern measure concentration generate wasserstein ambiguity around large robust confidence achievable wasserstein offer guarantee maker control display property tractable wasserstein ambiguity counterpart art wasserstein ambiguity fix atom bad program via effort optimum paper bad wasserstein ambiguity numerous efficient constructing attain otherwise attain asymptotically wasserstein ambiguity set polynomial performance result theoretically main contribution wasserstein ambiguity coincide pointwise finitely generalization maxima concave bad compute modern finite convex approximate dimensional bad expectation reduce wasserstein metric affine function indicator polytope indicator complement open polytope parametric side linearly wasserstein ambiguity bind optimal validate numerical uncertain fix subset ambiguity substantially reformulate tractable elegant regression list tractable robust practically realization discretization technique ambiguity set one closed bad various measure solution come expense believe evaluate bad function subject ambiguity kullback attract attention portfolio distributional asset nominal focus kullback leibler ambiguity offer model flexibility chance constraint involve respective classical chance nominal rescale probability moreover robust ambiguity set leibl ambiguity fail offer paper generating ambiguity center indeed kullback ambiguity assign continuous kullback leibler ambiguity irrespective fail contrast wasserstein center distribution far elaborate fall scope drive optimization spirit robust seek pass prescribe hypothesis wasserstein ambiguity view fit training rest proceed drive introduce ambiguity establish performance bad expectation wasserstein ambiguity reduce program develop linear programming quantification problem derive extend scope broader whereby two denote conjugacy preserve indicator define define dirac probability distribution represent view stage interpret spirit solve partially observable past realization comprise view govern support drive throughout dependence constitute object govern evaluate hope tight feasibility seek performance constitute bind depend respect datum amount program loss problem argument whereas leibler replace wasserstein metric divergence modify ambiguity conceptual fail continuous variation distance conclude generate continuous rule meaningful choice kullback ambiguity variation positive contain irrespective kullback leibler nd leibl ball inner wasserstein ambiguity ease notation throughout function pointwise elementary j sign real arithmetic whereby dominate focus pointwise maxima remainder convexity close concave mild much generalization subsection case distribution constitute dimensional intractable demonstrate program leverage wasserstein modern distribution wasserstein metric constitute dual pair linear another describe wasserstein represent plan ingredient subsequent bad equal conjugacy represent conjugate support wasserstein bad case expectation eq law construct result generalized moment argument inequality operator dirac introduce auxiliary allow exploit maximization restriction result express conjugacy substitution program reduce virtue extended result continue hold reduce singleton constitute maximum elementary infinity minimax apply ij imply proper identically coincide closure map assumption infinite generalize nonlinear constraint necessarily concave constraint admit theorem immediately clear bad expectation evaluate whereby formalize approximate reduction program stress experiment instrumental decision stress test wasserstein systematically program case bad expectation eq irrespective decision wasserstein attain supremum highlight evaluate extend imply contrast evaluate regardless low mf z q conjugacy imply proper low perspective apply coincide theorem lagrangian evaluate convention duality appear show dual minimax fact feasible simplify objective reduce vanish whenever evaluate least reduce prof coincide term supremum statement marginal dual wasserstein mass upper bind feasibility conclude nk trivial equality construction bad notable program radius wasserstein distribution force objective thus simplify emphasize attain general attain supremum discuss admit bad distribution existence problem amount within wasserstein radius attain
tx sx lemma weak x x assumption strongly monotone establish three lemma scalar satisfy negative integer next hold eq inconsistent read read plug strongly generate q inequality take expectation side desire ready derive rate quasi monotone monotone q assumption straightforward problem goal implementation parallel machine ram code eigen library operation code publicly author website parallel operation affect cache addition parallel load finish every core assign feature either iteration number figure core reduce imbalance load explain core give ratio imbalance grow nearly number core load delay small work well core parallel epoch htbp news speedup time parallel implementation regularize speedup imbalance core conclusion coordinate sure strong assumption preliminary illustrate traditional parallel parallel acknowledgement organization paper would thank grateful helpful discussion l asynchronous agent machine processor core randomly asynchronous special novel decentralize converge strong performance present numerical linear sparse advance storage rapid diverse area internet involve modern grow analyze fashion asynchronous processor core agent execute parallel next determine speed parallel request parallel continuously memory access asynchronous failure asynchronous propose novel parallel coordinate note find hereafter convenience widely equation machine convex km nonempty k strongly km linear differential equation include iteration proximal splitting splitting operator alternate method multiplier admm algorithmic solve random asynchronous mp k update counter whenever agent update iteration apply coordinate whose normalize cache use memory apply parallel due parallel establish appear state include generate properly weakly addition fix point quasi monotone fix strongly fix make assumption selection impose c essential tail leave employ coordinate discuss advantage disadvantage prevent agent general store global pass network secondary disadvantage generation require assignment event advantage user every agent power amount therefore assignment load tolerance numerically coordinate selection convergence furthermore coordinate nonconvex fashion extend continue operator example generate scheme compute sensor area point mention simplicity solve nonzero diagonal equation note give multiplying adding follow equation nk agent continuously kx lipschitz monotone assume monotone hence reduce complete addition generate converge solve nonlinear nonlinear solve ordinary equation ode consider differentiable tx ss q easier evaluate rather give program variable guarantee jk modulus converge convergence term comparison recent bound require decay similar assumption solve decentralized consider agent connect differentiable hold decentralized gradient mx iw I doubly consensus express di see compute iteration eq l poisson otherwise activation agent equality constraint unconstraine iteration q convergence agent decentralize select assign update agent consistent follow summarize view asynchronous decentralize nonsmooth following processing well regularize proximal operator forward backward average apply separable th huber indicate box jk guarantee assume strongly quasi operator monotone function differentiable modulus quasi operator contraction operator convex know think nd paper case projection splitting proximal component block evaluate huber indicate separable evaluating avoid backward backward splitting compute small update evaluate splitting constraint operator indicate evaluate speedup asynchronous nonsmooth consider close proper convex relaxed operator operator solve problem solution finer naive intermediate next special subsection discuss feasibility nonempty intersection intersection otherwise feasibility formulate minimization z implement update efficiently hold maintain random read compute maintain memory agent accord implementation computing reading parallel admm operator lagrange subproblem involve plug structure efficiently block matrix correspond separable need decentralize admm subsection parallel admm consensus optimization consensus eq q nonsmooth admm dual update select therefore arrive parallel h k I k algorithm survey al decentralize agent agent decentralize consensus introduce auxiliary reformulate write proper follow whenever activate present ie ki li update associate asynchronous associated iteration side communication edge derive consider adjacent activate period compute delay inconsistent correspond stepsize lasso follow eq logistic hinge penalty specify reduce fuse asynchronous number assign worker subproblem solve inexact asynchronous proximal admm consider eq differentiable proper multipli condition monotone forward backward splitting problem monotone asynchronous iteration eq admm method purpose definite choose eq update calculate substitute asynchronous kk ki kx x ki mp k kk solve programming problem solve pursuit svm classifier eq kernel function previously list assumption bound delay delay assume order actual delay independent update assumption quasi subsection
fix linear response quickly calculate demonstrate scalability learn gaussian simulate increasingly interested datum past paradigm practitioner complex practitioner quantify approximate bayesian popularity fast runtime scale major approximate multivariate information uncertainty interact family individual elaborate exponential define posterior perturbation previously derive particularly point parameter multivariate gaussian employ mnist handwritten produce monte mcmc even variance dramatically accurate magnitude mcmc wide theoretically gaussian number point fast influence mention unobserved dimension give model belief bayes factorize kullback divergence factorization obey rest denote exponential natural write scalar entry otherwise notation guarantee contain eqs across mapping factorize distribution denote I solution technique improve estimate log perturbation probability interior feasible ball fact generate far perturb p vector scalar perturb perturb success often derive interpretation individual approximation derivation q substituting writing arise improper eq q normal length variational factorize component case posterior correct variance estimate equality location appendix asymptotically transform fisher amount datum go detail unknown include nuisance nuisance mixture assignment treat nuisance directly compute inverse impractical covariance able sub divided grow variable field factorization factor eq sized identity respectively finite mixture gaussian model investigate analogously much bayesian posterior perturbation convenient formula covariance add notational sensitivity dx nm refer derivative connection covariance version parameter posterior belief assumption sufficient account correlate cause proportional q detail value use influence score note covariance require covariance draw covariance influence divide three main nuisance parameter also distribution use perturbation nearly result perturb inverse take appendix mixture constitute application may illustrate efficacy multivariate gaussian normal multivariate dimensional mean component employ trick pz nk nk augmentation component multivariate univariate nuisance distribution influence speed gibbs augment function implementation heavily algebra datum world mnist handwritten digits principle center intensity keep evaluation project onto subspace training separate handwritten result keeping calculate expectation posteriori expectation count classification majority label test measure test stress feasibility practical covariance estimate sampler treat shape real interpretation generally interested marginal uncertainty pose standard restrict regime switch sampler avoid prevent mode simulation component dimension uncertain point cause mis perform sample calculate mh produce truth derivation deviation particular alternative deviation covariance closely mnist take second whereas explore detail eq linearly polynomially also experimentally sample fast though preferable grow length vector product involve sufficient redundant parameter simplify though perturbation argument require interior nuisance normal replace sized sized directly cubic scaling simulation scale slightly vertical examined demonstrate mnist derivative score score find new numerically calculate influence point mnist point range concentrate one score perturb dimension mnist covariance calculate second process score mnist score two approach practically indistinguishable simulation fig see pattern datum depict moderately graph assign sign change effect rgb seen overlap distant nearly ht upper influence mnist label random particular one dimension recall mean wish score value respect influence point figure correspond axis logit component component different mode mostly handwritten influence amongst component variational demonstrate estimate scale point quickly calculate score influence multivariate traditionally hope work model allocation prove factorization abuse write whole vector exclude exclude aid sometimes explicitly component let denote th product define way intend make multiply lemma additional always via eq suppose collecting way k allow exchange expectation expectation proposition proposition exponential family interior feasible ball radius within origin define cf approximate track true mean vary equality eqs depend write part factor dimension dimension zero analogous lemma know exactly follow x posterior multivariate normal let index final constant invertible invertible equation stack assumption follow interior exactly log interior derivative e covariance exact draw connection correction expectation asymptotically transform fisher matrix formally argue variational block unlike covariance
marginalization significantly rely rigorous suited alternative bayes computational load posterior parametric unknown form encode hyperparameter assume accord gp distribution conjugacy gp tool trend encode gp greatly choice choose likelihood point multimodal happen point integrate could beneficial obtain marginalization average range robustness marginalization marginalization treat integrate predictive computed unfortunately analytically intractable weighted approximated independent law hyperparameter new hyperparameter marginalization multimodal hyperparameter posterior far propose simplify tune load hyperparameter ability handle monte carlo smc sampler closely popular smc particle smc extensively understand apply parameter bayesian audio smc sampler respect internal smc application compare demonstrate marginalization computationally demand dataset target marginalization marginalization previously literature implement depend tuning scale slice propose sample solving approach possibly adopt hamiltonian monte carlo chain carlo tune curse problem propose approach smc connection pl marginalization make transition marginalization require particle particle smc idea construct density easy reader familiar section construct sequentially py n give evolve alternative prior geometric path guide sample smooth weight usefulness eventually particle particle residual resample decrease resample invariant sample proposal adjust otherwise repeat update increase mix smc online another easily point transition decrease design choice proposal concern proposal brevity adapt maximize markov explore part center rao give covariance py p accord sample sample interval size use choice piece summarize compatible toolbox computational evaluation number smc number mcmc transition evolve high instance experience often could beneficial particle decrease represent probably probably couple dot stack introduce spread lot close opposite cluster probably carry smc monte importance solid line red dot grid computation axis equally thresholding adapt vice versa absolute bound adapt theorem justify vary particle integrable far let nn sampler adaptive particle eq neither induction lemma benefit compare common method world different well affine function square noise measurement good repeat run seed give similar apply adaptive importance marginalization fast comparable might competitive curse sensible however note load substantially decrease hyperparameter utilize conceptually figure deterministic clearly initialization cause computational alternative although initialization illustrate marginalization competitive hyperparameter base online application marginalization efficient hyperparameter sampler problem fit inverse seven degree robot arm map joint total end use separate involve hyperparameter point subset select ignore propose hyperparameter point report prior variance method regressor standardize divide well otherwise
define rbf suitable learn dimension distance wang label active criterion reduction exploration exploitation initially explore annotation acquire exploit perform refinement density uncertainty hyperparameter complex automatically measure surrogate seek make overall unlabeled discriminative operation example minimize stem unlabeled pool error feasible demonstrate superior criterion field serve baseline cope must consider subsample selection assumption explore evaluate unable perform refinement statistic label information within cluster also representation differ advantage ability refine boundary review detail pool base active initially know pool weight space node supervise learn harmonic energy minimization harmonic laplacian diagonal entry matrix part matrix except label conditional distribution learner learner quantify predict true multi class case base marginal sample integral produce error true expect greedy error combination unlabele q add calculate label provide select expect risk remainder risk criterion seek framework example size address approach expense exhaustive desirable hierarchical criterion note previously similarity correspond affinity similarity radial depend histogram set choice unlikely dataset want region dimensionality reduction non similarity conditional probability interpret pick variance center value neighbor choice enforce criterion give calculate send oracle want specify sufficient node adaptive unlabele every represent bold use evaluate contain evaluate bottom set expand advantage cluster walk transition summary cluster differ strategy seek reduction first unlabeled child start proceed expect expand active child add expect oracle hierarchy top bottom cccc ccc dataset rand entropy pca mean despite evaluation ht around represent vary run toy illustration advantage refine two need ask oracle low hierarchy toward occur tree reach depth make unlikely alone open make hierarchy potential respective graph near initial query hierarchy code produce inferior graph graph constant node elsewhere knn bandwidth segment mean outperform baseline compare algorithm seven baseline entropy approach strategy empirically perform summarize area interestingly outperform computation greedy necessarily optimal encourage set hierarchy observe offer performance high variability iterative propagation prevent expense marginal illustrate subset ccc segment depict present scale linearly soon impractical query perform accurate al human generalize slightly accurately well par complexity matlab default combine pick oracle big keep interactive validation variety situation supplementary far across dataset lead area across run plot apart hierarchical construction give graph compatible graph perform graph choice part file effective propagation future work either exist learn inductive would also account effort never future information online ep ep token good semi expert inconsistent dataset new evaluation variability release circumstance active costly dataset practical active vary sparsity dimensionality run human expert match tune supervise specify classification image massive expert suggest unlabele choose give oracle unlabele pick query quickly train repeat query unlabele therefore popular image vertex encode label feature select inference benefit influence criterion decide image naive put individual receive label method construction good crucially exploit building construction hierarchical allow ask unlabeled image heuristic overall curve significant hierarchical cope dimensionality balance exploration good refine decision boundary specify benchmark construction establish across dataset cover thorough active learning tracking categorization object semantic segmentation video segmentation human automatic body relatively facilitate interactive perform far active learn propose voting feedback user however unlabele pool interactive co informative current wang annotation supervise labeling
sensitivity confirm case asymptotic behavior default diagram focus diagram score forecast tuple forecaster respective outcome compete issue set prediction convenient interpretation empirical probability average empirical expect compare difference namely diagram expect intersect unless support analytic score fortunately establish compare well argument quantile empirical forecast dominate dominate n n x omit functional case score right vanish I jump similarly case slope change slope median give example issue forecast artificial estimation forecast diagram panel forecaster world infer dominate remain give rise relation scoring reflect consider joint low empirical superior economic dark unique share attains attain share score diagram economic setting diagram empirical score visual appearance stem score depend forecast supplement diagram difference autocorrelation west view underlie band pointwise nominal forecasting united states survey bank datum source use motivated mean forecast another survey choice except fourth figure forecast diagram show curve score survey intersect suggest survey whereas survey explain well match survey band fairly relate rich forecast cover forecast forecast probit short interest period economic choice find original probit probability forecast gray bar indicate tend probit forecast period diagram row attain band score difference small confirm superiority probit forecast partly probit demonstrate probit improve information play forecast switch introduce et autoregressive hour ahead forecast wind wind united states refer specification terminology datum period forecast period forecast bottom quantile forecast exceed outcome nominal forecast calibration forecast ar wind nonnegative quantile identify suggest superiority forecast benchmark forecast diagrams elementary confirm mean expectation functional forecast event aspect interpretation problem differential perspective argue forecast score specify quantile scoring target rather merely specify functional relevant class consistent survey specify specific function whenever way another scoring participant forecast platform inform scoring start competition receive available utilize community situation international weather loss apply center behavior routine diagnostic evaluation rank forecast center decomposition reliability extension diagram connect huber traditionally estimation quantile asymmetric piecewise choice respectively contrast logarithmic infinite lebesgue half generally raise focus interpretation paper criterion consistent extend generalized quantile mixture functional study whether representation score rule assign predictive proper variable forecast apply answer extension probability forecast general feasible member lie negative encourage widely rank equal forecast event threshold simplicity assume quantile invoke relationship integration quantile evaluating weighting scheme family scoring motivate justify european grant reference mm mm generalize quantile economic measure quantitative finance journal r probabilistic weather forecasting road journal american verification interpretation forecast weather journal convex journal mathematics binary structure work predictive accuracy journal business economic b square probability outcome misspecification working department economics california wang risk prediction journal economic j evaluate forecast journal american association american forecasts quantile scoring business economic journal forecast journal e calibrate forecasting american association hand classification american role forecast management p fan global forecasting competition international journal forecast huber j estimation location quantile quantile convexity quantile order probability forecast verification tool probabilistic forecast management relate specie economic management science proper forecast ratio situation weather review forecast verification weather decision forecast value asymmetric west positive semi autocorrelation consistent matrix volatility comparison volatility journal forecast paper economic relative economic journal verification guide nd j forecasting power yield business journal american association method statistics quantile david eps forecasting application journal research thompson weather forecast weather production possibility inform journal economic theory partially inform forecast bayesian journal american economic forecast market economic e coherence permit subsequent immediate quantile h hx straightforward consequence every eq increment unique turn bregman variable representation hx consequence x finally increment mix proposition elementary scoring associate coincide argument right vanish latter expectation handle analogously elementary strictly proof complete forecast c cdf density analytic score forecaster scoring quantile scoring elementary scoring xy accounting scoring event mm scoring ranking receive functional evaluate forecast scoring purpose hand score score functional parameterize probability forecast constitute important quantile along function admit economic interpretation threshold decision give forecast dominate sense preferable consistent empirical average element score call diagram comparison compete forecast key word phrase economic sensitivity forecast forecast past broad develop forecast nature predictive quantity forecast reason make report situation predictive distribution potentially set key compete forecast score consistent class score prominent expectation alternative classical show condition score functional subgradient arise forecast basis forecast broad forecast ever ideal compete forecast scoring observe among special correspond predictive success obvious reason prefer raise question many alternative motivate regularity x write important forecast score preferable elementary see represent oracle future empirical forecast let consider triplet compete forecast forecast respective versus display show forecast wind wind versus along pointwise band forecast preferable section generally quantile apparent readily interpretable sense represent expectation forecast special correspond level remainder organize mixture representation relate economic functional differential aforementioned diagram forecast comparison proof defer quantile review forecast emphasis consistent first introduce denote measure borel lebesgue follow right continuous rectangle yx cost forecast regular jointly measurable rarely evident implicitly notion set functional mapping functional moment generally correspond quantile value specifically functional map one quantile scoring consistent relative eq forecast admits score optimization attention functional quantile mean probit logit class consistent score quantile mild scoring form decrease prominent arise piecewise similarly level convex example arise estimation regression representation quantile parameterize decrease function function subgradient general neither uniquely version finite point leave decrease let hand everywhere function purpose family regular scoring function quantile apparent every admit member admit form satisfy h g furthermore member admit mix unique hx hand respective pointwise qx ex measure assign finite interval quantile asymmetric lebesgue e measure asymmetric square lebesgue special case exponential homogeneous line lebesgue remarkably case distinction elementary measure consider sense member mixture representation elementary extreme true admit average build therefore family need similarly denote class situation fully toward deal level cdf exceed threshold economic meaning functional far house suppose put score consistent respective strong scoring order relative continue inequality strict borel finite quantile score sensitive quantile every score functional relative representation sensitivity scoring scoring sensitivity apply strictly relative suitable closely et al characterize regularity compactly score rank representation notion follow seminal prediction consider distribution forecast probabilistic cumulative outcome respective tuple utilize measure language sigma cdf value measurable quantity forecaster mm z mm denote cdf specify joint tuple perfect sigma variable ideal forecaster unconditional outcome ideal uninformative sigma forecast extract focus quantile exceed forecast forecaster forecast case identify tuple forecast information x x extension might case quantile straightforward forecast sometimes prediction specify tuple cdf represent forecast define notion forecast forecast turn forecast set suitably penalty score proper rule therefore careful scoring induce proper score predictive probabilistic forecast dominate relative rule turn quantile regular functional quantile forecast outcome prediction forecast definition forecast outcome forecast x notion respectively essentially forecast dominate another inferior type case quantile dominate preferable least inferior predictive consider functional quantile straightforward dominate recently show rich dominate result carry include limited quantile perfect sigma generate
plug prove pp remain next l fact number conclude complete section lemma nonlinear dimensionality factor forecast forecasting index high predictor extract forecasting deep explicitly state sufficient forecasting correctly presence nonparametric forecasting extend asymptotic direction method multiple forecast component index forecast rich economic finance include forecast production forecast use asset outcome high microarray vast effective cross think vast dimensionality predictive turning curse dimensionality vast predictive reliably response portfolio management testing predictor leverage development forecasting problem forecast focus similar forecast predictor first principal pca underlying factor follow many point relevance forecasting lead improve forecast predictor similar fashion predictor take forecast generalize partial fundamentally forecast extract forecasting point model especially thorough often additional gain paper forecasting factor volatility estimate factor significant return effective effective class nonlinear cognitive formation pose challenge estimate work completely forecast introduce favorable call forecasting introduce seminal arbitrary unknown forecast independent unobserved put way forecast relate unobserved goal forecasting go dimension effectively reveal index enhance greatly forecast study improve especially forecast one advance method incorporate deal identify present technique well reduction aid factor high dimensional regime size organize forecasting fall demonstrate forecast simulation conclude remark appendix framework forecasting forecast unobserved index forecast function factor predictor loading drive response ease lk orthonormal model mention forecasting forecasting illustrate advantage scalable explicit kt input via model nonparametric target predictive index word reduction direction span identifiable subspace span throughout dimension direction index canonical matter principal linear forecast forecasting explore predictive embed common factor create index unobserve nonlinear forecasting forecasting situation effort tailor factor fundamentally limited forecasting traditional focus covariance forecast optimal forecast nonlinear forecasting utilize consider inverse salient feature impose ks thus estimate direction investigate practice inverse obtain predictive conventional follow conditional information underlie divide slice direction observe orthogonal eigenvector positive orthogonal factor natural estimate predictor since sequel interestingly exactly loading constrain estimator equivalent estimate eigenvector converge central yield consistent predictive index unobserve factor construct large eigenvector index make forecast loading constrain extract factor eigenvector sample counterpart yy influence introduce double refer slice observation slice form analogously show alternative way estimate loading example procedure characteristic model lead equivalence forecasting include determination slice factor estimate direction point smoothing may slower show rate convergence matter sufficient forecasting long estimation propose determine eigenvalue distribute forecasting model many latent determine literature estimator eigenvalue enjoys grow consistent conditioning unless vector represent norm large value large eigenvalue first detail forecasting function loading loading linearity impact correspond center contain reduction impose strong generating mix dependence consistently loading residual every establish estimate inverse curve norm far imply convergence index assumption enable obtain fast direction large eigenvalue eigenvector eigenvalue give direction linear factor regressor forecast affect forecasting extend predictive use forecasting coincide forecast consistently condition predictor let view forecasting find estimate effectively onto dimension fix cross dimension regime methodology applicability common factor vast run forecast nonlinear directly forecast link misspecification central subspace weight response normality specification link fall central specifically converge sufficient direction model normally forecast asymptotically direction multiplicative check fit suffice employ nonparametric factor forecast predictive belong subspace reveal dimension subspace power contrast entirely contain estimating try effective direction capture drive comparable significantly conduct assess forecast target linear plus prominent example asset could book ratio forecast aggregate market return specify I let predictor combination predict loading draw standard ar draw simulation standard take infeasible principal regression forecasting typically regression sample suggest ccc ccc sf sf pc simulation sf forecast principal pc result summarize table sf denote forecast predictive denote component principal pc perform poorly table sf five regressor sf consistent whereas due poor target economic influential examine financial generating e factor response plane algorithm forecasting involve strict subset irrelevant factor three suffice factor evident correlation estimate factor variance median correlation factor replication next principal section examine sample direction span ensure factor loading meet calculate invertible still forecast coefficient evaluate measure use reduction calculate forecasting direction cc forecasting square replication deviation ccc ccc median percentage replication use predictive extend matrix predictor sequel replication observe series increase term pick sufficient forecasting pick sufficient direction consistent evaluate index build forecast interaction sample report purpose principal regression extract first principal top denote principal direction relatively low interaction two pc predictor incorrectly picks exhibit large investigation sufficient forecasting dataset series take make forecast target involve recursive factor estimation datum begin forecasting category may target second category eigenvalue time small pick representative simulation sf sufficient single predictive index linear forecasting sf
error empirical risk dp good practice denote become risk proxy minimize limitation hence e approximation family express opt e est opt analysis sgd effort translate chance sgd reduce expect large limiting rather perform convexity last observation base neuron assign consideration consider stream sample risk equation datum descent gd update represent gain gd parameter matrix gradient calculation effort consider gradient available converge large importantly engine algorithm imbalance modify modify express imbalance run count datum instant derive analysis oppose section ease express represent far unbounded lyapunov increase w sgd output persistence amplitude boundedness prediction long bound true application apply streaming learning engine engine specification auto stroke mix air form cycle retain reaction rate air engine suitable c engine stroke stroke mm ratio lift peak lift degree angle negative engine control fm angle angle center include temperature manifold pressure flow pressure temperature air engine indicate angle net mean high pressure reading please refer identify variable dynamic operate boundary appropriate require model engine operating envelope variable capture steady conduct naturally condition vary fm every engine make amplitude pseudo use engine frequency suitable consider acquisition pressure angle per collect combination input engine drop bar phenomenon lead undesirable engine operating range conceptually constrain require without complete occur ratio result increase lead variation feedback operation temperature high ratio heat release rise rate engine phenomena input engine engine nonlinear dynamic use output represent past order respectively augment measurement convert eq sample classification definition ahead predict time parallel architecture explain exist use become convert model long prediction control requirement base engine variable cycle ahead action impact engine variable engine fundamental represent engine consequence engine input predictive control orient modeling exist identification slow eliminate hand purpose control engine condition approximate extreme learning different state os sg baseline baseline purpose baseline offline behavior completely offline offline produce purpose consist unit fix parameter cycle engine update sequential cycle ahead prediction compare step ahead represent recommend os layer initialization fair sg determine trial prediction robustness outside measure normalize square performing os sg initial sg growth keep os govern add leading consequence value os os sg implication theory result reflect summarize rmse incomplete training convergence aim engine may rmse os sg sg computation dimension datum one os win marginally accuracy marginally model ultimately feed predictive control framework step ahead generalization parameter well demonstrate sg linear baseline adopt nonlinear identification engine summarize prediction experimental instant allow several step see outperform offline online behavior operate note limitation sg model sg os converge stability consume system os tune properly initialize predictive operating engine develop prevent engine towards dynamic operating envelope engine model offline paper operating envelope sg operating envelope common unstable operating envelope manually unstable depend engine heuristic engine number stable unstable imbalance imbalance imbalance cost heavily sample os sg compare classification offline previous include justify adopt nonlinear offline train effectiveness online capture operating envelope control engine temperature pressure envelope measurement history dynamic sign indicate future q section learn experimental engine define label consider sensor fm fa engine cycle process engine update cycle step ahead ratio nonlinear approximate initialize extreme learning covariance consist unit study portion use initialize sg weight handle imbalance imbalance datum ratio number majority instant imbalance conventional misclassification bias classifier majority label predict label undesirable metric skew correctly geometric ta classifier accuracy equally class result imbalance table performance model perform well identification accuracy problem achieve counterpart os perform offline completeness sg slight train os prediction accuracy sg slightly os indicate sgd online subtle os sg accuracy indicate predict well sg tune fail tuning class predict htbp gm os sg model os sg unseen operating envelope within operate manner envelope engine fm instant record datum point predict engine operation unstable mark predict plot dot instability dot indicate misclassifie understand fm also understand fm plot whole os sg engine fairly amplitude inherent experimental fm bad choose engine fall limit exhibit dot work predict unstable available accuracy os clear predict dotted plot true sg sg inferior predict unstable evident set sg compare os stability lyapunov sg involve tuning suffer sg develop operate envelope engine suggest generalization os sg sg advantage design sg appear comprehensive perspective hardware explore operate range development upon department project direction pi account work unite usefulness use herein specific constitute united author herein reflect united stochastic article gradient extreme develop sg stability stability estimate identification nonlinear reduce os square order demonstrate algorithm advanced engine system online imbalance operating envelope engine indicate art add reduction effort stochastic extreme online online imbalance lyapunov homogeneous operating envelope prediction engine engine homogeneous reduce consumption compare traditional highly operation advance nonlinear control challenge factor contribute include absence direct narrow issue engine physics model significant associate cost engine calibration key engine engine several operating effect far engine limit engine operate engine engine operating envelope engine state engine state engine require pressure respect engine state represent engine temperature pressure mixture production engine engine narrow operation operating envelope operate engine operate engine significance operating envelope introduce operating envelope crucial insight engine load condition use enforce envelope could enable engine diagnostic event lack engine normal monitoring emission control engine detection essential meet requirement envelope alarm engine article detail art os stochastic gradient derive along background engine section discussion sg conclusion thank setup output feature include capability get minima iterative well capability label problem
step bit set locally constant implicitly rewrite set drop ease partial side element column transpose column column eqs respectively simplify chain l respective ns complete derivation similarly active row modality non active optimality condition section example ray laboratory md mail dictionary use input advantage feature fusion sparse input sparsity enforce multimodal dictionary simultaneously multimodal dictionary feature code multiclass flexible fusion modality multimodal three application multimodal recognition face recognition counterpart computationally equipped achieve dictionary multimodal sparse individual fusion fusion fusion aggregate different classifier decision individual train classifier fusion study fusion mainly due stack suffer curse concatenation among noisy redundant classifier limitation improve classification attract researcher successfully acoustic structured dictionary usually stack class expand feature joint show assumption simultaneously multimodal dictionary modality sparse construct suffer become demand neither discriminative face recognition dictionary compact dictionary svd aim adapt achieve dictionary adapt call utilize rather reconstruction minimize incoherent train dictionary minimize atom small unsupervise dictionary reformulate scale factorization solve recently tackle supervise solve compressive sensing majority drive dictionary applicable single view dictionary common learn application recognition view dictionary exploit correspondence dictionary pattern share formulation atom error fusion heterogeneous multimodal learn extract typical template multimodal feature template represent modality localization multimodal multimodal dictionary jointly modalitie encouraging utilize dictionary atom independently utilize among modality focus dictionary present overview propose framework different enforce modal classifier major contribution multimodal task drive algorithm fuse heterogeneous train classifier enforce modality code optimize binary multiclass classification unsupervise product version bi task multimodal solve framework fusion modality feature modality pattern modality performance multimodal multi multimodal counterpart efficient sense still rest organize supervise source information review joint sparse multimodal review propose drive multimodal comparative several benchmark conclude bold bold letter symbol vector column column represent index row index index form index element x ij widely compressive principal learn orthogonality condition statistically convex set atom dictionary exploit unsupervise often draw train reconstruct input robust feature code ground truth label associate input measure task formulation classifier parameter emphasize task drive dictionary show set set optimize classifier representation level fusion source multimodal dictionary different modality multimodal regularize correspond sparse alternate admm encourage encourage modality enforce modality present reconstruct add extend within result source fusion multimodal dictionary potentially represent supervise dictionary adapt unsupervise multimodal dictionary unsupervise multimodal derive characterize modality sparse code frobenius joint sparse case optimization assume draw use project orthogonal algorithm typical problem minimum practical minimize reconstruction modality level optimization parametrize observe take discriminative convex twice binary binary label belong multimodal classified sign monotonicity intercept easily add use bilinear learn multimodal regularization replace frobenius norm bilinear rich classification need careful fitting multiclass vs vs handle vs set softmax loss softmax regression define vs multiclass turn change coordinate rest optimal test classify heterogeneity modality impose sparsity reconstruction relaxation sparsity multimodal replace intuitively group dominant modality small encourage reconstruction modality design modify multimodal problem active row define update non ds rest unchanged multimodal dictionary ar face multimodal choose quadratic multiclass multiclass vs select use atom compare value use except use positive select heuristic constant anneal iteration whole try retain empirically selection considerably also note competitive cross comparison mkl multimodal classification modality include right face ccccc modality svm svm lr faces pose illumination expression seven sample portion use optimize five modality face modality ar modality show pixel zero dictionary dictionary atom mkl lr classification dictionary mkl rbf kernel equip classifier result multimodal classification way utilize modal algorithm namely multimodal fuse zero enforce row sparse denote multimodal norm unsupervise multimodal denote algorithm prior algorithms modality fusion well fusion right modality agree modality highly multimodal dictionary improve recognition performance fusion art evaluate propose supervise multimodal dictionary mix level fusion aggregate score vote independent decision obtain modality fusion compare include classifier mkl algorithms ar performance moreover achieve performance classification joint prior number atom ar cccc test sparsity equip dictionary dictionary confirm reconstruction achieve dictionary perform discriminative dictionary keep per dictionary moreover available meaningful unsupervised multimodal sub dictionary approximate stack final indicate indeed formulation enjoy especially number relatively recognition task real application hand fit expense discuss dictionary size typical multimodal fig increase advantage state compact dictionary dataset contain face use training classification different table modality couple modality reflect lr svm lr propose dataset consist expression record span head height camera subject present manually protocol view angle scenario pose handle multi modal divide view test way dictionary size class individual modality performance multi face recognition supervise learn outperform fusion study application multimodal dictionary learn one tune dictionary individual mkl modalities subject gender corrupt modality four modality modality subject overall remain transform preprocesse modality input atom dictionary modality split table achieve dictionary multimodal fusion fusion fusion modality fusion modality different cumulative roc originally recognition competitive modality recognition perform multimodal performance extract among modality comparison joint sparsity
total intermediate probabilistic latent seek dimensional latent equation white component iteratively prove model pca principal datum projection vector intuition behind first make guess principal state find give log parameter expect repeat method converge reconstruction analysis present base principal per show need iteration complexity communication state complexity work show pca without pca store intermediate redundant computation redundant tp library eigen matrix svd probabilistic analysis complexity eigen covariance cubic complexity prevent dataset time dataset dimension addition communication complexitie two high communication cost still method pca efficient assume small typically two evaluation however reveal employ communication use prevent promise pca large analyze pca show computational communication scale dataset indicate two eigen intensive complexity cubic multiply quite hand show svd pca performing suffer communication promising pca recent design scalable outperform often algorithm assume computing comment limitation support dataset analyze complexity consider software library report help researcher pca library characteristic design pca day sensor web site value business information machine volume machine learn new challenge distribute execution machine may intermediate carefully bottleneck regardless compute complexity communication consider metric complexity terminate intermediate analysis library library linear algorithms parallel machine help software library knowledge environment use report matlab name include compose letter indicate star transpose respectively element rest organize distribute execution analyze eigenvalue base call svd analyze pca algorithm summary conclude analyze two metric needed terminate well state pca node exchange among total intermediate complexity phase end phase must produce execution start amount delay increase execution bottleneck delay speed platform use code exchange intermediate storage virtual communication total abstract detail hardware architecture give target dimensionality summarize follow center computed subtract matrix compute multiply transpose multiplying none fit memory eigenvalue eigenvector put formula column eigenvalue cholesky factorization sort vector principal result intensive complexity eigenvalue decomposition size complexity incur substantial make describe section programming language language extensively use include scalable implementation include pca mean give several dense library framework svd algorithm optimize offer speedup memory usage matrix explain use compute provide converge apply singular compute qr qr matrix singular assume second value singular accord singular formula principal point operation either cubic dimension disadvantage routine intermediate overhead bottleneck qr computation result matrix intermediate matrix intermediate element store sparse application however implementation library take employ svd library obtain principal many mean subtract sparsity svd prohibitive provide svd slightly variant work quality rapidly iteration exceed value solution avoid support enhance quality eigenvector fix initial choose enhance describe recently randomized compute approximate matrix compute svd stage compute approximate via svd stage al describe computation compare randomized accuracy rest stochastic require orthonormal matrix contain importantly target compute matrix approximate vector extra add great computational set without change discussion term symbol step compute describe approximate comprise regard one requirement output compute slowly decay I vary accurately motivation behind singular value high power
influence assess sign color vi modeling assess sign color reflect class ask question influence pseudo unsupervise vi training fig rwm similarity influence vi determine hyper control direct impact result rwm critical never result almost together parameter much center attract rwm hyper variance shape shape consequence rwm similarity set consist five generate uniquely sign density large component small rwm different impact component rwm question vi three learn result rwm similarity training span principal project visualization normally first rwm rwm svm set normally rwm standard rbf two j rwm capture gmm base approach advantage rwm implementation question rwm always lead exploit property realize library cope psd parameter rbf penalty lin two dimensional parameter contain shape set sigmoid kernel htbp lead heuristic combination small generalization kernel straight minus cut good combination gray circle color along line svm rbf cf red line gray circle fig search span set uci machine repository correspond exhaustive circle parameter combination heuristic work rwm rwm kernel parametrize rbf define rwm unsupervised rwm partially rwm kernel gmm see rwm shall rwm underlie component process varied vi svm rwm colored fig show classifier accuracy slightly component htbp vi contain vi rwm depict rwm information model rbf consequently rwm yield namely rwm state experiment rwm parameterized rbf htbp value often handle categorical sample categorical one scheme extend rwm kernel eq weighting value respective encode categorical checking dimension equality also categorical part component identity matrix rwm behave encode categorical compare kernel gmm kernel svm call matlab adapt cope multi visualize kernel set benchmark mention detail summarize htbp visualize behavior rwm take five artificial suggest uci machine learn repository mixture email normalization five conduct fold apply exhaustive density underlie rwm gmm data rwm gmm first da fold color correspond algorithm whereas remain use kernel rwm black line decision boundary gray colored rwm gmm correspond locate indicate large svm worst surprising label usage derive unlabele significantly furth kernel relatively even generate produce shape data gmm rwm mixture rwm svm overlap clearly interestingly rwm well set set gmm kernel straight rwm responsible two assess new numerically conduct publicly conclusion conduct heart page uci low rwm gmm rbf comparison function parametrization average fold significance significant least state kernel higher high however correlation vector classify table limit accuracy svm combine small accord highlight set degree value nan hypothesis critical difference significance rwm kernel gmm rbf kernels classifier rwm rwm average significantly rwm combine vector regard need cm rwm rwm heart page block seed rank experiment increase also reject hypothesis rwm combine kernel confirm cd show svm perform svm kernel significant show svm rwm kernel perform good average advantage rwm rwm slightly gmm kernel column table htbp cm rwm gmm rbf rwm heart seed two experiment label svm completely supervise supervise classification rwm gmm datum reject rwm belong top perform classifier svm cf close table high small rank comparison rwm kernel state rwm gmm rbf experiment bring column combine structure information presence label rwm may kernel rbf rwm perfectly rwm gmm kernel c rwm kernel gmm c heart page seed rank evaluate run intel processor fold cross rwm gmm rbf solve unlabeled rwm train test density rwm size small function building comparison rwm gmm time estimation time much estimate offline train svm small rwm take comparable building matrix htbp r time rwm gmm rbf train heart page seed rwm rbf rwm whose estimate vi step give rwm mahalanobis distance sample rbf allow learn experiment optimally especially label reason combination yield validation experiment additional regard solution user advantageous rely good exist density shall technique parametric parametric generate clearly rwm cluster vi offline unsupervised manner besides gaussian anomaly third gmm vi influence also precisely gmm appropriate sampling adopt cope restrict isotropic rwm set article avoid discussion psd definite clearly formal always semi use experimental rwm g another numerically stable weighted mahalanobis rwm data key advantage follow kernel label suit semi due rwm way easy fact svm implementation provide heuristic svm easily adopt encourage investigate future approach cf e svm build step cf variant vi cope set theoretical rwm kernel density expect model rwm kernel machine need besides radial basis tailor assessment structure space inherently contain mahalanobi derive mahalanobis rwm basically two responsible capture structure kernel many advantage rwm easily algorithm sequential optimization know kernel rwm sample machine mahalanobis supervise pattern sigmoid polynomial support weight application time specific task attempt modify matrix mean essentially identify hide generation describe embed sample article datum training mixture case mahalanobis distance similarity measure space classifier dissimilarity similar maximum call rwm similarity influence determine matrix term respective rbf rwm similarity define rwm svm build rwm black solid result illustrate rwm produce input sign green circle recognize sample gmm component gmm overlap information gmm model show sake euclidean rwm rwm similarity consider information gmm case label build vector rwm fig regard fig curve rwm gmm distance rely fig illustrate classifier accuracy test essentially minimization boundary regard use thus line sample rwm decision become nearly shape curve svm rwm rbf kernel rwm advantage supervise outperform kernel capture structure laplacian svm regard training implementation rwm kernel extension rwm parametrize rely strategy give overview relate define rwm propose rwm respective property datum set finding rwm particularly thus focus aspect set typically amount unlabeled instance sample output conjunction aim consider capture unlabele improve many algorithm implicitly unlabeled call claim major idea idea move close similarity membership purpose descent soft classifier svm differ step mahalanobis metric svm train solve quadratic boundary conclusion underlie try place decision boundary svm implementation much lot major mean classify unlabele available without classifier regularization empirically label shown yield supervise generative model view unsupervised cluster adapt known algorithm algorithm well consider information unlabele parametric mahalanobis rwm base rwm define investigate integrate explore extend input contain model start input training model sum rule densitie conditional motivated application eq
concept formal neighbor cross fold context form cross correctly object fold object fold run base classifier whole prediction keep classifier building formal concept formal minimal classifier count upper object process near select metric form yield concept mostly ignore select concept classifier rule base building neighbor fold validation array train neighbor kx x extent concept machine gb uci digits p cm rbf c logit knn bag decision sec sec sec sec sec sec sec sec sec sec cm rbf logit knn p bag adaboost iteration sec sec sec sec sec sec sec k classification classifier bagging svm outperform case turn bag dataset digit paper underlie classifier bagging stack turn base implementation bag multi class continue direction explore impact metric base attribute importance type classifier investigate condition preferable bagging improve thank high school economic influence study gray briefly multiple classifier system describe improve accuracy label correctly correctly classify assign object formal idea multiple classifier well machine appear name mixture expert fusion base early formulate decision vote add probability majority vote tend similarly great boost compare bag boost stack paper present type recommender base classifier classifier label correctly correctly classify organized follow bagging stacking provide toy synthetic describe well many one application learn impossible bootstrap multiple source replacement may overlap result usually accurate single classify prediction prediction aggregation bagging change result unstable classifier misclassifie boost combine boost start base relative instance evenly account also receive multiply meta chance less hard appear overfitte test rarely classifier formal concept construct one address context concept chapter concrete formal lattice concept serve rule frequent mining obtain formal let toy synthetic comprise test set attribute class train try attribute classifier run leave classifier far fill object correctly object train label let classifier refer classification context lattice formal context toy draw lattice diagram build lattice keep top top classify object find neighbor distance use ham near object set neighbor maximal intersection set classifier concept object cm p
period ensure slowly newly specify boundedness usual gaussian mutually assume eigenvalue decomposition j algorithm assumption define define eq ensures refer frame subspace slow projection direction frame change great rely actual along change particular great direction drop time also nonzero therefore necessarily explicit direction direction part say special whereas old mutually mean let background around mean autoregressive thus q q quantity slow change example every frame mutually set video foreground static move always reach scene frame go top bottom scene appear fig denote distinct value ni nk object since static one move mean move support disjoint condition move one direction become vector notation vector correctness give explain algorithm subspace new estimate get use hold corrupted bound video translate subspace change enough necessarily frame along direction value fast change tighter require choose many pair background fast need foreground moving imply diagonal proof let become nonzero allow r variance matrix knowledge explained enter foreground way enough direction subspace direction assumption assumption use triangle show matrix theorem basis corollary need nonzero entry outlier seem counter since magnitude whereas expect low nonzero try simpler simple explain really small affect low outli assume assume mutually everything else set conclusion hold proof follow exactly fashion treat extra follow three fact place denote condition far relate online completion mc require maximum eigenvalue entry parameter turn delay explain require frame follow retain vector initial short video mc another initialization technique adaptive mc model place slow eigenvalue gradually explain slow eigenvalue increase foreground discuss constrain long entry g r accurate early often video zero easily current appropriately none moreover exact recovery analyze fast need storage complexity need importantly highly entry easy allow ti pattern every support constant strong reason result j instead correctness generalization secondly correctness estimate subspace automatically know algorithm bound advantage develop lemma detection false correctness analyze algorithm knowledge prove approach recursive assume something use subspace piecewise sparsity weaker weak detailed simulation demonstrate thing available recent parallel recursive track underlie convergence optimize second explain subspace discount projection observe reconstruct via update recursively retain historical observed converge optimize initial knowledge subspace use idea introduce automatic work provide go thing subsection insight simplification subspace lie slow change onto orthogonal complement rewrite support recover nonzero ls e q argue small recovery far pca modification projection need p sum estimate see standard uncorrelated argue close condition become argue compare large perturbation subspace apply bind address issue considerably pca change form threshold change frame project change p phase pca reason pca see pca however even argue good perturbation small explain subspace estimate change correctly detect detect direction detect within short occur direction recovery exact present recovery eigenvalue occur interval interval detect show use input u whose eigenvalue recover method estimate rest remain formalize subspace detect occur within frame subspace recovery exponentially pca follow hoeffding lemma subspace bind hoeffding need use matrix j definition summarize definition pca j detect kt kb kb lemma bound appropriate conditioning see give dominant term expression thus recall define event j notation r sometimes j complement precise u j j kk k condition u previous eq whose eigenvalue check detect whose define later bound bound conditioning model use imply slow along bind decay definition theorem hold tt condition satisfie furthermore condition satisfie eq perform direction add decaying exponentially successfully false change imply frame definition respectively definition pca accurate correctly successful detect get bind notice imply accurate subspace chain assume prove lemma precede lemma proof theorem j lemma k j u j kx case proof fact j u j j j inequality fact event imply use u x x lemma p thing estimate number correct j k u j proceed observe combine u u appropriate fact conditioning lemma k define k j j rest remove etc everything similarly p j e first observe form independent lemma need condition corollary hoeffde hermitian random hermitian ii b hoeffding condition u apply use assumption recall r apply q probability x k combine lemma remark apply recall lemma proof give definition noting side satisfie let various eq cauchy schwarz find term apply term proceeding much tight need summation fact q apply apply corollary use use get expression pca prove corrupted entry time change reach bottom start vector start column schmidt last column subspace occur draw uniformly v show noise orthonormal optimization first perform prove support exponentially step program fail average simulation code matlab run processor black zero simulation plot section subsection result prove proof indice small integer respect process exceed new appearing old object consist index index start assume let index eventually walk object may come nothing general index contiguous moving move object long support minor scene reflect back start move direction hence bernoulli prove time ensure tail move move claim ensure static move least index notice object k third every frame figure behind first generality top u fact index number leave simple state motion start I I si exception set exactly correctness result modification algorithm remove estimate subspace newly subspace accurately prove one assumption cluster frame key weak able currently valid tool autoregressive allow obtained apply algebraic sum modification explain conditioning past value tool also analyze background approximately model apply modification video tool recovery noisy analysis due assume simple pca expect simple partial obtain correctness online accurate initial knowledge slow corrupt one quantify need give assumption matrix parameter mention section besides various problem correlate interval construct mutually subset I model assumption recall index word plus number l u mutually disjoint model u u u l line plus proof proof special case kb rhs increase assume theorem j thus proof difference use assumption use bound cs begin bound cs j follow j j j j u j jt definition follow assumption theorem k k sum vector schwarz schwarz line side edu paper partly nsf grant theorem example theorem pca mc define separate low true use important video try slowly analyze mc open develop modification online online mc main contribution obtain mild obtain short video surveillance change least every often pca tool frequently reduction compute small orthogonal call principal variability relatively accomplish via outlier pose pursuit batch solution assumption solve norm appropriately scalar program later et amount batch performance need recursive online assume outli separate video foreground background layer automatic surveillance streaming foreground move sparse image usually gradually move move forest subspace change dense valid initial background initial foreground recursive structured impose change part vector entirely complete column track foreground object intensity know significantly different support simple nuclear norm subject amount mc problem miss entry possibility infinity quantify set problem complete surveillance application short background miss mc include convergence optimize second old recent another result differently neither estimate pca
algorithm build ensemble repeat time classifier result rf pool combining approach vote vote class final weighted vote exist classifier training subset divide vote datum use test frequent c train benchmark multiply weight rf precision label failure decide weight rf rf sum high score datum result class failure ensemble classification compute apply ghz intel processor gb take benchmark control continuous equation roc threshold classified failure decrease true grow positive different pr display sensitivity curve great correspond average precision well imbalance figure obtain benchmark appear first two benchmark due aggregated day incomplete begin approach display roc pr curves benchmark individual roc categorical predicting failure failure actually identify appear expense precision plot clear dependence value classifier become less failure imbalance plot clearly critical obtain good roc pr describe ensemble line roc pr maximize two figure display failure failure identify label failure failure misclassification depend position latter situation failure appear time machine implication relation classifier study assign point hour would whether different impact vary failure display next decrease misclassifie misclassification failure actually negative close verify assign incorrect label class irrespective real next failure next true positive tp false tn fp benchmark look misclassified positive failure classified failure misclassifie negative classify negative failure panel divide tp failure correctly failure classifier distribution plot many positive misclassifie failure moment suggest recognize failure classification failure moment approach fact hour large especially vs benchmark event hour divide failure next machine fraction entire day end still failure fp failure tn negative failure fp compare tn many give failure hour publication google trace include goal characterization statistic node identify high level heterogeneity system especially compare grid user profile usage shape characterize trace usage various management example trace perform model study use google trace far validation early simulator parameter simulate job status load prediction cpu ram history load future divide load level failure prediction year application file email etc monitoring error report introduce fall element tracking error reporting failure job event concentrate scale failure track resource failure failure distribution inter job cloud naive failure job name trace amazon ec run scientific application reach performance setting point total point similar benchmark never job failure reliability availability svms network nearest algorithm outperform reach precision analysis perform neighbor forest anomaly cloud select relevant principal correlate component identify outside range study cpu disk relate relate control house high production trace failure trace failure failure mention concentrate result study result possibility controller center model happen various technical require change build collect streaming compute aggregated feature correlation window previous window store dedicate basic negligible event aggregated parallelization time average window second newly datum second window time stage time feature feature speedup cost storage cost currently mb original raw gb day translate gb day day time last day need store analysis day require gb day new use google engine train eliminate need ensemble entire ensemble take since independent minute provide study classifier take negligible expect take minute parallelization negligible amount cloud scenario however storage resource require monitoring cluster trace extraction cloud platform ensemble day tested overlap day length trace repeat produce benchmark platform useful obtaining feature sometimes process month worth daily cost varied area precision curve false word identify failure classified look failure failure achieve datum extract ensemble training parallelization also explore machine obtain machine failure present suitable day scenario simulate benchmark would train overlapping day benchmark day failure day training account time ensure test live reason voting secondly predict cloud google grateful google regard trace center ever reach ultimately computational intervention limit setting goal rather management control build update path operator center study public collected build predictive failure google amount characterize many classifier hour false component live analysis publicly website google trace failure modern center internet cloud services device power utilize million user service many aspect daily availability center availability manner user current automated management tool limit allocation monitoring capability operator stream display optimistic per situation center ever compute technique center complex characteristic building control loop compute try system state undesirable capability undesirable action place center vast corresponding course numerous internal external include power management server removal network modification modify electrical change physical server storage device benefit modern new generation capture centre political environment towards failure study google event management month period employ google platform massive exploratory suitable allow contain reasonable combine ensemble rf unbalanced mainly occur rf initial limited tree rf combination day result day false day comparable failure study field contribution modern center adopt generation drive towards extensively develop potential drive secondly tailor subsampling bagging weight provide quantitative evaluation run website public describe section issue drive controller argue conclude publish contain several table monitor status machine task day gb evolution task usage gb interval feature change third correlation window start cpu memory disk table consume require aggregation step amount process gb hour window per feature hour tb similar platform google trace report machine cause failure machine software update event cause distinguish discussion google investigate way distinction suggest interest failure perform software ensure event failure failure use relatively threshold hour require typical base threshold total event failure predictive sure precede hour window completely remove completely fact feature add two feature entire last gb data negative positive certain less assign failure
calculate updating occurrence initialization incoming event I n joint n calculate adapt operation operation cluster merge sum merge cluster eq gram track symbol must substitute count new gram include boltzmann pattern formally boltzmann consist binary update rule connect represent initially boltzmann implementation consist consecutive symbol binary node initially nodes symbol softmax rule temperature converge boltzmann machine boltzmann zero symbol connect state reach rule rate yield aim threshold concatenation symbol strong cf iteratively unit create represent length length variant machine symbol boltzmann retrieve implement call act operate model machine change removal merging update adjust atom newly pattern node represent annotated classifier predict analysis annotate annotated evaluating agreement annotation suggest pearson chi square rand index evaluation since natural set partition generate annotation derive x xx pair annotated rand since rand baseline partition gets maximal ari partition ari annotation clustering draw number ari establish alternative one entropy measure system ari evaluation feature entire symbol segmentation input sound wave transform symbol occurrence include contain index equal symbol partition ground segment yield annotate generate symbol cluster table repeat pattern length g annotation audio low duration unsupervised category automatically annotate record simple slow country complex audio annotate audio website consist entire chain symbol expectation audio processing system give repetition length reach cb noise robustness cb receive determine next symbol symbol annotate symbol generate explain annotation bm ari constant assess scale average ari partition length gram reach perfect ari event repetition cb seem converge reach ari high event increase slowly involve window detect frame vector range grid tb ari length ari ccccc table ari mean audio event window test annotate result expectation evaluate module predict tb cb cb cb work lot cb result sequel assess evaluate extraction entire symbol stage use symbol annotation use evaluation table detect ari conduct tb cluster versus analysis window length column cluster window measure c ccccc task event symbol annotate tolerance ms configuration yield ari ari tb row audio website prediction performance adopt optimally annotate symbol find calculate annotation iteratively entry thereby establish connection row annotation column entry maximal entry htb horizontal map event indicate black line match due tb ta ta fig fig annotation cluster map ta pattern capture annotate match wrong initial line red square symbol symbol pattern occur annotate match regular process prediction middle event event wrong pattern gram perform correct prediction tree sound website sequence gradually amount versa mixed yield sound begin website sound event analyze sound third sound gradually system predict next sound event operate audio start perform sound event pattern currently limited sensitive context prediction alignment combine incremental cb limitation inspired imagine human machine may novel idea machine finally stop machine suggestion currently associate speech research university unite european ph music study engineering institute des sciences france school write scientific several international project van research speech music analysis ph institute des study mathematic pure mathematic brain interface paris stanford scientific paper grant music play music create present segment cluster predict audio unsupervised adjust dynamically flow segmentation detection discretization incremental g drive sound event symbol extraction symbol sequence gram newly introduce conceptual boltzmann machine sound respect rand ari entire ari music retrieval adaptive music novel music music automatic either symbolic audio base pre train classifier cope new label every new appear severe flexibility system new human mind novel unsupervised paradigm concept base e tree input deal decrease sound similar use unsupervised n gram couple split symbol count system prototype learn begin reasonable predict symbolic predict distance pattern cluster root build operate audio segment stream initial unsupervised clustering build maintain set sound result sequence gram phase adaptive learn music sequence document online cover gram previously method conceptual advanced overview system component segmentation introduce rand condition module give audio example support website fig four extraction incremental giving sequence incremental drive symbol discretization cognitive node distance align font execute em execute center inner draw corner segmentation xshift input block feature block right feature incremental xshift em anchor north yshift anchor north yshift anchor yshift line bend anchor shrink bend anchor south sequence explain generally applicable employ domain difference soft induce transform complex th build distance actual bin complex spectrum amplitude phase two precede map bin quantify stationarity summing bin consecutive frame yield detection median ahead window length control order eliminate occurrence smoothing window define length subsequent analyze coefficient cosine frame dct represent sound clustering receive symbol symbol important state system cluster arrival symbolic create continuously build hierarchical object assign create object leave node represent model heuristic use utility version extend cluster previous contain deviation feature vector cluster specificity dimension utility quantify specificity instance dimension standard deviation dimension specificity ii specificity utility upper maximal minimal control discrimination incorporation process count operator create
alm around mac around simulated mean general mac evaluate multiple effect construct square alm example alm mac exhibit behaviour mac tend well alm example apply alm mac reservoir case appear alm exhibit due nonlinearity reservoir chen total anonymous valuable help comment suggestion acknowledge integrate realistic thank matlab toolbox mac maximization iterative algorithm iteration construct prior solve maximization update update pass knowledge correspond function mac th e mac assimilation update k k unknown deterministic determine subsequent criterion determine functional may choose view conditional view enkf reference therein optimal dirac assume addition q denote iteration compositional define bayes functional construct q equivalently suggest pdf also note impose compositional functional term subject condition system observation error make rule nonlinearity non construction beyond current mac operation histogram alm mac repetition visualization horizontal axis figure calculate difference initial day alm mac right alm box alm mac model mac alm alm converge visualization vertical axis logarithmic step iteration along horizontal box horizontal median bottom range determine individually plus panel ensemble alm mac bottom alm mac difference dot indicate model member column alm nd column mac rd estimation average score ensemble alm c mac plot different step alm mac production top bottom estimation ensemble column ensemble alm nd column mac rd iteration red dot measurement forecast ensemble column matching rd comparison production history water production plot estimation leave mac bottom panel alm mac box difference step layer reservoir ensemble reservoir alm reservoir dot indicate location field alm leave field use production production rate top middle p column ensemble alm nd mac iteration red dot datum forecast respect ensemble st matching profile nd column water plot difference iteration step field ensemble alm middle mac field production year cross validate history reservoir alm production year validation production p case initial alm nd column mac rd column final step dot historical forecast st ensembles nd rd vertical figures nd rd separate decade figure water example eps mm international institute alternative iterative smooth formulae derive adopt approximately solve minimum mac alternative understanding analyze behaviour aforementione insight illustration compare mac system reservoir even assimilation way assimilation algorithm ensemble kalman enkf sequentially assimilation collect observation simultaneously ensemble smooth example variant enkf widely reservoir assimilation history matching problem engineering application es reservoir assimilation investigate recently es include enkf reservoir benefit avoid reduction circumstance latter computer es certain method formulae follow formulae involved adjoint enkf error call es lm discard lm alm es mac apply solve cost involve obtain kalman enkf behaviour aforementione thus iterative es example implementation base illustrate performance alm es alm reservoir reservoir simulator certain assume mean covariance ensemble suppose j en ei iteration formula focus define follow root product es make enkf error appendix end formula I I I total number iteration perturbation enkf satisfy call hereafter covariance simulate randomized likelihood lm sense ensemble update iteration I positive derive line accordance expensive j formula algebra suitable iteration consistent technical enkf approximations final compare eqs formulae distinction call alm alm start average reduce value stop consecutive regularize regularize assimilation similarity es alm implication also ensemble study approximate gradient minimization concrete prove solution cost square e use enkf also alternative interpret reader deterministic reservoir match square minimize weight convenience discussion model ill pose undesirable e uniqueness large might solution become weight assign datum term solution tend extreme approach tend coincide apart relative choose covariance correlation model space choose comparable instance threshold reader choice straightforward often iterative algorithm construct linearize I I positive scalar convenient want far simplify square eq simplify weight later intuitively provide control taylor valid I previous gradually presence bind prevent use analytically sequence locally noise level adopt technical prevent fitting wish let stop comparable often course iteration stop iterate pre scalar discussion b change consecutive iteration implementation number step reach early alm control alm experiment extension issue take account firstly algorithm typically assume stay sufficiently reality addition discussion order derivation exact must certain efficiency put practice likely iteration lead high iterate order increase extent execute iteration essentially track optimization theory study ensemble smoothing alm examine study iteration square namely distinction call mac mac point cost jacobian matrix may member jacobian algebra see formula j c one I j I formulae alm iteration mac general differ construct possible member close ensemble work coincide result mac mainly differ alm different centering circumstance model mac alm identical case distinction may alm mac thus alm stress although superiority one broad theorem intuitively alm mac optima superiority case make use around order theoretically sound computationally expensive around neighbourhood strategy ensemble around mac incorporate term onto ensemble beyond kalman formula aspect carry numerical discuss similarity es alm focus behaviour root matrix step decide iteration start parameter constraint alm mac total number alm mac mac explanation mac cost mac member taylor alm degree nonlinearity nonlinearity iteration ensemble member alm accordance member initial condition alm panel member ensemble spread ensemble member alm slight alm mac low alm spread alm monotonically decrease iteration mac reach minimum around final ensemble may certain local distance truth small apart sampling member contribute depict ensemble member please box indicate vector I whose observation ensemble compute mac ensemble alm difference step initial figure nature tend mac toward decrease result tend reservoir simple reservoir water reservoir vary spatially md channel md background water location mark rate day bottom bar matching water simulation day plus certain noise day production bar enhance channel eight statistical measurement include sample define neighboring see area maximal body extension body fraction volume total evaluate ensemble assimilation measurement variance statistic also run alm mac include assimilation close tend follow level need enkf sign convert matlab toolbox version ensemble alm mac addition parameter adopt maximum coefficient alm mac third f respectively see despite update alm mac final alm panel mac matching reference score final mean correct figure seem alm mac channel match certain seem alm alm mac alm stop early relative mac stop iteration step despite alm mac convergence speed final alm deviation std mac final alm follow performance water production rate ensemble alm mac compare obtain mac improve alm well mac history initial production water rate reference highlight ensemble alm mac water production tend visible alm seem final mac plot iteration alm mac one ensemble situation box observe rmse value enter tend alm alm decrease slight mac alm comparison figure normalize I step ensemble mac normalize difference flow non tend difference history match mac field dataset contain year production decade decade production decade history production production decade history match matching control production historical production water cut historical pressure production net ratio consequently alm use model provide mac drop member alm mac reservoir indicate layer realization reservoir one mac although final alm mac see seem result difference way function hence subsequent previously box plot
generally speak use system smc maximum learning recently develop importance smc strongly choice proposal match size monte variance resample mcmc complementary research distributional inference particle filter particle construction limit propose box automatically flexible proposal assess kullback smc leverage literature outperform proposal include community performance translate complicated handle parametrize challenge filtering smc approximate model briefly smc importance sis resample sir comprise space hmms markovian notation sampler simple distribution distribution convenient take supplementary normalise weight recursion sis elegant compute pass I suffer severe distribution become many sequential sir multinomial particle weight replace particle importance sir require trajectory importance particle need let represent index particle collect employ resample previous smc supplementary proposal critical employ produce trajectory quickly posterior proposal care optimal choice proposal time posterior intractable filter fail incorporate current house employ distributional extended kalman filter inaccurate poorly behave outside apply neither criterion introduce new adapt distribution limitation proposal parameter explicit adapt objective four main sample global lie advantageous exclusive fourth derivative approximated negative take final step smc unbiased intermediate filtering bring advantage derivative time trivial update update proposal proposal full perform maximum update usage similar proposal j p contrast employ analytic distributional loop global particle special previously rich proposal go work base smc literature optimisation available option train however example proposal proposal generate smc place filter variant use apply briefly flexibility neural network generally technique unsupervised setting wider hide multi modal due uncertainty sign latent experiment interested modal diagonal three md recurrent help recurrent improve spirit variational proposal proposal dynamic state adaptation proposal generative supplementary sequence bootstrap nn rnn implementations guide implementation acceleration significantly bootstrap term ess rmse estimate simple md indicate multi modal proposal outperform recurrent transition dynamic rnn interestingly cut converge correct box plot estimate ess ht rmse std mean std rnn rnn f md rnn md ess marginal rmse system consider drive white ode consider infer cart orientation noisy location supplementary material significantly directly admit md model successfully proposal ess mean angle md learn higher ess infer often loop prominent example particle chain latent context proposal metropolis mh accept form e walk smc sample use smc full detail model section follow random model md md setting number particle small enable significantly reason quickly model adapt costly adapt particle slow burn mix propose dataset different hour simultaneous lstm layer output fed dynamics supplementary lstm generative optimizer bootstrap tune standard comparison approximate smc particle bootstrap three marginally well state state approximate marginalization smc c c rnn bootstrap similarity free employ refine exclusive variational integral approximate posterior spirit way except entail approximation proposal accurate therefore proposal lead advantage energy severe kl avoid compute entropy prove problematic approximate tailed variational employ refine context extend adapt proposal smc long contextual descent outperform
evaluate carlo estimate trace draw equal component carlo trace satisfy q multiplication appeal expensive absolute interval knowledge loose low first section section algorithm definite subroutine estimating give matrix initialize chebyshev eigenvalue g remark multiplication matrix input theorem ready present determinant generalize singular interval initialize tc equality singular easy problem obtain counting span determinant general power small compute require multiplication matrix bind input value quite straightforward fact become condition number chebyshev require degree I sign multiplicative determinant zero state corollary input singular follow eq give limitation count span undirected graph span one classical counting problem application model denote degree let laplacian spanning correspond give follow algorithm q supplementary material theorem choose error chebyshev intuitively approximate chebyshev polynomial bind minor length chebyshev q chebyshev notational convenience plane analytic hence rate z hence observation version eigenvalue eq estimator give sampling know interval proof ready definite imply combine theorem follow experiment first generate random row five uniformly distribute matrix symmetric position entry sum add run scale roughly determinant error comparison relative c complement taylor mean determinant run compute cholesky decomposition complement use inverse million author algorithms accuracy taylor expansion fair number report chebyshev expansion superior accuracy propose field graph capture property extensively application computer spatial definite sparse graph grid million matrix node four generate use sample j likelihood report likelihood provide track spatial million log presence let j z solver linear thin two interpolation fit value tool g solver eigenvalue computation matrix decomposition important computation admit often infeasible set linear logarithm exact computation cubic furthermore easy multiplication numerous thank chen chebyshev complement hold corollary c mn provide vertex vertex tree complete proof claim positive machine determinant involve cholesky cubic number prohibitive approximate matrix call chebyshev efficient multiplication multiplicative error propose cholesky compute million variable scalability learn extremely model increasingly attention prominent optimization randomize one variety machine compute determinant precision also variety machine include variational addition learn form adapt involve bregman become determinant probabilistic recommend determinant cholesky cholesky cubic thousand aim accurate definite sparse million literature compute diagonal band filter count approximation score stochastic gaussian taylor expansion
importance weight active learn unbiased hypothesis particularly go terminology relevant explanation exposition equip dual space dual shall entropy proceed stream one step calculate probability notice point step calculus component exponential proper loss formulae estimate instance estimate bx tp bx labeling initialize tp bx ty technical receive evolution risk aggregate different collection excess eq jensen subgradient build put inequality lemma stream eq approximate bx mention approximate get choose allow flip report coin turning good hull combine majority decide query order boost bag assumption assume hypothesis hull version base possible combination datum weak learn boost admit somewhat algorithm build fashion run next generate random look margin label point budget decision along weak characterize dimension threshold expense far label dataset query table report datum stream report sample see stream note large convex aggregation see stream suffer loss large seem imply comparable unlabeled believe excess c c dataset budget rate mnist comparison query four scale expect stream mnist scale sublinear stream query clear eq excess upper scale discussion know expect large demonstrate trade make small additional unlabeled round choose fair budget report rate budget inferior possibly budget experiment stochastic mirror use unbiased excess aggregation experimentally direction tradeoff error query help tune excess guarantee establish need key state result normalize mirror simplex dual differentiable strongly dual pair definition equation q last old increase sequence sum sum definition mirror fact replace put together equation get assumption conjecture ex learn aggregation good aggregation make want query see mirror descent risk aggregate return risk demonstrate uci passive query accuracy passive machine physics biology web finance availability great machine discover problem supervise require unseen label domain easy unlabele hard speech recognition require supervision passive access label domain sample distribution sort empirical return whose collection much rich model ensemble view perform aggregation gradient via sequential boost base model learner aggregate boost capability final aggregation outline aggregation hull whose excess risk hull small remainder assume access underlie define give label unlabele query label study convex good aggregation access stream sample I introduce mirror shall show one mirror iterate aggregate excess risk aggregated number essentially mirror descent stochastic simplex simplex regularizer mirror follow step excess bound consider essentially pass mirror base algorithm stochastic bx stochastic mirror construct unbiased iterate length stream return excess risk convex decay algorithmic mild dependence desirable mirror descent introduce algorithm
version appendix power wikipedia collapse initialization dataset hybrid hybrid min wikipedia dataset topic implement lda tensor whiten fast baseline collapse gibbs wikipedia detail word reach set hold dataset document pick testing mix solve dd report hold spectral different hash length accurate table lda collapse gibbs iterations spectral lda different length collapse comparable hold run collapse initialization collapse topic already perform sampler x also report collapse build page hold build randomly pick topic get topic obtain table collapse initialize spectral lda much shorter category corpus usual wikipedia dataset email v norm wrong min specifically explain symmetric upper triangular note u u u u u subsequently exactly nonzero inner product accelerate alternate square experimental toolbox widely consider alternate least popular tensor decomposition maintain iteratively low rank good nevertheless index hash km bt ki v c ij similarly eigenvalue show cp different roll asymmetric kk ambient tensor bottleneck one c hold computed operation list wrong min exact compare various synthetic use matlab toolbox fair plain accelerate algorithm sketch length powerful review lda decomposition lda propose detail propose fast list ik parameter lda dictionary particular topic topic sample parameterized lda rd robust third know quantity algebra q extract vector procedure specifically find whiten orthonormal complete tensor decomposition perform moment v w w word moment co exact moment accelerate mention section help lda computation whiten bottleneck bottleneck come sketch tensor decomposition computation w pose challenge w sketch technique remainder efficient sketch mention efficiently dd w j km number word per document decompose slow application decompose efficiently sketch efficiently word decompose whiten sketch share sum document number speed sketch operation efficient computation w decompose consequently trick compute definition true variance I consequently chebyshev obtain prove I notational simplicity omit consider permutation everything demonstrate preserve product lemma state inner variance hash right prove hash tensor hold assumption real tuple pp b l expression r case definition l l wise l r r automatically generality tuple l l concatenation finally position combine get immediately add everything term equation easily refined section demonstrate noisy overall part v initialization randomly sphere eigenvector lemma approach exact ground noisy come tensor present noisy separate magnitude noise eq separate u u u tt examine numerator use assumption I v h yield eq prove term first vector since u v u subsequently noisy satisfy chain triangle eq principal matrix u level imply lemma plugging section upper principal orthonormal follow v ib bound upper manner everything section present appropriate hash detailed eigenvalue robust tensor tensor randomness independent follow condition I satisfy check tensor inequality addition hash assume induction lemma lemma hash yield index tensor hadamard conjugate n inner product matrix case tensor rao product n b reference theorem lemma cp decomposition wide learn randomized decomposition introduce tensor novel tensor explicitly tensor encounter alternate also design save combine idea whiten tensor power iterative technique fast quality method uniformity cp decomposition sketch domain modal relational important tensor decomposition cp decompose numerous variable modeling estimate mild importance interest world tensor fall tensor sketch tensor power tensor update decomposition dense tensor framework modern training attractive guarantee propose construction tensor input tensor factor empirical tensor sketch operation operation sketch force computation sample take express involve multilinear combination power alternate whiten directly compute sketch tensor tensor force th contraction addition directly empirical learn huge tensor contraction sketch symmetric tensor application hash avoid operation though hash computation depend property hand previous magnitude practice dense tensor processing propose randomized degradation accuracy competitive result tensor model implementation small collapse time gibbs within sampler much iteration propose efficient optima accelerate burn numerous work implement execute one implementation procedure whiten result significant high many bad moment decompose alternative carry decomposition batch extremely suffer paper handle batch therefore variance randomize tensor expensive sensitive particular uniform handle adopt require sketch pass inner take tuple simply nk I I em c n propose batch reduce noisy tm stand v sketch tensor eigenvector present bottleneck robust power u speed sketch approximately u nb hash go detail key idea behind tensor computation tensor inner approximated tensor whose build consequently approximately compute operation sketch u completely satisfactory tensor us I u f prove inner eliminate right side contain entry detail defer space run time exclude significantly improve method method analysis order plain method ccccc tensor factor tensor contraction tensor design style sketch build idea hash symmetric entry etc build hash random permutation address issue rademacher complex domain rademacher divide entry symmetric otherwise characterize
bind focus solely risk highlight whereas highlight exceed always define weight make equivalent form relate q margin notion place majority chebyshev reach correlate sake focus ignore distribution moment second moment second majority vote random first follow straightforward connection margin gibbs disagreement majority vote line proposition show condition consequence prove proposition twice risk vote small risk show average always risk pairwise concept mm note random qx inequality consequence moreover whenever large independent even large arbitrarily close increase proposition consider distribution eq apply motivate relate risk clearly individual predict majority vote great criterion boost algorithm quantity appear moment qr learner uci round study majority per dataset see however figure generally strong almost suited characterize quantity contain insufficient black r l heart ab letter letter test validation vs round sign well criterion every sign nan criterion evaluate tool adaboost dataset however split example contain remain example adaboost empirical majority classifier low bayes risk stop bayes round select early perform validation number round validation finally vote low bayes adaboost pay set compare risk majority well criterion test binomial suggest task validation perform suffer cross conclude surprisingly criterion boost pac theory estimate base recall classical pac bayesian majority vote classifier third pac contain act bayesian bayesian chosen observe pac theorem gibbs classifier rely quantity kullback leibl distribution note obtain present general pac convert bind risk majority proposition define pair loss allow pac theory pac entropy measurable equation jensen concave equality share present bayesian loss pac theorem specialize expect risk similar pac loss kullback point different relative weighting especially situation good note negative logarithm measure leave side f f easy many variant pac among kullback leibler bernoulli probability shorthand notation order loss let binomial fm fm f however case verify pac still apply obtain next risk derive theorem independent swap present classical pac pac majority vote find interpret corollary indeed gibbs corollary well pac replace corollary slight bind gibbs obtain inequality omit great classifier kullback leibler corollary gibbs risk eq multiply factor methodology bind trivially valid otherwise consequence close follow paragraph compute turn since hand convex constant show application pac package explanation load graphic explanation terminal need macro ltb lt lt lt lt lt lt ltb lt lt lt lt ltb observe intersection limit pac pac bayesian section expect disagreement definition binary disagreement define closely relate notion two usual equation define x q h either correctly rewrite margin risk rewrite bind prove define kind pair tuple pair loss pair pair new loss define recover directly equation explain classical pac theorems corollary multiply present rely third risk another disagreement corollary pair new bound majority supervise another semi q joint vote see definition q mm equation f ij pf pf mm mm finally e hence give pac bind risk vote distribution q always strictly two bind bound bound therefore suitably apply corollary replace major tight however achieve label affect huge tight tighter supervise large unlabele corollary obtain label follow pac obtain solve mm pac unlabele last section require approximation another extension pac enable new instead pac valid result tight bound simultaneously loss pair inspire real value q ij f ij notation ij f ij appendix give exploit increase definition p convexity ij p q q ij ij ij ij proof straightforwardly subsection notable former distribution variable kullback shorthand notation corollary apply inspired contrast base list distribution dm multinomial three outcome ij outcome totally notation mm ij ij ij convex ij ij ij f f ij k ij f last prove pair directly q swap ij kl ij since new tight notation indeed reduce achievable property bind correspond consider pair later valid idea equation pac supremum assume empty numerator denominator positive supremum mm attain subset pair closure constraint supremum closure imply trivially supremum attain case valid mm supremum equation probability valid q q slightly pac bound via remove define replace obtain theorem pac bind concave equation decompose nested maximization optimization twice risk need maximize kl black correspond marker vertical upper star marker tight bit pac replace optimize bound show bind present far obtain use adaboost uci contain dataset split training testing boost gibbs majority variant pac conjunction terminal option use load package graphic terminal need graphic macro ltb lt lt lt lt lt ltb lt lt lt lt bp bound ltb ltb ltb ltb ltb train ltb round boost tight pac obtain substantial improvement almost pac see pac bind testing round boost continue decrease drawback due fact denominator margin form moment slack unfortunately majority vote pac structural leibler distribution priori theorems surprising attempt minimize surprisingly kl poor regularizer attempt localize dependent prior inequality term provide notion contain kl term posterior regularization action inspire pac divergence hyperparameter restrict surprisingly latter use good assume possibly infinite set introduce suitable name introduce uniform prior align say finite set uniform quasi pair equal low upper rise kl necessarily small pac quasi theorem corollaries pac bound getting achieve loss gibb align need change term one fact result expectation f posterior distribution pm case distinction pac theorems section step linear posterior note version result theorem classical align posterior pm equation proof rely pac bayesian straightforwardly proof reason align term follow deal pair instead loss result case theorems shorthand define recall pf lemma pair distribution align j cf change expectation ij qp change appendix f f ij pac pair give df require property result obtain rest derive use inequality inequality finally therefore posterior give rise pac disagreement pac mm q corollaries direct application theory set overcome difficulty theorem section classifier depend problematic view suppose surrogate linear classifier induce turn similar support equation curse restrict posterior isotropic classifier base consist pac minimization sample pac theory directly deal involve notion conversely approach measure deal access refer information I I ks imply give sequence message compression priori observe either compression strictly compressed since message simplify framework message string bit sequence sc weight output sc really general depend compression therefore sequence gibbs risk mm pac vote sc follow issue draw instance belong sequence bias replace factor compression version average compression compression simplification allow character key reconstruction output independence simplify set describe complicated rewrite sc compression sample replace mm kl theorem present pac compression reconstruction output sc result calculation present precede generalize exception minimize bound compression generalize self output sc note proof reconstruction disagreement function pair obtain align let reconstruction output ij calculation p eq majority compression reconstruction self reconstruction rise sc always output vote margin vote gibbs risk disagreement express follow vote proof identical pac compress pac theorems direct pac express bound disagreement form rely justify vote design minimize express quadratic program case vote vote majority risk situation margin remain maximally uncorrelated tend use quasi see moment margin force become justify pac dedicated quasi posterior dedicate posterior always lead uniform minimize pac pac justify vote shall restrict margin quasi uniform vote value minimize reduce margin control random forest idea margin justification type directly consider recall form moment majority vote first attempt minimize without give instability estimate ss way restrict section upper bayes hence accord bayesian effect prevent compression necessary kernel next restriction vote let self quasi uniform distribution give vote nf f q ix nm nm nm definition point rise vote pac together restrict uniform overfitte consequence posterior margin uniform exist majority vote let quasi eq q quasi quasi distribution vc know capacity vote capacity relate degradation produce instability explain instability uniform interestingly equivalent margin constraint equivalent pac always represent margin call self quasi distribution empirical finding translate program qp remain turn qp self column qp represent definition quasi minimize rewrite qp qp weight recover sure solution qp uniformity property n stay give program attribute calculate normalize rbf svm empirically even significant among experiment implementation scale black test vs context interest learning algorithm handwritten digits split task intersection set example avoid binary result binary dataset table show handwritten context binomial context outperform uci help answer well dataset remain risk algorithm heart letter ab letter segment test l vs l binomial explanation statistical classical perform represent product task amazon user language review convert dimension consider least chen show r name book comparison test l explanation context test sign draw majority vote maximally uncorrelated sound adaboost classical binary sentiment highly gain handwritten digits context observe pac pac majority vote moment fix moment validation close basically severe degradation pac former small value value individual decision implementation maximum depth high datum pac corresponding pac multiple uci dataset construct remain example testing figure tight risk majority vote value tight nevertheless pac guide rely validation expect pac bind pac justify hyperparameter majority vote moment side chebyshev mild moment empirically vote theorem estimate risk pac allow function pac property kullback leibler pac together give end nice solid perform compare svm regard ever nevertheless machine supervise datum enough sophisticated framework several adaptation structure co armed bandit reinforcement national engineering discovery grant perform grid universit e innovation du comment vote anonymous jensen side markov obtain note straightforwardly kullback kullback leibler bernoulli kullback distribution straightforward expectation vector independent consider value probability success mm step one generalize lemma countable possible generalization give martingale sequence expectation value correspond nh h h give independent give take integer n combination extreme prove induction induction hypothesis denote vector element last term couple eq hessian semi eq section inequality pac bayesian generalization require provide countable moreover pf let tuple vote x probability binary number next pac bayesian self align note change expectation align obtain obtain expectation jensen inequality f f self f side lemma jensen f f therefore theorem convention universit mm analysis vote binary particular introduce risk vote average extensive observation training contain learning end allow sample analysis basically minimize reduce quadratic aside theoretically achieve vector vote pac theory mathematical approach empirical experiment bound algorithm art majority vote firstly bag known majority
make tend dimension fix use stability dynamical system chapter section number converge interior need close approach capture eq et adapt px px px result step expand around py follow part apply px px except exponentially small function critical mode density risk exclude boundary spherical cluster mode show despite variation distance perform high dimension difference component randomly replication achieve component separation far draw weight mixture unit measure mean replication vision much nonparametric mode dimension develop mode risk density region even several bandwidth regard believe possible low fail part boundary merge separate cluster chen mode nonparametric mode risk cluster cluster risk even beyond core idea assign mode find risk cluster cluster core portion moreover hold cluster core condition literature worth expand dimension might get estimate function type hierarchical superior rather simply find mode clustering cover case outline use estimator outside core noise consider risk mode introduce related density region several cluster tree density local eigenvalue denote usual eigenvalue denote ball give brief detail need terminology good theory point regular non hessian critical point maximum maxima minima starting satisfy ascent manifold manifold mode figure useful exclude point disjoint flow critical flow density perturb particular result lemma finitely derivative vanish follow function cluster core boundary core compact finitely critical another define cluster mode mode point cx x projection exact c let gd c mx finitely remark lemma exponentially apply theorem exponentially small beyond core theorem core fail risk separate assumption spirit noise assumption use high classification mode part capture cluster say boundary tail multivariate normal tail continuous derivative risk
thank triangle side know literature positive minimax therefore proposition whose section suitable regularity negligible principal affect rate consider infinity uniformly presence distinct label advantage factorization regard probability measure mode associate intensity intensity assign proximity view sake notation generalize probability one concern consider real asymptotic condition pc volume thank conditional pc combine asymptotic behaviour dx word see argument versa concern fact term unimodal particular unimodal sense unimodal assume large mixture illustrate bivariate gaussian joint factorize specified belong approach mixture view algorithm maxima form estimate pc part attention notice theorem fine different providing attention tuning enough avoid curse care latter choose fraction large criterion criterion decay provide curse dimensionality view approach nonparametric estimation full parametric see gauss moreover finite procedure strategy g graphic library contour criterion fulfil might sufficiently fast applicability theorems decay greater interested reader slightly modify adapt cluster procedure one linkage procedure estimation exploit diagram implicitly suffer bandwidth selection pc bandwidth identify axis parallel estimate depend smooth bandwidth identify axis orthogonal oriented pc fast decay well unless provide estimation choice bandwidth phenomenon nonparametric remark concern pc study difficulty neighbourhood spurious mode explain help detect look great grid within fix recognize alternatively multivariate namely pc connect thought display cluster procedure classify algorithm sample curve label nearest mode curve equivalently mode pc shift shift loo ray lin yu reference therein pc label curve illustrate berkeley growth phenomenon bring kind dataset year curve smoothing fit discretize detail ram curve curve instance regression classification cluster aim exercise retrieve gender subject posteriori assess performance remove phase curve step recommend merge amplitude variation removal performance agree second principal eigenvalue estimate concentrate pc explain total remark limit may decay provide justify eigenvalue type remark conclude analysis first useful besides eigenfunction mean minus suitable multiple take weight display j appear monotonic describe fan effect integral second eigenfunction appear eigenfunction mean curve perturb subtract illustrate main matrix comment behaviour section smoothing structure apply automatically group connect perfectly group group mainly cluster correct rate retrieve subject apply micro sub assign cluster illustrate factorial prototype automatic multiply modal modal figure relatively segmentation detail recognize whereas separate reduce composition variable correct recognize outperform competitive algorithm curve analysis appear homogeneous suggest reduce repeat pt retrieve gender subject algorithm homogeneous interpretable group interested whenever tend zero cumulative process span boundedness derivative depend fx linear center latter fix first eq fix concern asymptotic proposition proof proposition converge sure monotone convergence order eventually away algebra thank thesis notation drop hilbert since clear argument follow denote denote assumption norm thank iii np finally behaviour similar hold member get thank choose choose computation allow negligible author grateful hold university thank discussion visit towards grateful density cluster la le di proposition em hilbert value rigorously exploit fact asymptotically proportional principal pc depend pc core define parametric joint pc application hilbert exploratory tool technique whose reveal structural difference collection sense heuristic multivariate context orient primitive back wishart region connect joint along explore author cluster easy depend difference contrast absolutely drawback connect reference therein worth local density identify region look maxima family research reference apply go framework curve surface contribution multivariate always due see survey worth propose inspire seek propose new inspired illustrate density orient dominant oriented issue surrogate derive briefly see reference tend intensity theoretical tail deviation focus asymptotic estimation non evaluate convergence see well effort widely discuss reference framework study instance break choice center behave worth candidate play density finite consequently knowledge characterization direction factorization hilbert process determine decomposition component particular besides concern operator smooth show term lead dimensionality univariate turn quite restrictive intensity independence point principal independence independent implementation drawback independently drop independence expansion simply optimal way coefficient explain candidate orient motivating implement obviously density multivariate procedure involve instead one convergence obtain constitute intensity assign proximity real apply well functional connect go introduce concern asymptotic theorem application collect clarity four part asymptotic tend two effect behaviour asymptotic negligible intuitively close rate zero effort provide go basis consider sufficiently latter condition since purpose hilbert speak tend convergent assume infinity tend eigenvalue worth moreover convergence contrary behaviour drop extra theorem claim proposition arise result suitable choice term hence plug condition highlight suitable operator whenever algebra worth note hand side converge asymptotic hand eq exist instance guarantee vanish choose exactly approximation dimensional formula moreover hilbert author metric orthonormal hilbert space worth explain strictly play resolution fine despite fact intensity intend besides extract unless additional remark reduce hypothesis latter argument one density exponential normal pp whenever
align pool balance preserve texture face date subject contribution traditionally align similarity feed cnns however alignment rotation overcome limitation propose camera cnn base identity unlike cnn fusion patch feature form powerful grey extract grey length network convolutional fully layer use verification train inter intra variation far propose add supervision improve big discover selective robust work propose another face recognition learn use cnn contain around subject publicly motivated train deep cnn table train convolutional filter avoid texture along architecture powerful layer typical cnns clear architecture choice rather motivate evaluation benchmark cnns trained cnns cnn architecture since extensive training improve discrimination detailed subsection design good architecture layer avoid design cnn architecture adapt architecture size cnn medium cnn cnn convolutional fully connect filter cnn convolutional activation linear relu softmax layer exclusive rate batch fix table conv st pool st x conv pool st conv st pool c st conv c softmax softmax aim make class separable often face verification extraction hand fed model method verification task intra extra hypothesis base map respectively discuss computing first explain model six face recognition cosine correlation achieve recognition cosine among euclidean city cosine grey vs colour cnns grey colour grey colour impact type comparative face grey colour grey colour colour contain significant augmentation flip augmentation technique face recognition use evaluation little analyse impact test pair fusion fusion learn score score compare flip score improve performance fusion statistically fusion preprocesse step pixel usually normalise space original normalise motivated feature normalise cosine recognition rate dimensionality reduction learn feature dimensionality storage crucial mobile device scale separately feed face capture spatial train separately fusion fusion improve face implement fusion corner different patch original evaluate performance network face learn fusion network face face fusion actually fusion idea hand pattern multi dimensional local network well fed compare accuracy database improve face show compare method art method feature dim face convnet convolutional neural attract lot attention field work rigorous empirical evaluate architecture cnns face fusion recognition powerful greatly fusion metric factor cnn subject support union horizon innovation agreement contract ep audio home contract ep l union project acknowledge use ac uk uk ia ac cn deep cnn achieve recognition cnn focus cnn architecture rather reason conduct evaluation recognition easily cnn unlike cnns train architecture architecture compare architecture evaluate identify useful property dimensionality traditional exploit crucial good performance fusion make source publicly consist face alignment face important extraction phase face dramatically unconstrained environment face cover complex intra variation pose illumination ideal unconstraine environment last cnn deep unlike intra notably top three face report database face achieve cnn cnns recognition stem fact gpu greatly cnns generation effective despite promising cnn unclear good experimental cnn make task object recognition face align similarity transformation pose correction conduct alignment object recognition cnn choice publish cnns database available cnn contribution component cnn face system
per extract descriptor feature implement descriptor bag word use square normalization baseline classifier achieve also one kernel baseline classification rate literature reporting feature sophisticated geometric volume solve multiclass flow probability h py p k il r note geometry l f gr f ij g j dx gr f tr r ii ie gr w r p f summation convention basis orthonormal invertible derivative let directional derivative direction ii ie b r k r j r j smoothing penalty check g assume class notation subsequence leave q compact x k contradiction summary claim hold aa n h speak pointwise length maps convergence follow I k b follow jensen inequality k proof hard plot evolve decision boundary geometric representation framework amenable rbf initialize boundary leave column vertical department mathematics wu computer science university geometry multiclass propose geometric find measure volume intuition overfitte fast hence gradient flow move initial towards minimizer function establish mild multiclass compare probabilistic training label new label multiclass py plug get h f balance datum erm empirical balance erm complexity enough pixel human image change face regard measure fitting geometric method finds iteratively curvature priori assumption field point move show initialization algorithm bayes experiment multiclass uci repository green dots simplex value unlabele grey dot show position simplex flow htb example evolve inside simplex method towards deviation training boundary show additional contribution multiclass fitting term method consistent produce numerical result introduce knowledge current literature formula erm term closely plug regularize smoothness decision term paper geometric support contrast support follow regularize erm scheme scheme allow treat multiclass case simultaneously seek flat difference goal reduction flat minimizing towards flat volume dimension curvature associate geometric gr f l approach map remain flat impose penalty consist ideally vanish e distortion distance penalty ii md field necessarily exist map open neighborhood vector theory g curvature intrinsic curvature curvature r curvature correspond inefficient volume tr calculate geometric explicit formula volume penalty gradient field convention repeat penalty l f e ph pp flow approach functional speak hope estimator field recall neighborhood flow line tend bayes multiclass follow regularize gr gr k mt nn appendix flow sequence independent critical g p close treat practical choose radial la g ij ia plug rbf total evaluate center rbf reasonable field function role specify rbf extra summary learn summary input training every accord actually proper automatically distance vector way enforce geometric take center project tangent
thresholding pair experiment although admm run short time show converge admm test reach stop amount different three row experimental left much perform however class thresholde fine mean potentially speedup compare three term go supplementary solve group demonstrate effectiveness although focus group idea also extend admm graphical quadratic lasso thresholding solve graphical extend identify estimation precision shall fine partition institute precision gaussian model propose novel algorithm identify graphical lasso subproblem superior individual scheme split much fast thresholding validate feature extensively distribution estimating world formulate solve take advantage sparsity also screen develop detect entire precision matrix study relate distinct class underlie statistical power aggregate class formulate joint graphical non among fuse solve node base class number similar graphical lasso couple thresholde joint subproblem algorithm uniform decompose precision distinct split effect uniform precision require partition exactly graphical problem subproblem admm multiplier graphical partition scheme group efficiently fuse supplementary material bold letter k denote covariance precision matrix lasso sparsity pattern satisfy union element equal element partition denote strictly fine strict refinement denote describe element two submatrix ji concentration graph exist partition obviously base merging correspond divide kk k class feasible fine follow partition feasible partition first edge mix union one least contradict feasible paragraph contain contradict component pattern include typical graphical formulate represent penalty encourage structural pattern lasso entry precision accelerate graphical screening variable precision correspond computational efficiency divide group shall meanwhile method thresholding e employ partition method fuse node employ thresholding partition concentration split b connect divide disjoint without increase even separately another partition non uniform except white color identify feasible uniform partition kf ki ki theorems non supplementary material proof partition pair must hold feasible partition j jk condition non uniform covariance thresholding uniform screening utilize j uniform partition hybrid algorithm feasible partition generate good condition global toy three class three divide hybrid thresholding prove file hybrid screening algorithm feasible satisfy section admm admm minimize augment k admm iteratively variable insensitive require eigen k solve require eigen shall eigen decomposition upon update updating plain non thresholding detect
certain negative backpropagation training proceed sample ensure indistinguishable domain straightforward sophisticated architecture appropriate state art learn discriminative image let notation network parameter label prediction note preserve theorem class loss training layer saddle previously saddle estimate equation stochastic sgd comprise fed predictor domain gradient would convenient procedure sgd accomplish introduce associate act transform backpropagation gradient level sign pass precede layer use package propagation backpropagation multiplying update define domain classifier backpropagation multiplied effectively implement pseudo forward define run implement sgd sample well source section note result present subsection obtain stochastic consist gradient parameter crucially usual direction maximize respect minimize classification mm represent dot study variant problem one source contain example remove unlabeled toy share architecture hide layer neuron procedure keep regressor hyper execute algorithm source risk regularizer domain regressor toy boundary regressor nn relate look four detail show boundary predict label nn two decision boundary perfectly affect pca set map dimensional project cluster lie tag four crucial original column b nn happen conversely opposite corner resp locate pca well suit classification problem regressor precisely source classify otherwise regressor discriminate hidden representation prevent explain domain regressor nn learn domain generalize topology hand nn regressor capability seems roughly capture rotation angle neurons neuron regressor line observe nn group straight boundary classification however able roughly rotation observe regularizer prevent kind produce plane vanish one way domain datum hyper variant call tuple parameter unlabele example classifier reverse sample classifier multiple value select reverse architecture validation use early stop validation accuracy initialize reverse network nn svm l book book book book book c svm network machine svm amazon review pre process compose book encode star rank star unlabele separate target procedure logarithmic rate hyper training nn hyper among logarithmic present hyper criterion datum part algorithm poisson nn svm domain representation target improve representation art brief unsupervised source sample learn space reconstruct original svm reach propose source target different objective describe representation corruption number layer execute procedure concatenation layer search binomial performance report foundation claim toy confirm proxy various representation run nn section estimating source representation construct equation split subset equal large range low error firstly compare lead raw nn layer tends increase representation adaptation clearly low experiment one hand notice raw clearly help theory use parameter value improvement provide pt ccc ccc cm mnist source mnist number sign mnist mnist l difference non background adaptation perform row label da da considerably cover black amazon target sa l sa outperform state art extensive evaluation alignment sa adaptation synthetic number address testing view house synthetic digit latter image window vary include digit number orientation background stroke color degree choose structured clutter background backpropagation technique target label contrast sa accuracy probably information reduction case mnist mnist test mnist significantly appearance stay avoid minimum use obviously direction mnist equally diverse reasonably mnist appearance separation domain feed cnn solely network difference explain improve mnist scenario opposite sa perform either mnist da synthetic sign similar number due obtain image simulate target sensible datum unlabele domain additionally provide reveal label per add predictor change method beneficial set thorough verification semi supervise left work office datum office three distinct amazon discuss office spread large crucial successful fine cnn pre imagenet make comparable mean classifier previous work task commonly protocol adopt label source unlabele abundance unlabeled method previously state adaptation especially amazon domain slight moreover switch classifier apparent serve regularizer task identification people camera person person probe person disjoint illumination pose low make problem difficult human descriptor descriptor match commonly identification recall probability descriptor probe descriptor image several paper however drop descriptor handle evaluation camera constitute significantly identification report cross scenario training adaptation improve identification descriptor set camera camera appear contain capture every image view person two refer include extensive serve source descriptor probe probe correspondence p domain experiment serve view camera exclude image camera datum domain domain cccc cccc cccc mirror calculate similarity score correspond mirror image camera person average experiment architecture incorporate relu max pool dimensional descriptor within cnn middle share convolution training similarity feature batch batch domain adversarial architecture include convolutional relu include fully layer include intermediate representation verification descriptor predictor parameter pair train momentum schedule dropout concatenation output sized cm cm cm p eight hardness annotation adversarial ensure match source distribution iteration adversarial consistently performance pair far demonstrate adaptation descriptor p experiment propose approach feed annotate amount da domain backpropagation training behind predictive uninformative target implement new feed introduction simple gradient layer flexible benchmark sentiment convenient adaptation almost architecture backpropagation towards demonstrate experimentally feed architecture descriptor refer consist position product optimize em simple gradient update crucially opposite adversarial parameter hide adaptation propagation backpropagation f regularizer I f j network construction lead update use minimization discrimination minimize distinct update constitute special domain indicate index adversarial label pair potential strong gradient domain quite however observe rgb rgb different figure small domain mnist involve adopt set baseline package office domain adaptation identical layer bottleneck branch somewhat adaptation anneal progress follow restriction train et de universit universit universit institute science region new representation adaptation suggest achieve prediction make discriminate implement ii shift domain adaptation behaviour feed result backpropagation stochastic implement effort deep package sentiment analysis art standard achieve descriptor task context person application domain adaptation learn synthetic analysis person identification machine obstacle progress neural advance state across variety task datum training deep suffer shift datum come fully sentiment review movie classify review product book shift mapping domain target compose approach domain situation domain unlabele unsupervise label supervised focus hard generalize semi rather straightforwardly unlike work focus process decision make way feed network applicable target domain motivated adaptation representation domain identify origin observation focus combine invariance jointly discriminative operating predict minimize feature loss label classifier classifier update work classifier encourage course crucially three appropriately compose feed layer algorithm gradient modification momentum generic create exist architecture backpropagation rather layer multiply scalar backpropagation adversarial architecture architecture part label predictor success adversarial well sentiment image task traditional mnist office benchmark obtain finally person good retrieval apply domain adversarial label train loss demonstrate considerably often obstacle learn develop exploiting generalize focus situation similar example sentiment want distinguish review one type movie want book domain try generalize unlabeled review book approach classifier large body exist classifier linear recent denoise stack denoise autoencoder representation corruption transfer control domain suggest include work towards neural descent learn objective adversarial network confirm toy datum performance regular sentiment reach take rely explore many large literature focus mainly recently increasingly study notably exploit principle robust domain approach sample seek map source one aspect match dissimilarity space axis attempt distribution accomplish modify measure separability perform autoencoder gradually replace source improve separate representation learn autoencoder jointly considerably office adaptation deep feed tune easily available related building network way minimize discrepancy architecture minimize mention arise early dissimilarity use suitable applicable focus feed network minimize datum potentially rkh indistinguishable compare office adaptation arguably seminal optimize note representation word posterior regularizer task argue learn objective optimize reason part publish conference extend incorporate bring terminology depth justification extensive sentiment version descriptor person identification relate unsupervised adaptation feature approach source target aspect approach mean kernel reproduce axis distribution approach accomplish modify feature representation geometric use base separability target change among way autoencoder gradually domain improve simply train autoencoder actual learn autoencoder perform domain jointly single backpropagation simple conceptually achieve office benchmark unsupervise adaptation adaptation label target domain feed label target building network sample way discrepancy discrepancy focus feed network regard seek alignment achieve domain adaptation datum explore however recently non become increasingly neural notably denoise paradigm contribution domain another complementary robustness cross domain literature hmm also inspire addition sentiment classification optimize rely representation propose fair set result directly linear classifier distribution assume representation classifier note adversarial adversarial instead model possible source domain domain label draw I build information tackle challenge target error notion method justify
ordinary wind sensitivity capability reference initial observation three deviation height magnitude wind magnitude wind background background error covariance model follow error average magnitude reference background error wind magnitude reference wind reference ensemble mean covariance initial kalman cycle hour uncertainty wind synthetic multiply wind covariance calculate average covariance last per hour version long create several var background ensemble smoother investigate future assimilation three assimilation window interval assimilation window window regard spin assimilation synthetic observation available observation two window window accord experiment scalar day force background covariance lead covariance error solution rmse rmse trajectory assimilation window carry routine toolbox optimization process stop process iteration converge experiment use ensemble member assimilation window empirically tune setting keep fix hamiltonian keep number constant scale burn smooth approximately approximately cm p fix evaluation require window iteration optimizer evaluation hmc depend configuration burn number drop stationarity adjoint evaluate desire step hmc smooth adjoint rejection criterion run make cost equal assimilation window ensemble reject hmc case state collect run assimilation var hmc smooth assimilation summarize total forward run var scheme window hmc smoother roughly hmc smoother handle calculation computational hmc smooth replace burn suboptimal var course decrease impact assess technique increase smooth could acceptable information sample immediate consequence background error assimilation smooth monte uncertainty smoother build together calculate mean analysis moreover analysis ensemble begin assimilation window popular covariance analysis hmc smoother require adjoint var var must hmc smooth practical investigate strategy enhance sample smooth nonlinear gaussian acknowledgement laboratory reference construct smooth assimilation hybrid hamiltonian smooth time methodology unlike meet mode smooth variance conjunction assimilation window numerical compared ensemble assimilation da combine observation consistent da large great da gain acceptance first control variational three strategy variational find posteriori successful assimilation ensemble enkf root filter ensemble kalman filter efficient implementation kalman enkf provide variational identical error var enkf instance advantageous smoothing var assimilation time var update assimilation window ensemble hand posterior time assimilation window tangent adjoint challenge inherently var quantify matrix inconsistent scheme enkf approach var analysis statistically require additional observation unlikely ensemble ensemble uncertainty offer practical yield mcmc provide generating chain whose stationary distribution distribution mcmc gold limitation considerable state accelerate continuously hmc accelerate sampling hamiltonian generate hmc high good knowledge hmc da ill pose posterior approximated nearby anneal process hmc non solve four datum assimilation monte name smooth extend distribute hmc smooth ensemble distribution time smoothing carry sequentially assimilation distribution begin window assimilation background distribution assimilation present attempt reduce chain ensure independence hmc hamiltonian several important var framework system uncertainty water equation sphere moderately multidimensional test part assimilation hmc discuss direction review assimilation var initial assimilation background initial produce window analysis assimilation window background operator generally observation usually dimension determine initial discretize partial differential realistic numerical methodology propose small tangent jacobian dynamic initial solve adjoint k adjoint tangent linear adjoint challenge dimensional complex practical improve prior prior probability sampling give perfectly reality new contain observation act observation error background observation point perfect bayes operator functional compute observation mode var characterization uncertain dynamical system calculate infeasible approach describe directly act smooth system totally var enkf lead inconsistent uncertain dynamical system kalman represent represent matrix column forecast member ensemble q matrix forecast adjustment kalman calculate uncertain dynamical past solve smoothing smooth ensemble kalman employ state kalman enkf repeatedly write interval smoothing ensemble specify time condition available later version interval interval smooth computational fix interval single beginning observation highly violate show hmc smooth herein capable handling produce hybrid method ensemble trust mcmc distribution require distribution often consider impractical drawback take burn discard sampling target independence sample usually drop intermediate another drawback carlo member rapidly require control survey fast multi modal fail mode carlo hmc limitation mcmc hamiltonian consist momentum ordinary equation term hamiltonian advance solution exactly reversible common position advance hamiltonian time satisfy careful good accuracy practical consideration split evolve small sub flow exact flow hmc variable variable variable choose sampling joint distribution hmc variable position hamiltonian system serve markov chain kernel probability hamiltonian distribution position mean independence momentum momentum gaussian know view auxiliary acceptance rejection summarize hmc probability markov target auxiliary discard accept reject continue distinct propose q stage three stage energy length hamiltonian step tune covariance momentum choice converge sampling take identity ideally know approximated stationarity hmc sampling state fast generate parallel algorithm mode smoother simultaneously account assimilation initial assimilation window analysis distribution seek distribution potential energy var functional coincide var sequentially forecast analysis forecast assimilation window begin forecast forecast window propagate current background begin current include error considerably enhance analysis assimilation herein forecast hmc initial current assimilation give background state distribution follow calculate base forecast fix forecast ensemble generate ensemble initialize good suboptimal drop number helps propagate assimilation begin assimilation window initial forecast assimilation window use final provide assimilation test hmc smoother nonlinear sampling compare conventional var move similar use two local act initial choose
periodic arrange base particle diameter magnitude non scale denote box box formulae ratio volume particle height homogeneous mixed initial two particle carry contact contact coefficient detail contact mm mm profile flow depth depth red circle denote coarse scale track illustrate log block label denote scale order field essential proper coarse irrespective question scale make valid coarse length individual particle strongly thus lead spatial coarse suitable scale ratio steady simulation flow reach steady whose steady flow fig steady profile average time averaging thereby volume fraction flow fig fig ratio see plot depth profile coarse ideal scale scenario simulation scale average similar fig volume coarse width depth denote fig strong statistical temporal ensemble average probably particle particle scale similarly slightly compare fig label range particle latter usage illustrate construct coarse volume fraction fig fig average denote near base particle effect particle nearly decay base surface pressure pressure thereby large alone momentum velocity contact show sub particle length denote observe near base smoothed particle denote understanding mixture scope paper publication nevertheless spatial coarse scale particle cc steady mm illustrate centre steady spatially profile plot temporal averaging thereby dash spatial average steady range towards investigate issue spatially coarse grain field thus previous spatially average coarse carry depth concern temporal particle sec rather sec approximate correspond steady see fig fix interval min total define interval begin gradually result spatial averaging spatially average alone profile density increase increase fluctuation gradually fluctuation quantify error spatially observe proportional steady flow average average coarse smooth boundary solid dotted line steady denote temporal window profile large alone temporal contrary effect denote effect similarly coarse scale produce field block section apply expression datum steady flow particle underlie phenomenon predict thereby step application cg state system channel describe particle segregation happen see fig vertical centre particle coarse flow particle segregation dynamic particle alone consider investigation simulation approach spatial coarse spatial coarse result spatially profile result dimension average time average sec fix interval average temporal spatially average time focus size illustrate particle profile fig effect different averaging scale fix scale effect two old spatial scale invariant rectangular invariance region purpose something pick tracking flow one fig track fig average average take exist temporal block coarse scale invariant flow coarse scale average field cg thereby temporal scale average construct track flow depth analyse display plot exist irrespective average rectangular dominate fluctuation exist spatial coarse scale nevertheless invariance different coarse expression sec flow needs specify temporal ii similar steady flow range temporal flow technique average coarse flow applicable system investigation flow discover spatial scale coarse grain fig region fig imply exist coarse spatially average field coarse grain unknown validate shall develop formulation use approach micro macro coarse available processing reduce store etc couple pressure release like support project rgb rgb validate model micro momentum discrete position micro restrict micro macro flow average method advantageous interaction force construction boundary stress force determine component mixture segregation require average efficiently investigate simulation molecular dynamic etc illustration generate simulation mixture rough channel coarse flow source coarse part solver predict validate micro macro method momentum stress particle force include approach interested reader therein micro macro call average coarse advantage satisfy equation mechanic boundary spherical coarse contact shape ii contact contact instantaneous often micro macro situation therefore generate equation coarse deviation concern coarse see flexible use discrete molecular particle datum coarse successfully allow flow boundary analyse flow convex shaped flow material nature material crucial diverse designing extensive carry material past static material failure static spherical nature material particle shape external etc phenomena particle segregation tool arbitrary solve law interaction break particle contact computationally nevertheless million particle mm represent flow magnitude life flow environmental environmental flow simplify material closure relation law etc assumption parameter determine give flow call calibration year focus multi simulation application include mixture channel particle segregation application particle segregation often tend distinct alone flow channel eventually particle end free particle appear near need property material motivate need average coarse recently steady alone besides extend multi particle density channel focus upon topic steady flow technique ensemble nevertheless coarse average alone flow necessity temporal coarse field extract average field coarse cg derive case channel sec flow steady length coarse scale scale determine averaging flow illustrate spatial average spatial averaging field invariant main finding circle ex extend spherical system extend coarse formulae classical law mass momentum thereby expression stress particle mixture multi point construct distinct coarse formulae formulate media flow flow water mixture ice concrete deal partial variable mechanic dirac delta particle density field sec function integrable real function benefit later fraction define eq thereby expression spatially coarse grain unclear concern coarse expression thereby briefly reflect characteristic possible need possess certain ensure positive momentum delta w result differentiable efficiently manuscript polynomial three cut radius range coarse polynomial polynomial allow average gradient integrable analytically direct different make delta coarse choose coarse mass density momentum velocity field derive section coarse derive equation chain derivative coarse grain mass grain momentum follow momentum notation arrive mass balance hold particle mixture velocity ratio momentum field grain velocity mixture continuity exclude momentum velocity satisfy balance law balance momentum state expression stress momentum derivative force particle expand term represent index q second substitute force density force stress rewrite branch illustrate substituting expression allow force along vector identity rewrite contact length branch within sized receive big moreover contact force independent velocity particle I substitute stress field thereby total stress field contact stress field stress type expression consider fig coarse fig magnitude arise contact far derive mass velocity force
descent solver svms however systematically investigate incorporate posterior monte subgradient non max margin discuss detailed balance several hmc subsampling efficiency subgradient langevin method svms devise effectively integrate variable far hmc mcmc converge target posterior fairly previous attempt carlo langevin carlo stand also extensive max organize hamiltonian stochastic subsampling propose hmc version mixture svms experimental bayesian svms svms bayesian become increasingly big application overfitte account adaptively infer via growth fortunately scalable inference carlo readers review particular gradient prove dimensional space optimization technique representative example include stochastic langevin manifold high posterior differentiable little systematically doubly log arise vector svms max margin smooth differentiable metropolis hasting walk efficiency space sampler datum efficient inverting benefit view extra stochastic subgradient extensively differentiable objective many svms lasso regressor extension none systematically investigate idea subgradient markov chain systematically mcmc posterior generalize hamiltonian dynamic subgradient able detailed cost subgradient replace unbiased anneal subgradient empirically previous attempt subgradient hamiltonian carlo langevin little distinction distinguish hmc like hmc technique subgradient theoretical scalable paper organize review hamiltonian subgradient monte carlo sparse conclude hamiltonian dynamic mechanic describe momentum hamiltonian energy definite decompose hamiltonian eq carlo hybrid hmc classic combine besides conventional bayes posterior problem represent u conventional typically write give convention interest position hamiltonian sampler differentiable potential hmc simulate euler conventional stepsize discard augment sample one langevin momentum discretization retain invariance distribution deal u subsampling langevin extend hmc hmc stochastic posterior subset size noisy turn algorithm lot review hmc euler introduce fluctuation follow stepsize polynomially decay mh hamiltonian dynamics central gradient gradient hmc method hmc formally hamiltonian subgradient hamiltonian ordinary energy posterior non differentiable accumulation laplace hamiltonian assume hamiltonian hamiltonian dynamic hamiltonian stochastic subgradient mapping another turn back primary dynamic hamiltonian hamiltonian equation indicate subgradient potential energy differentiable hamiltonian keep property hamiltonian mean flow energy flow transformation separable hamiltonian regardless property metropolis subgradient hmc hmc subgradient subgradient energy ordinary subgradient initialize discretization stepsize analyze hamiltonian detailed balance subgradient potential mainly case zero implication hmc dynamic may result energy construction save space sample subgradient hmc draw approximation detail go sense converge hamiltonian subgradient follow subgradient hmc either euler leave balance subgradient leave readily subgradient langevin replace log formally simulate noisy subgradient exist recommend decay save mh correction langevin proposal stepsize properly subtle thus discretization stepsize scheme properly make relatively adaptive subgradient paper stepsize experiment scheme beneficial fast tb stepsize dynamic likewise version hmc generate via iteration mh correction adopt anneal properly generate sample subgradient inference svms dataset binary naturally svms interested classifier per unnormalized py ic regularization hinge differentiable log adopt subgradient sampler eqn eqn tb stepsize initialize tp latent mixture capturing consequently learn instead one description extension infinite svm mixture associate assignment work distribution gibbs classifier discriminative function hinge adopt build variance sampling wishart conjugate augmentation infinite usually fix simplify reader detail input subgradient partial probability introduce hmc step gaussian parameter subgradient hmc develop mcmc crp use randomly draw calculate sampling subsample inspire doubly doubly classifier doubly assignment subsampling formally take subgradient q gradient save assignment improve efficiency alg stochastic parametric svms extension max margin svms bayesian achieve still sample high dimensional baseline synthetic consider stochastic dimensional observation distribution input sec augmentation sampler obtain accurate sampler stochastic rather stepsize omit mh correction besides diffusion number sample burn give mark visual mcmc accurate almost indistinguishable method namely uci dataset binary space training use test stochastic metropolis good justified successfully decay stepsize diffusion carry validation discuss detail choose various respect time mcmc reach moreover subgradient augmentation method draw mini walk metropolis subgradient observe method mix fast dimensional rest testing walk metropolis sampler three sampling walk metropolis set choose stepsize adaptive respect running scale see ten fast might particularly curse decay well flexible stepsize decaying give analysis sensitivity subgradient reflect tradeoff general often computation efficiency consideration right subgraph e curve dot mild green serious typical right bit accuracy mild benefit result mixture svms doubly subgradient hmc f contain test use stepsize final efficiency doubly hmc greatly improve svm use stepsize hmc significantly also stochastic hmc currently svms still remain sample gibbs systematically subgradient apply linear svms mixture svms doubly subgradient posterior prior extensive empirical study subgradient mcmc improve systematically subgradient mcmc large laplace svms piecewise hinge g volume ordinary hamiltonian subgradient deal effectiveness mcmc efficiency draw subgradient differentiable posterior class see hmc subgradient method bayesian posterior investigate detailed balance subgradient hmc generalize dynamic efficiency correctness test well future subgradient stepsize acknowledgment acknowledgment go acknowledgment final reference supplementary differentiable potential ordinary hamiltonian dynamic construct smoothness countable infinite smoothness energy satisfie polynomial interpolation existence ordinary would hamiltonian dynamic see g solution generalize hamiltonian eqn correspond equation flow differentiable finish show determinant jacobian determinant finite except eqn write hamiltonian detailed balance hmc similarly hamiltonian subgradient hmc volume euler volume subgradient
internal split split expand fashion visit terminate empty present sequential generative process denote iteration empty tree assign split split location child iteration leaf internal child node iteration assign iteration terminate applicable expansion encode node expand expand expand fashion input initialization j pg j briefly propose limitation present pg sampler distribution bayesian sample gibbs loop tree associate condition remain j j j propose sample condition j summarize choose move split right child child leaf thereby leaf randomly internal node parent child parent child issue tree compute hyper encourage affect tree computationally involve deep likelihood subtree affect computational propose sampler illustrate section sampler next address concern propose ask complete tree indeed see particle markov carlo dimensional pg sampler instead sampler smc smc particle current pg nn contain residual integrating likelihood posterior top particle approximate complete substitute filter however leave version smc leave refer detail build filter decision stationary reduce clutter old tree write ccc end sequence partial particle old particle e tc particle generative process model decision potentially deep old hence whenever particle associate partial likelihood nod leaf node step smc typical smc resample old particle proportional resample resample none contain node stop loss return sampler summarize pg per iteration sampler explore likely reject decision internal since internal leave subtree empty pg training particle old pr w ct tc tc tc tc tc e tc ce ce w tc stop pr split normalize tc w tc tc tc tc tc tc tc tc tc tc tc experimental pg sampler contribution work efficiency algorithm popular box interested sampler run particle early sampler multiple condition equivalent inference residual label sampler mix consist lead tree true heart mix create depth control hypercube vertex offset generate generate vertex hypercube explain increase default leave mcmc illustrate material trace plot pg converge mse tend leave train indicate sampler posterior behavior sampler dataset compare computing ess mcmc ess discard iteration burn ess generate differ additionally ess ess likelihood ess ess pg ess ess pg pg sampler c pg pg hypercube consistent prior dimensionality dataset training point single characteristic c sampler ess ess three sampler achieve tree similar ess sampler increase observe pg exist pg sampler pg pg real pg unlike move pg complete confirm dramatically true consist pg well pg prior tree backward sampling show pg pg future direction bl fellowship college international fellowship receive ep agreement popular linear bioinformatic demand prediction size metropolis hasting chain local change slow mix fit deep present gibbs pg particle filter tree change individual tree pg propose complete fit pg sampler international conference intelligence cp volume ensemble heart broadly prediction reduce wherein explain remainder extremely additive serial like tree additive tree additive extremely popular variety include protein dna automatic spam detection drug additive avoid overfitte fitting structure prediction time inferential credible well measure variable time predictive comparable forest network mcmc particular introduce local tree expensive large consider subset however move poor produce inaccurate poorly chain use scenario user focus pg use purpose decision partitioning input align tree rooted finite collection exactly except distinguish node child child without child child parent denote split denote location tuple decision leaf give leaf
could good estimator outperform sample target propose covariance concern riemannian compose offline asynchronous perform subject classifier distance riemannian possible consider setup stimulus stimulus modify signal frequency result henceforth filter trial belong trial center input center compute center predict algorithm find offline brain nonetheless asynchronous one trivial introduce period eeg record observe record eeg overlap epoch n consecutive record denote epoch hold slide process continue reach epoch identify enforce negative occurrence criterion trial ensure detect user present good flexible offline online record eeg channel reference right hz hz hz hz setup stimulus minute trial record stimulus record varie record minute trial second detailed trial channel time stimulus remain vary effectiveness evaluate term classification matrix investigate bootstrapping replication assess compare trial length second affect accuracy estimator increase attribute trial producing appear shrinkage affect epoch direct independent shrinkage estimator epoch maintain correct large small condition matrix condition ill condition second trial singular b eigenvalue increase eeg length integrated vary indicate baseline integrate improvement ability complex apparent difference favorable long trial length compatible shrinkage offer well inferior hence significant covariance quality computation consider training center figure show matrix tangent star visualization principal figure riemannian riemannian riemannian remove highly allow remove outli center filter offline center lowest illustrate center estimate affect filter bad poor cluster riemannian could reject close far mean central b line black star riemannian riemannian subject riemannian part first evaluation delay online subject display signal filter stimulus blue hz hz visible synchronization second axis previous trial high trial power decrease capability low trial stimulus frequency axis line signal high low eeg plot subject low representation rectangular row trial hz hz hz band ht accuracy almost trial second crucial synchronization stimulus confident second asynchronous delay eeg epoch second ht hz hz hz show visualization tangent matrix classify stimulus attribute classification offline art offline opt show take response take online contain online show improvement acc take trial classification delay offline end trial second rejection epoch delay long trial classification accuracy optimize online moreover asynchronous strongly interaction offline offline opt acc acc acc delay acc delay c delay work efficiency riemannian classification feature rely covariance eeg signal apply investigate robustness verify estimator yield high accuracy online stability speed epoch perturbation curve misclassification eeg past riemannian determination offer powerful include adaptation riemannian dynamic successfully covariance mean report paper attribute eeg riemannian time bring like thank geometry brain computer interface asynchronous brain computer steady visually potential true mm geometry brain computer electrical engineering des de france la paris universit france paris brain interface signal yield study eeg allow common source variability electrical biological construct invariant matrix might advantage definite comprehensive review tool could matrix thorough conduct main contribution propose classifier riemannian space subsequent assessment steady visually computer visually potential human interaction rely capability computer scientific eeg dataset inter individual lead hand intra adaptive subject several signal vision work try maximize one minimize principal canonical technique covariance eeg cca aim also classifier largely handle euclidean space space reduce riemannian manifold inherently account approximate euclidean condition feature riemannian partitioning build emphasize brain outcome art performance version steady visually potential subject concentrate frequency focus brain stimulus subject propose performance brain response acquire external implementation phase apply dynamic criterion speed system allows dynamically determine trial riemannian estimate point similarly adapt manifold laplacian adaptation pdfs matrix interpolation filter datum riemannian invariant complete manifold point reach finite last kind instead adapt riemannian apply eeg eeg assign class close classification filter component back riemannian another unsupervised filter removal reject lie beyond computed window robustness affect riemannian divergence two different class correspond cognitive matrix algorithm eeg metric eeg selection spatial filter application riemannian geometry mention mi paradigm mi subject
component variance gaussian quadrature express use integration define reason introduce filter next quadrature approximation fix sigma usually enable relationship quadrature however often beneficial square consider sigma point location one square quadrature approximation sigma analogously sigma formulate sigma quadrature respectively prediction sigma propagate sigma dynamic sigma propagate sigma point measurement filter measurement think formulate sigma smooth sigma weight quadrature smoother start filter sigma k n propagate sigma k n k q cope additive augment filter lag analogously reference smoothing affect weight equation discuss choice choice sigma point sigma transform gauss machine choice make gaussian quadrature computing close cf turn computed sigma immediately possible depend observation one sigma variance variance respect close moderate optimize bfgs covariance choose mat ern covariance least chance quadrature sigma could similarly linear polynomial regressor form gaussian method evaluation classical integration see special detailed polynomial use find covariance positive definite covariance multivariate polynomial select evaluation weight become furthermore exactly classical method weight quadrature set theorem e se fig nature transform interpretation polynomially move diagonal datum transform shape clearly polynomial fit flexibility quadratic flexibility certainly see outperform integral show use rd sigma point integration rd spherical sigma process quadrature se covariance rd spherical integration use rule sigma place intersection coordinate origin bit divergence plain vary bit state slightly bad growth often follow extended gauss filter quadrature smooth point quadratic variance bar figure see quadrature filter low smooth close evaluate target track dynamic iii linear dimensional noise angle measurement sensor write q sensor two rmse error sigma outperform sigma quadrature gauss practically sigma point htb quadrature smoothing sigma criterion polynomial order well gauss kalman process well outperform previously sigma point filter e orthogonal polynomial inherently generalization connection quadrature multivariate function form orthogonal hilbert polynomial orthogonality denote function expand polynomial product deterministic gaussian covariance eq theorem algorithm section proposition remark corollary gaussian sigma use advanced kalman sigma method interpret quadrature suitably also multivariate gauss integration spherical discuss sigma polynomial method numerical gaussian integral regressor approximated regressor sigma integral predefine call point multivariate certain exact particularly multivariate context turn regressor quadratic closely relate polynomial gaussian process regression approximation regressor perform much selection weight function particularly useful smoothing kalman filter integral form sigma approximate example multidimensional type quadrature base integration transform sigma interpret belong numerical integration conversely quadrature interpret case sigma point taylor interpolation gaussian integral polynomial numerical present process quadrature connection sigma multivariate integration sigma kalman filter filter gauss kalman filter see case suitably generally quadrature gauss rule symmetric integration special criterion sigma error conference article analyze process linear article connection extend symmetric rule sigma selection well provide kalman filter k kp non state filter page smoothing see filter filter linear kalman q initial respectively update sequence smooth non non smooth smooth smooth moment matching comes transform additive transform moment variable point smoothing method generally approximate filter sigma point call refer weight sigma lead form sigma correspond unit sigma transform recall dimensionality algorithm unit th axis sigma sometimes computation concentrate direct method special g form process gp evaluation integrate gp regression predict test j contain model set q gaussian prior parameter equation want approximation joint covariance point posterior everything describe typical dimension conceptually practical typical independently gaussian quadrature idea point integral scalar simplicity interpretation quadrature interpret polynomial function aim still link due integration integral
others framework forecast decide time tracking produce use period assume historical activity week level material metric mean mae estimation produce ar current table summarize estimation estimate outperform period column table regular since cm whole c week week al ar naive ar naive ar naive naive blue ar grey background reflect suggest suggest panel period regular week regular paper week week year htbp train panel estimate close show post uniformly square absolute correlation correlation partial r avoid root absolute potentially alternative well publication internal lead available historical necessarily inaccurate week report website period period outperform accuracy metric essentially change suggest table challenge us article website counterpart change update affect estimate search website day month estimate despite change stable et result superiority term accuracy compare track google search methodology lead accurate access input google use incorporation key enhanced week activity year show reflect correlation integration series prevent spike series remain significant result increase query week average week smoothness substantial understand google historical complement period slow response activity current effect correlation activity google search detect public reaction investigate movement towards activity increment increment naturally smoother uniformly behind competitive evident globally select google autoregressive close figure tendency level evident wave h result self correct week series google search domain peak across term week important note algorithm update gold activity immediately activity display superiority method rely behavior google search user methodology change place future easily generalize spatial disease social amenable search moreover framework strategy track internet service twitter normalize volume google independent raw volume divide volume volume comparable available website www google trends date standardize normalize query zero year google estimate impose invoke kind penalty penalty dynamically week window week find hyper hyper however since window dynamically datum week cross validation need hyper combine validation autoregressive google tend unnecessary section still remain hyper considerable propose detailed specification hyper square rmse mae target increment two co movement hyper parameter explain aim test methodology respect come improve variable level transformation google time formal x py normal eq year window date specification hyper restrict propose google search autoregressive lag restrict validate google autoregressive lag term autoregressive restrict validate penalty google lag l l penalty google lag htbp partial sep lag specification autoregressive lag specification sep term autoregressive lag elastic autoregressive lag period prior third fourth rmse mae relative naive summarize specification apparent perform penalty exactly redundant penalize counterpart besides specification penalty error absolute similarly penalty outperform square net penalty elastic show restrict global validation restrict zero drop use want remain specification penalty google term lag table give flexibility improvement fixing process separate deviation well impose decide restrict report new incorporate publish report subsequent week available recent inaccurate presence model schedule report week typically delay week week activity week value eventually week train activity sense future beyond week incorporate training week metric activity aforementioned schedule outperform absolute train main indicate l whole week week website week year year study article example activity http www activity week available http www available week activity study robustness website http www google com trend variability input common cover version despite considerable suggest incorporation extreme focus entirely series formulate input insensitive datum good portion period google raw volume disease low quality uniformly datum mean mean absolute htbp mae correlation increment different common period partial cm cm accurate public health decision save propose track google search publicly online statistical previously google trend even quality publicly available capture people flexible scalable powerful temporal big activity pattern million internet service show great big set epidemic predict price movie google digital disease google activity traditional collection center considerable regard digital lead theoretically sound estimation still outperform tracking cause united ability availability activity duration remain clinical track lag http www lag make data forecast activity recent year internet google yahoo
admit form evaluation section fix modify whereas place mass neighbourhood optimize rate view integral vary depend expert general first conjugate introduction correspond special compute numerically conjugate max regret b become overhead rate know impossible reason achieve bind put neighbourhood issue instead desire regret bind depend put amount mass cost rate price slightly bad prior put constant mass density proportional run finite section adjust integrate quantile constant motivation suggest obtain suboptimal factor careful happen cv cv integrate improper highly integral e process would remain improper yet turn improper define still desirable equally numerical stability improper prior improper expert combinatorial combinatorial concept combinatorial reflect loss edge play expect loss p simplify algorithm concept expert upon proper notion combinatorial come strategy fundamentally suboptimal range mirror improvement exploit exhibit matching domain expand range efficient combinatorial obtain quantile range expert suboptimal find r k tv whole variance inside expectation thus identify mix turn mix exponential module weight combinatorial mix new variant component prove regret component module aggregate module learn predictor summarize proof find linear defer regret express quantile cumulative coordinate v discrete discuss discretization mis instance point grow number imply miss affect factor exponentially uniform grid arbitrary vector regret factor loss quantile switching share aggregate space yet another loss bound order see multiply ambient intuitively try learn separate rate example dependency combinatorial component case reflect integral potential function whole way optimal raise perspective explain rate issue future work find low available issue aware bound hold range round adaptively tune take characterize pareto optimal expert regime assume loss normalize normalization adapt question incorporate proof section otherwise let two plugging implie maximize non hence interval consider suppose x numerator everything q satisfied consider take left side choice complete go improper careful jensen bad expert arbitrary imply expert follow implie put everything together together use everything give minimize component instantaneous mix regret generator negative satisfie bregman mix regret hence equality design weight tell ensure exponentially spaced rate v side care weight expression improper look involve contribution form outside use range argument around fall directly c w design decision make adjust difficulty general combinatorial task satisfy minimax rate datum popular formalize bound notion sophisticated exploit one paper new construct show prove incorporate quantile core instance prediction play assign incur dot product good without specifically expert round respect every ensure tight loss yet ask case indeed line improve scenario line obtain stand kind like typically line independently parallel reduce dependence expert whenever expert perform occur construct expert call reciprocal prior mass bad expert guarantee obtain sufficiently consequently quantile imply closely bound prior study two technique develop prove bound denote reference excess loss specify common type variance factor small weight happen grow linearly obvious like imply expert losse unique good vary rate prior call quantile improve combine quantile improper efficient operation round applicable sophisticated instead expert action theoretically online subset schedule tree communication fix etc reflect sum coordinate loss natural example component family analog combinatorial quantile method combinatorial combinatorial quantile keep combinatorial regret average entropy expert expert follow concept instead straight case call collapse coordinate separately bound easy expert achieve scheme learning build monotonically decrease prove multiple rate diversity aggregate reproduce certain expert motivate second extend
online h continue define next hypothesis regret translate aforementioned online statistical prediction essence ask mechanism ability mechanism may points limited budget though observe portion parameterized budget assume post mechanism pricing price transaction transaction mechanism signal note difference step set suffer regardless guarantee randomness give algorithm arrival strategy price induce obtain arrival price expect brief sketch price state regret constant strategy general prove advance knowledge regret focus slightly agent pay cost variant tractable bound low variant apply problem obtain pricing minimize importance first important strategy hypothesis randomness lipschitz set meet regret mechanism improve factor pricing factor several rely quantity measure financial difficulty understand answer four question also explain aspect examine regret aspect descent correlation value converge expect side alone cost compare regret pricing obtaining point decrease vice versa arrival equivalently burn unit simulation full paper success along burn rate turn guarantee continue true main setting follow pricing choose give knowledge algorithm implication cost contain run case small enjoy regret improvement reflect hope vc capture instance one depend capture candidate case scenario expect approximate example z guarantee run weak knowledge deep investigation present easier apply unlike pricing intuitively pay decision private propose simply draw price strategy pricing exist price cdf pricing give h illustrate price arrival benefit obtain f cost regret depend expected budget regret equality cost cost cost fix bind datum immediate pay meaning obtain less datum long meanwhile strategy price give mechanism mechanism simply run aggregate final mechanism run respectively specify online statement state regret price main show variant formally transaction rather price minimize subject budget upper theorem give regret show hold main begin easier suppose pay take bind minimize lemma subject form normalization sequence key price mechanism optimal drawing distribution exceed remain price actually induce want draw price state mechanism mechanism pay arrival pay difference main bind cost give theorem open classic mechanism separation complexity cost know advance every provide zero flip otherwise biased coin usually tail tail coin distinguish require gives extend heterogeneous idea begin benefit expensive expect regret tt mechanism h datum handwritten digit feature digit depict mnist handwritten digit http mnist ask digit digits digits task drawing otherwise therefore misclassifie give implementation descent include train randomly dataset baseline every naive offer every budget knowledge far comparison adjust instead initial future reason use base pick price cost reason na I implementation compatible strictly explore implementation third party mention direction price mechanism agent reveal explore propose contribution pricing hold pricing mechanism interface box depend rely guarantee interest good mechanism one hope direction know marginal correlation unknown broadly motivate set crowdsource consist label mechanism offer price obtain label build importance paradigm acknowledgment discussion participant week theory science foundation recommendation express alone regularizer algorithm randomness choice h eq could quite consider however importance importance let probability clear expectation condition proof outcome algorithm follow regularize convex regret guarantee reference q take side separate q weighting hold implement assume underlie regularizer expect depend expectation choice let th wish choice previous expectation condition equal outcome algorithm thus every pick sequence equal zero otherwise let respect regularize bound upper reproduce regret h come fact since last inequality get complete q probability arrival tune later summation budget constraint take point pay completely goal summation step optimize sequence pick budget choice expectation randomness lagrangian eq imply complementary optimum case simply complementary c budget tight let call valuable complete lipschitz take expectation statement pointwise include advance budget third note extreme specific outcome suffice furthermore budget upper guarantee expect improve beyond pricing knowledge appear upper obtain subtle simplify proof budget decrease constant choice parameter valuable point summation regret let achieve subtle argument derive summation constant plug set may even guarantee case prove summation bound summation give lipschitz eq meanwhile k easily approximate jensen line q line algorithm online every regret coin suffice prove biased tail require coin algorithm coin output tail tail biased coin set budget round coin budget coin coin behavior coin except guess bias correct procedure coin bias everything tail biased coin round coin loss meanwhile expect regret tail inequality hypothesis large small coin actually coin need budget exceed online regret well regret follow still hypothesis coin fix coin point either tail coin irrelevant regret proof order loss either thus theorem mechanism expect constraint eq proceed budget plug term bind constant cost consider draw induced probability amount spend density amount spend arrival arrival budget simplification hand side satisfy difficulty quickly statement split correspond regret use function obtain expect still modification give argument analogously eq pick knowledge jensen plug observation rgb chen mechanism datum hold task model cost challenge past future mechanism classic resource budget robust many guarantee significantly due active analysis cost constraint couple regret order behavioral artificial address edu edu edu edu interest drive generate algorithmic tool accuracy make extent value leverage apparent million competition netflix foundation google ml thorough precise reduce beneficial must marginal paradigm imagine interface sequentially greatly label learn learn cost indeed world task obtain hold interested agent seek hold agent heterogeneous correlate mechanism compatible optimize efficiency inherent question scenario consideration company store patient medical public yet company offer contribute private record heterogeneity patient correlate content medical disease know website order target customer offer social facebook profile customer statistical hypothesis unseen hypothesis member point consist encode predict theory attempt characterize task inherent usually quantity classification difficulty capture vc achieve budget round cost cost offer randomize price long learn select offer minimize limited budget capture randomness depend quantity rough rough similarly resource constraint quantity also simple bound continues require rough stay post predict interact pricing agent price agent transaction agent transaction mechanism output completely design problem mechanism price predictive goal mechanism minimize final input arrival agent actually mechanism agent price reject obtain agent transaction emphasize focused pricing intend develop note implementation straightforward accept transaction could imagine learn disease act patient proceed
htb track subsequent object density tractable filter filter filter recursion close solution filter proposition target time birth next multi fig respectively grow filtering generate component new density component component I predict k h discard small review recursively filter sequentially proposition since direct sum density respectively k j jk k I stage disjoint label birth track factorize depend mutually exclusive predict separate short track run shortest augment set birth track track track generate update l compute rank intuitive drawback truncation survival birth portion component hence computation number rank cubic approximation truncate density involve track birth track track subsection joint update separate prediction procedure involve truncation preserve filter performance instead aim filter one current filtering specifically derive I essence extend include association identical except survival track denote base association follow establish simplicity target multi multi next k filter proposition track birth track measurement j extend k desirable association simple determine without update via eq depict j h h rank optimal use enumeration track valid association stochastic distribution extend proportional corresponding allocate extended association assign difficult solution sampler marginal computed prove generally contribution state element p give q simply state ensure track z conditional pseudo sampler directly rank htb k kk k h z h tt h h algorithm gibbs initialization length roughly distance true one good initialization gibbs either start track sample term sampling length therefore complexity present alg pm fast target gibbs generally assignment update counterpart fair rank tracking probability clutter rate numerical scenario adapt straight duration target origin pair target track velocity target k h process survival b x b p clutter intensity average false fair comparison maximum figure cardinality distance localization htb cardinality equally similarly second sampler except clutter scan performance number hypothesis trial e probability target gibbs expect pick target rank approach fig average time order new conventional construct track require truncation due elimination inefficient intermediate strategy provide superior counterpart method theorem axiom theorem condition exercise paper propose multi bernoulli combine prediction update perform rank assignment truncation drastically superior propose extensive study set filter refer estimating number trajectory drive application lie heart diverse text popular track bp systematic systems foundation development novel filter density filter design computer pr cell sensor scheduling rv zhang label multi lead development object attractive conjugacy family propagate forward filter update operation result shortest component implementation intuitive highly inefficient specifically truncation rank assignment truncation perform separately predict component negligible weight rank assignment inefficient truncation procedure innovation predict original implementation filtering filtering rank admit chain innovation generate component generating advantage first cubic exploiting characteristic background label present prediction gibbs conclude remark section rest g unlabeled one etc space represent use convention kronecker delta argument etc inclusion indicator single dynamical label space define integral incorporation identity object bayes multi suppose target take space finite evolve
computing sift cpu times imagenet descriptor gpu state standard patch gaussians point subset multi correct negative incorrect evaluate roc thresholding feature space six combination test combination denote combination take place train stream deep stream nd nd part follow iii branch branch vi reduce branch branch decision layer consist branch shorthand use follow filter pool layer apply report architecture also briefly conclusion architecture something indicate network network closely importance matching help achieve score sift art interesting try learn pooling utilize pooling stream well come image patch pseudo also experiment tested replace branch descriptor test branch extract descriptor compute distance network descriptor never image patch benchmark number comparison sift bad network filter subset filter fig display filter convolutional part layer filter worth part basically mean learn two patch match patch train choose contain sequence produce pair baseline choose image matching compute produce art descriptor choose cost channel extract patch estimation batch patch result pair point descriptor match distance case branch hold worth computed lot similar mean treat connected layer branch network branch location forward cost would require channel image pair network compute subsequently unary pairwise mrf set grid qualitative mrf visually verify cost network well channel architecture show estimated depth map fine detail exhibit sparse network eliminate quantitative comparison focus plot ground across threshold depth six threshold pixel plot pixel across threshold curve winner take mrf optimization local descriptor consist change gradually technique patch input size factor context descriptor branch extract patch test cnn architecture boost suggest great architecture compare patch conclusion already simply good distance outperform imagenet raw form cnn adapt extremely problem show precision average transformation precision curve material lack among architecture superior far accelerate architecture stream resolution turn significant boost importance compare consistently quality result far actually consider paris est fr manually compare fundamental opt appearance study architecture specifically adapt outperform ec project fp across fundamental vision subroutine variety range structure motion matching building image characteristic two factor final appearance include viewpoint variation illumination camera fact need rise many descriptor sift huge vision manually design descriptor unable aforementioned appearance patch gain access generate software question automatically aim generate patch manually design instead directly annotate inspire advance deep fig interested address explore architecture exhibit advantage train raw patch match non database readily contribution manually implicitly wide baseline illumination variety neural model network benchmark dataset show significantly art descriptor manually design learn efficient descriptor sift recently method descriptor descriptor pooling dimensionality recent involve various performance convolutional descriptor imagenet dataset show convolutional descriptor sift derive sift imagenet consist narrow broad appearance include baseline already mention neural image impose limitation colour patch increase however state exist choose patch exception describe patch give size may dimension several way architecture branch adjusted training flexibility much network next hand maintain notion descriptor simply two patch input image directly feed convolutional layer convolutional relu pooling layer give module consist fully provide compare two patch test patch break convolutional layer x relu suppose increase inside decision discriminative might observe technique architecture turn consist layer layer relu activation shall also contribute accordance architecture enable central resolution stream receive resolution patch low resolution receive original stream process use architecture describe use make stream improve match central patch twice high resolution stream implicitly put help
ap schedule throughput discount indicator define true inform state mdp practice state ap energy accordingly appropriate decision node active mdp extend state characterize action node active node belief whole system state ts otherwise unbounded space mdp section schedule certain simultaneously active ts ts state ts equipped transmission simultaneously rf transmission state node correspondence node normalise rewrite normalise current scheduling ts since active ts evolve belief monotonically increase proof belief ts belief state sake clarity throughput ts ap interested scheduling accord throughput optimization bellman function sum respectively expect instantaneous mp belief mp rr ts send bottom list ts monotonicity denote permutation contain node position denote permutation vector vector concatenation operator solely ts node accord mp ts belief mp mp permutation position position position position symmetric monotonically j ff boundedness throughput bound regular pseudo symmetric swap use decrease order mp monotonically map apply study structure ts mp ts lemma show difference different difference value differ certain mp optimal ts ts ts swap need scheduling mp mp value contain node order former kn optimality pseudo particular belief position value element monotonically un lemma function correspond ts swap q map p induction already mp assume mp optimal ts mp ts ts lemma hold prove complete study throughput state optimality study reward probability lost immediately energy available transmission ts expect energy available transmission node belief probability monotonically affine node schedule discount throughput pseudo denote node process node function order change function value pseudo swap j mp process node send queue queue monotonicity mp continue scheduling non note argument prove mp swap w backward provide mp attempt channel mp prove perfect channel error I channel detection optimality horizon discount horizon discount upper horizon problem horizon reward study show bound scheduling ts scheduling exactly ts relax impose ts relax belief steady probability ts throughput hence bound maximum node impose therefore solve polynomial section numerically schedule mp case rr cyclic policy regardless history schedule average throughput repetition unless state channel ts node period horizon case figure throughput channel throughput increase number increase throughput value mp curve effect throughput clearly occur throughput channel reduce throughput quickly capacity high performance scheduling policy h figure throughput different amount energy notably mp policy ts throughput high throughput fall due state low reliably action mp scenario mp scheduling access system model optimality node energy simultaneously process state suggest scheduling require scheduling schedule wireless sensor device storage node decrease p p p belief n nb monotonically nb np z nb mb z nb first increase induction respectively first hold regularity note distinguish rs k rs k boundedness belief km ks w j rs first similar node state must ts distinguish correspond case kn respectively use induction summation equal boundedness p r j n j I either w j un follow immediate reward ts belief ts belief belief km element belief value belief pseudo function pseudo value function equal ks value ns ns ns ns un j un j w p un j p permutation hold combination belief belief ts boundedness lemma monotonically increase series induction trivially realization four since case remain denote hand definition conclude n n p l l p k p w case I pseudo side linearity need ns ns remain swap induction receive universit ph college centre wireless college thesis european center carry universit cover device cognitive making receive ph electrical school research university stanford university electrical department associate university transaction communication student college center co conference co european school interest theory theory emphasis joint energy communication nd uk wireless equip device study ts probability availability channel arrival model ts ts node ap scheduling policy throughput study armed bandit scheduling mp compare numerically multi access scheduling observable armed wireless network machine sensor energy technology extend constrain energy availability energy scheduling policy wireless wireless network offline availability online assumed realization communication decision mdp mdp dynamic numerically many state mdps prohibitive much scheduling order avoid important characterize behaviour scheduling case knowledge statistical scheduling learn scheduling wireless ap equip device slot ts probability availability model ts
optimum expect insensitive limiting value note optimum panel maximize different value essentially threshold concern negative appendix value panel well alg nevertheless believe general alg interestingly measurement suppose estimate estimating sign provide experimental validate recovery sign measurement alg small coordinate assign alg sign result report scan predict theoretical alg top coordinate even measurement experimental setting time report medium error bit bit scheme multi threshold parameter bit reader variance actually variance example I value much estimator bit far reduce sensitive threshold alg estimate harmonic label measurement well compressed hardware sign reduction transmission retrieval current satisfactory require measurement decode scan sense use tailed design bit provably fast current focus nice relax recent direction n let l convenience ij I ij term q computer science nj usa stable projection compressed sign measurement stable projection become measurement reduction propose decode conservative sign experiment relate projection accurate recovery technical bit stable distribution cs topic research mathematic engineering design sense sense e stable distribution g signal pursuit omp develop compressed sense study literature compress many hardware sign transmission appear bit sense accomplish goal show length decode desirable scan coordinate projection scan effective conservative well stable norm stream cs signal stream fashion signal advantage I scan extremely heavy tailed nature recovery procedure significant recover transition vanish major disadvantage heavy tailed require storage substantial follow first reader want excellent book cauchy summarize scan sign signal generate formula measurement coordinate algorithm reasonably study assume reliably measurement sec bit alg paper simplify analysis practitioner burden simple report coordinate increase measurement sign intuition reader interested performance please eq probability
within domain dimensionality transform represent dimension zero represent augment particular correlation cross classical correlation cross method computing section error cross cv error explanatory regression inference matrix sample second matching success weight web small source validation generalize unknown observe appropriate regularization denote true fitting true cv rescale properly value compute cv numerical show error accurately diagonal column sum vector example similarly match matching rewrite respect weight say correlation although explicitly throughout generally well error find minimize symbol distinction thus k strong extra uncorrelated avoid laplacian spectral introduce regularization stability regularize properly nonnegative write matrix optimization denote cholesky eigenvector characterize part eigenvalue correlation part k correlation type correlation explain hold factor b k kb unweighted rescaling define consider section consider theory section match arbitrary weight omit notation omit although matching actually depend fit let proportional comparable adjust split repeat several matching cross error formal error weight weight trial success symmetry match cv fit cv compute several example time replace rescale k elaborate account structural convert grid add randomly element nonzero triangle mean domain diagonal proportional adjust grid scatter recover look match fig matching become observation appropriate choice choose value look choice agree circle plot rescale constant rescaling right hand change term fine evaluation column total nonzero assume x ik n ik ik x op op assumption describe eigenvalue zero negative however observe hold accurately look cause expansion p element greater express p define respect ij bias eq eq also fit cross change two equation p part pi ii ii p ignore extract part equation replace substitute substitute p substitute simplify look also write g jj ki jj ji ji ji jj jj ji ji ii get let k k ij jk b k kk ik k kk kk c kk kk rearrange formula last ij ik jk jk jk first lm lm w lm lm w lm summation follow lm lm f lm lm lm lm lm lm order role play give pe rewrite ik g ik ik ik kk kk kk note note derive expansion like omit define describe express lm w lm lm lm lm lm ij lm lm lm lm lm f lm lm f lm lm lm lm particular expression give use eq ij finally take w lm g lm kk h lm kk lm kk lm jk h lm jk lm h lm jk compare acknowledgment discussion grid mm pair nonnegative matching define transform matching matching matching weight
closeness input file likewise class name super noun use string string al information word specific mention string string appear versus string vector name insensitive mistake name instead word type mapping class uniquely opposite multiple string distance two string class corpus distributional evaluate occurrence semantic context define class usage argument detailed table observe repository distributional similarity previously describe corpus similarity occurrence contexts repository distributional similarity use entropy p pass occurrence definition would method count occurrence method occurrence statement size would count occurrence mean one purpose relation way define language normally common package organize infer belong structure hierarchy extend implement type package relation class within since extend implement website question user extract nlp package study text exclude stanford noun conjunction dependency total librarie code repository source datum source file interface package repository abstract feature sim string sim code sim refer software name additional aspect commonly noun attempt select mutual pair appear rarely corpus hard noun score noun highlight bold font see software frequent variable name method name class repository class contrast term sim type display report cross coordinate corpus distributional similarity corpus code note significantly corpu baseline accuracy code report feature code feature distributional similarity extract conjunction additional manual label similar coordinate term classifier corpus predict classifier predict pair full classifier whereas code text table top ten coordinate successful top hierarchy class indeed top label match pair class package belong code common exact visualize aggregate entity determine edge color entity determine centrality degree community indicate note prediction highlight connection group package one highlight within package basic represent simple entity map implementation library code extract usage class location corpus distributional achieving prediction add entity software f label predict build code connection common usage department coordinate term entity refer examine technical domain entity suggest couple lead improve statistic context appear validation dataset dramatically predict pair discover semantic text critical system variety nlp application temporal parsing examine semantic thing share et normally compare corpus statistic associate entity entity appear context domain world object name plausible nlp statistic couple world lead discovery discovery entity potentially corpus website user answer question development label coordinate attempt pair distributional entity library code distributional similarity additionally base repository able calculate organization demonstrate cross accuracy accord human high predict towards software entity software method application language domain capability application software domain token visualization similar work semantic relation discovery discovery approach certain lexical relationship pattern analogy relation second approach contrast approach normally recall extract language mapping sentence include physical world entire align supervision relative rich complex software code constrain discovery software adapt software enhance task
use first fitting practice typically thousand comparison less experimentally favorable balance analysis machine favorable function tensor nonconvex empirically sufficiently structure tm robust efficient require favorable tradeoff polynomial network introduce polynomial function draw unknown formulate reproduce polynomial specify numerical precision couple approximate underlie kernel nystr om approximations incomplete cholesky fall inherent namely knowledge target exploit target significantly small remain another supervised rely modeling consist recall nonzero exponentially al attempt relevant attempt learn sparse online select form next fitting polynomial computational cost use neural network guarantee learn layer reach predictor careful al cubic manner cubic class machine relationship factorization coordinate success recommendation application linear size length homogeneous polynomial represent polynomial involve contain power high drawback machine evaluate fm operation computational claim order order use approach polynomial exploit concentration directly dimensional random approach polynomial homogeneous polynomial sign map polynomial hashing transform feature outperform require large combine either approach motivate machine section polynomial degree polynomial machine correspond decomposition recall array inner tensor treat rr comprise polynomial equivalence homogeneous us attack tensor factorization rank feature coefficient predictor correspond tensor span tensor factorization predictor two comprise define vector coefficient implicitly search low target polynomial machine attempt low rank drawback try ensure represent machine impose polynomial machine propose instead fit machine low tensor since obtain minimizer proxy kernel machine objective machine couple machine empirical observe machine expect risk indicate efficient locally optimal avoid variant tensor machine equivalent formulation fit tensor machine measure noise rademacher take theorem train datum observe risk risk rademacher main rademacher tensor grow converge optimal class rank constrain lie accord surely rademacher rademacher spectral tensor proof world dataset demonstrate solver bfgs provide mark schmidt investigate use quasi newton solver use provide fit tensor tm tm respectively influence tm accordingly initial well set batch tm contain choose time reasonable original matlab conduct intel processor ram nonlinear algorithm publicly characterization basic feature euclidean ground slice binary forest govern interact iteration tm epoch tm layer width major record running mean performance tm batch dataset evaluate machine fitting fm available unclear plot algorithm relative median third detail tm across reliable tm converge tm significantly machine might expect nature fast algorithm tm batch tm low h ccccc tm batch tm learner year census forest rna framework solver tm model require fit solver long parameter error tensor machine tm batch tm census fm well small slice relatively factor census interestingly census lead census error assess vs slice determine vary tm batch tm considerably pattern mini update dataset explore tm comprise pair item website subsequently visit classify point remainder also fix epoch table classification grow figure tm give tm second evolve target tm decay theorem constrain rademacher satisfie proof straightforward calculation establish complexity specify structural theorem definition drop subscript rademacher euclidean slice replace sum
gaussian divergence flow optimization covariance derivative rewrite q simplify solution consider vector unit direction write solve must term three fortunately analytically quadratic formula yield put optimal four solution root choose situation choose identity root ensuring full belief next conclude optimal obtain component spherical begin note form scalar univariate align translate axis paper probabilistic dynamically incorporate stochastic arbitrary probabilistic maintain belief weight unlike incorporate small evaluation thompson computationally tractable transform flow optimally update theoretic principle intelligence cope even datum stream bioinformatic problem rich computational minima overfitte work formulate online cast enjoy guarantee filtering belief optimal overfitte prohibitive belief prescribe calculate take specify velocity field track answer mid conservative measurement prediction integrate parameter uncertainty update prohibitive well unknown possibly exploitation trade precisely state art method illustrate modelling flow field close discard belief input round belief parameter round vector consider regression network specify weight different layer supervise provide minimize predict rate vary perceptron need offset advantage flow flow flow belief shifted scale utilize leibl mid entropy principle show close form let transform vector mean respectively express span unitary equal eq small choose select find act subspace flow diagonal spherical diagonal unitary match posterior follow case independently constraint pick hyperparameter deviation preserve form scalar unitary align update factor previously produce landscape desirable flow flow transformation small typical spherical unconstrained covariance unconstraine described numerically maintain positive way predefine minimum training calculate calculate flow correction vertical transformation rotation subspace sample contrast translate basis independently spherical act radial rotation full unchanged diagonal flow force shift converge update intuition briefly trivial update show sample weight center move decrease occur first finally move correspond diagonal third flow regressor importantly sgd effect regularization sgd regime consider application flow binary linearly dataset model parameter sigmoid binary kl become sgd diagonal flow spherical compare draw isotropic assign true train label online test round predict give noisy update report mistake differential entropy online sgd outperform single optimum flow entropy equilibria invariant gradient model flow several classification dataset sgd bayesian langevin dynamic exception combine margin regressor input model sigmoid generalization mistake pass average remain additionally test inverting set dataset instance characteristic classify uci contain categorical expand forest instance classify take winner balanced feature eeg discriminate census aim exceed eeg sgd give good regularization avoid heavy scheduling draw keep choose eeg experimental classification within fall exploratory beginning outperform generalization compare come regime monte feedforward belief flow sgd aim choose avoid architecture test well digit basic plus background patch black image digit split dataset error plain sgd dropout sgd dropout pass choose unit update sigmoid gradient kullback output example either sgd dropout throughout online discarding average plain build minima attain flow difficult online
assume moreover heterogeneous address ca partition context match define set corner locate origin hypercube estimate matching relevance contexts hypercube hypercube tradeoff hypercube hypercube locate assign tn tn p ia I I ia ia ir ia ib ir I jx jt j htb x ix n kt kx k htb pt j j p pt j I j operate user phase another content relevance select score exploitation choose relevance score minus another request score user number minus score user time confidence content user user training come contrast help build accurate relevance score exploit current keep function describe time content training phase let context hypercube request content keep estimate phase counter update number content exploration exploitation identify choose give exploration content content second high training high exploration exploitation minimize time describe increase call value affect accuracy hence phase randomly content explore identify candidate similar hence balance possible reward report pt j empty select empty imply hence action set empty exploration randomly select request explore exploitation select matching relevance score matching collect exploration exploitation relevance score time take time context union request relevance match set account cost net relevance minus maximizer randomly select feedback I operate let content explored explore user exploit content user I regret l constant exponent horizon hypercube context phase content hypercube except phase phase contexts exploitation phase train train explore ca integer sublinear averaged increase context similar give final time call divide round length instance instance begin round modify overall user user almost parameter appendix technical user particularly provide service user quality recommendation low type user analyze feedback content feedback unknown give feedback large case feedback available algorithm reveal eq p proof report miss however user real online distribute mining spatially pattern uniform intuitively loss choosing minimize large carefully distribute begin differently adaptively estimate arm single hypercube context fine region explore focus space ball distribute ball context structured address basically learner learner space exploration exploitation learner principle hypercube hypercube denote contain mutually exclusive active depend time hypercube learner activate hypercube divide context explore exploit arm exploration control arm exploration exploit arm reward arm different function inaccurate reward learner expect reward reward learner learner train classify keep learner learner send activate hypercube learner train learner enough always therefore reward explore update select arm k tt reward exploitation step activation different scenario learner case train learner much high hypercube active let high hypercube high active hypercube hypercube arrival regret due hypercube hypercube q lemma whenever hypercube suboptimal arm give suboptimal set suboptimal hypercube hypercube regret suboptimal exploitation e omit outcome exploit arm choose event suboptimal suboptimal th equal chernoff hoeffding active hypercube combine hypercube choose hypercube hypercube hypercube exploitation hypercube suboptimal exploitation eq remain optimal action determine much hypercube lemma bind level hypercube lemma level tight desire regret bind form arrival worst bad minimum case locate inside hypercube learner learner correlation learner learner characterize extreme time intuitive q order intuitive arrival form context adaptively time order start split case arrival start begin regret order regret large case parameter problem regret logarithmic difference c arm active hypercube level hypercube inactive hypercube active small activate activate arrival memory estimate currently active activate reality memory hypercube arrival single hypercube requirement require trick multiply order content score context context exist speed change content call capture drift learner learn dynamically change observation relevance group round window round keep separate relevance context window round round w round passive call sub sub round overlap fig round round passive sub round round operate begin modify base passive learn active round form horizon stability assumption use action round sub round matching accord content matching mean relevance passive matching take fig result relevance round sub round start already depend passive exploiting whereas spend start sub past observation context current relevance assume past current length impossible sublinear regret theorem run denote large stability time proof appendix online decay round relevance action suboptimal matching decrease round propose cost choosing affect news article yahoo page compose recommend content iii click relevance equal recommended content assign content also access network content divide hence process different number click click match contain contextual rating student randomly arrive content music rating music reveal dataset scenario content randomly click next news article day news article stay popular several content implement centralized run select content user run control divide md divided provide service exploit performance user algorithm compute matching assume select high history observation explore good matching action probability adaptively create ball action index ball action ball high context context simulate set value find range well see differential service context achieve learn algorithm percentage exploitation phase different simulation expect increase action explore accuracy exploitation explore hybrid table avg length equal single parameter value click value good average click em hybrid service characterize learn match relevance characteristic content suboptimal content dynamic depend approximately learn match content another user increase immediate chance user switch denote logarithm hypercube p center symmetry hypercube ip matching action hypercube relate suboptimal time hypercube sum regret regret matching lemma slot exploration slot content match ca sum realize since I first process sample reward good sample bind use artificial event suboptimal content th time exploitation phase select suboptimal exploitation otherwise lemma bound match suboptimal select yield outcome need run parameter pt pt last upper running similarly inequality ca time choose suboptimal probability result near upper next come sum bound lemma q difference expect hypercube imply slot one suboptimal near action miss feedback matching lemma hold estimate accurate exploit exploration content match feedback feedback denote number training user binomial feedback therefore basic due relevance regret due relevance round balanced omit time set suboptimal eq net slot incur show regret show due suboptimal matching characteristic due matching regret sum r r electrical engineering university california receive engineer middle east degree electrical ph university ann research interest arm problem mine receive university electrical fellowship award van van electrical engineering california interest network theory online communication distinguish communication transaction journal topic signal paper award transaction circuit system technology award cite award communication journal circuit system award compression streaming international hold lemma van diversity content source news medium etc diverse content content numerous key challenge accurately content evolve preference propose aggregation demand content content characteristic preference contexts content aggregation priori online match bound speed importantly operate efficiently feedback preference evolve illustrative highlight system distribute online armed bandit web tv video news aggregation generate source interest often responsible mining numerous source find evolve source request request content user characterize value preference assume arrive sequentially requests either another connect gender query content etc device phone match suitable content source content change match learn matching explore content match help learn characteristic content source call content application business news aggregation business collect source recommendation specific need music music content instance music facilitate share music collection user g model framework test dataset aggregation music type direct indirect visit website indirect user request content receive indirect achieve match current I obtain request trivial collect way vast user type dynamically user content content content jointly content build bandit notion content matching give characteristic user content characteristic sublinear regret preference characteristic slowly change achieve change characteristic remainder organize highlight difference describe decentralized content aggregation content matching scheme regret optimal scheme unknown static characteristic distribute analysis user numerical online conclude recommender armed bandit recommender characteristic user maximize recommender learn preference online base true relevance user propose relevance score moreover due nature online phase phase account link long run characteristic online learning centralize recommender preference characteristic item exploit past recommend content collaborative similarity user examine user user high relevance relevant content match content prior know drift distribution concept drift concept drift take account speed drift window concept drift deal ad manner dynamic period popularity may type popularity due event certain type popular certain gender shift case content recommend ask another another payment make display incur website payment token assume payment return content mostly ask justify whenever user obtain benefit content recommend ca popularity increase pool content strategy
mu claim mu mu mu mu therefore mu mu mu mu mu mu mu mu z gradient regularity satisfy function lipschitz probability curvature convexity analogous cauchy schwarz nd line term cauchy line third term put inequality q curvature combine obtain q q f freedom bind expand q calculation show proof prove let respectively minimize regularization minimizer affine constraint q auxiliary equality augment lagrangian spectral norm ball otherwise rewrite transformation I ia variable form decomposition fact update iterate singular soft multiplier converge I corollary definition descent semidefinite semidefinite method mathematics signal arise derive relaxation combinatorial promise runtime current handle problem thus considerable scalable semidefinite programming relate family nonconvex program surprising effectiveness classical explore recent area signal process classification gradient stochastic optimization lead remarkably efficient attack large problem build closely program linear optimum rank find matrix interest generalization sense optimization np hard alternate isometry rip least semidefinite ii n orthogonal solve addition applicability affine connect semidefinite decompose squared residual take phase retrieval develop descent optimize initialization contribution concern descent recover probability bind potentially carry alternative review detail analytical result proof experimental relate present semidefinite rank descent invertible minimizer minimizer hence nuclear norm next sdp coincide nk isometry small furthermore attain word find satisfy affine semidefinite constraint minimum rank solution positive semidefinite coincide ignore comparison nuclear nontrivial affine rank replace norm already nontrivial minimization perform singular thresholding expense prevent problem recently propose project eq bx rx rip heuristic suffer subsequent propose alternate square algorithm avoid svd factor rx uv uv b uv iterates linearly converge considerable semidefinite scalable provably convergent exploit goal vector magnitude author global phase value see illustration method generalization flow turn initialization course present initialization spectral start unbiased fact z sm z v yield rate converge gradient hold nevertheless local achieve subsequent constant replace tb minimizer gradient tb I kk main sketch matrix note define distance recovery gradually denote ratio exist probability proof supplementary material step give regularity enough property mu mu show close iterate condition event small specify expectation regularity next suppose z z v eq finally valid sufficiently nu constant r n I number scale conjecture could experiment relaxation justify observation use admm approach simplicity scheme could compute partial ghz intel core per three affine nuclear svd update compute multiply vector partial require gradient via gradient dense affine transformation dominate remove overhead cause small dominate enjoy low conduct slow generate take frobenius define f approach regularization fast three three scenario general entry nuclear step value choose figure nuclear significantly tb ix recover refer successful trial transition rank confirm linearly connect case program constraint recently retrieval develop descent procedure conjecture sufficient condition significantly broadly technique semidefinite
good variable produce parsimonious absolute much attention year xu lasso least regularize solve become initial lasso yield level lasso bold letter face capital face letter denote transpose represent absolute design vector component vector result minimum notational indicate variate mean centrality follow identity stand respect indicator distribution restrict shrink study analyze application error risk level information depend assume effect non sample priori restriction test vector imply restriction linearly restriction known theoretical consideration tested eliminate redundancy description see analogy ol estimator small risk may bias inconsistent plausible follow preliminary take acceptance rejection critical value distribution proposal upon use test base computation everything easier highly level nature simplify making shrink toward major shrink double shrinking consequently combine stein shrinkage stein stein shrinkage fix equivalent investigate respective performance cc restriction rather many situation e q bias mse weight distributional k rl nz c hz rl c hz hz th h value chi f hz th hz class local rl qx q c c h ols lasso estimator setup non chi h n rl accord estimator proof follow w procedure generate assertion dominate large relative increase present decrease nan c ccc ccc htbp ccc propose estimator al study examine specific clinical measure receive description variable description remark specific response age seminal percent center second restrict lasso preliminary stein shrinkage sl shrinkage variable table min estimator average fold validation fold divide datum subset predict version specification repeat show error sd estimator construct decrease confirm visually demonstrate variability estimator paper impose restriction preliminary shrinkage preliminary lasso stein type sub size propose improve dimensional situation estimator analytically numerically configuration error classical centrality degree misspecification vary dominate perform get efficiency near present analyze validation deviation shrinkage stein dominate error sense conclude lasso estimator fu know th q vi iii obvious obvious fu making prove iv follow immediate proving eq implement fashion fact algebra bias significance ann l instability selection fan j penalize zhang zhang generalized statistics ann ridge estimator j study united j prediction york pt orthogonal fu asymptotic ridge subset ed usa test stein united statistics introduction integrate pt text york j diagnosis ii journal xu iterative ann regularization variable elastic axiom claim conclusion exercise theorem remark summary school school mathematic statistics university abstract suppose interest relatively respect absolute operator suggest restrict lasso class selection restrict performance propose estimator risk show double shrink lasso word double preliminary stein shrinkage primary draw regressor response vector goal response ordinary square ols tx standard estimate ol regression unbiased large coefficient reduce subset component regression scad net fan li variable objective intuitive large exist cope selection backward elimination method derive
use former repeat pointwise optimal limit eq ahead bellman prove number leave side repetition contradict complete control convergence ahead establish engineering south school technology rapid city email theorem corollary optimal dynamic horizon cost investigate control problem optimality follow uniqueness solution bellman furthermore iteration result ahead study pi scheme sometimes dynamic program value vi pi load per iteration vi pi remains adapt control convergence analyse pi control viewpoint two establish pi compare implementation variation pi know denote positive integer dimension number positive denote select I minimize minimize control policy word hx within initial select domain compact policy asymptotically define boundedness meet select continuity bind compact value admissible within conclusion value policy admissible trajectory convergence give two control admissible bellman nonlinear policy form approximate origin start respective eqs answer two lemma admissible eq evaluate bellman equation bellman contradiction contradict step policy optimal solution show pi equation former lead eq pointwise decrease value converge limit policy pi hence bellman per theorem everywhere admissible give requirement asymptotically respective pass vi establish former monotonicity state trajectory utility order need vi pi conduct guess former merge vi find reference guess subject evolution vi admissible policy guess vi pi seem similar load vi significantly pi vi pi eq pi vi aim analytical admissible converge q inequality monotonically therefore eq assume monotonicity claim prove
report respect one cg yield iteration infer start concrete possible proposal hasting mh algorithm scalability gp apply concrete allow initial acceptance ten run mean deviation line mh cg threshold step size start gradient batch computation decide fix advantage solution initialize system propose finally hardware parallelism even capable quantification gp without impose quantification primary interest gps calibration computer model report direction langevin scaling ideally scale gradient fisher compute unable due gradient despite demonstrate distribution covariance relevance determination covariance possibility extend gp g gp gps spatio temporal aspect calculation algorithm improve present acknowledgment mf quantification uncertainty accurately langevin distribution negligible need marginal advantage gradient solve system unbiased unbiased enable scalable sense quantification impose number gps offer possibility datum function underlie behavior quantification primary interest necessary accurately argue carlo involve store complexity set gp covariance matrix e space compute common several machine approximation usually subset application gps name extent approximation quantification uncertainty application regression quantification primary interest avoid propose langevin draw require computation solve iterative conjugate cg product idea put optimize despite complexity complexity solve slow practical compare speed fast yield gain large unbiased algorithm stop highlight batch alternative contribution scale batch dependent complementary aim rather iii complementary approach noisy likelihood gp regression cg obtain determinant remain calculation log likelihood far transform unbiased unbiased likelihood employ carry inference parameter gps computation day parameter hardware knowledge attempt enable full uncertainty reduce vector impose gps motivate infer gp variant present gps demonstrate methodology gps marginal conclusion employ function determine covariance whereas marginal comprise noise bayesian forward label require q posterior analytically necessary resort approximation tackle approximate sample draw mechanism accept reject log cholesky cost require inverse multiply computational complexity scalability gps unbiased log space practical determinant obtain negligible bias briefly describe adapt gps idea stochastic gaussian way transition langevin parameter except stochastic optimization local transition langevin proposal require reach langevin enough acceptance therefore possible avoid introduce negligible langevin phase monitor gradient define gradient eq langevin produce motivation log marginal require q give yield unbiased version consideration methodology involve solve much easy solve conjugate cg popular carry without store complexity variant speed computation section regression apply concrete datum uci repository compressive strength concrete describe tb threshold number iteration system cg algorithm idea calculate solution minimizer employing initialize iteration refine cg gradient characterize orthogonality conjugacy namely mutually orthogonal iteration remarkably cg implement trade accuracy theoretically cg lose condition cg take sample gamma rate distribution reasonably number encounter inference gp covariance slow numerical quantifying extent apply gps impact solve need calculation sake brevity efficient accuracy show cg versus versus obtain algorithm back cg double cg implement precision iii relative absolute tolerance parameter calculation compare large double calculation offer low lead achieve draw conclusion whether implementation hardware order would cg converge would reduce solution would cg convergence cg yield possibility approximate expensive propose introduce solve cg well make convergence speed inner considerably necessity cg need fig accuracy version single version reduce standard cg iteration count inner inner cg gain iteration different experimental might
rate fully local focus global sample turn provide guarantee alternate iteratively keep optimize require rip incoherence establish geometric formulation convergence version gradient batch full heavy burden stream random initialization fast computationally update lastly manifold include optima conjugate trust name instead rate analyze output understand reliable empirically operate manifold giving empirically initial slow local angle metric principal close evaluation offer angle angle u discrepancy discrepancy column zero one angle cosine zero frobenius discrepancy measure discrepancy span subspace angle angle discrepancy entire sum square start principal angle accomplish iteration come represent state rely novel initial random initialization theorem improve behavior reach reach isotropic iteration enough without figure though identically admit tight leave second require though experimentally behind version rate dependent regard improvement per state theorem phase uncorrelated imply tu tu u determinant strictly analyze prove determinant increase toward non attract stationary global problem point strictly great determinant monotonically stay away convergence norm discrepancy proof theorem standard gaussian use theorem may guarantee tu bind discuss support rank subspace sample uncorrelated identically distribute recover incremental constrained take manifold justified section scenario norm slowly minimum frobenius must create orthonormal vector choose initialize subspace characteristic initialization iid bernoulli quickly low determinant fig htbp fig provide tight fig expect rate optimizer expectation situation tight average trial run initialization iid reach compute corresponding take entry noiseless empirically small noise illustrate convergence generate low datum orthonormal coefficient bernoulli fig finally factor recover fm w subspace unitary matrix unitary simplify quantity reference lemma entry zero identical discrepancy share take result strong monotonic frobenius construct orthogonal matrix span give orthogonal complement unchanged multiplying allow follow thus replace first leave frobenius orthogonal determined term complete equation difference maximize take span prescribe incremental globally improvement minimum sum perfect justification analysis motivate seek future expect theorem rewrite let follow ready finish main simplified expectation side consider random determinant discrepancy discrepancy prove increase determinant novel initialization recall define orthonormal matrix gram exist pick jensen lemma complete reasoning give theorem expectation nonnegative incremental zero mean uncorrelated distribute happen phase loose number iteration require local region minimizer many reach trial monotonicity tight require accuracy hope deep understand proposition context success rank factorization instance scenario seek natural incremental initialization also span neighborhood match tool process numerical method impose orthogonality constraint computationally algorithm attempt speed actually therefore guarantee svd convex several algorithm solve expand publish kind gradient convex factorization algorithm gradient suited problem regularizer svd contribution incremental dimensional subspace special factorization orthogonality span algorithm incremental rate part local minimizer match convergence application stream subspace track medical communication environmental science extensive discussion variety
limit version sequence convergent point van convergent recursive nonzero digit expansion interior recursive convergent recursive base condition hold cell condition convergent without generality convergent recursive representation digital via fold potentially space even copy index complement denote consecutive rule enough context make split integer volume recursive volume geometric volume cell net nest recursive split volume set weak geometric half interval enough point weak net eq q maps convergent recursive preserve section lebesgue measure end lebesgue convergent recursive split nb c aa thus equality n ia proof extend multidimensional uniformity let convergent basis transformation base uniform preserve element go geometric net require digital convergent transformation let nf adapt wavelet recursively represent explain depend f sf sf fold integral leave unchanged version haar wavelet adapt base split cell turn product f kf lebesgue theorem converge convergent everywhere function unlike nonetheless know multidimensional recursive basis narrow entire study average net digital map split base proposition simplify property elementary orthogonality property define must correspond plain improvement plain carlo net put coefficient particular otherwise give smoothness rectangular take leave unchanged extension series rectangular bounding point shorthand later use interior anchor set set figure integrable anchor mf jj convention version calculus circular disk center anchor rectangular origin exclusive center diagonal region extension anchor restrict account appear make extension appropriate extension mixed interior form ignore useful fundamental calculus though partial continuous everywhere move arbitrary region let might eq convenience replace subscript keep substitute introduce rewrite q vanish assume empty fail non interior region like include domain great multi continuous extension function non interior use domain one require boundary measure fundamental apply induction suppose hold shall fix therefore induction continuous apply integral integral induction complete hand side certain smoothness require extension variance product section recursive ss u jj u convergent recursive diameter define extension close boundary measure regardless whether place alternatively supremum essential supremum argument smoothness either know integral smooth index depend make put write equation diameter box finally put large ratio rectangular similarly cell parallel angle triangle condition therefore algebra j consideration point randomize base convergent gain net zero regardless whether know may suppose substitution test see get plug desire result integration fold root square rmse plain might attain advantage net net attain really size effective compose well use smooth component move triangle however call induce infinite uniform result notably demand limit average nest uniform stochastically square discrepancy split could relax deterministic construction fail weak net net proved extend nothing split never dimension become extent nothing retain diameter split assumption develop integration accuracy plain triangle product space may numerical product space special interest point transformation net recursive partition integral unbiased smoothness fold quasi quasi unit cube region cube space plain unfortunately composition smoothness integration product triangle triangle triangle reach incorporate shape relative lattice like construction attain discrepancy van generalize construction unit replace van sequence digital net digital net order quadrature compare survey randomize graphic transformation map throughout whenever integrate plain monte finite net uniformly digital net variance via net primary behave integration allow dimension background material net present geometric split van construction product present product domain domain rectangular domain rectangular domain latter result compare plain net dimension conclude tensor simplex constructive ordinary case plain number root variance improve usually uniformity via discrepancy q dimensional sense account numerous construction interest construction net definition integer precisely net discrepancy box approximated box digital net base subsequence base integer digital net handle half track triangle valid prefer zero point split basis digit split panel label new new right look algebraic description case describe describe digit rule obtain multiply operate plane recursive fold split collection exactly member call say level member recursive split split need cell split cell level split figure arbitrarily result version van appearance level latter similar transformation triangle yield get shape triangular set split disk disk limit angle split angle disk
map partition principle mechanic cell product division imply reverse evolution sensor property conjugate transpose related arbitrary index existence derive equation possibility single illustrate possibility partition orthonormal formalism follow row integer moderate number make become consequently partition obtain eq example present music estimator sufficiently besides resolve closely operator pp e j life formalism quasi rao available extended operator geometry long consecutive half sensor spectra rao ideal spectrum detect ideal spectrum side fourier transform superposition magnitude present result eigen briefly describe eigenvalue rotation partition element array characterize invariance input create partition array subspace ne u computer verify performance consider source frequency sensor spaced total array angular limit identically process possibility due possibility take operator snr peak sharp operator easy lack peak explore matrix coefficient square spectra vary last spectrum music music last fluctuation paper source coherent drawback know hence eigen paper coherent high generate distinct version resolution operator angular position source formalism mechanic channel diversity suitable array elementary order prove power computer implement simulation matlab operator r r rp p r n rp p thm element set compute angle arrival moderate give possibility dependent elementary keyword array angular spectrum resolution possibility next third section elaborate description use concentrated negligible acquisition locate region direction permit linearly noise medium model ergodic process receive instant angle operator half signal additive model matrix array lot characteristic focused angle arrival decompose four channel second subspace equation presence variation due presence power diagonal block give situation suppose array comprise hundred question operator
denote transform model integrated average underlie wind window smoother characterize coherence illustrate link process coherence weakly continuous accord unity linear average theoretical use process introduce move case convolution square integrable separable convolution convolution suggest attain nontrivial coherence framework notion coherence examine random spectral define possibly define g interpretation gain phase process function asymmetric covariance shift angle exploratory suggest exhibit visually assess amount multivariate value matrix real cross spectral gain testing spectral give develop ern marginally describe mat ern functions mat ern class ern impose ij valid mat ern popular continuously path interpretation index act control mat ern interpretation analogous interpretation link behavior interpretation function mat ern correlation simplify covariance ern note constant imply linear band great seem suggest interpretation smoothness amount serve illustrate function common perhaps suggest parameter induce coherence parameter particular examine distinct small high behavior non coherence complementary share illustrate coherence concern cross yield flexibility parsimonious ern imposing produce inferior fit version mat ern class coherence bivariate parsimonious mat ern constant close empirical illustration random field realization high bivariate mat ern spaced grid low pass pass filter low pass filter bivariate bivariate mat ern range smoothness low b pass panel suggest complementary cross coherence low high exhibit positively coefficient panel pairwise filter contour e mat ern linear compete build combination univariate matrix strength dependency uncorrelated uncorrelated process useful follow unity exactly coherent multiplier simply case yield gain relative contribution mention spectral variate observe grid fourier tf natural asymptotic uncorrelated spatial framework increase asymptotic ever direction series complementary asymptotic sometimes call domain asymptotic point ever fine anomaly anomaly forecast hour location region day day day empirical yield pass compare various forecast calculate day validation filter marginal denote smoothed smoothed coherence lead substantial increase longitudinal band coherence appear relatively sensible great south pressure horizon begin build mid pressure long forecast frequency substantial forecast scale horizon bivariate mat ern forecast physical spectra statistical behavior spectral density kk suggest additionally across frequency appear empirical plot follow ern spectral density day forecast band minimize coherence estimate day tp forecast forecast decay decay almost exactly h horizon fitting parameter horizon cross covariance day whereas marginal day idea hypothesis substantial strongly word table grid estimate guarantee mat ern sufficiently flexible field area spatial height pressure science regime anomaly unite low surface core stream anomaly temperature interest science height anomaly represent pressure anomaly height vary day qualitatively bandwidth pass calculate square day pairwise coherence coherence pressure level moderate coherence frequency behavior pressure apparent play crucial role formation high coherence frequency capture one explanation frequency assimilation anomaly level pressure constrain observational weather anomaly frequency band pressure approximately km typical substantial shift pair model utilize value shift spectra notion gain literature multivariate spatial yield physical relative amplitude process analogous process extend coherence phase smoothed exploratory tool function useful detect readily capture coherence interpretation multivariate mat ern insight future research may coherence process multivariate perhaps manuscript process covariance spectral continuous symmetric fourier df additionally cross involve calculation involve include vector bb ib ib ib detail admit dm ij representation complex fourier f dm eq z fourier transform immediately lemma lemma constant frequency acknowledgement helpful development national science foundation grant dms ex plus minus ex corollary coherence increasingly rely alternative viewpoint model develop coherence multidimensional see band coherence fundamental construction suggest interpretation parameter mat ern class smoothness index frequency imply dependence smoothed illustrate interpretation forecast pressure examine insight difficult detect formulation keyword square coherence spectral density stochastic nearly development spatial recent review relatively explore pose construction flexible explore empirically dataset compare cross know krige fundamental open extent construction theoretically follow quantify spatial dependence gain multidimensional question previously lack sufficient suggest insight mat ern direct interpretation manuscript correspond develop autoregressive move rational know squared interpret quantification relationship sciences numerical weather forecast version generate forecast consider state surface level pressure daily forecast hour show coherence diagnostic forecast band involve pressure phase highlight pressure level difficult construction construction pp stationary j obstacle process develop flexible value nonnegative say nonnegative nonnegative
naturally specify procedure traditionally think setting parent generate body reinforcement powerful sequential decision paper encourage transfer tool extend drive advance model sequential change view lstm connection direct reinforcement develop training sequential imputation approach imputation horizon direct cover show successfully model sequential imputation loop generative implement mdp train model policy use motivated examine qualitative quantitative difficulty mechanism imputation baseline significantly baseline develop benchmark imputation imputation model special case direct gain popularity relative undirected counter part reason rapid available computing view precede decision investigate form indicate may factor eqn arbitrary variable conditional restrict exchange eqn permit approach interpretation take search available indicate compute control stem guide use policy direct interpret finite process terminal encode mdp place I initial draw train autoregressive variational low dirac delta training describe previous paragraph guide trajectory generates guide rewrite distribution z prefer define x px term eqn become variational training direct generative interpret search eqn enough model precede g author tx px non horizon tx x abuse sum trajectory integral target reverse stochastic transform write xx generate tx trajectories tx guide log eqn tractable construction basic reverse process eqn trick guide trajectory terminal start material subsection derive bind trajectory distribution capable learn primary trajectory recursively represent hide visible lstm lstm input connection author indicate affine lstm govern guide trajectory guide policy x z diagonal give affine lstm guide govern affect q read primary px primary policy p recursively eqn diagonal variance state change conditioning add feedforward alternate indicate affine lstm turn construct train care variance feedforward layer examine good upper benchmark alternate fine tuned fig show provide code pc imputation concern density expand cover standard generative shrink regression imputation policy mask complete step initial state reward initial policy reward trajectory px pz pz pz z definition imputation maximize log producing discuss train introduce guide produce trajectory x qx px I define imputation trajectory partial imputation sec lstm base c feedforward network relu primary generate trajectory feedforward relu represent step primary construct guide similarly policy imputation information incorporate guide give likelihood guide policy primary guide feedforward relu indicate train monte roll appendix provide full implementation code primary add second lstm include read policy read policy execution state r w tc imputation update add tag lstm govern subset parameter test read affine govern transform trajectory policy paragraph update ts tc distribution read guide observe compute backpropagation imputation three convert miss test completely removal pixel source unconditional test four type imputation baseline sample use hold test add lstm lstm model step lstm add jump autoencoder imputation template template matching imputation run multiple reconstruction input match template significantly outperform lstm outperform direct use naturally objective template imputation imputation valid likelihood fig high quality modal behavior swap non imputation imputation provide policy train lstm raw tune lstm jump close loop overfitte sec mm mm present view direct model reinforcement guide grow sort imputation unconditional imputation unconditional modelling show train comprise million search outperform appear qualitatively g improvement describe px x p qx tx x tx bind trajectory produce start
histogram panel aggregate green area window code red three digit rather feature pixel nearby train small traditional topic use highly contiguous grid generate grid document window topic usage therefore window cg count considerably histogram individual dataset embed room window window overlap grid window magnitude capacity thousand traditional require advantage count classification visualization question remain train small overlap evidence indicate direct imagine counting imagine contiguous create grid learning may suboptimal may coherence minima arrange coherent answer bring contribution family hierarchical collapse mathematically maxima important especially variational learn count consistently traditionally standard deviation contribution quality outperform evaluation study participant name name tb name name name pi dot name name name name n name pi dot solid w name pi dot w name pi dot name pi dot l pi w pi l pi k counting count grid counting stack count dot circle represent parameter represent grid grid dimensional index extent index grid grid generate bag list word value grid count window dimension bag pick grid location inside place collapse sum variable write bag count single window bag number capture occurrence explain single counting grid multiple thus large highly window prior location grid turn inference fast sophisticated train topic model join force neighbor explain cg tend exhibit topic move away mean topic go gradually shift grid attractive minima grouping certain break suboptimal fig digits window contain even however nearby add digit rich one component variation three peak location combination create contiguous stroke distant likely illustration build location map derive feature count add top location grid particularly another grid layer linear mix inherent nearby feature around peak also slight shift leave layer model mixture cg terminate uncorrelated arbitrary stack sake brevity generalize count grid utilize sum training cg show allow omit index expect vocabulary grid location word formula act higher firstly introduce factorize posterior true grid write entropy algorithm iterate status update grid reader directly collapse second variational presence summation window big resolution peak hierarchical stage stage smoothed deep incremental document contain word illustrate acquire document tuple retrieve last depend total sample ambiguity without relevance tuple individual document tuple uniform document employ sample stable complexity grid document overlap arrange grain statistic greatly improve despite strongly outperform bit around third despite allow correlate topic enable lda process index tuple perform task originally coherence topic coherence six order subject outli topic five high word six subject target often fail correctly detect lda small instance question perform sense actually topic meaningful human artificial lda would coherent one grid instead pick sample pick choose start select location respectively procedure wikipedia article amazon http www result show euclidean user able interestingly even pick show worth outperform cg model benefit dataset www com mc compose class previous subset vary similarity compose similar complexity classify document use show l cg lda possibly cg grid sparse intersection document clarity capture intersection topic intersection rather rbms often stack deep though change intersection modeling optimize advantage visualize end application expert uniform dropout grid first section main mnist digits cg bag represent location virtual appear time image learn portion cg assume window nature rather nearby relate overlap window learn mnist digit window rather nearby window digit index relatively rich posterior general hierarchical grid paper build stack cg stack deep course derive architecture grid previous specific illustrate stack counting grid put grid grid total top grid general network place would conditional factorization link link layer formula evident location act specify joint joint intractable inference resort firstly factorize posterior assume factorize multinomial grid location free energy h variational last yet last third add variational convergence distribution status reduce top place cg window token update posterior variable therefore employ place inference utilize cumulative slow individual detail section qualitative measuring micro originally topic coherence original task six user find model word select word six coherence micro slightly micro grid select average start started select cg wikipedia amazon http www word lda cg e cg lda cg cg lda
well impose bt weight use orthogonal joint derive easy bellman bellman equation uncertainty bellman analogous robust guarantee project risk bellman contraction w unique solution projection space orthogonality give point project law large order implement iterative algorithm repeatedly solve inner problem intractable section robust trajectory probability nx x aa empirical kkt section solution optimization problem obtain empirical inner material section effectively formula coherent thus shall derive saddle risk measure mean formula kkt multiplier saddle solution analyze gradient case envelope risk sensitive bellman equation report neutral material generate stage function q coherent neutral wise cost function probability saddle policy become impossible sampling unfortunately risk neutral bellman stationary policy action mdp exact value intractable summation compute approximate project address trajectory x np calculate use nx indeed structure case may replace transition np nx phase measure transition induce probability x policy j j wise approximation bind reader supplementary imply policy follow decrease increase sample p probability function close thus want infimum attain compact infimum attain strong markov empirical statistical limit show q tx nx tv quantity bound repeat claim strong imply bellman unique point whenever let corresponding multiplier x kkt x skew objective magnitude get recall function since affine equip kkt kkt x easily x second sufficient condition hold analysis implicit sensitivity kkt optimization know x sample x p exist probability complete identity q follow write notice maximum note every element p fact argument one x stanford electrical engineering department technology remark theorem sensitive method cost coherent risk accept finance research spirit dynamic value reinforcement generalize extend previous consider involve maker various application finance operation research objective gain popularity variability discount reward markov process mdp objective apply rl var risk actor percentile optimize view take preference another rare highly influential coherent desirable measure satisfy measure satisfy term financial coherent sequential mdps another desirable programming style property end optimal recently markov coherent measure work present rl coherent risk generalize focus coherent total discount return markov coherent formula coherent convenient gradient coherent risk programming policy coherent relate actor algorithm generality sensitive sensitivity variance optimize studied study dynamic risk coherent measure static coherent planning robust approximation suitable g rl style mdps robust part investigate stochastic system trajectory dynamic dimensionality motivation risk actor outcome sample event parameterize ease restrict finite without omit brevity space cost realization order cx state distribution discount parameterize denote draw policy real risk z intuitively risk ensure asset also intuition asset risk refer reader coherent risk show exist state worst suitable coherent risk envelope sequel risk measure envelope paper canonical convex programming formulation satisfy envelope parameter envelope coherent affine equality twice differentiable envelope know form risk hold risk semi account temporal structure dynamic measure take stochastic primary measure issue consider world less tight consistency evaluation risk illustrate multi optimize inconsistent reader risk insight markov particularly mdps length markov coherent measure coherent policy note coherent risk transition px aa depend sensitive random risk correspond discount cost cx cx cx trajectory induce mdp parameterized mdp coherent risk define hope neither tractable risk complex try calculate interested trivial analytically static correspond trivial case case devise suitable descent sgd learn structure dynamic think estimate static static coherent policy parametrization define assume requirement satisfied application management financial engineering survey fu fu value lagrangian denote write presents subsequently formula particularly hold point eq application dynamic risk expectation gradient return coherent material assumption wise develop sampling compose calculate sensitive update actor analysis follow highlight refer reader version supplementary material challenge space large dynamic curse exploit mdp modify algorithm robust actor main thm sample involve estimate trajectory mdp trajectory next estimate analysis actor incur supplementary illustration importance designing criterion risk trading agent return return third asset pareto widely financial train e train policy asset vs training return lower expect policy counter intuitive rational case stochastically dominate static risk gradient style combine convex thereby sensitive improve especially rare event conceptual explore maker preference flexibility cost variability dynamic importantly coherent theorem naturally relate make mdps markov maker able take sense principled trivial scope potential risk misspecification note assumption optimization duality assumption family absolutely saddle envelope result write trick equality assumption define f contact expectation gradient eq envelope theorem expectation trick back naturally base estimation let denote function randomness lagrangian recall e saddle lagrangian lagrangian set empty bound value continuous enough l chain definition lagrangian constraint l point page sequence p finite condition wise iv vi show derivation theorem n furthermore objective constraint interior follow interior l contradiction n n saddle l n must must saddle term guarantee
mutual information able automatically balance population ratio class mathematical interpretation mechanism mutual similarity truth baseline relation hold fig relation joint cross modify include divergence fig goal mathematically machine human identify machine novel learn linguistic argue exist unified behind purpose extend conjecture description study computational function drive law various ia ac cn position evaluate adjust selection briefly study theorem unify one come thing nature mathematical machine construction increasingly develop goal description principle loose reflect fundamental universe suggest interpretation mechanism relate subject mathematical principle mechanism mechanism mathematical principle belief principle critical brain purpose briefly review empirically measure base conjecture information processing novel distinct complementary great necessity describe three level learn engineering application study show address decomposition basic methodology novel fig level problem level give four level learn identify problem linguistic reflect description language expect cognitive science information notation include process design implementation utility subject concern complexity include realize evaluation selection adjust behavior machine adjust intelligence flow problem four neither exclusive exhaustive fig illustrate context learn scalability cost provide loop intelligence critical benefit utilize show example adjust level intrinsic methodology offer power four novel perspective learn dataset classification compatible error classification whenever target wrong unable goal another character describe un similarity linearly separate circle htb example learn linguistic similarity original semantic link namely direct describing linguistic inverse connection opposite direct distinguishing reflect difficulty direct way linguistic inverse way call ill target selection compare study learn systematic generative target selection advantage application machine provide drive law rule cost wu translation chinese classify accord rule dataset english describe chinese refer search consideration derive two principle support bayesian machine need shannon introduce concept mass variable variable call fully asymmetric list criterion discussion h joint symmetry information symmetry kl z dissimilarity machine mathematical principle margin
cnn deep feature mean five layer fully feed hash generate code please local normalization classification hash add reduce invariance capture subtle distinction connect layer diverse toward appearance hash hash represent composition function bias term omit binary computed single label label label hamming however multiple preserve essential point keep neighbor hamming distance semantic semantic database calculate share accordingly last share none label obtain sort similarity level truth ranking ndcg measure ensure ndcg ranking construct surrogate directly try practice rank triplet hash length h distance disagreement incorrectly rank svm rank definition ndcg top score well reflect rank retrieval system pay wish predict treat inspire modify level database eq ndcg normalization constant ndcg suffer assign wish hash give encourage sure decay sign optimization relax eq logistic function facilitate rewrite hamming distance inner objective observe function actually summation triplet loss triplet hash code mini batch mini batch image randomly retrieval six type word base sift histogram histogram wavelet texture block dimensional discount ndcg map use retrieve evaluate calculate similarity within position rank actually mean average big weighted number effectiveness propose influence performance ns ns weight adaptive weight quality ndcg expense performance less relevant layer hash toward appearance utilize semantic similarity b b hash illustrate ndcg use performance hash capability exploit usually compare activation pre train imagenet activation feature recognition show activation boost margin construct deep feature representation hash code utilize supervision fit hash advance superiority fine retrieval I evaluate well entropy achieve semantic rank supervision well preserve semantic structure label cca worse unsupervised consider similarity relationship fine supervision unsupervised layer hash attempt activation cnn even bad hash cnn paper employ hash preserve rank supervision jointly mapping binary code problem nonsmooth triplet stochastic descent effectively experiment demonstrate outperform hashing method term rank quality acknowledgment work program china cb china ia ac cn rapid hashing receive interest retrieval effort compact preserve however hashing semantic yet deep rank hash preserve semantic multi convolutional incorporate jointly learn hash avoid limitation power meanwhile guide surrogate loss solve intractable nonsmooth superiority hash large content base retrieval attract due storage hash aim code maintain similarity hash locality explore hash hash mainly metric structure datum spectral hashing euclidean semantic similarity preserve semantic structure label hash optimization base learn hash image multiple similarity measure normally require handle well hashing extract like representation representation code deal relatively structure semantic novel base ranking image view framework term hash use neural cnn hash rich feature learn hash supervision ranking list derive query database stage surrogate result stochastic propose couple compare activation experimental semantic significantly outperform rank semantic multi convolutional ranking supervision apply ranking image organize discuss semantic hashing formulate optimize conclude roughly divided category independent datum dependent preserve focus iterative quantization cca utilize cca minimize quantization pointwise guide learn preserve similarity assign classifier hash code uncorrelated motivated structural hashing propose pairwise upper binary approach learn feature semantic preserve hashing alignment similarity euclidean hamming use basis triplet ranking preserve relative triplet capture use triplet supervision hashing minimize hamming space discover deep scale column hashing combine boost
document car publication retrieve output new publish car car query wikipedia lda well dimensionality dimensionality space document output almost equivalent dimensionality similar representation fast traditional collaborative challenging relative content informative scale content digital recommender allocation lda belief low document carry public benchmark digital medium provide online platform article come toolbox tailor evaluation concern net latent lda bag conceptual query dimensional net toolbox implement comparison ability due architecture representation retrieval visible hide topic must dirichlet distribute comprise lda package conduct platform reading acyclic bipartite layer ability deep autoencoder da reconstruction consist visible layer layer rbm rbm input partly pre document count count rbm layer rbms execute rbms apply gibbs update give visible hide logistic sigmoid unit visible visible unit visible binary bias visible unit visible softmax softmax unit value hide document bias learn gibbs joint p document query possible proximity neighbor momentum weight initialize variance bias initialize epoch fine batch line search define deterministic perform comparison compare output output consider output internal input output train dataset corpus article category use category connectivity category wikipedia corpus wikipedia business consist document document wikipedia business provide category wikipedia lda model topic measurement bad topic evaluate evaluation wikipedia business higher dimensional measurement fig evident document number outperform twice size two identical show limitation visualize ht pca
start gradually beyond evident increase monotonically capture attribute include go popularity tb recommend article red use abstract string document text search manuscript figure list recommend article top proportion form count year publication recommendation construct five article red recommend cite recommend pairwise cite proportion test recommend recommend topic likely allow article exhibit proportion pairwise able topic quite pairwise link article topic article recommend exhibit mix small proportion look citation count article count capture degree popularity rank citation alone compatibility mix article sense article accumulate huge recent short period highlight offer article relevant row average document right year estimate one comparable publication interestingly per year rapidly publish later showing tendency bar proportion arrange color represent average figure proportion proportion increase transition citation count rate red article topic citation blue phenomenon demonstrate citation topic assign low citation rate citation count mathematics molecular biology mathematic molecular biology raw citation mathematic assign high citation average article issue tackle field citation activity share scientific impact publish citation publish special citation area citation network arguably mix citation connectivity content connectivity citation detection community citation content probability article membership topic citation within domain publication article citation topic variable adjust indicate likely cite locate domain raw merely account activity improvement link helpful scientific field propose enable predictive bayes acknowledgement national fellowship university impact scientific article bias citation properly field evaluate article derive joint probabilistic amongst lda mixed membership blockmodel individual article citation citation behavior recommendation account pattern fitting method control measure article compare researcher award consideration recognition indicator valuable various consider author reader unify journal journal enable article usage activity recommendation web twitter facebook count raw impact article index attempt author publish bias use impact scientific article accounting factor citation publication journal know relevant factor variation certain social typically cite molecular biology compare raw citation address procedure normalize count respect standard recently receive belong scientific model assume citation scientific influence article level citation account potentially useful scientific article name derive joint text whereby article citation citation belong article position citation compatibility research article relevant unobserved quantify profile journal since publication account citation network relational content information combine establish relational mixed membership blockmodel scientific detect citation intra community detect citation integrate community introduce variable article citation due compatibility topic act understand citation citation model article article citation field search paper technique keyword method relevant article citation count may yield citation scenario article recommend citation metric adjust citation citation rate intra citation relevant identify adjust citation count article external provide article citation monotonically fully besides framework fit develop efficient posterior real citation often massive analyze interaction scale square develop variant subsampling network iteration optimize variational objective reduce storage detect community pair informative set define mcmc space adapt link assume close suitably adapt model requirement scientific science research benchmark high physics paper model study join article date absence publication significantly organize introduce alternative recommendation conclude combine lda act adjust model generative probabilistic detect assume vocabulary represent vocabulary document topic vector probability topic document word draw assignment word slight abuse hand mix within data node community group node membership blockmodel group membership indicator receiver element refer generate text draw topic link em mixed pair indicator dd dd citation relational pairwise combine identify community topic suitably incorporate would improve em draw proportion position topic j ij topic receiver dd dd document circle variable indicate corner link cite violate world citation research affect citation cite high cite author attribute cite variation topic generation link latent variable draw citation blockmodel element one document topic citation factor cite citation probability topic characteristic citation topic proportion attribute area take link lda identically issue whereby datum content connectivity citation multi assume latent rise tractable distribution simplification topic pairwise probability weight bayesian logistic minimize structure laplace place word parameter log text lda annotate entity realize entity blockmodel link ball document instead random offset proportion cite document connectivity due pt sample document corpus pair perform success denote set successful update ij update half update optimize gradient ascent parameter denote take pass write stochastic ascent replace unbiased step satisfy lie column example stage estimator variational outline algorithm lda baseline lda follow regression link covariate take hadamard th text separately serve baseline method modeling text account link structure assume consider explicitly imbalance lda lda link variational lda implementation suggest author link initialize link text application article researcher search look paragraph keyword text link mean recommendation intra citation citation nature come citation scientific article reader reader preference amongst article document training fit perform document proportion iterate convergence update assume knowledge link document true variational posterior hold ability predict give rank model fit proportion document rank hold document actually predictive rank fit evaluate also able blockmodel perform lda lda computation topic proportion topic document mean citation cite element interpret citation visualization vocabulary inspire retrieval examine subsample add subsampling publication time take add package vocabulary us cpu divide fold fold training topic minibatch tb predictive rank runtime predictive rank cpu cpu hyperparameter predictive attain improvement close subsampling time maintain around topic concentrate fit fold blockmodel estimate close dotted agreement repeat blockmodel run quantity correspond agreement greater slightly right count trend increase citation capture citation citation high capture mix characteristic account citation among topic popularity document set illustrate incorporation performance example document segmentation classification cite organization rank give table rank low take account lda index high improve rank improve performance lda article recommendation represent total cite display font activity topic width arc total arc come infer colour origin visualization blockmodel word citation activity figure tends dominate topic citation hand citation individual article citation topic figure tends cite nearly besides high tendency helpful vary area even physics competition rank first cpu times rank cpu provide
learn begin size time thereby da consider easy expectation expectation big da sequentially achieve visible least one update let ideally practice sample da relate corruption set visible belong da da replacement require equal autoencoder distribute accord f distribute com class pixel instance comprise voxel water intensity scalar grey template parametric http www ac uk image accord variance even number imagenet comprise category collect apart hierarchy comprise million broadly http www image five category imagenet database amount million image center theorem lemma corollary corollary corollary unsupervised learn denoise depth interesting guide gap mechanism backpropagation objective incorporate two mechanism denoise autoencoder deep building upon level support empirical evaluate unsupervised success last year practical treatment theoretic concern generalization formulation relate ensemble analyze quality network choice fine proxy categorization question ask analyze importantly line like establish rate consistency output recall optimization specific analyze objective start nonconvex describe gradient analyze deep denoise autoencoder dropout net analyze recurrent activation subsequently interact seem influence concept type corruption importantly structure influence dropout estimate idea depth size corruption certain convergence certain choice last year dropout follow notation describe objective backpropagation dropout denoise autoencoder dropout reconstruct corrupted correspond learn dropping unit corrupt version unit bernoulli focus loss layer dropout sigmoid activation bias handle use sigmoid linearity activation minimization via gradient noisy k iteration randomize allow either priori train offer theoretical exist implementation analysis serve result da corruption compute gradient kk rp proceed bound architecture correspond convexity function estimating size discuss repeatedly da include supplement number let layer nn serve give dependence requirement refer proof backpropagation adaptive convergence impose restriction may lack study gradient present address general nonconvex oracle believe context serve da potentially widely convolutional net loose supplement choice ensure expectation construct presents nn run randomization fold layer layer require fold surprisingly fold need small minimum idea base nn work autoencoder closely section present da insight da correspond da autoencoder rate maximum da step size eq knowledge remark result relate da corruption autoencoder denoise also interest observation follow da ideal stronger visible layer weak support minimum show necessary provide evaluate bind interaction estimate denoise autoencoder instance da depend structural statistical characteristic correlation across exploit size unlabele need chapter consistency representation ensure gradient fall sigmoid estimate via refer predict decide exceed imply suggest present trend choice extreme right lipschitz scalar choice main b vs da vs impractical sample fewer empirical several suggest recently show deep pre corruption da since da module bipartite minimization da equivalent include carefully whenever setup little room da unit rarely thereby requirement seem explicitly disjoint share visible ensure solution summarize autoencoder remark small refer lemma ex choice decrease improvement increase infeasible combination dark blue exist improvement da da feasibility observe increase fix increase infeasible combination leave half denote dark general case drop iteration layer layer dropout iteration optimal size dropout instance dropout layer net address via comment hide bound reduce observe correspond averaging may little backpropagation therefore layer distant layer phenomenon partially obtain layer da dropout broad regularizers da learning fraction available rate dependency compare gradient denoise da trivial early analyze architecture present sample root interpret consider hidden layer length dropout rate reduce retained unit derive setting hide expect retain layer dropout layer ensure easily length predict depend check accuracy many satisfy generate bad bound present convergence backpropagation ensemble maximum dropping half mnist imagenet refer imagenet present interested plot htb gradient vs multiple color vs layer third expect sub asynchronous inherent supplement top mnist imagenet normalize respective h gradient call four trend show stepsize decrease stepsize gradient optima computation gain imagenet test bar error dropout gradient black blue gradient vs layer imagenet show trend trend three figure correspond decay layer black vs fractional da gradient strong black green red figure gradient decrease green reach increase blue report refer deep gradient help distribute vs least refer twice master via pass initialize run whole iteration expect decay strong attribute factor trend rate show imagenet increase speed increase rapidly fall vs distribute da give denoise dropout influence denoise b gradient corruption da denoising agreement depend da dropout efficacy use choose corruption rate convergence exist strong denoise context framework construct backpropagation net analyze interaction scale convolutional recurrent also boltzmann nn iteration let lipschitz recall eq sigmoid f u aa bb b aa ba bb ab aa bb significant start noisy gradient w k step denote rearrange sum estimate inequality randomization construct noisy stop criterion random however point iteration update markov process take expectation last expectation last eq q finally constant monotonic resp term decrease balance substitute stepsize stepsize q change nn instance require markov q sense
use problem dimension strongly regularizer induce exact gs efficient gs connection section generate regularize dimension difference performance strategy gs cyclic selection substantial lipschitz sampling narrow gap remain gs gs rule non randomize zero away compute gs rule coordinate seem improvement gs rule despite cyclic gs plot number gs advantageous perform label dataset dataset connect high base supervise implement efficiently optimization normally cyclic coordinate cyclic randomized coordinate gs gs constant gs rule randomize applicable approximate gs randomize similar justification exact optimization justify exact coordinate expect block parallel dual ascent successive boost algorithm strong like thank anonymous gs idea rely bound constant column column max give gs compute structure exist unchanged element change update expensive update modify total q differentiable denote strong rule variety notable least non process gs maintain contain product vector contain value store max allow gs cost reasonable cost cost structure update depend update gradient cost relationship convex imply similarly strongly imply strongly two relationship norm along equivalent derive strongly convex gives conversely strongly hessian h line scale constrain quadratic stationary combine give obtain fast convergence gs occur value exact gs one tight bad gs coordinate combinatorial graph maximize much particular case mode alternate alternate mode must eventually cycle one node cycle weight burn period repeatedly go mode final step finish several consecutive maximizer constant burn burn period implication away fast descent gauss eq dual strong put logic appendix establish relationship different convexity square norm use establish q expression nonnegative vector use less section additive gauss rule choose assume generality progress lipschitz continuous norm prove dependency subsequently give less rely show follow show lipschitz q subsequently continuous gradient substitute apply inequality progress hold eq imply inequality recursively although small usual notation need gs rule state turn gs rule analyze case lipschitz imply notice define contain notation gs use gs optimality eq notation unique relationship gs reduce q lie show gs first min gs progress gs min bind progress subtract give gs apply gs progress consider gs derive add subtracting note select gs upper would choose gs use continuity gs strong gx make substitution q side rule gs method eigenvalue correspond norm proximal indicator gs eq gs thus choose zero obtain even gs rule either hand gs rule progress clearly turn show gs possible rule proximal value function gs rule progress ratio gs choose obtain progress ratio bound substantial margin gs find counter able produce counter gs rule detail implementation gs offer runtime hardware rgb significant recent application randomize begin achieve gauss suggest computational selection rule comparable gauss give rule showing coordinate exact sparse problem propose gauss rule fast rate analyze gauss proximal substantial optimization seminal give coordinate minimize random coordinate well gs nesterov randomize later paper randomize contexts expensive suggest use context gs practice suggest class descent discuss context gs rule gs strong standard smoothness assumption randomize restrict fast show usual gs optimization provably fast certain sparsity result show benefit exact update optimization variant nesterov randomize lipschitz strategy variant optimize separable non show case perform update mean coordinate descent minimize express element family include core machine lasso svms solve form include quadratic propagation assignment continuous graphical general gs expensive however often gradient max implement gs randomize two slightly base facebook detect disease example friend friend gs degree maximum thus gs optimize gs inefficient star instance problem implement time maintain efficient gs rule solve near although factor general gs function product solve wise lipschitz twice differentiable base uniformly alternatively choose coordinate large directional bind make positive side uniform sampling subtract side notation progress imply gs definition obtain gs rule descent fast rate cyclic selection rather rate gs lose avoid e side q make use fx fx gs square norm one gs rule gs obtain extreme guarantee future iteration choose variable graph gs alternate large show gs structured maximizer consecutive maximizer consecutive implication edge conjecture value form correspond non fast review gs gs scenario relax backtracking proceeding lipschitz special fast gs extreme differ sample fast context section sampling factor nearly large benefit rule selection extreme gs lipschitz rate gs fast lipschitz gs lipschitz gs obtain fast lead call similar argument place obtain convexity constant appendix thus always fast rule lipschitz close minimum gs quadratic use rule optimal coordinate size iteration boost rate apply mi improve strategy require interesting gs near residual denote solve refer return justify approximation gs gs rule special power interesting rule lipschitz constant formula thus towards gs problem rule compute gs inefficient practical gs
topic topic disjoint ranking permutation general item center permutation permutation partial within probability permutation permutation row lemma conclude rank approximately separable combine paper approximate prior full consistently eq form select roughly apply k note sum give center ranking complexity except proposition novel comparison inconsistent user probabilistic share key insight connection comparison insight advance separable discovery separability appear restrictive outcome latent world extreme ranking provably new empirically competitive diverse application system pairwise comparison item inconsistent record web transaction click predict pairwise comparison new mixed membership comparison ref ref permutation pmf center heterogeneous inconsistent noisy mixture capture heterogeneous multiple furthermore capture factor consistently generalize perspective fit observation provable polynomially component mix pairwise comparison model receive attention yet theoretical guarantee learn extensively unclear view user comparison latent topic topic discovery provably separability geometrically inspire work ref ref generalize separability require pair item prefer probability restrictive formally separability arise set preference provably generalize angle establish reference computational allow user htb component pairwise provable pairwise provable available provable combinatorial vertex available top provable pairwise vertex provable full extensively study setting decade ref rank ranking component cluster heterogeneous preference type attention pairwise comparison full ranking provably correct base handle impractical within user view alternative pl study model relate validate adopt membership perspective permutation permutation inconsistent guarantee motivate another mixed ranking topic pl summarize relate separable discovery consistent discovery topic separable separability many topic perturbation date establish provable guarantee perturbation go augmentation improve ratio provable degree approximate explicitly derive separability provable similar separability strong user dominant full satisfied considerable preference rating influence share population ref come different mixed membership perspective organize introduce approximate separability summarize step demonstrate synthetic sec htb describe process universe item population assume un independently distribution outcome comparison denote order dispersion parameter permutation normalization comparison ex ex rank token z ex mix characterize order time item share reduction formally pairwise compare ex item prefer sample behavior ex topic enable let ranking define ki I ex prop infer prop infer item entry therefore pairwise ranking correctly hence ranking prop note topic document compose word topic weight sample ex column mixed membership topic observation pairwise condition ex model thus prop section family mixed membership rank membership ex dimension model ex key separable come consistency favorable enforce total ranking precise geometric occurrence matrix pairwise estimate splitting user k topic co ex separable discovery ref rank exactly separability order definition ranking separability recall high item arbitrarily close prop propose rank approximately negative separable e order separability order refer novel pair separability uniquely reference ranking seem share next model scale sufficiently fast negligible fraction satisfy approximate separability draw reference ranking permutation ranking sample dispersion ranking would small property prop supplementary loose separable separability approximately geometry row pair circle region angle exactly novel row circle separable row perturbation ideal ideal hull point hand non close form novel detect approximate novel normalize solid extreme probability row strictly approximate ideal become extreme separability solid angle close hull form pair solid angle angle correspond angle consistently approximate isotropic asymptotically estimate pair identify prop specific weight prediction inference ex main detail alg alg estimate regression scale alg prop alg ranking equivalent rank define component tolerance reference ranking I p j jk novel ranking precision I j k k k j k pairwise sort computation run proof loose parameter order moment rank consistently ranking propose algorithm fail normalize form detailed supplementary provide prop note complexity spread hardness smaller require achievable identifiable validate assumption demonstrate preference experiment suggest specifically projection ex measure ranking since align ranking distance ex ground reference ranking movie rating parameter normalize reference ranking depict estimation vary dispersion ground ranking reference ranking carlo comparison dataset rating due public partial viewpoint suggested frequently rate split convert rating rating rating ignore prior dirichlet evaluate performance log approximation likelihood phase compare topic tm setting summarize tm htb ex c pmf tm train rating aggregate properly test purpose optimize rich rating rating rate convert training movie rate tie train rating movie
email concentration invariant parameter vary isometry singular digital processing compressive identification compressive pursuit recovery match orthogonal pursuit compressive sampling pursuit tucker impulse response transform discrete cosine operating curve pearson external basis least absolute shrinkage selection negative output compressive spatio wind forecast autoregressive autoregressive portfolio prediction direction switch artificial neural root square square operator spatio ann least terminal routine wind power short forecast wind present incorporate datum inspire compressive sparse recovery dimensional structure collection exploit cast forecasting recovery signal propose east compressive spatio wind speed improve short forecast widely wind energy grow world global another total wind power year reach capacity wind make balance integration service load forecast one directly forecast wind convert wind wind generation wind forecast wind forecasting method group vs ii probabilistic forecasting paper short point forecast temporal wind forecast neighboring forecast spatio forecasting method later incorporate wind forecasting introduce probabilistic et al speed advantage markov model prediction wind aggregate graph spatio regime switch wind direction study various forecast error methodology density field wind power comprehensive review compressive cs usually exist collection weather forecast end cast linear propose algorithm forecast measure weather east york result considerable advanced spatio temporal forecasting wind benchmark conclude remark model variable present autoregressive generalize condition suitably interested reader signal block concatenation eq additional design due flexibility recover block computation recently topology uniform assume generalized target relate high target sparse concatenation zero generalization block block correlation adjust prediction wind speed east wind speed higher state speed datum weather report east york fig depict study red locate subject wind profile low area correlation wind mainly time simulation period wind throughout year compare spatio temporal wind forecasting forecasting simply forecast improve persistence advanced capability capture nonlinearity wind speed series wind behave sub band subsequently sub carry mode speed reconstruct high band wind perform recursive period hour ahead forecast compare ht spatio temporal forecasting method spatio spatio depict incorporation improve forecast ht ht new obtain every hour equivalently step hour recursive prediction speed prediction predict wind speed recursive continue effectiveness forecast wind forecast list speed measure moreover calculate spatio good example reduction persistence reduction spatio temporal ann st st tb illustrate calculated horizon confirm spatio temporal
conceptually requirement evidence meet derive example present address causality eqn work concern notion drive confusion cause measure direction influence assign rewrite eqn pe yield alternative pe expression gives require probability parametric relationship investigate use counting notion causality infer might drive causality zero conditional eqn former two cause appear influence two bayes use series mean series occur assume cause causality determination intervention absence cause perform action rather lack absence assume cause identify causal causality attempt remove assume cause eqn issue account time see dynamic assume assume cause causality argument causality address article term causal causality relationship time cause effect high assume address positive cause assume cause cause assume yield cause assume cause difference causal inference effective probability cause assignment series use calculation cause effect consistency causality cause must natural assignment assignment px assignment unobserved two effect probability g appear appear accounting must shift shorter counterpart single calculate library assumption cause cause observed mean observed cause cause cause agree intuitive causality unobserved cause cause incorporate averaging cause cause mean make cause calculation conclusion would useful cause show cause effect cause assignment series initially x effect length use cause effect assignment weight naturally conceptually account causal influence cause effect causal follow time noise could five see discrete calculation address estimation calculation eqn instead relevant py noisy simple noiseless weight tolerance large enough probability cause become tolerance motivate noise little calculate size become find tolerance sign compare know level time causal require become zero series signal observe calculated agree intuition e expect result depend library specific pair assignment calculation eventually causal inference agree intuition thus three value calculation way observe cause pair causal series cause rare effect impulse cardinality would tolerance imply weight median course agree library reason believe inference increase basic observed cause effect I calculated cause algorithm complicated cause complicated conceptual cause assignment set usefulness tool inference test directly intuitive understanding drive system dynamical specifically consider drive periodic impulse amplitude response drive ht cc large show tolerance increment intuitively show meet short increase example always inference agree intuition know deviation mean bin deviation bin show method ccc deviation bin yield expect part series independently exploratory assignment useful causal assignment exploratory causal analysis example different assignment near expect lag appear dynamical create synthetic dynamical eq instance eqn observe noise level use tolerance domain spread reasonably cause assume period counter peak assume drive poor value insufficient reliably cause eqn lead intuition relationship drive response signal tool conceptual restrict linearity datum dynamic example agree intuition tolerance causal inference generate nonlinear similarly intuitive ht agree causality tool point limitation causality exhibit behavior system eq pair couple model introduction convergent causality vice difficult justify instance seen see strong even strong present paragraph support calculation ht cc along domain use calculation standard tolerance intuition figure x enough implication irrelevant domain effect determine causal system calculation standard provide intuitively ignore causal serious usually seek causal two strong bivariate causality causal try drive relationship series see explore calculation eq directly intuitive despite dependence intuitive causal intuitive domain mean cause effect inference agree though consider calculation implies also seem imply case imply counter imply unable identify drive situation drive driving occur interaction imply autoregressive result consider cause insufficient previously analysis effect test cause include assignment weight assignment calculation case calculation part exploratory must assignment try understand assignment expand autoregressive definition extension article understand exploratory causal dynamic pair repository repository datum assume relationship temperature series daily expect temperature tell small difficulty exploratory analysis determination cause effect tolerance thorough calculation tolerance bin histogram close library bin deviation cause set purpose article series detail comment regard confidence exploratory however convenience take simply causal cause effect pair show point tolerance domain imply inference agree causal test pair assignment highlight determine decide inference depend max truth weather collect include collect national center b cause effect assignment could domain set l sample assignment plotted weighted cause assignment show inference line algebraic set algebraic mean aforementioned set weight linear system periodic impulse example eqn apply causal relationship eqn observe inference symmetric increase towards tolerance library length causal eqn library spurious use assignment library length spurious example imply relationship calculation care causality involve investigation experiment article explore exploratory proof dynamical system observational alone field physics analysis involve technique entropy te te tolerance attempt make causality token causality technique method include acyclic temporal logic causal connection broad causality leave general logic interpretation framework regard interpret cause assignment assignment stronger interpret
value predict uncertain prediction leave weight assess uncertainty relu squared covariance show mc dropout lastly fig blue colour represent half deviation confidence plot none capture mlp predict mark dash clearly sensible increase uncertainty effect predictive uncertain behaviour capture mc figure increase uncertainty increase relu stay relu covariance appendix whereas relu different different dropout dropout initially optimisation uncertainty interpolation repeat experiment relu network layer setup segment minus various interpolation show interpolation miss gaussian squared red green blue mean relu dropout increase uncertainty miss uncertainty capture deviation error bar uncertainty number forward draw number mean neural mnist dropout relu operation usual dropout train iteration reference scatter scatter uncertainty digit axis scatter forward pass softmax fully softmax predict softmax softmax output digits input axis image rest fig predict look envelope envelope class input softmax softmax softmax reasonable return middle high would ask fairly moment confident reinforcement learning receive various aim time try low reward pick instead great agent decide advance rl make network action agent state lead game agent human greedy explore uncertainty estimate use converge code world point angle ahead depict one different angle different reward reach look white approach batch purpose experience network step initial relu rate weight decay descent momentum batch original implementation change q burn additional dropout linearity dropout uncertainty greedy perform single propagate sample average blue batch plot batch move thompson reward within batch burn batch bad batch move greedy interpretation model uncertainty demonstrate deep reinforcement development exist new additional burden uncertainty estimate assess corrupt incorrectly high pixel change considerably space corrupt lie increase compare uncertainty variety gp approximation like thank dr chen mr dr mr van mr wu comment european fellowship remark ex ex ac uk tool gain attention bayesian offer come prohibitive extract away model computational accuracy exploratory dropout uncertainty task mnist biology name tool ever mlp know dropout convolutional network however many field towards classification confidence uncertain prediction softmax softmax confident point classify pass softmax reflect take appropriate result high uncertainty classify happen post office sort responsible uncertainty reinforcement value quality action often agent estimation explore thompson dash line solid area mark ignore offer reason uncertainty come prohibitive perhaps surprising often change optimisation dropout interpret approximation know avoid tool exist dropout mlp extract away often exploratory different dropout represent extensive assessment task different architecture important task mnist concrete lastly uncertainty set reinforcement similar deep reinforcement learn know converge place weight study extensively computational variational inference inference inference neural dropout ik similar well variational indicate unit layer drop linearity extension approximate place distribution map deep appendix variational divergence full kl sample bernoulli identical scale obtain dropout approximate moment predictive empirically l monte pass network average derivation uncertainty estimate multiply uncertainty obtain variance equal forward pass mlp precision embed ratio set modal approximate place matrix result layer modal mlp predictive pass exist mlp result dropout often
ssc b ssc b sc b normalize undirected node compute use convenient algebraic assignment recover subspace cluster ssc noiseless datum step optimization matrix ij e desire self property similarity make connection early sep obtain similarity could sep various regime albeit ssc subspace complete sep cluster list remark corrupt extension noisy present present concern require degenerate make algebraic generally mild example surely generate fix linearly position assignment truth permutation cluster identical property connect position point require reconstruct linear hand self exist connect otherwise contradict eq contradict identifiability position could drop relaxed notion identifiability union union point start repeatedly increment assign new subspace sep minimal minimal truth truth cluster subset certain regularization achieve structure intersection intersection otherwise address advantageous compressive likely yes evidence iterative hand shrinkage would reduce formal treatment idea suggest regularize degree freedom good generalize well everywhere ssc answer minimal union subspace span point order point sort subspace previously fix ball body hull give cluster restrict ready consistent condition design matrix noise furthermore self satisfy every highlight consequence feasible general reduce noiseless show nonempty noise level addition differ maximum differ cluster component pick apply pca point connect robust potential procedure subspace merge robust restrict problem nice complete proposition later notational x proposition equal follow low sign u plug first long hold construct range contain point belong subspace theorem yield eigenvalue nonzero connect component natural point noiseless input assumption regularization satisfy self c contain necessity imply c imply result contradiction order product note dc ij x dc argument substitute upper get bind inequality due imply result desire contradiction rv cluster separation belong would r merge never mistake subspace ssc year show noiseless processing step discovery additional data point condition provably ssc deterministic lastly ssc address often advantage ssc research improve ssc empirical evaluation singular u u q rgb corollary affine abstraction statistic recently line recent guarantee seminal subspace ssc extent justify motion face get condition ssc obey self property ensure subspace cluster together sufficient correct thank issue post mild general position robust bound margin subspace application physical law subspace human body illumination model cluster membership reveal source datum much wide application fall category image compression identification identification modeling study privacy movie recommendation algorithmic subspace back maximization method plane factorization early theoretically justify past decade spectral recently cluster ssc arguably due elegant strong provable guarantee condition connect use assume union subspace handle separation edge subspace self sep drawback within connect segment subtle originally partially address reach segmentation general position counter position long graph sdp condition previously sufficient condition exact sense subspace sep exact hope drop notion subspace subspace overlap completely identify ssc clustering lose ssc regularize reweighte robust ssc deal within intuitive cluster number might interest subroutine vector noiseless dimensional subspace point union intrinsic ground assignment permutation deterministic additional underlie subspace noisy consider white provable subspace progress theoretical regime beyond original definition may contribution list assumption weak model additional instance semi place sphere polytope non value subspace algorithms assumption subspace capital refer applicable ssc highlight table understand result optimal sep subspace substantially overlap subspace cluster
create dimension test effect transfer letter pre imagenet list extract region image first pixel body extract state art bag document previous cluster feature pool pyramid combination horizontal recursively split recursively bag original split vertical bag result dimension h dimension classification descriptor ensemble descriptor act performance representation ensemble represent document create cnns basic benefit distance descriptor sort list pca enable keep memory table accuracy cnn ensemble cnns perform achieve well worse pool cnn perform ensemble region cnns margin cnn compute first outperform every imagenet cnn improve imagenet descriptor suggest category gain spatial pyramid pool perform well interestingly descriptor representative retrieval seven different signature image similarly document content may lead author pca remarkably loss reduce dimension compression cnn perform art document image feature cnns extract cnn alternative document show training enforce unnecessary train approximately cnns trained show representation exceed acknowledgement discovery grant hold helpful discussion acknowledge use retrieval use cnn scene net capable abstraction explore confirm superior cnns compression cnns transfer well analysis enforce unnecessary training label collection contain document image across category cnns visual letter write motivated structure document image digital library document store process optical character tool indexing pre analysis graphic index image stage analysis arise fact correspondence document spatial header body spatial template often similaritie article form perspective circumstance classify retrieve intra variability inter challenge object classification current state inspired cnns presents extensive cnns deep cnns retrieval learning cnns object recognition surprisingly net significantly focused suggest capable region add past base image power structure business assume document distinct visually component business letter typically date extent document document letter fit configuration template transformation drawback manual template document document definition flexible structure herein treat document document bag histogram vocabulary document potentially feature position result geometric successful classify document template less domain template recently attempt bridge gap feature pool several stage whole proceed small small global pyramid categorization mind retrieval type represent researcher representation learn research domain concern structure geometric configuration toward goal base document building decision convolutional report well report spatial pyramid matching yet apply retrieval learn area computer recognition cnns currently performance margin cnn domain traditionally ill suited detection grain recognition cnns grain object relevant analysis field challenge distinguish ii powerful therefore grain object challenge major cnns fine cnn recommend train problem challenge regularization technique cnn effectively potentially information train train unnecessary entirely seek whether insight document cnn retrieval form abstraction low therefore extract near cnn vector light previous paper follow first evaluate deep toward present design compression cnns cnns non document transfer explore strategy embed ensemble cnns interestingly retrieval basic perhaps available new label structured document graphic element share cnn cnns additionally explore different initialization first cnns entirely feature weight train fine implementation computer input process stack layer convolutional vast hierarchical organization responsible feature classifier convolutional neural network activation geometrically invariance cnns datum add image architecture scale specificity cnn beneficial make treat region differently align capable region specific automatically cnn train activation near cnn dimensionality reduce involve distance query descriptor every descriptor sort sort document account cnn image aid grain discrimination category letter illustrate figure consistently short letter full address learn automatically classify document learn idea cnn cnns region dependent extract total four region header cnn train entire base build descriptor concatenation compress cnn extract region illustrate full vector use network classify goal transfer take share structure facilitate cnns initialization initialization strategy cnn alternative pre complementary training put challenge object category extract imagenet challenge fine tuning target question transfer imagenet feature document feature address whether initialization challenge result initialization document cnns seek usefulness transfer feature unseen version collection resolution public record american seven document label tag tag image tag version list image collection label work relate ten image letter category present full dataset category collection
prox encounter far use projection bregman unable use bregman divergence exactly problem project divergence maintain simplex optimize structure factorize marginal polytope bethe tree strong negative bethe entropy strongly interior marginal polytope consequence definition bethe project simplicity intuitively appeal oracle average objective gradient modify nesterov acceleration technique convergence solve stable belief entropy inexact marginal define mean field inference non well unlike propagation composition first order loss appendix energy find also work problem experimentally good tb example oracle distribution c crf max tb input example clique structure max seek crf respectively mrf parameter highlight index possible datum family convex prevent conjugate duality relationship family iterative procedure present crf structure variational parameter surrogate crf give surrogate gradient break show crf I clearly parametrization measurement amount equal gradient marginal truth marginal loop doubly simplicity implement learn derivative notation case sometimes overall doubly solely converge yield recall experiment update parametrization ht f energy dd variant ht nlp citation string author etc closely segmentation soft labeling constrain predict name name number last name measurement would enforce hinge constraint dual style crucially rely hundred soft impose hinge inference tune development ignore measure baseline high optimization difference local energy match dual dd programming whereas plug energy experimental configuration relax expectation preference map demonstrate algorithm use underlie analyze gram class mm next recognition setup equally fold train validation result fold achieve extremely clique state excellent clique unique people easier solve mind convex local energy intend lie standard image marginal sum marginal give expect unique word intuition global non eq variant differently approximate fouri simply multiply pointwise non linearity baseline input order letter vocabulary cascade motivate marginal length th distinct train energy add chain version note structured cascade give structure cascade course dataset different arguably much create logistic allow choose length map variant tune fourier much expressive indicate local global important method cardinality model dramatically aggregate represent via graphical node successfully time variable correspond count location poisson rate proportional infer patterns map perform likelihood observe count hard provide additional alternate surrogate procedure expensive solve inner loop applicable synthetic code solve learning framework tractable model marginal possibility gradient substantial work problem domain agreement grant reproduce recommendation author reflect learn reasoning response distribution bind structure show minimizer mrf clique structure section configuration constraint implicitly since bethe inference crf distribution behavior citation inference local disagreement modification along rough proof case energy heavily significant self contain work associate bregman composite euclidean update distance barrier visit build regularize average different minimize associated bregman mirror descent original algorithm conjugate duality actually slight tb energy function prox first similar energy convex convexity average online convexity mirror order converge stationary large tell norm notion order differentiable surrogate surrogate composite euclidean surrogate admit establish composite mirror descent strongly bregman surrogate gradient strongly convex smoothness proposition directly bethe entropy bregman unbounde corner polytope domain neighborhood lipschitz practice mirror barrier iterative never polytope effectively purpose minimization intuitively plausible iterate corner polytope constraint learn satisfie asymptotic follow note asymptotic choose lipschitz bound set smoothness constant rough convergence energy heuristic effective future believe examine parameter contribute barrier prox g l large accelerated procedure fast polytope measure bregman us cs david edu solve structure soft lagrangian semi broad objective capture statistic maintain tractable provably inference non project generating bethe structure achieve task novel inference procedure providing highly collective graphical apply show dependency graphical relationship often dependency word phrase cyclic dependencies nlp constraint sentence likelihood token predict marginal pose clique marginal tradeoff inference score clique us enforce way work objective optimize parametric entire linear enforce property whenever euclidean project bethe generating passing maintain iteration convex convergence abuse terminology objective generalize algorithm convex marginal framework utility preference motivate repeat call see success impose similarly constraint expectation distribution supervise expressive domain rather algorithm solve parametrize use implement black box experiment demonstrate power generality achieve discriminative art learn word algorithm improvement inference structure apply let define conditional ns capture joint clique go model field expect sufficient eq specifically go often dependence combine clique surprisingly approximate structure upper parametrize compactly mrf analysis technique convex benchmark undirected graphical particular convex joint marginal tractable factorize induce partition clique marginal product full involve simple base energy base augment clique include tractable potential clique repeat outer tensor product node non generalize allow
proof angle combine come order use prop face set interested analyze eigenvalue matrix index r distinct zero add laplacian graph graph definition side complement small eigenvalue laplacian square angle use consider model mrfs capture technique belief propagation minimum submodular insight powerful specific divergence variational produce furthermore message confirm scalability quality benefit potential probabilistic central provide foundation make uncertain general problem one amount attract community notably propagation size involve optima inference process assign like equivalently bernoulli set indicate concrete show task want foreground pixel traditionally indicate foreground define quantity pixel foreground employ view set make e function submodular implication approximate emphasis submodular special dpp modeling diversity tractable even ise provide optimize partition function submodular lead problem interaction slow impractical computer show problem problem algorithm handle indeed inference image hundred thousand insight agree mode secondly connection namely specific light type log image segmentation message lastly image segmentation demonstrate exist technique provides formally say submodular set add without arise measure typical ise task introduction edge connect neighbor pixel neighbor preference assign place neighboring pixel segment penalize weight attractive behavior model go far would modify potential concave concrete example assign segment assign otherwise segment function modular see analogue function say modular arise factorize q evident modular vector modular interested polytope q modular add restriction empty though many inequality optimize question configuration log minimizing result fast know combinatorial evaluate expensive ground perform well wolfe solution minimizer barrier perform computation normalize literature compute technique common optimization variational quantity optimize technique modular analytical function submodular lower modular idea parametrize modular optimize minimize inequality separable polytope divide solve error marginal frank even map require submodular costly convergence frank wolfe method minimum contribution surprising result crucially objective consequence submodular minimization point substantial performance gain seek optimal equivalence extract map marginal exact marginal point algorithm become demonstrate extremely parallel attack partition employ factorize parametrize upper marginal distribution end turn measure enable prefer quantify dissimilarity argument pick interesting minimize factorize probability set completely prominent example kl divergence interest enyi factor minimize factorize factor minimize infinite factor current sequentially I one alternative guarantee factor generalize setting change passing describe follow norm arise structure sum belief propagation vector base base problem exhaustive propagation size send receive message projection parametrize separable polytope divide solve factor follow differently every store extract factorize incoming message variable step coordinate incoming message formally see descent discuss describe possess message parallel important maximal connectivity new coordinate depend extend analysis extension variable message pass linearly specifically optimal initial graph h h h b specify whereas offer marginal dynamic range segmentation marginal ground truth segmentation compute exact ideally wish proxy quality area roc curve ground truth classify pixel roc pairwise interaction unary potential potential shift pixel unary potential belief fractional fast relatively converge minute pairwise variation less test leave validation generate grow boundary foreground accuracy boundary std avg std column curve auc fourth column deviation precede htbp compare accuracy aggregate roc approach auc whole image challenge boundary poor alternative attribute confidence verify optimize non iteration lastly order around qualitative characteristic result marginal method minimization low prefer side propagation confident four exactly concentrated around around strong prior low mainly unary procedure last two preserve boundary benefit variational inference log interpretation reduce minimum making tool optimization available approximate inference show factorize return type immediately useful model pass exploit connection strong natural approach rate lastly challenging demonstrate marginal produce variant move high potential become intractable inference variable bf hence
langevin target mala hmc proposal preserve distribution us proposal mh general proposal acceleration evaluate ar include acceleration technique algebra use first lemma give function theorem need check similarly jump transformation obtain first hold show easy v eq eigenvalue matrix eigenvalue entry factorize equivalent spectral decomposition l ht apply show hence section proposal proposal discretize langevin diffusion mala discretize hamiltonian dynamic hmc splitting analyse efficiency ar gaussian mcmc metropolis hasting target give current draw target kernel state approximation matrix give vector draw ar vice versa radius metropolis langevin mala ar discretize langevin hybrid discretize hamiltonian far analyse although identify useful designing choose gaussian mn proposal ar proposal may ar mh accept idea quadratic approximation objective iteration concern chain monte ar target proposal target ar target scan accept reject redundant algorithm showed accelerate ar efficiency ar mh accept reject step normal ar proposal gaussian proposal hard analyse question efficient accelerate gaussian require equilibrium interested sample mainly arbitrary burn integrated autocorrelation length reduce proxy independence per autocorrelation article case feature mh correct simplification analysis keep transition change analyse accelerate analyse accelerate moreover every local distribution approximately normal simultaneously ar restrictive see several mh analysis test behaviour quadratic function gaussian case accelerate accelerate ar proposal mh accelerate normal use idea solver observe example fourth mh case mh discretize discretized hamiltonian proposal ar replicate exist absolutely continuous inverse unknown prior hyperparameter conditionally hyperparameter involve brownian purpose splitting similar make solver algorithms operation product infeasible directly section analysis jump mh ar process proposal section apply langevin dynamic see proposal splitting assess conclude remark provide ar converge spectral process q radius satisfied substitute spectral radius matrix symmetric formulae see since efficiency average acceptance show walk metropolis rwm mala hmc require expect nm lemma algebra symmetric mh define orthogonal stop spectral theory algebra coordinate algorithm transform lemma analyse need lyapunov central limit theorem expect variance theorem x cumulative distribution equilibrium eigenvalue define normal cumulative matrix e dm n g ix iy I eventually lyapunov c mcmc usually integrate thought sample markov give see unable directly splitting depend concern proxy expect successive chain eigenvector precision eigenvalue mala depend study finite jump superior mala rwm mala burn like gap eigenvalue kernel determine rate aware analyse langevin mala accept reject mh correct converge usually mala wrong target wrong depend radius slow per mala require multiplication mala langevin diffusion discretization euler positive langevin differential eq motion time current proposal target ar mala table several mala langevin mala langevin call identify langevin correspond avg v symmetric eq q symmetric yield proposal target redundant analyse accelerate performance mh proposal would theorems fix algebra mh mh moreover proposal mh splitting expect size constant equivalently satisfy mh algorithm normalise eigenvector although langevin diffusion particular gaussian also efficiency consider expected jump require independent sample mala compute multiply assume another fit proposal hamiltonian see e treat state particle momentum accord particle proposal solve hamiltonian particle modify hamiltonian proposal hamiltonian l block vector time mala immediately hmc process still process split hmc splitting hmc imply hmc target proposal hmc iteration eigenvalue system alternatively may momentum attention try theorem matrix precision simple coordinate hamiltonian hmc correspond coordinate mechanic split precision matrix hmc reveal algorithm avoid restrict extension hmc matrix balance optimize convergence rather extend independent mala suggest alternative numerical suggest variant infeasible designing proposal mh challenge distribution job make hard mh algorithms focus ar proposal high new criterion evaluate ar process proposal guide construct proposal ar process
assumption limit small pixel refer parent shift invariance pixel sharing constraint mixture conditional take covariance variance dimensionality neighborhood far introduce factorize additional sharing neighborhood derivation multivariate supplementary describe memory spatial lstm cm pointwise product depend memory lstm memory precede state forget sequentially read produce hide vector pixel hide fed factorize predict pixel px ij recurrent much large region pixel far stack image c boltzmann rbm try weight sharing rbm et al autoregressive unit et al sequential manner draw bernoulli treat make difficult mean one video optimize al try step pixel contrast try pixel intensity heavy well modal momentum bfgs except early stop indicate recurrent augment conditionally whiten let pixel causal conditional whitening replace dependent evaluate change variance jacobian neighborhood ensemble pixel improve simple trick produce ensemble without need transformation leave invariant ensemble k simply mixture image yield ensemble argue leave natural invariant nevertheless boost dim gmm deep layer em dim layer layer recent image patch sample berkeley dataset strength capture correlation follow rgb turn account discretization split image contain dc live bottom pixel discard validation patch evaluation factorize pixel pixel fix find outperform single deep model table outperform ensemble knowledge currently density dataset try compute pixel pixel lead explanation intensity zero indicator neighborhood pixel treat infinitely image bfgs pixel causal neighborhood epoch range backpropagation log analogously average likelihood directly ensemble approximated evaluate lc layer able scale axis xlabel neighborhood ylabel log legend align legend anchor south east font xlabel near ylabel major font style axis mark densely width color coordinate mark solid line mark option color red green two comparable transform rate dc jacobian transformation two comparable sense patch would independently highlight result achieve gray large benefit patch rate apply large capture patch factorize approximately gmm frequently dataset van contain dataset pixel linearize intensity evaluation account discover section several model large patch patch model simple procedure correlation leave stationary statistic model pixel recently multiscale image greatly improve hand yield add lead layer recurrent par ensemble previously result dataset improve simply causal increase instead likely cause cm figure figure cm cm figure cm figure cm figure cm cm cm cm figure cm cm cm cm figure sample train texture illustrate capture recurrent kind capture capture try pixel pixel select purpose train epoch range pixel model correlation marginal periodic although reproduce periodic well suit failure likelihood indistinguishable region model correlation region miss sample resort miss initialize candidate one large sequentially overlap pixel metropolis proposal via accept patch joint density costly gibb introduce recurrent insight generative superior performance quantitative important abstraction collection factorize version large neighborhood parameter perform block video long network apply natural prove generative conceptual model abstract level author foundation challenge partly extend hundred pixel range dependency number problem recently generative short memory modeling image arbitrary tractable art texture synthesis see progress lee drive improvement supervise potential
hoeffding product lemma distribution noise easy reader paper integer linearly draw simplify notation generalize parameter recover definition resp distribution kullback return resp failure cost reduce solve negligible bound sample product transform remain return small bias advantage average large algorithm sample bias part except hoeffde mb kt main prove dimension natural distinguish yield uniform variable sample hence distinguish thus crucial gaussian elimination sample produce sample block iterate consecutive eventually sample obtain sample ultimately consecutive later adapt improved modulus switching vector merely may perform round produce essentially reduce balance decrease complexity round costly add round contrast make modulus hand maintain decrease allow balance modulus switching point view entirely idea result later combine sample repeatedly attain repeat sample quantization produce center modulus simple proven depend index point center without improve modulus might code partition k presentation setting specific exist true later coordinate define fix terminate distinguish equivalent terminate run accord noise independence uniformity add get noise indeed hand optimal amount baseline list empty final bias balance convenient auxiliary decide dependent parameter choose failure induction lemma choice part superior hence solve time choose parameter fulfil amount get kb kb bx ik finally already quantization else amount error find bound previous discover work correct except return negligible lattice problem factor goes use author incorrectly round p q reach actually assume part solve possible work big final need propose heuristic exponent sum sample independent lose aspect negligible practice associate opposite value coordinate second step sample center whose center simple pick thus short ever gain somewhat add cumulative vector gamma amount suppose fixed proportion formula sum within ball radius able much technique keep proportion instance twice fall sample notable newly coordinate previous completely ability expect norm minimal count within complexity factor bernoulli practice transform high match another significant improvement linear quantization quantization center basis decrease besides available help entry transform sample try amount available impact practice continuous modulus multiplying reduction assume variance correspond keep vector test complexity bit operation use multipli optimistic complexity c reasonable optimistic error optimistic distance thus solve repeat solve close access free lattice follow lattice last stem assume modulus eq lattice oracle lattice call n since bind remove solve apply large solve reduction solve polynomial zero lattice vector shorter impossible q b need prove intersection ball radius divided volume ball solve infinity negligible failure use call oracle lattice oracle law uniformly center origin return complexity k probability let summation bb definition n polynomially previous chebyshev previous polynomial prove sufficiently easy operation integer order circular generating let axis align cube length radius sufficiently solve solve solve independently trivially sum possible cube radius include ball subset problem q failure failure therefore break lattice slow nm nk generalization gr hard lattice theorem justification claim prove assume efficient use solver distribution either sample reduction distribution integer uniformly whose sample else distinguish sample distinguish follow uniform uniform counterpart rank result modification uniform accord return remark dimension take coordinate switch feed output uniform therefore statistical efficient distribution take error resp solver let uniform poisson property hash bias negligible failure hardness apply introduce failure error apply particular round direct solve lattice nn recover significant bit use principle coordinate radius ball radius contain sample ball add output add element coordinate clearly coordinate maximize fouri odd bias distribute small fourier introduce time lower universal introduce bias stage determine approximation use fast fourier transform recover continue get final repeat whole distortion first iteration input large small inequality reduction except bias lattice lattice short inferior lattice stop call basis proportional v dc algorithm lattice basis vector ba cb binary since count clearly difficult lattice subset sum density lattice complexity find remain precisely form column lattice coordinate inferior select nd n b binary apply require exponential number disadvantage vector preserve decision give deduce horizontal axis represent modulus classical gr hard colored contour attack choice vary increment q distribution failure choose n negligible apply switch distinguish output vector failure new except union n np fr paper interval introduce variant rely quantization generalize fine modulus significant gain front exponent dimension introduce variant require solve analysis require break security solve subset independent hardness decision n nb ne q concentrate given come average approximate bad show reduction modulus finally
component scatter plot expert fit propose quasi identical line correct tune cc right right predictor actual training experiment skew ghz processor gb fit bottom right except figure expert differ see one htbp c c estimate component bic give criterion poorly bic correct correct number component correct suggest ccc ccc ccc k aic aic bic aic laplace mixture expert htbp e value anomaly choose criterion bic aic except provide evidence htbp l ccc ccc ccc c bic aic anomaly normal skew suggest tail suggest heavy tailed noisy datum infer model successfully confirm evidence normal alternative propose successfully include tend aic poorly analyze obtain version future mixture natural future extension regression rather universit france universit france model heterogeneity cluster expert component group group observation asymmetric behavior heavy use expert fit introduce normal expert issue possibly skewed skew skew respectively develop dedicated em monotonically maximize log present experiment carry propose term show usefulness change keyword expert skew skew em expert study statistic fully proportion density expert analyse different context cluster usually expert expert well outlier may affect paper attempt overcome propose adapt deal possibly recently regression maximization mm normal skew normal skew beneficial asymmetric namely asymmetric univariate skew natural robust integrated develop propose recently robust univariate mixture sufficient asymmetric mixture skewness tail skew mixture skew expectation extension bayesian multivariate skew robust univariate regression laplace regression mixture expert mixture regression mix proportion mixture robustness deal asymmetric attempt overcome limitation deal asymmetric tailed contain use skew normal commonly skew normal accommodate asymmetric tail regard outlier skew tail unconditional normal expert mean covariate maximization model maximize model expert show maximize case stable log monotonically step maximization parametric non hierarchical expert expert model organize maximum estimation em derive estimation technique show perform non linear model perform conclusion mixture expert context including consider regression aim covariate via explore unconditional distribution modeling took distinguish regression regression generate hide categorical random component observation mixture proportion sum k although aspect difference consist conditional model model softmax expert analysis modeling proportion covariate proportion logistic covariate vector mix proportion densitie expert conditional proportion regression normal expert follow semi parametric case log likelihood observe respective log step em calculation follow posterior estimation update complete expert expert vector coefficient consist analytically linear proportion update reweighte heavy sensitive skewness propose normal expert fit propose expert tail skew expert skew expert component describe integrate follow skew skewness density pdf cdf skew normal skew denote skew hierarchical introduce skew skew mixing assume skew mixture extend skew framework conditional distribution skew skew expert skew parameter skewness expert component skew normal obvious see skewness stochastic representation derive representation skew normal representation skew covariate follow multinomial I I hide label th observation stochastic representation lead inference introduce iff th respective vector maximization log however framework maximization maximization algorithm dedicate variant mainly several maximization divide sub space sequentially one coordinate block algorithm complete z I I propose perform start cm step function expectation eq ik label correspond membership observe correspond hierarchical show mixture bayes expectation calculate analytically calculate parameter function adopt extension consist maximization decomposition step calculate skew skew maximization close reweighted square respect iteration newton update gradient hessian take maximization expert solved analytically calculate analytic calculate maximize k consist equation update skewness skewness update correspond standard characterize handle skew tailor skewness contain heavy tailed affect sensitivity outlier tail mixture propose robust normal distribution describe representation handle accommodate tailed univariate location mahalanobis distance gamma give gamma bx distribution give hierarchical express expert extend component mixture form univariate mean linear proportion expert expert location degree vector k approach inference procedure let hide iy follow categorical variable multinomial hierarchical present mixture distribution hierarchical unknown maximize perform maximization describe maximize vector e expect function latent complete complete observed estimation n ik ik membership probability easily step maximize km update iteratively mixture calculate kt mean analytically update note ml component multivariate algorithm modify may calculate degree freedom equation scalar root mention constitute tail derive describe single model give consist take maximization cm degree expectation tail affected skew mixture expert attempt accommodate heavy tailed expert component skew normal respectively skew hierarchical representation cumulative cdf freedom introduce random variable normal univariate skew freedom skew hierarchical stochastic skew give skew expert first introduce framework skew component skew proportion mean mixture see robustness approach flexible accommodate tail skew categorical label component generate skew say skew mixture expert skew hierarchical maximum model perform dedicate consist variable label hierarchical representation log k ik start cm step complete log current see conditional ik ik ik follow expectation namely case skew propose ik ik note adopt approach expression rather carlo mention exact expectation place maximize provide maximization carry update iteration parameter calculate tm tt expert provide update skewness update maximize show degree fix update calculate equation find component update hand degree update predictor expert therefore model training via prediction predictive q compute expert give k variance model describe normal variance expect propose case expert calculate respectively eq follow mean model mean easily thus expert variance perspective assume component skew skew expert interpret mixture dedicate algorithm provide represent fuzzy partition membership respectively membership apply maximize posterior base problem form one aic bayesian integrate observe criterion maximize express observe complete convergence correspond free proportion transformation covariate univariate covariate work case expert linear regressor covariate correspond univariate covariate dedicated evaluation simulate evaluate clustering implement matlab mix initialize randomly equal initialize partition fitting initial membership replace robustness initialize randomly skewness randomly stop log process likelihood illustrative value output I expert partition mix cc fit normal toy analyze experiment impact generation accord model generate mse component estimate one error average trial l table obtain three decrease confirm property mle mixture decrease pt mse parameter mse component estimate c mse one sample addition previously figure quantity counterpart plot show minus twice true middle plot true function middle show counterpart bottom em show mix probability clearly estimation correspond additional propose generalization expert
simplify positive q fairly verify complete controller sampling finite population specify time assume variable unknown controller policy expect sum outcome equivalently regret lack asymptotically horizon keyword sample armed bandit family support population population controller f discrete indicate controller bandit dependence controller grow equivalently restrict bandit ensure bandit bandit would bandit bandit bandit surely identify focus unknown kullback generalization part policy bind specific eq give policy policy achieve maximum optimal first show asymptotically policy construct bandit number index index current estimator eq choice n therein automatically satisfied condition see c condition essentially k f optimality policy often take space potentially bandit herein bandit sample early include asymptotically consider derive policy achieve include poisson distribution belong exponential far bernoulli thompson optimal sample likelihood respectively maximum policy indicate would lead n equal eq q break tie arbitrarily ta satisfied fail optimality technique insufficient verify may negligible tie arbitrarily remainder optimality additionally horizon bound remainder refine somewhat work convenient bandit take bandit recall follow sub optimal delay briefly present result infimum complete remainder eq q inequality apply interval take I proceed bound three observe hence observe indicator term account possibility second q inequality ti minimal span follow finite joint simply convenience observing result point convenience constant nice refine strong remainder complicate particular suffice take building take bandit value
also connection contraction property hilbert kalman contraction map metric thompson indeed fundamental property endow kalman filter prediction dynamic incoming order internal semidefinite discrete algebraic kalman invariant converge step show covariance monotone limit line formula contraction convergence kalman iteration flow contraction hilbert contraction understand assumption strict convergence kalman iteration combine iteration indeed ii underlie metric contraction riemannian hilbert metric drawing link contraction property hilbert metric metric definite metric pointwise filtering end cone matrix pointwise introduce strict positivity contraction base hilbert banach space cone empty iii k mx x hilbert cone positive say relevant triangle metric discrete expand definite strict contraction hypothesis contraction thompson discrete expand hilbert positive kalman q far mutually kalman observable component hide countable common use setting countable interested parallel transition countable state space measurable qx yx qx stochastic process value common space transition namely formally space hide determine equivalently x wiener process work filter measure initialize induce normalization pointwise map split ahead projective forward filter let composition three map iii expand hilbert see amount isometry thesis composition operator kalman span variable kalman iteration see gaussian literature see combine contraction briefly kalman kalman model emission constant connect kalman filtering recursion kalman recursion emission virtue priori give virtue
expression parameter partition free energy complexity familiar degree freedom regular aic suggest frequentist penalty singular contain fisher finite aic propose frequentist singular pearson hypothesis hope asymptotic well likelihood g detailed description objective frequentist weight posterior distribution parameterization understand model name weight prior perspective important significance resolution objective machine consequence weighting increase also complexity dataset information weight rather interval I true prior maximize optimize analogy predictive unified discuss directly maximize far prior also understand although strict improper rigorous long history successful improper jeffreys despite considerable approach propose improper prior perspective uninformative since scenario describe state interpretation uninformative parameter maximally uninformative sense maximize content interpret demonstrate regular desirable short come weighting interpretation believe desirable mathematical necessity three principle statistical objective bayes probability interpretation maximally uninformative information aic regular asymptotic limit free ad hoc complexity complexity applicable bayes methodology formally bayesian unknown second formulation begin gauss inference although arrive conclusion summary use principle implicit indistinguishable parameterization model understand maximally uninformative establish prior connection demonstrate regular bayes criterion introduce observation parameterize parameterization probability distribution bayesian realization parameterization introduce marginal partition nee normalize marginal parameterization posterior bayes update meaning initially prior free q closely performance discuss ability bayes model predict convenient formulate model information laplace assign exclusive equal prior lead unclear mutually exclusive exhaustive uncertain origin parameterization cutoff volume specificity gaussian mutually exclusive intuitively know difference sufficiently distinguish small observation value always resolve exclusive divergence assume indistinguishable exclusive sum exclusive write volume drawing volume fig simply time information learn machine drop dependence keep purpose retain factored sum make definition density model depend describe divergence fact need propose precise density first bayes q indistinguishable discussion indistinguishable unity improper name unbiased improper long give eqn eqn eqn predictive last probability connection meaning analytically continue temperature identify free angle parameterize physical average gibbs follow understand model indistinguishable indistinguishable illustration take key avoid validation true interpretation wish interpret code parameter careful validation posterior normalize integrate cutoff determine total strictly convergent interpretation respect uninformative wish code informative localize intuitively expect uninformative prior result content uninformative since gain maximally uninformative argue uninformative analogous use code understand maximally uninformative locally possess maximally uninformative regular special
certain predictor fold calculate use likelihood compare choose baseline global trade considerably recent decade international increasingly global air network price customer air trade express forecast decade attention surprisingly little air poor business air consequence significant service level reliability deviation arrival customer cause delay production service incur storage handle cost risk customer risk routine deviation refer survey name management strategy empirical study air chain literature interesting phenomenon observe datum include etc show figure figure risk observe datum clearly distribution positive well around concentrate day peak largely fail later usually international hour gap thus transfer gap peak detail empirical study primarily focus arrival delay review delay positive concern assume delay unimodal adopt ordinary least multimodal observation ols air predictor unimodal normal distribution mixture delay need develop new model contribution accommodate multimodal risk introduce art bayesian tool rapid development decade accelerate ever several year diverse finance reference therein estimate independent identically observation arise give parametric bold assign discrete finite process adopt specific formally importantly computational conditional create genetic develop variable risk characteristic range cover risk risk within allow relationship risk predictor include etc explore way reliability demonstrate risk estimation ol dramatically ols fail level importantly ols risk insufficient management drive powerful general assessment receive risk management attention practitioner co business company assessment risk assessment risk must execute update experience involve record long experience business detail carry quantitative quantification analytical management step develop negative event assessment assessment consist estimation hazard occur term notice distinguished impact part chemical use environmental conceptual risk chain work focus call advanced computation correctly implication alternative management develop tailor service customer service resemble capacity risk component risk assume particular bernoulli risk introduce organize introduction air challenge question exploratory lead introduce posterior gibbs propose several model operational conclude future detailed checking air public operate necessity brief explain motivation standardize examine air four short chain air provide company origin date detail piece service pick share connect map term refer customer air service use service upon request request several economic certain percentage include combine leading enable participant reliable entire air operating plan develop monitor air allow control major ground etc see appendix member management implement different control confirm create share describe along customer agree plan agreement essentially combination profile duration completion define system alarm correction take responsible party back meanwhile exception system illustration end keep party customer directly compare service customer direct include predict risk risk reliability help address customer volume demand variable month decision duration description customer provide customer next elaborate demand differ dramatically across air service level factor month demand weather e trend air service finish month approximated piece fail capacity large usually valuable high analysis help reveal dominant substantially across variety factor connect strong predictor variable duration number greatly reflect send early onto early weather available allow distribution company datum update five north sa south figure see depict service available around column figure direct service variety impact service cc l risk predictor table facilitate cause delay failure company datum helpful delay code delay day code appear denote use exception provide motivation mixture model decompose two part mixture model discuss selection provide ols aggregate histogram empirical accurate inference first datum usual multimodal mixture rely investigate risk demand decision categorical one stream double kernel popular mixture model joint dirichlet normal response normal express prior b specification involve carlo limited capability high require algorithmic proposal need carefully adequate gibbs update conditional superior tractable strategy augmentation specifically observe correspond replicate replicate drop replicate observation help indicator gibbs classify category represent gamma conjugate q n xy x indicator multinomial probability multinomial conditional l x variable n beginning equal augmentation scheme simplify following augment prior due similarity updating scheme coefficient j posterior easily generate slice discuss finite value conservative many utilize explore need label move mode set mode negligible probability framework switching greatly improve appendix explanation move consider parameter location mixture case global observation rough argument scenario prior weakly lead improve improper discuss prior transform convenient parameter simplify assumption component mixture dependent constraint baseline build hierarchy heterogeneity small lead set especially sure yield size describe characterize limit risk fitting inference range quality assess gibbs sampler iteration iteration burn matlab ghz intel dramatically improve efficiency rely recent development prefer diagnostic adequate lack fitting validation check visual cross check sample capability overfitte log strictly forecast log follow appendix model ols delay replicate ol separately check specification expect something column resemble test ol mean histogram represent posterior solid vertical dotted ol largely hour delay recurrent hour ols datum solid dot deviation predictive true check fitting level histogram draw real interval narrow predict capture location weight predict cm p p p mean interval thing range deviation day normal narrow flexible parameter range variation certain present distribution narrow estimation name great heterogeneity measure among confirm aspect close inspection reveal whose coefficient impact base ol significantly huge estimating independent mean playing peak ol detect impact considerable summary table hyper heterogeneity huge risk aid decision use provide demand variable help preferable service help price customer demand requirement choose service service initial start customer expect customer customer interesting several generic aid pick arise risk neutral delay early proper delay certain example deviation function expect loss analytical short figure present expect choice service service play dominant normal service estimate offer price different increase price sensitive price unlike business solve integrate set integrate practical certain decision service reliability critical interest replace capacity pricing generate comparison baseline certain effect thus type impossible baseline effect plug level sample distribution differ location peak allow comparison offer much rich comparison metric meanwhile rich comparison single truncate risk plus minus initial delay applicable zero otherwise author argue short incur incur author wise international air analogous schedule calculate ratio effect exclude thus calculate intrinsic service baseline
lf ef lf lf ef lf ef lf ef lf ef lf ef lf ef lf lf ef lf ef lf ef ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf vary learn inductive self clarity ep lr omit ccc l co train ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef ef lf ef lf ef lf ef lf ef lf ef ef ef lf ef ef ef lf ef lf ef ef ef lf co train round ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf lf ef lf ef lf ef lf ef lf ef lf lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef ef ef lf round ef lf ef ef lf ef ef ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef lf ef ef lf ef lf ef lf ef ef lf ef lf ef lf ef lf add round decrease amount observe omit sake brevity suit image lr fuse round baseline ensemble strategy powerful result descriptor extract train convolutional neural network consistently superior highly tune art visual feature implementation cnn image annotation convolutional extract last leverage three discriminative cnn alone see apply four scene four significantly discriminative cnn baseline mid level feature want substitute unsupervised kernel pass exploit unsupervise lf multiple used view experimental scene scenario plot confirm suit label image prefer label htbp cc qualitative class instance add round correspond part fig add round belong current green box red belong image add round pattern find occur see moreover classified increase confidence move leave class decrease right namely different training randomly contain number test experimental ten label similar plot show suited prefer available remarkable improve cnn htbp ht ef lf round ef lf round ef lf round ef lf round ef lf round ef lf co round train report height test image decrease classification confidence classification unlabele classifier scheme early fusion test scene three different experimental inductive test co outperform art class training round result image add label mid cnn art effectively leverage discriminative feature boost I e co co iteratively build two view view learn combine conduct experiment recognition feature combination extract ensemble unsupervise couple result semi supervise analysis supervise supervise account machine particularly instance many semi generative co difference classifier datum independence view success view technique work exploit way first representation unsupervise label train build sub separately feature strategy co unsupervise component schema fig view single feature recognize concept classifier exploit accurately recognize argue scheme together unsupervised make effective co build unsupervised component assess conduct set datum set scenario unlabele test coming set verify efficacy also unsupervise couple logistic semi result show outperform learn sake brevity discuss supervise book co verify web page basic idea train classifier confident concept co annotation recognition traffic analysis speech recognition retrieval image object individually conditionally label show co error support view detail year success framework observe automatically effective training boltzmann auto encoder network notable method conceptually occur unlabeled mean widely purpose vision lead variant representation briefly use vocabulary extract similar visual counting occurrence visual vocabulary feature unlabele learn noisy multiple accord representation implicitly embed kernel combine information issue recognition fusion bring complementary improve retrieval modality fuse fusion early position whole early usually refer single refer combination stage obtain comparison universal strategy prefer give task fusion semantic indexing get indexing fusion mkl support svms single computed svms mkl learn determine different different former choose correspond multiple different fusion consists label l view early fusion early ef use ef fusion lf unsupervised learn representation lf ef l ef ef ef lf lf lf lf ef lf view co train classifier regression co train view e ef ef l lf lf iteratively confident confident pseudo classifier ef ef lf lf ef lf round training choose vice us call candidate unlabele belong unlabele ef confident confidence extract confident q choose view pseudo label complete outline em ef ef lf lf ef lf ef ef ef lf ef ef ef ef ef ef ef ef ef lf lf lf lf lf v v l v ef lf lf l ef lf ef ef u lf lf parametric projection unsupervise train logistic latter another scene scene divide scene category category image category imagenet datum self current version imagenet come pyramid histogram orient gradient rescale scale pyramid neighbor use learn ensemble train plain projection learn prototype pseudo label indicate belong image sample prototype prototype large keep classifier prototype vector feature project follow logistic classifier unsupervised ef fusion lf vector make single ef lf kind compete classification co three scenario label semi training classifier unlabele take imagenet unlabeled test set evaluation average precision recall test scene e random label split strategy built label concatenation available ep ep specifically test train classifier operate fuse employ ep lr lr lr differ pseudo add round maximum add two co classifier number per scene five svm result report show ht test baseline variant available htbp ccc scene ep svm ep lr lr svm ep lr svm st svm
four predict display capture corresponding provide technique employ c mesh bs bs ls region analyse level certain hour around united air environmental bs simple spatial intercept spatial version bivariate spline stationary represent b location basis table basis direction representation employ cross score table bs log score bs g generally bs bs g representation yield small stationary bs display fact score b high stationarity selection aic beyond stationary predict united states bs display prediction present south corner north east corner north area stationarity htbp great triple directional coordinate directional directional derivative directional integral inner product calculate precisely area define note triangle basis construct easily spline coefficient spline triangle contain bivariate function test leave identity fact q th entry diag diag diag representation spline follow lx lx lx product integral e f bivariate spline suitable easy j associate cc eq piecewise result prove general note recall bernstein approximation arbitrary df let x fact cauchy schwarz next eq constant similarly put estimate length edge proposition approach bivariate spline field computational bottleneck gain partial use element new polynomial representation degree infer bivariate numerical also keyword spline non field especially structure additive name flexible extensively statistical computation typically order computational approximate bayesian assume integrated nest fast inference mcmc mat ern modify nominal correlation marginal integer determine underlie noticed mat ern fractional innovation white unit q g require suitable construct element gaussian piecewise carry dramatically closely finite recent propose field basis function rate finite refinement underlie alternative high degree polynomial element polynomial order multivariate employ conventional finite easy piecewise polynomial various spline show efficient conventional element solve spatial thin place concentration forecasting spline advantage piecewise adapt efficient spline review link establish approximation discuss extension stationary several numerical conclusion discussion proof space eq polynomial degree continuous possible continuous spline triangle form spline construction locally calculation appendix triangle coordinate polynomial call bernstein polynomial triangle spline basis spline eq find sensible set least square gaussian operator bivariate mean least square diag diag b recursive solution sparse make mass diagonal next application mention element previously bivariate theoretical section integrable finite bivariate spline span finite sequence edge th appendix spline converge obtain derive bivariate spline field triangle clear able spline high magnitude matrix approximation bivariate projection bivariate basis base solution covariance spline result show g norm edge spline decrease fact illustrate bivariate spline show extend stationary parameter location vary slowly domain representation basis guarantee domain associate see easily minor modification bivariate representation square solution locally interpret mat ern field local mat ern automatically conduct simulation spline finite term parameter ern model integrate nest laplace approximation brevity bivariate denote bs bs bs element study bs l fitting surface surface space surface mat covariance whole fine equally spaced surface nf four different include element spline triangle associate cpu denote recorded weight complexity operation stop function b function mesh present second bs bs l level different surface efficient reach low bs l bs bs bs right precision dimension surface reach mse high level bs bs l basis function compute bs generally gain bs bs mse surface previous bs degree level around reach level bs low level bs reach require quite neither b l bs reach require basis l efficient level function low b note shape surface comparable gain bs cm study bs prediction national center de choose cover relatively near three bs rmse location aim surface suggest estimate surface close validation employ embed predictive log six figure extend
fourth b rule inner eq jensen denominator plug mean identical update inequality algorithm k coordinate inside bind q term since divide side inequality prove lemma trading ingredient decomposition exploitation first reward explore decomposition schwarz deviation w regret exploitation regression matrix constant response feature true vector thus write subgaussian independent subgaussian reveal universal lead exploration round give exploitation regret failure union union event since exploitation challenge part phase exploration small probability follow lemma analysis algorithm characterize consider context denote feasible action round eq vector action action apply hoeffding realization round random randomization leave dependence randomness easy since hoeffding reveal round regret bound last randomness action regret round q us characterize regret action cauchy appear give account randomness algorithm notice bound deviation regret insight regret stop round two round regret round associate round bound q threshold set tn w probability bound hold proof know reference rather first bernstein type hoeffding assume value hoeffding due subgaussian fix surely probability hoeffding adjoint self almost rgb rgb pt learner receive action crowd search domain analyze enjoy linear transformation explore learn use achieve unknown algorithm explicit enumeration consequently computationally fully set partial great recent motivating observe prescribe patient work internet user reaction click article content framework learner observe take capture aspect learner maximize many round application decision internet application recommend information rich learner tuple receive composite simple play additional feedback unless relate refer rather feedback derive contextual semi bandit goal build contextual enjoy hardness composite interaction compete mapping composite tree neural net access warm idea identical efficient run logarithmic contextual bandit linear composite encoding valid move still composite feedback feature however still stage algorithm choose empirically scale call generalize contextual rich setting form grow body work bandit refer bandit majority contextual work generalize bandit contextual bandit require explicit enumeration instead impose linearly linearly work bandit assume sum generalize reward attempt strategy cope space linear bandit reveal include action unknown playing assume relationship reward composite action crucial action context let mapping weight policy one length bandit weight learner play round context learner class instantaneous success regret eq avoid enumeration exponentially access henceforth comprise contexts weight oracle composite structured mechanism composite action define marginal vector contain round aggregation composite avoid unbiased define counterpart nonetheless empirical context typical weighting scale typically desire set mix action could achieve uniform composite smooth uniform action play efficiently action indicator well composite action probability maintain non place default policy smoothed project always capital letter composite action letter action contextual semi bandit weight exploit failure zero kp tt tx ta ta l ta p structure bandit policy use smoothed play action distribution interaction op feasibility look thereby place ensure variance bind policy ensure exploration amongst exploitation tradeoff encourage place good encourage main op modification constraint action obviou simple action action constraint lead tight weighted improve regret modification reward estimate without equation really modification modify reward influence universal algorithm scale feedback additional relate result fall exploration direction restrict affect efficiently optimization oracle bad discrepancy contextual setting style contextual aware involve qp ks computational bottleneck solve focus subroutine classical contextual update violate equation add policy constraint long violate shrink weight involve calculate easy together decrease potential update shrink execute construction bad alternate update prove theorem horizon failure tx ta tr ta ty pa ir ia remain round context first explores explore thing happen action feature accurately expect policy horizon subset simple display round reliable weight vector well policy take inner policy remain play analog setting would action round estimate reward exploit exploitation thing round least square line use explore condition behave policy exploitation ensure phase difficulty condition round proceed also upper exploration regret accumulate intuition reveal least regret guarantee sublinear simple feedback epoch greedy greedy action hard exploration relate minimum enable learn weight regret exploration reveal ahead feature action classical bandit ignore algorithm shorter leave sketch proof detail intermediate stop exploitation term true stem deviation provide provide first term straightforward grow exploration suffer may eigenvalue exploration round ta follow cauchy schwarz imply cumulative equation explore uniformly subset round hoeffding sample concentrate feature round immediately translate accumulate equation come argument come deviation composite observe tuple simple regret provide regret algorithm leverage feedback scale action arise regret feasible contextual combinatorial bandit bind obtain avoid composite action partial feedback many application call beyond transformation hope address question careful inductive op first deviation variance estimate use suitable variance deviation spirit theorem argue version union tp schwarz inequality prove lemma lt bound one game empirical round martingale moreover use schwarz claim equip main note since hold policy set straightforward prove round achieve trivially choice deviation kt p deviation constraint policy induction definition lemma
form vote monotonic function much answer vote majority worker odd weight rule assume among class odd confident grow linearize theoretical desirable weighting monotonic exclude label property focus worker label aggregate refer worker problem analysis aggregation select worker require unfortunately weight majority voting majority vote unbiased label probability depend worker term gap vote I weight majority function worker give assumption coin worker coin model bound score coin natural unfortunately true unknown unbiased I tend informative answer side worker balance encourage worker ii simple estimation directly term worker favorable confident concrete question number answer worker question estimator interval defer modular modular worker pool question constraint worker sort select rank evaluate worker find worker achieve set objective maximize actually pareto exist feasible improve result intuition small rank perform good worker naive top worker algorithm tends select top rank specific practical experiment worker complete question platform ii task subset worker distribute answer iv final label though worker derive em worker ratio worker top worker worker algorithm aggregation perform bad omit plot clarity trial algorithm item randomly collect estimate item trial store average base com question know truth worker finish question typical question year internet create widely include use worker select worker htb task identify wikipedia option highlight sentence refer truth available complete typical example follow file run framework page runtime option http wikipedia http en wikipedia http en wikipedia http en run ask contain b worker actually supplementary worker achieve worker show worker true objective optimum different unbiased confidence optimize margin confirm select mainly worker reliability close truncation prevent go worker improve pool increasingly less bad estimation match decrease reliable worker select crowdsource labeling demonstrate simultaneously worker future advanced model supplementary material guarantee weight weight vote aggregated potential introduce ambiguity label event relation provide voting weight apply hoeffding hand straightforwardly depend thus valid say e desire thus move term construct lead follow lemma straightforwardly interval base form note collection variable hoeffde bind meanwhile cover least optimality give treat maximum worker select worker worker globally configuration value global optimal worker optimum maximum yield exactly therefore globally worker mention multi score worker actually multiple pareto theorem improve objective htb liu department computer science california crowdsource worker pool maximize accuracy natural worker allow analyze typical crowdsource worker simulate world able quality worker perform sometimes budget crowdsource possible collect human intervention large cost micro amazon crowd human task short payment unfortunately pay worker label low expert redundancy crowd answer answer much individual worker sometimes crowd diverse weight properly aggregate answer large body deal diversity method often majority voting answer majority weighting account use answer von liu liu necessarily well idea bayesian able procedure inference however worker never perfectly add worker extreme completely random answer attempt label able perfectly zero worker dominate quality worker recent empirical aggregated number accurate assume pool test gold question label aggregation want maximize budget worker I worker high
interaction environmental location assign label office achieve spatial understanding rather precise efficiently facilitate carry mostly modality informative place exploit two map knowledge new environment map insight place problem raw encode environment feature raw opinion hold fully exploit achieve high encode spatial consistency spatial proximity field corner contain enough office room robot mixed class input paper edge relationship information recursive input input contain field view represent increasingly classified raw automatically impose adjacency similarity map map feed learn form semi step predict label make tree carry remainder paper organize introduce construction tree make supervise validate section environment demonstrate effectiveness camera classifying place al al extract feature et place base range easily sensor vision nonnegative nearby contain include clutter fed label environment al classifier class furth address solution place specify unsupervise deep include object natural recognition discover extract success unsupervise end feature locate research characteristic consideration consistency feature process invariance work implement graph model utilize base effectiveness embed reduction paper view rise classification field boundary propose construct multi classification follow represent generalized topological successfully general meet meet adopt resolution level number layer denote layer denote sense local represent assign connection describe adopt angular l e r le le l implement demonstrate build low layer explanation contain without child connect obviously carry neighbor end detailed fusion otherwise eliminate high layer layer apply times th process illustrate figure move node decrease elimination illustration layer map node distribute high structure abstract consideration compose compose compose red eliminate layer recursively generate describe construction generate state end respective integrate give mapping layer carry achieve transform frame assume knowledge robot pose virtual generate ray angular range dimension different interpolation fix fact completeness proportion measure uniformity however interpolation range interpolation apply pre range keep sequence input follow v fr end eliminate layer black preserve reveal connection illustrate red position blue environment ray sequence one input obtain predict tree figure correspond maximize intend tree structure parent range reason layer number leave root factor compute classifier input sequence give lc c l ic j ic j lc ic I tree denote label optimize label tree structure decision consecutive layer layer always optimize confidence optimize optimize label tell change child confidence advantage optimize obtain optimize label leaf evaluate leaf clarity classifier separately construct classifier predict node assign label layer show child decision firstly initialization upper tree respective node label finally compare optimize figure datum testing capability automatically learn indicate raw label thus omit note map semi rich spatial classification convention denote way denote label give illustrated figure firstly feed red difference scan consecutive difference rich extract practical sort feed stack auto encoder build deep decoder learn encoder stack sigmoid decoder reconstruction weighted encoder impose classifier network parameter random discard pre preserve auto encoder regard work softmax learn stack auto belong whole preserve softmax consistency regularization fine tuning cost follow last layer respect input two cost cost cause error classifier impose hide learn weight regularization term detail construction build step firstly weight employ adjacency force close regularization input euclidean weight closeness input connect belong office although force close however keep validate end multi layer conduct datum international environment include centre university technology research centre artificial intel state robot collect maximum horizontal view place information class therefore target number office room room laboratory among six leave many utilize testing
k mle therefore limit easy approximate small ease illustration strictly abc whenever refer dc dynamic dc figure exact mle corresponding course hide simply gamma shape schedule set dc decrease iteration observe rate satisfactory keep last draw take corresponding error return good variability high dc accuracy find variability growth model model result cite reference explicit solution w wish preserve positivity conduct b determine since deviation observational normalize stability dc usual keep decrease small threshold increase schedule last acceptance comment dc relevant beyond acceptance stochastic markovian issue exact abc dc obtain system determine estimate vice versa let subscript maximization taking transition time likelihood z pz z obtain error asymptotic use observational time value estimate error also identify difficulty space abc dc thresholds acceptance rate determine satisfactory notice concentrate posterior maxima though strategy integrate approximate initially sampler beyond mle enhance however simultaneously criterion highly become increasingly unlikely accept increase abc dc produce reasonable inference even work value keep fix vary furthermore start mcmc mcmc start abc mcmc stage use possibility metropolis propose stage methodology rely within partly value value increase free differential equation present methodology estimation applicable main question since abc produce generating compare accord well inference unable rough locate mode conduct approximate basically mode switch likelihood propose draw use sampler realization datum might dependence go work assume model method stochastic unknown think measurement assume unknown measurement ease notation inference analytically distribution implement draw posterior carry chain monte carlo mcmc embed mcmc procedure see bayesian method strategy lead term consider enter enter state write independence nature latent integer copy stack time say generic individual latent imagine series result px k gx depend index integral write result mean choice sampling call generalization general proposal distribution convenience vx px initialization fix generate calculate acceptance large enough produce stationary distribution discard obtain mle returns covariance mle choose degenerate mle simplification take call expression simplification solve ready density deal posterior posterior increasingly deep mode existence fix start value enable rapid help analogous issue occur proposal difficult model process example trajectory result quite distant value posterior rarely simulate blind exploit many tune smc proximity forward smc smc large enough mle abc later abc phase enable surface acceptance approximate ii explore reach typical peak point value start iv complete maximum course posterior locate hence increasingly explore surface schedule rate reduce drastically iii value stochastic volatility model adequate equation unable go dc choose kernel generalise student freedom scalar weight ease read produce dc free dc give use start integer independent value denote conditionally correspond corresponding calculate step execution version abc dc cite propose keep fix remove anneal walk explore high enough metropolis face difficulty subsequent discovery identify mode possibly abc threshold etc mcmc accomplish implement current kernel perform interested generate calculate go check jj accept abc generate jj previous schedule otherwise abc parameter proposal random function exploration surface small abc
drastically I unseen input convolutional reduce albeit pool layer input side figure desirable convolutional high network input large convolutional constitute vision problem effectiveness vision research restrict neural network drastically require image exploit symmetry galaxy exploited version sharing exploit symmetry explicitly additionally image interpolation whose align row pixel galaxy classification raw vote excess galaxy galaxy raw interpret preference brain apparent universe bias brain galaxy probability contain rotation variant bias bias exist reduce refer map layer aggregate extract viewpoint reduce time centre galaxy informative viewpoint extraction modify width conv width width height minimum rectangle minimum conv dense width cm height width height height conv width anchor font base font anchor font anchor base align anchor anchor north east west east south edge west edge conv edge conv conv conv north west east edge west conv east edge south west dense develop setup overfitte behind successive set five viewpoint convolutional implementation draw black width cm input cm preprocessing viewpoint extraction averaging average augmentation augmentation convnet convnet section preprocesse augmentation convnet augmentation west convnet east west describe provide dataset answer evaluation image competition competition reveal image score image platform split real model network learnable million high capacity strategy unchanged dropout reducing parameter exploit prediction first reduce input middle background fit square approximately image rescale speed pixel subset operation object either angular measure object allow centre image far processing effect though rmse make competition provide format galaxy website colour colour considerably despite artificial intend nevertheless model correlation train instrumental example randomly rotation angle symmetry shift pixel relative direction limited centre random rescale colour adjust describe eigenvector factor adjustment first four transformation collapse together mean augmentation computational augmentation randomly perturb image never augmentation extract corner pixel centre right corner patch constitute shape image red outline outline total overlap corner patch extract galaxy centre corner patch patch view colour allow affine avoid interpolation indexing also image fidelity augmentation viewpoint minimal array rgb architecture process stack four convolutional layer position separate max pooling fourth convolutional stack maxout output maxout instead relu reduce maxout convolutional prove architecture manual competition million convolutional relu convolutional relu convolutional relu relu maxout dense maxout dense last describe initialization bias see incorporation constraint produces convert pass linearity normalize categorical obtain follow normalization however decrease predict rescale question ask ask question ask question unconditional probability answer decision see incorporate consist constraint must high level question result addition network purpose averaging network differ make slightly include layer three filter dense layer filter layer train minibatch nesterov momentum gradient decrease neural performance perform million improve convergence learning decrease decrease million million first output network ensure layer manually proper bias positive getting region although strategy layer improve uncorrelated computed affine image rotation horizontal unweighted average train see average fashion increase train result average total aspect library allow gpu acceleration effort able automatic simplifie train perform cpu image package training take reproduce win galaxy report perform variant error table metric score galaxy challenge average across transformation average important fast practical prediction generate million combine large impractical public transformation network competition fundamentally capabilitie interpretable fashion classification high participant predict count classification match probability classification fashion cause discard easier interpret agreement galaxy participant affect entropy option entropy minimal entropy answer equally likely select entropy range agreement maximal disagreement maximal condition use probability predict relate evaluation bin answer prediction bin average perform network average graph circle show accuracy agreement classification accuracy confidence show horizontal horizontal option overall indicate level agreement galaxy participant decrease low accuracy achieve perfect agreement arm low agreement near confident useful determine able trust expert could small annotated manually expert greatly expert confident accurate majority would allow largely assessment smoothness question classification accuracy practical q would manual input level annotation dataset analyse evaluation thick horizontal dotted chance number image included indicate able various precision recall score individually score classification list strategy classification least galaxy participant question number occur frequently category exclude galaxy effect attribute generally precision rare rare answer example rare construct smoothness disk yes bar yes yes shaped arc medium arm traditionally neural often treat box informative sometimes try convolutional image layer filter interpret visually layer filter individually three channel separately weight channel filter sensitive sensitive pattern phenomenon train neural look radial image b viewpoint unit convolutional note image still apparent activation except third reason third layer input layer pooling map layer map map map pooling map map map viewpoint upper visualize neuron learn activation type sensitive maxout type visualization clearly discriminate galaxy scale invariance observe direction seem multimodal galaxy activation value image across centre pixel depict turn try replicate participant tend classify image galaxy answer though answer answer galaxy web interface seem feature b loose arm tight look evaluation get idea strength rmse value center rescale varied fairly various way figure motivation additional center galaxy classify round b b b b fine grain architecture exploit galaxy project reliably predict aspect galaxy extraction enable quantitative galaxy scale exploit symmetry art win galaxy challenge win average model collection galaxy source code publicly hardware prediction highly reliable confident grain scale survey perform analysis research future vision scale galaxy paper provide modern though train model effectively improve dataset galaxy annotation recent galaxy take ensure generalize slice dataset number without survey annotation crowdsource possibility raw inspection include structural automate radial symmetry occur interesting architecture deeply small acknowledgement thank van anonymous valuable feedback acknowledge david help galaxy thank capital financial competition classification effort individually grant aid circle thick fill rectangle corner gray draw black corner font department school university st mn usa measure galaxy key requirement formation survey digital survey result availability wide galaxy traditionally mostly inspection train time consume attempt automate able level galaxy project successfully crowdsource strategy answer question unfortunately increase availability galaxy neural galaxy symmetry international annotate galaxy galaxy participant able reproduce consensus confident highly accurate make collection image expert manual greatly reduce large result survey processing galaxy galaxy shape property age formation galaxy course galaxy formation evolution probe physical survey galaxy complicate relationship environment deep survey start take surveys survey result availability million manually impractical individual build automate galaxy reliability scientific galaxy project accelerate crowdsource classification galaxy member public contribute classification web platform entire annotated follow two development automate feasible large primarily although network decade recently research available technique unit regularization possible build network section description reliably annotate galaxy available success galaxy classification size augmentation sharing average become day galaxy classify image deep coverage crowdsource expect expert classification logical necessary convolutional galaxy tailor efficiently exploit image several image classification model international competition base annotated galaxy project first place information enable study galaxy galaxy galaxy challenge discuss convolutional rotation invariance report analyse finally galaxy online crowdsource ask colour galaxy project colour classification project participant ask question determining ask question tune answer total galaxy sign disk smooth disk star disk yes bar centre galaxy prominent central galaxy obvious dominant q odd yes q end completely end arc end galaxy centre tight loose subset question many classify directly pixel convolutional task complex statistic detector hard recognize complex necessary feature selection representation participant challenge literature apart whose galaxy classify galaxy predict classification make galaxy grain task network question data galaxy bar strength star merging etc development crucial galaxy exploit galaxy however besides neural define convolution operation generalize pixel locally invariant affine learn representation convolutional feature invariance encode subsequently learn symmetry ability symmetry input image similar spirit work major effective computational idea discover consist hierarchy subsequent layer abstract build transformation optimize neuron compute combination follow
dropout entropy test epoch prove sgd unbalanced verify despite train sgd balanced unbalanced initialization curve exactly unbalanced considerably sgd case error range plot display observation often fast sgd cifar sgd also implicit regularizer generalize similar role deep dropout figure poor unbalanced except fast generalize path outperform sgd achieve fast implicit analyze look plot epoch provide stepsize momentum term could compare geometry relu suggest geometry geometry beneficial concept deep momentum heuristic enhance believe method plug hope also regularizer perhaps rescale relu path sgd certainly rescale might simplicity choice mirror appropriate seem non convexity acknowledgment nsf award intel discussion cifar cifar cifar cifar mnist cross test title claim choice training rescale sgd descent wise regularizer max regularization easy lead gain deep question generalization issue deep heuristic train training architecture slow open many initialization stepsize momentum inherently tie geometry tie descent norm exp least tie corresponding potential example gradient divergence view regularizer therefore regularization implicit performance align inductive driving geometry geometry geometry desirable enable fast also regularization link appropriate deep focus relu activation incoming edge factor yield invariant seek inspire max norm regularization maximum incoming seem inductive decay max max discuss measure express regularizer sgd regularization rescale classification feedforward network compute direct acyclic input output apply internal unit compute activation depth direct length unit define network hide relu homogeneity rescale change rescaling give node rescale edge weight rescale easy rescale compute f rescale goal minimize step update form descent stochastic sgd mini set rescaling affect rescale call rescale rescale rescale start rescale weight vector separately remain rescale unfortunately invariant opposite invariant update gradient poorly unbalanced network might regularizer relu activation homogeneity effective main relu norm unit rescale regularizer among rescale surprisingly feed forward rescale network efficiently compute forward see vector total path equal weight along regularizer establish path involve nest rescale approximate path rescale interested derive exactly instead update partial derivative follow call normalize direction regularizer let e know path sgd approximate respect regularizer whether path rescale next prove rescale rescaling neither incoming however incoming moreover get ec therefore similar argument path rescale however calculate forward backward step mini setting mini time batch moderate runtime typical mini batch hundred thousand show sgd balanced unbalanced commonly conduct benchmark handwritten digit cifar image
observation provide web survival probability state two state mechanism possibility covariate individual absence individual record individual observe covariate always mass alternatively covariate random correspond covariate presence model generalise covariate observe reporting incorporated omit far fit optimal aic statistic pt shift binomial initially four lead bad omit covariate stationary try two estimating capture assume parameter obtain probability state ii time distribution specify shift negative binomial time geometric markov display remove dependence probability retain though constant survival house absence distribution aic aic htb survival difference survival markov semi predict high contract covariate suggests recently recover duration explanation misclassification model distribution population assume stationary htb inclusion memory specification within incorporate specify attractive increase level increase number previous model state lead notably regard interestingly fit incorrect order markov refer disease status possible biological regard absence follow structure accurately markov crucial efficient order become literature usage almost certainly development modification handle probability computational argue distribution cover capture scenario series capture house word infect immediately individual suffer model markov recovery immediately form include event partially recovery directly event semi state transition observation process need assignment advantage specify capture separate able via penalize extension site status conceptually straightforward curse remain notably semi probability specify probability however non trivial move covariate modeling within development acknowledgement like house thank anonymous comment gray consider traditionally schwarz model first chain though fit biology specify order stay one specify expense significant number schwarz specify semi generally shift distribution expansion apply order result semi tractable markov schwarz increase selection procedure use semi important state spend state feasibility house state keyword markov multi often population uniquely mark record subsequent individual record marked individual lead study particularly individual covariate dynamic individual covariate status disease status life covariate refer state also observe discrete history may history represent observe time state recover individual time schwarz review homogeneity involve spend time individual already process stay one realistic particular state alternatively correspond infected time individual non geometric biological process follow mode distinct exponentially two infeasible memory semi model yet parsimonious specification model spent apply semi markovian shift negative shifted simply distribution process markovian state probability covariate fitting propose general semi markov specification flexible memory markov memory manuscript describe formulation analyse capture house correspond absence section individual covariate let potentially observe initial observe relaxed covariate underlie recently long typically interval occur drop subscript clear live extend hmm capture recovery present conditional essentially decompose state conditional survival modify deal scenario recover recovery decrease individual capture initially calculate sum possible initial mathematically fs calculate hmm define time shift shift flexible specification mixture estimate equally easy implement extend hierarchical survival state formulate semi state essentially mean markovian markovian represent arbitrary semi model expand state advantage hmm hmm applicable markov integer first chain state semi expand k observation role responsible time state semi kk geometric x jk j k zeros different interpret follow survival transition aggregate govern diagonal determine probability specification give length spend assume initial capture dependent state restrict equilibrium initial capture equation assume stationarity state alternatively stability stationarity aggregate q specify analogously definition diagonal entry diagonal aggregate representation approximate semi time spend tail definition make arbitrarily small sufficiently finite geometric ensure appropriately summary approximation arbitrarily accurate choose semi individual simply capture history routine know optimisation confidence inverse bootstrap underlie criterion aic investigate simple markovian semi semi three binomial poisson mean covariate state specify specify simulated individual capture event correctly
obtain theorem detail defer sketch iterate random eq core combine event q sketch sketch q bounding size recursion remain recursion sketch constrain algebra lead hand du subtract term triangle eq vector smoothness lipschitz hessian obtain inequality substitute rearrange find claim convex function book define update whereas proof unconstraine self function classical newton sketch involve fx prove high randomness newton tangent cone cone gaussian sketch sketch newton newton sketch phases magnitude constitute fx fx complete dividing phase phase c next gradient necessarily convexity acknowledgement office national science foundation grant dms mp microsoft fellowship independent optimality feasibility define algebra basic optimality feasibility subtract sketch consequently definition piece finally guarantee e imply construction satisfie g yield fx perform observe fx subtracting factor newton final repeating conclude claim proof lemma accordingly nesterov combine c hessian relation proof newton v lead basic optimal feasible consequently add subtract gx remainder proof broad give twice collection constrain cone tangent cone follow gaussians consequently apply bound inequality cm theorem theorem ex ex berkeley california berkeley department electrical computer science department second newton sketch perform newton self super convergence probability number dependent substantially newton extension equip illustrate program regression generalize program modification newton slow see another tuning size optimal smoothness optimize g twice method convexity conditioning whenever self suitably provably newton reason system pose challenge million common issue approximation quasi newton form hessian gradient computationally example bfgs scheme book disadvantage weak method restriction super paper propose sketch projection projection hadamard sufficient sketch regime low sketch moreover show nd convexity unlike hand strategy consider sub partial prove quadratic condition point arbitrary constraint sketch barrier iteration pre specify begin background classical measure include version constrain illustrative convergence theory devote convergence unconstraine setting additional aspect defer begin section see background convex uniquely eigenvalue f assume modulus initial newton modify globally convergent choosing size backtrack search procedure apply central development book number typical constant give restrict normalize base generally base entry distribute terminology matrix theoretical perspective well sub perspective disadvantage multiplication multiplication consider multiplication perform class hadamard fourier hadamard fourier respectively random vector diagonal multiplication another sketch independent take canonical leverage sub background width randomize gaussian width banach space theory cone sphere width substantially cone calculation background sketch number guarantee constraint motivate sketch current iterate perform take simple root square efficiently instance suitable hessian root newton update precisely sketch isotropic generate iterate recursion realization simple eq intuition isotropic sketch original analyze form additive sketch lead analysis setting dimension newton update dimension lead intuition newton apply lp polytope barrier denote inspection twice version hessian sketch require barrier refer central compare central ordinary newton polytope trial dimension middle sketch sketch sketch black interior step point vertex represent optimum optimum sketch central newton covariate case model count collection covariate glm convex user enforce case least square well set objective function guarantee explore return set sketch must order good behavior sketch geometry term tangent cone fx recall gaussian width sketch tolerance constant square root dimension width case achieved unconstrained substantially problem illustration constrain measure smoothness twice differentiable objective define lipschitz convergence sketch newton initialization bind probability linear convergence specifically rate step total guarantee notable depend sketch illustrative example portfolio linearly program section newton solve different size calculation appendix suffice sketch dimension quantity require theory convergence newton different consistent suffice case newton iterate super sketch logarithmic curvature constant practice theory local take iterate origin seek classical appropriate self appropriate backtracking begin unconstrained discuss sketch update equip exponentially high strong unconstrained optimization convex bound impose include logarithm affine transformation result accurate impose sort analysis scale depend backtrack self fx approximate via convergence newton parameter step nd case close barrier constraint solve many hessian highly instance constraint usual simplex hessian barrier problem structure frequently arise regularizer ill inverse regularizer regularization e norm strategy sketch retain previously provide step include choice line sketch dimension start function parameter matrix f f dimension depend iterate sketch reader recall let barrier implement sketch suffice see self barrier section barrier sketch newton barrier interior particular provide rigorous bad precisely problem form exist unique trace g optimality barrier successively update also newton method lie heart barrier fast newton algorithm provide different dealing sketch apply tolerance sketch backtracking update lead arise particular application barrier sketch g newton sketch sketch upper iteration instead interior solver discuss sketch particular various newton contain barrier sketch flexibility partial strategy sketch primal generalize enforce include enforce sparsity nuclear enforce differentiable norm variation enforce smoothness suppose sketch current iterate computing iterate solve cardinality effective apply homotopy lar optimality start focus choose suitable sketch function suffice sketch size constant us example u r u typical sketch sub newton sketch project intrinsic cone sketch semidefinite program illustration ia semidefinite semi norm establish definite sdp pre regularization encourage relatively rank solution standard self barrier psd barrier sequence hessian barrier hessian sketch first ij exact classical sdp interior solver consider portfolio problem find
constraint measurement identical maximize residual repeat variance small unity small unity estimate constraint residual area indirect element indirect diagonal error covariance analytical solution robust estimating variance difference indirect early dr constraint whereas describe simultaneous covariance flow assume need constraint element identifiable standard provide variance six obtain clearly indicate constraint estimate eqs subspace angle estimate flow difference estimate angle accurate knowledge rmse known show reliable drive obtaining estimate deviation knowledge sd section dr far demonstrate possible identify constraint require accurate estimate measurement identify derive connection pca dr measurement measurement relate pca constraint error covariance apply simultaneously constraint apply dr system note apply pca matrix identify relate measure relate measure reduce project dr measure relate variable balance argument derive eigenvector small rotation reduce constraint estimate identify bring process determined singular describe section hold difficult order great force conversely investigate datum partition small choose subset linear combination go estimate constraint indicate unbiased even constraint great estimate give orthogonal value subspace estimate show overfitte make recommendation conservative overfitte heuristic demonstrate effect deviation value example consider demonstrate concept apply different assume great model five clearly assume incorrect unity equal systematic identifiable value variance unity singular value obtain assume increment unity unity true model observe theory noise assumption nonlinearity consider ambiguity select clearly order choice lead introduce estimate marginally increase order overfitte lead overfitte primarily regard statistical useful denoise pca method steady state entirely measurement iterative pca covariance simultaneously necessary extract exploit constraint perspective dr integration fig historical process dr technology fellowship cccc control institute technology dr pca high preprocessing technique develop primary relationship lead unified dr collaborative datum extend partially incorporate partial principal derive consistent estimate develop technique software package plus software package chemical benefit apply dr estimate material typically dr constraint derive first principle material correlation specify additionally covariance derive historical multivariate processing popular principal method primarily develop regressor variable chemical engineering monitor diagnosis regard multivariate statistical technique pca author pca relationship simultaneously process interpretation apply purely measure difficult measurement technique matrix require rigorous diagnosis pca diagnosis modification incorporate partial impact incorrectly estimate model actual constraint application pca tool organize section introduce dr identification section partially matrix discuss criterion model conclude process section dr steady discuss measure know define exploit linearly constrain process operate water stream measure flow operate sample draw steady state steady relationship q label constraint subspace error instant error denote expectation minimize regard error estimate using normally distribute measure steady operate dr steady apply independently pca operating apply arrange value give follow illustrate flow flow consist six balance write order flow six stream measure noisy simulate noise rate variable base variable simulate steady normally fluctuation add flow flow eq steady normally random value steady steady base sde dr apply rmse computed table rmse report rmse indicate variable rmse dr rmse feasibility measure refer system partially dr provide value estimate dr variable uniquely estimate observable redundant observable dr describe variable label label matrix construct get constraint variable define q unique estimate obtain redundant observable algebraic completely book redundant dr measure redundancy visualization ease flow flow stream measure add connect stream flow contain whose flow flow balance reduce flow subject reduced technique measurement stream eliminate non flow original cycle flow rmse redundant accurate improvement flow compare obtained reduce dr component view describe steady pca linear uncorrelated principal variance pc variable hence uncorrelated variance pc pc eigenvector obtain datum scale eigenvalue orthonormal eigenvector remain diagonal whose diagonal element root eigenvector order magnitude pc give variance pc pc heuristic pc retain look sharp eigenvalue heuristic book alternatively denoise technique retain pc viewpoint importance article discover underlying variable identify author explore analyze measured covariance matrix generalization matrix simultaneously section observation draw steady datum lie subspace relate apply retain pc equal give orthogonality pc eigenvalue process constraint focus eigenvector retain identification concern small main constraint variable lie correspond orthogonal square identify subspace lie square identify matrix prove n sample matrix steady due measurement n make imply systematic true value true assumption steady linearly steady eigenvalue recommend operating eigenvalue orthogonal constraint corresponding n row matrix furthermore knowledge examine small know normally use result conclude error normally distribute identical variance mutually derive asymptotically unbiased note differ singular substituting estimate give orthonormal eq orthonormal eigenvector q identify obtain identify purely error different also purely estimate satisfy estimate closely pca dr may rotation pca differ desire constraint precede identical constraint pca furthermore identically distribute may note dr identification transform appropriate apply triangular transform let transform n apply transform estimate corresponding estimate constraint singular value part relate constraint datum equal provide systematic size numerically check small unity error constraint constraint assume error variance transform order singular four nearly original rmse rmse obtain pca
operate would main strong use lstm unit gradient descent initialization gradient training pair observe overfitte e understand purely intersection hull coverage self intersection fail simply report present area net mistake come aligned mistake hull input affect true sequence processing step update convex overcome problem describe whole modification focus point side create net lstm attention key inherently variable length half length uniformly effective single able degradation even satisfactory learn simple lstm attention give lstm fail attention lstm net lstm net net hull fact triangle coverage triangle case permutation net accuracy middle symmetric hard finding implement unclear would capacity solely feasible small importance good provide reasonable show unlike hull decoder unconstraine net sometimes decide produce row show optimal feasible net try beyond seem able could train paper describe architecture token correspond net learn work sized yield sized something sequence attention previously memory base output location neural assumption try sort combinatorial acknowledgment le thank help final google berkeley google brain introduce architecture conditional position address sequence number target input sort sized belong mechanism attention attention decoder use select member call net alone net us dictionary learn generalize beyond hope encourage broad discrete recurrent neural network rnn function three decade limit available introduce paradigm remove rnn decoder contextual information possible rnn domain art core language processing parse execute output limitation input output satisfactory model approximate purely fashion input rnn code generate rnn output feed generate content attention equal contribution architecture net effective deal fundamental length dictionary softmax distribution net distinct trivial algorithmic involve geometry generalize net learn competitive small drive approach approximate baseline section sequence compute I pn learn maximize example short p ii end input reach symbol termination independence assumption rnns symbol rnn typically lastly state feed step recurrent note become perform significantly well sequence hull output input nevertheless simple reduction describe modification us combinatorial dictionary softmax size attention follow u w j softmax propagate copy correspond attention mechanism note target problem output could rnn map back prediction long video follow protocol coordinate convex hull case sequence associate set hull ht hull c hull token understand difficulty combinatorial drive approach solution number consider position special represent plane every empty exact solution example triple token
wide ill probably various method five close typical class exploratory contour depth remove depth report assume define boundary neighbor outli locally tend typical outer relative extend near many influence several subsequent work come neighborhood isolate early detection outli score mahalanobis instance locate fall unified linearization method randomize pruning resolution suffer curse model increase suitable great handle extreme typical invariant outli detection datum low subspace outli reduction process help explore vast solve unconditional across attribute increasingly recent year contextual outlier song et generative response outlier although share similarity song al representation mapping parameter exploit decomposition make large require expectation step limit outli testing utilize piecewise probability individual improve significant sensitive outlier outlier joint infeasible section briefly learn assume free outlier unseen include outlier phase probabilistic multivariate instance unlikely detail outli detection outli accurate probabilistic relate define different datum study extensively multivariate able automatically assign tag keyword document posteriori purpose outli assess classification second equally assignment output assume response separately suffice world among important build among define via univariate product parent output variable directly depend model relation among response variable representation generalize component empty decomposition output circular view status work many like conditional vector probabilistic naive logistic building use estimation py regression use section present apply testing identify conditional phase unseen phase like score metric towards pseudo likelihood decomposable structure estimate likely unlikely dimensional response multivariate along set scoring metric transform test previous outlier exist outli method sensitive datum pattern utilize identify model propose piecewise individual useful evaluation outlier unsupervise nature outlier dataset assumption data process assign probable observation small portion outlier comparison fraction influence building model create conduct consist part realistic context eight outli six competitive adjust number wrong outli three show multi include video image label text categorization clinical patient consist context table label configuration c medical clinical biology plausible context find everywhere example label irrelevant tag clinical diagnosis patient inaccurate diagnosis gene sequence simulate outlier follow sized c medical multivariate outli rd run evaluate datum refer use use norm svm outli fair comparison radial fix penalize logistic cross lastly metric individual response input train auc curve true positive auc high table mean equivalent paired level bold follow level statistically superior mark produce score outperform four outperform produce well rest local understand condition information handle unconditional attribute efficacy due analyze benefit x different group score gray show bar significant good work even method testing actually improve dimension e sparse instance method three video annotation categorization labeling table characteristic control adjust clinical response space outli auc pr auc auc pr pr curve conservative auc sensitivity outli detection pr method axis indicate auc pr axis indicate use color dot simply dot pr superior general intuitively dimension hard outlier figure start bottom plot gradually increase usually auc dimension increase obvious auc rapidly baseline invariant dimension exploit posterior help outli special variable review exist outli detection multi method outli multivariate transform unconditional solve effectively accordingly five score usa outlier expect particularly define combination paper outcome space transform detection outli unconditional outlier rely classification probability probability output score use detect outlier multiple outli show artificial expert outli datum statistic anomaly useful annotation preprocessing help remove noisy irrelevant utilize beneficial surveillance disease clinical monitoring despite huge exist detect unconditional outlier ordinary response label unconditional method easily positive negative want detect suppose unconditional outli detection annotation due patient rare correct incorrectly image modern assign image image label outlier become one moderately respect patient include become consider child unconditional become apparent detection seek response outcome unconditional outli detection seek instance detection dimensional output pattern multivariate output correspond fall keyword incorrect detection particularly challenge pattern detect build classifier use discriminative briefly observe space express treat account output output method define separate due procedure together outli instance dimension score statistic outlier output image process label label randomly chance dimension network attack keep decompose covering help
preference threshold literature grouping divide classification sorting depend maker dm sorting assigning predefine formally predefine order category set criterion criterion relation category profile limit category characterize assignment alternative category criterion profile one four partial criterion criterion partial index profile threshold case weight coefficient computation partial index know profile preference threshold cut lambda permit assignment alternative relation two optimistic assignment procedure profile category main parameter preference follow phase formulate input application take american census approximate record synthetic census ds report research cause difficulty analysis facilitate record miss ability generalize play important role alternative importance lambda lambda increase consider substantially lambda parameter case namely link situation dm interesting preference machine propose record linkage matching criterion solve record application start initial demonstrate application good show performance experiment confirm record linkage preprocesse matching say measure well scheme search lc machine ordinal classification propose linkage preliminary experimental show correctly identify machine scientific concerned allow computer learn intend recognize make closely relate field mining recognition artificial intelligence theoretical computer machine learn type supervise generate map recognize supervise unsupervised combined impact environment translate guide principal statistical linkage matching network short propose record time linkage answer challenge machine record criterion linkage provide classification performance application light development context pair record heterogeneous maintain distance linkage describe record linkage use record section preliminary simulate remark conclusion generally speak record unique record source linkage methodology record two file file record linkage linkage solution record file identical match linkage file information record linkage big answer well understand suppose want age suppose follow cccc name age road read furthermore contain address age h st b contain unit probably match modern record linkage begin et odd ratio rule match year yield computer idea computer sciences operational research mathematical linkage theory demonstrate optimality crucial probability file formally file probability agreement comparison set match classify pair analyst decision three
specialized regret go bold face let finite space begin concave domain observe nominal convenience call definition agent action ta tf ts make identically unknown allow decomposable aggregate objective constraint handle define policy randomize policy policy context appear context nonlinear strong compete policy equivalent similarly define p rp vx rp expect reward enumeration impractical purpose access employ contextual bandit previous max policy max since optimize frank wolfe repeatedly ft consistent terminology average amount regard special refer constraint aim average nontrivial problem study call contextual bandit break component vector maximize vector budget form never allow budget nothing get take round need certain lipschitz detail bind call bound term optimum need scale regret feasibility problem reward ta ts measure algorithm share bandit change necessary constraint completeness allow ta select op algorithm policy class proceed length begin compute right policy ideally concentrate order probability large enable computable define bandit technical dealing constraint describe problem denote action take reward observe take select action way mixed choose complete every v straightforward unbiased reward empirically sp define regret obtain q op empirically get finding return smoothed assign minimum op regret small constraint one compete compete constraint derive implementation op call average due achieve hold trivially come mixed sketch give true second op high give op every empirical regret mixed section maximize ensure feasibility error alternate tradeoff however appear regret suffice regret well budget sufficient give rest intuitively optimization lemma appendix constructive small variable f rp instead construct idea algorithm section new combine op recall equation factor norm policy good policy estimate achieve sketch step use op lemma additional parameter new actual constraint op regret regret amount bounding term rp rp rp rp rp rp aim maximize ensure return however budget stop budget resource fully violate ensure happen early compete ta ta add regret would objective capture quantity suffice easy precise property lp similar general discuss property applying require great equal suffice outcome play uniformly general problem early lipschitz sure budget violate enough constant set aside entire follow round pure exploration aside amount run time full actual component budget essentially accounting budget appendix real v e version chernoff support least b feasibility require solve op op nontrivial sampling input mix op current minimum probability proper default pick schedule allow ta op op op describe main easy combination policy eq coefficient op tp convexity op version feasible feasible early side q jensen inequality trivially feasible op op problem solve descent assign weight shorthand op describe instead h e k p use loop start call loop initially compute ix remain solving sequence concave elsewhere denote elsewhere p maximize number time apply section constant since prove represent specify one suffice show mix need another choose compactly schedule quick place increase p record history round v statement hold epoch round epoch choice lemma get first epoch ix furthermore definition reward let mi sum union bind choice note e mi v mi tp tp definition rp p j side tp rp tp rp tp assume event round definition v v get substitute get round choice event epoch tp k universal constant fix immediately equation event observation op constant assume event tp tp induction epoch epoch triangle distribution tp rp tp rp tp tp rp tp rp tp rp tp follow fix epoch round epoch distribution tp tp inductive always inductive use fact therefore whether part epoch tp k inductive hypothesis simplify matter whether combine gives ensure inductive ready hold epoch follow mt r jensen trivially epoch round epoch tp q tp k c substitute next show th component recall easy see hoeffding martingale sequence apply triangle far get regret dt trivial dt convexity follow trivially always suppose rp satisfy exist rp lagrangian program duality get l l f optimal variable program statement concave gradient small hold epoch epoch policy tp tp induction case rp rp tp rp z rp rp tp tp rp rp rp substitute side z dp fp z dp rp z tp p tp rp rp tp tp inductive epoch round fix epoch policy event step ready theorem hold suffice whenever projection appendix linear mt mt jensen rp ft r rp rp inequality jensen dt rp use condition p tp apply bind detailed bind equation bound condition rp ft ta l otherwise budget p bp c contradiction regret algorithm loss regret regret order consumption round hold arm estimate tp da pick policy support observe eq component solve relaxed policy achieve maximum
behind entity compositional worth domain name contain average always cc entity randomly average word vector initialize work everything bilinear entity relation ideally simple horizontal translation traversal red expect parent square dotted red large compositional entity think intuition incorrect reveal query error along edge training encourage incorrect however close discrepancy path phenomenon empirically well path traversal operation path entity rank confirm relation compositional training precision divide path length box show compositional decrease precision decrease little irrelevant knowledge completion embedding reasoning knowledge compositional technique improvement help compositional compositional knowledge completion answer reduce incorporate compositional incorporate path approach use regularization walk social introduce answer query incomplete technique vector show compositional answer completion key path represent perform believe great stanford university stanford stanford cs stanford edu compositional question however fact query suffer error compositional training improve ability query compositional training act novel regularization reliably base basis reasoning question answer know suffer incomplete coverage distant entity elegant space control force fact reason compositional hope parent ability compositional query ask generalize propose path entity traversal drive vector transformation present basis broad high implement edge traversal interpretation encourage modeling include bilinear three two finding first compositional answer substantially base somewhat surprisingly answer query compositional regularization exist formal task answer entity relation knowledge graph triple triple answer query entity reach eq evaluation detail candidate answer incorrect answer completion candidate answer query motivate present technique compositional describe illustrate experiment entity relation bilinear likely use motivate technique adjacency matrix entry entity easy count relation positive q interpret recursively begin entity apply traversal new reach point traversal apply traversal much learn bilinear compositional scoring q score model every perfectly present optimize naturally suggest compositional training membership traversal explicitly encourage notion vector query different vector sequence another traversal transformation preserve traversal objective completion query path length compositional substantially answer base completion insight one completion scoring vector membership operator traversal handle visualize compositional bilinear diag bilinear relation view naturally matrix neural reasoning optimize initialize size validate query length form relation bilinear initialize bilinear diag inverse initialize gaussians entry bilinear yield performance code consist section reasoning subset exhibit subset bipartite person source entity contain relation perfectly correlate inverse relation edge easy inverse triple exclude query extremely amount training overfitte base follow sample entity mm relation current next via practice repeat except plus dataset remove appear training statistic train show compositional training substantial edge demonstrate inferior outperform state art numerous use query include answer rank normalize rank account candidate quantile answer quantile range average quantile normalization important predict gender query receive rank mean quantile several query match trivial answer query exclude query cc ccc ccc red red h vs compositional quantile percentage cc c single vs compositional query exclude compositional improve model show surprisingly compositional improve completion across bilinear term bilinear deep look query divide query subset source path never see training query explicit traversal subset
good web multiplier kkt theorem axiom rgb liu rapidly crowdsource crowdsource crowdsource worker minimax conditional truth unique labeling worker item difficulty measurement principle principle variety multiclass ordinal crowdsource minimax world costly considerable research semi recent year crowdsource service amazon associate collect domain drop dramatically enable large low cost provide worker lack worker overcome worker manner voting assumption majority worker good obviously assumption reflect imagine worker confusion worker correspond maximize worker stay item many item difficult worker may minimax conditional crowdsource worker ability account item ignore reduce measurement measurement conditional aggregate collect crowd derive regularize minimax overfitting generate label propose extend label present ascent empirical crowdsourcing report present principle aggregate multiclass primal show minimize kullback leibler divergence index assign item denote belong label true true worker approach build upon tensor tensor refer confusion tensor observe item tensor refer class worker worker worker item worker worker row match row assume observe unknown attack simple worker item enforce count worker enforce confusion item counterpart illustration c item item worker worker item item entry label class unknown intuitively understand entropy worker connect lagrange multiplier tucker kkt combine constraint yield q although mathematical understand intuitively worker worker worker worker confusion confusion substitute labeling equation dual minimax obvious extend define stay class true equivalent sketch moreover deterministic crowdsourcing collect collect limited count fluctuation probabilistic label item uniform either ask item overfitte formulate replace match approximate fluctuation generate label formally minimax true label subject relaxed worker slack fluctuation slack positive fluctuation normally distribute central motivate entropy objective generate regard deviation labeling turn log jensen inequality maximize conditional restrict slack say answer worker law correct answer note percentage worker additional problem express somewhat natural labeling equation consequence objective involve also worker independent involve comparison worker mathematically objective principle event formally item item formally obvious worker worker choose nonnegative probabilistic labeling equation immediately instead requirement result requirement multiclass ordinal ordinal web product since ordinal special multiclass label previous section worker multiclass labeling summarize adjacency assumption formulate introduce adjacent observation ray breast cancer cancer screening rate worker formulation parameterize worker relation ordinal consecutive integer eq subject constraint exclude hold draw shape center true observe give ordinal us equation ordinal label label section trivial check ordinal label indirect value ordinal label set choose label region partition example table define constraint sum equation similarly define set constraint item discussion constraint restrictive result disjoint degenerate thus explain ordinal write count item belong worker outcome label reference observe outcome enforce kind dimension match counterpart equation enforce count kind dimension empirical counterpart partition four write multiplier multiclass equation worker item confusion structure ordinal class expect ordinal write item constraint choose problem q worker define first event principle worker independent independent arbitrary worker reach ordinal labeling describe simple coordinate ascent minimax model regularize either ordinal ascent expectation maximization aggregate vote iteration give current estimate confusion estimate label close experiment multiclass label gradient ordinal worth unnecessary exact intermediate ascent suffice reach initialize repeat regularization true subset refer validation choose choose fold partition crowd label finite confusion item plug leave average likelihood go choice large average parameter simplify gain motivate square magnitude dramatically number super linearly scale relate generative ability confusion represent worker label worker item probabilistic item q generalize jointly estimate model task write worker usually coin coin assume propose coin achieves rate fix essentially belief propagation update assume worker equal coin achieve achieve spectral impose beta confusion true generate logistic full include belief test illustrate response model item location trait person ability variable response incorrect item mathematically difficulty response item trait correct special theory measurement model adapt integer scale let person location refer rating rating latent minor label item labeling difficulty easy worker confusion generalize multiclass worker cumulative standardized normal worker z unobserved parameter think crowdsource reader make crowdsource entropy propose apply texture field multiplicative mechanism crowdsource worker answer question skip error datum crowdsource publicly detail contain image crowdsource worker worker label image error worker amazon student price bin create student decide bin student systematically bias tend present two sentence ask check hypothesis sentence infer sentence pair annotation ask
wolfe probabilistic variation capture spirit wolfe derivative accept I direction strong wolfe condition write exactly bound strong wolfe replace limit overhead black box inner loop precede introduce six far decision eliminate search noise level runtime objective form one problematic learning rate wolfe threshold problematic annealing threshold new probabilistic motivate value free upon back envelope computation achieve compete empirically rarely observe either clearly condition scale eliminate scale ensure range digit line search take division cause seem notable deviation exchangeable variance batch overhead approach already statistic square beginning search amount finally empirical expensive summation simple overhead batch run average batch necessarily converge fast estimate separately capture project line search along weight demonstrate step size mis scale fit line propagate iteration initial initial search next search line search search put extremely loose control neural net nonlinearity mnist use cifar dimensional search deal univariate subproblem task empirical evaluation aside computation independent practice optimal effectively remove exploratory rate tuning mean std repetition central nuisance sgd potentially schedule decrease theoretically decay rate empirically decay often exploratory schedule find cifar respectively first sgd sgd decay schedule probabilistic fig epoch learn error base start quickly reaching report kind architecture without decay outperform search optimal one exploratory come lead start single let overhead objective ms mnist instance closely search search first thus optimally additional plot raw vs error dataset rich picture show choose tune progress use elaborate analytical convergence search widely accept noisy design combine principle idea user complementary combine evaluation reasonable quickly optimal matlab implementation available publication article additional network architecture cifar mnist plot evaluate batch sgd constant decay search enhance sgd keep line smoothed unstable black instability control regardless reach development step vanish size costly remove dash decay dash dotted line search initialize vary search accept course optimization nontrivial level gradient sgd instance accepted gradient noise level line smoothed magnitude search converge indeed mnist cifar dash green horizontal decrease search start slowly appear simple nontrivial objective pick cause jump minima already line performance drastically figure show relation encounter course symbol color little cifar instance circle behaviour sgd suggest line sgd typically prevent truly beneficial sgd search ensure stability efficiency formulate construct probabilistic search combine structure notion surrogate belief wolfe condition effectively remove rate highly multivariate gradient batch neural network regression arise exchangeable loss I limit distribute despite popularity inefficient main even noise sgd well individual step former adapting address meta newton direction auxiliary et adapt none size grow individual ht optimization search direction finds follow acceptable insufficient decrease point gradient exclude curvature wolfe strong free subroutine conjugate bfgs free search deterministic optimization easily small become space efficiency construct objective cite search change direction adapt cite present search essence explore reach operate univariate along scalar gradient xt search search likelihood eq regard three surrogate formulation wolfe termination allow discrete point evaluation subroutine requirement fulfil wiener process zero gaussian semi irrelevant regard give whose cubic generalize spline ill case observation crucial typically generic inference gram measure
result good semantic hashing net code autoencoder preserve distance encourage quantization autoencoder fast note binary autoencoder stack restrict boltzmann machine rbm binary output decoder autoencoder rbms differentiable involve normalization different nonsmooth binary autoencoder combinatorial code optimization rbms approach ignore approximated relaxation truncation possibly hash code auxiliary coordinate able break optimize parallelism armed optimization efficient particularly encouraging give autoencoder objective intuitive code nonlinear hash mapping straightforward much towards hash function mac framework nsf award helpful simply involve dual variable global optimizer optimizer small vector additional optimality optimizer possible tight condition complicate compute minimizer relax global minimizer solution relaxed optimizer relax global optimizer binary particularly interested computationally comparable objective besides arithmetic evaluate sufficient fast global stop relaxed minimizer training code cm thm thm thm thm hash binary autoencoder electrical computer science university california false attractive map low hash constraint autoencoder reconstruct code auxiliary optimization easy decoder optimize precision recall code hashing problem hash hash fast database space use factor hardware operation false positive negative retrieve verify ground still year code try capture notion thing reduction note binary real optimizing learn hash run reduction procedure filter pca thresholded minimize thresholded projection jointly mapping threshold optimize code binary show joint binary actually carry reasonably general solve complexity focus mac describe derive mac carefully binary carry function several reconstruction entropy show optimize autoencoder mac nonlinear sophisticated function hash basic locality sensitive hashing lsh lsh outperform dependent specific give unsupervised define either achieve space example essentially binary relax thresholded code variation eigenfunction obtain hash classifier spectral label optimize instead embed parametric thresholded threshold relax try nature code threshold subset number binary code couple learn hash code close binary autoencoder quantization fast hashing obtain code apply pca seek rotation make code latter base find continuous laplacian spectral continuous local minimum optimization function thresholded pca code optimize binary autoencoder relax code semantic hashing use autoencoder consist stack rbms threshold encoder round encoder backpropagation forward ignore round broad composition encoder effort apply mostly encoder pca code vector bit write tt act bit n code layer nonsmooth difficult exist nearly everywhere call later optimize pattern hash hash fit code bit filter ba optimize take separable break nest functional equality penalize coordinate minimization introduce I equality note binary augment increase eventually function optimally reconstruct individual reasonably still parallelism describe result step encoder classification decoder one operator iteration decoder decoder reduce optimize step stop convergent differentiable objective binary problem instead vice valid choice solve mac autoencoder stop change minimize n let prove even set function exact set ba independently ba end r ba lb minimizer limit set small ba ba g iterate stop parameter regression necessary equivalently simplicity multiplication binary hamming objective number misclassifie separate label classifier perceptron solve closely misclassifie optimize margin plus slack code surrogate make local optima generalize well maximum penalty constraint linear optimum warm initialize previous note decoder train independent np practical intensive computation parallel spend make good cholesky square precision error triangular l minima since depend speedup triangular e henceforth enumeration iterate optimize form early initialization far binary hash population intuitively hash code use equally preferable bit distinct hash ideally bit bit vector spc integer code normalize work large code usage measure real hence code use number available number distribution code entropy large available code entropy induce dataset hash preserve neighbor crucial necessarily precision recall code easy code necessarily pick hash half half half generate decision cut impractical hyperplane per internal distribution axis thresholded principal long hyperplane contain half hyperplane orthogonal thresholded generally see generally projection code projection approximately gaussian mind useful hash important ground size code indeed binary reconstruction retrieval cifar image ignore wide contain extract subset contain image sift image sift feature use retrieve neighbor either ham hamming evaluate minimize ba hash runtime code reconstruction b precision hamming purely ba mac approach pca find wide wide result ba dominate reconstruction precision expect worst hash ba competitive method mac inexact use warm relaxed qp fig cifar criterion objective surprisingly warm start ba fig warm solid line relax early warm good relaxed warm relaxed initialization result optima almost binary size eventually converge almost warm likewise fig alternate enumeration vary course runtime middle inexact step model learn remain unless enumeration reconstruction relax bl bl bl bl bl bl bl bl bl minutes runtime mac optimization ba step optimization warm vs initialization cifar matlab loop iteration processor observe scale particular parallelization rough ba cifar bit speedup nonconvex result code mac guarantee improve leave unchanged validation tends generally schedule double skip past occur schedule well seem thresholded locality sensitive hashing hash spherical hash use sophisticated near ba hash minimize use ba depend neighbor report small ground query retrieve near hamming neighbor neighbor curve recall result cifar retrieve image size depend
modality maximally space wise co vector space dnn ordinary follow learn representation produce fuse predict leave parent contextual leave parent category share bilinear across leave account audio visual fine grain contextual think share pool operation formally overlap joint propagation rule bilinear dnn share leave leave parent set bilinear softmax sharing keep error bilinear softmax layer bilinear softmax audio propagate term pass network bilinear influence give completeness diag diag equation keep control frobenius norm u j bilinear softmax sharing rule factor architecture initialize vocabulary architecture bilinear architecture audio visual fuse architecture dnn architecture improve average posterior three gain posterior bilinear deep multimodal audio modality demonstrate clean acoustic speech bilinear dnn audio modality technology di mit edu com com present multimodal speech modality automatic deep train separately fuse space deep network audio alone achieve phone clean vocabulary audio model visual channel phone second present deep network architecture use bilinear softmax modality bilinear network significant phone yielding per speech pose often carry information effect multiple party human read order enhance recognition clean speech help automatic audio video train build visual audio information enhance work visual show visual indeed scenario multimodal multimodal find modality interaction framework dnn audio restrict framework deep learning modality validate section training audio consider joint representation phone bilinear modality network bilinear posterior well clean speech organize vocabulary extraction visual fusion video video add central frame visual audio dependent cluster refer phone multimodal second feature note classification canonical modality would model audio feature objective stochastic joint visual hide network keep deep fused final layer fuse achieve alone achieve visual carry build fuse substantial audio visual audio visual task interestingly per h audio alone alone dnn softmax separately modality bilinear dnn dnn linearity sigmoid unit shown consider simplicity exposure assume
particularly rnn cause composition property dark transfer know include hard weight balance relative soft target respectively hard sample conventional force teacher transfer study dnn rnn target dnn risk fit fitting largely model soft hard target reasonable target refine target transfer conventional fine target easy however information soft refinement informally firstly discriminative dark conventional training approach restrict boltzmann rbm auto simple stack dark function discriminative though structure possess totally orient train layer pre train complex clear focus pre view discuss sample learn one dark regard train fine tuning interestingly regularization view pre closely training essentially place reach discuss train acoustic experiment noisy profile standard largely gpu dnn start construct plus train provide frame window lda dnn dnn architecture involve layer layer equal gmm entropy sgd dark transfer dnn teacher rnn rnn lstm structure dnn empirically fa speech train tr fa fa target train dark transfer dark transfer target employ soft soft target rnn target role hard soft target role pre regularization empirically htb fa dnn hard rnn rnn soft rnn soft rnn hard rnn dnn baseline much devoted momentum rnn inferior fa rnn additionally interpret suitable report rnns rnn well largely solve dark rnn system obtain dnn dnn rnn arrange rnn fa learning accuracy close set compare dnn baseline indicate soft fa well cv improve sense combine target pre fa improve confirm hypothesis role rnn confirm two dark knowledge rnn bad cv confirm high generalization dark model knowledge pre involve complex train deep rnns investigation probabilistic edu cn recurrent rnn acoustic automatic successful rnn highly research dark model teacher idea model use transfer target simplify combine rnn without scheme hessian dark gain success powerful rnn rnns rnns long speech signal back inefficient difficulty dependency cause nonlinearity vanish address architecture memory successfully architecture odd recently variant hessian successfully rnn problem computation demand recent momentum rnn reach performance address difficulty rnn e optimal e g simple powerful rnns work logit dark involve rich target training research focus term complex ensemble model employ dnn large employ transfer train research try teacher treat teacher smoothed step fact extend rnn task database improve organize dnn dark basically train dnn play teacher target posterior probability identity deterministic hard target target temperature formulate introduction target training rank information class reflect additionally apply class additional teacher additionally information class soft informative knowledge hence need appropriately task dark soft boosting also complex soft lead smooth objective function compare intuitively soft arbitrary hard
simplicity chernoff conditioning give eq take probability event remove submatrix line exponentially entry part split diagonal concentrate subgaussian coordinate hoeffde subgaussian equality complete estimate subgaussian random identically early easily kernel upper close low bound generalization formalize sensitive row concatenation estimate hamming attribute private estimate attribute achieve privacy operate mechanism every ridge regressor attribute achieve attribute operate time subgaussian individual vector subgaussian weak subgaussian part dominate computation distortion privacy becomes imply comes draw match proposition comparable thereby hold high natural minimization also lp program context handle complex z minimization acknowledgement grateful helpful discussion theorem pt author method extremely popular technique use many analysis practical performance develop implicit function return feature kernel evaluate kernel matrix entry dimensional main tight commonly application bound need privacy privacy definition regression perform outperform domain privacy distortion technique assume realistic scenario bound release restrictive recent year development kernel practical range multiclass ranking outperform heart notion value power define could nonlinear kernel evaluate introduction nonlinearity optimization ingredient build formally kernel ni jk j asymptotic kernel form establish matrix kernel require matrix property recent mostly focused bound input draw distribution satisfy subgaussian spectral construct roughly spectral high assumption subgaussian kernel correlate exist directly overcome argument combine subgaussian norm matrix role rademacher analyze conjugate gradient value establish application arise database want attribute code age literature partly medical operate linear attribute approximation attack privacy loose goal privacy record assume know public mechanism attribute private consistently reconstruct attribute privacy mechanism add privacy mechanism attribute private setting ridge privacy magnitude imply attack attribute attack release unlike attack analyse goal property matrix focus asymptotic infinity let symmetric random kernel f df limit matrix asymptotically kernel limit recently investigate development traditional area geometric compressed sensing understanding utilize release global property database privacy information database privacy notion privacy tailor differential roughly outcome lot differentially various application objective release follow complementary seek distortion release sensitive attack attack reconstruct accurate attack privacy differential privacy translate distortion reconstruction attack first consider context mechanism random database direction close privacy bound marginal linear parameter table per attribute non notation bound attack fraction arbitrarily attribute private show extend release problem point look singular however attack analyse subgaussian analysis dimensional subgaussian thereby applicable subgaussian analysis hamming letter transpose euclidean denote frobenius identity sphere center origin z independent brief introduction theory kernel book let empty set hilbert space reproduce hilbert space df refer allow computation know explicitly trick allow solve note infinite mining rank kernel pa ab polynomial dimension frequently radial x control locality indicate vice versa extension formally subgaussian subgaussian subgaussian say subgaussian dimensional marginal subgaussian subgaussian random variable arise naturally analysis spherical variable fix dimension subgaussian convex isotropic isotropic isotropic subgaussian subgaussian subgaussian random subgaussian random constant use net subset net cardinality follow standard x large kernel triplet entry additionally subgaussian need bind vector random center subgaussian dc subgaussian take net claim c establish spectral random subgaussian denote obtain kernel diagonal independent entry deal dependence entry provide follow center subgaussian p pn dc split diagonal let represent part norm bound substituting assume variable subgaussian fix j j whether easily simplify lemma c take union
win heavily study optimal rely social agent choose choose well whether player question learn order exploration range agent bandit characterize payoff strategy bandit agent plane intelligence payoff laboratory multiple parameter choose social intelligence interactive game game player aim maximize payoff round agent square integer write payoff bandit round store piece time obtain obtain information piece bandit move exploitation bandit provide bandit exploit agent previous round bandit randomly agent information obtain old bandit exploit bandit bandit necessarily receive among bandit intuitively upper hold agent observe payoff agent learn agent multiply represent player game agent game advance sequential round agent denote player store three piece bandit information player choose round start player observe agent environment website agent round store piece bandit interface show observe bandit would payoff bandit information large bandit obtain good bandit good report room student mainly school science subject room brief experiment sign experiment document game start round min subject subject among subject reward top subject environment could environment approximately addition relate subject ask three game environment randomly experimental experiment subject game environment optimal mean could player know know round denote exploit payoff choice estimate player game value remain assume piece information player information exploit expect bandit quantity expect bandit summing divide expect round payoff obtain assume player continue new payoff round front vanish payoff one zero otherwise obtain payoff payoff exploit large might optimal expect payoff per obtain bandit payoff round change account age bandit compare determine action maximum payoff bandit choose round conversely hold expect payoff choice exploit player highest choose choose learn condition player maximum payoff round payoff expect payoff per pi agent bandit exploit four simultaneously round strategy simplicity agent term tb thick solid boundary dot beyond plane thick boundary lower leave I great delay bandit might dotted boundary comparable payoff exploit good bandit exploitation trade social dotted noise social trade bandit previously small learn instead try good bandit exploit exceed intelligence intelligence intelligence subject choice observe intelligence human subject comment intelligence choose good observe exceed intelligence advantage estimate perform experiment reasoning player everything relate game experimental human subject analysis subject calculate environment divide round average payoff subject represent subject pi agent cc subject great expect intelligence fact h pi increase pi value pi pi obtain bandit however near intelligence subject could see agent find change provide bandit payoff payoff pi increase pi depend bandit great bandit hand obtain bandit payoff exceed agent bandit big performance pi depend large great intelligence payoff payoff low region pi good succeed obtain bandit observe intelligence variation examine make strategy linear multiple performance predictor kind proportion move round number round predictor agent program subject notice frequency former predictor include predictor regression average payoff round linear htb intercept n negative suggest effect observe rather two trade develop interactive player option bandit exploit environment scope exploration making observe payoff optimal restriction knowledge exploit observe plane strategy intelligence intelligence intelligence plane optimal I experiment intelligence effect intelligence subject proportion factor proportion agent make believe human factor mind elaborate make basis agent option round maximize
entry note I merge discretize well gaussian gaussian quantify amount distribution suppose parallel axis induction handle proposition general th lemma induction recall structural integer let variance independent positive take confidence use k guarantee return round describe original preserve minimum shift shift determine try integer express onto consider hypothesis select require take access om output stage distribution differently use lemma sample kolmogorov distance discretize tell result lemma rescaling get within true tv tv z triangle subroutine select remark fact definition pt pt mit mit mit sum independent multinomial discretized multinomial applying requirement factor minimum eigenvalue distance significantly cover term dependence particular result multinomial nonparametric multidimensional family ball perhaps characteristic bias towards bin mathematically distribution support basis specify understanding question via gaussian behave discretize dimensional exhibit variation hard multi limiting quantify finite dimensional assign general multi matrix typically tend provide bind distance discretize establish old use stein multinomial summary distribution discretized random covariance desire interestingly direction arbitrarily sparse add direction approximate provide intuition mean nash equilibria player share say player depend action player utility otherwise show total f approximation equilibria whose cover intuitively nash profile player affect payoff anonymous cover interest interesting feature discretization cover exponential discretization polynomial provide asymptotically space equilibrium anonymous game polynomial cover consequence theorem improve cover sample motivated application cover cover make cover contain form see least count count probability namely polynomial obtain namely polynomial view directly learn ok kn generalize poisson sample run vector binomial correspond multinomial random recent establish already work complexity exploit connection optimal pose projection onto vector poisson binomial indicator discretize support light one like aggregate heavy discretized gaussian light small correlate even polynomial unclear pay dependence projection onto vector behave log unimodal exhibit mod modal bernoulli give thus respect mod projection permutation vector identify show multinomial discretize independent multinomial roughly explain heavy explain light light heart proof approximate poisson issue application arbitrary decrease structural technical lie avoid latter cost version procedure shift equal sufficiently far round combine argue round result original variance partition eigenvalue span logarithmic repeatedly partition sort vector logarithm central fall bin vector poisson structural detail approximation comprise several sparse gaussian sum gaussians equal discretize quantify induce detail cover advantageous discretized characterization achieve exponential reduce dependence size naive one moment first profile leverage result size cover remove dependence exploit cover characterization multinomial sum discretize independent dependence cover discretize dimensional dependence challenge candidate discretize gaussian independent suffice purpose unknown easy variation intuitively answer yes multi feasible sample access case moment discretize dimensional movement mass aware broken arbitrarily vector round matrix picking round near minus notion discretize care gaussian live disjoint gaussians direction non simply non consecutive zero diagonal add matrix result multidimensional multi form ignore block difference sample structural state gaussian describe multinomial close preserve round multinomial minimum least main replace sufficiently far original motivate operation note central minimum covariance matrix perform round guarantee summarize round matrix efficiently start fix consider coordinate move light analysis relate careful coupling binomial single long approximately round eventually far analysis round lie relate multinomial gaussian preserve round plus start bound total discretize careful partition merge discretized gaussian assign cardinality discretize leave obtain original sum discretize preserve rounding preserve round overlap dimension merge repeat merge leave structure preserve rounding would cost care discretize gaussian two bind must previously merge discretized clear indeed swap merge describe pair cover structural poisson discretize multinomial grid vector cover overall size cover moment technique component write sum derivative multinomial drop total parameter evaluate point derivative multinomial two matching moment profile roughly derivative close count moment whose lemma section dynamic two lemma mention use style take remove cover try possible guess partition sample mean vector convert search spectral cover semidefinite identify take consistent sufficiently variation learn guess diagonal acceptable formalize component within fill coordinate block gaussian variation cover distribution sum mean following guess accurately enough order discretize suffice match section look fix prove real range affect direction multiplicative second additive direction error direction component additive error sparse due show covariance challenge significantly matrix close cover read ok contain close good pdf
acquisition result trees acquisition acquisition forest grow end incorporate acquisition cost forest grow greedy minimax split optimally cost intractable greedy approach output respect optimal cost classification low feature acquisition subject cost risk constraint acquire also low generalization theoretically characterize random forest superior curve full training extensively cascade belong acquire generalize multi propose however reinforcement learning capable wide supervised study risk framework cascade complex leaf sensor node various forest tree collection constraint tree low cost tree operate evaluation notion despite problem prediction pair learn budget user budget acquisition cost use example cost make problem iid subject budget forest learn cost write follow rhs rhs feature trees forest motivate tree feature budget forest strong feature acquisition cost vector acquisition label monotone helpful measuring mostly iy pt rt fs fs tt rt fs node build tree subroutine tree return subroutine greedy return leaf searching classifier minimize outcome intuitively feature reduce cost choose partition recursively apply allow return predict predict majority acquisition cost tree subroutine hope reduce maintain main broad class max classification pair internal leaf direct along path example reach acquisition denote cost incur path feature contribute subsequent acquire incur aim decision give cost criterion bound cost loose max number real leaf feature max function decision tree enough optimal strategy subroutine function negativity example set always scale outcome classifier say path contribute fs fs tree tree associate subtree root min fs fs fs maximize inequality choice fs fs sr fs fs fr fr leave subtree root leaf child max construct achieving example admissible feature optimal inductive induction verify base induction subset choose cost reduce choose child choose feature choose algorithm pick show fall admissible call paper object proof neither monotonic entropy entire therefore traditional max admissible power small offer advantage pair please detail conclude discuss implication subroutine build leaf meet random forest tree high acquisition forest constraint meet conversely yield due add meet illustrate toy figure circle triangle upper figure rest classifier draw leave evenly equal either useful reflect plot reduce choosing reduce set choose towards feature classifier contrast appendix subroutine call opposed child split lead cost split cost setting emphasize adjust threshold cost figure synthetic achieve compose belong fix feature integer range range label respectively carry unit cost correctly classify every max early stop comparison one show high prediction massive acquisition explicit achieve use fraction meaning example feature plot understand cost art configuration example exceed among run standard deviation bar world clear method error less yahoo consist document document relevance document algorithm take query acquisition extraction provide yahoo query precision predict sort rank predict irrelevant document reveal appear increase precision run tree leaf aggregate leaf node class sum show fast cost build thus require user search task goal distinguish signal validation forest number meet budget choose achieve every point budget feature achieve contain choose high whereas decrease acquire believe partly distinct categorical highly cifar datum combine initially test budget outperform curve budget comment observe achieve whereas low terminate early budget budget fast setting powerful incorporate forest algorithm issue incorporate strategy limit acquisition cost matlab rf default setting forest replacement grow select set rf show tree number feature example test achieve cifar quite rf yahoo even high c tree forest rf cifar rf j accord polynomial admissible singleton integer exist show summation index involve leave sum dominate another power various study building compare uci repository assume feature cost unique single instance label
content indicator reference corpora scope content corpora york article tweet twitter author article project public text extent wikipedia mixtures many language majority english attempt entry phrase length inclusion corpora process cross execute splitting list piece experiment likelihood distinct union accept list code experiment truly phrases code positive truly phrase code positive discover upon perform average spaced plot operate characteristic roc fig area auc expand list twitter wikipedia live present miss discover gram phrase length corpora material prediction accord consider gram likelihood frequency date live correctly discover reference add phrase produce filter corpus analyze phrase twitter far away reference corpus wikipedia filter filter gray likelihood reflect discovered horizontal dotted number discover filter filter short list take red dotted vertical gram effective miss list filter observe output perform cross day http rate think http video I I well change twitter background video http facebook video http http think video http go channel http http could live stream daily back thank rt http filter corpus frequency filter automate phrase definition manuscript pilot program lexical table phrase highlight filter summarize look lexical table closely corpora phrase likely name corpus consistently family know corpora phrase hand general filter also corpora pure twitter corpora extra english expression highlight rely beyond extent phrase tu ne syntactic english ne straightforward construct language indicator language fair phrase predict form define dictionary rise list well corpus twitter likelihood integrate auto correct possibility construct syntactic indicator part whose possible precisely present sec scalable understand language text aim lexical organization fall family partition importance universal applicability interest appropriately applicable research order categorical g phrase primary lexical unit object employ word phrase online collaborative dictionary develop miss phrase entry short list lexical extraction expand knowledge english shannon appearance symbol shannon assign word place occurrence probability find production english frequency still early modern language guess phrase roughly equally measure spend arise make people actually say size availability text shannon generally information association extract extreme many aside information theoretic syntactic scalability make shannon ic associations length shannon gram gram predict word length relationship sentiment ic word however gram special representation corpus level plot line frequency gram consideration concern shannon context appearance property window gram recall gram define page appear gram seem text scale parse practice sentence boundary significant new format five apply gram frequency express also advance gram corpus text grouping occurrence lexical unit informative word produce informative text quantify text frequency pmf phenomenon relate informative power applicability frequently human capable infer previous scalable partitioning since balanced underlie word frequency length norm generalization phrase length word removal set collection removal phrase joint phrase form produce page phrase external relation semantic external context interest actually development phrase internal context pattern removal phrase phrase phrase analogous removal pattern phrase length clear semi formalize template formulation difference restriction contiguous word I phrase mechanic secondary partition weighting context phrase accomplish secondary partition process follow relate phrase phrase observer sub phrase phrase retain sub phrase probability proportional preserve phrase conditional q utilize work eq define preserve context beyond point document normalize convenience derivation expectation section back line densely draw white lc lc lc lc cm gray lc distance fu gray mc gray leave right cm contrary gray contrary node cm contrary fu fu eps eps fu contrary deriving mean definition phrase
notice grow clustering grow infinity seem uninformative illustrated fig ari dataset improve c rand hc km ap b notice automatically detect span art topological maximally filter practice compute filter thank cluster build retrieve co distinguish return interested extent apply display market take lead meaningful practically odd trading day day partition partition obtain price return stable conclude highlight leverage lead fine novel could machine series describe relevance walk scale website lead area finance aggregation give article cl design rgb capital management paris capital management paris pre distance learn algorithm work identically distribute process split dependency metric synthetic benefit series market website field metric advance art many claim fair mention combined behaviour difficult conclusion fast develop restrict scope series identically mathematically subsection present similar copula namely classical pre dedicate synthetic correlate different also financial series market whose dynamic model market stock stock market cf website available future direction methodology usually step pre article study distance exist measure process classify two quantify divergence distance copula ignore property discriminate motivated perfectly correlate return normally risk hence illustrate benefit primarily finance obtain variable propagation mean cluster grouping object object cluster different cluster cluster dependence differ dependency stable stability desirable perturbation obtain resample spirit preserve researcher financial notice poor representation thus capture variable perform distance yet suit consider value discriminate obtain expect equal variable perfectly discriminate distribution actually half appropriate comparing grows take blue distance dynamic wavelet pattern distance take distributional random absolutely cdf split marginal distributional mapping transform represent follow element replicate seminal theory apart yet random let particular case hellinger quantify dissimilarity two measuring express implicitly hellinger trivial verify separation axiom yy u addition monotonic desirable unit device model bivariate dependence copula perfectly want discriminate two gaussian dirac function follow proxy tailed capture apply parametric realization continuous distribution statistic yield coordinate multivariate copula underlie approximated realization permutation function ix hx xx I distance use parametric use estimate suggest mix reflect information cross
show aggregation follow finite quantifie method nice stage prior knowledge beyond bounded enjoy rademach routine argument empirical happen minimizer mean otherwise model specify deterministic theorem focus present clean abuse dimensional sphere radius star algorithm statement outside optimality interior inside strictly contact location three range maximize argue extreme geometry cone rearrange prove extend claim h h bind immediately deterministic star multiplier empirical term start discrepancy multiplier rademacher inequality bound unbounded former statement function require tail excess surely contraction argument remove multiplier appendix offset rademacher offset rademacher localization phenomenon beyond contraction ball assumption condition analysis quadratic expect function multiplicative ball somewhat phrase low isometry isometry say depend mild behavior ball heavy tail assume ball condition plus comparison small satisfie ball arm isometry control via offset rademacher isometry exist absolute constant stochastically dominate rademacher requirement mild offset proof appendix extend classical offset investigation summarize show excess star high appropriate property estimator offset rate cover contrast radius way offset end offset empirical theory offset eq unbounded armed upper offset complexity class probability union cover objective process star shape isometry critical statement offset fluctuation offset process control second offset large obtain term radius concerned offset rademacher complexity gram conditionally loss offset interpret transform expectation order symmetric aggregation offset rademacher class define offset bound eq observe due fact finite pass offset rademacher star hull class case offset star hull offset rademacher complexity rate initially one offset define online regime parametric also estimate vc easily plug upper bounding offset offset minimax define offset rademacher complexity matching take star hull lie eq invoke support state prove stand respect term jensen rademacher observe unchanged precede contraction upper bind combine uniformly hold q jensen bound operator respect copy expression q expectation drop back sign excess later probabilistic exist term rademacher isometry bounding last term chebyshev whenever regime eq write claim q move unbounded n use proceed exist write argue term element triangle positivity bound keep diameter indexing proceed high ball let sphere choose compare supremum rademacher ball inside apply isometry conclude probability offset bound restrict within radius denote minimax excess uniform cn proceed exactly replacement conditionally upper hold offset lemma equation probability valuable regression offset excess inequality risk recover boundedness determine regression arguably substantial generality class verify bernstein condition relate variance increment localization phenomenon optimum analysis large part heavy obtain tight excess especially unbounded sided control tail control mild analogue localization offset online supervise supremum offset lower establish nature offset supremum high convex star estimator assumption even offset provide intuitive rademacher let rademacher index stochastic offset rademacher capture magnitude quadratic act
kk later term time begin decay continue number coefficient depend introduce compute store call consideration return purpose detail neutral bridge eq facilitate put fisher proposition recognize simulate bridge neutral mutation step complicated appearance address follow subsection x alternate combine monotonically converge take pointwise evaluation hard actually employ approach strategy triangular coefficient k property analogous drop convenience coefficient multiply alternate simply provide member decay use lemma analogue property respectively combine proposition order amenable k alternate q odd monotonically converge bound j sufficiently explicitly summarize convenience also simulate v z algorithm fisher diffusion candidate employ reject impose continuously differentiable z detail express recognize simulate give xt algorithm skeleton e find numerically minimize bound easy since table selection candidate result length parameter efficiency great extent length prohibitive nonetheless still feasible simulate collection short string together long diffusion mutation except path run mutation total number per path poisson simulate number l r l l c improvement underlie algorithm vary inspection permutation start may improve refine must compute recall dependence summarize evident coefficient algorithm value various exception observation relevant grow quickly know instability expansion unable separation implementation small fortunately much simulation apply distribution bound point product possibility simulate neutral fisher diffusion even dimension fisher dimension neutral equipped sigma algebra weak topology mutation mutation reversible stationary dx evolution dirichlet product simulate simulate exact simulation fisher diffusion exact neutral extension interesting currently wide perspective believe develop exactly process brownian function treat show subsequently decrease monotonically follow finite hand drop routine soon subsequent right hold exceed express mode separate two continue hold substitute j rearrange maximize note last inequality compare note pair head hence md md subtle increment use get proof immediately proposition helpful discussion department mail uk cv mail primary secondary keyword phrase simulation process diffusion bridge population span department fisher evolutionary widely population finance simulate fisher diffusion difficult know formula function drift simulation key approach yield exact simulation fisher leaf tree application evolutionary mutation perspective model simulation great chi inference serve evolution genetic variant randomly diffusion dimensional sde drift coefficient evolutionary recurrent govern natural fitness individual number copy allele trajectory become available dna evolution fisher finance evolve discount price signal filter evolve simulation transition exact transition kolmogorov quantify empirically however another recent diffusion process use algorithm motion construct path recover admit process brownian start process goal less require necessary occurring realize obtain path infinite determine simulate simulate simulate xt j tb u j b skeleton point finite skeleton restrictive use certain brownian motion
equal vector easy return factor perturb order zero convenience show rough roughly sum get work offset corollary conjecture assumption symmetric symmetric plus eq operator equal eq vector sphere vector nd determine single matrix recall noisy side maximum calculus skew matrix zeros th section noise orthogonal coherence intuitively map satisfy natural simplify analysis derive eq q observe turn form first technical follow vector randomly union establish desire line cauchy assumption q incoherence stage ik jk component another basis symmetry sensitivity lie union center radius permutation combine find term whiten useful whitening simple decomposition
greatly different symmetry allow integral analytically weight numerically completely integral increase accuracy mcmc perfect expect convergence markov chain mix intersection curvature plane ignore isotropic take account curvature volume reduce eigenvalue blue curve integral geometry weight paper red top figure thing worth curve rejection curve compare integral geometry smooth achieve benefit part variance weight geometry variance high intersection occur rare reason search small subspace dimension huge intersection volume difference angle curvature aware eliminate validity curvature motivate curvature mathematic concentration geometry weight intersection volume dimensional blue depict red volume close intersection volume exponentially logarithmic curse volume traditional avoid curvature volume variance volume informally cauchy rotation measure formal discuss situation point unit inside cube choose plane isotropic center cube idea choose isotropic orientation use origin orientation technical issue relate effect cube lack spherical natural conditioning cause subspace greatly much completeness spherical euclidean theoretical application fix non usually haar special generalizing euclidean geometry bit careful case issue poisson subspace g ds haar measurable drop plane element wish life infinite measure intersect volume proportional subspace introduction formula volume spherical gauss curvature manifold euler gauss theorem gauss curvature form connection intrinsic curvature jacobian determinant gauss determinant hessian manifold express tangent space gauss theorem usually relate curvature euler characteristic gauss way relate volume curvature come curvature sufficiently allow many weight measurable unweighted histogram go infinity unweighted slow term greatly variance hence greatly convergence cauchy obtain reduce intersection figure probability subset finite volume constraint accord point unbiased jacobian tm observe integrate x formula measure layer apply illustrate measure cauchy formula exchange hold theorem ds dot say dot differential manifold green dot instead weight greatly cauchy individually manifold apply first algorithm isotropic mcmc oracle dimension generate isotropic dimensional spherical qr decomposition solver unnormalized sphere radius full metropolis reweighte sphere unweighted conditional I whose gaussian h tx standard normal jacobian oracle search iteration random isotropic origin heavily nonlinear unnormalized xx x unweighted correctly accord find compare geometry greatly mcmc geometry differential symmetry haar depend choice moreover constant regardless generality uniformly intersection w w convergence identically slow give wishart determinant iid normal nonlinear solver intersection introduce great traditional ideally correct chain randomization pair great corrected require great randomization nonlinear weight slowly traditional weight great dimension manifold event high intersection however intersection assign point factor simple jacobian depend sphere constraint intersect much intersection intersect sphere curvature level manifold intersect plane pass near slice slice small density gauss curvature slice exactly reweighte gauss relate curvature volume would intersection heavily orientation respect mean converge absolute gauss curvature intersection gauss curvature general unbiased distribute guarantee unbiased distribute determinant inside originally intersect orientation projection complement denominator radius instance compute analytically gauss fact total always always volume sphere converge need uniformly without introduce curvature beyond curvature cauchy curvature uniformly manner function keep vary first haar orthogonal tangent plane suppose operator gauss satisfy boundedness curvature weight algorithm condition arbitrarily intersection occur cutoff fraction average intersection curvature manifold uniformly cut likewise volume curvature cutoff convergence manifold gauss corollary form sake completeness far beyond prove suffice plane ss measure euclidean restrict increasingly neighborhood geodesic cube search treat assume orientation subspace curvature location small remainder proof consist part place extend entire k ball radius tangent ball independent short line contain denote orientation write surely equation fact conditioning independent remain event symmetry formula coordinate origin rotation subspace span multiply rearrange expectation side q expectation respect condition intersect rhs exactly orientation tangent write place equation observe eq poisson wish think subset uniformly form symmetry volume assumption numerator denominator lipschitz denominator cut countable compact jj almost every must q side q nonnegative term expectation subset follow use indeed boundedness everywhere pre gauss together improvement step jacobian connection possibly derivative heavily solver support center step metropolis reweighte restricted sphere curvature original applying row also determinant haar expectation determinant definite polynomial matrix algebra form curvature random determinant numerically perhaps carlo also easy volume algebraic theorem always rare alternatively one might density density certain region search cause variation unless briefly new gauss interesting introduce serve introduction motivation curvature manifold chain algorithm gauss gauss hand may quantity pre generate unbiased intersection volume argue gauss case order point section estimate curvature equation ks implement represent information ability divide euler characteristic high estimate possible euler characteristic well euler characteristic respect statistically ss property gauss curvature manifold general say nothing assume statistically nothing quantity ss like know locally second pre attempt local make guess characteristic intersection would order probably hard implement reason nonlinear solver newton may topological riemannian manifold gauss curvature curvature another differential partial pde define say integral equal product invariant pde idea pde attempt curvature form curvature pde differential form vice versa whether elliptical implicitly argue exponential volume corollary gauss curvature involve sample imagine random subspace imagine subspace intersection volume speed intersection different volume volume theorem metropolis need many converge intersection volume cause reweighted volume avoid variance intersection volume volume volume vary greatly depend sphere fact intersection sphere curse volume wish sample sampler gauss intersection exactly deal variance intersection volume increase exponentially close although subspace radial direction accord haar spherical represent small spherical let affine subspace accord result deal spherical spherical concentration say variance volume exponentially dimension probability volume use generate show yet geometry make soon show haar dimensional great spherical radius exponentially dimension subspace convergence traditional within intersection volume gauss curvature regardless exponential volume geometry generalize algebraic manifold formula theorem volume algebraic manifold long analytical argument curvature convenient subspace degree bx algebraic intersection arbitrary integral absolute gaussian derive formula intersection corollary additional vary much direction manifold corollary beyond scope paper know large perform principal random matrix include unitary ensemble point process large eigenvalue limit limit model converge limit condition eigenvector statistic algorithm case discretize iid stochastic discretized matrix knn normal cutoff cutoff due decay eigenvalue decay like discretized operator already iid constraint simplify nonlinear solver subroutine approximate instance metropolis temperature randomness subroutine randomness search step isotropic random radius deterministic nonlinear solver start intersection weight independent approximately accord fx iid normal I weight solver subroutine introduce solver probably intersection solver would normally compare weight scheme use simplify numerical implementation beyond rather purely deterministic solver metropolis contrary section traditional scheme together plan perform numerical metropolis solver briefly explain fast rejection eigenvalue deviation since condition rare probability event big oppose equal value since equal event dimensional manifold remain situation eigenvalue dependent reasoning come week majority involve neighbor hence test general situation week conditional dependency even probability would geometry histogram blue agree probability rejection black weight traditional histogram skew either blue greatly bias intersection point h histogram use rejection sampling six would cause rejection metropolis subroutine geometry agree approximated rejection weight extremely skewed histogram probably theoretically traditional weight nonlinear solver intersection skewness especially simulation event chance find unless search indeed tell vs sec algorithm hope rejection would day make reasonable amount subspace subspace rejection integral fairly close obtain blue weighted traditional much skewed implying reduce skew solve restrict plot error histogram case skewed right obtain weight
regret discretization supremum norm attempt start contrast complexity analogue discretization expert time suppose tx ft ft depend invoke suppose order indeed satisfie conclude view hold term absolute ignore constant minimax respect balance choose euclidean loss additive logarithmic exhibit imply tight turn field gradient restrict still gradient descent constructive mention function exp infinite barrier self barrier protocol local bound independent together bind take boundary appropriately importantly sequential cover hilbert class ability hessian barrier give simple yet due appendix regret hence crucially self regularization ensure gradient linearly surprising lead ball loss offset sequential complexity problem offset rademacher complexity via scale sensitive cover supremum take entropy expectation deal drop indeed outcome p tv eq suggest key tight keep uniform give concentrate first addition tree notation optimal sequence reasoning specifically subsequently probability surely choose take argument side lemma random variable probability proceed fashion become become shorthand sense course scale zero element eq supremum choice n h lemma term conditionally cauchy far q indeed clearly q w eq q statement proof closely refer define recursion proceed induction induction proof induction fix tree accord sake contradiction aa construct root rest gradient barrier without loss inverse consider reproduce completeness ellipsoid statement acknowledge grant dms assignment loss expert upper term sequential complexity factor bound loss logarithmic employ bernstein intrinsic assignment expert parametrize hilbert ball discretization barrier interest study observe loss allow history forecaster expert specifically consider singleton set contain far may view forecaster optimal extensively alternatively forecaster slightly adversarial fashion forecaster markov expert time invariant case act formulation interesting affect minimax regret eq shorthand forecaster nature upper attain last year root analyze bring assignment study rich class unbounded choose later let thresholded class truncate function check minimax modify via sequential dependence bernstein sub tail behavior expert well static obtain set discretization style expert unit supremum set approach contrast employ idea dependent notion technique attain optimal section loss square matching open attain finish introduction sequential assignment theory code mostly exact case class connection compression interesting algorithmic compression method thresholde state lemma abstract view sequence notational definition appropriately say respect set mapping purpose constraint reflect minimax regret stochastic key difference bias coin logarithmic dual approach worse bernoulli range tree supremum random index mean supremum indexing eq possible act display bernstein crucially consist q collection infinite recall sequential number sequential respect value depth cover denote become tree eq readily identify immediate q section theorem sublinear sequential finite define respect equivalence thank summary complexity sequential covering number match balance day upper quantify soon control sequential covering number could directly many calculation say value
eq policy decision bandit give asymptotically interesting different uniform prior time indicate left table produce regret random bandit additionally interesting small variance indicate exhibit tail policy superior largely science foundation nsf grant du joint classic z observe induction q complete room improvement bind arbitrary power influence result utilize instance similarly event normally simply proposition equivalently eq rhs simplify limit complete convexity suffice relevant sufficient c c dc h consider problem sequentially population specify sample assume normal population expect outcome total equivalently lack simple index additionally controller sample unknown controller convenient define maximum bandit additionally paper largely non policy controller bandit n pseudo take due controller controller would expect follow eqs introduce context per nr constructing modify along two play winner derivation strong say however slow well controller make trade turn strongly policy existence present pt width additionally therein give policy exist fast primary motivation population considerable bind imply guarantee population sample logarithmic therein population I policy achieve motivate definition convergent within convergent n blue arc pt nn establish sufficient therein express conjecture open conjecture appendix fail I asymptotically optimal technique establish insufficient lose establish demonstrate thompson sampling achieve horizon provide remainder thompson paper conjecture optimality depend probability tighter possible paper chi square bound demonstrate second giving worth improve use version eliminate term simplify take infimum complete define dominate linear follow prove make good achievable choice bound yield tight still optimality growth convenience value regret basic define quantity q follow express indicator account term let third recall chernoff bandit u I ti maximal optimal bandit hence aside would play optimality essentially successful conjecture observe dependence integral extend
empirical truncate spectral b range employ backtrack implementation detail alternative main work appropriate perform descent q size conventional progress objective backtrack line repeat constant definition mainly calculate product tm product step extra incur backtracking compare set algorithmic parameter determine truncation threshold backtrack fix employ backtracking adopt extra herein fall range table proceed understanding start uncertainty presentation let direction helpful intuition identity average direction unbounded consequence non negligible influence issue one separate individual directly whole absolutely sufficient ensure truncate truncate obeys account bias truncation look direction sufficiently align reasonably angle away towards step size appropriately regularity fundamental rapid procedure descent specialized sometimes rule plant nonzero reasonably contraction z z otherwise finally connect former guarantee graph distinction stem pair fix simultaneously stationary point truncate objective neighborhood suffice scheme section report practical applicability numerical conduct current concrete parameter unless employ initialization iterate iterative refinement series concern free design independently draw claim return success rate trial iteration ambient dimension find experiment code describe depict success indicate value sake empirical default success rate suggest fast exhibit behavior experiment demonstrate stability varie vary snr cf I generate accord show scale function match stability scale slope predict mse vs snr section prove theoretical absence noiseless mainly truncate similar argument return obeys exceed constant suffice state local contraction consider noiseless case monotonicity reasonably attract geometric rate hypothesis everything constant proceed event begin truncation two fact universal bound prove eigenvalue ft illustrate immediate consequence cauchy together two demonstrate eq satisfying follow event resp statistically independent close inspection reveal quadratic facilitate interest follow inclusion prove well imply refined provide tell establish suffice uniform form formally derive measurement condition constant expect second decrease convenient inclusion reveal leave right explain influence truncation discard reasonably please recognize term right side I share nonzero rare upon rigorous first subsequent non move second rise influence satisfy fraction put quadratic rate constant recognize necessarily q come give amplitude inequality pick appropriately get simplify precede restrict poisson carry broad nonconvex objective result continue hold within neighborhood time sharp concentration truncation truncation might randomness leave future power rank imagine wish known problem computational hope develop modify maintain operate truncate spectral initialization successively measurement low evident precede add scheme concrete application imagine instance image align pair denote cyclic one shift efficient make q quadratic moment respectively make outer I I proceed conclude homogeneity suffice indicator function proceed orthonormal resp resp identity arise since I gaussian inequality deduce obtain net n discretize unit eq lipschitz guarantee arise place unit arbitrary putting complete make convenient lipschitz definition apply probability unit cardinality arise demonstrates claim difficulty handle indicator work auxiliary function purpose tail I inequality sub indicate sufficiently proceed uniform control sphere follow lemma put deal observation basically tell none consequently fix recall eq soon large everything come make inclusion control one I substitute collect useful observe sufficiently besides hoeffding yield sequel first vector pair distance obey packing argument inequality take union r derive convenient numerator denominator stochastically simplify presentation affect I enforce group consequently existence equation fortunately put satisfie k later light collection precede notational move many vector notably remain argument proceed apply bind markov together vector set consist form lemma eq probability thus consequence come lower define bind proceed identity computation truncate state proposition truncate obeys homogeneity case imply non isotropic sub gaussian deduce besides repeat argument omit justify condition tt let obtain addition indicate consequence far union conclusion claim apply effectiveness backtrack search contraction keep noiseless difficulty optimize constant boundary size backtrack notational throughout I truncate evident start scalar get simplify observe plug two identity yield I term consequence combine gm secondly follow mi gm mi I mi put together yield backtrack seek satisfying criterion taking argument one criterion omit acknowledgement support nsf grant award foundation nsf thank long manuscript grateful many flow rgb problem system equation start compute nonconvex approach distinguish feature notably operate fashion drop careful quadratic time soon exceeds extend nearly example random quadratic hence title square imagine set form know priori nothing phase sign product nonlinear nature alternatively pose recover magnitude boolean example equal letting indicate one formulate system check np complete physical sciences technique ray arise record intensity notably upon object form however intensity measurement leading magnitude magnitude spatial depth motivate line think record intensity always noise noise shall pay poisson reason variation optical imaging noise impulse seek maximum denote outcome poisson unfortunately log surrogate propose particularly vector choose quadratic trace relaxation scheme performance guarantee many aspect achieve near optimal exceed applicability another high paradigm iterate nonconvex promise suitable successively rule namely iterate exact mn presence formalize advantage hope achievable convex relaxation enjoy spirit propose novel adopt subtle informally proceed stage guess observation remark firstly data varying correspond result well recommend either take determined backtracking instance appropriate take stage vector product desire truncated gradient detailed specification defer reader practical illustrative impossible sign evaluate solution represent sign signal real shall throughout straight numerical concern square b solve arguably popular least square cg go condition equal ideal cg fig show cg iteration cg design observation applicability image digital code set discrete diagonal entry delay mask illumination quadratic code generate band green separately carry equip ghz intel core gb truncate gradient total cost color band recover display iteration take extremely concern noiseless numerical extend draw poisson model independently snr snr var mse e solution phase reveal addition give away mle cast program illustrate plot incur extra db loss ideal mle reveal phenomenon please precede promise exponentially recovery noiseless complexity nearly minimal mean square offer finding assume tractable shall use absence noiseless size backtrack universal estimate specify explain take make precise truncation threshold take appealing equation optimal since one measurement I cost outperform provable enhanced refinement stage proceed mean operate upon contribution control take away compare movement broadly must guarantee estimate represent claim backtrack eq q least estimate specify constant poisson exist event satisfy reader material universal simultaneously noise stronger noiseless prove use inform complexity rapidly logarithmic put way arrive guarantee snr emphasize e even approach formalize derive fundamental minimax error minimax obeys eq q infimum numerical measurement proportional energy plant theorem achieve vanish match optimality careful reader naturally normalize importance optical employ detect sensor receiver typically practical black apart nonconvex procedure phase iterate alternate favorable fall theoretical support except call attain exceed
correctness step must atomic processor put write cf algorithm execute master worker set worker master return average master fourth execute master cf memory master worker share processor read processor memory snapshot algorithm share master worker worker master processor access global master perform update possibly date gradient pass back independently processor share memory master master processor evaluation mini finish update apply asynchronous overall optimization time involve read clear reading processor use execution depend processor cyclic delay mini batch processor update decision receive termination satisfy establish property optimizer eq iterate average converge residual slow step correspond inequality tell side negligible mini batch processor algorithm therefore processor parallelization furthermore update roughly quickly processor mean speedup depend advance easy optimal minimize second inequality q reduce obtain serial mirror descent size master worker algorithm interface library although argue section atomic flexibility environment text categorization document span decide related classifier regularize token assign document document document scalability document tolerance meet figure speedup accuracy speedup average number asynchronous mini regularize smooth iteration algorithm run vary closed rate penalty negligible speedup experience confirm instrumental argument recursion suppose iterate number gradient problem subgradient bregman plug equality know rewrite left side equality result generate rest convexity recall obtain rewrite error seek quantity turn convexity norm substituting simplifying complete relation assumption iterate subtract leave sum precede inequality drop left conclude strongly convex zero norm see history last expectation side imply inequality prove theorem since increase completely integral verify substitute guarantee assume describe clearly ready multiplying use sum dropping yield term hand substitute definition tt simple type indicator se mini batch optimization powerful paradigm art mini batch cyclic order worker capability delay cyclic leave slow complete asynchronous loss suitably strongly iteration negligible near speedup worker expect confirm implementation arise signal processing expectation loss possibly nonsmooth term elastic stochastic descent nonsmooth approximation develop mirror composite explicitly account stochastic cite inherently place processor access whole happen unable handle amount cause develop able split processor therein one simple point recently processor compute gradient span processor drawback rest run processor paper propose mini regularize overhead synchronization processor perform update gradient similar asynchronous mirror stochastic asynchronous mini interestingly show delay compact set extend value contribution regularization running iterate around algorithm size function compact set prove average iterate time vary residual rate improves previously know delay mirror vary long establish iterate processor rate asymptotically strongly optimization serial review essential formulate mini report natural number include endowed definition refer distance modulus generating bregman another strongly respect simplex bregman function q motivation usual convexity throughout generality indeed scale stochastic support nonsmooth extend eq differentiable denote unknown situation occur application machine learning application time identically impose optimal q continuous effective possible include set
adapt intrinsic validate help even represent euclidean space correspond interesting practitioner include variety measurement uninformative distance powerful technique learn notion distance emphasize spurious measurement decade leverage domain notable mahalanobis quality explicitly prediction task popularity study attribute dataset theoretically practically vary uninformative measurement change formally modality develop two pac popular empirical two framework use objective small optimize base objective mahalanobis cluster comparison proxy optimize prediction incorporate hypothesis learn interesting example regime metric quality learn metric help structure framework lemma absence assumption generalize previous light early uninformative weakly expect metric formalize term intrinsic metric way intrinsic refine framework variation minimize erm jointly observe bias expect intra balance erm algorithm regularization metric efficacy criterion benchmark indeed metric adapt learn weighting remove arbitrary literature minimize notion underlie want metric metric label base reasonable way early explore regime popular quantifying amongst point class opposite weight yield short distance pair distance constraint rise error denote want become generic loss compute weight mx upper limit optimize computable criterion look keep amongst total amongst opposite class variant low limit distance rather hard opposite triplet focus relative distance triplet draw discuss variant triplet triplets metric maintain gap opposite comparison neighborhood affect performance make distance comparison act optimize explicitly incorporate insight retrieval incorporate learn principle constraint formalize framework consider hypothesis shall hypothesis real weight space good study ideal minimizing size definition discuss sample sample grow sequence pair sample mm bound convergence unknown lemma q note I mm ms conclude sufficient never dependency necessary distance error bound weight make distribution make bad classification metric explore effectively complexity vc dimension real complexity error excellent b pick hypothesis class line key achieve sample note find hypothesis hypothesis class complexity lemma absence specific datum pick function vary degree content must solid concept generalization emphasize contribution spurious thus reflect quality individual feature measurement learn performance intrinsic norm feature refine canonical metric framework start follow weighting metric sample loss frobenius quadratic weighting complexity consider class weighting help yield discuss automatically account induce data distribution still base feed recall feed hide arbitrarily enough incorporate hypothesis feed feed forward specify metric also criterion compare uci benchmark dim dim dim unknown synthetic simulating regime large uninformative uci synthetic covariance matrix entry draw set drawing ambient dimension uci split validation setting pick rank coordinate average near uci notice dash noisy introduce uninformative quickly unweighted poor interestingly consistently high whereas yield regularize solid improve performance remarkably degradation classification noise robustness show regularization encourage complexity noise metric generalization optimization framework instance pairwise distance complexity sublinear characterize specific likewise ability triplet partition training representation perhaps work similar erm criterion erm metric bound generalize lipschitz loss dependence alternate weighting distance result lemma loss result lipschitz importance structure formalize intrinsic metric rate tune typical focus complementary representation arbitrary hypothesis class partly success base regularization regularization design high measure interested bound I second uniform width depth let h satisfie x hx hx rademacher complexity class choose expectation exceed level failure well regular simplex later assign show consideration minimize metric restrict weighting simplification note solve binary classification vc bound detail weight pick belong f x mx mx x rate optimal equality ii occurrence observe note let quantity return follow note satisfied imply moreover would suffice height vector q vector definition define unit moreover non empty bi mean centroids knn n set point map p j ip ip x constitute random uniformly independently suppose sequence value function note inequality maximize select well distribute eq h class matrix value minimize cover call note observe I combine namely form fx bound failure probability pt spectral cover volume know v construction universal suppose value bound distribution find see shall pack generic value class domain value let cover resp packing minimize cover resp maximize packing follow maximal x hx determined net case distinct b apart packing
identity set easy get combine monotonically lagrangian function second equality fact convergent integer x k x k let k hence boundedness convergent relation q q k k let continuity optimality moreover x admm implie hold optimality condition get inequality hold monotonicity apply side identity k x k bound convergent far show fact I suffice solution replace k sequence complete block admm x imply immediately remark define apply globally range cover conclude free satisfying range motivated fact block parameter global convergence admm counter show block impose look sufficient however usually admm block also iteration yield add inequality increase sequence inequality k yield kf convexity k monotonically sequence yield boundedness give boundedness sequence k reduce k k apply three hand imply furthermore hence moreover x third k k sequence corollary zhang method multiplier admm apply structured optimization superior admm extensively literature prove admm implement ensure chen al study admm usually require small difficult compute small admm still parameter solve commonly cover keyword minimization square structure arise processing computer survey particularly separable minimization usually admm certain place block solve eq lagrange multipli dual convergence admm study extensively literature nice block parameter restriction prove block admm admm particularly attractive solve admm solve block variable q lagrangian convergence chen et show far impose block stable pursuit robust alignment semidefinite great convergence restrict relaxed chen lin zhang restrict bind sublinear convergence admm admm study variant block require strong boundedness constraint lin zhang admm lin zhang far propose approach convergence strongly trade penalty affect block admm suffer alternatively opt modify classified category class step add admm jacobian gauss manner restrictive also affect arise effort acknowledge admm admm probably restrict value convergence parameter free give great globally convergent term regularize square next decomposition seek decompose certain decompose fitting admm solve et advantage subproblem easy especially subproblem zhang admm regularize statistic lasso zhang reformulate numerical conduct note lasso vanish interesting example share literature interested component pursuit aim wise formulate corrupted respectively form admm surveillance aim extract surveillance frame surveillance find move foreground pixel restrict add physical et molecular pattern discovery identification reader component pursuit rank sparse observe small set measurement q note unconstrained interesting compressive measurement similar paper globally
fluctuation observation km row red solid red blue dotted versus subject subject accuracy repeat procedure relevant last trend probable mode notice appear first mode maxima compute choose standard second change mean deviation trend characterize essence difference people patient develop analysis time contain part series use decomposition low component apply heart interval ability extremely slowly strong ability conjecture activity partially support nsf fa cm new iterative decompose series extract mechanism underlie show many measure time key word outlier heart interval heart many heart attack health exercise complicate apply classification application commonly focus aspect series dimensionality randomness etc tool use technique include restrict deviation empirical mode series transform characteristic nonlinear representation great success biological medical sciences engineering texture chinese main purpose time two modern come belief contain reflect basic system variability mathematically represent low frequency frequency motivate frequency quantitative wavelet second intuition come perspective lot decrease statistical perspective set represent motivate time careful practical examine pure structure motivation series analysis learn redundancy support classification heart heart disease heart failure heart decrease heart failure literature propose analyze heart name incorporate allow purpose fold enable diagnosis kind heart health mainly decompose secondly outlier pure informative interestingly filter denote limit operator iteratively roughly speak noise cubic spline connect low envelope cubic spline connect use lack foundation new pass generate mask convolution iterative convolution rigorous mathematical foundation mask finitely crucial method wavelet trend characterize profile signal detail application need extract priori without priori get proper statistic dependent heart interval illustrate construct application record interval decompose function previous heart heart motivate statistic large deviation statistic term outside statistic characterize maxima compose amplitude series hour heart period activity think people motivate idea split whole suppose series correspondingly mode denote th quantile total component fundamental mechanism individual may trend trend represent three description list deviation st st deviations rd rd standard subscript subscript statistic series less omit compute maxima notation part diagnosis find irrelevant firstly almost diagnosis eliminate eliminate svm feature size small might feature lead inaccurate refine eliminate least feature repeat iteratively conclude diagnosis essence make heart conclusion datum heart series hour activitie activity period classify slight iv severe use method heart people patient people patient
restrict context acknowledgment grateful thank anonymous conference whose remark answer air office uk ac uk theorem output prediction allow q interested computable list computable besides make infinitely derivative side infinitely differentiable side satisfy list computable loss prediction corner eq loss later computable check smoothness obviously computable strictly measure probabilistic intuitively regard get function essence attain loss smoothness intuitively corner new typical repeatedly simplicity set number suffer say place precise proof consider trivial spherical function fix small call intuition behind prediction whose sequence long finite stand define section never arise replace random prediction identify measure universal say respect probability applicable computable whereas former computable randomness element ignore uninformative object equal within randomness respect prediction continue ignore informative randomness respect function special use log theorem follow quantitative computable role ignore coincide randomness previous randomness prediction e proof prediction notice pass translation moving map l computable proper loss function function result q computation eq spherical sign explicit simplify loss fundamental expression condition eq suffice compare criterion see back lemma criterion give partly check kf taylor convergent hand randomness log grow simplify spherical function fundamental cutting end correspond easy check case restrict restrict coincide title least typical ask question say log lead randomness respect loss parameterization replace impose parameterization requirement ensure randomness existence computable necessary yes intuition behind behind notion set suppose curve straight segment point set canonical namely tangent line cf stay transform correspond say selective preferred spherical
hdp infer double manner estimate double embed also use synthetic continuous speech represent sequence categorization task speech recognition whose acoustic manner acquisition child continuous segmentation boundary speech give isolate direct knowledge word problem solve list process direct acquisition access speech recognition current automatic speech modern language knowledge distributional well acoustic represent knowledge linguistic corpus however access acoustic raw acoustic speech signal human discover continuous speech et distributional co rely relationship speech child detect co entity consider distributional speech accomplish month old solely distributional seem age imply fundamental word segment distributional fundamental distributional help segmentation viewpoint acquisition consider language finding distributional explore discover signal distributional unsupervised learn directly double organize word feasible acquisition acoustic develop newly probabilistic generative hierarchical section section present hdp extend hide acoustic sequential computational kind method decade program recover boundary source segment maximize text sequence improve include process sophisticated calculate gram word context treat infinite segmentation method account nest language letter gram embed word gram backward boundary mention recognition error learn recognize knowledge knowledge acoustic recognition become overcome word method occurrence enable robot word multimodal show cognitive raw sensor human et al interactive interaction unsupervised et enable robot linguistic communication speech behavioral viewpoint basis online word concept build category acquire name multimodal dirichlet increase co occurrence multimodal categorization show category update categorization co occurrence name pair mobile robot acoustic word selection criterion carlo localization localization robot result ill error solely speech signal al unsupervised et et outperformed experiment lattice text discover language iterative report improve propose jointly word learn sound recognize acoustic ill recognition distributional acoustic train manner insufficient constructive acquisition raw hence unsupervise acoustic acquisition acoustic categorization transform continuous include hmms acquisition use category learn acquisition categorization overlap sound al word effect lee al discover proper sub word acoustic unsupervised manner language lee discovering letter sound rule automatically determine acoustic several study simultaneous acoustic language small method simultaneously integrate acoustic propose enable acoustic find descriptor parallel technique viewpoint point segmentation acquisition mutually theoretically integrate acquisition acoustic integrate theoretical author double analysis view unsupervise discovery raw regard double represent double structure structure period discovery become double et al double hdp hdp nonparametric extract motion sequentially model generative recognition categorization letter hdp hmm unsupervised terminology letter latent basically conventional newly conventional apply drive double conventional purpose mining topic respect drive drive compare raw driving letter conventional raw speech background mention paper double acoustic assume infer variable double unsupervised novel double hdp double generative hdp series potentially extend hdp name hdp contain language basis hdp word hdp next latent word basis illustrative overview hdp sampler briefly hdp hdp extension unlike hdp conventional markov hdp explicitly model duration hdp breaking super distribution emission hide next semi super state hide determined duration super categorical super super time tt assume emission efficient base construct gibbs hmm super pass reduce cardinality duration super order backward filter constant double structure extend hdp super state fundamental th generative hdp generate letter furthermore latent letter letter output word language lm respectively latent sequentially latent letter word hmms duration explicitly latent time letter th letter latent draw duration duration duration latent hdp latent letter draw emission distribution map letter word generate assume datum double viewpoint language acquisition review compose machine hdp sequence letter transition probability correspond letter regard conventional hdp consist inference generative inference acoustic simultaneously sampler hdp letter language acoustic structure continuous propose unsupervised machine overall hdp sampling adopt instead naive sampler sampler backward sequence backward sampling procedure make message super hdp follow super state super transition super represent obtain st emission condition duration state easily procedure calculate backward message pass hdp word st message occurrence become partition duration substitute message hdp look complicated efficiently latent latent letter recursively formula message calculation backward procedure employ letter hdp super iteratively use backward message please refer original hdp concrete letter word sample accord latent word word generative model regard super state letter sub hdp letter sequence sample ordinal hdp word letter sequence latent sample latent sampling resample sir define word kp represent hdp way hdp propose sir procedure employ proportional sample model letter letter I update update letter sequence parameter acoustic update state sir sampling accelerate result sir acoustic sample hdp hdp overall sampling initialize initialize message initialize super state word pz super state ss model sample letter sequence super word sir basis hdp time variable analyze word manner validate propose infer latent double apply hdp synthetic series comparative generate five word w word sequentially th letter letter poisson parameter emission index emission compare word pair represent follow six word letter seven emission comparative hdp hmm average fig trial result work appropriately gradually probable increase contrast speech acquisition double viewpoint precision adjust rand quantify ari estimate letter ari letter conventional decrease conventional ari ill ari ari gradually latent variable show generate sequence top show letter latent word infer word estimate show procedure work estimate tb eps bt c effective double embed series evaluate method applicability datum ask sentence use ie five five five sentence ie ie ie ie ie encode data size shift datum frame hz language set number seven hyperparameter maximum letter seven duration emission dimension conventional hyperparameter hmm similar possible hyperparameter conventional gibbs iterate seed trial open speech engine dimensional feature speech acoustic speech conditions dictionary encoding unsupervise conduct discover model unsupervise propose software fourth word contain ie use acoustic contain acoustic manner label dictionary base acoustic letter speech letter indice bt c lm conventional check
train recent work adopt deep supervised encoder decoder pair entire information convolutional encoder relate unsupervise semi loss add loss fully encoder train loss jointly nature depicted q intermediate constraint loss construct convnet phase encode decode convnet forward pathway part pathway feed opposite encoding pass convolution relu pooling pooling layer complementary preserve add pooling within pool pathway operation pathway basically place right position region encoding pooling figure loss indicator sample input encoding reconstruct decode successful architecture stack attempt useful widely sub manifold one manifold carry identical perspective experimental amount discard decode pathway fall convnet go purely unsupervise deep auto architecture deal miss joint flexibility ease switch loss collapse indeed common auto identity cause case avoid code must jointly introduce supervised scheme tend task label pre take account recognize generalize pre bound henceforth restrict training drawback due supervise improper epoch latter offer control interpretability play reason add list prevent otherwise work properly secondly avoid situation reconstruct middle upper regularize intermediate intermediate reconstruction shape light statement digit cause change equivalent nearby meanwhile major component sub direction introduce invariance sensitive basically sub explain digit exhibit rotation range supervise middle b right architecture layer mnist supervise semi mnist label unlabeled size sure uniformly several round new dataset form report along fix identical well computed choose hyper train union training validation set configuration digit kernel denote pool layer pool region jointly regularizer regularization architecture include dropout connect dp fc train without basically dropout model report aside follow softmax encoder denote tool fine entire encode softmax write c fc convnet pl present publish besides supervise label basically use regularizer use improve dp dp fc knn na knn na na dp fc task bayesian zero construct unlabeled class effect regularization show well publish get supervise architecture lastly drop add mnist become highlight address separate whereas merge step wider likewise tune validation induced performance reconstruction help display classification couple convnet semi dataset label remain unsupervised unlabeled video edu novel architecture stack generative pathway unified essentially net convnet couple objective include convnet produce feed position feed decoder desirable mode learn desirable property generative train wise stack feed pathway manner fail mechanism unsupervise another boltzmann boltzmann rbm kind encoder deep rbms procedure sampling tend inefficient main stack mapping implement feed forward pathway conversely mapping implemented feed back generative pathway e reconstruction deal rbms category sampling tend complicate inefficient feed reconstruct good invariance desirable mapping layer convnet invariance max subsampling idea approach layer complementary pool reconstruction model consist feed convnet couple feed back stage auto encoder convolutional layer relu follow layer max next complementary switch incomplete information feed
svd orthogonal matrix negative eigenvector principal quantum feature quantum picture mapping quantum pca way component quantum representation quantum eq representation compose eq representation representation imply inner product factor representation principal linearly classical respect linearly perform measurement return yes answer answer probability return output q positively classical likelihood new quantum digital image analysis classification copy digital support national centre st national institute mathematics technology quantum von quantum image paradigm computation obvious quantum computer construct field quantum rapidly many create analysis measurement behind training pca divide signal variability mostly noise lead quantum classify encode quantum measure quantum image processing draw quantum elementary system basic choose hilbert vector represent state combination operation join state system big eq also quantum system column hilbert orthonormal represent product iff outer quantum measurement outcome assign corresponding measurement request measurement operator first state execute propose recent year lattice representation intensity encode real quantum serve position pixel encode quantum inspired pixel image computer responsible encoding encode vector kronecker product responsible enhanced digital state form sub deal discuss publish recent classical already quantum cosine wavelet technique quantum state example author circuit representation processing number quantum author projective store operating processing basic algorithmic
expand detail rough google compare several image variant google aspects factorization approach scene factorize add region add scene factorize greedy dataset greedy add attention scene metric moreover benefit region base attain previously show except art also qualitatively two dynamic abstract mean visual influence generation pixel first pixel distinction foreground focus background small ie focus region highlight region highlight foreground please refer visual scene category significantly predict scene scene hold scene lda topic topic drastically correspond regard hold slice topic scene impact generation scene exploit structure contribution process generate attention visual introduce scene lstm scene system popular attention scene context combine model intelligence advanced project via department laboratory contract nf c additionally nsf google award fellowship award nf reproduce purpose view conclusion herein represent express imply partially cb image dataset provide patch example patch classifier output patch adapt describe vocabulary less discard token begin take denote start sentence well size token begin sentence adaptive model advantageous effective especially scene factorize multiplicative optimize jointly regularization dataset ensemble observe minibatch give minibatch take day gpu validate k table metric table rough google google images sentence proportional assume occur less probability model system top retrieve sentence length quality cccc several compete need typically compare image evaluation metric rate return sentence image sentence low perform less par good group fig recognize object image interaction localize patch scale softmax output focus function softmax feature word mean visual illustrate determine sum foreground attention focus ie also region well sequence fig show sentence go allocate word occur patch experiment contain go slot match patch rule match match patch choose match learn learn match inside patch semantic significance modal red cat em em em em theorem proposition university china edu california equally em comment send progress image salient meaningful paper propose exploit parallel experience impose alignment characterize novel introduce context language generation specific benchmark contrast several improve furthermore attain recent generation image greatly vast visually information language nonetheless image shown describe salient meaningful attain leverage crucial vision learn represent rich visual localization capable generate sentence progress remain challenge task sentence far probability decision represent language model information encode visual fed predict sequence govern select text cf arguably start understand object reason object focus salient generate relationship linguistic determine sequentially order word sentence keep salient secondary information generate follow exploit parallel structure sentence conceptual diagram correspondence detect like word generate align experience region impose alignment characterize mean share visual description recurrent neural next also another novel contribution scene specific context encode place activity people model word scene instance unlikely scene rather context affect word recurrent neural differ detailed defer localize scale visually salient object represent ground concept unable grain collection stage patch word correspondence region word parallel contrast publish either specific context combine attain art compete follow detailed sec sec image generation long traditional pre define template detect retrieval image retrieve sentence generate sentence recent language learn log bilinear propose multimodal recurrent architecture visual feature extract rnn image detector word patch generate
deep argue approximately way benefit clearly illustrate benefit value classification without upper bind incur n careful benefit preferable result cost analysis assume label probability explicitly obstacle strategy nontrivial literature major style calculation argument mistake correspond spirit ensemble decision vote stochastically mistake oppose et cause order example n motivate near primal max min advantage order j eq strategy unable choice useful suppose optimal minimax outline example effectively present analysis aggregate formulate pac analysis learn manuscript enable appeal nontrivial strategy without error allocate aim argument future proof suffice game payoff maximize predictor predictor performance raise nature progress satisfying extract every advantage leave slack first example v payoff check kkt example select n procedure constraint I desire proof play regardless n suffice predictor force play primal play play v minimize maximized nature I value irrelevant set call force therefore constraint prop yield variation nature trivial without think nature adjust raise budget budget payoff proof remain budget satisfy little n w game duality use reasoning throughout substitute game keep mind order regardless neither identically example prove p contradict strategy purpose lemma bad simplification predictor predict optimal depend play chance incorrectly incorrect give lemma thm thm conjecture thm definition em california california consider advance derive minimax rate prediction set pac rule distributional readily predictor allow application binary classification drive rate concern classifier encoding approach averaged choose predict vote vote extreme fairly slight variation rough argument true suggest gibbs traditional pac spirit focus average weighted paper well aggregate consider predictor nature sum game example nature choose predictor maximize correlation unlabele constraint correlation predictor therefore clearly combine ensemble question motivate contribution predictor game minimax pac minimax predictor average quantify enjoy extend predictor early scenario formalize rule predictor unlabele example tt argument encode predictor predictor jx correlation choose true example prediction average predictor rated bad predict prediction immediate make gibbs average identify outperform make true ensemble effectively impossible without outside minimax prop minimax game without game defer label blue minimax simple training statistical compose hx h kl kl observe label choose use pac set distribution convert p classifier scenario well game label token incorrect probability uniformly training randomness benefit voting exp h low due benefit disagreement many nontrivial significant fraction g extend class rate part source robustness pac bayes work part admit would achieve notable generic extension online replacement classification game predictor treat relative early predict output rest I randomize cost suffer nature gain incorporate
cc cc pt sketch sketch recover also reasonable datum pixel white cccc digit stock sketch sketch sketch identical mix center run h f sketch sketch f r observe sketch uniform extensive comparison recover speed digits text important bias component topic pca complete south frame transaction principal discover pca h c order w parameter word appear pca algorithms sketch match closely pca mixing sampling well small show r uniform sketch comparison factor sketch also bias toward large look stock complete table show stock appear pca sample discover validate corresponding gene multiple principal since c gene symbol top occurrence gene respective cancer co eight eight gene list characterize gene incomplete disease identify additional report suboptimal construct suboptimal sketch reveal popular toward large work svd leverage follow score square norm row row matrix rank project low project space span preserve digit onto top datum top respective separation three component leverage sampling digit stock rank sketch leverage score range optimal sketch possible sketch incomplete mention good feature identify pca column research remain pca getting point optimize natural look datum minimize objective maximize theorem corollary one matrix sketch data one particular principal sparse text biological financial sparse incomplete drop projection preserve equivalently reference principal classic challenge interpretation combination original variable significance biological financial desirable factor interpretable small feature incomplete recommendation privacy preserve get carefully pca provable work demonstrate mn dimension effective rank often kk kn component solution principal optimization non entry sparsity input pca np hard heuristic provable typically address get top principal component address know incomplete datum sampling else pca sketch instead datum perform try optimize measure result solve closely approximate capture much sketch completion sample high element sampling sampling solve solution datum quantity give speak ignore multiply stable price far incomplete sparse thresholding denoise observed matrix perturb principled small quality sparse principal component algorithm input benefit fold summary show optimal sparse find component np approximate take heuristic heuristic practice may able sample rather establish recover outcome reasonable likely really really negative bold bold denote element k xx give fluctuation sophisticated element pca reasonable necessarily variable lipschitz setting maximize bind surrogate set tf singular von trace fact simple incomplete small zero happen large noisy setting treat ij fraction energy lose zero datum create truncate satisfie eq show appropriate signal element recover near keep particular spectral wise element way bias high rhs upper gx fx fx f f follow step set instead keep toward proportional element element trial sketch wise sampling index entry note unlike deterministic small intuition expect outcome probability define element appropriate deviation suggest choice summarize create sketch select simplified version least stable away suffice sketch tolerance use two depend performance pca hard heuristic heuristic next six
quantitative quantitative feature qualitative mapping quantitative example qualitative store qualitative along compare experiment consider qualitative quantitative method backward elimination lc lc perform lc ap complexity pruning standard number five fold pruning fold node run respective deviation experiment induce tree example remain approximately example belong aim ccccc example heart breast cancer bc bs bs ap lc ap ap lc bc ap ap lc lc lc ap lc ap perform quantitative ap design split qualitative qualitative feature purpose ten five accuracy ten report tree tree ccccc feature qualitative set misclassification cost split discriminant minimum node splitting bank deviation produce accuracy dimension produce comparable matrix reflect split matrix whereas dominant empirical obtain method perform tree furthermore capable classify quantitative node quantitative node one example complete class eigenvector since parallel one reflect space complexity finding reflect node decision region homogeneous particular tree cart algorithm space split boundary boundary potentially simplify limitation induction new tree utilize series consider axis split reflect datum appealing classifying iteratively partition disjoint region dt tree terminal non terminal call consider hyperplane sub hyperplane obtain recursively reach homogeneous region terminal node misclassification play classification aim accurate depend node dt split include split split partition axis parallel desirable align feature hyperplane split feature appeal boundary align axis split split decision boundary split arbitrary shape easily noise induce differ study tree therefore become increasingly dt dt non node decision tree specifically attribute reflect eigenvector reflect mechanism reflects reflect reflect axis split axis reflect original search enhance classification problem class eigenvector explain propose version class dominant eigenvector terminal available find covariance whereas dominant eigenvector split coordinate axis reflect parallel separate already axis parallel split hyperplane find algorithm satisfy misclassification child user equal algorithm mp pi ji ji h construct reflect good parallel hyperplane ji ti grow raise question search axis size split twice second
abundance report reconstruction spatial coherence improve generate mixed figure abundance fourth include detect part face part return localize sparsity comparable sparsity abundance element numerical generate trade sparsity remark reduction powerful important nonnegative factorization nmf extract localize technique identify sequentially particularly suited incorporate localized look approach comparable state hyperspectral matrix sparsity imaging dimensionality technique tool well take interpret nonnegative nmf classification air emission rank nmf look via combination nmf impose basis g pixel intensity interpretable imaging hyperspectral image cube provide scene hyperspectral sensor vary material energy signature material certain hyperspectral use material pixel hyperspectral cube convert dimensional vector stack row signature pixel mix signature pixel combination nonnegative signature contain signature hyperspectral cube dimensional nmf matrix signature pixel signature row represent th th pixel figure hyperspectral cube road kind row abundance map abundance pixel unfortunately difficult np non rank reason nmf refer recently allow compute sequentially nmf rank time try localize feature factor residual compute nmf advantage pca mild pca sequentially factorization priori part enhance decomposition part successfully hyperspectral imaging modification nmf reference therein precisely propose add abundance localize art particular hyperspectral prior localize image describe introduction nmf sequentially dual fix lagrangian relaxation uv scale vice versa note trivial satisfy contradiction generality base optimize variable update lagrangian scheme write vice versa u uv tm convergence share similarity rank incorporate neighboring coherent desirable contain isolated pixel tv adjacent pixel evaluate spatial pixel column correspond pixel indicate neighbor account preserve oppose smooth spatial information incorporate feature contain imaging translate fact pixel abundance author incorporate decomposition entry approach incorporate sparse nmf information process add compose relate classical least residual sparsity abundance improve coherence balance relax formally ht small uv locality matrix w nx bp z z bx lipschitz fx w nx original give close subproblem iterative perform require operation cost lagrangian multiplier nmf sensitivity surprisingly critical role mention many classical completely prior supervision tune quantitative conduct nmf accelerate suggest author nmf sparse add norm abundance sparsity abundance hence nonnegative nonnegative use quantify percent amount abundance normalize normalize column term trade three measure factorization lead coherent fair sequentially generate sparsity optimize zero handle refer processing method detect hyperspectral abundance hyperspectral show correctly detect widely mit matlab code equip intel cpu core ram gb code assess hyperspectral technique consist precisely see effectiveness test variant conduct experiment show wide abundance extract abundance figure indicate seven abundance penalty penalty spatial vary lose contour sub figure b constraint sparsity abundance seven basis vary observe spatial parameter detect figure locality high pixel abundance element improve take zero spatial coherence influence supervision necessary localize introduction extract well reconstruction coherence display abundance quantitative algorithm sparsity spatial follow nmf generate localize image qualitative confirm quantitative nmf low since focus coherence bad spatial despite give solution unable figure abundance image surprisingly coherence return sparse different abundance map material abundance sub generate coherent surprisingly provide material rather noisy dense add increase spatial coherence achieve trade sparsity low show image set right top
comparison preliminary preliminary stein stein shrinkage lasso characteristic simultaneous exceed conclusion make efficiency carry test estimator paper shrinkage classical error analytical asymptotic risk carlo behavior propose application life organization contain stein improve lasso section setup discuss detail application estimator demonstrate life conclusion multiple regression far know preliminary shrinkage depend lose vary want estimator classical least shrinkage belong restrict minimize l write tune explicitly yield n pn solution estimator selection computational later angle efficient glm estimator good nearly unnecessary threshold set zero datum multiple restrict vs condition n value estimator optimality assess mse stein give note replace inherent change sign value define estimator positive stein eq define improved estimator six penalty shrinkage stein shrinkage lasso estimator eq stein rule estimator alternative center estimator pn q alternative equal coefficient remain take ns thus se chi eq next compare take whenever le difference preliminary test lasso relative estimator study mse relative le conduct sub test partially linear regression study full shrinkage setup simulation distribution consider simulation indicate nan set hypothesis realization variance parameter subsequently least square preliminary read secondly generation setup accommodate function translate response generate least square consistent neither stein dominate follow life pre set center predict interest use regressor le improve preliminary stein regression validation fold validation validation randomly aside term remain model predict observe predict varie run average deviation standard deviation analyze seven represent covariate species km km km area km covariate convenience specie figure h min max r corrected display validate deviation summarize error well le notably le estimate population life percent mean capital city square response summary table correlation predictor display present study observe predictor rr correct sd le table give average average follow large error visually yet highly error widely seven predictor price national people armed force old people employ data matrix highly average variability small plot prediction error demonstrate max sd l correct le propose estimator stein rule preliminary lasso stein shrinkage stein study compare configuration size variance relative estimator vary degree misspecification table configuration dominate mse among uniformly neither one efficiency estimator near decrease preliminary stein average le outperform picture outperform life try correlation predictor moderately estimator equally case estimator would set among predictor h c c h c h c h
base prototype calculation dp approximation likelihood work entirely considerably work due asymptotic interpretable recover outperform benchmark dataset behind formalism agree cluster closely discrete categorical select draw point maximum mle denote belong p uncertainty across cluster force close extend insight enforce common whether categorical log modeling make separately intensive elegant k repeat center choose point assign create select feature consist binary dirichlet avoid priori hyper underlie exchangeable probability tune start explicit categorical respectively whether bernoulli parameter assume prior categorical cluster index categorical beta value formalize small variance select around select draw independent log detail pair elegant categorical differently term control would turn cluster provide supplementary categorical feature feature take select categorical draw initialize categorical assignment compute generate cluster choose feature low categorical nd along feature outline center need center follow asymptotic cost assigning exceeds would information feature guide feature select specify constant later modify recover exact implication dp mean supplementary denote feature aspect estimate often need application minimal tuning statistical turn via prior readily informative overlap away introduce covariance inverse vanish asymptotic hyperparameter absence asymptotic thereby computational available resource selection trade enhance feature benchmark feature per cluster experiment facilitate specific subspace result experiment synthetic subspace categorical disjoint include comprise evenly split subspace independently bernoulli cluster overlap second third accurately dataset comprise evenly gaussians unit respectively cluster disjoint add isotropic distribution standard modify cluster contiguous additionally completely subspace contain may subspace allow modified dp cluster method indicate select real dataset two compute frequent normalized labeling divide cluster assignment lie close henceforth fair bank spam c c bank comparison dp dp determination categorical extend retain dp retain comparison outperform extend importance entropy selection art benefit accomplish art unsupervise besides dp bank highlight global selection finally besides time spam execution second attribute benefit mean style oppose require intensive spectral feature datum asymptotic set vary code website various show derive retained b binary categorical particular dp mean assign distribution find shape ensures assume uninformative conjugate I gaussian categorical contribution contribution categorical categorical nd assume draw independent global categorical likelihood beta provide posterior equivalently simplify quantifie select since uninformative contribute simplify specifie equivalently maximize minimize simplify cat k quantify change cluster constant must enable feature thereby bernoulli discrepancy control mean joint cluster log eq first k nd nd nd analogous asymptotic eq data mean cluster initialize mean indicator compute pt nk generate k distance within point assign otherwise cluster start assignment successive objective equivalently write mean characterize uncertainty try thus force point come together ensure point absence regularizer singleton cluster thereby lead trivial case uninformative conjugate contribute negative retain letting obtain compute dp objective recover dp imply get derivation automate binary draw bernoulli feature proceeding section dp c reproduce contribution proceed contribution different also avoid value underlie noting set q put everything objective
expectation update equation mean maximization log lead precision equation unlike monotonic improve performance become large instability kronecker prevent iteration posteriori contribution rescale often algorithm side hypothesis side nuclear minimization variational completion normalize square keeping row linearly draw form choose compete algorithm repeat measurement repeat simulation vector realization independent compete algorithm use follow toolbox estimator compare factorize eq block induce vb reconstruction vb nuclear require knowledge compare rao rank setup mention valid lower technical fulfilled estimator absence verification hypothesis experiment figure plot well sn respective side second good sn experiment consider nuclear norm robustness study improvement nuclear region vary confirm noise deal completion measurement consider inferior measurement use show sn typically find vb investigate vb find improvement sn experiment varied result attribute vb arise away relate rank type side laplace conjugate model maximization equation name relevance name simulation precision side outperform nuclear estimator though nuclear outperform completion second order around since expand minima integral em help derivation regularize result occur equation formula find occur remove penalty update lemma learn low determined system rank rank relation justify kronecker structure matrix parameter numerical inherently popular sparse reconstruction setup reconstruction regularization regularization bring type rank penalty literature eq matrix mention nuclear penalty literature optimization exist compressed sparse solve nuclear reweighte solve algebraic approximation convex priori signal noise absence priori preferred capable measurement prior posteriori estimate type information hyper sparse reconstruction form machine bayesian learn gain popularity ii bayesian pursuit monte iterative via technique low help characterize hyper treat follow precision determinant sense estimator derive evidence compare numerically learn convex aware evidence hence unable organized learn sided matrix derive side compare machine reconstruct measurement measurement eq main learning lead vector latent assume laplace lead couple solve repeat maximization type concave convex solve conjugate appropriate two example rank penalty follow penalty c nuclear base penalty wishart scalar instead prior easily right precision question stem estimate side otherwise develop sided sided precision enhance model random matrix relation l notice evaluate bring suitable low function nuclear establish direct connection indirect space respectively interpret skewed comprise correlate skewed correlate column relation presence highly r ij strong auto mention qualitative sided model side base unable capture side precision estimator amenable
appendix building propagation snapshot projection model snapshot architecture highly architecture literature heart decision rich enough relation fairly broad example consequence take action provide exposure attempt target failure mode motivation expansion allow agent state agent endow depend agent capable seek instant operating pick act randomly domain guarantee knowledge snapshot restrict endowed realization sensor sensor action agent name precise x accept system various available differ nothing control mind invoke set purpose control outcome outcome action reflect viewpoint impose restriction enter moment must precise principle restrict example action intersection set outcome action force interpretation action generalize every outcome equal move contradict opposite aside generalized set admissible define matter action tt regard simple example endow mutually exclusive atomic action leave correspondence pure observation oppose evolve structure represent two restrict interested interaction set coherent fact duality median embedding provide consider unit along integer length formally environment agent action enable exist sensor realize hold exist relation indicate reach relation product relation equip snapshot derive mind notational agent task predict tt jointly decide action agent invoke record complete complete represent observation tb define see assign reach possibly position serve recall snapshot tb direct path path implement update produce propagation explain snapshot update propagation use graph tool turn rest variant expand record vertex visit sensor snapshot snapshot observation first u v fashion corollary ts reverse vertex zero implement prohibitive process time network plan kind ability action ability behind form lead direct point snapshot order allow sensor necessary formalism carry sensor aside future contain abundance sensor environment action consequence consequence apply propagation mechanism predict outcome provide snapshot among sensor fidelity perform planning snapshot sensor rewrite thus sensor q whose far expand figure large geometry geometry geometry propagation ability immediate theory greedy decide target characterize region represent desire may considered action guarantee possible may select tie break completion lemma directly motion planning euclidean plane absence approximate path point next kind arise presence occur determine selection weak capable matter motivate follow model complex form implication record class incoherent sufficiently review homotopy requirement place consider example kind example realize equip collection position identical I vertex simulation equip label vertex adjacent belong remain snapshot sufficiently statistical nature learn agent weak set complete could space vertex adjacent short prescribe attempt sufficiently agent implication responsible overall planning example b circular model integer action operation subtract relate simplicity complete structure without system let specify target origin accommodate intersect separate circle satisfy geodesic set pass yet constraint action enable demonstrate strength target agent signal example sense information regard transition absolutely topological provide reference planning snapshot stage threshold seem plausible however cause threshold relation principled exist sensor function become must failure adjustment mechanism evaluate agent system paragraph introduction close loop control suggest range multivariate human mechanism absence current motion simple seem possess decrease fix point environment single simplify sense stay put specification first failure action behavior different stress result precede agent suffice structure associate requirement close control exposure relation necessary offer vary representation precisely map patch integrate map record annotated learn known topological presence topological valuable information loop closure otherwise observation extensive effort notion topological map hierarchy allow plan vary scale leverage topological efficient structure motion planning geometry sensor family algorithm know employ neural pose cell engine underlie cell field dense configuration make also represent system observation cell connect place evidence recently introduce encoding spatio context drawback cover intersection guarantee combinatorial duality turn idea leverage relation among geometric recognize well snapshot entire close snapshot agent capable fact snapshot introduce sense quantify reward g reinforcement innovation planning capability variety enable even contribute topological representation encode facilitate drive sensor inconsistent model architecture result improve connectivity snapshot capability immediate day neural network demonstrate ability perform include symbolic sensitive hierarchical structural feature physical even feed forward network show capable control token environment merging architecture capable match human play raw output internal representation maintain encode symbolic term solve simplification realization controller direct neural code hope snapshot architecture provable symbolic reasoning understand purpose whose property model mechanism expand stable threshold line characterization organization code perturb structural topological constraint analogy obtain coherent snapshot take nature analogy investigate collection threshold strict geometry snapshot could snapshot architecture may expand hand symbolic architecture discrete operation completely nature maintain evolve snapshot size quadratic sensor plan propagation architecture order weak set structure characterize half account equivalence provide agent duality symbolic space interaction dynamic formalism rigorously symbolic planning efficiency snapshot architecture certain find exist architecture represent predicate predicate course lies propose attractive principled analytical computational compound symbolic abstraction relate duality weak appendix predicate symbolic abstraction clear snapshot extension acknowledgement air office fa foundation duality go successful envelope positively term provide review element support memory overview provide intend current duality necessary formal actually job mainly result elegant exposition duality weak necessity structure weak endow call say negligible negligible element negligible weak two preserve denote relation formally construct symbol set fix intersection addition equivalence order notation stand relation derive generator relation compact set partial satisfy empty identify symmetric operator translate pointwise respectively realization weak empty point relation endow obvious denote realization structure identify duality base construction selection denote metric fixing explicit isometry hamming thought skeleton cube combinatorial complex diameter vertex say incoherent pair dual skeleton skeleton illustrate whose diagram form set realize augment complementary question implication implication record observer endow cube proper correspond face redundant use question coherent subset planning sense put vertex weak characterized interval define vertex fundamental quick finite coincide well connect graph triple modern generalization presentation another state precede dual finite median formula coherent determined majority vote value strong recall say deduce subset p aa say median intersection family pairwise convex convex subset induce graph subset vertex subset median preserve map underlie space odd via finite dual practical offer understanding geometry aside view categorical duality category possible element satisfy us realization tell weak nest pairwise space restriction cube relation face incoherent vertex finds remove improper element exercise tree nest robot capable move suppose sensor position say turn turn turn form set path whose please note choice point order matter correctness discretize sensor imagine description appropriate indicate exclusive case ignore underlie join sum external union endow iff iff abuse notation identify natural representative easy proper element cube product path appropriate value dual precede coherent vertex agent incoherent agent vertice spread circle agent capable question arc position question agent sense symbol agent agent result v observe representation relation difference clearly advantage deduce must note vertex white vertex form incoherent family nest example give none fashion symbolic category quick review notion refer reader category one introduce major unnecessary connect every assignment rather let easy median preserve map appropriate yield composition notion category construction together duality fp correspondence ff composition correspondence duality order statement theoretic speak aspect cover interpret term geometry conclusion translate boolean deal survey contribution category theoretic application recall fact proper restrictive business verify duality weak map coherent negligible weak denote median making indistinguishable concerned obtain weak set flexible structure easier evolve dynamically possibly since sensor binary pair sensor equip free aa nothing special never observe transition write implication believe imply correct vertex realization call realization motivate consistent rise maintain record discard incoherent view organization state introduction contribution connect non positively detailed non positively metric space suffice hadamard generality collective effort graph skeleton median skeleton develop paragraph topological space realization collection cc vertex result homotopy circle hand back example explain qualitative threshold possess observer model realistic extend graph situation define absence solution include cube cube embed convexity canonical piecewise thus although graph describe dual geometry dynamically real set motivated analogy idea kind update capture identity map observer yet regard nature pair maintain set represent observer identical belief dual underlie dual g strong complicated symmetric definition satisfy identity loop consistency constraint rewrite form q verify identity anti pair end turn trivial case exactly pair none proper vertex path particular contradiction therefore cycle evolution trivial snapshot empirical snapshot indicator hold satisfied compare conversely presence suffice snapshot write form trivial weighted unit pair satisfy iff snapshot proper fact selection coincide counterpart generality one choice conclude consistency decrease must selection finally snapshot triangle name dissimilarity observe undirected edge induce connect nothing equivalent chain mind whenever connect direct metric inequality conclude iff triangle actually structure need convention observe assertion observe lemma imply turn proof summarize progress snapshot triangle operation point iff edge acyclic let propagation eq map preserve suggest weak yet contain therein prove insufficient support planning study close point projection inequality v lk know gate pair non empty subset gate apply proof consequence gate exist equivalently kind reasoning empty lk lm uniqueness force study technical necessity explain recall notation order identity coherent coherent aa aa show coherent iff disjoint ba aa coherent complete also coherence vertex define coherent observation weak adjacent coincide range characterize follow weak correspondence point projection coherent short element replace step iff algorithm projection projection kt leave suppose precede proposition disjoint corollary converse case already coherent hence intersect henceforth lm u lemma mean propagation recall j jt jt tt since precede second precede self capable degree planning explore problem beyond produce viewpoint space formal notion ai notion space draw category fundamental able formalize early come equip engine notion extend define automatic category allow notion snapshot capable handle observation boolean despite obvious current maintain control evolve control learn sensor immediately mind duality problem motivate possibility together theory brain discuss introduce approach gps base human processing yield advance understanding role visual space dominate become numerous include attempt effect reason comprehensive perhaps evolve advance become fully base machine use operational sub goal reasoning effectively problem solve search flip serious memory absence form symbolic common follow concern uniformity formal capability unbounded resource reflect management difficulty discuss deal regard search leave ability intelligence return system disadvantage cognitive phenomenon limit modern course imagine provable property become setting human ideally bridge enable extraction problem conversely like formal connection cost storage cost planning implementation life emphasis symbol entity ability set well dealing mean player eventually self subsequently produce intelligence character organization collection word category quick introduction cognitive architecture category whose finite category object whose median preserve map specify space state suitably equip sufficient map agent universe realization capable operator present present sense simple basic cognitive space equip snapshot specify coherent collection atomic input loop produce heuristic heuristic combinatorial distance take automatically model serve solution agent work exposure snapshot architecture avoid symbolic symbolic abstraction way symbolic abstraction snapshot addition agent maintain bank already term engine present rise goal pre total abstraction encode aside future cognitive form symbol base agglomerative reconstruction architecture use reach search pattern formulate production happen substitute g snapshot architecture memory nevertheless implement snapshot architecture weak snapshot analog employ say agent observe resolve unless boolean exercise care topology result advantage snapshot drive another snapshot drive agent duality explicit formal extra example duality pose preserve snapshot ready learn one argue reasoning place agree seem wide periodic sensor expand space admissible include sensor statement importantly algebra quantify come necessity convenience quantify natural meaningful class operate algebra value think real value uncertain propagation would construct snapshot way recurrent code code cell stable al word proposition code analogy code stable pattern raw geometry perhaps possibly general consideration reader might characterize convexity theory evidence viewpoint section thm thm thm thm thm electrical systems school engineering rd pa usa school engineering university rd architecture capable support learn absence information agent enough ensure sensor requirement quadratic execute complex agent internal minimal every class subject agent state capable homotopy state provable property memory structure symbolic discrete positively rich convexity cycle obstacle memory human memory seem functional hierarchy system vs split scale address science action task explore map intelligence architecture one stand formal notion space memory system comprise history format argue architecture notion domain vast discuss agent universal learner optimize gain suggest result insufficient broadly formal advance provably intuitive property environment encode generic minimal obtain develop arbitrary encode observation whereby atomic provably correspond near projection generic sense equip sensor action behavior give instance interact environment natural power generally accept must support enough account exact abstract planning require representation eventually account transition review obtain description absence strong sensor obstacle rather impose precise characterize exactly small effective object call snapshot keep track state collection quadratic history implication atomic cycle crucial architecture formally cat see skeleton snapshot update encode transform contribution informally provable architecture absence encode impossible distinguish skeleton chapter rich topology chapter quadratic sensor storage quadratic time pick action learn result walk limit planning action search process height chain implication provable appear contribution briefly review topic arise distinct intelligence present explore relation implication trivial ambient avoid collection geometrically represent fundamental planning topological literature planning membership reduce storage plan planning model play role traditionally euclidean generalize strong convexity enable cost greedy demonstrate oriented topological use encode causal symbolic generalize formulate self evolve pairwise intersection record necessity planning idea encode fairly specialized additional principle available signal interact result control mechanism may realize simplify simulate analogy intersection activation sensor sensor topological mapping competitive necessity pose general formalism ability plan essentially flow construct maintain allow curse guarantee sufficiently rich account class approach come largely ii result topological shape basic drive agent efficiency mechanism agent necessarily state possible control early stage feasibility mechanism agent gain choose formal weak geometry space repository elsewhere discuss observation numerical implication claim iv contribution control validate extend discussion result literature appendix environment sec integer snapshot snapshot new snapshot update direct satisfied follow snapshot weak iff direct path snapshot construction snapshot maintain frequency trajectory agent try trivial snapshot snapshot say trivial constraint snapshot snapshot eq choice vanish justified evolution snapshot observation snapshot snapshot snapshot obtain snapshot snapshot k characterize snapshot snapshot trivial return define snapshot accordingly imply acyclic acyclic henceforth utilize paper restrict attention endow agent trivial snapshot
form provide influence simplify influence bound function consideration eq er rao bind fisher mle point design robust behavior hard set come include local regularize coincide property high order advantage nonconvex loss consistent loss suboptimal viewpoint high scale oracle unless exponential proof paper existence local proof eq q rsc inequality old eq scaling imply element imply inequality give finally existence define interior program argument adaptation theorems primal define c subgradient condition local point rs apply interior program satisfy imply equation zero subgradient condition fundamental mean condition restrict region sum obtain q oracle implying imply u u fu w u w covering argument let cover triple furthermore analogously imply hand side arithmetic mean eq finally invoke concentration result average imply take union least plug inequality far note argument establish inequality inversion relation return assume desire selection imply plug proof eq bounding cover hold complete construction finally minimum program condition minimum norm regularize apply program rsc apply inequalities eq recall interior follow identical left derive remainder rsc provide local note combine q feasibility hand lower bound optimum inequality inductive rsc q together q give combine inequality conclude side hypothesis scale conclude complete suppose iteration simplification rsc denote hence hence index inequality eq inequality eq complete provide technical proposition establish statistical section conditional condition bound variable I desire sum sub main supporting provide subsection general proposition everywhere definition lie triangle theorem truncation well particular lipschitz truncation eq expectations function truncation provide hold eq note quadratic unit exponential proportional guarantee w inequality inequality extend domain replace accomplish follow fix inequality finally proper rsc cauchy eq gaussian finally q proof gaussian condition denote homogeneity inequality property calculate define process gaussian calculation expectation lie inequality lemma function inequality nm inequality apply arithmetic mean define define event inequality define eq eq analog lemma cauchy schwarz side sub I average exponential parameter proportional hence version put piece arrive notation define event I proposition modification replace every follow arrive familiar inequality remainder identical proof fairly consequence eq condition satisfy u p nn careful inspection reveal restrict attention exactly restrict rsc al imply satisfied hold estimator appear body paper rsc imply conclusion stable eq may old second inequality consequently exhibit exponential ordinary note hand hence eq finally solution eq lasso cm ex ex em department school pa applicable contaminate tailed covariate fairly loss curvature within radius minimax lasso nonconvex place fact equal correct support case immediately nonconvex local loss useful consequence optimization regression regularizer nonconvex possess outside descent initialize linear point region optimum convex regularize regression obtain increase efficiency result finding ever robustness statistical scene box toward quantify procedure notably huber huber estimator property class theory construct high mostly g paper light estimator paper globally arise possess curvature new curvature linear type ordinary view normally ordinary sub least square converge whereas usual assumption show covariate weak covariate normally distribute estimator inconsistent observation contaminate response exist wish extend dimensional estimator estimator version dimensional estimator set high robust deviation contribution provide condition optima statistically presence condition strong convexity true conclusion strong convexity previously traditionally condition loss robust function interest restrict convexity main provide curvature covariate agree least sub study estimator distributional consistency estimator question contribution estimator advantage nonconvex huber dimensional reason nonconvex convex justification viewpoint function rise nonconvex cauchy prove regularizer scale sparsity normality number use correspond dimensional sense regard nonconvex strong provide technique construction extend optimize propose solution devise region inconsistent even dimensional stationary within consist separate situation nonconvex optimize obtain sufficiently initialize nonconvex rigorously second curvature successive iterate lie stationary statistically suffice huber covariate optima consistent optimize huber optimize possibly nonconvex estimator literature note involve huber step optimize loss step paper resemble technical regression estimator paper category addition notion optima optimization regularize go beyond composite gradient another relate fan al develop robust huber strictly estimator analysis huber still relevant provide gradient apply step differently tune accord distributional additive noise reveal choice function albeit factor analysis convex cover nonconvex suggest consider primary alone remainder organize basic concern regularizer concern robust distributional proposition concern estimator conclusion oracle conclude variety brief proof proposition contain supplementary universal constant write simultaneously write restrict gradient subgradient provide background generalize cover theory eq I function program general setting regularizer may nonconvex include feasible scenario convex imply stationary certain statistically wish function outlier misspecification classical regularizer appear program encourage appeal review estimator treatment basic concept book cite define observation estimator q always choice error function exist first check exist everywhere degree freedom heavy distribution nonconvex desirable view explore cauchy maximize check although third always turn result intuitively equivalently q outli contamination literature exists completely eliminate give estimator nonconvex expense estimator measure article et al whereas estimator describe outlier covariate concept intuitively behave motivate large estimating follow define sequel allow distributional covariate e form weighting estimator consider choice take indeed effectively elliptical likewise close g influence hill estimator around effect leverage term variance function note take equation reasonable see remark estimator form finally regularizer analysis composite objective function satisfy property amenable regularizer scalar satisfies vi say amenable everywhere define amenable penalty fan li amenable due mcp amenable amenable amenable regularizer oracle point discussion normality concern general statistical stationary restrict regularizer next interpret consequence generalize proposition covariate hold high lastly provide establish equal nonconvex amenable feasible slight minima interior local maxima require satisfy rsc rsc note impose outside radius rsc use region nonconvex cut behave main condition stationary local region function rsc amenable suppose eq stationary contain distributional covariate error come play rsc prescribe scale one local rsc alone fact case robust stationary actually oracle truly actual neighborhood around global lie ball omit local program suppose rsc condition guarantee section regularizer rsc within min state unweighted similar assumption well give twice differentiable ball amenable addition proof build upon develop simple radius completeness modification necessary obtain form previous concern optima oracle careful local optima essentially optimization previous simultaneously huber upon nonconvex concern estimator wu extend allow grow prove normality convex program oracle nonetheless convex standard estimator number apply normality sample unweighted case type normality amenable program v provide derive slightly modify useful estimator composite guarantee region rsc hold denote rewrite q composite iterate stepsize soft thresholding q iterate take descent near close enough denote remainder strong convexity satisfie slightly relate taylor rsc exactly rsc repeat restrict smoothness condition fairly mild simplicity scad mcp regularizers appendix composite iterate linearly rsc amenable successive composite descent obvious radius iterate remain expect hold rsc result composite outside rsc cause stationary point outside region proximity ensure trajectory nonconvex estimator derivative estimator bound view efficiency long guarantee composite inconsistent optima nonetheless theorem stationary radius optimize robust even converge statistically function convex output initialize dimensional consistency produce optimum appropriately composite gradient converge stationary final consistent agree oracle use amenable penalty optima single initial asymptotic efficiency property finer grain al optimize order possibly method mostly justify composite estimator step theoretical efficacy importance step result throughout generate model simulation consistency robust various failure minimax n standard stable suppose scale problem lasso establish proposition ordinary yield estimator n rate cauchy scale run level equal huber regularizer huber loss two huber initialize penalize yield consistent curve align error plot rescale b p huber cauchy dot loss size tailed huber cauchy robust yield consistent predict proposition normal also statistically rate losse significant normal represent contaminate constant value otherwise rise statistically also ordinary robust upon run relaxed distributional mean cauchy scale run trial huber initialization theorem huber difficult see nx exponential large huber yield statistically relatively slow simulation large huber run cauchy path huber panel huber choose distribution optimum preliminary log initialization plot error red roughly huber sublinear convergence initially convergence locally within radius indeed plot outside rsc converge unique global tolerance implementation cc show path huber loss iterate enter initialization huber slight perturbation local restrict green initialization converge predict iterate point need proper initialization statistically huber random initialization green blue initial iterate satisfie
ignore validity linguistic law linguistic law k kind law unit kind type law link law measure length term linguistic write text law law corpus large letter refer frequency text representative law law see name word database recurrence range autocorrelation lag entropy size length law lexical network law good linguistic ref historical state accord th type frequent word modern analogy tail motivate frequency formulation map attention part intend describe example idea variety see ref therein word word database word book increase token end draw english article separate dot trivial database strongly law summarize capture text linguistic list quantitative observation motivate corpora question address law determine around allow law discuss linguistic law law ref notion scientific law quantitative obtain special validity science probably work scientific ref rule violate law straight forward identification linguistic law statistic theory linguistic law notice law affect production meaningful short persistent text strict text sufficient law linguistic law syntactic law role statistic quantitative distinction language law conventional language law nature universe modern physics discuss law refer interpretation statement law frequency decay interpretation collection text vocabulary size text mention unlikely law modern physics point predict corpus determine magnitude law include possible statistical subject linguistic law detail linguistic law language degree corpus ii relevance law quantitative rich principle entropy sec argue linguistic law fig availability linguistic law linguistic translate precise address describe law compatible representative fundamental discussion importance three list linguistic law visual inspection analysis linguistic law widely fitting often combination transformation law straight axis logarithm law visually valuable fitting scale fitting uncertainty distribute justify quantify goodness insufficient evaluate unable assign suit rigorous central fitting assume validity search account correspond multidimensional parameter law kind log ml power distribution law review article ref g cut law third list regard gaussian fluctuation assume j maximize fitting comparison form compare likelihood function criterion calculate average validity linguistic law value compatible linguistic low strong violate computed fraction realization assume linguistic law first kolmogorov linguistic law second third kind fit scale plot correspond english wikipedia representation formulation linguistic law conclusion fit law scale linguistic formulation likelihood compute case frequency frequent count contribute observational quantity occurrence count count point large dominate fit ref fit straight log across statistical either frequent fitting frequent asymptotically formulation law assume datum fit law describe fit reflect different weight high case surprisingly vary database large computed bt word draw ref failure surprising nan statistically previously alternative description publication ref generalization assume assumption fluctuation assess validity unclear negative validity violate text letter obviously show word fluctuation usage write likelihood violate affect analysis book thought approach account correlation small approach come method law straightforward exist show position book agreement generally asymptotic value generation linguistic law sampling affect model unclear extent bt article solid line dependence reveal scale ref law text natural relationship generative process ref range text skew recurrence law fluctuation underlie nan need nan word typical consider every probability global frequency usually text formulation frequency lead nan implicitly explicitly derivation figure connection law usage reproduce fluctuation observe particular fluctuation vocabulary scale linearly central limit ref taylor different structure book claim linguistic valid close inspection claim chapter critical linguistic law evidence support argue linguistic law sense selection compatibility datum compute statistical test plausibility choice original matter picture straight application linguistic law good description strict law linguistic law capture see unable text existence additional process ignore describe violate write p value linguistic limitation necessity able linguistic law incomplete meaningful linguistic allow fluctuation generative relevant ultimately model text interpret explanation linguistic law despite attention fluctuation scientific linguistic law fully long range variation fluctuation estimation text quantitie linguistic consequence retrieval generation law use independence applicability law artificial text fluctuation generation text impose constraint apply law linguistic rejection emphasize law rigorous test reference assumption observation treat case nevertheless law fitting scaling law acknowledgment appendix list project book filter supplementary remove symbol letter string symbol consecutive space english wikipedia filtering keep symbol separate letter law appear fig unique word word type word count dictionary b book word word wikipedia success wikipedia size large rare word strongly database universal language fit availability accuracy fluctuation fluctuation much simplify
architecture linear relationship cell fundamentally pathway trace plot large scheme also effective mode efficiency effectiveness attribute hamming scheme change across mutation hamming sample large exhaustive enumeration impractical block require significantly effort simulation fail lead scientific model discrete object regression analytically material would observed response covariate contribute explanation observation model problem quantitative trait concern nucleotide phenotype observation response covariate explain redundant perfect consequence set challenge truth range mcmc sampling massive block sampler conditionally block size hamming sampler ham block gibbs sampler block hamming bottom cpu times integrate autocorrelation effective ess estimate compare plot hamming block indicate able identify relevant frequently inclusion strong dimensional simple hamming ball sampler integrate time ball sampling block require exhaustive enumeration latent probability sampler particularly effective example utility involve covariate application hamming factorial hmm chain represent discrete whose correspond hide length challenging comprises rely approximation sample conditional sampling scheme easily become condition three different ball give balance radius block strategy big application hamming sampler performance standard sampling provide implementation actual toolbox currently computation trivially advantage hardware graphic processing unit hamming sampling update also scheme evolutionary finally believe many explore field conduct develop description tumor deconvolution example pair read variant allele site distribution allele p ki attribute tumor prior hierarchical equivalent automatic selection tumor population specify probability ki ki simulation si tumor deconvolution sparse response total small represent follow variance assign conjugate gamma hyperparameter scalar distribution obtain density c g hyperparameter hyperparameter prior inference si typical sequence interpretation presence absence different different row ki x k bernoulli parametrize additive k w whole determine px ball step separate gibbs p x I ff time normalize forward pass ff bs time inference model si factorial hide markov support uk research new grant ref mr trust z university introduce hamming markov involve discrete iterative polynomial generalize conventional big datum control statistical illustrate generic algorithm statistical across include modelling typically rely mcmc posterior object proposal explore effort distribution state receive example classic wang unobserve value discrete matrix conditionally py intractable exhaustive entire metropolis gibbs sampling subset q exclude sub vector allow resort hasting major difficult possibly incremental lead local mode exhibit address mcmc high name hamming employ auxiliary slice slice slice significant spectrum block strategy express novel scheme computational ball enumeration realistic approach panel ball vector hamming joint factorize auxiliary indicator normalize ham I maximal hamming column hamming ball consist whose behind hamming ball sampler augment admit marginalization recover target hamming step update alternatively hasting accept reject q q generalization scheme latter radius become enough si hamming sampler crucially ham conditional summation admissible matrix inside hamming cardinality enumeration inside hamming would slice element necessary ensure scheme ergodic differ e step draw observation necessary ergodic figure illustrate ball hamming require subset specify address deconvolution mixture efficient block could factorize p factorial pool dependent divide may use operation sequentially precisely split sequential scheme hamming incorporate iteration special equal purely ball scheme algorithmic illustrated detail si find time hamming block block hamming scale accord applicable hamming ball sampling flexible control ideal hamming hamming update shall outcome actual beyond simulation circumstance advantageous flexibility balance hamming ball conditional denote maximal distance allow si b discussion alternate
auxiliary particle define empirical dirac conceptually step step put emphasis particle step target distribution simulate correspond well suffer dimensional setting dominate importance proposal degeneracy distribution inference fairly forward adapt limited space proposal graphical coupling simulator instance couple sampler backward construct efficient adapt proposal arbitrary latent space derivation class simple key presentation orient class degree w iw qx x replace obtain consistent motivate auxiliary explicit relationship particle specifically make justify properly weight properly pf take px interestingly construct target possible density point wise access class approximated procedure typically correspond internal carlo unbiased member function place proposal distribution validity interpret sampler qx x qx u properly furthermore standard example refer appear generate correct implement modularity fact nest mm execute categorical return properly properly repeat precede return inference let denote wise sequentially resample weight often kx x kx particle notational give kx probability z kx draw approximate form loop resample particle step equal z analogue initial accordingly access k z multinomial probability z proposal weight replace procedure despite condition denote point converge accuracy procedure leave work relate ideal procedure modular sample use generate properly uniformly definition path degeneracy sampler time improve procedure use backward simulator smoothing algorithm particle pass backward approximately uniformly w categorical probability x assume unweighted particle conjunction procedure particle appendix direct use chain standard special imply proper weight interest recent couple problem variable sampler component internal sampler construct way require estimate nested algorithm relate develop implementation spatio temporal dimension distinction sampler comprise simply correspond variable regardless sampler easily particle effort construction call particle furthermore match ease essentially cubic markov implement target distribution dimensional particle smoother review however inconsistent approximation systematic mention validity mention increase three bootstrap distribute knowledge result tb ess resample filter filter latent measurement dd state sampler fully adapt proposal constitute target level properly sample operate bootstrap proposal actual sampling conditional datum exact filter standard evaluate kalman independent involve reflect mean intuitively correspond resample eq effective particle resample step computational bootstrap present display conduct experiment agreement trade possible maintain large probability improvement block imply running give around perform satisfactory particle final study measure location look north year decade spatio define compare region model location year rectangular essence figure consider configuration relaxation c north c north north number estimate site rectangular method level target distribution operate proposal structure problem agreement receive uncertainty illustrate three level coincide visible year particle keep low proof concept particle attain thousand close challenging project contract contract proper weighting theorem manuscript turn square relationship serve article section consider manuscript htb international conference france methodology require
solution tensor mode condition projection mode multilinear conditional subproblem project vector line correspond differently unit eigenvalue identity size include lagrange multiplier indicate orthogonality vanish non substitute get maximize eigenvector n multilinear respectively eigenvector calculate large eigenvalue limited rs rs fixing without maximization idea motivated model fix freedom impose orthogonality reduce variance rs model differ start dependency strategy generally subspace also summarize rs remove set c order extract capture variance effectiveness relax tensor subset face subject challenge subject probe binary face subject rest random repetition rate follow set split repetition rank five exist full select sort classification recognition recognition much worse include variance study save ccccc cc pca face std repetition highlight top bold easy rs compare well rs five least rs size overfitte top highlight consistently pca outperform rs rs outperform still face extract rs face capture rs face clarity sort variance semi orthogonality discuss sec moreover though capture less consistently rs less surprising maximize rs rs iteration convergence evaluate method get experiment relaxed help rs improvement achieve rate rs rs outperform average face performance rs improvement rs improve recognition rs improve rs rs control experiment relax multilinear pca explanation rs investigation multilinear pca setting name multilinear relaxed rs learn tensor capture orthogonality impose semi orthogonality capture feature achieve generalization fix start projection recognition show rs good overall compete addition effective semi future learn rs feature mode separately acknowledgment grant special grant remark china edu component pca learn multilinear extend multidimensional tensor tensor multilinear orthogonality paper propose novel tensor impose capture orthogonality generalization rs fix vector increase variance datum rs compete whole relaxed effective classical input pattern tensor video tensor datum monitor tensor break address multilinear tensor directly main base dimensional low rank generalize generalize side sided projection reconstruction multilinear extend high base tensor minimize greedy uncorrelated multilinear maximize successive derivation low mode usage orthogonality tensor build n consist tensor projection consist present rs derive successive conditional introduce relaxed start orthogonality follow available sample kronecker product consider pm project projection
minimal inequality tell give enyi degree concentration variable minimal value copy imply weakly collect reasoning precise give heuristic give value graph replacement imply two algebraic connectivity note sampling main estimator figure value various observe observe fit well estimate case greedy sampling sample chernoff hoeffding follow inequality kullback leibler divergence mean kk expectation proof lemma os adjacency I need u directly inequality divide bernstein inequality heavy degree discrepancy variation bind exponentially space approximate depend surely part light heavy u iv u iv u u n v bernstein e nn union next property every degree least edge discrepancy property probability property ab b da da bm ba eq right side side b replacement qualitatively simulate independently plot pair contaminate sampling op database video assessment pair live attractive dataset pair live include video distortion compress simulated ip transmission wireless video video comparison therefore comparison ground obtain pair video show experimental video live database interesting collection increase without replacement ranking assessment total publicly live live distortion totally number internet pair occur pair comparable three score pair live database stability exhibit replacement dominate practical initial transition point graph random replacement random sampling performance sample large gap vanish performance rely comparison make easy adapt situation adopt greedy may gap scheme vanish enable reliable rating helpful tool exploit pair comparison xu national china china national program china grant cb cb grant project crowdsource extensively pairwise pairwise via sample estimator estimator stability graph limit vertex finding compare greedy initially replacement item replacement computationally trivially world analysis crowdsource os enyi internet growth crowdsource crowdsource employ community crowdsource researcher participant economic cost laboratory researcher internet conduct computer approach test result control control experimental propose randomized conduct accommodate combinatorial aggregation incomplete inspire statistical rank theory computer mechanic rank norm triangular flow harmonic flow perspective provide general model pair possibly incomplete crowdsource sampling replacement sampling replacement pick whole regardless replacement pair chance possible simple sampling replacement pair os stochastic start edge uniformly dependence experience could design crowdsource exploit topology clique os r enyi least necessary rank edge collect maximize equivalent maximize small nonzero algebraic unnormalized cost greedy prohibitive large effectively algebraic connectivity os enyi collect internet crowd passive benefit collection trivially weakly viewpoint simplicity generality situation online rating crowdsource interest paper greedy attractive crowdsourcing trying scheme theory experiment paper scheme measure value enyi replacement estimate value value associate random approximation increase sampling replacement random replacement graph consider analytic conclusion support base compare recommend stage initially replacement recommend computationally trivially remark future term crowdsource crowd distinguish public crowdsource benefit include crowdsource amazon probably popular provide seek internet crowd task request website besides diverse rapidly idea internet crowd provide break platform million community bring crowdsource ranking ranking avoid member item website create survey either help image document recognition mining game worker complete micro crowdsource researcher find expert aggregate non expert could crowdsource reasonable ranking rating pair comparison widely variety science machine rank centrality draw algorithm able aggregate global provide pair e uniform angular ranking map pair may receive comparison partial comparison apply combinatorial flow flow rating harmonic flow base active subsequently categorization video co knowledge gain effort active problem rank rating reducing must collect scoring reflect euclidean sampling approximate np minimum arc active complexity rank vector rank apply crowdsource scenario rank small number sampling maximize nonzero arise subject analyze sampling pairwise discussion voting choice rapid growth spread internet crowdsource technique scenario typically pairwise choice use scale purpose preference look lx pair item lead feedback arc set problem complexity benefit square global ranking extended clique triangular subgraph admit decomposition satisfie flow satisfie locally globally local characterize triangular cycle involves long arise cause sampling without unnormalized algebra laplacian characterize subsection sensitivity score perturbation give parameterized system
weakly lemma positive ml concavity likelihood concavity model identify imply asymptotically normally establish convergence estimate procedure likelihood use solution present theory option wish correspond maker ten option softmax temperature observation parameter explanatory estimator decision simulate make figure explanatory draw gaussian response quasi newton converge convergence observe estimate represent solid represent horizontal theorem plot distribution compute great closely distribution converge importantly statistical amount biased insufficient bias parameter correspond choice choice option treat vector interval imply ensemble parameter repeatedly simulate hold explanatory black mean parameter option figure apply parameter estimation figure value total explanatory draw accord response scalar behavior mean confidence represent true dash around estimate clarity omit figure show estimate width interval scale true dash line value repeatedly hold explanatory confidence address problem objective nonlinear nominal value linearize stochastic make multi armed bandit problem option call analogy slot option probability solve decision pick receive reward option maximize value reward decision agent reward must arm reward reward arm information tradeoff machine bandit subject machine show excellent armed bandit algorithm know case attribute subject structure design stochastic human human armed bandit would belief facilitate design system design solve reward ir part maintain depend belief decision introduce assume belief reward parameter belief reward spatially embed arm spatially reward interpret absolute belief complete confidence posterior select compose agent option respectively belief identity choose heuristic function value arm temperature schedule assume inverse distribution decrease softmax quantity linearize recover near nominal relative prior fix value include deviation simplicity exposition inverse denote nominal root follow get element diagonal element deviation must imply upper lower small bound value linearization valid variable linearize heuristic q explanatory linearize define form provide parameter describe simulate algorithm various figure parameter linearization linearize true converge linearization linearize objective correspond true true horizon confidence implication robust linearization realistic empirical study horizon statistically guarantee amount get depend linearization true value algorithm sufficiently reward gain regret initial effectively make useful except uncertain noise confidence estimate width order magnitude display exhibit away confidence interval precise value simulate true estimator converge observation value repeatedly parameter imply linearization point grow dash value form repeatedly simulate confidence imply simulate weakly informative linearization algorithm line linearize grow ensemble repeatedly compute line panel normal confidence interval much estimate omit linearization local effectiveness sensitive nominal linearization linear linearization fortunately aspect unique objective estimate know linearization estimation linearization point result estimate linearization intuition choice linearization show broadly linearization relatively insensitive linearization behavioral intuition parameter base datum subject experiment section fit experimental select nominal linearization linearize model fitting behavior review run bandit tasks nj usa protocols participant amazon web task platform participant play could point goal obtain part arrange grid reward choice report game dynamic reward structure participant task stochastic participant value option arrange thought landscape reward landscape landscape landscape landscape flat dimension follow along landscape landscape sophisticated strategy landscape task task rate cumulative approximately achieve remainder low subject landscape landscape high subject outperform frequentist subject quantify wish stochastic make high subject landscape task level four landscape combine identically iid apply estimator subject subject subject value performance fitting matrix iid population four category individual subject table population four column deviation parameter deviation nominal compare standard deviation high likelihood original comparable subject clearly differ level noise uncertainty value decision represent great placing factor encourage value explore help reward subject region allow answer question subject task performance category separately deviation difference landscape confirm subject precise side word distinguish subject match human linearization objective difference cc cc e regret growth reward landscape high significantly surface cc power cc law motivated decision make make function derive use formulate important softmax make objective nominal point parameter perform linearize use could true value depend parameter represent prior variance easier readily hold extend generalized logistic softmax objective procedure nonlinear linearization fit develop human subject science technology develop provide quantify multi armed bandit facilitate principled development machine corollary example department university nj towards human contribute systematic infer make behavioral softmax figure human make softmax making derive likelihood asymptotic distribution show nominal fit credible limit human significant difference relate variety decision option scenario agent receive maximize example air traffic controller select reward option task challenge especially reward air human enable much research decide option condition lead make empirical define option model option operation differentiable operation softmax operation plausible operation goal decision objective explain observe q form constant decision softmax make relevant several behavioral decision making process identification seek design step determine determine equivalent call step rigorous softmax fitting infer intuition represent develop present estimator algorithmic credible armed setting qualitatively reproduce experiment infer belief class since commonly motivate pick two option option rate increase discriminate slope explain decision option scalar parameter picking option softmax study particular explanatory belong multinomial regression class softmax logistic value vector multinomial generalize explanatory objective appear restrictive locally human decision work fast parameter ensure converge instead likelihood convex imply estimation optimization contribution condition maximum parameter convex condition matrix operation derive rao another contraction operator block product analogous hadamard develop composition estimation procedure general nonlinear nominal linearization datum remainder define softmax define softmax review iv softmax converge vi model parameter softmax model apply linearization fit predict expect know explanatory explanatory height posteriori estimate unlikely ml solve problem frequentist true framework standard summarize answer depend concavity statistical q identified yes concavity observe fail vector mi unable value follow weak ensure design estimating answer mild regularity expect hessian limit hold use n permit tool interval estimate test obey er study likelihood condition reduce optimization problem condition ml concavity operation product product prove operation hadamard matrix value real block block size denote block hadamard whose define e thought analogy hadamard product rao compose sized hadamard product two semidefinite liu analogous rao let partition square compose block preserve hadamard special rao
tx f ta next particular consider gradient demonstrating aforementione unchanged iteration iterate unlike similar however rather biased update iteration end gradient iteration highlight result motivate schedule update store cost however since storage finally store low expense slow optimal update epoch storage frequency optimal I epoch iteration denote straightforward new combine specifically schedule show schedule exhibit storage depend likely incur computational per iteration hand cardinality I ss concluding framework incremental second platform analyzing help asynchronous designing sophisticated special setup assume gradient estimate iteration ease exposition epoch case epoch replace randomly brevity quantity epoch quantity q epoch size choose iterate schedule immediately obtain linear specify value theorem corollary setting satisfied sufficiently epoch similar ready asynchronous version capture make manner key algorithm read iterate read schedule schedule iterate update incremental algorithm schedule update processor hence change iterate correspond maintain counter track denote iterate delay integer capture parallelism typical read time section key asynchronous sparsity convergence norm depend small frequency warm asynchronous analysis asynchronous epoch asynchronous schedule consider rate epoch asynchronous variant calculation give fast since run sparse linear speedup asynchronous case complicate unlike epoch iteration positive epoch modern processor constant asynchronous variant schedule give free asynchronous compare decay version dataset convergence implement algebra operation eigen website normalize lead lipschitz constant choose recommend speedup asynchronous speedup define runtime speedup achieve surprisingly speedup higher furthermore low speedup report experiment compare stochastic particular variant describe performance variance empirically verify asynchronous variant figure complexity epoch versus runtime core outperform qualitatively similar version see outperform observe b sim news right right dataset core descent develop primary provable obtain asynchronous variant like asynchronous obtain speedup typically encounter explore analyze variant bregman depend index clear expand epoch define follow follow schedule substitute fashion third substitute particular gx apply choose strongly inequality eq following use recall algorithm term expand manner equality simple algebraic calculation way directly nature second repeat triangle gm fact lipschitz finally last definition equation gm inequality third inequality get gradient add substitute equation definition get index manner define use follow follow manner insight differ change follow fashion lemma since epoch turn identical since combine theorem detail fact mention manner q recurrence notation ease last delay particular substituting bind get q use substitute bind get q bregman use follow inequality constant eq follow fact linearity negative bregman remark sum exploit suitable stochastic reduction thereby sgd although advance scale still process sgd key new asynchronous parallel method provably inspire influential two core framework discussion ii asynchronous parallel formal special key broad understanding attain linearly processor concrete illustration asynchronous reduce
study de cat response lead ability cat pl knowledge support generalize nominal response select fisher cat response category independent belong prove mle selection consistent go infinity mle ability efficient significance nominal full capacity choice item comparison treat false second cat nominal major address cat allow answer cat efficiency none operational program allow traditional paper become reason program decide cat mode clear provide environment reliable ability error mistake long feature cat propose cat argue impact response cat carlo simulation foundation cat rigorous assume multiple item response algorithm give previously wrong setup allow previous item however experimental cat select information accumulate scale cat response conditionally response pmf give previous answer ability maximizer conditional give observed decision cat incorporate need regular cat nominal probability switch correct wrong item pool probably organize property section cat cat establish property finding study illustrate conclude response cat quantify denote response item category nominal number satisfy follow identifiability determine c simplify datum nominal recover pl difficulty particular imply likelihood take depend positive upper first consequently item draw bank whenever bank useful result gx illustration jointly universal cat item category response govern nominal response define scalar I select sense however parameter cat response measurable response structure cat able fisher information obtain mle asymptotically knowledge try adaptive nature maximize level belong course number restriction exposure benchmark performance expect asymptotic sense call estimation ability process conditional log likelihood response nominal response item unfortunately root acquire q cat item item resp value strategy resp acquire response focus asymptotic root item normality maximize adopt heavily martingale score nj item moreover therefore measurable vector follow martingale n complete establish consistency selection selection let item strategy proposition martingale martingale law around consistency guarantee fraction last remain suffice j continuous infinity recall rewrite q subsequence follow establishes complete second jointly establish information maximize strategy maximize strategy denote start show show need nj therefore martingale increment variation g obtain eq cat response allow choice go e previous impose result item unlike include though item correspond formulate correspond item complete previous algebra response attempt item algebra response contain cat assume govern every I c pmf nominal govern nominal conditionally independent assume observe item specify completely left probability determine nominal response whether ability beginning case cat possibility random denote indeed formally reveal stop select information say vector characterize value measurable cat accuracy final maximized maximize fisher nominal current provide response maximizer conditional likelihood item strategy conditional probability follow define response cat every property go martingale score selection increment finally complete response proceed choose previous measurable respect measurable martingale respect square q consistency without item martingale moreover strictly martingale follow taylor vector response event j prove consistency claim need go ratio strong consistency continuity x go strong condition normality selection indeed regular cat need eq item current regular cat nevertheless q theorem particular information number case probability difference indeed martingale stop increment martingale limit take lie follow ratio go application theorem cat distinct since review item study illustrate result cat category follow interval pool analysis respect item recursion possibility item rmse summarize ability square exception circle response dash square line achieve standard cat interval cat result validity illustrate actually c cat dash square response dash information ci cat plot I allow first design cat nominal response belong consistency mle fisher binary nominal reduce pl one indeed assume unbounde rather item bank moreover mle heavily proof general nominal response cat design response propose estimator strongly
recover eeg contrary measurement produce analysis nearly zero theoretically compressed isometry adapt rip property equivalent signal sparse put zero draw zero analysis eeg signal recovery recovery eeg signal sparse representation incoherent way coherence super eeg third eeg signal approximately signal eeg system channel process jointly eeg slightly dictionary channel way channel support generalize single straightforwardly code correlate signal preprocesse eeg analog analog eeg signal sample sampling channel eeg signal measurement coding exploit tensor channel eeg less correlated low compression motivate channel eeg signal eeg find newly eeg singular nd try piecewise exploit structure enhance signal recovery exploit channel signal simultaneously encourage method criterion optimization transform multiplier analyze eeg eeg recovery recovery simultaneous achieve error mse mean cross eeg paper organize section exploit structure system simultaneously multi eq put sequentially reconstruct compressed exploit low structure formulate singular variety method eeg signal structure channel eeg measurement formulate simultaneous low nonconvex norm sum e sum newly programming besides step process core processor computational decrease experience acceptable eeg group detail material recovery eeg signal kind gap candidate measurement proper recovery simultaneous matching simultaneous greedy pursuit argue nd eeg recovery cs eeg dictionary wavelet subsample quantify value quantity mean value imply formulate eeg channel vector eeg signal reconstruction variant form percent mean index similarity eeg mit eeg database http www bin intractable without anti intervention channel international eeg position eeg recovery segment channel segment eeg eeg channel eeg eeg frobenius take segment eeg compress reconstruct reconstruct omp fig omp different gap gap slightly care accuracy analysis computational complexity b b section interior admm mse cpu interior optimization value interior admm outperform one speed admm fast rest worse acceptable recommend admm candidate channel eeg nd dictionary eeg exploit recovery compress eeg recovery norm encourage use low solve nuclear optimization criterion exist eeg cs channel eeg show optimization van deal single channel compress eeg enforce rank channel eeg alternate eeg reconstruction computational candidate compressed sense method enable successful compressed eeg sparse compressed consumption wireless eeg multiplier admm recovery rank eeg take signal wireless central eeg frequently
algorithm demonstrate text network lead speedup importance continue grow big enable wide amount increasingly machine significantly fundamental couple important machine spread machine big fit storage one require storing keep machine likewise distribute inference efficiently furthermore server face communication iterative nature algorithm produce huge amount traffic document specifically bandwidth memory bandwidth potentially bottleneck achieve large amount process constrain fraction need communication key pattern scale
three validity nonzero element goal separation nonzero two number sample correctly reject normalize arise nontrivial regard support support nonzero insight whether adequate reflect total correct value classification large measure correct decision reject incorrectly classify correctly classify measure correct block give value induce give insight approach classification correctly incorrect classification nonzero
frequency proportion share across let atomic measure conditionally identically base draw precisely atom hdp share place location detail describe come population population segment boundary segment pick origin distribution prior proportion correspond population stick impose order atom atom efficient describe monte mcmc update turn probability slice allow measure update make forward filter backward hasting proposal update model metropolis detail matlab implement scheme measure hierarchical dirichlet process model sequence index dna sequence assign chinese restaurant hdp auxiliary form hierarchy infinitely simulate address sampling
provide obtain positive th q exist configuration besides leave investigation parameter configuration dual stepsize constant strength convergence decay use propose uniformly uniform sampling well adaptive consider erm handling extend wider allow several regularize conduct competitive sdca sampling provide fair dual
channel datum manually identifiable pattern formalism wavelet transform via fast wavelet pyramid decomposition forward inverse analog wavelet cascade consist filter pass uniquely bank end decomposition detail scale fine detail coefficient vector
compact well map situation nonlinear comparable space computation complexity suitable natural term structure matrix past randomize embed locality sensitive come suitably matrix efficient achieve complexity matrix entry variable define require reduction matrix space transform denote
formulation algorithm calculate let q use remain unchanged however converge figure viewpoint soon totally influence intermediate estimation relationship monotonically decrease decrease first figure smoothed pattern
basic property kronecker apply define problem apply cca recursive correlation combination uncorrelated determination tensor constraint decomposition tensor canonical x pp pp th suggest classification collaborative linear instance linear end kernel extend case projection project high feature map dimension infinite accord follow pn proof appendix trivial follow least positive definite cholesky decomposition similar rank recursively maximize obtain pp mr instance complexity straightforwardly offline complexity dominate accord complexity determine size respectively small effectiveness image annotation follow label used percent datum specify accuracy structure least
need mapping unstable accuracy physical function unless efficient inexact exist closed inversion target I term memory become propose span reduce rank alignment impose plug product matrix reduce set training benefit requirement practice raise accurate empirical solution differ depend adapt eigenvalue generalize project span eigenvalue notation projection eigenvector complement projection residual k ns n squared
compare vary conditional satisfied relevant undirected separation dag separation supplement structural emphasize apply much wide furthermore fact undirected cycle connect direct acyclic connect structural criterion simplify detail partial property directly translate consider implication random describe determine polynomial result describe inequality come gaussian graphical monotonic necessary condition structure occur analysis various translate miss choose select designing survey science building search may application design markov procedure see etc component vertex underlie supplement detail
label baseline quantitative first dnn recognition denoise recognition evaluate mnist add clean accounting clean testing correspond noisy testing use denoise baseline autoencoder quantitative method baseline competitive baseline cc b lrr dnn cccc lrr wiener dnn task output prediction boost classification total belong unbalanced purpose answer camera helpful fold validation fold testing fan deep leverage svm
vector directional difference estimate throughout reasonable eventually speed epoch take approximately hour ghz intel figure observation efficacy outlier part formation perfect image manually mistake difference symmetry efficiently mind momentum synthetic dataset might predict sgd extra progress low qualitative say momentum free cg progress progress exist determination
riemannian inner exploit account symmetry tucker framework riemannian manifold nonlinear algorithm end representation comparison outperform across address tensor estimate generality order tensor tensor operator I I f r r mode unfold n nuclear term unfold generalization lead applicability especially necessity exploit tensor algorithm tucker unconstraine study uniqueness tucker build upon suggest manifold matrix completion connect tensor symmetry novel optimization tucker manifold develop list art instance implement toolbox development
true true section corollary edu cs private answer statistical high database fail first smoothly privacy privacy answer connection lower purely answer arbitrary standard laplace gaussian mechanism achieve bad case guarantee factor privacy preserving enable rich database individual guarantee privacy significant influence control privacy upper
simplify lagrangian q constraint marginal cross
network anomaly choose newly sample weight vector define anomalous across satisfie xy w ty xy ty anomalous vector eq follow direction far ty neural anomaly detector activation follow ty xy ty anomalous
interest discriminative take input depict confident th hierarchy representation input categorical make back partial respect objective matrix composition correspond bias apply dag structure recursively consider left formulation diagonal span play recurrent allow prevent vanish recursive neural net composition gate lstm allow propagation benchmark sentence phrase visualize project pca qualitatively paper detail review sentence review overall sentiment customer product classify customer
bootstrapping algorithm estimator within subject lagrange f px xx df worth involve minimization subject idea behind et average suitable obtain lagrange multiplier rt r rr th order respect bootstrappe df treat size k rt tw rw rw multiplier bootstrappe balance unbalanced estimator lemma entire second use et study bootstrappe well
guarantee paper interest understand remark enjoys kernel question possible norm foundation supplement metric measurable family z rademacher sequence ss proof section proof prove family definition expectation term rademacher rademacher metric making entropy get smoothly function separable q difference cm g choice bounding existence word tr db
curve red curve sd sd sd sd sd sd curve dash correspond replicate dash color black nk component curve replicate plot intercept plot gray dash correspond black solid dash correspond proposition section height support research business usage business replicate usage usage give usage condition time thus nonparametric estimate state error frequentist world technology operational
consequently sign importantly transition spectral spectral almost vector opposite sign sign community generate trial std std std std fraction fraction network noisy edge randomly critical value eq n surely fact
show tradeoff term frequency come term depend recovery graph signal choose w step w ki ik signal frequency come size graph contribution column similar leverage evaluate column random component bandwidth algorithm bandwidth tradeoff expectation discriminate
lexical semantic future take account development embed try thank anonymous comment le thanks helpful self parse supervise parse iterate ir start rich parse tree achieve et al parse supervise common sentence percentage token system parse produce model dataset unsupervise parse generally use attempt tackle parse mention nevertheless aspect would use third state parse cost overfitting make disadvantage linguistic upper performance annotate performance
global find close sort I eigenvector summarize positive lp j feasible modify follow point generate necessity proposition matrix although function constraint become product two scale solve similar reveal solution assumption global minimum assign limit generate robust gauss arbitrary matrix update block convex term diag diag unbounde immediate implication unique point stationary application loop surrogate easily impose structure update problem constraint update demonstrate impose covariance
deal vocabulary size serious nlp point normalizing neural solve tree use representation approach instance metropolis softmax although mh unnormalize input unnormalize great efficiency efficient name relate length enable improve use deep undirected machine extend
fit evaluation sample exercise bt variance reliability purely ranking sample unknown bt self assessment student ranking perform ordinal pairwise per exercise consequence time insufficient accurate estimation inaccurate decrease error approach outperform point us reason amount self collect student around study enough lead assumption artificial whose agree assumption show work reasonably model major
call aspect make surrogate classifier build theoretic density approach easily translate language additive show family distribution attain zero also related entropy aim schwarz divergence sphere density simply big
eigenvalue decrease point decomposition flexible situation parsimonious estimate parsimonious propose formulation parsimonious formulation analysis parsimonious gmm propose algorithm mixture case type issue parametric possibly type penalize criterion bic likelihood use compare namely mixture model describe well adapt realistic recently parametric process chinese restaurant crp principled cluster offer principle jointly cluster derive restrictive chinese restaurant base assume generative dp first dp ng representation variable underlie occur atom dirac probability atom atom independently base hence dp property among process dp add give generate dp add dp example density multivariate compose matrix inverse wishart dirichlet property make crp dp connect crp share partition distribution label ii crp provide infinite predictive
hoc member privacy security economic security network content member run laboratory group researcher recognize nsf award student award receive bs electrical engineering university college institute technology early department electrical computer engineering interest detection education unite engineering contribution fusion distribute open claim proposition mit edu technology edu continuously person device collect analyze behavioral mobile device period novel due duration modality absence restriction large environment organization user device likely come modality soft visit location device gps
sf tf f z mentioned naturally simply recover extra log prior knowledge play zero sum define matrix like strategy player player prescribe payoff player player action drastically use kl bit distribution produce dynamic game letter refer player use prescribe irrespective I prescribe q priori w sequence action
true reporting probability expert outcome observe expert proper rule report expert score logarithmic rule divergence scoring euclidean derive
competition participant offer daily side effect new use produce forecast use summarize environment evaluation scheme present conclusion regressor successful result use series implementation library benchmark methodology series auto moving originally describe ar support
use thompson innovation true allow theory quantify certain thompson elegant fundamental martingale bayesian armed bandit current state reward next current reward distribution property thompson essential markov martingale property proof shorthand expectation notation shorthand underlie markov martingale underlie time furthermore assume frequentist regret smoothness almost surely regardless consequence see avoid posterior much small value tend thompson
insufficient variation terminology al view database experiment use recently receive literature drive gene expression profile gene module reference therein representative profile activity profile relevance retrieve require feasibility profile model dataset learn relevance infer slightly retrieve likelihood store experiment database introduce instead query learn measure suitably dataset beneficial extract characteristic query way dataset importance
theorem square definition role pair require provide dataset cancer gene task come dataset merge dataset datum point five retain ct dataset relative ct slice validation time obtain dataset prediction six preference united price target price date date day date date price price etc normalize measure detail dataset convert day dimensional vector via answer proposition conjecture axiom underlie dimensional dimensional regressor sparse coefficient deviation light dependent
admit equal cluster sufficiently study iid vector rotation support take summarize lp small center recovery disjoint two different ex yes exercise theorem ex lp theorem conjecture summary recovery separation give recovery recovery dimension
percent community mixture forest cover type landscape close open system forest cover forest exceed forest exceed cover less water present water cover follow multiple system classify build cover forest component comprise landscape ice ice throughout never year water either water activity temperature problem framework additionally approach involve specify look detect type shift form bayesian collection sense put emphasis explore change index series temporal detection break ii method demonstrate feasibility process free spurious due change method identify step metric use cover
map cluster final unique state respectively decompose temporal light cone cluster system follow serve weight quantity use reconstruct predictive unique set light light nonparametric replace final weighted mixture reconstruction attempt spatio temporal forecasting real world material state experiment frame hold experiment effectively consist slice frame
find sensitive initialization fine little progress tune behave training architecture later initialize local appear fortunately good fine sufficiently mini batch fine tune note unless layer deep fc second conv fc layer place nd evaluate single view region whose short imagenet number layer speedup conv conv unless decomposition single evaluate layer layer unchanged involve solution decrease nonlinear consistently activation relu relu indicate relu substantial portion activation conv rate conv conv conv degradation speedup pca little degradation quickly small need drastically speedup ratio whole experiment conv speedup conv conv conv conv conv conv conv
likely outlier affect collection rarely cluster exist set qualitatively five normal normal simulate three give shape convex separate cluster half separate split convex examine technique ct hereafter seven half normal intend repetition simulate visually true cluster calculate ai datum assign calculated software describe randomness repeat mean ai repetition ct first familiar microarray quality really red one white drop hard implement dropped ct rarely perform assume cluster describe eight example choose du implication straightforward choose eight six technique convex three generate normal apply seven set normal leave roughly merge htp seven clustering
boundary none far ii entry ever replace iterate satisfie ever error replacement original theorem give claim verify inductive iteration auxiliary score eq recall moreover obey precede c rise long replace valid continue repeat increase turn establish low limit minimax proceed construct base notational simplicity index respectively permutation abuse value satisfy rank informed impose alternative generate probability verify error bound bound accommodate introduce l generalize p come assumption arise version arise start iw w yield hypothesis location location bound rely bernstein simplify bernstein probability know explore rank aggregation consistent rank reveal preference popular pair item top quantify gap rank rank return item
whereas diagonal generate block argument extend kronecker measure state present subtracting recall kn u kn tn algebraic evolution regressor rest recall stability radius argument eigenvalue triangular focus step size satisfy guarantee estimator kn mn rely necessary combination negative negativity apply iteration ensure stability although derivation sufficient mention simulation become show instability justification constraint result instability framework norm element stand trace entry q expectation side denote tractable replace steady kn tn sense
iteratively add fashion train mnist black digit overlap intensity add great digit suggest piece visual image view house preprocesse house image two visually rectangle indicate attention patch digit writing size h consistently extract image centre house highly realistic fig reveal lstm read mnist cifar challenging draw cifar natural cifar diverse
supervise task regression task share task learn minimize neural purpose labeling firstly stack auto allow build feature trick output order hierarchical pre output first relate architecture vanish gradient trick layer backward fashion start original pre layer learn output structure discover allow incorporate apply fashion auto encoder mlp reconstruct
I minimum achieve note naturally compute matrix system use efficient system unconstraine theorem contrary property allow linear gradient subsection condition depend value contrast h chi small figure write use illustrate function orthogonal scalar basis system condition uniformly linear obtain finite illustrate drop fine mesh representation form illustrate small coefficient without theorem localize accord theorem localize iterative illustrate diagram present pyramid fine soon apply scale space write interior node similarly element ib kn n require operation approximation compare present complexity robust lack regularity pde rough involve uniformly support air force office scientific award fa computation department office office advance scientific material extreme contract de theorem method rough method discover decision theory identify operator incomplete pde hierarchy elementary gamble orthogonal pde enable compression
model enhance recurrent unlike feed forward neural rnn history summarize hide rnn decay less frequently lstm replace recurrent equip principle discussion digit translation art benchmark encode human automatic translation translation digit capability context lstm train global cell lstm track distant indicate train music rnn fail chain inherently input carry merely structure lstm may capture
estimate variational table show trick yield dropout substantially high dropout early additional estimator stable compare top layer stochastic epoch epoch regular separate efficiency modern gpu optimization I take per efficient speedup test error choice hide unit variational equal adaptive counterpart difference especially variational dropout dropout network come part beneficial dropout kl divergence seem prevent dropout efficiency global translate noise locally instead globally obtain trivially low extension dropout infer
knowledge augmentation broadly posterior assume exponential field specify family copula parameter condition separate dependency red fit blue alternate third field right fourth set free variational aim free energy term reward variational joint mass crucial posterior dependency use copula particular dependency dependency factorization cdf say I densitie information transform among bivariate copula define
question convolution significantly require secondly multimodal section image treat semantic component match compose question far natural cnn employ illustrate field share utilize cnn capture rich composition word sentence layer convolution pooling perform sentence unit map layer segment convolution convolution unit parameter unit share window slide begin cnn embedding word high composition representation question pooling process follow convolution representation quickly make pooling select
option input termination history reward begin execution option hierarchical option bt whole e g termination child return return call primitive perform interaction core depend aspect selection execution learn see action termination action termination learn result option guarantee condition division converge node behavior speed convergence validate action node version could execute trial iteration experiment divide room possible action room chance agent save room chance type chance must leave room
immediately inner loop main mini component kt f exactly sag work nevertheless impractical result comment optimally satisfied px mb b gd special obtain special case recover equal focus post choice translate analyze modulus moreover fact mini inner reasonable target guarantee epoch focus fix epoch sense minimize minimize recover mini mini target evaluation follow present formula attain less mini batch quantity stepsize work gradient
value procedure patient selector parameter display indicator vector patient well patient higher order figure illustrate tendency diagnosis accuracy thresholde true value spread easier accurately distinct iterative selector propose find selector upon compare alternate result loop alternate method approximate yet significantly less use acknowledgement author
feature vector sample symmetric outcome sample distribution xx constant learn xx stability linear nx randomly generate symmetric sample n define square achieve stability rate benefit become mostly par decay aim classify document belong hinge classification accuracy regularization log exclude rate prescribe determine original inner
question important variation reveal via distribution via suggest testing specific document cause patient infer ask generate newly sample decide generate observed closeness try entity different example generate author suffer distribution like whether one researcher tend infinity reference recent two independent sample unknown either high namely constitute graph show recently low take high maximized suffice closeness every error closeness application lead scenario suffice monotone concave element require support optimality outli collection test poisson collection consider direction consider support
fig test constraint surrogate induce zero tumor eps constraint risk extensive cancer algorithm algorithm loss onto therefore however use activate subgradient information elsewhere ms I ds I dd classifier classifier property linear bound satisfie conclusion cm cm theorem conjecture pearson
recover rank noiseless solve feasibility noiseless primarily study argue compressed study constitute choice present recovery rank eq q devote herein limited covariance obtain attention convex therein modification enforce obtain eq value receive attention recently treat nuclear regularization enforce work refine state impose without result prove concern complex semidefinite unit trace matrix consideration notable contact von low constitute enforce establish least square geometric upper indicate competitive world
distribution py fy pl expectation give uncertain censor observe otherwise censor pseudo expectation eq generality p f uncertain update point condition lagrange propose show terminal condition data life
label competitive incorporate kl tune newly htp c neutral entropy kl space mac financial top feature pool movie use positive unbalanced document expect preference case select label label feature without label significantly outperform incorporate neutral feature kl lda label among control unlabele apply constrain
one step system unlikely generate energy two dimensional eqs working limitation approximation molecular show boundary configuration visualization experience sequential main ensemble ensemble anneal proceed comprise follow initialization initialize energy eqs approximation initial new monte carlo energies pool visit new energy histogram annealing desire entropy namely determine compression small overlap successive ensemble anneal slowly fast risk anneal harmonic ground canonical ensemble inverse q kl ratio successive relative result geometric
frequency intersection bin know coefficient number per use cluster instead suggest singleton identify proposition code sufficiently high singleton identify dft singleton bin least operation please claim synthetic theoretical dft used phase instead arbitrary dft coefficient perfectly reconstruct dft corrupt successful support zero dft coefficient recover perfectly observe error recovery successful evaluating signal r rather feasibility promising demonstrate random support dominant dft practice cluster mr image reconstruct dft fix snr well empirically validate scaling sample sparsity time successfully ambient signal simulation setup dft coefficient random obtain varied corrupted noise db period bin front varied least plot average shown cluster decode singleton bin recover dft overall coincide reliable singleton bin point proposition discrepancy weak total front recover dft
nd order ensure match bl outperform convergence bl seem knowledge error implement build matlab use compute nd th approach remarkably positive band limit one b normal bl pdf kde kde estimator keep cutoff bl create time calculate test potential potential potential spike carry characterize glm cell train open radius period minute area spike
availability include require fit question km area rt artificial network krige method rely ability prediction approach residual simple content national france add spatial component give might transform apply mapping stock stock national preserve median map essentially model whether local krige area although neighbourhood issue regression could forest model residual opinion consequence spatial likely compare efficient state rely sophisticated rather datum modelling state scale solely include france country short certainly model new extent might occur demonstrate study add increase analysis uncertainty map solely quality complex many could improvement possibility improvement obviously candidate drop study
variety outcome describe routine predict miss entry four baseline art variant baseline sample rmse group validation running performance law exponent power predictor exponent residual predictor per exponent exponent sum square scheme use score velocity distance adjust velocity predict art model entry column percent nuclear completion nuclear propose rank completion follow local paradigm extend size circuit minimization circuit outline modelling circuit co circuit event distance close reader way completion predictor law bag power weight one line predictor rank triple weight radial ps aggregation bagging predictor approach aggregate approach completion circuit variance use notation matlab usual subscript stand whole also boundary affect original ht performance estimate denoise entry event close restrict index row except row ia I variant repeatedly rank event choice obtain component component correspond consider scalar sub sample four top validation completion singular compute rd singular nd rd st column precisely residual summary manuscript third summary obtain coefficient computation significance equation error prediction bootstrapping performance per consider error region
discovery proportion simplicity assume wish coordinate statistic false note proportion define structure simplicity nonzero stochastically normal th loading provide estimate square estimate follow estimator spike relax much weak proportional obtain assumption convergence rate attain second relax allow estimate conduct simulation demonstrate behavior eigen proportion constant generate histogram standardize eigenvalue low b c jj histogram asymptotic diagonal position stochastically observe report correlation element three eigenvector base repetition correlation theory b b j normalize uniformly result sphere normalize q normal distribution pairwise angle realize ccc
author threshold observe distribution pair distance formula triangle geodesic yes cholesky yes yes yes jensen bregman yes yes yes cm cm cm invariance rotation frobenius cholesky yes yes yes jensen bregman yes yes yes yes yes yes log infinite euclidean geodesic euclidean euclidean distance
price vector good assume contradiction minimax primal price gp price must induce bundle price price approximately lagrangian bundle price induce bundle satisfie assumption note v p detail reduce price bundle subgradient access bundle subgradient lagrange price bundle gp easily bundle perform project lagrange price return contain center subgradient x difference guarantee project descent know obtain induce project descent lagrange subgradient work concrete preference abstract unknown utility game choose action response reveal space bundle utility minus cost producing utility approximately maximize formally player action unknown u l function associate game action action induce tie break rewrite objective note optimize approximately algorithm utility function player need game concave lipschitz space follow space diameter first target action want learn q observation polynomially action action
multiplicative function cosine value hadamard employ discrete multiplicative observe algorithm multiplicative dft denote calculate present ccccc transform
optimizer sigma variable repeatedly p pp integral partition unnormalized distribution define follow direct measure eq hold scalar hold insight bind old bind possible integral right several upper quantity provably convergent polynomial finally bind partition inequality imply
aa n aa aa aa possess therefore max notice conditioning index bind obtain part eq use combine union two obtain q dominant require certainly notice precise fast dimension reduction feature method moderate variable via variable reduce discard substantially screen crucial fast selection algorithm fail method establish consistency particular dominant concrete screening subject design limitation challenge facilitate improve decade field focus recover
affect space observe quickly propose chance accept problem fast region original problem classify datum
state experimental table achieve lc well lc improvement respectively classify dictionary fast lc randomly acc acc sr lc ht compare lc art sr shown compare lc perform among method confusion misclassification error store high setting category give per used evaluate pyramid lc lc improvement c lc ht examine performance
addition turn technique virtual also introduce virtual symmetric anti case anti pi simultaneously pi pi unitary eigenfunction operator five operator consideration eigenvalue arrange eigenvalue operator eigenfunction belong since reference analogously adjoint orthogonal q operator accord share pi preserve common pi pi either anti symmetric
worker worker time worker read worker obtains share worker guarantee precisely one modification component atomic parameter share share memory early miss analyze asynchronous parallel share select update analysis assume early memory practice physical consume step reduce trick ask multiple cycle update update write q coordinate compute index express index iteration allow jk might practice probably asynchronous sg asynchronous basically serve abuse mini age inconsistent read result hold
year next execute initialize extra iterate pass reverse label year title nan format title max sort order sort begin execute begin execute call execute end pt pt xt thank title title width width ex ex sect mark pt conference plus minus pt sp reference page em theorem condition principle outline diffusion achieve bridge develop mathematical simulating path important widely phenomena broad economic finance life computational class markov jump diffusion markov sde denote instantaneous diffusion brownian motion compound process jump jump coefficient typically ensure weak naturally simulate path infinite random avoid simulate broad jump simulate tackle simulate jump path represented sde jump diffusion diffusion purpose restrict impose coefficient simulate induce construct simulate condition simulate jump construction appropriate measure
imagine ml process relate entire loop relate solution enable rapid solve problem material context formalism general cubic neighbor hamiltonian unit density center section seek machine output solution vanish gap level self chemical advantageous entire remain
monotonic reason explain heterogeneity low whereas link nod link consider probit include dyadic probit except fit rgb psd status age row status col col col age col characteristic appear indicate fit model regressor multiplicative effect bin goodness fit discrepancy proceed examine regression effect interpretation multiplicative proceed identification association multiplicative compute effect ordinal characteristic association effect via plot u status status age formation although multiplicative additive model associate office binary extend accommodate latent provide treat nuisance ordinal approach simplify specification general ordinal computation use level model dyadic record dominate dyadic nature heterogeneity lead dominate lead scenario dominate unlikely able dominate available plot age particularly dominated ordinal probit
transformation usually easy arise abc evaluate proposal accept define alg implement hasting mh rw mh smc proposal view stationary algorithm far characterize behavior induce assess pseudo mh function pseudo find objective paper could proposition expression statistic jacobian note es need r suggest parametrization make change employ lead broad class abc constant closed proposal restrictive extend assume exist intractable reason density metropolis construct tail abc simulate enough simulate sample standard draw complete importance weight necessary update argument genetic analyze role except
dataset unbalanced structure require choice despite tend uniformly comparable growth comparable general produce growth curve process glm np thank provide dataset provide le di theorem proposition study curve rest probability mixture factorization small ball approach principal datum computational attention propose base kernel density estimate introduction supervise unsupervised role base call require development deal belong space introduction refer book consequence orient classical multivariate implement suitable put tackle follow put coefficient underlie process instance technique another aim refer principle ball associate depend term reflect underlie
would response overall impulse impulse open proof criterion expectation define parameter eq iterate estimate two problem respect collect obtain unconstrained solution consider derivative expression spline factorization find reasoning consider ml argument theorem definition conjecture se advanced learn contract identification regularize
discover rule mining consequence generally event high proportion proportion contain input constraint rule greater great medical still interested occur ignore rare support medical health outcome unlikely support unless low left support constraint detect lift association chi square significance thin gender code parent read code code level item medical thin total gender read mine medical support minimum propose refinement unsupervise signal drug interest health
margin multiclass way extend theory complex output develop multiclass pac de france universit universit al tight vote output provide generalization multiclass label output majority
contradiction constraint lie edge polytope rgb straight ex ex ex minus plus minus paragraph em claim corollary edu study predict linear partial constraint generalize predict rational agent unknown reveal mistake learn algorithm learner program thing rational objective change control program may day partial learner systematic dimensional information single fix study capture follow utility decision day day observe price bundle learner bundle price face round constraint optimize broadly objective change constraint predict agent constraint unknown
lebesgue density satisfy consider piecewise polynomial partition orthonormal hold lebesgue form orthonormal polynomial sup suitably control localize piecewise sup orthonormal localize histogram existence interval polynomial localize sup localize convenient set integer l give explicit strongly localize assume q jj admit localize
develop leave algorithmic closely study past recurrent symbolic symbolic system deal link symbolic symbolic network design simple recurrent generalize promise truly count mostly pattern data gr tackle work architecture internal symbol generalize network gradient able learn simple context context unit choose linear count constant store amount recurrent investigate mechanism context memory oppose store new al lstm build roughly resemble
show learn solve example tree learn determinant nuclear dictionary involve fits k n empirically degenerate solution shape gaussian difference variance phenomenon impose shape model hyper incorporate joint learning still wishart gaussian pdf derive term please wishart learn dictionary address parameter properly initialization rough consistent ml much rough contribute still please k learn require manual annotation particular corner similarity initialize orientation determine although initialization transformation difference identical statistical overcome learning shape compose linearize problem actual fig effect matrix fig determinant well tree shape obtain connect localization liu find shape shape assume independence liu learn reduce summation node take span root optimal configuration tree cope variation parameterize pose change index part correspondingly dictionarie z use dictionary belong overall dictionary component similarly dictionary ideally learn first face image pose subject learn dictionary separate set recognize select fully automatic pose experiment face pose neutral expression
sparse statistical establish connection view dictionary coefficient center draw moment center lead kk th reduce mean dictionary pm original e characterize compressive k specify important heavy tailed tradeoff rate signal increase
matrix reason store secondary storage pass call would pass deterministic single cost performance trade conditioning invoke round original state one pass conditioning run round separation non along matrix compute size svd n p size ram increase block conditioning ram round replace embed low rank running result substantially condition practice trade real application subsection embed method relative basically meta os embed present four category discuss discuss method first structure distortion embedding p follow distortion embed ps nm solve subproblem terminology low distortion distortion qr decomposition condition low equivalence similar result qr rounding condition c ccccc name pass er er er er er ct qr n mn qr qr compare trade conditioning qr take round well conditioning trade theoretical certainly affect distortion embedding distortion preserving embed ps nm embed closely j l point ss global mapping construct subspace scale simplify improve moreover scale maintain latter storage although os geometry vector os arbitrary dimensional subspace transform n ff eq apply transform include well transform able refined spectral orthonormal approach whose normal random g dense use multiplication embed compute transform start algorithm give orthonormal call product matrix sparse approximately hadamard projection simplify subsequently refine analyze name preserve chernoff sign hadamard scale dimensional uniformly essentially combination dimensional eq distribute although might matrix store like issue embed run order polynomially exactly stream extremely write matrix column independently subspace heavy base norm orthonormal base algebraic subsequently
imply move generally outside temporal clutter phase calibration slowly across calibration ccc clutter clutter filter whiten traditionally however pca component number sample sample unstable problematic reliably estimate filter inverse clutter span clutter clutter filter project orthogonal clutter effective parallel acquisition clutter since require relatively free enough target partially target especially problematic online implementation take inherent space kronecker covariance kronecker plus noise clutter iterative excellent applicable essential matrix p minimization derive principal identifiable advantageous kronecker appendix hermitian lr initialize
depend three merged applied affect desirable receive current trend distribution appropriate homogeneous among sake final protocol storage several stream homogeneous database use storage formal framework mining system refer slide window section sequence characterize briefly reduction merge let monitor incidence city city whether incidence disease city one report merge report incidence upon composite picture able handle requirement algebra framework monitor impact composite map approximate map map design like language language operation area combination language grid offer image processing combine mathematically prove control combination wireless application forest temperature temperature change also strength collection processing sensor life infeasible sensor capability summarie average sensor construct base observation summary rich goal cost use may wireless sensor simplify pruning send
subsequently correspond optimization demand make direct impractical many problem pixel perform pca apply feature deep auto standard rl xlabel frames ylabel success legend style axis cs anchor south bar cd plot plot bar display average success rate error deep together tailor success around solve quickly graph learn far behind truth rl solution achieve rate trial frame auto red fail explain auto
correlation graph sequentially matrix matrix row identically dispersion vector elliptical decrease matrix spherical assume assume common dispersion dispersion pre change post change respectively take value different realization random maker either sampling change occur
perhaps nuclear near rank constrain selector analyze propose weighted depends sampling empirically study minimization rank noisy constrain least estimator loss approximate matrix genomic matrix smc subset goal reconstruct whole observe genomic integration introduce model study association expression ensure adequate one goal marker power power phenotype practical feasibility genomic genome sequencing provide information genome wide certain wide genomic extensive genomic include analysis extent genomic observation suffer miss propose year include decomposition many often observation unclear statistically extend genomic integrate study extent genomic arrange structured submatrix miss construct rule datum significantly improve row whole value column
sample system output cast linear cast framework follow depend context kernel identification advantage kernel like parameterized decay generate impulse response namely impulse square mmse common kernel hyperparameter maximize likelihood follow square estimate output
agent bind marginal connectivity follow update agent truth exponentially trade communication private adequate agent highlight communication unnecessary agent interaction neighbor recover signal digit threshold consensus almost versus randomly observe involve load analyze group try world rely private signal sufficient information private adequate communication show
side figure highlight method mean test guarantee permutation approach f f among level permutation note lead conclusion focus description multiple detect match behavioral classical window potentially cover whole interval trial record implement enable permutation parallel window delay count return b set count multiple non period count significantly considered window train necessarily discovery fdr table precise package code set simulation assume process homogeneous trial delay correct permutation always whereas fail comparable result fdr basic value trial robust much permutation shown figure share correspond see formula self permutation perform overlap window run trial resp
big small root recent common pairwise distance grow balanced ensure small fall short path generation generation generation necessity tree grow produce vanish come generation upper bind decay tree prevent unbalanced grow grow unbounded second balanced quasi tree satisfy assumption growth node rate differ upper bounding uniformly determined contain parameterize iterate iid several certain gain tree tree allow node sub within condition survival grow conceptual match fourth balanced distribution typically bound certainly interesting node potentially result various subsection investigate social publish simulated tree two network
coordinate requirement distribution real dataset lead parametric case facilitate least scale approximate situation collection frequent fraction small perturbation coordinate perturb assume small upper approximate via perturbation coordinate adopt mixed analysis let incremental mixed coordinate q decompose direction fisher compare fisher n difficult analytically fisher bound turn direction fisher proportional equation calculation maximum incremental fisher inverse fisher constant scale square information analysis parameter information hierarchy high confident confident additionally confident neutral indicate hence implement replace confident neutral reconstructing tailor l verify preserve tailor coordinate begin draw result show ratio preserve ratio maximally preserve fisher distance surface denote half square parameterization ellipsoid center general maximally upon coordinate determine maximally preserve rao metric general
firstly algorithm sift descriptor algorithm cluster norm rewrite membership indicator cardinality element belong convex phase call code codebook coarse overcome yu al relaxed instead put regularization nonzero formulation turn another
give identifiability tight measurable identifiable mm identifiable identifiable identifiable identifiability monotonic measure identifiable identifiable contradiction l identifiable contradiction identifiable follow exist mixture b j view small value minimal heavily geometry tensor product space section tensor product hilbert space tensor hilbert basic intuition hilbert space tensor tensor completion
costly mt formulation kernel arrive problem describe one implementation mkl mkl solvers tailor hinge prove integrate module toolbox occur fast convergence overview mkl survey kernel kernel string kernel space tailor large considerably previous top core alternate weight improve variable precision initialize initialize optimality satisfied descent compute decrease pt compute weight module module illustrate table mt approximately later context multiple carry line maximization infimum attain boundary step carry solely fact descent objective presentation choice optimize optimize I purely involve support infeasible carry hold procedure solely vector one track coordinate way course explain rely inner product adequate computing row substantial gain mt argue aim express analytical solely thus dy mx x
image image send svm result summarize generative convolutional develop propose enjoy
let currently associate k max contexts belong create word np differ sense discrimination probabilistic observe context pair remain train vocabulary snapshot wikipedia corpus approximately million article token occurrence context occurrence unless hyperparameter amount manual skip np noisy word train initial regularization describe word quantitative et np skip gram skip show implementation size corpus
hypothesis contain algorithm linearly lp solver factor slow sort sample bring overhead factor close iii learning achieve piecewise piecewise constant hypothesis show piecewise hypothesis approximate gmm note slope curve log term perfectly ii factor close roughly obtain htb format n avg minus low high gmm avg piecewise beta table avg minus plus gamma txt avg time error high piecewise gmm x avg low plus piecewise beta txt avg minus low plus txt htb format error avg piecewise txt avg piecewise txt std piecewise gmm avg std piecewise linear avg error txt requirement roughly also robust essentially acknowledgement thank early thank lee discussion help dedicate recall statement run partition return j condition split three partition interval jumps create follow vc suffice equation triangle sign change therefore eq interval interval contain jump singleton interval jump cover assign simply follow negativity finally interval create merging jump triangle since triangle along start bound prove complete summing use pair merge suppose interval recall iteration apply empirical interval jj sign li j contain jump jump interval combine obtain complete require elegant tight polynomial unit change function analytic modulus principle since fix symmetry also choose thus therefore pz w claim polynomial uniformly bound interested polynomial integrate constant relate bound use bernstein let degree ready py
use pass whole determine value index operation separately partially sort turn sort index sort turn around also compute bad operation implement practice conditional test layer binary solution fix reinforcement reduce transition capacity bp algorithm generalization transition occur transition teacher around infer teacher find improve theoretical case point solution teacher batch classification fig maximum sample started fail eventually result decrease depend max
simple homogeneity combine possibly end sample kernel associate tolerance approximate simple schwarz control accuracy figure visualization discriminative dimensional red negative representative pick boundary define loss misclassification tight generalization misclassification loss margin svm amount allow margin q simplicity e must similar spirit perceptron produce
free energy free motivated entropy reweighte free energy temperature specify polytope hypergraph counting number henceforth refer energy restrict marginal polytope reweighte recover typical bethe choose reweighted algorithm choose correspond appearance probability span reweighte energy form point investigate likelihood mrfs demonstrate bethe energy guarantee marginal moment match marginal bethe maximize approximate mrfs reweighte investigate via bp approximation theoretically necessarily double loop bethe piecewise whereby divide small subgraph combine subproblem inaccurate piece bipartite unbiased utilize rank vertice mle estimate converge result improve method free energy learn variable minimize compact invoke minimax yield minimize ellipsoid slow sequel wolfe free energy bethe energy bethe work mle dual follow mle specifically unfortunately convex
notice carry throughout require yahoo likely page top derive consumption customer unable news display collect yahoo piece news indeed intend reach yahoo ucb news cluster quite performing probably inductive bias cluster actual meet yahoo reporting time get payoff retain world world dataset dataset yahoo theoretical omit relate regret simplicity result hold vector incorporate provable need advantage size translate practical universe tends determine major behavior one frequent behavior user profile uniformly positive suffice arbitrarily item I condition payoff
statistic explore large grid define also however generally bad notable method explore grid use mutual possible grid include dynamic sample test measure dependence variable either variant estimator estimator convenience define include analogue pearson different covariance point schmidt general reproduce hilbert space intuitive pearson coefficient however perhaps give search measurable maximized finding widely include randomized search linear ideal assess instance know present reality bad equitability measure aim good equitability relationship type assess small set general context noisy functional broad class strength coefficient determination ensure test along many include add distribution analysis equitability across sampling equitability characterize broad trivial equitability analysis include functional additive regime combination set correspond evenly along curve add dependent definition size examine evenly spaced generate realization relationship regard parametrize assess equitability setting significantly equitability present default parameter mutual equitability value test supplement section generally equitability quantify interpretable section equitability denote interval contain bad equitability analysis red plot case interpretable short mutual range along case equitability offer salient equitability normal strength quantify interval indicate red equitability interpretable reflect generate relatively weak question equitability respect represented parametrize whose parameter affect equitability parameter equitability marginal test natural ask direct mutual achieve equitability appear vs mi equitability plot worst interpretable indicate red bad list statistic sample equitability marginal mutual information correlation equitability independent variable marginal distribution version table equitability mutual sample equitability
de du fr une est e en pr l sup l analyse es pr trait es de la k jj pr des dans dimension identification un le de la figure des observation le par des et des dans ce de
singular eigenvalue reduce avoid undesirable appropriately specifically huber remark surprising despite outlier purpose indicate forward distribution isolate outlier spectral distribution finitely outlier corollary reasoning normalize line display scenario outli value n eigenvalue isolate isolated reduced font style densely yshift anchor yshift west fill north font xlabel ylabel near n bar major scale coordinate axis cs u st diag outlier insight bt font densely dash yshift yshift pt west xlabel ylabel width coordinate font style densely anchor west anchor north east font xlabel bar width pt false scale mark repeat plot
complementary four space extremely specifically high internal orthogonal tend well symmetric link design performance complementary log model future provide well corresponding consider fitting logit give taylor expansion around eq first proposition edu tr ny edu justification long claim probit despite similarity researcher dedicate carry study aim characterize similarity predictive equivalence model explore various way probit link
example consider graphical factor generate copy normal f ft f nan replicate choose estimate significance increment power clear indistinguishable indicate report type clear tb b graphical precision precision compare scad likelihood report
split constraint know assume gaussian show unconstrained point split two ccc middle unconstraine right approach spectral sl flexible constrain clustering via sl modify weight matrix edge connect link weight edge zero spectral satisfy user solve full generalize eigenvector correspond incorporate class sdp aim adapt code link encode
call importance analysis empirical replacement element sample replacement uniformly immediately use cardinality call replacement nf indeed expectation randomly complexity sign term appear denote I risk slack replacement subset uniformly inequality inside find sect compare new bind show
second need avoid summary create human still perform music classified summary remarkably whole segment duration counterpart summary sometimes discriminative also state summarize dataset size well song individually acoustic vocabulary besides classification also fast transform peak information retrieval kullback leibler semantic coefficient music evaluation exchange music information retrieval relevance point value transaction audio speech constraint lead performance segment leverage text summarize diverse appropriate segment make good music binary multiclass music task obtain full centrality performance summarize difference make state summarize music decade address music algorithm human orient summary people song summary entail extra besides non people generic algorithm however focus diverse
spaced graph interpretable statistical reader build case interval indeed tailed test distinguish demonstrates examine contrast mutual respectively extensive varie noise marginal compare curve summary curve indicate interpretable bad interval indicate analysis via nan hypothesis plot legend describe functional informally equitability dependence us relationship across broad relationship give conceptual motivate equitability different way motivate equitability begin though asymptotically detect deviation independence datum relationship absence strength relationship estimator robust detect relationship outside perhaps value estimator relationship computing parametric know relationship whether difficulty consistently propertie measure dependence approximate version consistent lead equitability equitability allow dependence approximate relationship formalize equitability interval show equitability equivalently state correspond different strength power distinguish trivial statistical independence equitability property fix detect relationship pass certain threshold across show threshold straightforward equitability converse low
bayesian combination kernel present assume hypercube original consider hyperparameter accordingly adapt towards area g pos intuition behind surrogate model could search region near many problem difficult near high variability important note initialize local kernel small capture could local scale smooth result less total
channel keep depend respect zero arrive closed output covariance weight feedforward finally formulate fully rewrite recursive scalar perform neural minimize online streaming phase update activity neuron activity local rule feedforward anti sign connection neuron except free cumulative neuron square end argue single neuron derive representation input activity recover feedforward weight present neural take zero arrive linear equation solve output component interestingly right feedforward fix descent single linear stimulus presentation repeat coordinate descent algorithm neuron activity neuron update algorithm update convergence always spectral whereas prove
stack queue input top read input form rnn controller output controller roughly rnn controller stack queue integer encode present start symbol end symbol symbol separate source integer encode symbol convert embedding embed separate mapping encode input embedding target symbol vocabulary symbol uniformly replacement source vocabulary ignore deterministic apply source sequence entirely generation sequence follow test symbol sequence symbol form source odd index th symbol form examine sequence context translation interesting
add local arithmetic multiplying add sum add bn add reader thorough discussion add introduce notion return simplify notation polynomial unnormalized boolean multilinear indicator summation boolean polynomial boolean expansion unnormalize rooted dag leave whose sum negative value child child root set indicator terminal indicator follow sum scope consistent appear another decomposable iff scope clearly unnormalize distribution sufficient necessary focus probabilistic semantic complete node define root compute weighted root root child product induce sf root htb x paper discussion keep boolean state straightforward discrete random variable plus bn size graph decomposable add time represent thm immediately exist boolean bn add bn theorem corollary simple bipartite terminal boolean sum node
exact quite balance moreover well standard relaxation recursive inspire tackle directly cut asymmetric cut propose information cluster contrast balanced cut competitive respect balanced cut recently factorization also amount learn encode similarity instance cluster partition partition cut graph vertex cut furthermore x b order sf aa set submodular extension f j submodular balanced correspond balance towards attain perfect
assume take deviation summarize demonstrate sampler comparable preferred factor model likewise compare model specify detect variance value suggest point probability demonstrate work well normal true sd c vs variance variance mean sd sd vs c number still correct replicate entire previous change replication iterate gibbs sample obtain collection number variance show
degenerate realize common format likelihood covariance bind eigenvalue invertible treat derivation ridge numerical flat sample algorithm mle entry adjustment find missing proportion square four grid square point miss exp exp points square miss generate estimate latter encounter sensitive point optimization appear convergence approach reason favorable approach exclude approach exhibit range large lattice case effectively grid preserved accuracy provide inversion last gradually space reduce miss zero obvious bias severe quick among high take entry slow efficient take second take achieve fast take huge functional compare fast estimation occur suffer convergence reasonable extend functional demonstrate facilitate propose
mean addition gaussian proof path almost follow simply almost surely rkh specifically closure expansion ft norm inner give functional regression range operator wherein dimension reduction fall consider rao measure binary control separation easily covariance covariance following maximize schwarz solve eigenvalue link study linear rao xt c
number obey family subset nr mi contradiction pseudo triangle result
radius also fact specify net n nf n u obvious gamma stick construction gamma time technical random back measure dr stick break representation surely yield variable laplace transform get arise identically whole stand q prove theorem sequence regression straightforward satisfy wavelet n nz nz get proof together quite abstract conclusion validity consistency kernel coherent type wavelet type imply belong surely verify coherent wavelet admissible ensure coherent straightforward yield generality metric easily check constant enough choose exhibit correct trivial assumption wavelet recall situation
mean rank tweet select case bottom bank mistake gold ratio narrow arise optimisation relate unseen investigate conclusion method spectral rbf lin lin mr mr move domain adaptation first half significant difficult training like previous loo train good tweet report process rbf hand rbf report spectral together classifier spectral examine cluster automatic determination ard whereby
orthonormal basis see detail call sub solve tn en ij theory property tt compute project compute treat nevertheless quantity correlate play characterize substitute obtaining tx subsection obtain easily eigenvector call reflect discuss aim relation particular since understand analyze square situation trivially moreover eq square estimator panel pointwise dot asymptotic latter see panel pointwise
element sequence unbounde claim shorthand quantity index initialization subset eq lemma vector cc condition event equality I step lemma coordinate separable minimax applie scad remain discuss base relaxation et reformulate optimization function boolean many standard programming hierarchy relaxation possible pairwise interaction incorporate constraint polynomial conjecture contain achieve prove conjecture penalty strongly result imply compute bad local minima descent interesting algorithmic broad rely give broad initialization acknowledgement support grant dms nsf
preliminary use calculation occur curse cut scheme frequent number architecture hide calculate average minibatch epoch current evaluate performance utilize calculate translation regularizer dropout technique neuron performance later different report configuration translation system model train different vocabulary table en fr vary vocabulary size help dramatically neural affect translation word english well achieve notable
group g aim disease modern datum center total represent expression brain cancer package package apparent dataset focus state right apparent comparison apparent misclassification rough diabetes brain na na na na na na seen work diabetes exception winner pp indicate technique explore performance variant pp robust rf discrimination replication summarize figure come provide detail later another aspect computation perform good operate infer predictive inherently line somewhat rely existence unstable pp huge spike prediction depict cause hand figure last right pp variant
take standard wise effectiveness spatial refer dimensionality cnns impose capacity bottleneck approach pooling reduction project onto frequency issue approach sharp dimensionality reduction encourage invariance capacity approximation loss window often represent well exploit uniformity natural power concentrate frequency high frequency minimal addition pooling permit map manner since truncation frequency exactly correspond resolution supplement pooling pool additional convolutional network convolution truncation batch well fourier basis
favorable insufficient regime bind q tight fairly bound simple analysis bit allocation allocation two weight need result weakly geometric geometric decompose remainder correspond remainder term bind ease write remainder discard given consider eq therefore condition codebook decompose term sphere fix observe k k complement choice k k summing lemma vector quantity suppose lemma define side schwarz lemma therein sum sequence sequence notice take expectation apply follow turning allocation assignment allocation go sum follow q give thus matter conclude let lemma specify hold
amplitude infinitely way cosine format amplitude frequency infinitely smooth general favor take amplitude quantify cc parameter unique locally theorem describe different form tt ms ms represent lt constant right hand universal focus representation follow generalize amplitude phase instantaneous amplitude take ft instantaneous call always positive constant harmonic behave harmonic error encounter ingredient processing concentrate dirac feature visualization
equal denote output eq function prove necessary lemma correctness main write feasible solution otherwise say
usually solve repeatedly adapt cost multiply synthesis algorithm computationally practice problem popular dictionary svd learn follow keep alternate enhance replace penalty importantly globally transform rest briefly derive cost demonstrating also show brief denoise conclude transform discuss w degenerate simplify positivity value help remove admit exactly penalty learn singular value typically little image denoise hand denoise function proposition hence encourage transform transform become condition tend scale specifically learn number close depend application condition invariance trivial minimization sparsity learn learn low minimum value equal pair sparsity exist underlie transform transform minimizer therefore sense model interesting admit solution minimizer pair row permutation certain setting learn poorly exclude crucial help overcome condition replace version recently transform weight penalty p extended previously solve keep step transform gradient low synthesis transform fact
bias fall network dnn acoustic speech recognition single work hmm specifically dnn produce phone hmm produce probable although desirable dnn train dnn force parameter acoustic acoustic correct force alignment word architecture unit softmax hmm target ms advance frame predict version acoustic strong train dnn acoustic hour example system achieve frame word predict procedure initialize find create diversity significantly outperform explore diversity model
construction marginal distribution variance precision integral inverse want marginal new joint transform advantage mean reduce transform joint q original eq compare add addition correction conceptually effect might seem
index u number activation activation bandit time sufficiently n n n n kn eq bounds aid take additionally convenient finite q similarly sufficiently observe aside finite bandit activate infinitely iterate time since apply n n proposition let eq proof proceed analogously iterate logarithm define hence relationship eq proof three come term inequality iterate bind far finitely time surely almost surely maximal u I last iterate simplification event constant combine fix recall index notational convenience
cumulative linearity gain randomization consider q k instance sake gain gain algorithm equivalently definite gd maintain projection old new set predict choose sampling belong cumulative randomly allow l g one easily deterministic include obtain force strategy act add hence adapt find adaptation
east west east west east computation posterior intensive activation distribution posterior greatly consist activation bottom weight top locally unit use softmax hoc however identify assume label unlabele identify activity represent bottom middle neural derivation activity interpretable crucially however gradient space rate systematic change point elementary minimal rate constant refer neural apply implementation benchmark handwritten mnist investigate weakly divide part proportion label label report independent unlabeled refer feed forward simply contain feed ff ff layer approach deep weakly train dimensionality unlabele feed layer formulation incorporate top regime availability take tuning subset tune free normalization randomly e
multidimensional beyond high frequency channel wavelet transform kind nonlinear iterate filter bank desirable transform subsampling transform subsample avoid drive discuss degree freedom appropriately strictly summation window partitioning subsample wider
dm uniformly exist finite integer would dm dm weakly acyclic strict dm policy asynchronous denote baseline eq integer generalize multiple dms policy strict possibly dm strict discount acyclic policy deterministic lead large game weakly acyclic stage cost figure dm dm dm choose dm dm game dm single I dms dm strictly well game probability dms dms patient discount factor error dm perfect equilibria equilibrium dm equilibrium equilibrium dms joint strict equilibrium
pdf subscript integrate power may several pdf sample structure consideration quantity term ip ip pdfs cross cross potential obvious force define potential done bring mass infinity field contains apply force ip sample derive novel blind separation bss volumes region bound hyper surface equality derivative pdfs pdfs result independence indirect bss direct target potential information potential derive field rip place point closed expression field square multiplicative pair place pair paired kernel affect choice bandwidth estimation computation rule help achieve two blind bss potential method function simply blind separation unobserve available mixture interpretation derive bss focused kullback base independence interpretation approximation statistic interpretation independence towards new bss bss mathematical solution always spurious optima existence spurious optima large bss balancing derive subgaussian source bss base parametric measure kernel ica independence may estimation actual source
test send widely metric click number email customer click place order ccc date avg avg lift significant order extremely hypothesis give online confirm add capture temporal customer click order suggest recommendation mining stream temporal recommendation attention recently et factorization movie music rating propose class moreover aim differentiable rank et al formulate temporal filter one wang recommend meanwhile diversity recommendation another problem make attract extend memory rank rather item factorization extend optimize orient ranking relate approximate extend factorization optimize like ndcg instead non couple convex optimize low reciprocal differ optimize rank bias rank propose filter partly class label classification still widely construct class bias class accuracy recommendation item matter
rnn initialize rnn many epoch day core analog nlp induce class induction pos operate token algorithms obtain embedding factorization ignore transition token embedding require stochastic token context token spectral method similarly offer mle moment efficiency rnns nlp include translation language parse replace term interaction however rnn careful stochastic scalability favorable preliminary use initialize nonlinear encourage work latent multinomial develop sophisticated practitioner start use nonlinear recommendation differ rnn good nlp consider observation dimensionality latent completely choose maintain either fix stable fit center eigenvalue less lag
separate slack margin modify convenience e elsewhere incorporate denote estimation become unstable x mn induce tr tr fit error explicitly degree fitting avoid constraint enforce validity adopt dag advantage dag employ topological order bilinear moment skip dag concentrate solve optimize hyperplane strong duality svm multipli j dag package eqn matlab many difficult learning algorithm let compute nj minimize eqn eqn eqn optimize feature optimize svms imply discrimination discrimination far instead method binary minimum maximize minimum generality false label identify sample whose assignment maximize target maximal separation class naturally class replace false irrelevant eqn maximize likelihood sample maximize hard formulation enforce false slack indicate margin denote eqn iy iy use kl define constraint dag constraint enforce order constraint please refer eqn solve summarize solve matlab step programming input eqn fix fix optimize eqn enforce dag
algorithm select whereby play reward k player cell reach total least improve total fair player propose mm mab player load accord learn accord probability distribution kt action select change receive fulfil fulfil receive reward update whereby non action k simulation configuration scenario version system simulator three bss macro randomly macro mobile
variable previous yield shrinkage onto nonnegative propose multiplier ascent term update per hold verify n output abundance datum technique sparse lagrangian recently nonnegative method admm involved metric execute different whose detail finally abundance reveal algorithm hyperspectral reason sparsity lagrange multipli parameter admm lagrange multipli regularization influence extend efficiency correspond value admm algorithm compete experiment root abundance stand reflect ratio estimation error window abundance abundance dictionary spectral band interval signature subject
also really xt recursion assumption eigenvalue part origin recall hence convergent approximation control well sufficient approximation control markov study algorithm weak requirement analyse future direction extend approximation actor theorem claim stability sa iterate iterate track solution define measure control
evident round mode adopt precision il precision representation matter consideration perform fix define scheme round probability round eq rounding round possess desirable round il irrespective round outside operation point format inner product also represent split produce number think enough precision sum product width worst rare low convert convert limit round fractional rounding mode error fix round mode round fractional adopt hardware product hardware hardware unit implement addition accept accumulation hardware overhead implement stochastic rounding simulate library fix hardware optimize apply
noiseless unitary even maximizer defer da zero formally column permutation non recover ica rise simplify suppose free ica scale e zero entry consistent notion optimality minimum signal latent signal recover kb pearson k fact desire convention let non minima set restrict investigation gx gx mm x km yy maximizer minimization first equivalence note change demonstrate recover extend ica source provide variant discussion ica appendix ica work construction
use word relation simultaneously subsection solver leverage pair relation like reflect example optimize answer cosine similarity index word analogy list apply optimization select answer form word pair question property distribute belong co information training word word accord possible candidate index choose sense close solver word locate word solver translation entity knowledge specifically offset embedding candidate candidate word offset close well question explore solver distance solver offset relation might solver co occurrence vector relation lie embed solver skip first conduct examine result word embedding publicly corpus text snapshot wikipedia process meta english word token unique vocabulary accord specifically
pc severe restriction pc reliability hundred restrict search true approximate explain work phase add variable remain variable shrink irrelevant variable prune irrelevant parent child datum children xt datum variable parent child parent child xy xx hill discuss ss draw strongly phase hybrid time appear idea employ sound identify hybrid identify hill bn begin empty continue recursively difference search add discover phase list explore change change maxima ever algorithm terminate score clearly false positive allow enter burden impose ss label synthetic eight benchmark size benchmark bn term various investigate well dependence match true benchmark implement code code pc publicly carry ghz ram running bit size c c output bn benchmark table repository experiment repeat skeleton pc cb measure false positive I divide edge divide number edge combination precision euclidean distance assess quality phase
achieve g r take negative positive derivative vice versa increase vice versa lemma proof loose curve loose low treat curve vary estimate unlabele given roc pr explain section figure translate pr optimistic loose yield loose applicability approach roc curve set estimate truth ranking simulation test independent fully label set enable ranking set negative positive negative produce estimation discard discuss remain supplementary
detail able preserve variable method margin gain lose independence admit upper dependence preserve topic copula margin know appropriate parametric margin marginal exhibit skewness marginal tractable inverse cdf mixture highly flexible ideally proposal recover jacobian term correction analytical monotonic flexible via kernel
initial stochastic mode search chain rapidly density partition state overcome pdfs development twice differentiable concave application pdf definite case pdf identify hessian property slice hmc embed state partitioning replace discuss context state handle convergence general case carefully twice violate unconstrained deal constrain mix sampler within slice capable deal therefore assign subspace handle relax twice concavity
number accuracy bfgs bfgs use gradient competitive intersection multiple iteration minimize figure sd iteration fast size bfgs immediately
training divide unlabeled vocabulary rank batch unlabele reveal select sample report ap annotate concept crowd build extensively annotation annotation histogram effective technique text content feature color wavelet frame randomly select video concept set size virtual character platform library point
store unnecessary resource stream observer find occurrence streaming four characteristic na framework benefit choose closed form sequentially test nan use surrogate unknown use average performance contribute contribute concept consistent present assume stable consistent change indicator eq thus mse synthetic streaming stream confusion drift scenario accuracy classifier drop choose cc algorithm rate algorithm report detection sensitivity rigorously compare color horizontal line grey characteristic
reduce potentially example realize code implement approach filter dt generalization character roughly speak irrelevant present environment introduce projective ps ps physical intelligence processing experience weight represent sequence observe activate walk walk walk randomize toolbox analyze scheme physical thereby relate artificial agent quantum walk walk potentially agent quadratic ps realize internal interaction dynamically increase well step see detailed learning trial make rl ps perform standard task grid car ps evolve experience exploit similarity network mechanism notion abstraction importantly knowledge generalization ps rest characteristic list learning ps generalizations agent scheme require curse bellman agent consider environment irrespective resource ability learn agent follow ps enhance mechanism detail analytical ps
select precisely question worker question like worker possible worker compatible compatible mechanism mechanism turn unfortunately compatible guarantee worker establish processing storage human average distinguish seven state human verify subsequent establish fine human verify incorporate establish option worker belief one situation totally follow worker belief lie wish full worker assign low mechanism compatible coarse mechanism coarse belief mechanism evaluation worker answer gold question payment worker gold question payment answer otherwise
image goal primarily image large image search color shape descriptor typical system database store query similarity feature fix low necessarily semantic call context intervention form rank ask identify irrelevant system retrieval rank process continue discriminate image retrieval consider image region attribute texture adequate satisfactory image several researcher precision cluster segmentation cluster uniformly effective database propose modify rf firstly novel feedback cluster relevance weight utilize content color shape occurrence
derive depict formalize sphere divide behave euclidean exhibit local either negative riemannian capable reduce least number spend away minimizer objective r independent q since embed riemannian riemannian hessian taylor approximation unconstraine semidefinite hence f q act choice tangent n translation translate tangent tangent manner restriction derivation restriction establish x generate adequate write sphere negative consist union sign vector consist center around vector cover step take similarly corollary always minimizer subproblem constrain boundary lie hence constraint force case condition various context page next riemannian trust region reduce descent trust trust subproblem point page descent lemma goal convenience state statement section suppose trust section trust l page np h region decrease assume region trust numerical carry current iterate w g c proposition h p trust decrease objective similarly region proposition w n decrease obey near minimizer convex unlikely lower bound decrease movement nevertheless trust take iterate drop next concern constrain trust give q trust subproblem page canonical section canonical exist cn w see page section np w definition constrain trust value eq numerical constant section claim carry next iterate decrease
good performance worse sag order magnitude reasonable top quantify require dataset naive sag storing store unary marginal mixed pos due sag application involve mini batch reduce mini batch need use twice pass requirement sag sag sampling crf extension examine large regularizer proximal variant see structure adopt sag might suit implementation acknowledgment like thank well helpful comment engineering research company conference provide institute cognitive proof strongly section subsequently second
feedback user parameter pure explore new standard regression overfitte insufficient provide dataset extract user click user characteristic click dataset span day ground test day day one query contain default search user example click information click day pass list click contain pass irrelevant relevant labeling document click click strictly relevant click unit highly document click short unit document associate click action pass click interested click

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

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