fact additive borel e theorem correspondence definite sequence spline etc penalty easily literature work unknown thresholded validation method bayesian preserve study classical lie linear use datum replace angle also write unit kernel commonly circular von representation study taylor variance lie two derivative thus close give support illustrate take c increase increase kernel estimate density large region peak raise kind smooth adapt base spline work regard comparative also know calculate detailed explain also disjoint interpret deal circular deal outlier two major inherently detail boundary support presence edge lose adapt mixture presence knowledge able break compact adapt spline point detect exponential join general partition unity unity topological set unit interval neighbourhood function sum requirement partition unity often extend construction whole unity assume index say open space index index support compact compact satisfy construction use analytic category thus unity exist unity help cover localization treat reflect localization lie enable store structural adaptation basis especially fouri spline function result estimation use fourier spline give adapt various outli edge deal function partition unity real life datum discussion follow conclude probability absolutely continuous haar aim regard estimation circle bandwidth fouri estimate comparative perform usual procedure method usual incorporate penalty estimate trade simulation detail give fourier common integrable article variable represent suitable reconstruct fourier analytical heat fourier consider modern transform integrable transform integrable bar fouri science fourier transform fourier translate convolution integrable transform convolution transform definition transform factor operation product give respective fourier transform clearly ahead know calculate moment coefficient power conversely know coefficient correspondence periodic say converse circle transform negative solution circle belong class definite thus theorem definite definite sequence absolutely continuous rise spline integrable correspondingly definite sequence come actually fit minimize minimize empirical datum circle u situation density belong inverse transformation note write circle minimization give run moment coefficient absolutely respect haar fourier penalty existence fourier inversion theorem coefficient density say problem spline format problem correspondence thus spline technique solve minimization empirical kx n zero q coefficient nx value show define q value see literature reveal kernel kernel curve rate calculation mean penalty obtain inside expression thus smoothing expression mean eq go infinity situation similar variance trade find nd value clearly approximation methodology able simulate smoothing define eq density perform simulation follow comparative spline technique bandwidth usual kde technique usual kde mode take usual kde spline smoothing comparison repeat simulation perform matlab consider bi method kde kde nj centre accept reject nj note test reject information edge suitably mid region function control jump effect proceed edge test false vs proceed exponential mle experimentally numerator chi square degree percentile chi degree freedom reject report cumulative chi square detect step vs testing vs proceed way cumulative chi square detection contain edge break contain local edge estimate explain unity support region create cover base unity smooth outside support partition unity say concerned region possess region local estimate section thus start estimate local lie support fit ix say likelihood denote repeat possess local adapt spline lie circle treat empirical fourier coefficient become come region decrease belong region remain procedure get estimate q choice obtain minimize thus final estimation local proceed highlight show aspect density comparative come uniform mix proportion come triangular execute broken work exact leave little proceed detection region estimate region final density show figure integrate square calculate spline kde identify point effect allowing apply vertical feature finally position local kde actual density far justify spline small kde outlier come come normal truncate take whole repeat integrated squared table local kde support detect case figure outlier perfectly exponential presence outlier create kde fail motivated work able spline kde consider successful detecting come point c fourier usual spline perfectly successfully mainly fail explain modulus compare jump detection moderately see fail far kernel rao come methodology group consider group uniformity degree upper gray depict kde depict almost perfectly extreme region identify local kde estimate smoothing belong would see spline constraint influence use multiply unity detect notation cover support estimate multiply final proper function family give estimate comparative study give new circular smoothing circle density work smoothing density develop spline try circle get circle transform derivative fourier solve spline result density density comparative application novel working profile outli detection done consider perform consider exponential overall consider partition unity local exponential spline lastly life well estimate kernel estimate solve spline negative comparative local focus fourier paradigm complete
version set tune smc preliminary run vary option general facilitate efficient accurate strategy sample iteration ease exposition variable wish fact require invariant set auxiliary conditionally ki pi define target note scheme proposition mcmc kernel upon auxiliary auxiliary accept vi ki possible auxiliary accord possess notation induce everywhere algorithm adapt theoretically hold procedure addition admit auxiliary primarily use disadvantage posterior slow dependent many might find tune example value level close together illustrate use deal toy compare implementation deal model genetic another case adapt entry uniform proposal normal walk compare implement development computation order several extension firstly adapt implementation rare event examine framework quite framework multi level splitting smc might practitioner familiar could sampler article scheme sometimes focus clearly gibbs model methodology feasibility development methodology apply thank work education grant author support grant work complete author department college result adapted conditional expectation e denote convention n define normalize line construction regard validity metropolis hasting kernel density pi p k metropolis procedure similar one may sum equation admit part consequence similar proposition statement argument marginal write denote independence ki apply university sg mail electrical college mail article process stop first naturally posterior advanced mcmc particle carlo smc embed within mcmc smc stop author smc level formulae nest set resample intermediate naturally intermediate scheme multi level proposal propose proposal methodology stop article markov stop reach boundary wide population finance majority deal observe stop stop datum cope partial good previous direction fully area stop efficiently process start terminate return getting achieve importance overview smc prove smc achieve popular traditional sequential monte describe distribution density know wise normalize smc resample idea introduce density sequentially term particle proposal success lie incorporate resample operation whose fully stop process resample take particle likely particle start result particle approximate long reach early author later split resample reach intermediate sequence idea appear formally interpret interact multi formulae naturally intermediate towards level systematic exist deviation adaptively path process contribution address issue law stop process dataset context employ feasible difficulty trajectory stop inference successive bottleneck static chain construct spirit apply update e stop bring possibility appear within enhance flexible unbiased addition via adaptive adopt article structure formulate motivate multi stop multi smc approximation strategy level theoretical level give find measure write borel denote total ji possibly class dimensional everywhere else collection stop measurable throughout process evolution empty markov empty define positive direct evolution vector assume restriction simplify exposition omit un normalise constant subscript explicitly stop process f likelihood trajectory process throughout assume density dominate measure design mcmc normalize become clear framework rather motivate figure particular start split choose continue evolve move genetic stop mutation genetic observe forward split mutation point join compose begin default chain first individual change mutation event figure confusion present compose alternate tree could consecutive mutation stop individual terminal correspond match space write carlo inference reverse simulate data sampling adopt weighting simulate backward two tree define stop markov definition initial terminal appropriately modify convenience reverse previous bottom tree obtain respect likelihood brevity omit current mutation stop integrate remark crucial shall briefly introduce extensive refer reader ease exposition present shall drop smc simulate probability g possess respect dominate normalise density g choice wise require knowledge recursively properly importance resample part density possess normalizing path obtain smc approximations normalize compute resample n nu compute smc resample step particle path particle backward recursion view smc approximation measure induce simulated sequence obtain smc approximation introduce use establish complex extended target enhanced stop process smc seem general stop provably much introduce nest denote implementation level generic reach path remain multi smc smc ultimately target sequence distribution eq finite value stop spirit generic intermediate density r dominate natural density nz define sequence density dimension compare grow increment consequence sequence addition nh hx base particular generic firstly presentation multi convention importance herein write smc whereby reject assign weight resample reach smc multinomial define normalize eq n compute weight smc begin show proceed backward process indexing root level candidate expect quickly lead poor one construction th resample hard result smc empirically case level advanced method next median rank sample aim next vary mcmc importantly sequence monte mcmc algorithms variable proposal sample lie order target providing insight question valid smc level seem obvious strong propose extension introduce three particle metropolis hasting metropolis sampler paper level sample accept probability pi ki pi p pi pi pi
vector gpu implementations benchmark single computation rip compressed sense repeatedly start large objective way batch give possibility matrix significantly multiplication small enough size limitation core gb ram core machine ram gpu enough big code store column iterate letter cardinality loading outline solve formulation variance formulation fx ax ax equivalent form x ax x ax n p x p n ax n ax ax x propose via yx fx yx xy problem subproblem close subproblem list z sa aa aa u elementary correspond objective function region maximizer objective result four operator give vector integer vector problem subproblem give formulation write select constraint v stop satisfied necessary resp however normalize maximum reach happen generalize compact work maximize subgradient nontrivial relationship apply formulation apply start formulation iterate correspond iterate convex start solution subproblem formulation iterate iterate formulation note iterate x start produce ax k ty v ty precisely penalize formulation th iterate iterate start consider individually identical q iteration start iterate ax precisely vector iteration start compute formulation develop approach scheme load batch need perform parallel multiplication product single g case library parallelization lead compare idea operator start scope parallelization vector product parallelization start choose computation algebra case parallelization naive batch dynamic strategy informally converge quality sp run obtain explain poor amount useful run number experiment choose randomly generate find effect substantial nevertheless employ different start point naive solve start run start resource architecture penalize hence besides choice htp axis vertical strategy core computer benefit run range speedup plot speedup especially batch size nothing continue already converge minor negligible speedup dynamic predefine batch list scheduling replacement variant look compare right notably com implement parallelization strategy describe use optimization implement processor parallelization computing real numerical large corpus remark instance core see plot show core consistently number core across core random cpu result htp marker setup high marker code linearly core slowly code gpu code right gpu speedup reach several penalize problem matrix focus table matrix file take second take second iteration table depict take perform treat use elimination preprocesse able expand code size sp gb gb gb gb gb gb gb gb gb gb constrain medium article appear york times article publish article abstract clearly sparse approximately million million exploit sparse component pc fourth education united interpretations pcs pc nd pc pc child play company player million percent student us team stock united st rd pc th disease child human primary tumor multi implementation intel mkl implementations multi parallelization interface architecture intel mkl library serial nevertheless implementation fast intel mkl implementation gpu allocation gpu gpu core cluster mkl communication formulation maximization observe case suitable function formulation study notably constrain variance parallel aim core efficient multiple starting pc variance speedup point achieve implementation gb fully acknowledgement support mathematic digital resource centre grant ep g support simplicity partially start research grant theorem corollary com principal analysis aim explain combination paper formulation compute single loading vector factor employ measure l way constraint formulation propose maximization al formulation parallel implementation code parallelism aim computation time serial code deal gb maximization compute big tool area engineer finance encoding feature aim combination principal pc column center extract first suitable measuring loading vector pc
model model word unnormalized rate regularize across compute estimate corpus smooth differential usage proximity hierarchy ideal applicable corpora author annotation document challenge offer dirichlet multinomial conjugacy likelihood smoothed across membership parameter explicit distribution label result challenge conditional word membership poisson likelihood tractable divide member conditionally allow inference factorize hamiltonian conditional sampler efficiently space leverage hessian allow intractable walk sampler hyperparameter membership augmentation affinity hyperparameter marginally exact loading explore augmentation parameter find entire comprehensive annotate corpora topic aid factor scientific progress biology progress enable collection topic allow quantitative evaluation list frequent word new intuitively list news use analysis include twitter unsupervised infer orient around axis create prior ensure discover document latent conjecture problem pt pt collection content infer interpret frequent word limit interpretability exclusive characterize annotate collection tree nod poisson convolution analyze annotated infer semantic word exclusive experiment amazon demonstrate summary score interpretable frequency summary produce model also hamiltonian sampler allow scale keyword categorical inference challenge statistic interpretable summary simple analysis correlation recently powerful however interpretable statistical summary value aid qualitative dimensional quantity scientific practitioner design scientific interest mind balance interpretability live high live link scientific interest two low interpretability maintain direction future text analysis motivate count document create content represent length organize specific leave news label hundred inferential discover content time content low structure express approach arise commonly parameterize loading frequency usage useful onto word list interpretability stop sometimes appear incoherent post meet expectation instead differentially across exclusive summary less likely content rarely form look frequent word corpus content idea usage idea variable capture content perspective base topic vocabulary stable usage word within suggest popular usage arguably lack across topic trade strategy word versus tackle generative poisson count unnormalize whose regularize lead word extension code interpretability category create collection focus use topic build membership conditioning hierarchy hierarchical introduce hamiltonian efficient design interpretable restrict correspondence supervise lda covariate restrictive lead associated individual infer label term word exclusive topic automate domain topic specific predict hierarchical poisson convolution collection whose hierarchy topic label topic vocabulary document normalize topic vector assume word similarly present represent feature overall corpus topic draw parent control child branch greatly word close parent express similarly learn equivalent similarly otherwise hierarchy occurrence equivalence count combine topic model proportion allocation document membership topic count poisson give draw vocabulary get content document human extra content topic content allocate active generating define topic proportion element constraint non since implicitly model two source generate receive content vice latent affinity document associate membership document affinity active membership topic inactive restrict topic parameter simulation generative parameter feature draw hierarchy terminal draw k membership parameter document draw di fw outline vocabulary word simplicity presentation terminal child level word vocabulary positively content univariate semantic document membership nuisance content topic parameterize via across topic usage also child force toward explore empirically topic arguably distinguish related topic require qualitatively difference important content measure diverse dimensionality discovery construct measure able low alone necessarily harmonic toward score harmonic default cdf apply membership hyperparameter matrix word count length scalable critical offer advantage group posterior membership hyperparameter hyperparameter break independent interface computation draw block resource document core conditional direct walk inefficient uncorrelated hamiltonian hmc hessian distant acceptance hmc work unimodal relatively curvature implementation initialization follow scan block draw conditional posterior hyperparameter latent set posterior explain conjugacy variance parameter influence group hmc variance parameter poisson covariate exposure rate parent relationship use hmc speed sparse conjugacy specifically cardinality second posterior affinity rate document sample covariate regression covariate simplify generate corpus rate v discrimination independently convolution specifically topic back distribution hyperparameter put average every draw burn word vocabulary unlabeled count unknown predictive without condition contribute size affinity factor likely base evidence alone analyze model corpus large collection great word content give special corner word alone finally baseline model month period hierarchical category remove redundant allow distinguish assignment category modify annotation document child document token stem frequent stem stem million token level divide fine content divide high level business news market economic bag category grain broad node branch market split market topic market divide soft present panel compare child market panel cluster gain exclusive topic appeal aspect strongly parameter least frequent word rate least regularization prior parent topic science technology branch panel branch shape joint posterior rare unable exceed usage rate toward frequent even rate frequent exclusive great topic similar quantile frequent exclusive word alone science technology term american program similarly topic almost research ph cr gold say year nuclear million water nuclear say million deal frequency average table top topic alone topic model neighbor topic examine red set solid case since incorporate list word everywhere market understand word rank across right panel stop extreme frequency exclusive prevent highly base index regularize logistic york sample balanced sample across split fit set topic membership train micro weighted topic weighted logit micro micro macro macro york l logit micro micro recall macro compare average dominating corpus label lda display accuracy sample maximize interpretability summary term frequent exclusive neither lda may offer quantitative illustration statistical support vector machine interested reader analysis hypothesis summarie interpretability summary regularization rate occurrence estimate frequency affect variation turn translate summary create effect light plausible less word word amazon human outline issue experiment dirichlet score diverse summary summary arguably provide interpretable summary obtain lda metric quantify diversity summary across summary whether present diverse word summarie interest rank rank score estimate leverage lda lda summary topic lda cb summarie cc summary lda summary combination score frequency summary contrast proportion word drop length summary half distinct concept reflect interpretability fit lda estimate rank increase proportion gain diversity summary dominate summary topic summary occur next determine extent summary contain word interpretable analysis summary frequency summarie lda interpretability automate method human result randomize conduct amazon amazon execute comparative three different experiment participant interact extract coherent summary human amazon produce summary interpretability outline measure coherence summary belong topic another intuitively identify summary express coherence coherence topic picture summary summary ask coherent state preference p word coherence score word along topic present ask summary come number interest strategy topic summary response figure coherence provide summary summary randomize incoherent include interpretability include option topic size result relative summary rating summary preference summary word summary low probability space improve model topic equal topic detect probability summary estimate use result interpretability summary increase show coherence absolute summary strategy word regardless interestingly summarie strategy rank indistinguishable size smaller quickly drop increase lda summary display rating three summary preference topic summary increase question average quality degradation summary number grow topic trend average coherence summarie number topic summary vary coherence column right column summary grow
cl cl cl cl cl cl fractional cl cl c armed bandit homogeneous moderately diverse asymptotically fractional tight addition demonstrate fractional fractional mab theoretically performance efficiency budget approach outperform demonstrate fractional counterpart cost performance density significantly differ relaxation cost similar easy order approach behaviour cost diverse cost density order greedy fractional relaxation give fractional three homogeneous moderately cost extremely diverse cost homogeneous cost within diverse randomly set number arm represent see test performance divide see fractional differ typically low regret value fractional typically fractional unbounded problem fractional counterpart similar homogeneous behaviour moderately outperform counterpart improvement diverse apart also outperform budget particular counterpart limited logarithmic bound paper mab problem phase subsequent good agent reward plus interval take order algorithm determine fractional fractional counterpart good separate exploitation provide theoretical prove differ well finally simulation counterpart increase cost average implication fractional practice work consist tight bound extend mab distribution dynamically algorithm rely expect thus real proposition corollary budget multi bandit mab problem action constrain consequently exploitation optimal arm repeatedly agent within budget policy namely ii fractional former provide latter less expensive prove logarithmic asymptotically good multi bandit mab originally present trade arm must payoff receive sequence reward arm high play agent reward sample choose optimal arm mab trade ucb mab incomplete end variety study recently number arm model exploration budget limit arm reward subsequent exploitation phase exploitation phase address crucially limitation application wireless sensor network sensor action capacity setting perform limit exploration reward exploitation take phase budget mab mab therefore mab efficiently deal simple mab overall budget limit suffer value value exploration inefficient exploitation give particular budget limit cost typically poor policy budget limit regret function learn exploration exploitation adaptively arm reward detail provide high unbounded determine set however unbounded np hard literature occur well reward unbounde solve behind budget arm form arm fractional set highest estimate reward approximate arm instead use order solve unbounded fractional analyse counterpart devise asymptotically differ well possible one numerically demonstrate fractional increase give first mab algorithms regret algorithm logarithmic mab bound present counterpart mab arm step denote reward arm exceed operation total arm exceed typically world application denote reward agent receive arm arm budget initial must total agent budget formally give represent arm since total exceed total let q order thus optimum precisely denote regret sequence arm describe introduce fractional challenge algorithms arm similarity unbounde reward unbounded introduce unbounded capacity item weight item exceed capacity one either problem hard however detail put differently reward arm know budget limit reduce budget mab estimate frequently thus explore greedy computational calculate update estimate turn approximate unbounded within differ fractional unbounded crucially fractional fractional unbounded arm within form agent fractional relaxation solely high ratio fractional randomly arm tc I tc also see version ucb computation order fractional computational show fractional counterpart asymptotically focus regret fractional define end asymptotically begin simplify assumption define useful ease exposition reward arm cost appropriate mean density simplicity minimal denote cost high mean note finite operate average always budget estimate achieve regret main budget positive e bound follow chernoff range nx far devise time I e guarantee show case regret average state arm q number prove lemma arm arm unbounded arm tc tc arm high confidence total step budget arm imply inequality drop necessary add feasible show add arm budget time count inequality great small cost arm great inequality arm implies last obtain equation give possible conclude lemma already abuse drop notation explicitly number consider right sc statement hold apply since give obtain I combine together give conclude denote expect difference number lemma arm
grow locally discard information comprise pool form leaf likelihood leaf prior proceed follow time smc explain section criterion update associate shift likelihood time predictive change next dt absence active data pool tree grow active split assessment make discard prior information grow move move stay close counterpart stay move stochastically action mechanism prior discard likelihood information leave pool leaf suggest additive rule eq data discard sensible novel child proportion active share parent active parent preserve total preserve reversible operation prior bring discard future lack location actual therefore newly must immediately dominate grow partition candidate hierarchical inference prediction sensible data point go concept concept formulate choice discard ad regression separately require argue ad easy al enable easily update smc learning procedure sequential heuristic common heuristic active learn gps alm select average certain case demanding alm require integral numerically lead exploration sample change rapidly alm disadvantage cope search evaluate simplification ad grid need preferred integral evaluate location discard al leaf special data variance give design expression discard active location integrate rectangular general previously grow exponentially reasonable accuracy leaf tree key analytical integration observe remain near high partition surface fast active allow illustration learned sequentially sample initial show panel round proceed smc update implementation leaf regime surprising degree variance small final end response change indeed high absolute heuristic predictive surface class greedy near largely literature fortunately analog low datum discard tend interior pool discard divide update ad node ad leaf version update ad need unchanged posterior change propagate discrete tree change particle keep limited cm tell discard discard result good rate accumulation historical information eventually streaming mechanism evolve additional point update effectively effect strength update recently oppose old datum leaf recursive conjugate irrespective family show theoretic enable streaming refer historical discard perhaps obvious context contribution past become discard discard benefit behaviour discard heuristic mistake surprising retain clear pool heuristic resolve lie beyond scope historical henceforth smooth replacing allow vary smoothly control surface responsible complexity simulation ahead performance follow first generate rmse dynamic basis ht probability via bayesian conjugate shape indicate away extreme change distribution repeat change datum distribution peak figure discard model measure height vertical complexity drop dt without discard incorporate dt mild first pool size rise complexity deeply drop retain level adapt easily novel streaming henceforth assessment paradigm wherein allow first problem old become increasingly effect fuzzy display left plot old way discard useful unlikely hard illustrate introduce dt discard discriminant streaming context mind area newly prefer alternative auc discard explanation consider way evolves show visually full online interestingly latter bottleneck auc offline exhibit comprise minute identify four prohibitive transaction status determine detail time provide inferior example dominate example remain dataset lowest suggest essential maintain informative prior information much although classifier application experiment give use flexible tool fully potential updating smc model preserve flexibility enable availability allow devise active feature incur prior enable result algorithmic parametric classification tailor massive streaming context technique automatic factor streaming heuristic laboratory university department college b business il massive become wide variety setting vast offline database stream single pass streaming context remain change online evolve stream parametric technique suited increase take stream modern parametric streaming performance inference incorporate information become without algorithms remain operate stream arrival massive operational become area scale case estimation formulae conjugate dynamic modelling sample filter variable quite parametric cloud dynamic track parsimonious regression surface arrive sequentially speak online require history summary essential parametric modelling wherein allow maintain operational cost new necessarily discard historical help response real value one dynamic tree review section allow sequential local adaptation arrive fit simple discard require work concern discard whereby discard partially informative flexibility new require access consequently discard come cost surprising employ scheme discard active heuristic active discard lead performance streaming datum temporal concept learn exhibit evolve formulae require modification adaptive whereby contribution past smoothly historical suitably construct reproduce remain real show compare modern alternative fast smc yield surface increase available specification partition refer logical rule instance node three leaf belong model leaf parametric hard seminal generative split induce via leave employ tree py py sampling change grow long integrate regression leave multinomial leave choice incorporate see move embed describe old new arrive set partitioning set insight dt tree evolve transition ty allow move pruning grow occur leaf build area move grow move among location choose random complete stochastic dt specification inferential mechanic replacement py outline step involve require parent nevertheless great distance govern particle appeal division ensemble assess effect compare method like cost dt may access
repeat randomly predict metric biology evaluating feature first coefficient tune fold cross exist mse mean biology correlation e lasso lasso lasso propose propose method solution lasso negativity lagrangian furthermore redundant propose real feature show promise real world vision bioinformatics speech extend investigate acknowledgment thank provide spam dr valuable international ms yahoo st usa institute technology university find subset responsible lasso computationally feature selection input redundant strong statistical dependence measure hilbert schmidt optimal problem many microarray categorization domain could either structure sample original transpose goal responsible predict allow assumption dependency tend regression irrelevant lasso large package develop compute critical limitation lasso handle optimization I wise apply transformation wise represent k nk vector output transform linear transform th programming hessian number much big number dimensional instability irrespective linear limit flexibility capture non statistically particular problem compute efficiently scalable dimensional knowledge dimensional clear interpretation statistical regard method highly preferable practitioner hilbert schmidt spam selection ccccc dependency scalability structure feature highly linear scalable linear dual spam hessian guarantee convexity although validity select exist alternative feature frobenius norm I k negativity meaningful addition gram select incorporate structured structure formulation non negativity impose kernel combination lasso selection intelligence rewrite ignore statistically minimize regularizer relevant output lasso redundant feature redundant redundant strong idea preferable remove negativity select allow hard interpret interpretability thus non negativity constraint kernel laplace permit moreover delta thus kernel input gaussian delta normalize kernel regression scenario unit kernel categorical kernel scenario tends poorly rewrite plain feature high shown incorporate negativity dual formulation expensive overall computation practical large g memory dependency choose term redundant redundant strong dependence output deal high feature possible intractable backward elimination feature potential approximated window window mutual maximize advantage feature selection maximization selection backward elimination solution regard continuously relaxed version feature continuous optimization problem memory bfgs bfgs many initial expensive attempt base computational able try problem originally feature perform qp hessian singular spam feature spam kx k kx x spam closely relate employ induce spam optimize moreover spam spam additive spam thus spam optimization tend expensive spam spam deal output multi label section experimentally synthetic selection code report computationally expensive dimensional solver solve experiment experimentally x multi variate distribution mean e datum additive rank regression spam horizontal axis vertical scale lasso horizontal vertical figure selection function fraction lasso select spam work poorly assumption violate compare number run observe lasso respect compare spam trick deal trick compare image microarray microarray experiment sample repeat evaluate classification use regularize accuracy feature feature coefficient width parameter fold investigate red absolute red reason decide th select
I f gs fourth put similarly stable respect input bind expectation also e f bc e e sf en sf en sf e bc monotonicity symmetry give rademach average training conclude double lem lem corollary lem lem microsoft concentrated solely handle supervise also classification w supervised adapt technique learn effectiveness real ordinal ranking real goodness conjunction vector recovery generalization ordinal reduce effort limit reason notion artificial constraint like psd paper successfully technique albeit guarantee notable exception define goodness goodness provably accurate classifier yet classification view point margin instead target similar propose goodness definition space psd learn challenge bound learn specific b appropriate surrogate generalization restrictive showing definition admit psd additionally goodness capture influential recover require sampling test time predictor result issue construction typically ease practice try datum theoretically give experiment baseline kernel learn broad use convert psd project psd perform approach expensive operation apart inherently store operation notion notion goodness give enjoy advantage exist psd ability give bound psd model adopt third supervise goal similarity supervise predictor domain similarity enough predictor enjoy property natural learn good preliminary learning fraction inspire similarity tie definition call weighted neighbor define goodness propose definition incorporate practice loss final say loss goodness definition sure predictor utility sure commonly psd kernel unlabeled least hypothesis e kk learn put semi goodness definition existence guarantee outline choose unlabeled predictor definition second learn generalization prove utility dd l admit psd psd us correspond notion kernel margin psd psd kernel learn psd treat similarity rkh adapt modification value ordinal utility guarantee supplementary material space utility ndcg ex ex ordinal regression pt solver primal spaced dataset quality label ordinal measure compare sigmoid almost case sigmoid refer criterion provable show restrictive psd focus problem identify influential aim predictor formalize typically small fraction influential problem provably sparse predictor empirically benchmark ordinal recovery outperform direction would function real social notion amongst microsoft fellowship award microsoft ex throughout prove statement originally present certain proof step allow give statement bound third technical help fx dd allow project euclidean various exist margin appear always clarity standard hoeffding argument ff x probability expectation apply switch expectation apply inequality p f domain lf unify analysis predictor however analysis bound cover number strongly respect lipschitz c l weight psd kernels psd exist bound weight k unit similarity prove generalization use variational technique problem problem domain h optima example choose p p I weight discrete problem actually contain discrete variational analysis practice act require label use fall average guarantee elsewhere generic specialized make presentation easy f l sf k bf ik f b stable change embed predictor ss sl jk jk ik I argument easily extend incorporate respect stable
eq form method estimation fisher result error distinguish additional close true determine task theorem coefficient error constant size estimation iii relation function generalization unseen observable predictive dominant dependency appear approximation enable calculate observable variable unseen substitution latent numerical approximation exist discuss ii estimation summation error method estimation estimation type eq estimation type bayes variant criterion immediately asymptotic form conclude estimation maximum training affect analysis bayes type result subsection bayes fisher include true ii bayes straightforward n variant type depict rational get satisfy integer type remain ii equivalent change indicate estimation joint target change enable behavior estimation search determination ii iii expensive successfully em commonly search model maxima method expensive integral ignore prior reduce iii show case expensive vb computation form reduce many vb present formalize kullback leibler derive maximum mathematically determine case accuracy approximate cross approximation form foundation prove first hold error another expansion rewrite eq remainder number eq term w taylor expansion accord w consider end us taylor remainder converge apply end corollary positive triangular matrix fisher symmetric definite assumption prove corollary taylor remainder term correspond use complete end proof theorem taylor expansion state complete definite confirm right complete mm theorem statistical science datum important optimal active however accuracy quantitative latent model method probabilistic employ observable variable datum model employ observable label concern unsupervise learn analysis hide process analysis limit many criterion evaluate present vb latent theoretical play important observable selection simple analysis true generate recently redundant range latent conventional criterion validity necessary unsupervised study sufficiently singular observable estimation table summarize target target example observable regular conduct estimation goal provide error measure first simple attribute mathematical leave follow three fall manner error explain prediction observable form conclusion observable target observable represent latent us datum dependency dot observable gray target estimation prediction three estimation latent observable gray item refer bottom refer type ii marginal unseen panel unseen type ii include estimation section present maximum likelihood
predictor arise sample location sampling see instance realize scalar column measurement scalar predictor subject longitudinal outcome effect usual subset independent subject specific functional functional index time index somewhat role longitudinal densely vary time effect q prescribe independent term reduce baseline point independent situation function rewrite equation association model sampling approach impose directly combine subject n kk encourage preferred via operator penalty consider penalty heuristic informative dominant dominant eigenvector large construct penalty dominant eigenvector prefer belong treat preferred estimate prefer compare prefer penalize longitudinal estimate subspace longitudinal constrain tuning penalize minimizer estimate minimize expression obtain q explicitly term value decomposition component example continuous impose inform near return sampling penalty q scalar determine estimate analytical property penalize discuss appropriate minimize tuning b simply fitting equivalent whether randomness device truly follow conditional estimator unbiased trivial see conditional unconditional expression v v dt discretized version smoothing ratio derivation knowledge allow regression limited index sufficient flexible appropriate propose decide unknown point confidence band band drop non preferred subspace current grid select maximize aic restrict property study compare second simulation study influence size functional tune coverage band feature summarize subject visit flat pre white predictor instrumental noise generate location vary use select discretized function prefer estimate regression decompose trace square bias denote noiseless method estimate regression vary simulation regression use predictor generate table draw c simulation panel column penalty panel true method principal basis panel use finding table display exploit exploit limit use feature surprisingly relatively precise l c simulation average aic maximize mse aic increase decrease vary primary assess variance relative contribution generality indicate great emphasis estimation process section list realization simulation ny ny scenarios iii mse display plot mse mse evidence scenario increase mse increase result however beyond result almost unchanged plot column use define contain function represent increase small mse hand feature similar maximize aic mse mse guide set standard mse increase mse decrease mse panel penalty middle panel average decomposition penalty display bias location true b positively approach panel middle panel band coverage dot base subject display longitudinal horizontal mark nominal describe matrix use define panel middle bottom probability notable coverage band cause large lead shrinkage result solid line span line middle panel ridge penalty external process simulation information peak display figure ridge eigenvector highlight appendix contribution ratio contribution external shrinkage towards display panel larger minimal observe gender residual plot pointwise band panel panel pure investigate association obtain longitudinal stage patient association evolve elsewhere response mr comprise spectra background profile cr figure decomposition include age baseline gender choose covariate see appear gender respect vs pointwise different lead aic interpretability hard leading follow specific intercept distribute model formulation score residual display purpose model residual obvious indicate lack display band aid spectra display reveal pure cr part regression coincide profile cr significant peak location pure longitudinal suggest find several associated band suggest structure longitudinal available period propose longitudinal derive valuable extend longitudinal longitudinal scalar predictor advantage dependent regression incorporate equivalence advantage exploit inform penalty smoothness spline constraint probability band nominal naive band bias method still relative contribution space sample size absence penalty penalty weight onto generalize ridge regression formulation informative prior formulation linear use criterion insight convenient way derive possible two observe express functional model simplify outcome important arise status patient collect setting author thank manuscript support health grant mh tr extension two across multiply ridge situation without generalize put estimation gs interpret tune order gs vector respectively w l w expressed far express w kk l estimate obtain generalize model longitudinal plus plus minus abstract longitudinal association scalar collect consist vary exploit several apply patient collect phrase mixed model pt minus availability storage collection part time course longitudinal function analogue connect covariate study collect regression longitudinal approach longitudinal subject regression consequently suit association predictor evolve extend relate functional estimate fit generalize impose inform quadratic extension longitudinal allow priori structure procedure within implement analysis response follow square integrable model denote unique solve regularization require impose smoothness expand
way capture ts attract efficacy detailed discussion comparison mab achieve display news contextual ts delay ts search ad search thorough know ucb ts run however provide weak guarantee namely context version problem significant regard open include contextual bandit amenable mab challenge novel martingale demonstrate ts generalization ts near contextual bandit payoff bandit additionally ts solve problem bandit payoff provide upper implement long efficient arm paragraph discussion theoretic fact algorithm work gap theoretic question contextual technique discuss context context adaptive play history I unknown reward contextual bandit problem arm play context optimal tt rt appendix norm indicate norm assumption bound scale free regret regret appendix sampling algorithm precisely bt bt bt bt compute nt bt computation thompson distribution play maximize reward thompson contextual bandit algorithm completely reward play bt rt td algorithm sample arm td time analyze notational thompson variate problem vector arm time step thompson algorithm tt td thompson without require thompson achieve regret bind arm notational change contextual name reinforcement name problem correspond cube bind arm advance complicated subroutine regret infinite arm assume give bandit regret set give none arm point maintain np arm sampling propose optimize linear maximize convex even combinatorial pay regret away theoretic low achieve achieve good regret find computationally achieve heuristic thompson contribution thompson bandit despite attractive use useful multi bandit analysis ts basic seem applicable novel idea resolve play arm high mean problem reward regret play basis mab variance estimate arm small mab proportional play every arm know incur also arm quantify exploration exploitation tradeoff inverse suboptimal arm play arm arm standard estimate large regret arm bound improves play arm deviation small context arm able distinguish utilize play small step play probability group arm arm standard deviation arm property enough concentration arm useful bind regret irrespective arm play play union context probability arm establish super martingale adapt super inequality mention early introduce notation notation also appear notation begin material define distribution difference arm rv p ep te concentrate around respective precisely tt v tt always time keep vice versa observe determine history context include identity arm arm true proof appear appendix use state utilize prove state arm reward anti concentration mean standard deviation provide concentration around proof follow arm arm choose highest play great arm optimal play ct b tt ct f p tt g p g inequality apply define tt ie super regret super completely ie trivially ready apply along lemma see notation begin hold pp pg g rt alternate remark thompson stochastic contextual payoff resolve open thompson mab ts achieve martingale technique arguably technique ts amenable extension gaussian analysis anti prove happen anti concentration exceed deviation similar prove lemma regret remain tight remove desirable believe would provide bound contextual delay agnostic bandit payoff agnostic refer setting slightly modify version r bt bt concentration anti inequality super martingale random martingale hoeffding super measurable imply value measurable conditionally information reward conditionally mention make eq q tt bt inequality definite therefore r tt bt bt bt b tt b bt alternatively tt union tt tt tt tt b tt b tt tt tt f g p hoeffding last use line result imply r bt b derive along lemma observe f ti respect new apply hoeffding inequality bound condition simple hoeffding tm tm tc x reduce clutter unit greater equal exist subset event occur tm tx tm tm ty tx tu ty tu tx tc tc tm tm
bag frequency apply sentiment document bag gram well onto window sentiment learn gram sentiment try rbm nlp task effective explore sophisticated rbm rbms difficulty natural vocabulary size rbms linear issue chain monte carlo units yield demonstrate rbms million word gram previously rbms sentiment classification application boltzmann expand year rbms image bag movie rating although rbms originally include observation rbm observation issue natural word vocabulary impractical vocabulary typical nlp vocabulary gram window training scalability obstacle use rbm processing directly address associate softmax unit visible gibb visible carefully transition training rbms million window capture meaningful syntactic learn induce yield even extract art sentiment describe boltzmann observation vector layer unit parameterize weight convert simple conditional give train visible scale parameter q visible fortunately approximate replace second carlo rbm observe maintain parallel carlo step simulate step chain eqs mini procedure belong class requirement stable outcome rbm representation construct visible group use index observation encode unit contain rbms illustrate unit I bias weight unit ik visible probability softmax nonlinearity refer softmax equal size layer language vocabulary visible unit expensive update distribution dominate activity select mini batch negative computation since contain gradient rapidly compute barrier efficient multinomial rbms gibbs multinomial address binary tree leaf external consecutive word wish introduce undirected strategy conditional rbm rbm conditional rbm deal boltzmann large open nlp visible multinomial operator sparse satisfying approximation learning sampling conditional chain leave current ji v visible unnormalized separately efficiency operator move speedup sampling rbm v efficient correct possibility designing explore metropolis marginal word corpus I outcome use sample constant setup gibbs make proposal simulate metropolis operator distribution linear space bernoulli dependence practice although poorly mix good occur well examine mixing use corpus vocabulary describe imply initial accord computation require offline purpose hasting converge target six randomly gram corpus metric variation tv mcmc tv across group figure break dark green highlight mix mix slowly benefit h use neural use boltzmann machine model rbm vocabulary tie weight word bag tie single document sized multinomial distribution word condition softmax incur notably address computational burden observation rbms boltzmann machine could large softmax although motivate nlp softmax boltzmann natural language task representation task previously different mention conditional last gram window approach training network middle use base objective contrast rbm use fill within window single rbms nlp rbm windows parameterization rbm sentiment useful gram rbm separate position rbm gram window would prefer word position dependent rbm inspire neural learn possible word dimensional projection encoding energy position let vocabulary see value gram representation bias energy refer parameterization easily construction rather train metropolis hasting joint train use observation find helpful momentum updating representation additional window text english corpus text source extract window boundary two sentence also correct relate replace digit word character discard vocabulary consist frequent word plus token map word publicly available task feature crf f use window unit baseline comparable representation try
cart subtree select motivate extension concrete take uci repository seven coarse aggregate fine aggregate per day concrete height flow objective predictor cd cd intercept water sp fit multiple regression water coarse amount decrease water one may conclude water predictor see interpret hold main inclusion interaction bring interpretation challenge instead control effect similar goal partition restrict show predict terminal node water water flow produce easy nontrivial affect jointly flow water great intersection yield concrete model three response ideally predictive variable bias extend guide solve fit data node observation sign th two perform small exceed threshold depend characteristic categorical goal classify sign pattern residual work sign instead response residual residual pattern residual predictor residual pattern contingency table sign group row chi test independence specific method univariate guide except function sum split mean sign node categorical mean end categorical create contingency chi test mean singleton k predictor select small step possible ability detect pattern interval reduce guide page split find consecutive total deviation categorical form computationally quick approximate variable describe procedure pair water concrete group contingency form sign group water chi fold cross description multivariate guide refer bar concrete water coarse water concrete high combination predict concrete water split guide three unit apply one sum mean result close guide guide univariate uncorrelated flow weakly strength carry simulation experiment compare bias guide concrete randomly predictor select left probability guide trial standard error proportional variable water sp aggregate aggregate demonstrate select split predictor multinomial equal base panel seven variance regression univariate guide different utilize response variable structure challenge absence effect variable mutually generate regression cart model mean response estimate pt half terminal also terminal expect univariate guide scenario scenario take relationship multivariate guide detect high guide interaction step second multivariate zero mean covariance independent experiment result except scenario guide scenario univariate guide notably twice average splitting wrong effectiveness greatly predictor variable challenge guide use miss group carry chi allow datum technique pattern search split value categorical additional missing value split miss split consider missing mean usually miss great select approach cart maximize square bias toward sample concrete randomly brevity teacher applicable grid may restrictive broad applicability motivate explain longitudinal black white time varied section fit partly overcome partly model complete near force entry black school intercept subject denote time applicable interval user algorithm node datum great number restriction linearity fit split show curve node eight give tree terminal curve five terminal tree contrary finding page distinguished see tend rate decrease trend imply simulation predictor longitudinal spaced q effect fit case estimate package compound almost perfect interaction guide except simulation generate appropriate simulation obtain point error q record table simulation good model guide make assumption slightly approach predictor child daily presence stress child child week status education health ordinal people regression answer question child association stress cause child day pruning suggest variable health status education child interaction day significant two separate health figure plot group child health find curve day health cause child vice versa fitting predictor feedback fit simultaneously predict stress child interval stress health predictor plot tree confirm stress curve solid vary monotonically fair bad frequency stress child tend decrease study period significance specify specification parametric approach often possess important feature consistency estimate increase longitudinal datum response assume compact consist correspond constant setup treat longitudinal total collection terminal node partition node contain pt assume ij terminal ni n follow condition sufficient assume algorithm capable choose right curve regression g condition first hand independence identically density everywhere condition constrain standard estimate ensure correlation error small write u ik ik u ij ik ij n ik u u polynomial nh fu imply nh n nh fu k hence algorithm tree longitudinal cart bias covariance guide bias likelihood select contingency trajectory mean trajectory nonparametric split set selection pruning number normalize response implicitly residual trajectory pattern assumption node quite powerful situation satisfied assumption autoregressive justify satisfied within partition though robustness mean easily evident longitudinal smoothed mean terminal realistic summary parametric substitute parametric parametric model unknown may model construction base small powerful sample predictor tree quite interpretable serve purpose identifying subset implement guide grateful anonymous associate comment suggestion lead improvement cart statistical support nf national health grant previous construct tree cart consequently bias difficulty cart guide treat longitudinal use chi square tree besides predictor result square example compare effect generalize condition asymptotic estimate regression tree regression partitioning set value predictor cart split ordinal categorical sum deviation partitioning continue node test subtree lowest select extend longitudinal often use measure longitudinal symmetry em difficulty space extend cart exponential response response transform binary value function computational difficulty every node cart square deviation mahalanobis different treat longitudinal trajectory fit longitudinal low curve fit regression coefficient predict predict trajectory spline coefficient mention component multivariate
proof omit choose dominate probability one provide estimation compare term term drop subscript proof odd scalar theorem obtain regard derivative accord q n x n stand hadamard let denote index use imply contradiction operator positive semidefinite similar problem class argument density tt ii unique contradiction entry equality follow diagonal result side thus write zero positive definite consistency accord prove ni rewrite side rf definition theorem em widely nuclear circumstance basis completion various completion quantum recovery paper propose rank correct establish error importantly quantify nuclear reduction substantial provide interestingly highly concept foundation rank rank simultaneously recovery capture low structure keyword rank consistency without considerable interest quantum state idea address rank completion np nuclear convex envelope unit spectral nuclear remarkable norm property rip rip incoherence matrix recover noiseless sample later improve count employ technique develop extend coefficient relative analysis besides noiseless address plan noise study nuclear completion literature nuclear technique encourage efficiency circumstance certain high completion point recovery trace norm base column marginal prior penalize satisfied involve concrete density quantum quantum state semidefinite trace trace norm completely nuclear fact much relative room motivate possible norm seek completion support low correlation completion include special fix due observe sample high recovery propose correction nuclear minus initial estimator choice optimization scale stage statistical attractive thank lar overcome stage naturally occur particular stage enough desire efficiency lasso important broad overview interested reader refer natural reweighted trace minimize rank extend lasso however completion initial valid numerical reweighte nuclear approach matrix reweighte square minimization extension matrix transpose smoothness subproblem encourage hard constraint basis involve difficulty inspire matrix problem non frobenius adopt discuss impact correction recovery importantly mild rank correction nuclear square estimator asymptotic consistency sense ideal analyze gain insight limitation optimization key analysis interestingly recovery criterion construct correction consistency broad rank unless rank correction three improvement advantage propose apparent involve theoretical foundation set matrix optimization introduce completion basis bind correction step provide quantification improvement devoted report correct conclude relevant brief denote complex matrix symmetric semidefinite matrix nn hermitian positive endowed induce n mean n real case matrix orthonormal column short denote transpose notation adjoint cardinality I let denote euclidean denote norm nuclear almost sure respectively set basis nuclear norm solve problem relative recovered practical matrix assume reliable index basis need recover collection basis coefficient eq identically distribute control unless general weighted follow copy e notational slight express present example correlation real hermitian semidefinite entry entry zero ie eq completion integer hermitian semidefinite quantum rectangular rectangular aim n introduce operator frequently operator convenience rest arrange x n xu rv rv rx situation matrix efficiency sample model extreme nuclear completely nuclear propose generate pursuit rank estimator correction magnitude closed operator refer operator subsequent penalization nuclear bind restriction mild correlation boundedness hereafter call correction term rank step nuclear singular certainly directly serve practically lack convexity extent available achievable suitable substantially offset penalization nuclear expect low outperform penalize function theoretical support rank capture choose penalize suitably estimator generate adaptive nuclear semi penalize associate constrained optimization n x idea penalize version next iterate subgradient convex compare correction step least different noiseless noise constrain assume uniqueness though observation purpose precisely constrain correction bring similar norm correction impact rank argument use define quantity basic relation matrix estimate problem emphasize penalty sampling therefore base quantity probability easy extreme large constant universal case sample rectangular reveal derive error bind frobenius operator previously absolute choose correction control roughly degree freedom logarithmic recovery important error fix bind therefore recall nuclear theorem nuclear recovery rank case ensure simply reduce nuclear norm penalize large easy true account optimality table demonstrate substantial relate magnitude increase relate upper bound slightly decrease find small illustration relaxed attain q proportional important nuclear least c bc find validate numerical consider asymptotic step expect reveal chosen parameter true set matrix follow elegant behavior solution follow sufficient consistency recovery path sequel hermitian hermitian notational simplicity divide term extend argument least square case rectangular system solution r semidefinite nuclear simply reduce positive semidefinite condition hold consider eq assumption consistency necessary next theoretical guarantee system extensively eq meanwhile may define n adjoint rectangular rewrite semidefinite rewrite result either rectangular semidefinite self adjoint definite immediately constraint q correction step hold correction positive completion consider structure entry fully corresponding correction constraint problem completion special completion fix basis control trace constraint correction redundant thus constraint reduce automatically importantly uniformly class completion class correction rank focus correction section error meanwhile suggest choose semidefinite need replace decomposition eigenvalue conduct exactly good choice operator twice continuously differentiable correction let spectral constraint mf omit suggest ideal overlap error rank recovery error ideal guide contain rank regard divide perturbation confidence shape one need approach rank singular choose certain exchange recommendation choice particularly nuclear penalize remark semidefinite reweighte al reweighte surrogate meanwhile correction n however notable broadly speak rank correction reweighte trace norm recovery different problem adopt proximal solve admm reader sequel respectively stand least square correction short randomly diag bar take true randomly diagonal hereafter true pattern note correction consideration report whole diag diagonal point show consistency estimator possess result solution infer recovery unknown advance monitoring search different matrix diagonal entry gaussian set norm least four rank correction function plot figure decrease together attain exception observation test chance subsection covariance different initial r upper diagonal double produce produce penalty choose attain small penalty see result desire plot clearly observe initial always substantially subsection reveal good small attain search find actually consistency strategy question correction step quality report result covariance matrix rectangular report small different know estimator value take second correction function covariance c rd pt ratio e e e completion true subsection weight rank number non partial fix double among noisy observation reasonable sample low entry bring recovery completion r bar test trace besides frequently system strength row differ assumption noise randomness may quantum solution drop trace trace constraint one recovery among attain step report table besides recovery report square fidelity closeness fidelity relative htbp ratio fidelity fidelity fidelity pt e pt rectangular completion weight ml bar ml mr take dimension scheme sampling respectively word leave much top bottom partial un set large rectangular recovery ratio low meanwhile advantage remarkable rectangular rd pt ratio e e e e
model compute cp vanish obtain discard finally representative sample universe universe situation seem simple reliability base introduction share may decrease combine category spurious every eq note immediate recall end assume moment upon atomic transform class precede long provide validity model change category level change could identification prove last uniquely thus obtain prove obtain get q solve obvious formulae part formulae involve fluctuation lead thus expectation value formulae accord e c minimize note introduction base investigate accuracy remainder section vary inverse model quantile absolute range value parameter model define actual error range half determine proposition note exhibit affect affect range five belong category statistical distinction accuracy contrary rather significantly attribute fluctuation finally error deviation frequency expectation number influence five error item item rough beta reasonably low item email inter agreement account agreement chance exhibit tendency high low depend condition reliability researcher agree view accept reliability upon reliability upon rate nominal model inter item reliability uniquely relative frequency exhibit setting nominal like medical science content analysis simply measure adequate occur reliability cope prominent measure chance correct agreement vs maximal agreement way chance correction take correction compute give rise favor certain behavior explicitly assign category choice assignment whether assignment may extreme assume category distribution chance chance correction assignment thus marginally influence category population distribution hand agreement category considerable literature e hard reach consensus category certain distribution chance agreement exhibit usually produce marginal marginal distribution similar proof contrast inter contrast intra reliability conduct way group individual reliability prescribe another score category example justify reliability rare category argument problematic design typical approach define reliability common category assignment correct item reliability item inter side effect reliability state direct inter reliability simulate evaluate accuracy section author might statement acceptable close look impact set category classify use inter item independent mutually exclusive exhaustive exclusive existence usually call category implication category relative category category category knowledge sure case follow base category cf certainly assumption field e medical diagnosis upon kind item education vs formalize item true fail recognize identically item random q atomic convenience let model subscript refer choose assign item assign chance rating process choose agreement call category occur cf first model distribute seem discuss inter reliability contrast intra reliability use parameter characterize consideration actually arrive situation situation assume indeed dependent put item outcome reliability randomness setup would would subset attention belong independent imagine easily distinguished subset necessity value reliability cope return aspect inter parameter observable frequency category r expectation condition expectation
though occur finance class mean optimizer rate convergence property simulation extensively statistical statistical ambient concept area dimensional much tool literature procedure zhang penalization concave penalty well numerical property fan fan unified zhang focus dimension although appear information criterion bic provide forecaster apply often case variance non variance transform many transformation ignore heterogeneous aside estimating often important parametric explanatory finance control high explanatory positivity capable capture vary order magnitude main contribution propose iterative optimizer mean variance estimate mean simulation show outperform analyze vector form contain index denote submatrix column index norm usual extension notational matrix frobenius common variance penalize estimation procedure introduction correct probability tend literature variance jointly study extensively set solver new develop author acknowledge log study however discuss principled way resort ad hoc optimize convex estimate stationary however final optimizer estimating find maximizer solve form vector residual reweighte estimator principle likelihood generalize second cast result unknown likelihood alternate estimate mind usage result satisfy property sparsity continuity support satisfy mcp penalty produce particular scad replace iterate establish theoretical non trivial observe stop iteration selection candidate aic likelihood eq freedom compare bic simulation procedure scad scad grow relate ambient large size establish correct selection parameter satisfy far weighted assumption establish convergence second weighted estimator normal generalize least know true ccc bic bic bic bic assume conduct study sample identification precision precision variance ccc nd nd bic nd aic st nd aic st bic nd bic nd average data iid normal logarithm associate normal marginally illustrate scad iterate summarize toy perform correlation predictor already visible tends pick complex select standard previous number observe aic select complex size furthermore iteration demonstrate benefit prove important country base community country current international forecast form return country country year forecasting remove variable group category balance form ar initially lasso lars bic plot residual bin fit residual apparent variance third bin difference country log response equality variance justified variance tune select metric compare mse log table outperform mse score mse fold address high control mean exist perform solve
table temporal baseline still baseline centroid regularize visualization framework grouping temporal use past introduce version static counterpart demonstrate regularizer intend preserve area concern visualization dynamic thousand node scalability extremely significant large equip grouping human challenge expression compute algorithm gradient jacobian obtain evolutionary design dynamic object adjacency involve ordinary static smoothed quickly adjacency drop framework adaptively time minimize mse frobenius adjacency view characterize choice q affect replace variance sample static perform smoothed obtain membership estimate cluster membership refine iterate paper cut algorithm reader detail acknowledgement partially national research office grant nf xu award sciences engineering research structure evolve typically addition provide preserve consecutive human evolution network visualization set present create static belonging step penalty increase sequence preserve map dynamic multidimensional group interpretable dynamic topic recent year application science real node removal old edge development low rank detection community closely dynamic open attract visualization provide insight intuition static static typically graph representation create graph include direct scaling present aspect remain trend poor structure create visualization treat snapshot stack result visualization interpret present evolve time snapshot challenge preserve human undesirable drastically position frame may cause reference visualization involve create interpolation constrain successive way make real create snapshot static time often due static network cluster set visualization encourage preserve map draw explicitly employ regularization dynamic stability many ridge preference stability preserve map design set modify function static node belong knowledge group evolutionary grouping keep member close help preserve node belong group often evolve fashion temporal helps preserve map cause human side good framework visualization incorporate employ grouping appear work graph selection importance temporal interpretable time indicate square dynamic discrete sequence snapshot adjacency choose edge assume graph simplicity time row dimensional position visualization norm trace column summarize multidimensional operate graph snapshot multidimensional family variety refer interested reader cost stress give denote weight maintain refer confusion matrix stress adjacency distance metric calculate two weighted weight denote dissimilarity convert dissimilarity compute crucial kk force special stress stress minimize decompose independent optimize iterative iteratively stress optimize eq stress eq derivative upper solution solve note consequence stress translation remove solve cholesky convergence iterative laplacian optimize weighted dissimilarity spectral solution eq easily function aim edge short placing heavy edge remove place location view remove translation mean correspond scatter differ constraint variance vary undesirable would scale number node generalization give exclude abc easily often normalize problem differ replace degree dot optimization th small exclude constraint normalization static snapshot snapshot visualization visualization interpret regularize framework define optimize stress nod member group choose node far previous control quadratic form let define introduce grouping add map grouping representative decrease towards node group place knowledge membership position position often refer draw literature maintain preserve propose temporal decrease towards unlike assign step thus node act anchor measure measure goodness topology prevent adapt trade stability next grouping penalty give modify stress eq q stress temporal group optimize write sized similarly desire add row column I denote node stress write minimizer result solve sequentially dimension use iterate attain iterate take ordinary equation rank compute short compute show fig factorization follow back substitution position position fix invariance dominate initial subsequent back dynamic second term penalty compact define augment add representative laplacian two write final independent henceforth modify cost consider degree derive optimal zero orthogonality become denote eq center combine unnormalized problem replace obtain scale static consist close appendix get poor multiple initialization center form assign appendix initialization interior solve initialized consume solve cholesky previous dominated cholesky choose applicable graph force static ultimately decision type preference often prefer maintain entire series tune grouping encourage grouping treat rank group belong supervise optimize function grouping vary eigenvector u addition prevent eigenvector well preserve static method generate network software explicitly control stability insufficient node experiment enforce dynamic objective several criterion stability criterion stability position position spherical center probability constant scaling amplitude logarithm exclude grouping cost maximize preserve tend specific rule q journal update include localize optimize grouping way create stress minimization penalty mostly several penalty visualization modify term modification optimize without anchor view modification grouping come explicitly stability static eigenvector initialize node position second compute smoothed perform constrain method force direct achieve implementation emphasis improve scalability extremely initial apply improve applicable incorporate sort apply result support website priori compute baseline line use previous modification method compute smoothed standard propose grouping baseline regularization neither summary statistic section choice snapshot optimize static energy close group calculate centroid cost respect even learn centroid respect square node consecutive preserve mention stability goodness display depend require tolerance lie ensure choose minimize choose pt stress centroid sbm learn static mit static ccccc centroid mit spectral table one lower since grouping might method centroid significantly low centroid temporal position static employ less employ add regularizer experiment detail block model sbm sbm group stochastically equivalent I form node belong sbm specify probability cd represent probability form particular node sbm membership assign remain unchanged assign group simulate create static centroid cost method average simulation grouping regularization centroid expect group centroid although temporal static hold structure alone temporal decrease benefit average fig plot static centroid much generate cost grouping centroid slightly method combine perform centroid fig centroid temporal cost change beneficial penalty strong mask undesirable centroid temporal cost event exploratory long period respectively fig distribute decrease decrease sensible node move significantly representative temporal slightly increase centroid centroid place move representative grouping regularization centroid temporal indeed show always penalty penalty grouping towards representative increase centroid temporal cost fig part examine incoming transfer student week rank individual house private collect week break process data preferred convert graph edge convert dissimilarity divide priori affect plot use step membership time poor job topology temporal example mostly student switch blue around fall break back continuously change preference observation student large compare mean student fig respectively top row correspond
evident induce rather something layer introduce model moreover complexity mapping reduce typical toy real world set dimensionality first process obtain stack pca vary usual latent basis create level stack gps depict hierarchy top stack stack respectively dimensionality learn ard find correct hide signal one encouraging indicate truth confidence toy unsupervise learn describe observation input simple create toy stack two gaussian process exponential receive equally spaced kernel gp create sample present equally spaced randomly leave rest effect range become traditional gp layer layer trivially learn actual obtaining split deep figure suggest deep enough less assign deal control ht toy show towards five contain frame motion consist digits handwritten digit digit represent gp range bayesian hide evidence layer quality mistake decide number depth plot fig ard final hierarchy abstraction output encode whereas high encode circle many conversely dimension parent two vary output level abstract introduce bayesian process mapping approximately allow discovery hierarchy describe human motion handwritten digits variational handwritten digit experiment give evidence powerful enough encode abstract could validate improve something past effort combine gps deep pre present consider layer deep architecture experiment simultaneous allow principled model use top hierarchy lead model process future across task jumps gp set publication show variational formalism gps acknowledgement research state foundation section figure table chapter chapter chapter section gps belief gaussian model input gp variable variational marginalization model layer belief fully treatment selection justified modelling neural model popularity complicate associate abstract human inductive concept question question structure justify useful applicability around binary latent rbm stacking technique amount importance rbm deeply small seem machine core interesting approach modern much back implication model family vector almost introduce perceptron mlp certain unit relate deep learn autoencoder dimensionality expressive variant sophisticated gps complex structure powerful base lead principled truly associate rbm rbm model parent output input condition conditional role significantly rbm input likelihood dependent function nonlinear integrate analytically gp analytic integration propagation book consider deep marginalization treat might include rbm binary scale exponentially intractable similarly analytically solve maximize respect combine gp posteriori model algorithm available exploit advance stack go show variational gp stack truly rigorous allow selection applicability deep different hierarchy cascade layer layer layer place process mapping gp gp regular rbm latent likelihood automatic relevance determination ard prior place gp dynamical extend process obtain approximation latent output flexible gps gps process input store seek unobserve responsible prior draw operating input reflect covariance mapping quadratic l infinitely aforementioned normally analytically omit clarity gaussian latent elegant treating input latent employ prior gaussian process take form mapping likelihood analogously architecture three kind observe intermediate latent unobserved prior constitute input focus scenario intermediate depict generative take involved play extend deep segmentation already layer number layer crucial priori unlike allow automatic also add step automatic relevance determination ard gps weight latent switch drive introduce treatment nevertheless analytically procedure available regard e distribution node whole cascade prior step apply jensen bind variational appear expand integral expand gp auxiliary augment p clarity x able augment tractable variational free back replace joint version compute analytically break logarithm grouping fraction divergence expectation associate leave find bayesian gp output quantity see expectation involve involve output main calculation demonstrate hierarchy I
name field extensive survey find considerable develop cp still concern cp decomposition agnostic optimize outlier cope determine noisy despite existence paper calculate algebraic specifically algebraic determination rank due component avoid determine correct less sensitive noise superiority briefly basic tensor slice decomposition cp call tensor denote cp decomposition want cp noiseless find simultaneous svd determine n rw w k equivalent tensor simultaneous iii coefficient notational difference write column fix form diagonal slice component whether cp include terminology whether tensor tensor probably case retrieve notion appendix generic cp cp replace equivalent prove rescale generic element presentation reformulate imply lie span presentation correspond existence invertible correspond combination exactly one full state together component slice representation v converge calculate svd candidate mode switch mode simultaneous omit algorithm center component shift cluster link cluster center pair triple unnecessary mode g majority vote vote closeness decomposition output rank determine center belong return simulated toy input tensor slice uv uv independently independently normal normal covariance estimation detect htb numerical analyze increase figure synthetic ls emission rise emission thus suit assess physical priori knowledge show emission auto detect excellent agreement spectra fig lack negativity spectra attribute scatter thus well agreement shape identify present determine decomposition noisy tensor argue unstable component demonstrate competitive art advance outlier use intrinsic low tensor oppose method fit prefer whenever proper emphasize structure
implication work scale descent bfgs randomly variety bfgs nd fundamentally literature curvature minimum l bfgs cause variant fail curvature near shaped I plus transformation though ignore operational suited storage discard batch iteratively receive lx rx base simple rule tx subgradient prove regret remark loss estimate sublinear slow growth package implement online whole cost utilize collect file communication baseline learning step manner adaptive minimize generic square solve tx check fit newly represents forget previous emphasize datum indicate past equally correspond fit perfectly concatenation newton minimize function computation computing grow costly fortunately differently approximate analogue eq update finally q coincide standard return pass algorithm evaluate cumulative remark incorrect evaluate precisely case change set determine subsample old datum roughly previous point serve anchor fit batch expense global regret generalization learn recent subsample routine associate start store l bfgs hessian initialize descent routine summarize process check datum tm run learn batch aspect partition significantly speed gradient run parallel subsample parameter make subsample quick check finally remark reasonably acquire choice matter compute directly situation discard typical gradient descent priori bad bfgs consider ad necessary order appropriately rank ad click time display combine classify property display search query acquire engine publicly learn unique ad description display user ad position ad depth depth normalize click training click rate build vector go regard experiment compare bfgs technology randomly high randomly choose one compute auc portion million basic conjunction perform approximately winner win exclude future set relevant model auc bfgs implementation relevant learning quickly heavily optimize utilize robustness job failure gray ccccc cr bfgs seconds auc sa accelerate iterative bfgs sa bfgs induce derive large less speed set capture interaction query ad frequently well accuracy bfgs relevant technology bfgs sa amazon bfgs add measure second auc auc comparison feature cr model achieve bfgs cr time fast furthermore sa bfgs achieve auc likewise speed enable require compute note speed bfgs tie gain reduce reduce bfgs force expense specificity tp gray gray ccccc cr cr bfgs bfgs second rapid model use new l bfgs combine question ask model explore discover sa suited underlying behavior market pricing change term usage sa l bfgs dynamically batch datum variable size furthermore statistical aspect bfgs bfgs affect learn new care flexible technology context relevant company friend family work company list five united microsoft windows internet payment system lead expert political micro degree university master science engineering experience technology innovation engineering management team microsoft massive storage microsoft team microsoft principal microsoft microsoft ever window lead range science science recognize machine publish paper microsoft yahoo spend decade building system processing microsoft work understand yahoo focus ad relevance school focus recognition incorporate massive semi machine master electrical engineering investigate broad question people twitter social tie publish cover medium national public work microsoft master mit work medium laboratory medium receive applicable medical imaging principal experience performance load balance build optimizer computer mit present enable rapid training example million sa batch bfgs perform balance old iteration
mine practical widely theoretic express observe another categorical pa degree association one predict evaluate redundancy variable call feature selection correlation filter relationship variable variable e exclusive neither capable handling way desirable computationally cart recursively subset tree multi variable theoretic mention select tree select theoretic redundant result redundancy separate set decision split split redundancy redundancy problem tree eliminate redundancy new similar select eps eps number make return satisfy order incremental refer select first relationship limited categorical analysis limited categorical tree section splitting leaf limit node level depth position decision tree level node level child md denote decision condition md decision refer md md show md tree level md instance maximize node mutual calculate md tree feature add redundant information redundancy introduce recursively node g tree build belong unless coefficient produce large select splitting framework regularize sequentially predictive expect contain non redundant advantage svm penalize form interaction extract objective redundancy node feature tree subset conditionally terminology call practice usually unknown selection feature instance induction expect commonly subset preferred preferred criterion classifier strong classifier capture may subset uci empty select add nb feature accuracy classifier nb stop add feature rf continue feature indicate add nb contain validate classifier prefer capable high subset regard instance class instance auto c cc cc cc cc cc c c auto uci benchmark database vs implement regularize framework tree eps eps rf feature selection set accuracy rf pair subset replicate well feature level denote subset show trend evident regularize ensemble loss feature note present tree ensemble though ensemble rf rf tree capable capture interaction relatively differently rf accuracy competitive use rf predictive capable extract performance select next ensemble simplicity datum set equal rf versus backward elimination automatically rf feature randomness run rf rf competitive optimum still cutoff computational efficient second capability model forest quality subset strong computationally deal categorical numerical variable etc provide practical acknowledgement enable regularization gain previous forest tree easily tree tree select subset strong categorical variable interaction effective training set consist instance variable significant role curse interpretability framework splitting node produce gain subset single build wide forest handle categorical miss scale etc many background tree propose regularization regularize regularize section demonstrate efficiency extensive experiment work feature divide embed filter
ap proof make extensive proof proof theorem condition measurable lemma arbitrary sense proof c mh deal condition obtain recall hence geometric ergodicity spectral fold iterate mass every x arbitrarily small relax statement assumption follow independent cumulative associated suffice ar consequently take small two example g generality q g satisfied calculate lemma computation common scenario intractable markov chain facilitate fail geometrically ergodic consequence reliability phenomenon typically tolerance bound ergodicity condition alternative asymptotic involve pure jump sufficient condition variance bound reversible provide bound geometric ergodicity mixture reversible kernel refer branch parametrize computation approximate increasingly diverse likelihood computationally prohibitive standard prior metric artificial commonly approximate computation value interpret slightly abuse simplify draw general version standard sampling avoid carlo methodology whereby iteratively stationary sum start latter ergodic converge geometric total iterate mt property geometrically ergodic motivated ergodicity approximate reversible theorem bound integrable course reversible bound markov seek control bounding geometrically situation use partial problem identify negative reasonably mild geometric geometrically addition condition ensure property meet g prior everywhere tail quantitative example background bound geometrically ergodicity reversible markov spectrum operator restriction order bound equivalent ergodicity gap geometrically closely aforementione spectral gap variation eq accord common accept reject variable densitie common dominating g case pseudo kernel parameter model smc evolve involve fix kernel simulation readily output evolve denote pseudo auxiliary balance verify involve fix auxiliary variable kernel stop concern value behaviour demonstrate nevertheless fail propose move locally suggest may subsequent computational effort particular place successfully ergodicity hasting proposal many relation provide assumption however output output kernel share write represent extended g satisfie infinite collection proposal broad common proposal collection bound geometrically mt follow addition far get move away proposal tail decay exponentially mutually exclusive hc theorem simplify general reversible indicate geometric ergodicity coincide lack bound appendix concentrate reversible irreducible invariant kernel measurable bounding concern poor irrespective value former implication upper almost almost uniformly establish uniformly ergodic uniform ergodicity little motivate correspondingly exclusive computation indicate autoregressive cover kernel analogue far asymptotic systematically geometric condition order mh p condition proposition control manner different difference dramatically condition condition regular contour obtain corollary decay super tail decay exponentially everywhere satisfy euclidean geometric former already argument state q np ergodic corollary acceptance suggest cover countable collection proposal modification state could conjunction mh mh mh invariant metropolis hastings mh mh justify geometrically ergodic reversible invariant implementation qualitative effort unbounded iterate irreducible q quantity rejection sampler gap require much three proposal poor initialize little mass chain compact interesting value comment support improper success series give proposal need rejection sampler pn rna supplement qualitative ergodicity investigate number consider truncate explicit spectral figure spectral gap range gap exponentially gap even relatively reversible converge slowly reasonably considerably p figure stable former require simulation result obtain illustrate well behave p appropriately need region figure constant tail might infer tail indicate appropriate inference behaviour mean value kernel informative n implement parallel core device graphic processing hand exhibit biology approximate set pure markov q consumption death pure historical article straightforward simulate process inter jump exponential sophisticated stochastic observation discrete approximate use transformation kernel horizontal correspond two deviation add horizontal line estimate diag posterior sample tight strong give also run give indistinguishable inspection partial sum reveal figure estimate sampler percentile use sample sampler estimate seem correlate perturbation kernel sum practitioner estimate second use p rejection proposal one around average reasonably exhibit tail sum reveal difference figure illustrative produce chain practical determine easily geometrically common condition hold bound geometrically ergodic variance bound verify systematically course
avoid reasonable nr observe e variance ml implementation em treat nuisance major package ridge newton iterative iterative simultaneous ml restrict g information ml variance inverse estimator realization estimator denote replace say detailed section consider effect full u section variance square mse variance assumption trivially statistic exact q denote chi freedom estimate statistic empirical e version estimate covariance notice eqn consequently make simultaneous g prediction estimate statistic statistic generalization degree freedom df nan approximate fisher degree freedom denominator expression df given see degree df approximation kk variable unknown estimate natural estimator k taylor estimate estimate evaluated estimate component provide variance gradient estimate sr section q consequently denominator degree freedom eq indicator freedom unitary argue say say mse matrix source bias comprehensive solution consequently mse mse prediction simple take square get variance covariance mse get mse covariance matrix second derivative natural notice depend covariance argument blue justification however point effect additional mse expand expectation assume order ignore degree suggest estimator nan scale overview make statistical conventional improvement effect suggest effect notice derivation distribution unique solution modification consider expression simple practically environment present straightforward get covariance parametrization matrix approach development education science grant valuable feedback banach international discussion implementation assume matrix z inference leave side particular assume shall expression matrix rule inverse symmetric far whenever equal get explicit present mse linear predictor give I decompose sr ji ij base xx ii covariance property mse also component empirical covariance sr ip scale eq compare expectation estimate case general statistic however derivation suggest component consequently suggest version use variance sr ip present procedure implement iterative step solve system calculate iteratively dimensional diagonal stop ml estimation evaluate estimate asymptotic otherwise iteratively rank q log estimate fisher information estimate similarly final solution detail chapter al completeness component variance component g define system equation quadratic iv easily evaluate vector form use solution equation denote eq define uniquely unless inverse appropriate vector biased overview statistical hypothesis construction interval fix effect simultaneously mix fix mse rao research mention experiment meta analysis medical well device tolerance interval application uncertainty recognize sort mixed improper usage still problem brief fix effect conventional simple pointing appear improvement generalization detailed technical description solution mixed
broad application private add list choice initial publish know application student plan enter college year year year area mathematics computer general science year examine herein student ten preference university college college college university college college clinical speech human tr college college mi primary college college european college place allocate basis leave student seven nine subject leave six leave overall allocation degree solely basis point choice accept degree publicly leave subject minimum entry requirement score leave college receive medium part student leave fair especially gain leave explore choose coherent focus choice people pick basis factor behaviour example application table illustrate influence health science choice appear course application include degree first choice area close km year education science college application system report review recommendation concern research report examine exploratory technique without preference order profile bayesian subject student fitting paradigm cluster manner additional mining college combination select oppose college influence choice degree influence intrinsic matter look student college degree ultimately factor degree choice year college office ranking college degree herein appeal account offer preference degree discover similar bayesian paradigm infinite nonparametric operate nonparametric process item index suppose partial ranking item denote notational partial ranking construct yet pick interpretation item let arrival item let arrive ml ranking unfortunately alternative item see joint probability ranking latent exponentially result instance take joint km inference gibbs sampler algorithm occurrence ranking differ article author full ranking partial item gibb derive choice item among partial limit item observation degenerate define unobserved item sampler update item unobserve allow ranking ad hoc infinite limit capture underlie space choice item college gamma gamma marginals atomic gamma construct chapter distribution assume poisson intensity concentration know evy intensity homogeneous l evy intensity play gamma interpret correspond describe extend arrive atom arrival item constitute depict ranking appear ranking dirichlet ranking partial ranking rescale partial ranking inter arrival ranking variable give arrival g ranking associate jump evy via theorem word deal nonparametric infinite necessary condition strictly generative subtle prior simply index infinitely many natural atom location impossible priori mutually item infinite sum ill approach satisfy sure well atom purpose define explicitly observation variation normalise measure denote ranking list unobserve choice introduce auxiliary independent probability list auxiliary cf appear list appear denote unique item among list finally occurrence take partial ranking laplace moment completely formula decompose jump conjugate derive conjugate posterior process random mutually gamma k measurable denominator formula proof gibbs sampler conditional analytic integrate except leave latent versa simply joint partial ranking rescale keep proceed construction completely simple sampler generalise completely propose rank consist nonparametric augmentation derive school ranking preference dirichlet denote concentration stick breaking nonparametric describe th atomic specify atomic use draw process unfortunately share come atom result ranking mixture degenerate large consequence fine preference ranking motivate dp different atomic dependence gamma atomic connect leave marginally root gamma atomic form measure shall consist formula law mutually gamma gamma jk define degenerate number atom inspire coincide define jk jk jx jx jk addition neither since law marginally law compactly model observation interpret control indeed large large cluster ranking cluster different group construction induce atom number qualitative difference hierarchical dp law marginally gamma marginal law individual measure gamma process alternative induce atom marginal share measure control infinitely share purpose dependence component atomic dp well allow similarity hierarchical dp generalise normalised observe rank trivially develop observe occurrence item list cluster indicator similar observe atomic atom denote ranking process atom jk jk mutually process atom dependent probability mass poisson sample work sampler derive sampler mixing g scale total unobserved update backward recursion independent jk mass remain distribution allocation model finally dirichlet valid collapse gibbs however cost average scale college office year flat hyperparameter run gibbs order point posterior sampler decrease cluster matrix record show student student cluster member etc science business business outside outside business finance engineering music engineering business degree determinant student table cluster besides concerned degree heterogeneity area college see number location cluster correspond science outside inside college college science university college college business software college finance university college college university college biological chemical software computer software net university city computer application university city college science national college business college college information college science medium student pick rather area requirement determine available place phenomenon student pick law separately therefore consideration look variability student quantify normalise average normalised cluster normalise table indicate variability choice within normalised student popular teacher education member teacher education variability student student three popular matrix reveal reveal belong degree require select specify major observe fairly exhibit share name marginal respectively correlation rather college often appear top relate variability bayesian base random generative biased atom derive consist nonparametric model college cluster structure observation choose mainly gm generalise generalise distribution discrete whereas component rating flexibility capture strength generalise precision rank generalise preference cluster coherence term support primary factor reflect intrinsic matter aspect third location far degree education live home extension propose completely measure preference location however totally density leave careful carry probability gamma process formula e poisson apply iteratively use idea calculate numerator denominator already theorem numerator technique evy poisson process divide numerator denominator characteristic functional iw e iw see corresponding measure gamma note update atom locate w iw complete proof posterior sampler easily exposition process atom general evy satisfie lead infinitely law evy naturally homogeneous proof hold evy probability latent moment exponentially evy give law mutually evy intensity z z kn explore nonparametric generalised gamma beta generalise gamma form gaussian process include intensity form recover stable moment generalise gamma gamma generalise homogeneous recall ranking consist rating one partial ranking l evy intensity
close two geometric good utilize parameter cover achieve entire facilitate simple error computation proceed eliminate high peak dataset reduce input greedy minimizing use could linear harmonic currently length parameter histogram nonzero space nk x matrix tn construct entry construct transform inverse j symmetric part lead discrete cosine integer nx use analytical recurrence see appendix qx qx combine recurrence relation approximation refer chebyshev chebyshev polynomials quadrature order convert periodic making substitution serve purpose shannon kernel analytical straightforwardly present chebyshev apply rate qx c fourier absolutely qx schwarz analytic slow series also inferior convergence series still list chebyshev approximation another strategy obtain problem pca care need perform scale conjunction multiple extensively hereafter rf pca dimension learn memory multiple kernel operate convergence rate monte suffer curse dimensionality decrease generally make suppose another aspect pca bring unlabeled accuracy amount dimension pca discriminate strategy pca training fully need core high computing unable datum singular rf rather center loading time memory eigen transformation pca dimension large eigenvalue denote ki td convenient hessian regularization regression compute core td retain pca use ridge n order compute need additional involve read sophisticated perform separately l rf needs transform still term td retain load memory rf randomization square worth core square regression scale extremely well core inverse hessian need obtain scale need classification imagenet ridge baseline conduct challenge imagenet http benchmark subtle observe conduct medium compare imagenet approximation intel ghz gb core approximation cut match truth subsequently match plus segment segment well skewed carlo quite trial seed trial seed final approximating perform reduce dimensionality kernel type bag sift gradient feature via linear svm library empirically library produce context dense feature approximate optimal approximate case htb chi chi skewed chebyshev pca chebyshev direct kernels htb c chi skewed chebyshev direct chebyshev pca exp recognize image generate pixel ground truth pair art train class create database image tractable conduct multiple different descriptor scale sift foreground color foreground avoid fair processing step report high regression latter yet avoid gradient since tune convergence potentially bias average trial different htb chebyshev direct direct chebyshev chebyshev nystr om ap sift foreground show imagenet million author primarily descriptor spatial pyramid could calibrate calibration output score class htb chebyshev pca chebyshev direct nystr conclusion go conclusion go conclusion go conclusion conclusion go go conclusion go conclusion go conclusion go conclusion appendix recurrence concern symbolic software compute b observe symmetric odd argument get solve cl integration identity trick cl k z acknowledgment author thank author author like author author thank author text text text text text text text text text text text text text text text text text text geometrically rate lead segmentation besides core principal expense factor perform jointly unlabele testing datum improvement imagenet method histogram tool visual descriptor recognition utilize histogram descriptor extract use near machine descriptor score metric histogram paper derive pearson test utilize object performance current datum million thousand comparison often facilitate testing approach devise represent two top fourier kernel approximate transformation nystr subset comparison follow principal pca represent nystr hard functions regardless able nystr om partial rf metric paper elementary enjoy term error previously directly translate classification rf kernel analytical chebyshev slow linear chebyshev derivation paper record research rf generate theoretical carlo eigenvalue fisher kernel imagenet large classification influential dimension consume nystr hence crucial max code spatial informative descriptor whole image undesirable recent propose successful semantic
validate shaped spatial cluster detection empirically method spatial scan detect spatial point similarly straightforward limit want proportion boundary p zhang foundation international partially identify shaped ratio measure discovery spatial discover shape heavy computation large many detect intensity compare specifically collect location follow alternative region location example scan scan comprehensive multidimensional develop spatial scan scan scan windows location test window window cluster modify base spatial scan methodology powerful power circle ellipsoid approach issue control false discovery fdr shape scan every individual region spatially method extend adjusted discovery take spatial make test maintain detect shaped method internet detection idea form spatial novel variability test threshold help maintain specificity comparison fmri scan multiple testing fdr identify spatial organize derivation binomial poisson normal validity study approach fmri discuss variable hypothesis alternative objective success assume resolution detection expect region location maintain balance specificity sensitivity detailed formula statistic distribution justify set discuss idea location presentation column sequence I vector aggregation likelihood theory likelihood regularity unknown denominator instead derive parameter distribution binomial shape square circle square illustration calculation next independent straightforward introduce notation leave detail k ij easy straightforward adjustment eq sure small hypothesis median mean hypothesis hypothesis formulae approximate test eq alternative ij statistic fdr introduce location variability choose cell five point variability circle circle circle circle mm mm circle circle circle circle mm circle circle circle region close identify average test consecutive threshold monotone region increase region exclude variability similarly variability threshold average neighborhood high carry location test statistic measure arithmetic k ts illustrate certain condition surely region inner non boundary total spatial region boundary black fill gray rectangle fill gray black draw rectangle black fill gray black draw black draw draw fill gray black gray draw fill gray black gray presentation show normality signal strength proportion theorem give justification outline statistic relation remark statistic purely region large threshold substantial part one boundary boundary variability good chance scale large depend method tend show scale scale even paper simulation aggregate method region likely false region scale follow use window simulation single control fdr scan save manuscript binomial example provide specificity simulation times shape region shape shape datum cccc shape shape percentage paper real location survey application signal success method measurement include specificity percentage sensitivity percentage pixel average number show alternative increase sensitivity specificity variability increase variability specificity increase dramatically meanwhile sensitivity increase cc cc shape shape specificity sensitivity specificity specificity sensitivity specificity std std std std std std std cc shape c triangle specificity specificity sensitivity specificity specificity sensitivity std std std std std cc shape triangle specificity sensitivity specificity sensitivity specificity specificity std std std std std specificity select control discovery use false sensible level lead sensitivity show specificity decrease sensitivity alternative specificity scale specificity spatial triangle performance three shape sensitivity increase even shape specificity spatial scan complicated detect calculated frequency detect signal weak chance detect region probability map shape almost slight effect corner shape identify provide detect adjust shape correctly identify identify relatively small strong signal figure similar map detect scan default circle scan excellent shape bad shape shape subsection window adjust alternative success local variability adjustment
bring series na complex interaction achieve purely na case still probability empirical attribute dirichlet implement partition combination depth criterion identical attribute whole classifier decision letter splitting classifier member cause give vector tree bag show classifier replacement change index object include thus build subset binary cope one fit introduce reliability tree view non attribute split great comparison threshold attribute possible category cart include effectiveness usually measure retain stochastic implementation correspondingly attribute category except two contain generation split classifier gain split building way subtle interaction usage attribute beneficial accuracy score class become less contribution final increase imbalance reach point whole vote large develop assumption equal rare cope enforce divide count leaf object bag obviously misclassification might raw score voting package function require pass version explicitly pass predictor frame interface fit find class aside even object false model confusion prediction specie truth add raw score r bias test external reliably assess accuracy bagging sort internal train prediction build classifier object e subset idea originally use trivially bag random calculate object confusion forest access raw prediction data score appear bag prediction use ensemble also bag moreover importance information misclassifie importance attribute attribute also attribute evenly even marginally relevant attribute ensemble call process element model importance length width diabetes cccc forest forest cv random forest tree error deviation repetition testing build range next minimal verify subset compose classifier performance assess repeat time cross validate algorithm forest good optimisation target depth selection give lead suboptimal significant approximation forest yet show practice variability forest almost use attribute certain attribute importance cc speedup speedup importance mean repetition consecutive triplet region sequence align lie st describe signal model spectra thus attribute signal interval frequency band class peak importance obtain case expectation attribute site location score band interval differ qualitatively agreement forest model completely possibly high code repeat make training I collect time certain object scale object forest importance fast even slow complexity usually
euclidean together q proof lead desire distribution quasi minimax optimal contraction result greatly work focus scalar difficult naturally vector contraction situation dimension another interest restriction explanatory resp principle conditioning ill one conditional finitely mass continuously estimate observation analysis brevity direction future research note resolution quasi approach investigate bayesian typically frequentist implement usefulness bayesian able avoid finite frequentist analysis instrumental model nonconvex difficult obtain guarantee globally optima interest result nontrivial since estimate pose stochastic major author department economics mit suggestion constructive comment associate anonymous improve quality assumption aid aim develop quasi instrumental asymptotic generate induced gibbs prior slowly derive contraction bernstein von type result distribution greatly triplet scalar random instrumental much may unbounded instrumental variable structural interest form potentially form statistical challenge difficulty ahead attract interest structural ill pose equivalent operator identification though hilbert schmidt integrable value pose roughly method involve assumption however frequentist perspective theoretical bayes quasi bayesian posterior approach bayes neither put prior likelihood quasi moment estimate moment nonparametric quasi put value formally call gibbs proposition ahead quasi build interest finite grow dimension role ill choose basis space wavelet basis treat smoothness old space unified series setup condition contraction attain contraction state standard asymptotic normality view bernstein von bernstein von parametric quasi posterior expectation attain finally specific quasi bayes estimator attain minimax closely work form assume gaussian gaussian conditionally posterior study distribution clearly present largely differ fs speak tie tie slowly largely different present estimate comparable paper unified framework scope detail prior differ conditional restriction unconditional moment restriction restriction restriction approach importantly neither rate cr derive contraction comparable formally derive important distribution notably contraction structural contribution paper considerably asymptotic property bayesian bayesian main construction suitable ill alone sufficient contraction additional theorem analysis ill pose stream research topic mathematics literature however substantially finite error prior rate fs fs fs result prior hence contribution build upon bc b bernstein von cover account none paper inverse pointed condition relax identification approach organize quasi bayesian main contraction normality technical limitation contain scalar z distinguished population transpose ratio let denote usual lebesgue denote let function norm denote denote singular value outline quasi say older contain conditional equivalently robustness assume proper available instead use quasi let n candidate quasi maximize structural however unknown instead suitable series consider use quasi proper put shall prior grow consider another material basic say approximate space become prior subset span prior widely quasi call quasi distribution quasi proper contraction posterior intuitively posterior rate contraction correspond distributional normality framework broad statistical page foundation gibbs open line research interpretation quasi posterior borel suppose q divergence proposition show quasi minimizer posterior posterior optimally average measure kullback leibler constant typically see sake take positive reduce provide estimator quasi integral attempt restriction argue empirical interpret posterior quasi characterize purpose space concentrate estimator form basis begin state b generally add file fix sufficiently j wavelet appendix basis notational convenience convention span denote j property basis base modification however wavelet suit periodic p c g thereby quantify ill ill pose definition mild ill severe ill pose density analytic ii example adjoint eigen basis case ii trivially assumption ii self adjoint slack study convergence past denote satisfy minimax rate pose moment bind pt replace c determine class readily possible assertion borel construct put exist converge pt importantly contract rate page line hence sense fast possible ill pose pose ahead indeed contraction quasi baye fouri two assumption positive satisfy sufficiently cn ns nb nb I suitably bind counterpart ill nj abstract section g p ng b ns contraction arbitrarily satisfied constant ill attain attain contraction bernstein von type radius prior p contraction j r fully satisfactory appearance ball state page dimensional proof work effect n j suitably devise contraction w establish sufficient condition condition ii see quantile think process lebesgue measure pt w bernstein von result state quasi normal center type approximate reason add nature quasi model center coincide approximate bayes satisfy gx c file thm page typical goodness estimator hellinger uniformly bound need three typically relative indeed convergence n estimator attain rate contraction convergence isotropic prior choice notational think ill suppose isotropic lipschitz qx b rx rx e kk take product isotropic sequence n file class prior construct grow lead minimax contraction rate isotropic know arbitrarily case isotropic convergence proposition case take product isotropic appendix file proposition play role grow thereby abstract cover deal allow place prior shrink grow proposition method slowly grow upon request supplementary material approach sharp contraction lemma lemma preliminary contraction proposition notational convenience technical characterize variation center multivariate increase dimension j p ir definite ok eq quasi contraction rate take generate theorem dependent empirical sequence q understand context tb ib last independent posterior distribution eigenvalue likewise event construct lemma iv db c n n nn n
problem link collection relation role link limited link prediction reveal type connection tensor jointly address latent factorization bayesian overfitte chain monte conduct world demonstrate improvement exist art relational become problem analysis link concern predict unobserved link pair observe network link via explicit relation membership like share interest develop treat ignore dimensional interaction predict multiple relation relational network define overall object pattern object illustrate left figure social network among link multiple unobserved indicate task miss refer link pattern fine network relational capture improve include therefore factorization consider factorization address challenge ignore previous challenge factor characterize object feature latent factor relation capture correlation among reveal distinct quality sparsity usually cause sparse employ treatment placing handle moreover parameter conduct world relational prediction accuracy effectiveness follow briefly introduce problem fully carlo experiment conclusion link entirely network factor stochastic relational link extend factorization model probabilistic task concern strength link literature link task link address several enable jointly collection feature entity object type multiple impact relation prediction factorization attract lot attention mining community use web email communication tag tensor present propose multi infer factor object tensor multi pair another status involve type relation pair tensor observe information relate relational predict link show link link pattern factorization observation factorization factorization unobserved assign latent pair object relation learn three factorization modeling introduce generalized term part tensor mean cp tensor factorization incorporate logistic map miss obtain rewrite sigmoid somewhat prediction datum tensor usual bayesian impose independent gaussian latent factor type latent place place predictive unobserved link pattern condition relation interpretation modeling link pattern specific factor correspond different example company less interact type distribution average constant value hyperparameter model minimizing follow regularize error factor inference tr predict link simultaneously however limitation map choose average ignore parameter limitation manually validation treatment estimation large solution employ distribution hyperparameter predictive observed generalize miss model multivariate consideration hierarchical bayesian conjugate precision precision inverse latent still latent multivariate subsequent wishart precision wishart degree freedom relation w weight indicator contain representation follow denote conjugate specify tune hyperparameter change bayesian next predictive equation resort predictive draw whose proposal gibbs variable parametric gibbs partition sample iteratively initial apply derive predictive conjugate make efficient procedure rule condition observe sample follow gaussian factor distribution form latent hyperparameter k p ki nu u type influence object still parametric form q conjugate gamma hyperparameter simultaneously conjugate form wishart form latent mean denote hierarchical relation choice complexity efficiency eliminate adjustment distribution initialize sample desirable discover pattern real multi discussion examine propose behave world relational country relational social consist extract relation among relation evaluation construct tensor country dataset international multi relational sharing site interact video network construct propose link prediction bayesian probabilistic gibbs latent factor factor obtained compare art relational nonparametric relational object latent relational slice tensor implement report examine relational experiment e set use auc robust test link repeat report result model auc performance model dataset posterior sample good show
regularizer assign noiseless noisy scenario snr scenario denote height output initialize iteration block terminate threshold art basis inner loop besides omp noiseless situation pursuit index mean noisy rate rate define successful successful trial experiment trial experiment ram sparse situation show rate number identical vary zero block intra varied trial fm fm omp speed block five locate zero block generate uniformly snr estimate support fig although slightly advantage well fm fm even outperform fm fm intra block wireless body area network health branch compress raw binary compressed sense energy consumption algorithm ica fidelity clean decomposition iii sense column compare recover cosine dct accord basis reconstruct original raw accord measured fig fm slightly recovery fast speed fact fm clean decomposition ica experiment notice ica recover reason coefficient sufficiently sparse recover propose life application outperform close fast among algorithm fig liu fan fu technology recognition laboratory national china com sparse measurement structure signal exploit intra sparse show fast derive much scale sense sparse intra marginal likelihood recovery signal find rich structure structure widely structure refer case nonzero cluster location information block zhang rao sparse superior recover block fast implementation likelihood thank framework structure intra block correlation conduct synthetic exploit omp however much throughout bold symbol compute build follow mean suggest gaussian deterministic confidence block capture bind optimization extend bayesian compressive sensing index rewrite I rewrite ii rule relevance structural investigate eq unity note require noiseless global cost global correspond regularization converge minima largely reduce although regularization correlation recover signal transform via transform coefficient may zero block regularization fm ar
change dynamical iteration method utilize study converge see utilize change text eq eq static horizon nothing show dynamic evolve standard euler simplicity euler approximate taylor become graph compute initial k ht propose method expense product linear could use set sense euler simply method study change flexibility beyond brief beginning relationship time hour day implication time wikipedia hour hour forward euler convergence change hour essentially equivalent converge minute iteration power precede change interval base smooth jump average investigate time time give rise reference extract score instantaneous rank summary average experiment rank node occur intuition normal page large topic news popular time figure american score frequently help euler series trend omit wikipedia month twitter euler period unchanged wikipedia copy database page wikipedia page page page remove wikipedia page hour raw page wikipedia graph aside vertex indicate degree dynamic page degree page visit graph generate seed extract tweet tweet represent use tweet vector represent tweet month dataset demonstrate effectiveness adapt page external provide insight page help intersection top j operation identical wikipedia similarity static cumulative measure combine external appear produce something top page reveal ability change clear find actor medium result demonstrate effectiveness identify page external page magnitude cluster series page similar trend page share pattern amount popularity tweet temporal adapting dynamically perform step ahead move average specifically exponentially series twitter twitter validation predict percentage tweet predict dynamic evaluate informally time time shift bottom approximately stationary compare across time series information forecasting evolve external importance treat incorporate change utility use predict evolve graph external evolve adaptation capture change external interest influence node generalize converge external stop effectiveness evolve twitter external page important graph variety science bioinformatics many study node something execute parameter model pick node depend chain stationary govern graph variation instance study dynamic graph closely utilize quantity
paper mention derive e provide approximation conditional moreover mn n recursive formula matrix define k tt taylor expansion diffusion neighborhood symbol respectively satisfy eq one go reason asymptotic theorem realization sde satisfy additive reduce emphasize evaluate one computational viewpoint efficiently van alternatively subspace case de preserve alternatively formula equation make evaluation predictor determine discretization prescribed error tolerance performance estimator exact order estimator adaptive histogram limit observation different distance length equation estimate respectively estimate initial value addition van pool respectively multiplicative parameter first illustrate moment simulation error approximate moment exact one table rate realization euler linearization noise simulation time take way time allow explore period interval distinct estimator set exact conventional large parameter signal sampling reason drift provide estimator parameter period increase estimator shape c c l l ccc r r r average estimator nh respectively compute ii estimator calculate specify estimator histogram estimator compare figure decrease u negligible adaptive usefulness available c cc r cc r table order r cc r r r vi equation conventional example order h respectively figure histogram confidence example estimator negligible thin show bias approximate conventional unable illustrate usefulness shown happen insufficient number adaptive accept acceptable explanation accept step accept ten modification method estimation complete moment scheme time observation asymptotically linearization scheme performance simulation simulation order satisfactory stepsize discretization observation decrease higher adequate reduce complete distant estimator also sequential numerical simulation international centre physics thank support work diffusion theory inference time vary covariance weak differential equation correction diffusion process business statistic diffusion method http http continuous linearization time volatility financial market linearization bit discrete noise letter linearization space simulation differential equation linearization comparative equation normality system estimation overview review third identification application water nonlinear dynamical system linearization statistical process comparative stochastic process time nonlinear stochastic equation versus kalman taylor dependent multivariate example compute length c example compute example sampling period example period interval c period n axiom conclusion corollary criterion exercise theorem remark pc pc e mail modification conventional quasi method diffusion convergent moment result approximate likelihood asymptotically distribute bias mention enhance estimator intend situation observation distant process currently intensive basic except analytical decade method focus grow literature rao al deal estimation series simple estimator derive additive discrete typically euler rao linearization approximate number increase method observation enough estimator display performance due simplicity modification reduction useful base euler whereas taylor expansion result specific sde demand second affected expansion propose estimator approximation moment mention observation completely observation bias finite observation modification conventional diffusion orient converge two consecutive sample approximate diffusion decrease asymptotically approximate detail new approximate enhance estimation give distant typical practical example local linearization present section implementation increase right continuous family process function mm smoothness existence uniqueness bound process kt sde quasi denote diffusion asymptotically quasi maximum likelihood paper rao recently k k k kt pseudo prediction inferential consideration want contrary time impose completely difference quasi continuously growth equation approximation weak sense positive definite choose quasi sde et sense mh stepsize accord construct likelihood euler linearization numerical conditional quasi rao left side give pair discretization go clearly include particular since design term first conditional weak imply estimator asymptotic go next deal time sde realization sde sde k identity h h k h jt moment imply first sde sde follow underlie sde sde mh assertion exact zero control criterion clear assertion mh sde decrease deal sde observation sde maximum expectation probability realization sde sde trivially mh mh space sde sde assertion complete worth approximate convergence state order restriction order close time measurement restriction consideration designed quasi study eq generate sde sde index error loss minimize
use multiple target retrieve via pathway pathway classification reweighte recursive elimination select pathway associate phenotype cutoff improvement auc compare test confirm via hoc reveal auc far individual figure stability b stable comparable schmidt breast cancer interestingly consistently show assess et behavior compare network method retrieve signature subsequently relate drug target signature associate knowledge previously interpretable signature test much affected integrate disease restrict target candidate disease additionally integrate gene target signature show association surprising rank disease filter diagnostic google gene expression absolute score show often disease phenotype significantly classification signature stability dataset moreover yield signature clear additional integration disease gene could enhance nonetheless improve performance signature stability appear effective integration disease disease significant computational might candidate consume machine algorithm mechanism include school like provide gene area roc curve figure fraction cv stability si signatures disease gene know drug effect integrate protein signature gene overview dataset cancer year schmidt breast cancer year survival cancer non cm fr gene signature towards well decade purpose important obstacle make signature biological purpose grow integrate molecular protein selection discovery compare reveal well signature moreover disease information candidate disease target decade gain goal reliable molecular patient effect avoid reduce drug cost diagnostic signature copy entity signature gained take diagnostic signature group patient predict patient forest combination nn retrieve gene signature interpret recently try pathway annotation interaction review fr hope make signature stable signature diagnosis centrality protein interaction differential expression association disease cancer rule error svms rank construct conceptually previously hyperplane pathway pathway extremely signature investigate gene target disease gene candidate gene breast dataset repository cancer normalize normalization carry clinical consider breast cancer cancer clinical treatment residual except et assign describe al protein denote consist differential gene suggest protein web page link expression gene node apply filter rank mapping use rule upper cutoff margin disease gene disease compute employ candidate proximity annotation space test combination disease gene rank well go go information information light go propose disease annotation addition rank gene rank call major tf target associate fdr cutoff target conduct disease tf gene bind human sequence gene retrieve
different change prescribe normal estimate quantity unknown magnitude vary procedure estimate change stop detect time window length residual detection choose alarm characterize average time delay detailed accordance detection detection run upper eq mean unit pdf cdf normal accurate even exactly change target without derive statistic I ask ask good statistic use approximation numerical quite provide convergence challenge characterize aspect approach early multiscale closely geometric characterize favorable multiscale particular magnitude coefficient asymptotically datum lie well depend curvature leaf node approximation result suggest approximation number estimate appendix complete show projection close counterpart miss omit denote notation coherence basis white gaussian constant q bind bounding theorem theorem show first first demonstrate detect tracking intrinsic space vary change method average metric denote distance parameter figure horizontal vertical percentage well fraction fairly consider ease study observation tracking demonstrate able coarse tracking slowly vary intrinsic identical tracking http figure line union sign past sample color recent dynamic keep parsimonious curvature track maintain stable leaf meet study intrinsic ambient signal phase element correspond spaced change entry tracking compare intrinsic fig demonstrate dimension however small error significantly force use tolerance approximation assumes distribute gaussian close numerically verify generate mc trial track apply evaluate theory different comparison mc miss mc miss mc delay jump subspace delay magnitude delay threshold employ threshold detection delay track delay delay subspace subspace single subspace c single demonstrate http edu l display define minimum pixel streaming state anomaly figure background image shape make detection base background detect ease intensity consistent demonstrate detect present accomplish far even suited change detection around yield occur c peak statistic tracking less reliable significant change statistic automatic idea typical numerous detection examine competition person transaction transaction dimensional transaction bad dataset generic anomaly detection generally transaction particular track detect label exceed recently detect person transaction isolate transaction large spike show mark near threshold repeatedly repeatedly stationary paper describe novel method online tracking tracking perform fast important subset ever analyze multiscale heart approximation manifold update manifold estimate adapt change curvature dynamic potential play volume stream arise monitor traffic paper result approximation correspond dimensional many relate dynamical change alternative pose interesting problem restrict subspace cost recall assume estimate hence term bias condition expand write since equal zero since ab minimize optimal tend second tend online asymptotically restrict sample projection u identity hence asymptotically uv gaussian vector matrix consider datum projection u hence w denote du u eigenvalue xx dx dx u upper sufficiently noise statement ex email edu email electrical engineering describe point scale detect data conventional address challenge model dynamic lie dimensional ambient streaming use track measure deviation series statistic deviation detecting change sharp dimensional include subspace tracking multiscale point online analysis experiment highlight efficacy detect otherwise slowly dimensional manifold anomaly change accept change soon stationary away dimensional result challenge wish quickly change social surveillance video structure traditional change method deal scalar impractical high accurately delay false alarm poorly develop extract restrictive assumption address univariate change high lie close ambient model non test reliable multiscale efficient calculate element miss literature manifold remain challenge batch design observation lie lie accept sequentially change significant rapidly special appear parallel tracking least incomplete manifold negligible curvature union core basically track underlie observe generate kde naive variant quite computation poorly stream datum hoc face computational allow spatially bandwidth increase additive assume work applicable broad align manifold kernel open efficient kernel shape kernel adapt time connection reduce preserve challenge challenge setting vary merging learn precise gmm consider gmm impractical high additional assumption ill pose setting geometric resolution subset encode straightforward wide surveillance modern collect massive video stream analyze volume collect huge quantity motion multiple connection cause failure phase activity mind detailed expert essential reliably detect salient scene detect explicit anomalous surveillance track motivate alarm evolve network detection attack characteristic network traffic characterize rapid primary contribution present online tracking analysis experiment organize describe multiscale statistic change theoretical numerical suppose ambient measurement lie white random slowly time track vary allow tracking residual varie slowly change projection euclidean norm simultaneously without problem determined specifie might project ellipsoid projection numerical mahalanobis data quadratic covariance mahalanobis cx however distance lie curvature near collection well mahalanobis hybrid mahalanobis subspace let projection coefficient residual eigenvalue mahalanobis approximate u u motivate mahalanobi eq complete side approximation avoid numerical instability cause divide mahalanobis distance measure distance x definition distance mahalanobis distribution available subset update close virtual child preserve c calculate structure new minimum accord virtual calculate parent close virtual algorithm index calculate mm du turn virtual initialize virtual parent leaf virtual create mahalanobis distance sample alternatively close virtual near computationally attention greedy readily however nearest tree virtual center whether tree residual eq structure prescribe tolerance approach update subspace tracking reason alone shape fu orthonormal offset instantaneous derivation update size subset close estimate write ignore accomplished solution recursively denote row order recursively guarantee orthogonality onto approximation obtain schmidt require extra cost continuity frobenius subset lack continuity track fast
total time multiple regression regression set eq avg avg vector note avg via opt opt opt avg opt opt opt compute leave derivation opt opt avg avg opt avg avg opt avg opt follow regression still complete transformed block repeat copy identify important identify multiple minimize opt involve need somewhat costly sophisticated unconstrained frobenius rank opt opt rescale opt section theorem construct eqn need compute approximation ratio k bind matrix construct problem eqn svd low approximation lower achieve perhaps sophisticated agnostic approximation ratio need whereas carefully target also present heavy fact simple lemma opt approximation last property pseudo full opt conclude opt rescale proof theorem n run time svd svd compute svd opt rt svd svd satisfy second opt opt opt agnostic carefully guarantee agnostic agnostic appealing rescale line briefly agnostic construction construct rescale opt solve eqn run compute vector rescale opt opt hold agnostic column whose except sequence look algorithm probabilistic guarantee previous bad bind exist randomize randomized may depend agnostic depend agnostic construction integer orthonormal whose theorem appear select last r opt theorem ratio regard conclude success agnostic unconstrained multiple bound r satisfy kk whose k kk observe opt integer integer exhibit frobenius span truncate svd construct bind suppose give rescale eq opt open size guarantee simple linear conjecture tight certainly size zero give air force laboratory support nsf dms nsf decomposition svd contain contain singular pseudo inverse qr factorization frobenius spectral norm frequent fact n provide proof lemma result lemma optimize spectral property matrix slightly abuse sample consistent v r multiply technique select column describe convenient matrix time satisfy achieve exhaustive total r procedure nr exist deterministic svd svd svd q take inequality inequality deterministic procedure svd let take lemma prove corollary constrain regression square ask whether suffice construct room improvement linear area robustness error contain essential yes vector meaningful full complex solve significant construction multiple addition identify considerable huge incremental approach distribute costly essential constructing arbitrarily achieve performance regression fit response sophisticated community discuss point feature raw q positive opt x least hold datum possibly weight suppose minimize construct also provide paper switch equivalent formulation background similarly element effective c depend seek say various formulation linear c pt thm unconstraine response eqn pt unconstraine agnostic thm notation dimension row response ratio constrain single eqn eqn thm frobenius unconstraine frobenius thm unconstraine response agnostic ratio opt opt nk multiple four opt constrain describe construct size achieve constrain randomize arbitrarily bad despite logarithmic provide concentration concentration break response seek minimize series row seek exactly frobenius norm regression column response seek vector trade quality writing equivalent regression construction multi theorem describe size k response frobenius spectral norm case theorem present size note algorithm show construct
say feature feature detector detector high low entropy detector high low selective svd mlp see b mlp omit generator initialization architecture mlp detector generators appearance singular detector fact norm detector feature generator also feature detector feature generator detector diverse strong correlation feature detector full spectrum mlp rank basis random mlp able patch cc mlp variance image observation htbp code hide binary surprising activation prior application normally distribute create plausible support detector norm feed noise mlp layer activation indeed binary binary activity detector htbp htbp regard behavior mlp explain mlp observe figure feed compare apply input responsible responsible effect mlp denoise reduce thresholding domain wavelet operation typically leave unchanged fix value zero absolute effect denoise always trivially achieve question preserve feature version feature figure input detector feature detector value unit look create clean noise present layer eliminate repeat activity activity hide layer absolute small conclusion feature detector high activity one feature detector high summarize denoise image preserve high autoencoder denoise autoencoder stack multiple sequentially layer provide layer optimize performance initialization supervise suggest wise contradict suggest deep architecture deep net gradient difficult abundance generalize phase impose restriction stochastic local fall gradient descent activation almost completely agreement regularization force hide restriction regularization information input preserve input preserve virtue denoise reconstruct preserved question well classical achieve form indeed useful parameter weight activation classify vector activation mlp rbms deep net stack autoencoder feature detector rbm visible learn appearance learn cc activation pre training supervise real unsupervised learn real activation activation mlp logistic sparse train deep layer patch detector maximally feature many mostly location image sense high remove detector activation see activation fact explicitly denoise fit regularization denoise easily interpretable achieve output layer patch output identical denoise autoencoder generator study mlp architecture generator mlp notice generator look mlp single look perhaps seem difficult intuition learn layer mlp mlp hide detector generator look detector output detector still two mlp layer two mlp interpret lead denoise htbp detector htbp detector output mlp clear detector feature generator identical corresponding feature detector lose layer separate basis detector answer single unit pass mlp input mlp provide question answer detect activation effect mlp layer hide correspondence detector mlp effect mlp detector high look clearly generator basis detector activation c name db couple db db hill db db db db form mlp image well patch arbitrarily well difficult code allow answer try image form word try patch clean slide region patch noise see image perfectly though image slightly conclude last approximated related figure full rank upper spectrum flat mlp layer imply diverse image mlp mlp db db db couple db db db hill db db house db db db db db db db db db image typically refer pseudo problem omp patch denoise slide averaging overlap ask mlp combination sparse image mlp dimensionality column approach denoise approach dictionary hide serve mechanism create code pattern maximize patch absolute activation neuron third hide neuron third activation layer activation layer input cause activation activation ii evaluate activation activation maximization patch neuron save image contain maximization well large patch pattern mostly center intuitively make covered fall away patch filter layer layer observe output exhaustive patch case clear patch hide layer detector look feature detector useful look less answer behavior set detector remove used performance replace detector htbp detector detector iterative choose mean detector detector detector row detector yield good interpretable yield detector yield yield bad look detector generator train make observation strength noise detector generator htbp detector generators detector generators generator look similar detector detector cover patch away agreement strong patch result provide explanation unnecessary large patch explain achieve result small see detector generator detector feature generators detector feature generator figure generator respectively type horizontal noise detector feature detector output generator somewhat affect visible noise generator sometimes type learn autoencoder mlp block generators detector learn mlp patch patch correspond neighbor patch adjacent patch filter reference surprising update connect neuron neuron input update show state image paper possible procedure part gain insight unit train vary size observation make conclusion regard denoise hide iii go increase architecture finally fine lead deconvolution address might also benefit problem benefit briefly feature noisy copy patch activation observe mlp internal work maximize activation often remarkably similar output cause true rise rbms force representation interpretation denoise autoencoder binary representation give rise representation rely sparsity deep architecture recognition also representation require criterion make argue representation obtain representation criterion image rbms denoise autoencoder noise type noise mixed noise achieve avoid activation image denoise patch achieve denoise denoise way toward gap discuss paper explain technical achieve good patch make denoise problem difficult require consume modern somewhat neural especially gradient descent sometimes science exist training lead poorly attribute initialization especially time consume understand lead good box merely observe output logic usually inspection certain interpret human hide maximize procedure qualitative train varied set encountered training initially degradation phenomenon undesirable explanation phenomenon gain insight mlp difficult hide layer activation insight denoise mlp hide contain patch patch input patch input procedure mlp average refer whereas refer patch denoise show art denoise show tracking process mostly many day time modern small training set one use dataset train mlp architecture reason superior test refer opposed patch denoise denoise denoise area patch test still albeit slowly briefly become bad overfitte abundance zero issue use either size hide patch unit see add wide layer beneficial mlp patch hide train patch five four beginning decrease later mlp achieve degradation performance difficult landscape mlp layer effect figure deep narrow difficult network good ask question image train architecture set either gain image full training degradation figure layer patch increase lead patch bad patch patch still degradation still approximately later degradation explanation size difficulty patch stochastic may fail input patch ideal architecture patch difficult remove size expect bad question happen patch patch slightly architecture keep vary patch size point degradation patch input output explanation good show patch layer create degradation combine patch layer degradation patch layer quickly degradation however architecture patch degradation performance comparison conclude deep narrow optimize hide layer beneficial input patch may procedure layer hide patch attempt least fine procedure initially switch learn suppose encourage low suppose htbp show indeed fluctuation addition test turning obtain achieve bm patch patch layer size architecture four never bad approach bm fast fast slightly outperform bm slow explain training well achieve patch reason achieve large patch become strong weak indeed patch large cause difficult ideal influence strength patch block lead mlp search window four progress progress use block match plain particularly begin advantage plain evident perform size window well mlp block mlp use plain achieve compare progress win test particularly begin begin procedure procedure helps regular rather match procedure size block without match procedure block later stage superior present achieve plain mlp stage achieve noise level patch employ step bm achieve medium noise important avoid achieve ask question insight mlp work mlp
knowledge series chen rely specific algorithm splitting approach transform convenient reason generic spectral short specific unclear specify spline average sum plug infinite must error sum suppose decay slowly number explain fourier bound simply discard value memory follow directly process modification method idea correction il il il il il g du du I admit cubic rather interpolation interesting spline otherwise grid give essentially fourier range q simply decompose fouri two fourier correspond noise close fourier term differentiable since mean quadratic computed point aspect next idea common theory memory reason explain introduction posterior assign weight eq approximate certain spectral rely heavily parametric class true notation great generality certain statement term say go infinity implementation simplify ignore uncertainty simplifie observe particular converge fast feature quadratic denote coefficient toeplitz obtain evaluate quadratic explain typically rely expansion refine toeplitz one obtains easily adapt far reduce rewrite dataset fourier need value cost typically perform monte practical ignore operation spirit nearly independent rigorously speak establish valid true merely indistinguishable develop section speed rather respective langevin discuss carlo sampling approximate discussion see k step walk conditional birth reversible jump sampler death birth complete death walk sample birth constant k death step many basic strategy could consider variant two behaviour parameter scale walk tuning parameter shall manually principle form mcmc review past distribution however less learn address difficulty comment refer difficulty explore modal posterior tend mode may paper multimodal posterior propose spectral dot dash fit fourier hard distinguish except unless large multi mode may correspond true refer reader one mcmc spectral solid line line line coefficient describe generic smc sampler backward detail former must easy smc operation sample multinomial assign value markovian kernel anneal geometric bridge discuss conclusion alternative may sequential adjust dynamically coefficient solve eq take default equation sampler birth death advantage particle motivated hasting step perform move observe phenomenon increase lead improvement case repeat close variance correspond particle seem bring illustrate phenomenon seem obtain slightly equal formal suggest investigation context sampler instance one use reversible step currently particle sum acceptance birth death coefficient run algorithm cpu second correction star compare sampler mcmc simulate give time minute sequence smc run start set pilot run acceptance seem alternate mode seem poor trace middle plot visit satisfactory also reversible jump burn period proposal calibrate past posterior significantly chain auto correlation respect post decay posterior range one mcmc scatter add trace five nine mcmc bottom plot post burn posterior evolve grow goal property procedure effect correction step marginal resp confidence band light grey grey true one correction concern marginal vanish sample correction particle grey dark grey band spectral spectral cpu smc unless sampler expensive dark grey plot bottom row fortunately correction see prior smooth density inference give previous section spectral top plot marginal moderate impact spectral essentially nuisance critical inferential purpose interesting difficulty observe sampler figure mcmc trace band bin plot bottom prior preliminary methodology either inferential view seem detail author literature rarely g temperature seem additional work practical package record per time convenience dataset side although empirical biased true bias estimator quite frequentist parametric return suppose maximum likelihood return plot spectrum order see side find strong evidence long marginal left panel mention briefly thing correction negligible smc sampler little variability marginal weighted smc band dark grey corresponding order dash work say could adapt little parametric point sequentially course dependent distribution provide sequential sample could smc distribution likelihood lead cost reason far thus interested instead smc sampler acknowledgement comment j conditional coefficient distribution belong regard admit representation l enough q calculation n p tf idea show lf lf l lf f step n supplement f supplement n n p control lemma supplement fr n supplement lf nj j uniformly properly let n successively lf n term note supplement term n supplement second kn kk without loss generality positive covering ball nk k nj enough q k nk ok nj integral separately lemma supplement supplement q combine supplement together set go replace abuse notation ks n cn cn obtain precise controlling terminate theorem theorem often sparse nature propose recover importance show importance vanish go posterior modal sequential sampler annealing evaluate world semi spectral reasonable counter several assume observe toeplitz vary specification separate behaviour short certain choose semi run methodology take short expansion finally spectral density exist computational difficulty arise bayesian model expensive compute entry reliably quadrature feasible metropolis reasonable context since pilot tune process implement sampling green reversible hard tune spectral possibly memory vast parametric approach rely commonly use frequency paper hypothesis bayesian inference finite frequentist necessarily entirely satisfactory reason frequentist discard see parametric provide unstable inconsistent misspecification satisfactory provide frequentist propose address manner discuss likelihood hardware propose fast likelihood scheme motivate step monte carlo constant front intensive sampling however multiply front small put thing many discuss literature discussion importance discuss sequential
privacy basic place main begin impose certain family consider base differential corollary privacy proof result smoothness usual notation g illustrate multi median eq lipschitz respect belong classification loss set loss compute verify natural communication strategy amount maintain proceed review problem form outline essential stochastic information iteration receive appropriate mirror descent enjoy defer privacy subdifferential corollary information privacy apply assume notation definition take important precede paragraph optimally locally private contain ball radius geometry function perturb belong ball notation constant unique achieve loss proposition precede paragraph optimally private theorem yield minimax focus second result specialized analysis previous analysis necessary appear require restriction actually indeed domain establish consequence corollary ahead perturbation theorems stochastic mirror obtain section theorem sense mirror additional turn result suppose ball contain radius state result section alternative computing definition setting differential communication theorem quantity differentially interactive begin let collection differentially definition differential corollary differential applying mirror assume privacy result interactive privacy long optimally private guarantee minor error interactive conditionally eq infimum mechanism method work together find subject apply class lipschitz loss state minimax restrictive simple optimize privacy assume channel differentially private universal domain optimization loss restrict small captures functional corollary sharp factor note theorem corollary dimension substantial theorem inequality characterization private minimax tight relate set dependent query gradient depend obtain convex theorem method set open privacy play allow noise achieve necessary provide sampling precede privacy increase corollary roughly rate privacy comparison divergence scale roughly though theoretic privacy explore give sometimes sufficient compact distortion channel conditional privacy distribution wish capture locally private optimally differentially locally private attain begin characterization reasonable focus suitably index unitary set extreme minimax saddle polytope constraint unique uniform attain saddle brief remark somewhat deep understanding theorem attain saddle maximize still obtain privacy maximum theorem characterize minimax ball maximum entropy attain certain proposition discussion section binary almost surely coordinate slightly understand scaling concavity imply bind tight complicated theorem characterize saddle mutual ball variable support replace proposition somewhat sampling accomplish define mutual appendix lie result characterize saddle non trivial attention result generality constant almost fix constant satisfy equation q accord satisfies definition technical remark simplify matter show proposition point assign initial flip corollary proof classical exploit due consist begin appropriately finite member mean approximate property deduce low test obtain low inequality mutual detail outline begin reduction error assume proof construction collection representative discrepancy measure q separation optimality nature choose draw demonstrate q observation bound possible minimize risk testing uniformly procedure observe variable measurable satisfie contrast uniformly observe le inequality become apparent argument provide theoretic le step constitute argument technique satisfy definition definition loss lemma mutual interactive differentially scheme present pack conjunction step final lemma turn precede outline exhibit minimax separate risk separate collection relatively base generate standard define linear function final separation risk scheme domain sign multiply complete median loss loss pair cardinality q define sample functional construction whenever risk minimizer collection functional precede discrepancy subgradient outline proof private available mutual bound loss mutual independence single randomize careful calculation yield final somewhat accord proposition subgradient appendix accord proposition subgradient appendix family information apply prove le complete median begin state statement theorem pack construction median numerical couple dependence pack linear lemma le obtain variation appendix multiply side complete statement part eq eq construct analyze norm median recall packing cardinality choose among define hinge strategy risk radius minimizer risk risk immediate verify separation directly minimizer attain construction require careful guarantee provide gradient radius provide complete characterize proposition turn mirror use note low large probability similar eq focus identical universal minimizing satisfy low mirror privacy general utility statistical number interesting access perturb view convex question whether dataset could insight release privacy induce lie compact perhaps fast estimation preserve channel alternative might know care may question appear answer especially wish privacy technique herein hope work pt receive arbitrarily q oracle minimizer amount consequently guarantee devote mutual proving construct sequence element information support set mutual inspection must let denote minor giving apply convexity q subgradient independent define scheme uniformly choose scheme q define entropy concavity substitution imply mutual addition value fix component subgradient entropy minimize q proof concavity proof subgradient individual hinge loss equal eq shorthand therefore construction imply similarly parallel take thus use marginal condition via algebra seek bind mutual generality add claim coordinate similarly similar hold e expression note recall expansion must sum case probability remain representation mutual coordinate symmetry marginally concavity let fold product exploit concavity inequality achieve mechanism ingredient distribution subgradient differentially identical private also let sample independently probability set scheme differentially private calculation corollary sampling odd strategy eq strategy apply stochastic bind note regular probability appendix classical arbitrary inequality conjunction g must point precise address issue involve precise regular probability markov consistent transition probability construct appropriate chain extend proof state measurable though subset moreover topology clear measurable mapping accord additional drop indeed since theoretic turn contain slightly outside extreme represent regular define random construction measurable version sure set see turn analogue lemma private support bad markov without private discrete maximization lemma establish completeness p vector choose constraint exist may lagrangian domain kkt minimize dual conditional q since must constraint kkt broadly outline guarantee distribution mutual reduce entropy minimal mutual optimality begin consider extreme extreme define extreme generality extreme technique convexity symmetry constant determine upper attain extreme establish statement lemma denote minimize mutual problem extreme moreover convex inspection since satisfy equality uniqueness solve write shorthand normalization expectation constraint q satisfy tucker kkt g derivative shorthand symmetry note x px maximum rewrite entropy inspection mutual minimization problem maximum saddle claim extreme point maximize choose solve maximization additionally simplex w g remark immediately focus verify fix leave take multiplier problem infimum derivative identification mass coordinate say loss write plug perform logarithmic imply equality arbitrarily along support extreme radius mutual plan sum write lagrangian solving guarantee I symmetry normalize appropriately thus solve nearly side whose root belong desire algebraic statement outline proposition satisfy support extreme reduce optimally private simplify lp finally map radius ball choose privacy channel differential privacy level differentially channel implicitly converse view abuse hypercube conditional vector exist additionally small possible perturbation cast mass differentially private channel lp p optimally private lp use problem satisfy suppose sake contradiction matrix similarly satisfied symmetry information information give contradiction optimal differentially indeed entry corresponding eq characterize structure solution linear proceed large specify lemma uniqueness due begin lagrangian constraint negativity generalize constraint hold vector satisfy vector inspection kkt must vector eliminate negative homogeneous kkt indeed inspection see question remain whether definition seek q sum symmetry kkt argument remain define statement lemma eq strict apply objective previous lagrangian modify linear condition similarly definition suitably uniqueness completely set vector argument complete proof identical mutual follow invert mutual show define denote expansion f ff expansion yield tackle note f berkeley electrical department university california berkeley privacy model sharp procedure consequence exhibit tradeoff privacy arise whenever aggregate individual wish learner quantitative exploit freedom whenever paper statistical theoretic advantage foundation development decade goal goal thereby determine third theory permit abstraction detail specific procedure allow fourth loss allow optimization theory optimization practice theoretical decision risk randomization privacy measure minimization independently minimize risk provide access give denote explicitly history go suggest privacy census presentation orient privacy key hope comprehensive survey reference member belong dataset generate statistic literature limitation linkage yield maintain privacy currently standard privacy privacy formally privacy probability differentially guarantee adversary know privacy information adversarial attack break researcher algorithm empirical differentially private add estimator obtain privacy statistical estimator suitably asymptotic stability perturbation demonstrate maintain privacy providing though intuitively tradeoff precisely impossible call useful differential guarantee guarantee sample give amount necessary work relaxed differential privacy give additive failure also closeness low bound match theorem involve similarity somewhat goal count number output use geometrically optimal notion utility provide accurately statistic substantially minimize guarantee drive force focus keep private many individual classical privacy true set privacy interactive sense depend private locally natural population give trust medical application perhaps status develop access internet activity search provide utility web wish search show setting locally coincide concept complexity query powerful model rely count query interested precise polynomial quantity broad risk sharp privacy datum wish nature differential privacy explicit private e interactive turn theoretic recall version mutual random measure mutual theoretic idea security survey important standard mutual privacy notion mutual say generate possibly minimize characterization estimator privacy constraint collection function minimax estimation domain guarantee collection constant characterize e moreover turn differential able constant risk exist constant procedure ratio universal numerical establish quantify sharp amount effective maximally decrease roughly differentially
square penalty rest organize group effect give devoted criterion huber criterion en technical random mean indeed sequel existence vector unknown minimizing unnecessary introduce intercept appear already paper fact penalty tend group attempt solve group lasso procedure penalty impose predefine precisely consequently group level see choose group next naive minimize coefficient square suffice large reduction small consequently penalty act scale fan li oracle p weight effectively penalty en penalty make penalty know enjoy notice penalty behave correlation procedure rely sort coordinate zero achieve unique integer l consequently en elastic variable see spirit create group group group avoid remove group penalty scale coefficient smallest individually penalty tend keep large heavy outlier huber criterion introduce quadratic describe quadratic linear take huber criterion minimize respect propose huber estimation huber coordinate problem run determine constant huber usually j instance ordinary root form huber criterion penalty observe type least classical bic minimize huber criterion bic minimize eq non satisfy group high lead estimation property stability quantify group net net goal theorem grouping penalty huber task future us remark since initial effectively quantitative penalty ridge compare net provide sign natural happen adaptive net initial grouping elastic estimator initial estimator adaptive elastic avoid ridge avoid du grouping effect property various fair study involve normalize explicit calculation variable normalize upper property enjoy replace huber loss two denote matrix suppose consider need see huber criterion satisfy distribution note hold absolutely lebesgue strictly origin stand distribute random result hold matrix asymptotic huber combined ridge ridge ad en huber ad huber huber en huber ad compare present study simulation insight provide paragraph performance inspired involve group correlate model intercept way form ny tn matrix compose diagonal third block coordinate equal coefficient amongst highly correlate group intercept nature noise intercept block gaussian mixture gaussian deviation block group allow group penalty divide light tail whereas heavy error model quantify robust absence likelihood square performance performance measure design design center stick theoretical fix framework size provide associated selection report table report constitute procedure indicator absolute I vanish amongst count I case go overfitte zero least u report aim provide zero zero correctly zero recall model closely estimation illustrate estimation figure concern hyperparameter choice regularization penaltie bic criterion describe grid method linearly space space huber study report recommend huber penalty remark tuning ridge cross validation grid linearly space similar protocol pick relatively grid linearly fold validation selection en surprising en impose norm square close example en behavior penalty zero correct zero comparison fact type reduce huber contrary en method identify huber zero zero zero figure coefficient good observe penalty lead performance term ridge bias figure expect ordinary square well performance especially ad huber numerical come cancer early elastic net eight clinical logarithm logarithm percentage score predictor name divided part observation study set performance observe allow huber ad lasso slightly let observe great huber procedure stable except ol associate variable en penalty en solve optimization package program programming impose rapidly convert problem reject version convex self well huber huber satisfie kind lead characterization number trivial interval guide manner however expression compute represents begin alpha mu l l w compute multiply derivative kkt eq last index belong become second cauchy schwarz get belong become cauchy eq imply normality step adaptation concern treatment penalty notation difference normality u nu nu limit together p p tu suppose consequently selection part suffice n infinity claim infinity tends infinity tend sufficiently respect differentiable entail z z l l large q depending since involved sequence since possible tend infinity tend concern since converge infinity u entail concern j j differentiable tend bound entail second expansion j vi stronger hold treat b j ai ax ax p imply convergence involve finite ensure moreover tend numerator tight denominator part page notion weak net theorem eq show convergence convex u ji ii vi convergence prove f successively theorem iii vi previously page ensure imply since deterministic p u satisfy intermediate piece li semi note semi value low infinity recall ai j set I
conditional ic simulate conditional lasso neither automatically scheme variable thus credible infer let credible interval multimodal partition density great zero ideally zero firstly density current mean suitable modelling mean sample mean large integer credible define eq number sample fix active increase six dataset six rr produce validate dataset pair cancer normal validate find literature validate expression study interaction generate intersection prediction cancer researcher derive broad contain tumor fourth experimentally identify interaction search also interaction validate interaction assess initial interaction infer interaction coefficient rr control parameter determine integer large value shrinkage fold cross cv separate broad estimate part select sample gene training error finally test gene datum coefficient value gene subset coefficient estimate fixing algorithm select gene choose rr threshold mark gene method constraint perform describe rr lasso across dataset rr similar interaction sparse zero see gene example infer identify exactly zero lasso several detect less rr able gene credible active credible interaction infer interaction density look infer confident c ignore density infer select density peak around zero peak calculate credible interval significance gene credible statistical significance shown experimentally experimentally detect c estimate rr identify estimate bar plot involve information red bar plot involve expression rr produce protein form computed magnitude around zero vice versa unless rr approach credible show target eight seven target plot trace vice versa imply agree conclusion prove rr performance auc uncertainty interaction bayesian method automatically credible looking advance perform prefer method produce effect protein protein simulate competition target acknowledgement would thank david stage manuscript target compose protein gene ridge lasso algorithm produce ht credible highlight experimentally validate gene value fix roc rr threshold ht ht ht square ridge bar red ridge bayesian lasso detect experimentally validate infer list gene credible b draw bayesian put interact black red credible credible computed lasso gene great highlight experimentally significance credible credible negative bayesian lasso six identify significance greater show highlight experimentally c significance credible credible compute seven great highlight validate significance credible interval active eight significance experimentally credible interval significance credible b b credible list bold experimentally significance credible credible credible effect significance list bold validate target validate broad experimentally validate significance auc broad cm cm liu engineering university technology china mail cn abstract rna important target useful understanding role potentially disease point estimation interaction lasso explore infer target rr four sensitivity specificity bayesian meaningful provide manually choose meaningful matlab google express process primary combine rna induce protein reveal report important role development give sequence identify interaction target complementary seed region show throughput datum greatly potential target computational throughput mutual partial least approximation etc method probabilistic employ carlo require employ variational density regression approach propose lasso negative employ infer point parameter non identification target non employ rr validate dataset representation thresholding produce unable calculate credible uncertainty significance significance significance expression represent collection expression interaction collection model target matrix model bilinear could since illustration model direct advantage consider represent complex multiplication thus negative part negative degradation gene involve example activation involve target algorithm algorithm briefly
nonnegative corruption corrupt clean noise handle error moreover concerned partial corruption portion kind portion clean traditional follow obtain constraint control tradeoff dependent portion corrupt optimize norm effective convenient additive nonnegative need decompose clean nonnegative difficult find instead work nmf decrease reduce follow update ms decrease I I pe nu value correctness updating rule z tf z problem update rule need prove positive rescaling substitute updating additive result part gained keep auxiliary second important z ensure thresholde e updating cause objective always converge present experimental position reconstruction deal assume advance fortunately able position missing locate experiment patch average experiment face image vector face randomly pixel additive face image scan nonzero position evaluate precision total algorithm knowledge handle nmf nmf gain try detect appropriate performance sensitive try poor probably partial corruption nmf recall face recall face sample explore large enough long set set face middle speaking algorithm precision large little recall indicate detect precision present denoise first simulate additive reconstruct handle position use image cause face face generate denote similar since mask new face experiment face mean percent mean percent pixel faces see traditional noise large position partial corruption enable norm prefer pure nmf subsection denoise natural affected convert patch set reconstruct original denoising show image show nmf traditional space truth material application prevent nmf algorithm handle require advance miss solid justification application rank automatically adequate application nmf widely etc nmf many face motion segmentation etc dimensional nonnegative corrupted position variant nmf treat corrupt unknown application prevent usage traditional variant nonnegative corruption position clean nonnegative position noise iterative justification nonnegative factorization recognition etc nmf receive substantial attention practical improve incorporate nmf propose corruption propose pca recover propose robust
map basic discuss relate dynamical compare range piece extend kalman batch discover identify dynamical structure fourth dimensionality reduction discuss connection among show spectral localization motion orientation range robot throughout know range associate handle localization motion standard gauss unknown robot parameter online filter map new robot linear motion though sensor informative enough robot range application exact multimodal gaussian inaccurate linearization lot range attempt enough extended initialization delay initialization parameterization accurately multimodal show track multi hypothesis much expensive instead maintain statistically mode approximately sparse approximate identification attempt directly identification learn carefully observable regression originally algorithm almost identification observable tb history bottleneck rank method discovery spectral state appeal implement operation contrast em consistent typically impractical requirement range spectral discovery discuss measurement give property make carry spectral efficiency finite bound structure motion problem vision recover rotation sequence pose discovery represent frame camera factor position camera rotation state inspire range way identifiability geometric information beyond give dimensionality view possibly nonlinear like multidimensional example use approximated geodesic contrast inaccurate robot integrate additionally measurement force sec block resolve ambiguity popular classical sensor feature manifold lie reduction nonlinear dimensionality network location think suggest dimensionality unnecessary reduction well simplify strong portion generalize sec discuss robot sec discuss discuss datum online discuss motion measurement measure robot insight robot n position recover factorization unfortunately transform factorization hope metric estimate available four position simulate orthogonal transform translation constraint show set contrast camera orthogonal nonlinear tb environment svd recover transform robot position position information recover high frobenius receive pair robot environment robot successive velocity factor key observation write easy position contain contain robot value four singular learn make either target angle successive pair location th location collect top singular c matrix suggest alg thin discard remain randomly sample estimate measurement top include datum show number location robot robot position robot true detail recover factor presentation although zero noise finite time weak proportion gain argument sec detail bound svd learn nonzero eigenvalue covariance receive practice rarely satisfied receive range entirely often outline relatively miss em use penalty miss structural second divide subset robot build predict read datum range next finally average locally interpolation work much well practice drawback many extension straightforward sequentially need motion motion require want derive latent location learn motion identification sec robot idea complete receive next collect substantial computer gb ram take simulator place environment robot move step receive read range perturb sample equal solve coordinate accurately path investigate map increase generate environment spectral measurement exclude true datum collect fig collect use wide range find range mobile robot stationary environment place throughout robot coordinate ground ground truth truth path accord characteristic km pose receive turn pattern robot km pose measurement worst depicted qualitative spectral localization square show baseline robot standard report rmse worse particularly initial portion also gauss matlab range subject optima intensive follow suggestion minute evaluation nonlinear run computation time empirically optimization time use initialization batch optimal start likely certainly tb last good opt opt opt batch opt path last spectral path spectral batch last batch good full c solution differ substantially previous essence minimum free algorithm robot system state real acknowledgement support grant support nsf position case position know hope robot position orthogonal translation rotation turn metric row know column exactly exactly dimension analogous matrix learn robot coordinate position transform coordinate two matrix high fit surface scale surface surface step linearly transform approximately linear coefficient orthonormal coefficient column select right r bring quadratic spherical start transform coordinate quadratic diagonal set write r eq iv surface coordinate else guarantee unique take set pick software automatically quadratic get quadratic desire coefficient coordinate expand know first solve last equation bring coefficient eq leave one vi satisfy learn metric advantage correct coordinate identically since function identically svd top small singular remain provide estimate counterpart two part concentration hoeffding show element covariance value empirical multiple kronecker minus constant derive bound
exp exp identify see exp exp exp exp fail noiseless exp robust vertex hull perturb toward hull matrix index maximum well interior convex hull column exp perform ill conditioning matrix oppose exp ill perfectly extract well exp oppose exp fast algorithm computational c exp exp loose partly consider bad structured value show average guarantee guarantee exp exp exp algorithm see already perform solve solve interior third ten middle five twice add hull second solver lp mention solver much percentage correctly show exp exp exp although large percentage decrease fast noise increase percent correctly extract always c exp exp exp exp algorithm noise condition far still make heavily rely separability assumption perfect practice preferable analyze hyperspectral pure generalize hyperspectral framework explain algorithm robust direction bound one tight improve feedback improve lemma nonnegative separability column hyperspectral mix pure assumption prove small perturbation generalize exist hyperspectral hence provide performance hyperspectral pixel robustness hyperspectral consist acquire scene measure surface pixel hyperspectral material model spectral signature pixel linear combination call weight image band entry matrix equal signature pixel material whose signature abundance define equivalently give nonnegative hyperspectral abundance factorization ill pose assume material pixel identify hull call write cone contain sense mining interpret topic bag topic separability appear require topic discuss pure separability appear row topic pure sense actually part document reference hyperspectral mix pure nonnegative assumption handle situation sense comprehensive overview hyperspectral identifying normalize column prove work input separable perturb robust notable et program suited deal pixel several expensive world hyperspectral namely subset column balance importance extract al convex problem deal large fast incremental gradient tune impractical huge scale problem paper family separability noise fast requirement extremely priori tune factorization recursive section repeat separable identify convex hull volume section desirable approach guarantee recovery also need matrix practice satisfy always hyperspectral imaging abundance uniquely derive noise pure propose handle algorithms generalize successive automatic generation successive pure pixel justification base pure experimentally illustrate result synthetic denote position th entry x cc decrease identity denote discard nmf separability strongly function permutation theorem j corresponding maximize give step project column onto complement select number stop whenever residual last specify analyze separable small perturbation assume original matrix separable pure pixel hyperspectral imaging see implicitly otherwise general ill pose one make generality column normalize construction sum column entry correspond hyperspectral slightly image contain background pixel zero signature hyperspectral image intensity part angle camera scene hence although contain material signature noisy account signature matrix global notice whose whose continuous tell assumption recover set correspond column w k w k rw iw jj induction therefore column extract project onto complement condition step extracted say loss full unchanged correspond extract affect go assume perturb noiseless original satisfying sufficiently introduce kf satisfie convexity r I rx kx analyze bound imply convex parameter w give eq fact lipschitz continuity gradient prove induction work perturbation convexity column denote maximize satisfie contradiction extract perturb convexity strict vertex lemma equation obtain contradiction therefore condition large singular proportional algorithm memory sensitive hull discuss previous discard outlier simple outlier describe satisfy noiseless row use abundance column quadratic program separable contain cf j easy column unique index must hyperspectral outli abundance large perfectly reasonable assumption convexity cardinality step correspond column let row show identify extract precisely go prove bound row prove since step prove derivation exactly omit denote also projection column span column finally bf relate derivation whose good choice along hand worth qr perform respect column successive projection use show technique robust able refer generation empirically hyperspectral explanation explain theoretically analyze data dimensionality reduction successive successfully take hull maximum unless logarithmic achievable polynomial advantage algorithm separable identify whose hull robust clear experiment hx choice limit large potentially converge go infinity check choose check use choose depend condition derivation bound precisely ball locally continuous convexity investigate derivation use fy hold gradient gradient continuous respect apply guarantee noiseless hull column consider matrix select fail norm sensitive pixel average author perform numerical justify give synthetic highlight synthetic hyperspectral reader algorithm propose compare test variant robust comparing perform step compute summing element extract step way operation total approximately operation eventually impractical matrix minus reduce formula operation interested column test implementation experiment turn seven hyperspectral time less half residual matlab intel core minima attain generate function sphere identify maximize separability column attribute column correspond generate column large let number vertex computational robust noise vertex noisy soon perturb identify small perturbation confirm noiseless critical conditioning arbitrarily noisy column isolate column could high column extract example critical hyperspectral close pure moreover reason correspond column step principal component see notice preprocessing set
kernel value query hardness depend concentrated surface hardness depend characterize reverse closeness closeness concentration pi directional concentration project point set figure spread ball diameter point well possibly ball diameter number radius index onto notion directional concentration indexing indexing indexing space partition many indexing scheme medium hierarchical indexing scheme branch addition branch bind importantly build select efficient desire indexing tree metric align split tree use representation induce make tree prohibitive cover way quadratic construction multiple moreover analyze extensively cover tree requirement theorem index integer scale decrease node invariant cover dp q finite explain infinitely many level contain dataset node store tree child child supplement distance kernel evaluation construction construction would tree implicitly point modification branch provable support branch branch triangle absence obtain max subtree present search object subtree root node possible kernel query subtree notational convenience give rooted query suppose possible figure cauchy invariant cover tree node level bound recursively j ht surface correct loose root object maximum object q supplement r node retain far remove branch alg maximum kernel value current subtree possible subtree retain node correctness cover alg complete present supplement sketch I approximate focus exact max max absolute relative care take follow end iteration loop alg good approximate stop iteration r I stop point achieve technique supplement space make bound explicit search directional analysis tree depth required result child encounter whole pruning I ir r directional bounding bounding ball ball hence query scale price solve search experiment top candidate evaluation perform tree supplement fix length mnist uci machine repository netflix yahoo music dataset length representation protein reduce input sequence summarize close magnitude except mnist quite clear attribute directional constant property protein speedup order logarithmic sequence size require completely multiple load whole memory constructed distribute inefficient high branch method compare obvious exchange framework indexing scheme scan evenly distribute master node single master perform scan parallel match node master return match total good match important actual master runtime distribute system us scheme indexing contain build save scan single every tree match return master report phase query logarithmic time scale constant runtime enough small enough yet efficient achieve scale scan extension technique achieve comprehensive general date first rigorous hardness provably logarithmic query speed practice tight analysis without directional constant abstract object graph kernel detail tree single explicit kernel trick metric dp call current level tree recursive call subroutine represent splitting form completeness root object bound denote angle lie level angle angle origin simplify give cover consideration r kernel search query find ram al draw eq simplifying present always miss top query replacement want rate k tree subtree root return max operation follow improve actual construction parent impossible experimentally verify store reduce number evaluation computation gray proposition claim section wide max search focus begin hardness problem notion algorithm object feature provably search well order magnitude speedup extension search present accelerate object object similarity similarity scan set ht similarity class fix protein text music financial kernel kernel trick evaluate inner product require neighbor match computer vision million grow make scan prohibitive appear posteriori well widely successful obtain representation preference ever scale million cost retrieval recommendation find dna sequence set object various maximal alignment preference biological matching implicitly letter like p crucial lose normalization normalize kernel
seven gibbs bayesian iterate time keep size report result illustrate figure represent gibbs successfully outlier model contaminate table contain estimate outli occurrence size present remove detect outlier detect outlier variability outlier illustrate detect correctly see conditional obtain remove biased typical htb true consider motivate regard ip server university indicate outlier remove respectively autocorrelation partial autocorrelation function accordance estimate detect poisson affect identify require outli outli occurrence respectively effect cause depend position outlier methodology higher derive distribution implementation additional model work operational mathematic project mat de mathematics commonly problem additive outlier integer consider show methodology use keyword problem model poisson autoregressive series intervention often occur effect framework gaussian series detect outlier intervention investigate author intervention yet albeit relevance inference motivation stem adequate worth model intervention effect estimation contaminate period additive wrong motivate focus autoregressive independently real motivate concern address page construct concern page find author show infer improved outli bernoulli arrival additive outlier occur otherwise identically contaminate magnitude describe estimate outli contamination distribution conjugate ga distribution q marginal describe full posterior tt x first markovian affect fx I therefore conditional outlier give denote vector full conditional markov chain distribution case conditional log concave rejection metropolis arm gibbs
general evy process need monotonicity process rich parametric fa sd simulation introduce evy therein normally intensity realistic volatility determine drift brownian motion time brownian motion model infinite activity index respectively imply risk neutral measure base price throughout time negative put price option noise level error due bid market simplicity market contain bid note evy dimensional triplet infer price accurate estimation evy pricing characteristic finite imply martingale condition curve plug theoretical result observation additional smoothing spline two might improve interpolation spectral procedure rely identity look convenient option identity shift lead scale jump therefore use allow cut frequency parameter fa separately precise correction negativity jump latter right hand pricing apply inverse option benchmark minimize measure employ square approach correspond evy triplet pricing formula l evy plug efficiently numerical effort determine minimization consideration term mention include lead correction hence difference residual square impose price derivative least small minimize calibration drive choice quasi optimality study perform well setup follow version subscript onto quadratic sense domain particularly high frequency frequency smooth transition improve like weight outside determined reflect smoothness estimate scale version flat top kernel suffice transformation interpolation whole estimation finally choose differently quantity let simulation european option quantile fourier additive consist level choosing determine measure interpolation account use aspect mention quadratic smoothing gain high weight noise frequency reduce illustrate spline function whereby increase jump tp level iteration spline zero finer decomposable infinite activity u nonsmooth yield eq smoothness away zero analogously coefficient q determined lead solve substitute know jump one function q numerically compact large truncation reasonable decomposable monotonicity monotone arbitrary decomposable normality estimator fa setup decomposable let treat similarly nonparametric part choice cut trade construct negligible approximate u logarithm term linearize error construct variance linearize error nevertheless asymptotic approximation error negligible exact vanish analyze linearized observation quantile q incorporate may sample equivalence find fu fu asymptotic equivalence approximate tu integral yield finite variance linearize error isometry variance central distinguish finite shape sd scenario pointwise linearization kx version quantity evy representation l evy triplet latter cut density estimate point bandwidth interval give denote standard confidence cut wide interval delta estimator confidence interval gamma simulate price relative european call price linearly application interpolation cccc ccc level carlo oracle cut confidence percentage true value sufficiently confidence range fa fall bit picture confidence evy density pointwise interval evy density k confidence illustrate true evy pointwise everywhere contain confidence interval far plot confidence estimate negative peak cf method european option index therefore price financial market accord put respective price option two seven day price b unknown sd focus option price sd option calculate square fit especially seven minor trading day confirm sd coincide figure calibration l evy jump positivity correction might look interval model suggest neutral stock pure process decomposable reproduce option activity jump diffusion index equal contrast investigation estimate jump tp sd residual option pointwise option sd trading section stability moreover trading day procedure option estimation option significant decrease hand curve significantly slight respect axis volatility time market activity activity calibration option latter jump measure shape calibration unimodal minor differ reflect jump obvious trend pricing model consecutive market misspecification pricing however difficult improve activity fa evy model admit determine variance linearize estimator set precise error european option price fa fa volatility trend activity decrease long evy misspecification forward high available market interest whose risk activity interval base linearize tu tu finite variance estimator linearization write brevity mx purely linearize give decomposable compare differ characteristic tu finite version define u view linearize x theorem proposition thm grateful anonymous lead considerable improvement research economic observe price option evy evy activity well self decomposable l evy interval volatility pointwise jump probability infinite base option index calibration trading day european option nonlinear evy frequently price assume constant l triplet jump price account tail appropriately evy model reproduce volatility fact shorter adequate fact
relate express interval extract either equation provide place represent volume trading integrate choose location follow stop iteration optimum adapt impose element potential city aim identify high identify area would open additional evaluate maximum market volume city daily market characterize market volume around million daily copy far city concern commonly economic sale copy economic conditionally sale datum market potential justify daily represent available consist daily day locate sale total daily sale volume available around copy volume location circle measure measure type customer day copy fix effective suppose customer suppose attractive potential go network zero area covariate covariate represent density joint stock induce sale covariate distance car distance comparable rescale c estimate provide discuss interval original run test global likelihood particular application bad strong competition report approximated consider coefficient positive sign low control run retain pt average area city estimate suggest highly spatially competition nearby quite apart measure market potential location equation market city provide copy maxima area global note city market estimate potential copy place locate maxima represent location additional standard area city b copy city pt volume relate area market interval market daily copy increase total market volume optimize copy volume market volume actual light economic daily improve volume additional economic daily guarantee additional market daily phase potential prove essential statistical spatial market sale uncertainty include spatially unit potential market volume economic daily city daily sale focus city characterize potential include study temporal fluctuation relaxation worth outside sale city drug disease spatially available aggregated form g privacy reason precise usual multivariate variance estimate give equation evaluation derivative distance matrix vector eq acknowledgment provide preliminary analysis always recover market spatially sale tackle way measure location recover measure collect interact affected measuring maximization inferential apply market city analyze order area evaluate market city give market trading area sale expect spatial market low maximize issue potential already sale spatially available aim market sale spatial interaction interact sale volume affect presence potential ignoring regard spatial trading perspective spatial spatially market random ever potential instance recover realization spatially case interact interact location measurement nearby measure miss sale spatially market city aim area identify area total city respect organize background spatial datum interaction case technical estimation spatial term turn sale receive attention solve new interaction potential flow interested reader refer seminal store often attribute product store main drawback spatial market potential spatial location concept suppose beyond existence market also reach product attribute store affect spatial spatial potential drive interaction uncertainty critical application provide market answer question denote market potential generic spatial company trade company maxima store near company wants evaluate market presence actual suppose three mean covariate process w completely characterize reason later paper assume respect variable location realization interaction adopt f nonnegative binary continuous monotonically decrease distance parameter equal equation directly model line rewrite element explanation namely measure equal added function particular contrary reflect fact influence site worth exchangeable measure potential action measure measure suppose effectiveness case satisfied sense potential second equally simplify model measure measure typical potential namely suppose order supposed figure potential measure measure measure location place collect vector covariate partition site miss partition elimination permutation proper partition measurement partition sequel
cubic trend work mechanic penalty extensive comparative penalty tool often consume develop penalty interested reader toolbox site material derivative function list table lrr drop subscript henceforth penalty set derivative reflect compare path path power family thresholde objective soft scad penalize function solution objective mc path either smaller penalize display glm state art solution glm penalty algorithm warm setup predictor default show short ode major compute ode solver smoothly track whole definition institute nc david department university nc propose high dimensional remain routine lar behave highly combination perspective rapidly posteriori hyper instability validation procedure corresponding correspond prior mainly extend filtering scale efficiently large dimensional glm operating apply several keyword model glm lasso lar posteriori estimation convex penalty ordinary solution via prominent last decade find across variety norm lar lasso popularity excellent ordinary square unfortunately serious estimation bias motivate regularization properly large shrink signal non convex convex concave penalty estimation hyperplane iterate support regularization produces iterate estimation penalty category numerically implement sensitive setting sparse demand lar important expensive challenge cross incur criterion criterion tune minimize penalize shrinkage freedom often unclear extend mean bic lose criterion issue general twice differentiable subset scalar parameter penalty general penalization assume condition symmetric iii iv decreasing generality fold problem likelihood precision commonly aforementione assumption vary net vary regression elastic net thresholded oracle scad via partial derivative scad natural knot act scad mc derivative shrinkage function frequently use development list supplementary material general framework knowledge gps rigorous fundamentally gps path specific homotopy piecewise solution penalize efficiently consider lar ode approach fit piecewise adopt paper move improve impose great track least mc penalty empirical imply parameter path shrinkage power induce exponential power double pareto regularization correspond utilize engine appropriately take penalize generalize trend filter generalize regression produce less toolbox regression author site web site remainder organize path follow discuss various correspondingly z index current define hessian active path follow necessary denote article optimality satisfie furthermore semidefinite convex directional derivative order necessary satisfy track stationary along path slide regime semidefinite stationarity minimization check optimality claim stationarity directional function j scalar convexity h tt remark directional bivariate directional derivative negative local minimum passing local go hill abuse terminology ht first serve illustrate secondly orthogonal design building coordinate thresholding rely path non objective fashion list material derivation find assume standardize depict jumps minimum iteratively update take format covariate loss around iterate eq apply thresholding sequentially iterate precede difficulty regression non penalty enter vice cause contrast lasso piecewise make along lar strategy penalty descent find success net article explore descent penalty determine grid tuning advance larger likely devise path strategy track path smoothly path ode z nonzero penalize coefficient hessian restrict continuous differentiable vector write predictor calculate variable independent suggest solve segment promise path care stationarity condition inactive due may reliable inactive regression become thresholde formulae step cause may along whenever nonsmooth optimizer start coordinate strategy summarize determine penalize enter correspond predictor become inactive penalize coefficient inactive penalize jump segment remark path ode give reverse sign termination termination path specific stop predictor exceed subtle poisson separation complete likelihood occur surface behave linearly along dominate mc scad flat infinity penalize appropriate square prior include un penalize probability ap appropriate bayesian penalize proper marginal analytically fortunately manner informative information path also double pareto writing place double pareto un particular bayes penalty pareto otherwise log path restrict normalize jk h unknown laplace normalize q suggests easily compute family exponential unnormalized coefficient approximation bayes penalize h regression laplace laplace normalize plug numerous application path recent regularization penalize shape regression nonparametric density parsimonious sparse non pareto scad etc path hard readily path ease presentation equality assumption trick regularization amenable twice differentiable transformation matrix full lasso matrix eq rank filter fuse trend full row section cubic trend demonstrate financial highly structured compute use cholesky regular variable example illustrate procedure mechanic logistic regression prediction penalty third trend filter simulation small setting display I cpu gb ram example site first concern cancer seven predictor logarithm cancer volume age logarithm amount score set observation path nine penalty penalty continuity cause along power unbiased achieve allow model along path informative variation path upper panel plot penalty proportion solution scale explain scad brevity penalty advantage high explanatory power avoid display evaluate squared error test set path show low panel penalty achieve well highly concave penalty power achieve along lasso moderately quite achieve prediction admit logistic heart set measure heart split path train cancer regression lead bias along path plot versus panel power evaluate test figure penalty able concave one h third illustrate acquisition article constitute company become seven covariate company list long market return finance theory indicate otherwise market market return explore possibly nonlinear effect predictor say code discretized circle figure bin nonlinear monotonically book ratio log market cubic covariate demonstrate use convex log discretize specify bin covariate piecewise cubic two end linear spline regression trend location knot choose display unconstraine path dot bayes mostly match regularize middle correspond model bin formal significant cubic effect term index explain linear logistic finance company unlikely company flow unlikely meet burden associate company possess technology obvious estimate company exceed evaluate performance bayes glm replicate determined canonical model explore
incorrect ol adjustment still error adjust bias subsection ol adjustment sample improve incorrect regularity technical omit motivate experiment imagine slope regression correction choose ols slope fix slope variance sample slope strict naturally conjecture experiment treatment interaction improve asymptotic incorrect ol conjecture confirm unless uncorrelated efficient efficient unless group equal covariate finite infinite finite increase regularity condition sequence population still preserve influence would see regularity condition generalize covariate applicable covariate avoid extra subscript population th treatment whose finite invertible population limit finite least square analogously let covariance theorem corollary supplementary variable eq assume difference slope regression seem analogy help thing adjustment far add add eq ol association adjustment design treatment group derive treatment covariate interaction unless uncorrelated asymptotic variance I estimator asymptotic precision asymptotically iii square pool ols tends fall two group z b ab ab subtracting adjustment add time pool adjustment group covariate value population nine subject nine generally run observation asymptotically material equal van difference mean weighted square iii usual adjustment adjustment covariate sufficient either must similar treatment usual adjustment adjustment even pool single covariate interaction subject outcome independently assign subject remainder treatment adjustment potential outcome opposite sign remark panel theorem limit replace panel deviation bias simulation pt adjust adjusted ol ols ol precision approximately justify accurate solve asymptotic linearity prove finite population q regressor th huber white analog huber formula show consistent asymptotically conservative imply weight supplementary material prediction error material condition great probability eq equal difference generalize design size variance also conservative analogous pp basic related problem pp randomization high leverage substantial discuss university support service service estimate year evidence service alone alone improve likely contact simplify huge percent student service service percent percent available service service group financial service service service estimate close school difference treatment interaction include school school predictor first year usual adjustment appear social outcome lin al pp examine social find value outcome standardized outcome researcher prefer adjust improvement way confidence strong margin interval assumes normally distribute small conservative adjustment iii technical simulation constant effect heterogeneous estimator conservative interact asymptotically percent coverage sample heterogeneity ols estimate bias hc hc hc classic hc hc classic hc hc hc hc hc hc hc keep datum actual experiment service service unbiased slightly simulation treatment treatment improve adjustment panel hc regressors hc remark hc mac page residual formula factor appear panel way percent standard estimator pp use hc standard degrees freedom fourth panel average agree adjusted adjustment yield inference without one limitation bias treatment heterogeneous covariate lead minor potential center bias balance constant substitute school covariance college square center high service service limit remark would order magnitude table bias negligible replace lead pp useful interact validate analysis limitation shape extreme skewness group size serious service roughly simulation inference goal check sample coverage interval weak strong effect permutation asymptotically ol effect heterogeneous literature remain valid asymptotically nan hypothesis exploration approach adjustment adjustment summarize empty highlight extent encourage researcher probably difference specification encourage strength conclude always adjustment reporting always randomization traditional believe adjustment health agnostic perspective theorem major contribution discover property derive adjust assume treatment argument detail acknowledgment small valuable david associate read berkeley helpful discussion david help early education theorem ordinary adjustment estimate randomization conventional adjustment lead bias sample either minor easily ol interaction include interval huber approximation illustrate evaluation student reason randomize ideally average assign researcher adjust characteristic adjustment tend argument regression correct researcher conduct social adjustment adjustment many influential assume randomization effect subject linearity assignment variability treatment adjustment asymptotic inconsistent iii adjusted treatment randomization justify behind ol perspective agree general defer argue sufficiently minor regularity ol full interaction huber white whether include distinction traditional ol property assign interaction asymptotic omit example write random justify biased negative second provide intuitive property ols adjustment incorrect view depend assumption similar precision subject infinite interaction generally adjustment may adjustment covariate interaction page source randomness infinite pp purpose population need address major concern help understand balance adjustment main result
iterate empirical counterpart provide mistake unbounded iterate exist penalty far constrain norm precede paragraph manuscript problem split optimization piece thank piece unbounded form core linear linear regressor correspondingly track support core split hard core core nonempty produce simply unclear core countable existence hard core somewhat appear let follow property say grow optimization proceed hand either set region etc impose mistake occur loss within force core r h nh property transfer sample crucially bound quantify exist every restrict sample establish order manuscript measurable property probability lf f general q provide useful logistic determine precede statement draw sample weight suboptimal correspond unbounded portion eq unbounded bind bound differentiable available correspondingly strong albeit h suboptimal l almost assumption presence finitely manuscript class asymptotically grant probability measure hypothesis l h significance consistency class borel denote axis align split take family classical theorem compact measurable function similar differentiable exist subsequence sequence suboptimal f manuscript basically consistency course elegant apply powerful class scope manuscript measurable lastly everywhere measurable everywhere follow everywhere contradict since turn lastly bind probability loss truncation since bound subdifferential correspondingly every lastly desire let rademacher b choice every rearrange use finite convex notation eq define subdifferential subdifferential integral heavy result cf zero everywhere fact conjugacy formula next notice give since continuous set define continuity dominate term conjugacy result provide integral continuous mutually conjugate lastly derivation subdifferential result supremum attain supremum zero operator semi convexity question lower since everywhere fact let take optimum p np entail provide transpose conjugate thus desire appropriate note significance act constraint lastly banach fr meet follow desire goal duality expression statement consequently term next grant restrict always iff dual lastly present optimum iff part subdifferential desire mean structural crucial define set statement exist satisfie everywhere exist cl metric direct construction crucially subspace orthogonal exist h n moreover argue projection countable take nothing compact attain provide remainder desire say sequence every representation every contradiction representation satisfy unbounded e define eventually similar due low cf assume replace h also minimize lie within compact perhaps subsequence since continuous cf attain duality measurable primal minimum since everywhere finish construction satisfying order invoke result material minimizer although appear depend proceed demonstrate suffice mechanism proper may duality attain infimum employ show subgradient grant rate step size index finish attain measure weighting extract suboptimal representation lie lie minimizer finish attain infimum necessarily optimum I achieve particular infimum attain generate possess logistic finish let exponential early imply henceforth low suffice furthermore eq state next define countable support positive example margin go key max class maximum solution enough henceforth one positive sample point example maxima since margin solely eq mean rescale wrong follow employ let restriction existence core section property limit exist monotone convergence exist limit pointwise measurable meaning ip thank dominate countable q nonempty always supremum grant continuity attain supremum relationship core tie analogously possibly primal core contain follow satisfie close countable intersection start intersection every correspondingly integer due finish whereby countable family consider whereby continuity intersection sequence definition consequently exist subsequence primal core always contain minimize infimum finally finite grant continuity q attain infimum existence equivalence core core elsewhere p construct meaning follow uniform thus subsequence thus exist exist suboptimal b set property core exist core take provide definition core per automatically since exists misclassifie arbitrarily example every nn example fall within within consider line segment thus every deviation twice low return form grant r mh comment fix grant correspond lf inverse failure various final statement x depend interpret whereby statement chernoff basic mf henceforth exist cf portion measure linear space deviation version vc mistake plug bind suboptimal predictor particular since grant sequence c lie since combine piece precisely cf provide simplify precede bound refinement let core provide core set large refer corresponding strong ff event guarantee de meaning bound large proceed next allow consideration restrict approximate infimum attain exist correspondingly cf lastly q crucially pass compact verify borel separable metric compact hausdorff henceforth measurable compact satisfies continuity compact imply uniform since outside form half open interval correctly partition side vertex partition arbitrary finish form align lattice model modification stage straight choose necessary massive proof study minimization input pass treat consistency boost class hinge loss fit scale complexity class surely binary operate feature label iff interpretability popularity rather try pick mistake intractable attention clear zero penalty margin specifically specialized logistic analysis scheme path positive interpret perform search provide way analyze parameter analyse full certain
bernoulli bernoulli inequality leave organized loss generality optimal suboptimal integrate want forecaster forecaster mistake reason arm play reward random arm th respect case event algebra order behavior forecaster bandit shorthand modify optimal satisfie step observe strong positive eq deduce q recalling n extract last result prove reasonably restrict except section ucb argument allow whereas require improvement subtle condition author take ucb light theorem gap towards gap replace hoeffding bernstein derivation theoretical guarantee second approach replace neighborhood ucb neighborhood line start later also ucb attain finite kl ucb attain kl strictly reward tend incur idea easy regret shall modification ucb get attain small numerical small order regime attain minimax always improve plausible regret appear variant arm typically quantity probability close expectation dominate concentration ucb analyze detail fact around theorem horizon impossible concerned strategy concentration greedy strategy play empirical time grow practice first distribution thompson proceed past I recently mainly prior reference therein behavior thompson property first fact attain essentially interestingly reasoning frequentist moment generate tail distribution become provide moment surprisingly essentially refine basic ucb order satisfy interest robust variant armed bandit make generation reward gain step simultaneously player arm adversary arm player loss gain step regret gain version sense translate set whenever proof simple achieve sublinear round possible adversarial gain deterministic exist suffice q key randomization play forecaster adversary get stochastic goal respect randomization adversary task first regret upper argue pseudo regret adversary pseudo coincide point guarantee adversarial fundamental idea exp exploitation number trick estimator arm namely next play indeed forecaster weighting scheme standard science weighting provide pseudo assume forecaster version instead knowledge pseudo exp non exp divide verify particular imply idea reader recognize quantity arise analysis two step step third let fourth putting obtain term easy conditional eq simplify uniform probability unfortunately exp define adequate task exp weight distribution uniform variance cumulative order say exp idea estimate loss core gain gain trick estimate expectation last use exp uniform gain estimate gain arm describe sake describe time horizon exp eq exp first prove exp q first immediately g thus key however notation let induce exp mix use inequality last third summing obtain come lemma union bind consequence yield derive regret adversary regret expect regret pseudo case difficulty integrate exp p integrate deviation formula value random variable show section reward pseudo forecaster average probabilistic kullback leibler aware technique prove bandit convenient bernoulli arm supremum reward generation reward internal randomization forecaster r immediately regret forecaster time bernoulli bernoulli parameter whose forecaster formalize kullback inequality forecaster arm use low expectation reward parameter forecaster five statement max empirical play forecaster play round law distribution adversary recall inequality immediately j I computation forecaster deterministic sequence reward forecaster uniquely determine play law conditionally stochastic adversary law adversary kullback example law precisely fourth step conclusion concave proof deterministic observe result forecaster randomization k realization bit chapter originally forecaster observe complete history problem strategy p detail analyze chapter many analyse logarithmic exp bind construct strategy inf forecaster idea first exp forecaster inf translation precisely inf probability tc tp regret take exp correspond realize inf mirror significantly refer reader pseudo significantly free forecaster action moreover vector include adversary strength propose adversary bad loss chapter logarithmic ask minimax prove adversary exp pseudo regret log maximal reward gain explored building author attain exclude variation arm apply bandit ingredient reservoir slowly bandit describe combine minimax pseudo compete consistently play define regret might big natural chapter compete policy switch play simple attain switch prove viewpoint exp safe well match regret describe stochastic adversarial logarithmic guarantee ucb n n flexible sequence reward relevant interesting strategy logarithmic adversarial armed bandit variation full feedback signal loss feedback chapter describe propose end round decide ask loss current know set minimax regret fundamentally algorithmic idea besides forecaster select round reveal speak simple coin regret model bandit undirected vertex vertex arm encode loss neighbor equivalent graph full feedback minimax regret factor number naturally click plausible display example monitor incur stochastic customer dynamically adjust customer item price item pricing item signal incur loss reader include historical recent problem arm contextual optimality mapping arm well optimum typically desire context policy contextual information forecaster arm news recommendation pool candidate news user website arm click article activity may detail application bandit general presence create variation combine reward nature available remark mention aware basic example contextual round mark forecaster must context arbitrary exp introduce bind adversarial bandit section exist bandit contexts instance first theorem identity subsection extend several immediate adversarial theorem mark variant basic adversarial set policy randomize context bandit seem role decide include set bandit recommendation forecaster obtain arm randomize kkt jt depend pseudo adversarial bandit introduce pseudo bandit eq weight exploration exploitation number uniform get probability arm probability loss estimate expert exp chapter set give order contextual forecaster exp framework even exp contextual setting run exp arm expert maintain exp come exp expert exp show moment arm bound exp exp without apply analysis forecaster use pseudo loss inequality similarly inequalitie jensen get note draw use proof besides similarly exp modification basic scenario arm mark contexts eq contain forecaster set find variant bandit nontrivial combination examine exp particular scenario exp expert aspect expert combine learning context theorem give regret improve scale idea explore exp class combine us logarithmic albeit dependency intuitively expert immediately exp hold play draw requirement bandit game play arm distribution kt ki ni tt exp different tp obtain observe statement reader give contextual variant exp exp assignment assignment mix coefficient mix achieve pseudo exp mixing coefficient satisfie along modification expert past forecaster way exist forecaster set run forecaster mixing instance expert distribution exp instance joint k bandit inherent exp improve construction mapping context forecaster supervise standard context supervise contextual setting forecaster time research generation variant contextual context draw analogy learn theory evaluate risk class arm view counterpart adversarial regret characterize supervised rest policy theory follow dimension satisfy arm uniformly play simulate exp policy exclude logarithmic vc price bandit information essentially vc internal randomization realization element function state class binary finite randomization chernoff least internal randomization quantify agree point remain remain randomly position g realization sample desire viewpoint variant protocol sequentially protocol keep classifier step side notation online denote predict label online protocol learner observe associate use adjust observe prediction bandit online multiclass perceptron matrix rewrite update correctly generalization perceptron number mistake notion infimum denote hinge iy multiclass represent variant multiclass observe p tp operate protocol multiclass true available correct inspire bandit w expect prediction mistake make sequence mistake multiclass perceptron bandit prediction mistake q need bind condition prediction otherwise hence ready mistake follow perceptron derive w n du jensen inequality see iw eq low statement assume bandit study reward parametric class non metric lipschitz remarkably slowly change reward covariate finitely dependence reward study model capture dependence reward model subsection adversarial p correlation expert contextual arm finitely use contextual bandit model important loss function hand sublinear bound infinitely time arm naturally bandit source introduction good choosing path path optimization forecaster simultaneously adversary choose incur pseudo forecaster internal randomization similarly feedback forecaster observe loss note far assume incur vanish extra playing rather original approach computationally us idea chapter analyze without generality play play remark critical powerful subsequent scalar loss without rewrite element arm discretized show appropriately exp describe strategy first geometry concern ellipsoid minimal volume contact contact convex set ellipsoid contact contact position strategy hull set ellipsoid play detail transformation satisfy select build play outer product matrix invertible tp quantity exp algorithm exploration correspond playing probability parameter correspond exp exploration support contact point exp finite adapt analysis exp take whenever hand write v low small introduce mirror descent problem losse convex end forecaster observe analysis section online mirror apply obtains let subgradient convex convex closure admit partial definition bregman f fx fx fy understand act proof q map original primal divergence primal exactly bregman divergence resemble geometry square inequality convex moreover simple space primal back tw rewrite convex imply set describe description outside dual differentiable exponentially average similar chapter know strategy take powerful compact convex obtain w q divergence fx domain step compact effective use bandit information extensively gradient perturb way observe estimator present restrict loss simple specialized performance true mirror mirror descent compact set rate initialize perturbation observe fx g relate bandit strategy forecaster forecaster arm coefficient pseudo theorem assumption pseudo compact convex replace use obtain combinatorial loss many fall rank planning semi feedback playing observe td namely observe loss thus full still bandit basic key tackle kind random table view perturbation play surprisingly randomization perturbation loss loss unbiased unbiased indeed directly concrete factor subtle start entropy set perturb note computation give entropy continuously associate potential regret instead reduce entropy potential check f u taylor expansion ready bind run potential specific potential improvement adversarial multi run choosing first inequality sx qx particular old loss bandit section action belong loss time feedback receive scalar obtain order logarithmic show one obtain show chapter another strategy factor base follow perturbation interior random draw rademacher variable unbiased ready suitable euclidean ball feedback play incur convex composition euclidean open regret second need first algebra fact inequality soon since triangle conclude suffice online feedback uniform improve last exp mirror mirror originally seminal convex online community algorithm online mirror explicit see mirror application mirror self barrier unfortunately drawback suboptimal dependency regret precisely attain mirror bound extract mirror development take go back inspired topic extensively recent account view propose mirror strategy proximal see viewpoint finally mirror prediction several see semi series adopt last generalization bandit feedback done derive introduce inf strategy understand instance list omit important briefly review assumption feedback bandit case minimax regret semi order match exp remark exp provably set chapter focus pseudo chapter concern bound partial result direction exp strategy algorithm similarly restriction adversary prove logarithmic long polytope progress chapter scenario arm loss necessarily function precisely adversary select value define consequence loss perturbation order study chapter agreement initially set face nonlinear loss problem kind view dynamic evolve evaluate loss play choice evaluate point forecaster regret investigate section similarly arm return unlike mean lipschitz unimodal necessarily keep thing pseudo regret analyze point forecaster loss extra choice neighborhood estimate control reason euclidean result strategy descent parameter convex rate initialize estimate introduce technical point play two sx symmetry compute side preliminary unnormalized consequence random differentiable uniform q satisfy hence equivalence relate integral sphere ball generalize immediately conclude tx provide smoothed exploit loss smooth take lemma perturb estimate convex lipschitz convex loss process obtain gr second moment exploit ready gradient order point loss feedback set rx immediately since loss conclude proof pseudo point pay feedback rate estimate obviously tx
employ high layer triangle soft threshold feature encode image patch effect order similar approximation code sparse code solving close example video author take solve code identify predict dictionary dictionary directly code quantity surprisingly uncorrelated evaluation experiment outperform proximal structure proximal extension operate dual space outline encode chapter soft code proximal outline soft chapter usefulness conclude offer necessary support focus proximal scheme variant dual tractable throughout chapter directly code applicable broad problem presentation reader familiar lagrangian justification discussion support find standard concern give quadratic equation introduce access refer variant central chapter problem iterate step converge method scheme scheme pick inverse form proximal consider approximate approximate arbitrary algebra show update amount multiply step ensure selection scheme specialized consider primal connection form q forming require minimization split ascent converge unfortunately problem problematic unbounded ascent augment parametrize lagrangian problem nice augment ensure guarantee dual feasible write dual proximal ascent ascent derivation connection proximal iterative difficulty gauss block instead modification lead since problem code feature process many introduction allow reconstruction dimension reconstruction alone unique encoding code uniqueness introduction phase phase representation encoding phase dictionary provide external source since reasonable performance offer condition want explicit phase formally encode collect dictionary write regularization reconstruction fit chapter element add positive sparse code encoding phase together quadratic easily optimization successive focus lead since sparse coding refer together term refer equation non whereas non sparse typically imply treat fix problem first three proximal framework fourth solve encoding work iterative soft thresholding name thresholding arise threshold encoding refer statement iterative thresholding fista momentum fista fast reconstruction handling problem chapter framework fista set scheme development discuss full coding algorithm share namely fixed objective still iterate present chapter encoding name nonetheless encode form lead rather alternative cause mapping vary solution cast functional descent weak learner start regularizer set force arbitrarily equation optimize arbitrary case iterate adjust size fista feature proximal choose available suggest feature threshold multiplier operate space value choice essentially arbitrary value expression long single encoding admm soft threshold perform iteration admm force factorization order inversion choice feature proximal add get feature sparse admm smooth newton overcomplete thus rank replace inverse numerically admm ridge take interpret elastic admm inverse encourage briefly unlike one element lead magnitude mean step feature arbitrarily dense compare accommodate sequel writing feature omit iteration find sparse report cifar experimental framework pixel cifar feature examine accuracy encoding library patch subtract library centroid include dictionary directly normalized smoothed random feature dictionary appear build training image extract densely encode dictionary iterations pool encode patch involve patch separate encoding alone repeat number different algorithm encode rather non regular coding give select admm well order maximize c admm summarize cifar notable feature exact run iteration performance fista admm nearly high produce multiply long iteration different problem equation vector problem aggregated column replace library multiplication figure fista actually drop seem interpret implication carefully figure reason expect lead performance optimize optimize contribute drop fista unstable regularizer threshold comparison threshold cite work discussion fista l fista admm design relate relationship reconstruction previous construct describe result previous experiment library cifar encode patch encode previous experiment correlation reconstruction give consider bad another feature experiment lead optimizer reconstruction beyond confirm include run reconstruction produce inferior ht soft gradient descent sparse encoding surprisingly successful broad framework four feature sparse encoding approximate proximal encoding image benchmark objective around minimize reconstruct patch dictionary would expect reconstruction demonstrate intuition obvious extension would thorough exploration insight suggestion evaluate properly usefulness report use dictionary examine problem investigation indicator appear effective smoothing connection elastic net understanding regularizer empirically well different coding extend induce potential author investigate proximal method definition recently certain feature achieve image belief net factor rbms rbms autoencoder several moreover intuitive put forward remarkable performance justification main realize negative sparse variant several sort ignore problem localization predict location report common tool introduction neural model serve purpose tailor specific provide encode construct way vector classify accurately raw pixel neural network prove tool representation major barrier require image technique
ensure sample variance estimator default apply power randomly power eigenvalue power eigen leave towards direction type analogous shrinkage optimal orthonormal basis author assume target rank aim excellent even relatively noisy adaptively detail compute basis corresponding eigenvalue low compute project computational randomization multiplication runtime normalize runtime stable propose stability shrinkage describe rank stability guess e basis assume approximately enhance th eigenvector assign eigenvector th direction dominate expect sharp eigenvector signal partition estimate value optimal accuracy bi validation row partition respectively result submatrix block omit modify complement cross frobenius submatrix training stability suggest fix top become estimate bi three block optimize arrive final formulation generalize solution base reduction sir outline nonlinear assume characterize reduction correspond assign basis subspace sir span structural impose applicable unsupervised first statistical visualization predictive modeling goal latter quantity dimensional focus g r rx sir regression sir effective data global marginal projection manifold inverse sir sir matrix estimate linear sir assumption problematic linear manifold structure sir adapt localize inverse explanatory dimension structure around suggest compute construct local index neighbor consider response within slice belong sir reduce truncated reduction encode grouping classic typical cholesky back canonical approach progress projection span datum prohibitive approximate explicitly recover rank restrict span subspace contain consider sir slice matrix construction tx construction full reduce svd sir number slice substantial respectively entry near fix knn nj knn pairwise point knn knn lemma dimensionality projection hence fix n generate matrix eigenvalue eigenvector transform ambient estimator runtime randomize fast approximation information reliably solution truncate generalize add control improved sample begin contaminated pca method achieve art require runtime scale input propose applicable lastly adaptively coarse default uniformly sphere exponential increment iid es noise signal noise rank effect large strong use input average exponential sample variability cost multiplication ht c ccccc error absence randomization well established operate tend complexity typically product runtime rank relative approach base notice runtime approximately increase suggest order top p rank implementation p consecutive simulation replicate q percentage explain focus noise regime vary weak scenario direction covariate direction assign factor spherical criterion correlation report third predictive criterion use square onto subspace coefficient proportion set different n examine size sir slice ten neighbor subspace value randomize method result pn randomization beneficial irrespective set low randomization seem sir term adopt seem ccc ccc sir sir rand rand l ccc ccc low noise sir rand sir rand rand performance knn reduction supervise accuracy slice sir equal first contain digital handwritten grey scale digit extensively past contain ignore dependency neighbor remove become detail appendix reduction method reference fix value rank neighborhood accuracy training consists select identical fashion rank digit sir limitation allow number sir tend small sir pca increase digits train rand acc pdf digits train rand acc microarray purposes variation process classify origin expression cell individual population china feature pre neighborhood slice estimation classification report accuracy reduction estimate size upon baseline projection sir strong structure gene predictive produce comparable increase suggest potential advantage randomize parsimonious reduction model massive modern paper problem analyze tool recent estimator implement popular parametric supervise method sir regularization randomization quantify randomization contribute practical utility interesting relate acknowledgement sm acknowledge wu sm acknowledge support grant biology gm fa dms sg acknowledge slice nearest step fix instead basis matrix dimension rank describe set project onto axis digit mnist com mnist handwritten grey digits dimension large centered training file image microarray microarray expression replicate use version genome array input background correct intensity thorough annotation annotation processing pre process expression individual nd mark annotation remove technical improvement outline idea randomize text manifold focus localize locality laplacian method introduce allow reduction embedding seek datum embed ambient infer manifold represent neighborhood adjacency le preserve adjacency association laplacian decomposition eigenvalue nd notice certain laplace manifold motivation typically computational advantageous linear projection without locality projection approximation non laplacian reduction specify space locality preserve projection state bandwidth exclude trivial direction order eigenvalue neighborhood optimal require generalize last derivation entrie sparse neighborhood construct detail algorithm neighbor capture manifold subspace default power iteration value embed subspace p projection randomize complex generate underlying data scientific application increase feasibility reduction natural massive play role visualization predictive significant impact research review theoretical instrumental powerful provable stability classic efficiently dimension consideration error consideration runtime information apply estimator generalize mean scope work implement truncate randomized independent propose number infer svd reduction provide attractive accuracy argue due implicit impose approximation adaptive svd randomize estimator inverse sir localize illustrate methodology algorithmic extension three pca sir robust estimate direction randomize reduction low serve core engine reduction reduce truncate generalized setting input consist feature response sir apply widely case provide outline extension develop manifold focus locality
lead bt ct transform regression regressor collect observable reduce copy estimate large estimate quantile technique cutoff I bt give observe covariance sx similarly typically il simple notational estimate j integral estimate result slope function optimization transform solve plug choice cutoff discuss pca basis basis fourier basis become drawback discuss analysis paper reproduce quantile example plausible hard see theory also apply interpolation observation understand nonlinear correspond regression case limitation shall discuss nonlinear ill interval monotonically construct explain iii xu xu xu carry project adjacent convex xu xu show sense derive previous optimality j beyond scope dependent mild restriction moment exist c xy xy xy xy xy xu assumption smoothness require ensure identifiability eigenfunction thereby follow uniformly page role flat quantile essential define il xt impose discrete covariance twice continuously differentiable control discretization depend vice versa relation integral kernel reference condition precise case satisfy discretization speak old continuous xt compatible establish slope suppose satisfied inspection continuously observable assumption discretization negligible seem quite example seem mild functional analysis supplementary file establish average criterion tend small affect estimation work well cauchy work look one find perform dominate bic work figure increase become well essentially similar comment figure supplementary file deviation c normal cauchy cauchy theorem divide three subsection prove supplementary file notational uniformity line almost surely let b mu constant q soon mu mu conclusion straightforward hypothesis mu proof minimization u proof distribution absolutely surely however absolute conditional right mu define step event mu mu complete verification lemma pick define n separately order xu x xy pick fix independent rademacher apply conditional vc subgraph class universal constant envelope assign q universal q show taylor xu q xu xu xu xu order evaluate bound independent rademacher independent apply conditional regular conditional function h proposition appendix bind measurable c I constant necessary order conclude order evaluate mu mu mu bt jt jt jt j jt bt dt mu j j u j b u j mn complete acknowledgment author mit thank suggestion anonymous constructive theorem remark aid b quantile covariate subject per increase size give unit quantile index likewise function quantile covariate slope estimator plug plug necessarily monotonicity quantile index consider estimator rate norm rate minimax kernel slope initially seminal kb attractive make estimate quantile material quantile functional quantile scalar covariate quantile model covariate suppose differ subject number sample allow quantile vary interval slope two time quantile quantile quantile index covariate slope slope pca terms standard estimating estimate estimator construct necessarily monotonicity conditional quantile estimator slope plug estimator rate minimax smoothness cutoff empirically namely integrate aic choose cutoff criterion limit simulation although criterion work increasingly grid multivariate large grid high view take datum refer reader comprehensive treatment early study functional linear reference establish fundamental linear derive slope continuously expansion slope function early quantile function quantile model establish rate sharp nonparametric estimation quantile function consider indirect conditional quantile possibly adapt developed invert estimate regression conditional quantile parameter check conditional quantile indirect direct indirect flexible offer contain value conditional useful appear say occur growth height picture predictive height know mean response growth datum quantile linear benchmark quantile quantile vector slope empirical ill pose cutoff level ill pose slope function conceptually handle paper nature nonlinearity technical view challenging build upon asymptotic estimator g arise essentially regressor estimation careful moment calculation additionally bring technical account covariate technical formal remainder
penalization powerful role classical regression popular penalization technique net regression covariate direct penalization dimensionality exceed incorporate structural importantly fundamental parameter regression well low number see affect regression illustrative example x z usual size rest inner differ usual detection imaging infer adjust b criterion bic along solution suggest truth solution nuclear singular nuclear size display nuclear matrix nuclear norm achieve substantially lasso might argue fuse lasso might piecewise signal numerous fuse lasso article family covariate regularization formulation within generalize glm variety penalization lasso elastic net scad response outcome highly scalable scalable effective select parameter extension freedom development hand aim scientific receive attention involve tensor multidimensional array covariate idea cp introduce fit tensor thresholding covariate case contrast fix thresholde non logistic work research completion aim portion concern covariate organize follow optimization degree propose discussion potential future section notation vector order mapping instance model belong simplicity covariate associate straightforwardly univariate extension model generality regularization glm likelihood bf square parameter indexing tuning penalty lasso ridge penalty elastic penalty scad penalty scad spline knot act signal penalty lead regularize estimate mc scad beyond lasso shrinkage comment besides penalty depend scientific w q produce e cluster eigen convexity essential study necessary sufficient furthermore subdifferential singular v v lead optimality regularizer convex local minima regularize program global minimum strictly u loss convex nesterov minimize state building algorithm v share b singular thresholding nuclear regularization singular u singular matrix thresholding ready nesterov nesterov attract regularization resemble descent first utilize search algorithmic incur improve rate dramatically optimal wide nesterov predict gradient force iterate initialize h critical role update nesterov whereas also base loss search irrelevant optimization ridge act trust region next iterate loss denote differentiable u u update dynamically obtain value decomposition intermediate matrix singular top retrieve efficient solver rank row lasso part minimization know nesterov iterate nuclear regularization summarize omit reader refer continuously differentiable iterate monotonically remark decrease guarantee potential algorithmic essential objective step fail objective iterate fortunately rate nesterov cover commonly convex nesterov reweighted square nesterov penalize least convex way glm step expensive rough potentially cut twice differentiable continuously v glm systematic b x cauchy v parameter yield well along regularization path criterion commonly practice datum considerable attractive criterion report penalty power finding illustrative shape synthetic compare vary lastly motivate eeg mention employ examine variety shape follow aforementioned regularization square signal disk exact see version counterpart yield signal lasso lasso power power lasso power signal compare rank generate covariate covariate consist percentage entry entry level rest zero binomial response systematic normal binomial former bic mean independent validation tune set rmse mis common table highlight bold face since rmse array coefficient brevity sparsity method power power power lasso power lasso lasso response penaltie exception non zero element binomial rank superiority predict signal insensitive contrast propose version insensitive sparsity agree expectation coefficient finally penalty yield lasso perform well agree useful eeg regularization individual individual subject channel perform stimulus match one association pattern eeg randomly subject control horizontal vertical axis curve distinct analyze stimulus trial result covariate indicate th power specifically divide full training fold leave far fold test misclassification error report leave see regularize counterpart lasso achieve well leave fold remark glm method regularization employ ridge maximized classification well fair solely train report testing error optimistic principal pca employ variant pca reason preprocesse eeg apply potentially information layer choose version determine number goal datum analysis primarily counterpart require tune modern area regression regularization crucial role regression complex propose regularize spectrum upon signal approximate focus rather nonzero entry penalization original coordinate jj often eeg fouri direct interest regularize analog concentrate problem throughout fmri array tensor extend tensor regression tensor analogous fundamentally currently report elsewhere pt proof lemma develop matrix fan fan entry order low statement fan singular let function order conjugate f f value v subdifferential inequality x x simultaneously u f equal decomposition b b b proof calculus gradient multivariate jacobian hold partial derivative jacobian chain jacobian fit value freedom denote usual square estimate v tackle piece singular right singular eigenvectors b coincide singular rule b p u b jacobian b b p cyclic permutation trace u p u p p b b piece symmetry let value utilize matrix chain b v shows b v I u yield next ridge response piece show
region select good function introduction weight feature feature entropy addition feature speed viewpoint redundant list construct line help redundant viewpoint line example remove fail might compare exhaustive fortunately extreme happen fail construct decide decision system q case indicate would execute remain base similar hard expert matter always competition working specifie obtain maximal winner construct exponential specify competition strategy max purpose answer question select repository database list pt ht vote game decision vote name library test information correspond three distribution pareto simplicity test ds test cost ds ds backtrack three backtracking step size diagnosis optimal always third run time backtrack propose step min backtrack voting table show backtrack namely backtrack size backtrack tree dataset sometimes indicate sometimes sometimes maximal compare backtrack run algorithm take real backtracking algorithm time information extension computation concern among sensitive reduction define classification classification boundary classification address extension concern symbolic partition transform combine concern different computed loss major model semantic application rough indicate precision rough concern probabilistic rough rough set neighborhood rough region rough error range range test information entropy meet misclassification decision cost misclassification value likely select weak less coincide compute reliable parallel unlike positive conditional increase change relation similarity entropy change minimal total meaningful identify definition change issue combination exist extension involve extension area research viewpoint concern test cost constraint limited backtracking algorithm experimental efficiency backtrack one effectiveness competition heuristic note competition viewpoint rough selection natural generalization viewpoint help identify meaningful aspect input research trend concern beyond rough set natural foundation science foundation china j technology project china grant world cost include limited resource shall informative constraint backtrack efficient solving sized backtrack scalable heuristic heuristic find theoretic rough set viewpoint new research sensitive constraint backtrack theoretic set employ improve hundred attribute rough jointly individually decision often address region problem minimal simple representation often provide generalization problem aim maximal minimal store free application resource take like select number develop relate identify address numerical observational searching subset preserve information nevertheless resource limited necessary constraint resource serve problem coincide constraint condition fall namely input new simple viewpoint furthermore viewpoint show selection rough rough minimal aspect viewpoint insight trend concern broad viewpoint attribute systematic backtracking medium backtracking algorithm obtain however deal see e reason people problem exhaustive backtrack respect heuristic complexity dataset employ function prefer low stop improve algorithm employ competition value open algorithm experimental backtracking sized obtain backtrack time fast another backtrack competition rest organize problem propose backtrack uci california exist feature rough viewpoint interesting problem briefly conclude review rough classical rough concerned new selection test constraint fundamental ds q object universe decision information feature partition determine brevity pos iff pos sufficient individually necessary preserve context decision name necessity attribute reduction pos pos may exist relative simple minimal objective constraint namely necessity meet necessity feature minimal word simplify view hierarchy sensitive minimal output see problem input cost external sometimes issue respective ds constraint eq feature subset q set element constraint brevity constraint meet select ensure test minimize construct read problem cost input bind objective important primary objective problem simple phenomenon appropriate observe constraint region objective secondary primary minimal subject problem meet essentially redundant serve constraint objective consequently coincide backtrack backtracking produce problem heuristic subset backtrack rough people employ attribute reduction partly form backtrack invoke global statement backtrack execution store tb store backtracking continue constraint violate exception coincide subset backtrack backtrack discard coincide need devoted optimization serve implementation repeat positive note remove ensure correctness
contain projection code maximize entropy partitioning provide select projection equally candidate entropy consume thus center simplification reduce time calculation obtain entry sort top overall l definition adjacent group intercept sort create binary data dimensionality mean adjacent intercept alg compute obtain step alg clear scale stage need code sensitive hash high near million feature dim publicly collect extraction dim publicly sift dim publicly remain return close distance accord distance average precision recall average pt seven algorithm near neighbor locality hash projective randomly locality hashing generalize space use shift approximate invariant code principle hash direction projective pca publicly code author anchor construct anchor superior use author nearest publicly three control group adjacent selection lsh projection method regard lsh cccc c lsh cccc lsh sift set three lsh low consistently increase fail code increase decrease code length probably later use poorly discriminative geometric projection advantage method almost outperform sift large sift present bit time algorithm method dimension consider projection lsh computation relatively slow explore fast algorithm test lsh method long analytical eigenfunction involve sift sift alpha alpha alpha sift sift control number group performance bit vary achieve change increase reasonable balance consider efficiency sift vary adjacent consistent separate projection critical redundant decrease hash call hash neighbor base hash locality hashing guide projection hash empirical study three set scale hash let proposition thm li fundamental mining pattern recognition hashing locality sensitive hash lsh hashing find table hash require precision recall address propose hash hashing regard lsh explore geometric avoid purely projective function agree extensive experimental world set locality cluster neighbor nn efficient build structure propose neighbor unfortunately bad scan dimensionality high intrinsic difficulty propose search key datum preserve hash random preserve one locality lsh offer time query sub linear however lsh random preserve therefore order probability object hash lsh many hash long lead storage full hashing affinity item item similarity code success small code fail increase hash call effective dimensional near neighbor regard extension lsh hashing try utilize geometric guide projection hash adjacent group one select entropy hash art review hash experimental result art remark hash hashing intercept hashing find respect hash locality sensitive lsh lsh relative map hash code hash lead storage space limitation hash propose pca hashing simple choose direction exploit affinity binary spectral analysis affinity consume hashing analytical geometric ignore anchor hashing overcome generate affinity efficiently hashing hashing stack restrict rbm ph supervision learn hashing fail rp detailed sensitive hashing increase length framework lsh lsh generate guide selection present toy illustrate idea gaussians plane ask encode hash code lsh projection hash projective four gaussians hash generate code generate subsection quantization recently performance search
regret cumulative loss loss typically notion sequence evaluated online convex mirror consecutive bound mirror regularizer regularizers achieve work corner different corner expert sharing share achieve set simplex proper trajectory corner mirror generalize set simplex notion distance rely generalized notion include case regret project mirror fix essentially technique online improvement loss parameter simplicity show extended linearization may cast repeat game forecaster denote simplex forecaster choose forecaster accumulate g goal forecaster expert choose rich state goal describe generalize formulation algorithm parameterize past make serve generalized algorithm share rate dt e past weight g discount see regret discount factor instantaneous forecaster sequence vector variation expert regularity dd dd traditional obtain share algorithm perform supplementary material generalize optimize tune tune corollary share eq exactly share expert almost well vector eq correspond time fixed forecaster advance norm valid order optimize maximal illustrated omit vector optimally proof theorem multiply examine first update substitute substituting summing sequence loss forecaster regret specialize modification forecaster study drop forecaster corollary adaptive regret forecaster change shift notion tight exp forecaster enjoy guarantee forecaster update optimally mention obtain factor regret sequence remain r bind theorem introduce discount old horizon close game theoretic set matter recent close mostly monotonic sequence decrease non assume requirement get satisfie u natural quantity something claim whenever monotonic regularity monotonicity know advance arise tuning discount order fix via contrast guarantee therein trick fix forecaster gain manner next must horizon discount latter weight positive fix tuning time basically know advance forecaster easily focus loss tuning parameter regret forecaster improve expert focus share update run update sequence l l online technique refinement substitute analysis section replacement state introduce able switch time appear naturally sequence upper union support generalize update generalize simple satisfy exist improve theorem suppose share vector sequence corollary material tuned quantity horizon trick parameter extend focus run replace description loss number define way update performance sequence loss constraint provide illustration acknowledgment acknowledge national grant exploration exploitation resource allocation use body simplex simplex forecaster bind different beginning depend remain modified u I proceeding claim next left side rewrite I I entail dc I w nonnegative since maxima nonnegative e satisfied one q real well understand optimally choosing get numerator denominator straightforward need use ss bound
linear price vector budget bundle bundle structural immediate pair equivalently bundle good operate ratio sort optimal bundle spend item ratio high probability op price budget intuition bundle ratio bundle bundle need bind good preference well follow observe bundle class j pr j pr j bundle I example pair less bundle know preference bundle modify utility agent separable derivative concavity marginal item agent ia v ia ib derivative utility first adapt function bundle characterize p ix bx maximum bundle budget bundle intuition follow utility observation optimal bundle derivative need infer bundle k initially li kx li kx kx jx j kx kx new thresholds j b budget maintain ratio derivative distinct equally maintain ratio discretization convenience bound analogously maintain ratio concave maintain throughout pairwise ratio maintain pairwise ratio derivative infer item bind analogously update update appropriately completes use item property ki rest step represent comparison prefer section threshold sequence ratio threshold sequence threshold learner desire polynomial threshold include conclude threshold bundle optimum bundle discretization bundle admit separable introduce get strong feedback complexity require interact constraint linear bundle agent else return bundle either bundle almost receive linear program enough cut fraction polytope probability polynomial argument polytope begin model agent utility say bundle bundle accept agent bundle propose instead inequality optimality return pair two price domain distribution suboptimal bundle result exist inequality evidence suboptimal pair price value approximately bundle ip jx optimal return bundle object budget enough object suboptimal exchange unit object unit constraint pair vector without generality bundle object budget limit decrease make value increase unit unit rv I v completes claim maintain represent initially sample example sample uniformly body convex polynomial time end add program volume stop learn algorithm replace new body confusion final return end future explain example mechanism bundle either reject constraint end restrict form terminate hypothesis succeed predict valuable iteration example number phase terminate return utility function body bundle return bundle suboptimal subset suboptimal suboptimal probability return bundle return bundle body prove claim contradiction pair begin iteration compute end bundle suboptimal probability volume less volume probability feedback feedback suboptimal pair new add fraction case loop example hold property cca happen less reveal reveal preference course tight dimension reveal preference dimension ill suited unit without constraint exchange beneficial optimality case bundle identically exist prove claim I complete follow item item infer low bound infer find p p j p il il satisfie threshold threshold object maximum increase time result bundle range stop strictly budget unit threshold budget unit satisfy remain unit threshold want must try pair algorithm lemma object v jx j nk jx nk update low thing claim similar lemma lower less union accurate point threshold least jx nk happen therefore threshold infer imply property threshold threshold lemma find prove bundle property case ix bundle completely optimum might something else budget completely compare remain budget object bound least value fraction object increase budget optimum increase j optimum v budget comparison q kk construction update begin prove low iteration actual value vector vector loop constraint claim reveal preference perspective price budget draw bundle utility price budget utility hypothesis agent future algorithm agent linearly separable population merely suppose day day day attempt utility budget predict optimally behavior reveal nice preference focus determine whether classic utility sufficient explain past future increase piecewise restrict utility make inefficient computational utility accurately utility maximize access behavior restrict class linearly separable utility nd polynomial learn predictive polynomially agent select bundle goal learning correctly price utility generally utility bundle price price agent budget maximizer price
measurable dirichlet stick break process form thereby inference need inference monte carlo bayesian collapse unlike lead base base function parametrize draw define reproducing embed therefore important observation truncation stick already excellent approximation version almost sure truncation nonparametric sure hilbert inequality arbitrary kx f kx yield te sufficiently work much small choose effect truncation level would employ hilbert kx x kx cast prevent overfitte regularizer qp identity ij kx f similar thus reader perform use conceptually variational approximate moreover vb require access require value investigate understand relate method likelihood effect connection k answer method hilbert embed dirichlet stick break powerful avoid need intractable frequentist suffer comparison benefit efficient good dirichlet explain nice come source property heterogeneity parameter regard extensively community existence
derivative propose unconstrained multivariate kernel explore applicability convergence simple flexible finite behaviour new drive finally proof th multivariate density derivative derivative multivariate taylor compute th derivative involve partial variate derivative organize hessian hessian usual convention x however clear derivative adopt define product naturally h estimator symmetric symmetric definite f ik see context density dd dramatically draw definite moreover issue loss severe unconstraine bandwidth derivative improvement integrate error quantity integrate minimizer f stochastic detail integrate iv f rf expand show explicit x wise application approximation minimizer h expression analysis kernel involve strategy quantity optimal density derivative propose bandwidth datum asymptotic cross validation density univariate name cross cv study detail seminal paper either oriented consideration estimation part write smooth rf k cv coincide h mn sum equal third x useful bandwidth plug pi formula plug univariate density vector plug selector criterion analyze estimate f pilot possibly pilot bandwidth square square pilot bandwidth r g square optimal bandwidth overcome involve successive kernel estimation basis bandwidth minimize selector derivative counterpart derivative estimation explore gs criterion pilot selector lead g g analogously plug pilot selector selector build say rate denote indirect appendix rate eq bandwidth proof defer hold convergence selector plug selector smooth cv partial e previous unconstrained cv attain constant asymptotic expression seem possible pi bandwidth slightly special term achievable bandwidth general unconstraine slight loss optimal unconstrained see similarity note asymptotic pi estimation exhibit rate cv selector noting spirit cv selector different lower indicate slow tend rate sample size tend attain high provide realistic take account ignore explicit formula latter finite bivariate examine closely section c c cv pi pi pi kernel matrix short plug initialize th pilot selector sample stage pilot selector minimizer selector eq derivation previous facilitate execution selector modification pi selector include simulation minimizer nr cv pi smoothed cross develop exist library bivariate display definition find htp selector square conduct conclusion decide selector selector give mixture density selector nr selector previously available uniformly line simulation cv selector though former cv attention variability introduce oracle complicate different occur check try discover true datum consistently estimate slightly property pi correspondingly examine nr shift direction density produce ascent follow recursively density recognize variant algorithm employ advantage normalization illustrate accelerate sequence density attain unimodal profile decrease therefore g estimator eq ng understand unnormalized kernel term shift arrange note reasonable equation lead recursively define eq adapt cover unconstraine shift point successive gradient take mean bandwidth selector support result estimate relate derivative thus bandwidth goal identification mind contrast take variable convergence reach however task seek determination estimator engineering point way cluster local modal specify advance discover start partition apply shift mode grey point automatically nonparametric procedure via compare cluster technique impossible selection introduce shrink closely relate shift shift median pdf density identify normal multiply instance due include since recognize three compare normal label nr bandwidth cv plug bandwidth label comparison along five bivariate density situation shape different scale intended explore since explicit modification equally component break bivariate define mean normally matrix break weight half weight component radius indicator picture figure sample em adjust rand ari correct chance version partition prefer ari value membership membership estimate assignment cluster accord seven plus ari ari nr cv pi break none shift pi bandwidth performance choice rate second ari inferior performance mixture acceptable scale nr inferior break intensive provide quick initial follow especially difficult unable situation contrary powerful setup seem reasonably study ad hoc comparable shift exception surely careful bandwidth shift conjunction selection apply tends produce spurious mode tail enhance due phenomenon chapter sim real form singleton group outcome shift algorithm identify less singular mean bandwidth singular naturally newly requirement similar call merging rate henceforth group uci repository molecular biology university represent feature protein localization site observation om pp I cp find original seven five scale cluster correction pi merge group shift nr membership performance configuration nr pi group datum remarkably ari performance poor ari give reasonable ari introduce consist eight chemical sample divide nine area region cluster compositional reference chemical measurement transform place euclidean principal euclidean major try discover sub clearly recognize either grouping region ari division area nr grouping method remarkable ari ari bandwidth shift respect although region divide several shift bandwidth assignment smaller show ari shift pi lead pi always derivative negative set definite reject distribution adjustment testing modal maxima bandwidth significant curvature selector figure record depth west date distance surface depth depth pi estimation region three mode obtain bandwidth grant author author france institute sciences work henceforth symmetric x partial derivative integrable develop section integral suitable exchange taylor expansion well define lemma r r h r integrate thus h h theory lemma function x h k k term involve express r r h h x derivative read desire return asymptotic real moment pa
set diagonal complete consider projection solve subproblem corollary pass generalize describe spectral project gradient simplest suitable one iterate discuss important fairly invoke typically fast ordinary projection formally thereby bb global extraction subsequence theoretically depend compute obviously operator include norm must fact due treat inexact main aim separable say entire preferable simple realization replace stochastic result suitable additional account potential projection result norm show running method ease projection onto ball refer qp experimental fp experiment core bit ram setting measure table difference qp fp qp competitive fp consistently although fp twice qp fp method qp fp magnitude qp r qp fp qp fp qp e e e e seek column parameter row group row entire eliminate combine column overall norm notation yield long reduce ordinary make use mini row subset contribute w objective whereby upon iteration mn w become bottleneck counter may restrict subsequence implementation mini comparison qp fp solver solve h million million r projection reach spend method sgd use fp reduce science spread method initialize rapidly accuracy start eventually get interestingly long attribute difference begin take general either sgd yield rapidly problem prefer extended matrix duality theory mix norm especially special norm overall problem norm suggest algorithm gradient experiment lasso instance class mix study projection g bound extend method euclidean operator explore regularizer remark joint structural lasso efficient surprisingly seem present gradient projection onto open mixed norm norm stochastic sparsity encode permit widely therein literature reader sparsity constrain problem cast follow nonsmooth regularizer alternatively prefer perhaps latter latter primarily admit optimization algorithm constrain formulation attractive include nonconvex applicable separable inexact projection realistic type remain simple recently lead example enforce especially choice use lasso compress version letting differently unless impractical mixed norm hard since move projection ball batch mixed norm open develop basic aim efficiently algorithm iterate gradient constraint computation projection operator tie projection projection indicator compute whenever proximity exploit lagrangian dual duality primal optimal key equation nothing observe equal accurately solve useful let define exist decrease differ assumption claim fx yx finally differ change interval unique therein root iteration recommend use rather invoke combine interpolation fy fy fy fy fy ball generic projection upper prove dual conjugate dual partition p follow old invoke h u p conclude q u
treat theoretic plug link maximum hilbert schmidt rkhs depend bilinear form moreover definite definite operator deal look property empirical version note definition like calculus spectral gram construct infinitely relate spectrum I satisfy eigenvalue therefore eigenvector positive consequence relation proposition focus attention spectrum previously follow find compact adjoint separable hilbert compact let enumeration nonzero extend enumeration theorem operator normalize infinitely matrix hoeffding take hilbert schmidt combine yield definite consequence convergence constant gram seem extend conditional normalization q x similar follow report entropy hadamard gram experimental setup similar pick benchmark shift exponential student double multimodal unimodal asymmetric multimodal asymmetric gaussian asymmetric analysis theory need entropy product infinitely reproduce space dependent gram operator numerical usefulness want show metric positive symmetry cauchy schwarz relate calculus reasonable hermitian nd exposition space countable necessarily finite consider generalize generalized probability h strictly denote sequence entropy average define partial ordering measurable say shannon htp p concatenation union argument refinement orthogonality refinement possible refinement generalize cut simplex positive function respectively come possible belong turn fx f ff minimum state reproduce reproduce q definite respectively tensor definite x x j definite representation look theorem derive restriction restriction f gx hilbert space set definite class infinitely proposition corollary section rao department electrical computer engineering department university usa edu principled way among theoretic statistic pose challenge plug operator reproduce kernel infinitely functional entropy matrix axiom entropy result difference integral positive avoid estimation moreover mutual information independence art operational theory base law generation probability regard interest probability advance available finite theoretic call two quantity estimation case quality continuous plug rather employ choose assumption parametric tuning computationally possibility bring incorporate smoothing capacity control mechanism despite difficulty suitable version quantity entropy serve unsupervise variable density random equation entropy functional shannon entropy suffice monotonically plug estimator c potential show special potential density reproduce kernel space rkhs idea already rkh bring estimator go establish employ positive without independent input important mention recently minimum span consistently estimate nevertheless similar unbiased near graph introduce error despite nice objective conventional propose differentiable introduce shannon differential work estimate distribution functional positive event evaluate kernel pair reproduce differ scope matrix evaluation definite available datum drive act presence gram information property propose functional choice define show infinitely subject suited informally motivate measure plug base windows mechanic operator definite functional set axiom closely definition entropy hadamard product infinitely particularly entropy mutual sum entropy behaviour matrix statistical gram relation arise dimensionality finally carry independence art reproduce space idea however decade popular one deal provide kernel practical datum problem involve notice compute first statistic hilbert schmidt measure motivate descriptor datum concept algebra probability feature family xt xt completion deal difficult impossible result give deal bivariate define product hilbert reproduce kernel hilbert kernel eq call definite abstract space let simply allow relation element element verify gram fundamental method order base interpret reproduce convolution integral correspondence reproduce space implicit potential correspond law integrable square integrable bilinear uniformly integrable probability density square kernel define rkhs note x bound albeit set verify eq take window integrate information motivate information gram concept property entropy carry generalization argument quantity information monotonic theoretic also method derive serve mainly aim structured subsequently example probability product investigation information would reader theory definite provide set axiom measure axiom completeness provide axiom entropy employ nonnegative definite generalization extension matrix use spectral definite real aa definite satisfie condition monotonic take unitary trace invariant unitary continuity calculation thus power eigenvalue simultaneously matrix eigenvalue respectively yield notice provide eigenvalue lot operational quantum von entropy extension also functional gram obtain evaluation positive construct positive mapping induce key matrix definition hadamard product hadamard product definite extend entropy representation pair represent two realization ij hadamard compute x compatible individual entropy impose expect joint entropy component concept say order doubly easy important convex define eq concavity convexity concavity ready hadamard entropy individual entropy positive entry prove preserve simplex first consider order respective eigenvalue order eigenvector order accord respective inequality extreme shannon remain observe joint contrast multiple positive nonnegative unit quantity nonnegative mutual knowledge marginal gain entropy entropy analogy unit base divergence joint inequality need extra condition element notice element simple focus
svm obtain primal sub translate bad convergence paper ascent different proof comparable nevertheless dual analysis show gap achieve addition analyze version minimization dual inferior neither analyse logistic shall bound optimality interested duality sdca superior early randomization important cyclic dual coordinate order slow bound cyclic inferior analysis mean cyclic ascent inferior structural frank wolfe sdca hinge convergence rate sdca lipschitz relevant gradient sag recently analyze loss match regime sdca practical explain loss type primal frank dual sdca sgd dual describe code parameter good choice gap terminate sufficiently iw w return analyze simplify assume follow lipschitz duality suffice total optimality take choose type advantage average easier implement stop duality gap average hinge however inequality consider sdca duality gap suffice moreover requirement sgd namely suffer loss problem behave like regime smooth bind like practical observation smooth still dominate sgd sdca however sgd look sdca basically much sdca sdca may take size sdca help natural sdca epoch perform modify dual end sdca introduce convenient notation denote primal dual primal objective maximize dual objective end modify sgd modify prove sdca employ sdca require extra sdca procedure sdca sgd initialization modify sdca iid sdca sgd initialization duality stage suffice sdca duality ideally show sdca sgd ideal result show step sdca hinge analysis sgd versus advantage sdca explain sdca sgd try refine sdca asymptotically refined rely complicated precise interpretation result use explain sdca sgd loss although pass small gram matrix arbitrarily bad argument dependency gram smooth loss example hinge convex dual rely refine strong everywhere smooth loss variable eq therefore deviation convexity svm close away mean nearly point assumption may procedure iteration superior consider three ns os factor remove slightly convergence duality gap consider sdca eigenvalue define small superior complex case duality imply convergence specify sdca loss sgd may epoch sdca sdca permutation iw hinge use procedure sdca loss sdca close hinge significantly confirm ia loss sdca solution hinge hinge step sdca solution smoothed hinge worse confirm experiment sub gradient fact primal formulation proof theorem increase dual duality gap lower bound therefore upper duality theorem similar objective duality strongly iteration update dual write strongly combine rearrange obtain take obtain eq side conclude duality optimality last equip ready small require duality require prove first part eq either imply need turn lipschitz rely fix lipschitz ready theorem tell claim choose rearrange obtain q vector randomly choose iteration draw optimizer maximize dual derivation basic eq obtain proof need feasible display inequality follow nearly identical focus valid update objective obtain q equality I rearrange moreover fact n sum prove hinge take dual denote gap perform dataset different sparsity paper physics collection detail sparsity ph convergence indeed run sdca vertical straight linear original sdca minimization regularization show smoothed hinge apparent value linear still refine slow observe refined figure smoothness rate convergence dominant effect zero hinge error smoothed hinge provide hinge apparent hinge one repetition sdca sdca vs fix cyclic see cyclic rate slow cyclic dual ascent similar behavior cyclic ascent inferior sdca sdca particular sgd w iw x ii clear sdca calculate primal sdca sdca sdca regimes sgd theory sdca quickly smoothed hinge primal depict epoch ph divide set epoch gap depict ph horizontal divide corresponding epoch duality gap round sdca smoothed hinge plot axis one axis training duality achieve dual repetition choose random repetition sdca fix cyclic order depict sdca smoothed axis epoch optimality
approximation expect peak furthermore show evolution posterior sharp form evolution vertical colored enhance peak minimizers minimizer estimate minima median popular noise variance function minimum addition illustrate active surface apply acquisition criterion relatively initialize random surface grid evolution estimate sample contour select sample dot minima dots show median perform response surface slow due inaccurate estimation black dot one global fail minimum accurate accurate two minimizer c propose active criterion learn optimize transform process plan accuracy computer engineering usa mail edu minimizer criterion ignore uncertainty process uncertainty minimizer tractable optimization criterion explore unknown space search quite costly aim active carefully order reduce learning follow interest need surface query oracle potentially quickly minimizer mostly obtain function minimum providing consequence often discard minimizer could discard effective solution acquisition criterion balance tailor find problem densely potential minimizer minimizer maintain enable multiple minimizer target function furthermore unique noisy objective minimizer function often utilize focus contribute random variable minimizer functional next entropy additional straightforward intractable utilize widely posterior pointwise estimate minimum equality definition minimizer fx x upper q covariance normalize proxy px broad two minimizer minimizer natural modal multiple lead ambiguity independent remove covariance acquisition estimate
utilize experiment calculate conventional method include success summarize result ccccc specific success time size less compare utilize consider orthogonality pc utilize ground truth verify evaluate method assessment minimization maximization sparse pc loading reconstruct error pc reconstruct variance reconstruct utilize method introduce utilize evaluation method pc extract nonzero element nonzero table figure illumination methodology calculate pc pc pc include setting pc achieve lowest compete advantageous easy superiority extract pc component propose small propose maximization view consists extract dna micro array employ extract adopt pc loading try time besides carefully tune pc similar get ccccc easy low pca demonstrates extract pc advantage dominant propose usefulness exist nonnegative formulate covariance j schmidt design respectively success introduce list ccccc pca success nonnegative pca utilize experiment utilize nonnegative calculation adopt specify pc pc fair comparison pc evident dominate low pc furthermore advantageous extract propose nonnegative pca calculation far verify recognition together performance application mit mit resolution utilize loading face comparison cccc depict nonnegative pc clearly exhibit utilize face among show usefulness choose non mit training rest image extract utilize pca method respectively project pc obtain four respectively fitting lr directly fit original easy cccc face face potential novel methodology method decompose pca sub thus easily effectively new extended pca certain extensive current sparse explore model heavy correspondingly difficult adopt much problem although convergence know stochastic optimization anneal computation far improve thresholding correspond equal indicator function q feasible easy attain kk optimal complete denote permutation soft soft tf optimal thus solution exist feasible easy optimum deduce monotonic decrease tf k number complete deduce monotonic decreasing prove please lemma imply solution attain present w I follow discriminant solution express mm lemmas conclusion appendix prove equivalently solution first sign zeros one region v position nonnegative otherwise pick denote get w j w positive nonzero element new get first fact j j p p conclusion conclusion conduct c hold tp p methodology original pca simple sub recursively analytical solve sparse pca computation extend pca nonnegative method maximization face principal divide sparsity classical wide successful application throughout engineering pc maximally preserve always intrinsic feature conventional pca suffer component combination loading typically zero interpretation biology gene loading sparse asset pc loading especially expect nonzero loading transaction cost much attention recent decade topic develop attempt certain post e rotation loading pca enforce sparsity advance simultaneously pca loading al net et call pc semidefinite programming sdp series pc factorization include et issue concave maximization induce extract block simultaneously e solve recently briefly additionally set solution number coefficient etc pc sequentially pc properly extract pc pc utilize optimization sparse simultaneously attain pc requirement nonzero pc divide sparse make greedy approach method appropriate sparse pc iteratively pc way constraint pc constraint complexity dimensionality suit idea method introduce series scalar upper bold bold letter low letter denote maximized maximization viewpoint pca formulate trace denote empty imply exist denote I note decrease respect optimum main idea close form attain update implement iteratively recursively update stop summarize aforementioned algorithm pc solve pc loading briefly stop monotonically step terminate update reach computational evident algorithm essentially iteration e calculation operation involve require around sort computational approximately dimensionality input evaluate monotonic optimize column low convergent evaluate formulation indicator equivalent unconstrained solving set compact generate assumption regular hold condition theorem see advantage propose methodology easily need pc
stochastic difficult pair give result filter decomposition yield true backward filter h estimate likelihood generalize decomposition numerical suggest new sensitive point forward backward filter filter may beneficial forward explore introduce extend backward target upon w dx dx n n remainder mix backward follow formula zero property schwarz deduce convergence conclude subsequent notation define q series schwarz expectation note jensen inequality corollary paper adapt deal fix depend upon section mathematic college e mail department national mail hmms use sequential smc filter expectation marginal smoothing approximation two filter smoothing decomposition investigate likelihood description phenomena time process unobserve observe hide n dx xx dominate observation I x k n n dominating hmm often state static shall observe term interest popular collection hmms smc parallel combine sequence increase state introduction hmms normalizing constant representation understand smc recent design filter algorithm approximate appropriately meet smoothing relative standard likelihood decomposition via first perspective second follow investigate filter backward proof joint omit smc idea parallel reason sometimes particle adopt compute un j ai un j return joint distribution technique smc generalize two representation define smc meet density convention essentially efficiency normalize approximate sometimes index adopt notation un l ai ai un n return estimate normalize one calculation p tx x resample time marginal ny ny undesirable cost x x dx backward respectively time ny ny dx approximate estimate detailed omit establish divide e normalize obey filter backward intuition vice versa w joint calculation reveal e ny z q appendix notation lemma univariate idea compare variance filtering allow one representation within distribution section x n r calculate via observe affect recover distribution resample run nine plot variability estimate
condition identify offer penalize concentration matrix matrix suggest penalty penalty trace nuclear necessity could prohibitive efficient attractive penalty rank respectively course attempt penalty nuclear computational integer equal except replace modification reflect adapt deal clear call estimate glasso lin also hereafter refer method short ex pt infeasible interestingly though computation thank combination recent advance compute lasso latent clear observe estimator datum em iteratively penalize current lk nk k recall eq therefore q note replace penalty become glasso solved iteratively correspondingly maximize glasso purpose recover uniformly pair location location variable connect entry covariance ensure typical latent left panel figure along glasso use provide insensitive report little variation shall brevity play critical role fine recover pattern roc glasso close necessity account interesting perform penalization method
refer cell available cd cd focus variable cd seed summary output fit component fit model summary fit single skew mean skewness mixture component matrix freedom five fit fm posterior membership row evaluate argument aic bayes show fit multivariate component five rest cluster rest omit aic em user suppose specify list specify demand achieve illustration datum body record component skew height setting examine skew component iteration iteration matrix skewness component proportion compare restricted version distribution skew normal skew skew use component constrain label true label reveal fm high fm give fit skew group true false r nan nan eps fitting fm mt skewness fm nan statistic fm respectively asymptotically difference hypothesis alternatively value directly specify last consider examine fm high skew mixture skewness fm significantly c specify fm perform datum fm specification fm fm fit classify seed x train contours generate contour grid cluster level nan scatter heat contour cluster nan coordinate either five default include plot contour require option specify label specify percentile contour contour require quantile plot bivariate scatter option fit specific mixture mixture contour plot length take argument argument colour contour generate contour fit heat return scatter contour contour fit fm point cluster r model fit suppose interested simulated datum dx fit contour model plot intensity plot contour component plot provide contain intensity know marker cell patient parallel intensity surface protein involve identification population perform manually visually separate interest approach marker develop method sample three marker cd cd group fm increase dot accord expert consider b percentile contour display visualization device support user interactive viewpoint follow code fit contour fit level effectiveness label manual take label permutation cluster result report minimum note remove evaluate fm superior method fm accord true contour fm dataset present package fit finite heterogeneous asymmetric form fit fm visualization contour major demonstrate institute result via restrict skew distribution gave skew slow higher intensive calculation multivariate benefit acceleration strategy acknowledgment support grant research correction helpful package distribution skew distribution implement em ml fm visualization contour finite skew flow various multivariate skew step paper focus mixture iterative usefulness propose application bivariate unimodal distribution institute comparison achieve misclassification rate analysis skew heterogeneous contain mixture distribution skewed model datum intensive past decade application scientific bioinformatics processing survey volume paper year skew exploit tool model multimodal asymmetric introduction skew sn study extension multivariate skew automate model density mixture normal family skew impose component expectation involve skew skew mixture model implement restrict expectation multivariate truncate variate central package mixture distribution em user contour fit density interactive viewpoint paper organize section brief fm fm explanation visualize fm present fm cluster dimensional vector skew distribution skewness eq corresponding cumulative p tt skew skew recent noting version incorrect consist incorrectly update degree freedom eq simplify make update rarely preserve working consume algorithm implement quickly mean attempt perform log initialize sample component th th create extract systematically across search traditional stopping criterion size acceleration describe absolute less tolerance asymptotic give fm fit represent represent degree proportion htbp cccc dimension description matrix skewness degree proportion function evaluate fm sigma delta skewness parameter
multiple way interaction reach amount hierarchical polynomial regularize differently algorithm typically good matlab proximal used online mkl identify variable code error feature mkl uniformly nonlinear radial validation testing repetition mkl polynomial regression use minimum polynomial report tailor mkl less nonlinearity evident radial section estimate repetition benchmark census delta segmentation uci regression build training set draw breast algorithm subsection protocol sample neighbor since method polynomial degree mkl result selection stock select equivalently case similar nonetheless bring provide choice method among plot clearly second derive definition follow example note mention easily start immediately second q proof procedure consequence general inexact algorithm proximity fast condition say inexact tf tf exist since regularizer composition ensure generate inexact share follow matrix operator trivial derive adjoint eq observe extend rewrite proceed hypothesis f prove observe operator functional satisfie euler arbitrary subdifferential use euler contradict also theorem construction subdifferential plug kind subdifferential subdifferential subtract inequality start prove hold ff f n technology usa di mit edu di university di variable dependence depend nonparametric key make derivative intuition propose new notion regularization reproduce hilbert method correspond solve estimator term analysis common bioinformatics computer thousand thousand dimensional regime feasible quantity satisfy regularity idea behind call turn construction interpretable interpretable situation multivariate relevant detecting largely motivated recent advance extensively linearly naive try subset meaningful greedy convex relaxation basis algorithm therein guarantee therein situation latter understand approach mostly additive function relate encoding among assume function depend though provide interpretable size initial notion additive idea suggest way sparsity regularizer essentially question derive actual issue reliably high ensure drive stable reproduce kernel hilbert tool question linear functional representation computation regularizer ensure derivative sparsity correspond concept space since correspond functional suitable rely backward splitting proximal third statistical practice mean preliminary appear conference discussion derivation proof consistency augment experiment paper organized section consistency extensive future functional precisely statistical measure minimize prior paper interested estimate relevant sparsity requirement norm often norm studied extensively understand reference regularizer minimize norm univariate norm respectively multiple kernel consider limit variable interact partially consider simple triplet fx general essentially propose attempt tackle problem allow discussion function interest paper discuss detail reference paper follow variable basic idea locally linear window depend locally select depend one partial select risk criterion include local thresholding backward step wise dimensionality variable emphasis quantify optimal pointwise improve see discussion recently estimator prescribe study careful regime cardinality globally relative scheme classical spline modern variation utilize capture notion partial notion variable consider follow issue first need variable partial regularity need depend support restrict linear capture relaxation functional encode classic replace regularizer sufficiently ensure good poorly spirit manifold propose hilbert rkhs differentiable latter ensure make strongly rkh allow computation partial potentially devoted algorithm remark entry subset comparison remark difference regularizer regularizer gradient version classical uniform measure regularizer derivative regularizer give force propose utilize root group point penalty note consider derivative differently localize correspond partial trivially pointwise define complicated uniform interval moreover although relevant impose basically imply open rkh summarize contribution main derivation provably convergent begin extend dimensional coefficient forward backward admit closed form proximity operator overall procedure multiplication thresholding derivative universal depend selection estimate consistency hold extensive analysis simulate work rich space correctly solve outperform set interpretability select subset hierarchical reproduce associate reproduce kernel hilbert start discuss regularize observing make tf standard minimizer divide part allow function minimization prove x denote derivative main outcome coefficient provably clearly soon explicit expression also unitary consist two iteration describe add size priori ensure experiment accelerate simple warm procedure discuss principled way discuss induce smooth practical ultimately definite function q reproduce basic fact pointwise induce point follow return thank rkhs allow variable importantly derivative return partial write regularize adjoint matrix differentiable minimization nonetheless computation splitting algorithm belong proximal large paper nesterov optimal first order since become signal fista accurate regularize functional modulus differentiable convex class backward suitably proximity term simple penalty use additive proximity discuss next paragraph briefly basic previous iterate function provide provide sometimes iterative backtracking require computation share rate well strongly accelerate scheme minimizer set fista tackle minimization formulation proximity operator subdifferential describe practically start norm unitary ball convex sum smooth indicator v easily correspond proximity indicator mention convergence guarantee thank structure possible see compute convergent accelerate outer proximity procedure iteration v initialization inexact proximal approximately split proximity inexact accelerate inexact still computation proximity sufficiently inexact convergence accuracy operator suitably control adapt define f accelerate split outer remark evident positively influence crucial minimizer minimum far stop third remark inexact generate belonging discussing describe summarize proposition update update iterative operator reasoning make size second since quantity v accord outer iteration detect evaluate projection proposition algorithm inner stop depend inner iteration result consequence euler characterization subdifferential observe subdifferential belong subdifferential subdifferential control evaluate irrelevant discuss regularizer ideally result show consider let restriction force derivative guarantee motivate add square spirit manifold approximation allow sample counterpart deviation give study useful since belong always precisely convex continuous rkh norm asymptotic explicit learn relevant irrelevant sparsity context partial everywhere relevant goal variable correctly assumption regularization safe filter variable sufficiently large relevant converse inclusion give variable work implication additive model prove achieve probability sparsity much total considerably source possible challenge interesting structured sparsity operator adjoint operator sense one penalty divide partition penalty obtain vector restrict orthogonal th form rewrite infinite reproduce reproduce hilbert one operator range practice empirical indeed structure still operator penalty projection penalty q regularizer conclusion highlight typically would difficulty arise operator scope paper subject future divide tuning always consider step procedure variable apply fix regularization validation validation influence discuss solution principle parameter stand show propose pose ensure reason coordinate aim penalty systematically investigate enforce degenerate toy coincide
bind argument first graph dependency graph whose graph vertex edge vertex give pick vertex graph define bind codebook hold sufficiently er rao sufficiently symmetry code substituting since eq get share term code eq choose theorem consider show interval r right lower fast grow lower write choose number study combination dictionary polynomial ensemble achieve rate exponent source show robust design source source within distortion sparse cover property ensemble goal achieve distance encoding lie denote distortion govern grow compare mean happen standard solution approach realization ensemble go coefficient section design one use encoder asymptotically attain compression relate recovery linear recover position zero recovery priori connection investigation follow er sequel denote value gaussian q large source increase divide side grow grow hold g first strictly equal see indeed solve verify imply value attain write q therefore large similarly taylor minimum expansion attain get claim large author regression definition corollary distortion combination column call sparse superposition codebook analogous recently communication channel encoding attain shannon distortion exponent sense codebook design gaussian square ergodic variance sparse covering I term matrix distortion distortion exponent sparse ne problem information code compression general source rate shannon superposition code compression distortion criterion sparse block codebook communication sparse computationally rich minimum proceeding case random low vector normalize otherwise mutual gaussian interval distortion length nr quality measure q distortion criterion near source distortion distortion give achieve shannon codebook selection storage encoding codebook grow base code compression widely storage certain code distortion complexity recently distortion encoding decode code consider code main achievable exponent ergodic source moment error distortion special I source result imply distortion exponent bit per rate robustness ergodic distortion ergodic source codebook attain distortion function sparse good much codebook I consequence dependent deal literature know exponent refined indicator codebook encoding decode procedure describe exponent state distortion feature make I random refined distortion level exist code design distortion draw n nm theorem bit encode achievable region lemma particular source distortion yield distortion performance describe theorem shannon rate distortion coincide begin exponent distortion rate optimal exponent pair supremum sequence distortion give exponent describe error grow length gaussian criterion exponent kullback sequence within distortion decay consequently obtain sequence variance great exponent characterize exponent source eq equation sequence level great matrix draw mean satisfying achieve exponent attain I er deviation q exponentially eq get pick matrix codebook vector equal encoder code q great restrict trivially use codebook negligible ergodicity regression codebook correspond l uniquely section overlap position sum
energy maximize set consistent satisfy apparent carry step similarity belief propagation equation several adopt algorithm sbm several belief instance way assumption graphical loop decay treat graphical fully connect interaction edge weak put decay argue belief asymptotically probability prove rigorously probabilitie I ij assignment product recursively term message compare equation belief edge bp simplify term message update marginal use bp fast change find use bethe formula free bethe exact graphical many mf free set bethe derive equation order modular network modular approach section ensemble know true assignment follow permutation would successful mf right know I unique maximum seek practical situation estimate marginal overlap confidence quantity agree large undesirable give reconstruct ht mf bp approach network node generate group modular structure rr rs group note possible mf overlap region iteration effort maximal bp converges fix infeasible mf region mf large average probability use bp mf plot fig superior mf well agreement region overlap bp mf equivalent region overlap converge contain group parameter close impossible phase overlap group assignment mf confident part bp agree evaluate mf confidence considerably large overlap region mf depend overlap reach close explain bp mf time middle mf need bp converge different initialize message converge run mf overlap mf different mf quality mf conclusion bp mf reach problem bp concern give region superior notably suffer maximization maxima get fortunately indicator run small volume grow linear considerable initialization desire large correctly recover class label bp whereas spectral serve initialization em bp achieve modular example bp well exhibit panel fig core modularity overlap group adjacency cluster conclusion search original always spectral modularity test fig marginal improvement important make case clearly region clustering claim average opinion average degree think end start point improve overlap em result example network belief propagation traditional mean field recent discovery sharp block inference provably control heuristic speed reliably deep feasible sparse difficult method block subgraph community short serve quick claim optimality spectral network technique find burden spread fully connect structure belief expect formulation may field connect maximally sparse case outperform albeit finally could decomposition select reduce conjunction expectation wish thank discussion aspect dim france support fellowship science foundation inference classical commonly approach belief contribution perform comparative na accuracy tendency portion phenomena particle consequence capability neural apply system pattern far result phenomenon datum research understanding ultimately easy map explain allow pairwise bind protein protein link protein operate protein network sense structure make guide aim similar consideration social network online datum discover difference often important detail infer parameter undirecte unweighted discuss consequence capture fluctuation dataset theoretical technique sharp impossible briefly expectation maximization algorithm conjunction introduce em mention infeasible transition particularly highlight discuss finding consequence inference node unweighte enyi fall link generate degree characteristic small length unfortunately explain fail density may entity property protein protein form involve interior would soon entirely social gender education tie node simple stochastic assume node indicate link easily assign multinomial specify develop multivariate analyze embed behind adjacency measurement dimensional feature row apply multidimensional analysis find direction onto spirit remove symmetric mean say transform modularity via modularity row sparse since eigenvector large magnitude eigenvector recently aspect rank eigen decomposition well frobenius norm matrix order decrease eigenvalue rank approximation eq row anti parallel model comparison density mix modularity matrix instead
propose parameter interpret posteriori map justification lasso estimate credible interval markov mcmc mix model towards even evidence bayesian literature concave log prior reduce recent lee bayesian interpretations net median series pattern predictor irrelevant dynamic modelling important receive attention bayesian adapt elastic accommodate bayesian type include magnitude latent follow dynamic construction recently rely normal tailored monte carlo smc generalize numerous sparsity propose bayesian consider standard regression response identically distribute adopt p independent interpret laplace prior express admit hierarchical construction normal generalize density write pdf concentrate heavy tail make desirable distribution law laplace law show performance regression chain also consider superior parameter represent control behavior time identity priori independence value slowly evolve sparsity sparsity unchanged multivariate evolve smoothly pattern semi one q normal turn use vary introduce sparsity likely sparse zero decomposition multiple overlap joint distribution marginally follow define predictive normal latent model stationary pattern independence pattern iid sparsity successive appealing literature desired gamma define pattern go evolve extreme independent sparsity tune alternate period see induce minimum autocorrelation non way note sparsity lag share pattern clearly autocorrelation share pattern statistical uncorrelated autocorrelation share evolve depend stock stock trading period value diag carlo particle hastings sample replicate particle weight replicate particle particle compute run conduct generate artificial observation red circle truth solid additive figure show map ground truth parameter notice control correlation truth fit implication firstly create smoothness stability secondly slight sparsity aforementioned iteration particle metropolis hasting moderate depend circumstance flexibility practitioner unique extend bridge replicate see drop step replicate fully produce induce shrinkage stock derivative alternative micro micro trading asset consider news horizon day day variability stock order magnitude stream bad news increase trading activity macro correction market stock source yahoo stock derivative stock study mention early massive price next asset technology index observe activity market measure interest stock index constitute core begin stock technology volatility trend figure contrary attempt plot filter time range moderate panel early market close practical portfolio situation largely home specifically home aim portfolio correlation immediately leave appear highly correlate market induce derivative market conversely two positively might choose house stock directly might decomposable clique
seven recently object non capable capture inherent ambiguity localization compare demonstrate appearance variation capable capture incremental principal construct tracking track sparse principal decompose motion cover object appearance motion capability complicated appearance illumination change head pose video scenario illumination pose motion vary video scenario scenario video scenario partial body variation scenario head video clutter car pose eight highlighted bounding box color show fig figure video walk illumination head pose track break nd l fail able successfully face video scenario toy affect illumination l fail nd th lose tracking toy inaccurate track toy illumination change car dark road scene clutter vary condition st track car illumination clutter break frame keep car inaccurate tracking tracking car video variation head th frame head th nd frames l track head video person compete suffer severe track person frame achieve head vary body pose begin frame fail track frame stay far away track sequence player piece fail track face nd frames frame performance capture face throughout fig sized traffic scene track car th frame car video scenario try parallel car partially car achieve nd frame subsequently begin drift break th able track car tracking contrast car tracking throughout video ball ball l fail ball rd inaccurate tracking result fail another due severe contrast continuously severe video drastically person track face nd frame begin break influence head plane rotation inaccurate tracking result frame th track inaccurate result frame show car fail car st frames track car th track car th inaccurate rd frame car situation tracking result video sequence performance pixel tracking error well evaluate call tracking frame video sequence successfully criterion frame pixel location truth localization ground bounding box successful otherwise video use h sec influence sort exchange order neighbor video choice configuration within close dct sensitive supplementary tracking achieve equal track accuracy utilize state method representation improve tracking make refer dct particle dct slide clearly fig tracking inference show dct particle obtain accurate slide conclude dct object representation responsible enhanced tracking highlight color eight sequence compute standard location video report report total sequence propose sequence inferior difference e head appearance motion body appearance usually color appearance greatly plane circumstance haar color video sequence dct object representation appearance appearance sample dct reconstruction encode robustness complicate pose c video car car ball car seq seq simultaneous tracking dct compact construct dct energy spectrum component discard construct dct successive dct dct video dct dct newly well lead computational efficiency computing store cosine dct dct discriminative base contribute evaluate test belong foreground foreground background reconstruction discriminative appearance e rotation use discriminative conduct visual time video include illumination pose complicated change effectiveness acknowledgment arc discovery fig author theorem li van wang center visual school sciences intelligence key laboratory brain like system university china manuscript tracking model change illumination factor video appearance sample form basis limitation easily b challenge paper model discrete set cosine dct energy appearance discard signal reconstruction update representation dct dct successive cosine dct discrete cosine transform dct video incremental dct frame dct dimension computational dct algorithm belong foreground object discriminative time track cosine transform dct incremental template tracking move object fundamental computer vision include surveillance human despite effort challenging appearance variation pose etc crucial effective object challenge appearance change popular learn appearance variation appearance reflect appearance learn appearance compute reconstruction project subspace evaluate likelihood test belong foreground object approach basis correspond inspire track cosine e cosine basis propose projection dct incremental lead type enable long tool encoding image video many promise video dct lead correlated mean remove rapid often dct cosine fix cosine simple construct dct computationally incremental tracking represent calculate dct dct encode temporal redundancy correlation sample compression discard still relatively low sample cosine evaluation compact independent wavelet capture adopt dft amplitude haar aim detailed wavelet transform although conduct function work minor modification dct propose utilize compression power dct construct novel representation dense dct dct compact measure loss evaluate foreground give set efficiently update dct successive operation dct dct video newly add well dct along cosine advance reduce dct predict foreground discriminative criterion foreground background dct reconstruction likelihood capture adapt appearance focus learn compact discuss dct review relate tracking claim dct aim mutually uncorrelated cosine function express manner vision face recognition retrieval video object video localization application typically extraction construct coefficient geometry illumination focus effective dct object visual tracking field researcher variety object representation discriminative representation object include kernel density subspace covariance tracking base object review elaborate em appearance change appearance use mode kernel e framework incremental learning achieve real li compressive matching pursuit vector namely track video view frame n cosine define dct high encode high coefficient low relatively principal tracking build compact maintain control degree tracking dct obtain compact dct approximate reconstruct high form basis section cosine basis matrix dct employ efficiently dct dct I dct visual object representation dct object transform incremental dct aim current z formulate stage coefficient axis correspond computational increase normal dct grow much fast incremental dct accord dct unchanged visual tracking remain concatenation along algebra decompose update computing view dct e dct dct employ use fast dct dct summarize n frame traditional mode strategy dct computation become computational dct computational incremental dct dct incremental dct z n f collect negative object region pixel perform incremental dct dct matrix compute compact discard reconstruct reconstruction likelihood likelihood optimal refer update training maintain element keep last set module selection dct tracking module spatial region object select region draw make sort set final positive set middle fig negative around show fig generality evaluate candidate belong foreground object appearance candidate point locality locality versa locality constrain dct h compute nearest refer n left fig incremental dct
encounter smooth uncertainty principle bayesian gps offline series size matrix scale demand although experimentally scale poorly dimension equation integral integration practical gp necessarily smooth gp slow function evaluation ode cause eventually require principled smoothing gp gp computation exactly linearization sigma representation integrating model induce code publicly acknowledgement partially support grant intel b university usa propose bayesian system increase importance signal processing function modern finding estimate parametric analytic gp demonstrate fail nonlinear smoothing machine filter latent history methodology smoothing dynamic propose gps robust approximate gps contribution article derivation principle smooth gp compute close filter linearization evidence nonlinear kalman filter kalman refer ability background smoothing introduce explain gps detail system sec run prior purpose article gaussian approximations bc xt sufficient approximations eqs computing include extension etc differ covariance require eqs derive follow smooth relatively detail gp dynamic process practical application form increase gp control assume h infer value input fully eq specify respect square automatic relevance result square characteristic se function easier justify fix se zero prior common retain great inspire correspondence basis along universal writing axis white convolution function jointly convolution function specify distribution random prior require everywhere use complete eqs see universal make function se implicitly latent eq encode assumption give gps assume uncertain observe use collect maximization estimate evidence automatically avoid setup g notation hyper analytic dynamic extend filter analytic smoothing account linearization require sigma smooth standard frame eqs covariance smooth involve linearization deterministic joint distribute x ij q noise gp target role q desire definition law expectation kernel integration turn eq ac responsible definition unnormalize gaussian integral product obtain identity obtain q full completely eqs filter necessary provide smoothing analytically analyze state stochastic contain transition originally function uncertain broad eqs relatively small consider numerical eqs robustness filter sir resample particle filter particle gp gp ground closely filter compare trajectory evaluate filter make analyze filter sir performance mean mae sir pf average randomly start per setup develop indicate robust filter significantly outperform sir green particle amongst filter gibbs filter sir pf filter express difference sir pf near filter depict filter instead sir matching poor linearization filter sample small statistically even tb mapping panel sigma dots predictive show sub panel area solid gaussian track gp anti angular velocity constrain transition moment acceleration state ode hold w
get q positively definite uniquely normally euler sde expression independent wiener uniquely covariance divided advantage disease uncertainty sometimes calculate addition sometimes sde severe disease clinical despite heavily life ability rare incidence research patient duration account regard take sde age show pair path single path figure region path lie dotted indicate path mean quantile calculate additionally ht big difference gender incidence ratio age hazard estimate burden total number obtain statistic specific path figure path b becomes describe derive duration duration person divide total ever respectively age path factor unity disease differential equation decade transformation accomplish disease stochastic equation rare disease preferable deterministic incidence fluctuation strong age course obvious figure middle respectively empirical observed agreement describe equal incidence duration disease easily eq pyramid mean incidence close empirically observe per year relate uk incidence high person year datum way inconsistent duration basic contradict although difference duration noticed duration etc global european country publication mention incidence help burden disease especially limitation application patient publication find disease long duration applicability disease derive consistent indicate institute diabetes death disease incidence basic incidence model model also death transition mean disease consideration number intensity incidence rate depend also ht consider disease disease depend age disease specific disease duration reverse incidence inverse problem cause death
entire summarize appendix current amount perform build entire tree efficient tree sub recursively reduce single leaf join tree recovery tree divide correspond involve perturbation edge perturbation edge perturbation correctly marginal guarantee also successfully nuclear assume return tree subset return sample topology adapt nuclear base test divide simplicity ease guarantee translate tree building spectral representative nj liu nj proceed recursively close additive ij ij denote diag nj design true configuration need p k subroutine reconstruction still choice apply liu tree add neighbor algorithm tree mutual information determine nj four configuration hide observe start identity perturb formula pa u percentage varied perturbation randomly initialize compare spectral varie lot stay one especially hide method lead state achieve choose allow resemble indistinguishable even topology size experiment topology randomly recursively recursive stop group variable join recursion control partition versus close obtain skewed shape path close skewed balanced hide perturbation vary construct tree leave partition imply partition tree show large try resemble work perform well discover stock dataset stock relate stock www finance yahoo com daily price discretize value visualization learn tree topology discover figure htb stock accord subtree international group subtree subtree subtree subtree utility service energy ed subtree american express see subtree financial american express car branch financial operate segment services services company financial service real hide algorithm term hold randomized likelihood tree state simulate validation select present tensor automatically liu however produce discover structure model pre specify quantity unknown practice joint table variable tensor spectral design exponentially simulate usefulness discover latent property tensor multilinear algebra allow address arise pt edu discover structure important discover require hidden contribution characterization design relation use subroutine recover tree condition error sample size compare stock dataset meaningful fit discover observe task discover understand potentially heuristic need determine structure provable guarantee recursively nj algorithm distance variable variable nj provably consistent resolve subsequently good repeatedly singular value joint table require advance number rarely number base th insight reveal community concern focus state represent tensor miss translate propose nuclear rank advantage compute since tensor subroutine latent tree structure divide scalable mild use compare alternative among agnostic hide latter improvement cross expensive stock grouping stock lead well conditional observed variable furthermore letter letter latent tree way although px parent r lead throughout assume rank learning topology discover structure tree topology neighbor observe variable latent figure ht style mm black fill hide inner black hide circle sep circle inner mm draw hidden hide name name empty mm style dot empty empty width dot dotted width style dot h dot tensor th rank relate algebra rd tensor tensor denote along tensor ht mm characterize property test although propose unfold tensor subsequent computation provide conceptual structure tensor exactly grouping variable group grouping matlab column rao assume correct structure factorize mm mm mm mm factorization different interact column similarly middle whereas similarly discover assumption suggest attain nonzero correct grouping variable condition verify rank discover structure due matrix nearly deal rank rank nuclear summarize three way matrix nuclear algorithm work base base bind meaningful nuclear job concentrate fourth nuclear measure measure
solve constraint agree solution lagrange original community decade dual increasingly computer vision flexibility problem enforce tractable correlation zero write following decompose edge sub problem correlation cluster identical optimize configuration optimize energy set recover optimum since maximize set simplified entail optimize relaxation ascent non smooth tackle objective like optimize solution appear complicated early highlight inequality violate describe weight use cut successively approximate constraint constrain cut edge become long edge simplify edge weight independent edge objective amount objective also upper constraint impose constraint sensible wise decrease appendix explicitly exponential constraint possible partition solve lp cut successively violate collection perform constraint break cut add batch find produce solution e constraint set fs x optimize general tight cut cut constrained edge individual cut problem second term cut agree motivated start find subproblem partition edge pseudo display lp equation whose cut cone straightforward valid cut valid cut cut segment cut exactly constraint impose cut appendix solve contains produce result thresholded take cluster cone unit cube cone polynomial lp optimization considerable full cut cut delayed generation scheme dual cut correspond primal lp tight partition cone allow access partition without relative speedup logarithmic computation produce optima much demonstrate cluster run contour region merge average contour merge perform globally return perform slightly final performance move segmentation segmentation contour perform well edge negative average perform range threshold result visible report measure merge successively low objective expect segment break contour could practice seem one explanation proceed merge new contour form average average simple averaging outperform substantial possible explanation greedy truly successful correlation threshold fairly still solution suggest combination via structured performance note segment small benchmark outperform optimal segmentation basis apparent application local biological imaging quality clustering exist variety exploit decomposition subproblem dual acknowledgment google derivation optimization equation cut let write standard far simplify expression define cone cone yield objective main lp insight compactly cut plane optimize lp define add constraint cut primal add another dual lp delay column generation scheme dual lp keep grow optimum find additional affect formal decrease without optimize satisfy without low cut indicator polytope graphs polytope compactly inequality relax discrete polytope relax lp constraint side decomposition collection cycle optimizer dual cycle cycle minimum energy zero constraint th cycle cut subproblem ec implication weight edge case update edge quantity tight produce satisfy additional parameter update repeatedly remain denote ce establishes lemma claim optimizer optimizer satisfy uci edu find perfect energy demonstrate segmentation outperform competitive quality pixel image surface come bottom contrast recognition pixel develop bank detector formulate markov pixel quite valuable segment scene understand salient predefine label partition pairwise similarity rich technique partitioning appeal segment function parameter structure trivial graph partitioning np et hardness cut hard difficulty correlation et segmentation score cost integer programming branch cut et correlation potential edge define linear programming lp technique new specifically weight cut weight correlation low output always yield tight bound range criterion dissimilarity graph specify dissimilarity edge vertex correlation seek disjoint minimize specify edge cut edge within let configuration valid triangle enforce express cluster energy optimum partition objective etc weight specify segment place weight example vertex edge place valid objective appear standard pairwise convert mrf unary next call minimal color partition color g example partition useful term vertex equivalent
guarantee hold least positive index guarantee p bind bind square recovery exist intuitive multiply analysis show nevertheless quantify build satisfie require vanish modified natural analysis around suggest strategy know estimator gap regime confirm section follow instrumental instrumental interpret snr measure instrumental design column covariate independently key suppose dependence move previous main dimensional understand correct allow avoid optimization enjoy support guarantee expect correlation however sub everything else interestingly knowledge instrumental directly advantage previous support recovery course quantity add error unbiased omp element correspond large somewhat surprisingly estimator advantage support knowledge distributional effect know result recovery omp support recovery final recovering column square submatrix row note unlike omp assume adapt h input I output design mod omp identify provide mod omp identify omp covariate obtain non however proof implicitly residual hold unable apply result I thank mod identify reduce dimensional exactly previous sub mod omp mod omp identify correct recover clean covariate reduce omp concentration miss seem error sub mod use technical lemma concentration statement zero parameter union corollary sub p substitute w z e w w observe sub n n w z proposition apply sub u u I entrie great mod omp miss illustrate high empirical corollary theorem correct rescale normalize clear alignment actual result regime efficacy different noisy covariate demonstrate addition fast empirical term recovery result set omp problem low identify correct corollary corollary scaling look vector estimator build knowledge corollary scale roughly straight line origin align trial plot b simulation knowledge instrumental instrumental gaussian corollaries proportional control build recovery scaling versus go noise magnitude roughly trial additive versus control recovery versus corollary curve align additive trial turn become roughly line align future noise point trial subsection study mod dimensional setting gradient additive omp use omp outperform project ccc omp knowledge instrumental error control parameter circle gradient dot mod claim mod omp omp enjoy formal omp mod omp iv estimator plot error performance mod omp method gradient recovery quickly graph contrast estimator excellent recovery error project precisely throughout case figure mod omp independent believe omp project gradient performance comparison mod plot correlate dot correspond mod omp gradient mod omp trial concentration random repeat statement convenience union assume sub kn kn n h rescaling assume u u u last thm thm pt corollary engineering tx completely noise efficient omp deal set omp dimensional seek measurement set recover regressor noise also improve guarantee omp compressed deal projection physical measurement often noisy corrupted standard include popular know noisy surprisingly corrupted give recently turn scale significantly even far particular prediction hinge sparsity thus provably subset recovery large call computationally focus fast additional good well noisy set dimension regressor exhibit regime standard compressed sensing setup describe dimensional omp condition high regressor recovery corruption aware compressed sparsity low computational noisy corrupted estimate dimensional estimate set without issue convexity move covariate recently attract attention make important cope practical recent exist handle selector mu selector follow let covariate selector thereby adjust effect modify lasso minimize w xu similar computed program interestingly project descent optimum method bound minimax covariate guarantee recovery weak recent simulation seem critical recovery considerably require performance demand prove theoretically successful clean covariate greedy prove method clean complete omp measurement response receive noise paper paper result noise seem weak consider clear contribution greedy omp noisy noise receive sense sparsity specifically exceeds regressor simple covariate covariate covariate instrumental correlate covariate noisy guarantee far available case instrumental variable aware rigorous require column high dimensional scaling greatly give noisy regressor result interestingly noisy noise recovery simulation require corruption advantage covariate indicate result algorithm seem promise covariate measurement g meanwhile gap quite paper paper exceed regressor omp determine low omp regime section illustrate advantage regressor low setting regressor dimensional dimensional case assume covariate link via observe miss following seek unknown setting basic definition use gaussian gaussian row simplify subsequent notation generality lose section general result parameter analytical across define sub
simulation experiment validation widely statistic estimate statistical square estimator reference cross selection cross validation fold cross enjoy computational classical small second useful surprisingly result answer question least cross suboptimal stay distinguish value intuitively reduce validation estimator risk aim provide two valid variance computation would like common least particular fold validation fold penalty constant unbiased fold method knowledge asymptotic lead estimation framework rely new penalty proposition paper asymptotic variance computation understand cross penalization depend see remark say necessary getting confirm experiment synthetic integer bt infinity reference study estimate estimator measure risk want estimator comparable form large call front drive criterion oracle inequality penalization criterion criterion ideal orthogonal projection follow reference definition necessary main train test n fold cross one thank fold validation criterion study name penalization resample resample associate resample penalty distribution invariant permutation cost exchangeable exchangeable resample fold validation jj n resample focus cover resample base loo x loo exchangeable exchangeable resample factor framework prove fold fold cross leave validation prove square lemma family exchangeable aa cross unbiased loo penalize lemma penalization free front penalty study cross fold leave exchangeable take penalization bm lemma penalization picture oracle result hold constant front prove oracle satisfied leave concentration establish real value random piece satisfy mt mx proposition fold penalty deviation term help discriminate value histogram collection subset lebesgue countable histogram span precisely illustrate result every since show nd study interval equal state hypothesis family hereafter space real piece space either let least q section take fold classical assumption term inequality loss drive asymptotic bias fold penalization interpretation model therefore lead upper unless use correct cross show oracle constant remainder regular analyze correspond validation much large simplify presentation set previous less simplify message first asymptotic inequality prove penalization procedure concern validation oracle square framework inequality compare consider oracle inequality leave treat lemma recover concern fold penalization either square bound infinity valid remove particular remain loss ratio compare performance several fold method minimize make set goal understand validation develop apply ideally prove procedure inequality probability alternatively review result validation good square bad model limitation accord set distinguished leave certainly must go take account variance already challenge prove specific challenge compute precise seem heuristic depend second term square different partition satisfying case solve necessarily yield precisely unchanged translate random make desire often mistake behave lead remark quantity tool clearly hold dependence small claim reasonable least equally intuitively minimal magnitude drive instead decrease third much say really pair penalization come back piece two span model performance correct validation improve take section constant multiplicative factor eq difference lie one behave like ideal cover theorem conclusion decrease cross leave soon similarly derive leave formula component clearly believe show focus fold leave write major cover correct c former quantity dramatically matter variance increment compute section depend basis prove section section illustrate theoretical lebesgue density sx sx weight gaussians collection model consider every regular histogram bin dyadic histogram two n j piece set simple flexibility optimal small risk consider show help reduce risk even consider collection histogram collection procedure datum oracle model setting experiment less difference risk two well drive penalization vx suggest multiply penalty multiply procedure independent measure uncertainty measure empirical divide report setting p report penalty already know table performance fold method consider result decrease increase large l go fold confirm help picture fold minimize indeed increase simultaneously variance validation depend least estimation fact perform much contrary fold regression explain fold perform fold fact fold validation coincide fold penalization multiply fold penalization phenomenon discard setting difference comment fold influence unbiased satisfy distribution independent index ms conclusion decrease significant leave distinguish set l figure close confirm appear indeed mn proxy similarly heuristic decrease translate well distribution around explain observed influence comparable report lead conclusion b computational fold penalization validation density naive exposition fold criterion estimator square framework orthonormal describe precisely occur avoid risk even criterion correctness histogram algorithm close complexity particular histogram discuss fold selection discuss model square fold hold datum definition hold dramatically variability note fold follow unbiased fold either hold x prove similar theorem quantify criterion comparison fold similarly role split criterion proposition paper inequality least square penalization procedure lead constant inequality prove stein practice inequality suboptimal constant oracle overall none imply strictly mention propose evaluation regular contrast slightly modify compare well regular histogram likely present performance trade square since see well decrease increase remove completely fold penalization variance decrease reach article provide concern bias fold resp fold factor resp simulation factor even literature behave address penalization answer least fold cross penalization want would depend want thank careful reading author acknowledge de project bs project program use repeatedly proof number right side schwarz consider b converse true proof eq p cp follow us eq fold always leave hand use imply statement prove algebraic show cv hand cv minimizer coincide lemma variable v I otherwise density separable let e
observe model intractable connection computation finding although consequence previous uniformly whenever ergodic uniformly ergodicity soon pseudo direction nonzero right bound pseudo chain nonzero gap rely gap proposition ergodicity pseudo bound modification remark observe establish necessity existence ergodicity neighbourhood ergodicity pseudo marginal existence possess robustness perturbation marginal pseudo metropolis primary implementation uniformly rate rate scenario support imply geometric together mild additional integrable establish existence lyapunov satisfy sub lemma distribution possess power moment establish walk metropolis rwm conditions ergodicity rwm existence polynomially ergodic situation e allow tail remark intermediate tail vanish tail natural average counterpart one fact situation one marginal theorem support observation efficient version go converge show geometric prove practice implication central theorem possibility asymptotic variance pseudo introduce measure measurable two fx gx start play order concave jensen desire order facilitate marginal reversible lead acc marginal qx gx quantity ax rx g qx gx ax consequence truncation rx qx last follow generic add x qx q bound scale average reversible measurable denote whenever fx apply lem h specific gap auxiliary define nonzero bound also imply integrable respect dirichlet kernel operator spectral gap pf qx decomposition bind imply concentrated simplify lem appendix exist imply fx w fy qx fx qx fx let satisfy definition satisfy coincide marginally together imply observe fact kernel reversible repeat marginal next support theorem different statement explicit fix x w may imply necessity existence consequently geometric ergodicity existence uniformity mh walk second case existence bind necessary bounding ergodicity may boundedness pseudo geometrically ergodic modification unnecessary marginal exhibit define leave gap existence spectral ergodicity reversible positive establish ergodicity require record pseudo positivity spectral follow rwm write version f qx w fx z sum share indeed scenario u z multivariate ergodicity soon marginal algorithm admit nonzero spectral gap marginal geometrically ergodic able find completeness counter pseudo yield toward atom geometrically ergodic define otherwise choose gap family distribution index reflect relevant scenario particle clear hereafter pseudo invariant easy weakly invariant invariant marginal situation ergodic improve convergence situation take integrated autocorrelation see pn consequence corollary suppose kernel weight uniformly consequence corollary unbounded geometrically ergodic formulate technical return check lyapunov suppose autocorrelation time finite assume exist satisfy g autocorrelation inequality generality claim autocorrelation n k n corresponding coincide marginally couple set n nx show follow inequality k n qp n ng old cauchy two probability n nx nx imply exist uniform nx qx qx qx third therefore let eq q qx u additional find go sufficient imply condition quantitative bound behaviour marginal result rely establish algorithm drift keep presentation simple set vx gx v write w gx u n w therefore metropolis hasting ergodicity pseudo illustrative develop relevant analysis assume ratio w word x ergodicity pseudo rely inspire establish ergodicity explore sub x ax ax w ax bx bx w ax v bx x fast cf suggest marginal uniformly state compact uniformly integrable pseudo drift toward practically small page quantify geometric drift indicate consequently ergodicity q record implication exist result drift away pseudo chain away geometrically ergodic suppose acceptance marginal away vx w vx qx qx rx qx w convexity imply lemma exist claim often moment record highlight straightforward pseudo drift vx w b proposition toward imply geometric showing sub qx qx recursively nx b assume establish condition fact condition marginal geometrically walk rwm exponentially decay contour existence rate ergodicity marginal limit rely moment condition section may appear general cover condition admissible corollary extend result beyond case condition sake clarity detail unbounded distribution ergodicity possible perturbation indeed rwm establish exist almost every geometrically ergodic throughout region almost acceptance rejection r moment supremum lebesgue simple throughout hold also inequality observe w result respect lebesgue continuously differentiable tail exponentially decay regular contour moreover satisfy neighbourhood origin drift pseudo satisfy constant metropolis invariant density proposal define constant eq vx vx apply fix lemma define must least polynomially allow establish drift drift negative presence establish ergodicity seem however condition behaviour follow write w large drift suppose vx w z integral partition intersection acceptance rejection set notational straightforward x x w u w crucial unity grow q z x imply w w vx vx vx vx x deduce regime vx r x constant subset x term yield drift u u c consider condition z enough conclude replace moment moment tend infinity exist lebesgue z possible rejection metropolis implicit assumption exp whose help gain intuition metropolis increment satisfy constant polynomial drift furthermore quantitie lemma lem theorem quantity prove give sufficient additionally increment satisfied drift depend marginal choose choose condition satisfied compact set monotone decrease tail sufficiently large due imply define easy tail ray conclude hold condition denote b upon appearance denote condition observe x u u z similarly z w w u w may lem ensure increase sufficiently large vx vx vx vx w let exist vx u w z cx z cx cx decompose sub obvious abuse observe x z u x x hold z x z x z z cx u u claim verify decomposition section case bound central limit hold function specifically relate drift constant reference therein ergodic drift reader due well limit take
proxy use technique low rather quality undesirable request approach facebook platform al build tool request statistical approach match request pattern could facebook platform support friend application profile read preference access resource limited request application request request et facebook find request information approximately build version page website find page list yield link page feed select permutation ac market engine additional page rate rating et us facebook application rating limitation data request attempt explore collect secondary request might incorporate global rating popular play role request list frequently depict frequently facebook name email offline friend friend access communication view storage phone call read phone fine gps location control tool device tool start phone read contact control video tool retrieve run application globally price tail high evident almost decrease slope pricing application amount area decrease price extra rating user rating logarithm rating rating except peak rating follow center score star examine average see high tend quality rate additionally extreme score less indicate score alone insufficient cumulative price axis start cut rating score total dot use logarithmic rating discard rating rating respective htb infer request application request entry request request input want matrix encode frequently request share request pattern boolean matrix concept factorization request z employ factorization binary likelihood statistically outcome factorization representation originally role request mixture mining request request correspond request application assume follow sx boolean request assign model entry modify parameter flip request distribution reduce computational ultimately terminate predefined input I probability request request particular assignment request assign request request cluster group request simple pattern suffice could lead overfitte contrary complicated might pattern significant individual heuristic selection instability analysis instability pattern subset independent produce clustering clustering able precisely two subset lot I subset datum difficult classifier second I run multiple arbitrary htb request facebook suffice facebook depict five mining technique identify assign sort common second disjoint member application request multiple include reflect conditional conditional estimate whenever score facebook color end conditional detail request false ideally positive false false positive assign occur request cover define false incorrectly assign application otherwise false false please resolve discuss facebook positive thereby small application rate reflect great application positive negative partially attribute large place application request fully high training set statistical request application htb request help quality user high incorporate could request fit risk positive high regardless request false high negative indicate suggest request recommend request pattern application incorporate market group application category purpose book reference weather category roughly request strongly figure illustrate relationship category category classified many application communication fall indicate category contain request application kullback report pattern always find one category informative check simulated request want application request exhibit pattern pattern reflect chance request high pairwise conditional eq quantify one request test request request application request pair htb conditional probability request circle red request low pair figure dataset please axis logarithmic also histogram bin bin significantly please total average suggest provide potential direction user rank result receive enough review possibility request would notion need subject research technique suitable detect aim quality otherwise without receive favorable request find request alternative create good obvious drawback approach manually manual analysis consume hundred diverse available bad corpus significant application quality partially influence request exclude application specialized request interested user believe reason rating request significant find stability essence necessarily require fraction application request use request pattern facebook fit request low application request pattern application indicate request part risk relationship pattern application category intel national science facebook party access private ability call security restrict application privacy study often understand request request system cluster corpus facebook application pattern request factorization facebook request fit complex request request pattern application suggest request user application platform result availability hundred thousand click end potentially choice look decision security hardware facebook access user social operate access privacy relevant resource request example application text user complete intend privacy security unfortunately demonstrate pay major know combination simplify identify request application method
usefulness change uncertainty lead correlation treat interface I minus minus optimal middle maximum figure interface period maximum amplitude interface marginally interface well uncertain interface come three thing geometry secondly facilitate change elastic position introduce specify position correlation field specify depend idea rather inversion natural soft opinion may mail difference devote use change reader field denote element depend implicitly define position denote denote operator equivalent mixed operator convergent separate angle inversion discrete layer use system become specify prior flexible incorporation advantage markovian introduce huge differ additionally geodesic quantify multivariate spatial specify different prior multivariate correlation cross problem inversion primary inversion propagation equation solve include yield well inversion result adopt text convolution shape wave relative difference wave elastic relative pressure wave velocity velocity wave background discretize discretized operator novel system partial differential flexible region figure appear comparative colour scheme min individual make sense without inversion importance choose depend heuristic covariance covariance define change field widely inverse correlate correlation specify knowledge discretize eq correlation elastic typically possible transform computing covariance together lattice sensible require stationarity storage another computation requirement pseudo differential pseudo differential closely short transform approach storage statistical mat ern covariance firstly noise mat ern computation operator induced cholesky requirement vector relatively quickly address field alternatively manifold boundary merely namely box boundary direct interpretability term specify condition retain usually opinion stationary prior point space sound merely desirable letting correlation vary flat define direction eigenvector stationary flexibility bottom layer stationary interface interface direct interface almost isotropic relate usual model white noise inversion article obvious specifie way strength operator inversion answer obvious investigate alternative criterion new model replace symbol local operator state polynomial markov approximate rational jx suboptimal discretization suit candidate eq stationarity convenience ern choice define restrict hard correlation field matrix inversion phase layer phenomena potential drawback possible correct correlation situation actually interface expert incorrectly lead geodesic discuss handle quantify interface possibly require suppose say interface bottom geodesic grey cover interface properly fair interface section investigate purely possible combine interface increase complexity model estimate range location many modify demand flexibility domain simplification interpretability motivate behave hand kronecker carry case natural way cholesky locally square possible define way get minor concern practice long interpretability triangular inversion hyper well natural way hyper identifiable field well wavelet denote possible latent treat correlation interface straight geometry scope enforce interpretability namely multivariate field matrix free specify must great three apply depend different model stationary interface simulation simulate multivariate density unimodal quasi estimate unimodal identifiable estimate eq much difficult reflect broad distributional recover come may come smooth estimating density test essentially noisy reconstruct field fidelity follow subsection reconstruction base identity noise second give reconstruction iid noise three flat interface priori correlation structure level function flat optimisation one reconstruct field attribute sum prominent use smooth exhibit middle model seem lead range
continuity probabilistic order express conclude mm mm corollary corollary difference estimation mm ac university ac du technology ac song liu technology song ac institute technology ac address estimate density computing procedure well without regard step error shot procedure separately usefulness demonstrate experimentally density kullback quantity element approach two separately poorly stage carry regard incur stage big error appropriate without seminal separately estimate learn sufficient ratio two density separate explore probability shoot various purpose change change extraction detection segmentation shoot directly separately estimate derive stable equip cross systematically parametric setup optimal distance show accurate estimate compare experimentally usefulness semi change rest investigate approximated section shot density analyze formulate set identically dpp goal use kde give determined cross argue kde density good nature cause final tend estimator tend smoothed overcome shot estimate estimating difference loss parametric gaussian kernel center optimal computed replace add objective function parameter derivative function analytically eq finally density investigate parametric basis q matrix imply parametric setup setup width consider parametric rkh prove satisfie utilize dimensionality vector regularity non achieve naive kde rate kde regularity negative difficult implement always regularity superiority cv divide n denote validation average hold implementation available make expression replace approximate average follow replace estimator estimator eq approximation nevertheless generalize bias eliminate impose yield regularization lower replace reduce see eq plain essential generally popular divergence illustrate distance kl implication density draw depict kl distance whereas divergence infinity infinity outlier possess direct depict outli run show profile divergence value mm mm mm finally procedure conduct density dataset assign distance nan ranking depict rejection nan outli outlier whereas rejection base keep two presence test depict outli outlier keep low rejection tend nan reject nan p apply supervised change series real bias balance reflect test recognition classify unlabeled test set wise input mix illustration match uci randomly label unlabeled class misclassification weighted run result translate datum series subsequence single dependent incorporated subsequence measure plausibility distance record environment display true manually annotate indicate score next provide orientation take bottom score much interpretable method directly propose framework solution systematically optimize normality finite case convergence also bias cause experimentally outlier kullback paradigm useful topic estimation explore application acknowledgment comment support program du song liu define rkhs endowed width difference exist member integer modulus technique regression exist constant
think equation term negative recognize obtain bind treat constant lower call q indicate select complete complete latent maximize constraint easily introduce hide select posterior lagrange easily incorporate pz pz pz nd pz nd pz nd pz pz nd pz nd pz pz z pd nd pz pz pz pz pz w I thank zhang point error manuscript information medium social mining recommendation collaborative filter paper establish originally early technical report estimation estimation hierarchical factorization equivalent show well behind topic vocabulary represent probability topic secondly topic original specification multinomial appear document generative although subtle need token vocabulary distinct differ token place appear multiple time imagine want decide term token document position want step multinomial suppose choose term generation token summarize token probability token document appear formalism lagrange multinomial optimize summation parameter apply token position choose word token position token position document plug standard posterior hide di pr di di di di di obtain give hide similarly token position term simplify simplify subtle need indicate token satisfie element calculate still lead write token variable follow auxiliary auxiliary jensen lower bind goal clear left hand side possible q previous way formulate see equivalent pz pd whole q whole maximization guarantee likelihood maximize useful really maximize likelihood likelihood maximize show
architecture communication inter machine scheme simulate architecture microsoft windows platform speed virtual investigate parallelization call know parallelism mean satisfactory aim goal reduce need reach theoretical parallel aim dataset mod operator stand division operation sure prove belong stochastic gradient idea entity perform instance represent prototype technique denote investigate scheme rest organize provide bring satisfactory provide consequently bring finally asynchronous adaptation latter fit communication cloud computing test synthetic investigation start parallelization resource start apply sequential subset process processor synchronization share unit local comparison typical scheme implementation communication instantaneous resource bring entity investigation satisfactory series sequential rewrite synchronization iteration resource rewrite assume distortion consequently iteration drive speed choice sequence exploration trade respect share observer section process exploration lead suppose seek evolution sample accelerate merge share version average translation result typical display speed obtain resource acceleration great phase frequent could attract would slow compute parallelization deal communication cost update delay delay architecture share computing hardware issue subsection follow remove synchronization reduce phase realistic receive share entity synchronization unit share soon complete dedicate share barrier delay performance result
originally feed via agent interact environment uk partially observable decision widely provide making provide slightly call decision going join basically interaction agent light make observable markov make agent environment notable characteristic plan uncertainty mention mdps computer among mention demonstrate interaction markov implementation remark development call former one basically factor benefit state factor two different writing stand observable factorize eq music compound simple make decision receive concept follow environment fully state whereas observable suppose simplicity beyond interval eq q action agent stand make decomposition decomposition relatively partially observable intermediate make make possible value q finally factor second testing
marginal equal input system respectively function probable modelling phenomenon determination kernel describe literature complex kernel allow reflect htp take depend point view convenient inference conduct use determine hyperparameter value likelihood experiment take web execution environment service simulation request proper work iv request v service htp input execution output system quantity cart matlab cart toolbox occur demand size exponential express likelihood htp environment allow point iii universal time unit iii represented box mae comparison cart represent square perform slightly sample mae significance cart share variance reject value reject statistically cart htp htp process attribute service regression field learn service gaussian index namely process statistically cart show develop result high experiment sum paper try fill acknowledgment partially european technology web service performance system attribute execution service repository stream request index square perform statistically compare prediction service modern network method theoretical consideration explore web service consist layer service service design service paradigm application layer predict express service repository g service service dependency attribute stream request technique detection use resource allocation fact internal unknown execution system random probabilistic seem consider application parametric regression system application web service organize sect problem web sect gaussian environment draw input execution maintain execution execution attribute spend execution call simplicity target output however know responsible process dependency input hence historical consider need
ode initial converge uniformly note arrive iteration avoid randomness stable equilibrium proof g sf briefly update track n update sf one taylor g sf obtain surely theorem allow satisfy still sequence difficult tune appropriately analysis sf exclude verify claim hold alternative sf feed arrival process customer leave probability service complete customer service node n ii distribution service node individual update customer cost minimum nn optimal measure measure method average run value indicate use begin network control arrival leave system service simulation intel second relation trial gaussian simulation trial clearly fluctuation side cauchy sf last indicate flat ccccc cauchy side sf sf axis side sf axis sf axis iteration axis indicate sf sf suggest run sf sf attribute behavior quite obvious plot numerically achieve zero plot independent trial trial govern run hence multiple fluctuation reduction fluctuation prominent middle well smooth occur distant gradient become provide exploration nearby c whether control parameter step tuple run take entirely second concentrate overall case customer choose cost queue length experiment quite one table trend specific time service exponentially ensure well constant vary scenario expect reasonable performance since decay gradually v however theorem simulation increase drastically pt c sf c sf distribute service mark origin popularity notion build fact generalize equivalent smoothed functional two smoothing ode significance demonstrate smoothing improve algorithm appropriate modification develop adaptively conclude note idea hessian develop newton random identically uncorrelated hence possible gaussian generate box algorithm variate use correspondence student distribution duality observation state variate na functional sf scheme efficient kernel include gaussian cauchy distribution among new gain decade attribute ability generalize exist smoothing motivate sf constrain project set ode numerically numerical author computer institute ad objective expression common engineering financial encounter financial noisy optimize one commonly originally approach algorithm develop control track objective employ estimation limited applicability provide system parameter efficient technique base sf stochastic newton involve long estimate parameter perform long average constitute approximation sf performance sf gaussian sf significantly tune size improve perturbation sf distribution smooth paper propose smoothing aforementione extensively mechanic gaussian family distribution smoothing sf unify control control convolution thereby kernel kernel derive smoothed exhibit smoothing sf incorporate procedure algorithm neighborhood optimum differ early straightforward simulation sf counterpart sf present sf gaussians smooth gaussian sf provide conclude remark technique gaussian unique measure minimize existence continuous procedure optimize functional verification trivial instance translate impose turn process general lyapunov say conditional py see parameter control field compact nn one depend function idea functional call call smoothed sf control purpose indicate sf sf also functional behave behave retain sf aforementioned function ng ng g distribution dirac zero concentrated mean fluctuation estimate discuss gradient rule simple show gradient long single stage parameter tune simulation gradient side sf simulate parallel pt gaussian cauchy kernel search univariate dirac delta ab list column indicate retrieve clear motivation study gaussians suitable infinitely kernel continuous value evy super finance distribution shannon expectation coincide usual notion retrieve gaussian maximize shannon eq q generalization first moment usual form q well probability gaussian distribution discuss kernel also covariance variate normalize constant limit fact infer apart special table circle smoothed algorithm sf notion gaussian case imply uncorrelated standard convenience center functional gaussian apply sf ensure rest smoothed functional two sided property evident g ng ng g ng claim manner hence result sided nature control existence sf sf consider respect side smoothed I ji write sequel identically random ergodicity assumption govern similar manner two sf eq sequel obtain satisfy nb must note quantity positive number average inner principle generally typically generation consist uncorrelated random describe project vector step ht fix simulation govern z nz sf similar except update estimate simulation affect fix vector generate generate govern update nz prove technique compare present gaussians provide variate consequence result key uncorrelated zero variance zero axis case first use integral beta expand integer get I gamma function result similar equation however define easy limit qx subsequent analysis sf update fast g sf rewrite iteration use p pn disjoint p quantity convergence invariant satisfy martingale difference variance xt exist ode well unique globally stable differential equation ode xt rewrite
node potential guarantee decay ise degree absolute potential respect observe short depth trade depth identifiability impose degree low representation observe distance bound information impose requirement estimation edge characterize presence absence correlation extent enough edge impose impose edge potential path low potential need depth cycle graph determine build roughly depth ensure distortion additive test choose appropriately establish correctly node close absolute distance bound consistency tends supplementary locally high requirement logarithmic require complexity graph locally like match fully apply distance form local implement improve complexity extension learn tree establish like graph extend discrete notion acyclic subgraphs subgraph encounter induce denote consider di j j v j di access distance estimate algorithm relevant assumption graph marginal neighbor q converge locally bind q routine variable decay degree node identifiability tree local ise edge potential use bound require similarly b impose parameter local span constraint imply compare enable reconstruct bound concentration bound distance merge node latent condition structural q pac reconstructing model graph large graph latent tree model notable require requirement decay establish converge attractive ise model without analysis require local convergence applicable discrete decay accord gaussian mixture step correctness tree distance di concentration distortion presence briefly establish correctness graph distance cycle correctly latent correct local correctness guarantee reconstruction learn analysis performance ensemble degree constant depth straightforward union vanishing scale uniform scale far reconstruct notion graph inexact let unlabele without latent page decrease test indicate generalization also evaluate occurrence observe select version capture degree graphical edge lda dirichlet bic monotonically decrease bic indicate complexity bic general involve employ propagation exact configuration bic addition coherence frequently topic model pair score good measure human coherence external corpus contain article bag vocabulary article top word probability vector select method graph ground hand bic overall normalize output show node edge overfitte threshold l na na na employ control cycle tree long cycle find short effective discover intuitive relationship link computer see tend subgraph computer program context modeling contrast design hide input competitive coherence find short cycle intuitively unable topic coherence suggest however model well indicate use model lda top word disk video window memory windows graphic pc windows software version disk pc space disk window file graphic format disk window mac evidence human power outcome market observe threshold tree another score threshold certain learn figure capture particular degree various group home group together however relationship threshold decrease add first connect index cycle decrease company chemical connect stock stock return stock return density stock initialization building cycle add decrease neighborhood trend stock predictive predictive edge prediction evidence however reduce degradation predictive effectiveness discover line early reveal learn established method decay derive selection finding reveal add complexity certain regime tractable latent thank berkeley discussion begin algorithmic uci david uci anonymous substantially appear remarks award award award nf award graphical consider characterize tractable develop provable locally regime ise structural n node node depend potential ise structural derive requirement implement provide output learn discover latent variable extension many form tree effective evolutionary give rise day specie financial modeling recognition vision learn g dimension consistent much small moreover despite restrictive topic encode topic reality relationship assumption nontrivial challenge develop area active neighborhood cycle regime practical tree latent like correlation implication decay immediately subgraph cycle challenge subgraph estimate merging matching estimate clear locally like graph latent introduce cycle model consist add neighborhood precise structural consistency page general simplify ise random obtain number require near depend minimum potential homogeneous variable match fully graph select maximum degree respect propose attractive parallelization dataset provide flexibility cycle variable output incorporate score bic schwarz estimate fidelity control cycle belief propagation experiment discover intuitive compare well generalize allow multiple greedy computationally expensive algorithm guarantee em base extensively mainly thorough overview provable guarantee paper inspire work recent classified group convex latent decay make connection explicit relate limit learn graph analogous study provable guarantee discrete high gaussian theorem et al consider convex relaxation easily model incoherence method hard contrast graphical estimate poorly dimension accordance page graph discrete edge variable mass p graph clique configuration clique serve normalize correspond family special ising ise subset node discover presence hide learn observe variable observe general graphical model tractable construction bind graph recently although tree like general global specifically constrain constrain latent tree establish tractable condition limit uniqueness weak condition correlation configuration decay neighborhood provide distribution markov potential neighborhood let graph former refer edge short distance denote norm say exhibit variation decay decay maximum latent graphical know tree factorization distribution pairwise distribution root root parent say recover distribution satisfy counterpart equivalent positivity particular ise potential positivity configuration give graphical subset structure model extensively study topic method additive metric structured joint consider metric metric special ising model logarithm operate additive tree path refer tuple denote characterize tree term characterize distance refinement robust test meet search occur side output test relationship infer involve child weak merge use modify grouping routine locally node observe node j j v ac bc relationship far break forest hide length merging method term add add local neighborhood consider neighborhood set construct especially attractive reconstruct graph long liu grouping develop latent section main acyclic approach challenge piece presence metric address decay challenge merge
often correlate differ order magnitude narrow local proposal propose underlie intrinsic thereby principled proposal automatically tune move possible mix chain chemical reaction obtain paper method compare commonly hasting study chain monte behave next overview markov jump approximation section mcmc x process satisfy q conditional state problem transition depend time rate expensive exponentially initial simulate next state exponentially mf complete respect time transition occur transition specify suitable apply transition solution propose trajectory transition time lead construct iteration trajectory observe demand system observe poor mix many sufficiently result demanding suggest simulate use exact metropolis master provide simulation observe result purpose presentation formal derivation reader requirement approximations constant jump fluctuation example chemical capacity circuit langevin equation match dynamic satisfy condition transition poisson variable transition large imply approach limit compute variate langevin columns return wiener differ due intractable augmentation type transition unobserve euler follow langevin dividing get eq see fluctuation equation eq linear differ stochastic taylor differential nothing finally collect remain jacobian fluctuation state magnitude negligible coefficient deviation thorough supplementary convenient expression approximate linear integral sense integral simplify imply multiply time assume trajectory trajectory state continue kind encountered measurement reaction inference independence applicable available stem normal diagonal need notice still exploit markov likelihood invert variance brief work concept mcmc require introduction scope metropolis hasting algorithm move accept context inference density involve proposal mechanism proposal multivariate normal therefore replace tune fast mix high mix markov hand deviation posterior substantially correlation adaptive hasting reaction rate scale simulation metropolis scheme scale independently target manifold mala langevin employ euler solve integration th symbol coordinate mala stationary metropolis hasting correct proposal local gaussian metropolis contrast tensor avoid discuss simplify derive assume manifold depend see mala tensor identity correspond euclidean probability momentum potential hamiltonian manifold energy energy simulating require reversible volume preserve generalise simulate detail generalise path across iterate tuning integration simulate hamiltonian hasting mix e h tensor manifold hmc mcmc discuss metric natural tensor fisher multivariate normal general time point integration derivative quantity ode system ode system equation sensitivity derivative respect ji operator stacking model chemical differ order mcmc allow operate unbounded unconstrained parameter numerical simulation determinant jacobian derivative follow prior incorporate exist suitable informed knowledge design want restrict chemical become high since enough prior restrict introduce normal strictly moreover note fisher use add near simulation chemical consist unstable unstable convert reaction time ts ess h know make explicit infer constant simulate simulate state discard trajectory observation trajectory mass rather trajectory perform drawing index independent point paper similar derivation namely system approach transition ode system density identity condition include draw hasting sampler scale adapt burn phase acceptance follow simplified algorithm initial tune achieve sample adequate sample table compare ess different gradient fisher mix good achieve system sampler system poor normalised ess mix contrary report marginal posterior posterior figure mh sampler similar standard around significantly system chemical reaction transition matrix consist specie appear depend fail system concentration depend posterior reaction constant reaction demonstrate procedure parameter namely burn sampler converge variance obtain posterior show along constant fail illustrate applicability methodology system also involve expression protein adopt model specie dna chemical transition degradation degradation reaction notation dna dna system state transition rt rt rt rt gene section model stimulus towards gene protein concentration gene model hill make protein refer auto single expression probability simulate discrete unknown constant interval take time independent trajectory discard rest resemble experimental observation finally also trajectory system unknown figure infer b sampling independent point chain gene firstly see provide accurate parameter mala perform explain correlation parameter make sufficiently degradation protein affect rate expect heavily joint posterior exhibit correlation trace auto gene true hasting ess ess mala ess ess mean ess ess hasting ess ess manifold mala ess ess l ess ess panel expression dash dot posterior contour sample p red colour bayesian important show although inference autocorrelation limit applicability stem discrete simulation manner simulate ordinary differential fluctuation analytic sufficiently diffusion approximation research metropolis demonstrate exhibit metropolis sampler strong reaction gene system geometric proposal mala conceptually provide good trade auto believe
extent spectra near overlap sec time sensitive lie little spectral mode sensitivity examine accord criterion work nominal fouri periodic frequency show movie embed separability periodic doubly degenerate periodic mode fig movie mode generate anomaly amplitude mode exhibit amplitude domain movie e along variability california current frequency laplacian suggest one low mode power power shorter upper north typical movie prominent scale temperature develop west family gradually short peak short movie key temporal arise sharp year periodic counterpart nearly base integer multiple year result fouri spectrum peak skewness frequency mode fine spatial activate amplitude small aspect enhance mode propagation scale temperature anomaly spectrum include anomaly composite temperature anomaly consist lead periodic window near spectrum comparison scale parameter singular mode mode anomaly north see raw year field determine mode lead frequency mode describe degenerate describe variability dynamical pattern highlight account nonlinear high existence variability variability restrict temporal lie lead datum riemannian capture rare event occur behavior truncate expansion show norm dominate lead hand clearly moderate amplitude occur embed introduce qualitatively pattern illustrate spectral gap separate mode spectral evaluate via versus parameter display fig evaluate embed cf fig line preprocesse subtract filter overview fact lead degradation conceptually preprocesse nonlinear neighborhood relationship extract north study entropy temperature field temperature anomaly field structure mode sec characterized decay obvious peak periodic spectrum obvious turn mode spatial experiment dimension depend crucially dimension window temporal pattern conclusion qualitative scheme series take arise like utilize partial high distinct mode linear differ crucially square integrable nonlinear manifold comprise coarse grain space tailor nonlinear geometry riemannian increasingly behave orthonormal address theoretic method assess assimilation k overview online release initial prominent g statistical temperature identify regime nonlinear application v advanced w preserve decomposition compare laplacian low frequency variability qualitatively spatio learn representation diffusion map reaction g intrinsic system diffusion et regime shift definition h f volume geometry series detect volume page r geometric tool harmonic definition volume page determination reaction coordinate locally scale lee york wu nearest search j temperature california extract qualitative g reveal nonlinear analysis von w university et data assimilation range forecasting part I grain simple quantify predictive coarse grain analysis generalize spectrum complex represent pattern exhibit manifold constraint notion smoothness require temporal belong hilbert span lead eigenfunction eigenfunction evaluate ambient singular fine graph important rare dimension overfitte criterion north year run besides recover process carry variance capture dynamical keyword spectrum acquire large weather forecast seven organize spectrum variant physical interpretability algebra detect portion dynamical building work operator use differ crucially tailor hilbert square ambient employ eigenfunction imply relation cf partial capture laplacian capability rapid transition rare efficacy north version mode contain behavior domain characterize five year period amplitude spatially mode enhance region anomaly dynamic little raw order capture pay characterize minimum beyond particularly prevent overfitte machine hereafter define exist fulfil eventually theoretic step familiar system delay indistinguishable differential large manifold incomplete thought euclidean e tangent construct inner riemannian effectively constitute adapt exhibit regularity pattern projection rapid geometry nonlinear produce ingredient replace datum specifically temporal mode well behave k associate define theory theoretic basis scalar product solution problem orthonormal span lead element derivative
eigenvector dimensional isotropic center multiply unitary thus row assume isotropic take probabilistic close likelihood rotation generalize model semi pca observation formulate noise coordinate remain part linear pn q linear toy comparison spline case introduction generalize look verify eq finally eigenvalue covariance obtain pca order combination knot position function formula drawback equation matrix rotation original datum perform inverse step p intercept directly non linear select among candidate penalize criterion classical criterion certainly observation either bic popularity compare pi measure interest tr maximized method index parametrization refer topic pursuit problem index normality complex scatter dedicated outlier coefficient matrix entry otherwise could iff neighbor proximity generalize near projection axis eigenvector article additive b spline select coordinate sample deviation standard axis near original axis htb axis use criterion select number point htb c dim bic residual draw library give coordinate sample standard plan display library bic criterion residual variance factor expect axis dim residual spline function library consist dimensional classified network network terminology article model select largely control regression parameter step datum cloud pca display htb summary table visualize point dim htb predictor evolution coordinate visualization prove principal model simple truly interpretation depend highly projection linear library simulate display proposition example section corollary cover property propose set model experiment simulate pca pd affine scatter principal linearity assume cloud principal start technique curve principal surface belong neural non linear model projection adapt point approach estimate model give hereafter auto function dimension construct auto affine precisely write equation principal surface linear thus pca study view situation projection linear however difference appear get tractable noise assumption seem estimate estimate extend moderately high complicated simplify inspire perform present seem linear organize probabilistic
clustered hmms ms despite suffer memory hmms need store evaluating time replace output consequence result appropriate assignment similar spirit bregman base divergence difference form random whereas validate point discussion experimental density automatic benefit cluster sc high synthetic similarly annotation favor suggest robust experiment run suffer delay number hmms standard em considerably section hmms consist input hmms hmms hierarchy level hierarchy first level input hmms experiment input hmms simulation cluster hmms various summarize hmms input hmms learn explain build hmms hmms recursively h form cluster hmms group center virtual hard cluster virtual cluster calculate hmms hmms satisfactory allow gmm general experiment good affinity integrate extend construct k mean hmms virtual particular observation reduce consistent life goal series sequence input hmms hmms hence cluster cluster directly cluster model texture dt series hmm modeling stage fashion variable immediately level hierarchical together dynamic dt mixture analogous h time gauss discrete multimodal metric measure cluster assignment whether pair incorrectly assign well fit time likelihood hmm expect generate hmm assign sc hmms assign explicitly optimize similarity likelihood consequence optimize intra wise intra cluster tb walk b motion motion series represent hmm cluster hmms represent motion motion apply sc cluster motion capture action dataset motion span class jump marker spatially illustrate build hmms learn motion state emission level contain em dimension repeat action dataset sequence consist time coordinate body marker capture subject emission e h sc random initialization performance dataset hierarchical color indicate ground cluster motion motion class jump together portion probably dynamic action movement shoot hierarchy together indicate walk highest create desirable return considerably rest experiment sequence properly incorrectly quite furthermore high level ccc rand quantitative comparison rand vs level particular cluster sc level overall keep mind similarity favor h addition novel center motion center conclusion effect table generally rand em original motion sequence implicitly integrate virtual variation motion sequence robust roughly time sequence favor validate still closed expression virtual approximate statistic without center h large significantly long finally report consistently cluster dt note rand sc optimally cccc cccc rand present physical two rand e level level outperform high center sc ground truth rand index metric h sc cluster hmms hmms hmms hmm noisy generating estimating noise noisy version hmm noise vary collection hmms number always emission sequence original hmms state transition emission hmms emission distribution set hmms differ emission emission specify experimental b figure bottom differ variance emission sc hmms cc result clustering rand expect log likelihood center hmms assign average hmms experiment different variance produce superior line line color rank song retrieve metric tag least usage tag fold validation tag sc song gmm annotation f time sc algorithms annotation look overall runtime music runtime sc h song level efficiently song tag runtime sc correspond set run significant performance affect contrary em strongly list regularization leverage song sc considerable cluster affect result annotation model novel cluster retain annotation achieve considerable sc computational demonstrate hmms observe outperform sc suggest former accurate center fact cluster cluster maximum sc hybrid contrary separate cluster optimize log respectively center may affinity finally generative outperform audio metric base pair test annotation h base contrast hmms gmm non time experiment outperform perform similarly music audio short frame investigate character originally motion character segment hz force numerically smoothed training half testing write character hierarchical character relevant gmm character aggregate hmms component use virtual virtual hmms assignment hmms determine mix similar temperature hmms desirable hmms estimate character character write character example classified character retrieval character example rank character sc hmms learn since vary variation threshold varied retrieval performance average tag include intermediate hmms trial initialization stop retrieval classification accuracy em sc list retrieval various consistent annotation retrieval sc metric center h representative relevant intermediate hmms hence relevant classification training fast sc hmms character efficiently actual perform retrieval reduce number approximate suffer overfitte suggest h consistently improve metric converge suggest regularization average sample virtual actual positively elaborate annotation retrieval section character character song audio tag datum performance evaluate lead slow iteration character control character intermediate clustered summarize provide good candidate center conclusion face information select hmms cccc p generation virtual virtual annotation retrieval training tag song level varied retrieval section vary similarly virtual music annotation retrieval virtual song level improvement experimental virtual sequence short positively reduce quality variational cluster respect distribution represent demonstrate efficacy h summarize annotation improve model virtual prevent learn virtual performance virtual sample stage hierarchical intermediate particular partition song song ms execute song tag impact depend slightly hierarchical may suit cover inspire datum intermediate intermediate h ms final plan extend gmm universal emission hmm commonly speech move complexity estimating background inference g similar plan hmms emission extension background model song motion acknowledge wish acknowledge yahoo fellowship national science grant grant china support foundation research support part appendix derivation maximization carry independently follow terminate maxima step involve hold distribution substitute reduce mixture weight collect weight constraint maximize state independently hmm collect summary consider appendix give similarly hmm collect transition maximize emission function index gmm give gmm optimize optimize hide markov generative condition hmms base collection hmms characterize representative manner consistent cope intractable leverage variational demonstrate motion capture annotation music hand show method variational reduce improve robustness hide model markov hmm probabilistic model assume double hidden evolve markov encodes encode appearance condition current hmms music online hand write cluster hmms representative center appropriately represent application motivate hierarchical speech hmms indexing reduce hmms large semantic annotation video cluster hmms estimate work exist cluster hmms operate directly accord pairwise manifold application would solution propose construct cluster apply cluster prove successful group hmms generate center give hmms hmm maps close suboptimal hierarchical estimation hmm spectral sequence instead hmms algorithm construct output hmms derive hierarchical input hmms input hmms hmm guide estimation em main em observation algorithm probability mixture reduce gmm cluster dt image indexing represent I mixture semantic extend mixture hmms hidden markov mixture ms marginalization state process tractable hmms formulation leverage derive make tractable hmms hmms representative class graphical leverage approximation general suitable center manner memory requirement hmms mixture large dataset estimation run intermediate principle maximum estimation average possible step provide prevent robustness compare full costly operation pairwise fold hmms generate center evaluate involve iii hmms improve work originally propose organize markov present follow discussion section emission evolve take evolution encode matrix px generate accord emission emission mixture multivariate covariance matrix specify efficiently observation sequence sequence hmm joint observation finally obtain summation state parametrize parametrize collection reduce clutter hmms emission probability could cluster motion hmms indexing retrieval reduce mixture hmms annotate hmms efficiently subset compact hmms hmms hmm input weight generate input likelihood sample explicitly generate instead maximize likelihood way marginalization input model marginalization gaussians long hmms leveraging propose kl hmms present derive markov likelihood b likewise variable indexing distribute base component group cluster encode pz mixture reduce virtual manner require generate virtual r hmm gmm emission gmm py b I hmm hmm state gmm c hmm gmm emission mixture gmm emission sequence px rp py py hmm b l z py j hmm py gmm component clutter short emission denote indexing component gmm reduce denote reduce finally hmm base hand notation e b sequence base component imply addition expectation take variable specify short b summarize include name short notation table summarize variational variational subsequently set virtual base draw entire virtual base reason space different hidden cause problem virtual sample hide instead treat estimate likelihood virtual reduce use number virtual start different virtual virtual eventually maximize preserve otherwise general deal estimation present maximization view step family log optimize variational respect formulation readily extension replace practice tractable ms derive observation eq observation kullback leibler kl variational approximate reach compute lower let take expectation side follow jensen convexity family probably cut broad context expectation intuition compose cost successively arise result nest lower hmm expect likelihood average exactly forward calculate expectation essentially emission another gmm emission hmms variable variational I hmm essentially mixture low first hmm state state state respect hide assignment gmm emission therefore q py log likelihood gaussian virtual compose nest hmms emission variational variational alternate variational subsection variational maximize particular first gmm emission base maximize hmms statistic calculate first form variational well maximize low gmm variational represent computable bind chain variational explain substitute carry independently pair separate break substitute q assignment base py r ms
parent visit count action initialize throughout maintain backtracking h exploration simulation sample set reach leaf expand action mdp determine discount provide leaf action node value e return diagram simulation treat forward explore number effort nevertheless visit often eventually h ar h bs ar h h ar transition statistic fully observable mdps transition reduce planning give address expand lattice tree reduce offer tendency concentrate equivalent limited reduce path previous indeed tree costly transition require parametrize action additional latent p p parameter end chain rewrite require parameter stop parameter state instead necessary individually action performance improvement mdps single example policy locality reward work optimal free manner use interaction acting translate pure exploitation exploit instead greedy policy bias high provide complex policy overhead g state construct h h h confirm efficiently supplementary exist give list planning exploration rather bayesian rl maintain transition sample execute idea combine finite quite difficult variant provide criterion suffer tree node bellman value value child node wang search belief non trajectory sample take optimal node sample adaptation search mdps fact bellman monte bellman wang tree expand accord trajectory albeit action selection decision promise upper bind solve et al step step precise turn hard translate sophisticated c loop bayesian mix boltzmann first standard problem comparison popular task infinite location implement sample set exploration domain vary varied factor depth search set domain large algorithm mdp grid direction execute small design collect collect give since last visit step execute give quantify fair comparison prior criterion discount might rl dynamic domain run multinomial grid multinomial describe collapse scheme ba algorithms transition transition inside probabilistic general collapse scheme section domain plan increase code branching parameter specify performance ba learning l rl include different test task require exploration scale well plan opposite build merge optimistic explore planning generally obvious trade branching depth greatly naive example drawback forward sparse bound expand depth solve successful monte carlo towards trajectory grid average environment discount run generative prior correct prior gain performance report infinite associated otherwise agent know reward return oppose standard dynamic correlate transition transition posterior approximation conjugate supplementary planning solve mdp dp state ba deal must belief affect planning sampling full many planning time behavior drawback policy hide reward period algorithm treat node armed planning theory bandit nevertheless prior principled principle encode prior work explore structured prior match class agent base rl show tackle approach poorly provably carlo tree explore augment ba expensive belief tree root sample rl comments ba apply bayes adaptive equation ba ba effective define history lemma lemma induction root induction hypothesis definition match node induction sample node occur h induction ari apply ba mdp simulation ba result intuition unnecessary root update root evaluating mean illustrate empirically cell table act arm act accord guarantee bayes index arm mean discount step root root breaking execute result root execute estimate update get expand action select ucb obtain count green dotted prior employ video relatively block agent exploit explore b rate find might sub bayes agent sub likely plan qualitatively match construct hasting transition notation location reward column rough correction row account observe column due potentially since biased get proposal parameter could introduce enough keep link probability propose accepted row accept reject finally sample observe omit minus ex mm david reinforcement elegant exploitation ideal unfortunately find policy paper bayes planning outperform rl benchmark avoid bayes model belief qualitatively previous work exploration process mdps discount dynamic discount short potentially costly exploration reward long exploitation state agent acquire bayes
unweighted opposite approach right lebesgue ease rest paper continuous bound away shorthand metric metric bx r volume dataset accord connect vertex neighbor versa distance carry graph length weighted edge weight path case short path short let connect parameterize define know geodesic path geodesic distance particular inequality geodesic path monotonically pass work monotonically region paper geometric unweighted graph statement interior distance rescale distance unweighted consider unweighted two one condition stronger satisfied small see prove couple proposition ad imply imply dense assumption say exist exist geometric graph x bottom geodesic divide dense sampling vertex ball fu fu apply triangle fu geodesic vertex write low fx continuity boundedness approximate ball hold continuity lead geodesic connect path write lt qx lt dt qx quantity definition unweighte approximated near neighbor bound eq pr min pr max bound ps ds px ds follow kx ps k ps es binomial prop get p low finally sampling decide keep parameter maximize success sample prove exist density increase weight eq q edge assign interested limit respect go distance distance guarantee distance determining distinguish regime call prefer along distant ir formally prove contrast scaling factor simple set weighted consider I density increase edge fix two essence sketch step adapt graph adapt general nearby say probability probabilistic criterion geodesic connect near connect sum along adaptation fu fu graph path part segment get write fx call intuition take example straight prefer go rather big graph prove effect special aware regime study manifold case difference unweighted underlie low distance geodesic unweighted last unweighted graph short path unweighted high unweighted heavily region region uniform semi either implicitly laplacian alternatively take suggest way author omit estimation show simple family assign edge paper path unweighte even avoid result seem obvious machine practitioner unweighte robust unweighted behave degree graph density graph short short graph decision largely drive consideration implicit consequence choice carry machine learning make use structure benefit remark corollary conjecture exercise rgb weighted build short path tend infinity unweighted graph function shortest weighted distance fundamental machine geometry near assume go question arise behavior short measure give shortest distance study geodesic extend recent consider completely power little regard second
instance fit summation distance summation multiplicative representation find seek exist median integer projective admit general speaking point importance fitting shape denominator numerator sensitivity quantify importance mean cardinality dimension euclidean space size polynomially problem scheme range functional pick subset assign appropriately pf sensitivity pp f definition certain shape projective many area small thus construct small remove analysis article prove bound projective total total shape fitting shape set hyperplane sensitivity answer euclidean roughly intrinsic shape consist shape contain sensitivity projective problem shape tuple contain dimension tuple total tuple approach sensitivity projective sensitivity optimum fitting computation simple median distinct sensitivity greatly proof projective shape project contain subspace constant usually directly method translate computing compute approximately sensitivity bound ambient bind point sensitivity consider way every ambient dimension small exploit f f ps negative construction get positive get shape fitting run heuristic fit heuristic modify via original item interpretation issue broad projective paper organization article establish various clarity omit sensitivity methodology also stream article summarize relate low section upper approximation integer projective result study crucial define fitting hold fp fs literature requirement weight requirement satisfy instance shape problem sensitivity understanding denominator sensitivity fit somewhat read skip definition instance system r p projective cluster power distance vc set dimension fit sensitivity fit projective problem sensitivity projective upper projective clustering depend integer projective integer depend point order optimum total sensitivity fitting quantify shape sensitivity contain regardless ambient shape fp pf object sensitivity upper contain fitting minimize let relax p qp r us triangle p remark function derive upper shape p bound sensitivity contain turn problem sensitivity projective cluster fix dimension contain b projective phenomenon somewhat general sensitivity high projective clustering euclidean contain dimension basis projective tuple subspace derive one derive proof power distance projective shape total sensitivity correspond center p copy I cc remark result projective subset point use upper problem shape fitting tuple sensitivity line cluster problem total shape function alternatively recent sensitivity line lie sensitivity projective sensitivity give cluster total shape r I vertical horizontal vertical horizontal nj ip I nk tuple instance sensitivity projective theorem dependence sensitivity generalizing dimension readily accomplish manner size notion row th matrix condition column span satisfy ij u matrix rank z step use upper sensitivity fit hyperplane consider fitting hyperplane total sensitivity hyperplane u x h I denote whose u md md total dependent sensitivity shape total fitting sensitivity set hyperplane hyperplane arbitrary
solution however user gain guarantee service minimal goal convergent satisfactory solution numerical realize payoff basic banach eq order strategy iteratively htp distribute banach guess slot banach solution banach measurement get noisy invertible respect non alternative mean iterate without several remain without feedback start impact player fraction fully feedback moment iv speedup deterministic function stochastic question investigation iterative procedure stationary equilibria game space contraction banach iteration converge rate contraction propose behave lipschitz pseudo however acceleration cognitive convergence nice mean game without examine equilibria application financial speedup satisfactory illustrate game market biology cloud rich equilibrium little would allow scale game huge therein mean behavioral less scheme large player player action player aggregate form action player slot simplify drastically partially distribute player function capability observe previous play logit distribute dynamic would term action payoff fully payoff mathematical know gradient ascent directly player numerical realize payoff employ pattern payoff combine distribute payoff strategy algorithm estimate want gradient gradient observe payoff scheme estimation adjustment hierarchical three algorithm combination multidimensional address regime reduce learn partially distribute question framework partially know payoff capability observe field boltzmann field response fully scheme generic player numerical payoff noisy mean feedback information player player scheme payoff overview fully game fully reinforcement promise learn heuristic bandit difficulty extend game balance maintain exploit gain explore update dimensional adjust infinite action conduct claim recently selection dimensional even dimensionality idea continuous space slot standard continuous action widely correlate equilibrium compact equilibrium finite work game number framework behavior player decision manner create field contribution follow game player learn depend field learn drastically analysis interactive case acceleration time gap satisfactory player fully field scheme satisfactory game reverse speedup improve solution numerical illustrated games service section speedup player formulation game yet game aggregate game function player depend payoff game action j constitute shot game wide economic market market price etc influence total center server influence center serve share utility cost sharing bandwidth share wireless influence interference delay depend aggregate profile equilibrium deviation ask well e compact quasi respect possess least equilibrium development equilibria area characterization nash topological convexity detail examine admit equilibrium good mapping scheme scheme memory unique element map game follow simultaneous mean require observe common j compute htp slot user aggregate banach iterate consist contraction map unique fix contraction exist real satisfying banach fix unique suffer drawback continuous resource share player demand unit payoff simultaneous good derivative interior clearly contraction banach limit convergent modification banach action response evolve go least banach lipschitz may point attempt take decrease rate refer read htp sequence slot observe due clearly extend htp response initialize slot observe aggregate f j htb big rate mean field response cycle scheme well know convergent say converge fast class banach comparable banach reverse convergent point property asymptotic important engineering result limit compression asymptotic generic strict contraction follow gap regularity class set mapping subset empty combination follow get constant converge geometric minimize integral uniqueness ode ode aggregate continuous steady response nash equilibrium lyapunov equilibria game call lyapunov game turn first reach phase say trajectory explicitly convergence trajectory reach set e go fast field payoff single player field banach eq reduce banach mean field know mean mean field strict empty convex learning find field mapping iteration field strongly empty fix suitable condition approximate point major concern iterate exhibit slow speedup learn mechanism convergence trace measurement solution speedup compression regime go infinity asymptotic window bound aim non speedup learn algorithm aim pattern gain outperform traditional scheme problem convergent exhibit user aim point generic accelerate slowly converge limit consideration I quadratic cubic speedup extend multi speedup basic newton method iterate newton generate locally root cubic update speedup map player classical fix point iteration speedup derivative fix bind fix point guess differentiable fast locally learn arbitrary high method smooth systematically high quickly converge degree fix converge field learn great nonlinear avoid computation gap let convergent convergence range transformation transformation frequently computation present newton replace derivative speedup obtain note method write scheme two sequence speedup order htp method guess slot observe compute speedup obtain variant use htb speedup newton speedup newton model base technique non speedup however computational speed iteration slowly cost iteration iteration one derivative iteration may newton speedup consecutive sequence reproduce convergent speedup fully player algorithm perturbation observe r j j j taylor expansions proportional second hessian payoff player difficult convergence estimate pseudo hessian independent field learn second realize payoff j w tr generate estimate pseudo action mathematical feedback player action result false consensus belief feedback speedup initial iterate payoff j action analyze integral scheme social optimum payoff sum divide significantly reduce optimize technique conduct limit limit next mean game compare aim guess prefer initial analyze experiment reader choose way see guess third guess besides would guess still stick guess nash would stick game simultaneously choose number interval target player number close target payoff study experimentally game useful number iterate reasoning game illustrate never stochastically dominate player choice iterate iterate nash feasible profile modification interior equilibrium target interior equilibrium satisfy assume learn explanation feedback game
department mathematics universit du universit de cp nj usa abstract reduction frequently natural represent curve argue though technique benchmark adequate indeed essential serial inspire dynamic component propose study illustration improvement entail static procedure dimension functional functional functional technical storage phenomenon life improve record high frequency extract characteristic possibly high specification analysis recent prove consequently field research community realization curve observation cube surface deal surprisingly efficient role arguably technique analogy multivariate rely decade take early later influential book work asymptotic et smoothing et robustness application include linear usefulness also recognize scientific chemical et reference cite paper section background though serial serious numerous either space daily transaction daily pattern environmental etc realization time series daily observation day parameter time study ar refer ignore dependence series context inefficient serial quite problem estimate traditional traditional adequate fact seminal recognize operate dependent hence negligible instantaneous predictive besides failure produce optimal static uncorrelated exhibit motivate development dynamic idea transform say individual mutually lead lag autocorrelation allow thank orthogonality dynamic analyze analogy reconstruct low dynamic version expansion dynamic principal first suggest series purpose similar setup methodology domain rest organize section application describe proposition asymptotic feature simulation study contain proof appendix c paper aim objective essential difference develop main serve approximation study mathematically quite elegant increment disadvantage behavior eigenfunction contrary construct remark work technical leave consecutive curve level underlie show reconstruction panel panel expansion component color notable completely symmetry addition daily average extent curve trend spike considerably illustrative reconstruction much well application static mutually orthogonal mutually orthogonality lag component contrast treat illustrate superiority orthogonal auto break et al detecting mean sequence functional project proportion account enough pc th eigenvalue converge roughly statistic converge score become independent obtain separate statistic aggregate structure hold converge still need non dynamic principal long let get ensure aggregated principal feasible series response say depend regressor intercept natural frequency parameter scalar constitute functional analogy involve operator vector mild score cross greatly reveal frequency th consequence regressor assess individually quantity version one may hypothesis justify dynamic retain impact tool technical detail integrable unit within consider take integrable function stand modulus use serve equip inner define possess curve define ex h ex operator kernel well quite consistently mean center preprocesse element eigenfunction orthonormal eigenfunction loading approximate instantaneous static perform observation take serial likely exist globally speak base goal introduce density operator analogy classical concept operator denote sense hilbert schmidt see condition provide spectral operator create filter section build block construction define ty shall filter q addition write emphasize operator filter play density absolute coefficient hc pointwise general consider element I collection frobenius equality thus understand g consequence every adjoint hilbert schmidt analogy admit eigenfunction eigenfunction assume choose ty desirable discussion wish suppose measurable discuss eigenfunction standardize imply zero hold conclude filter lag functional principal zero value satisfy assumption th principal component call dynamic multiplicative devote dynamic elementary satisfying ex hermitian real coincide one ex uncorrelated lag ex matrix tell analogue static formula dynamic square dynamic mention score contrast draw analogy static replace however curve reconstruction involve replace alternative filter expansion dynamic theorem suggest well define score stationary mean series operator impose preliminary integrated square estimator functional state show estimator satisfy concept usual lag estimator eigenvector need identifiability th large finitely essentially common ensure eigenfunction sign remain provide sign situation slightly since setup multiplication way fix eigenfunction impose identify long assumption frequency denote exist choose almost consistency variable define hold mt practical record often necessarily datum require limit common error tend specification omit technical exercise cast proposition necessary carry represent basis spline wavelet survey preprocessing refer sequel matrix entry span dx linearly orthogonal linear follow let operator p ij matrix operator q iii linearity I eigenvalue dynamic task multivariate put throughout lag common choice tuning estimate coefficient ms numerical simple well available power substitute replace eq expression create observe dynamic variance estimate incur consider reduction q quantity datum half diameter ambient air transformation perform heavy observation remove traffic intensity business software transform discrete curve daily represent display figure fourier black represent curve center empirical traditional operator tuning still leave element away rapidly justify element ht filter focus dynamic explain latter static figure entirely idea static score course loading appear remarkably another total get insight static say tu deviation effect attribute regard dynamic interpret sequentially advantage fact approximation single dynamic expansion study impact triple eq mean panel impact score mean curve follow panel concentration curve highly correlate indeed panel first dynamic static interpret static day increase decrease data trend roughly consecutive compare middle panel dynamic expansion simulation performance static variety simulate result static dynamic expansion performance measure computation implement along linearity satisfy v var fourier process mutually choice combination follow methodology spectral density tuning calibration lead moderate variation fundamentally obtain perform integration chose impose time moderate fast experiment time deviation setting thus basically dynamic replication systematically procedure finally contrast exact numerical integration require truncation lag little deviation matter practice explain contribution higher visible slightly static static static deviation multiply value principal component take literature functional sequentially serial happen instance process static setup reduction paper dynamic serial functional case reduce static serial dependence static quite significantly serial strong outperform implementation ii toy air iii simulation study bring evidence dynamic functional small cast rigorous show mathematically rigorous methodology introduce adopt specialized setup throughout denote separable equip space since frequency domain nevertheless series space mapping pd pl hilbert product fourier denote limit almost fourier eigenfunction eigenvector scale unit additionally unit circle exclude impose way expand eigenfunction fouri sense line stationary hilbert schmidt operator mapping assume hilbert space sometimes hilbert schmidt hilbert impose possess convergence easily show self adjoint establish condition random call I applicable series purely moment dependence verify many impose follow result trace
problem opponent space addition state agree def unfortunately intractable agent belief expected utility arise payoff opponent advantage select opponent payoff sequence probability payoff gap second version expect gain bind prior indexing gain experiment parameter environment action stage norm first stage finally measure choose gain relate environment consequently guarantee two adversarial must strategy obtain stage exploitation hard reason execute belief execute calculate posterior well know reinforcement confidence alg reinforcement thompson alg choose exposition restrict attention alg use q optimistic evaluation efficiently mdp construction belief stage belief type thompson multi armed problem confidence p sign convenient payoff payoff stage policy regret stage follow shape policy begin two type slightly main terminate action take stage terminate set payoff payoff function sample uniformly opponent knowledge maintain payoff select payoff explain estimate belief stage problem quite case opponent fig greedy policy payoff empirical linearly grow slowly adversarial opponent nature fact payoff greedy r r r r r r r r greedy r sparse reward capture act environment previous theory multiple arguably purpose another purpose gain without action mind instance overall force payoff adversarial opponent sophisticated environment arm begin opponent importantly thing noisy signal hard consider herein bind discount well opponent example problem task hard agent close game approach bandit essentially involve policy relate search bandit actually action experimental show greedy policy strategy sequence encourage visit frequently naturally suffer change goal exploration agent measurable negativity easily execution agent explore environment necessary thus setting define reward stochastic agent opponent act accord apart formally describe link bandit factor mdps examine reward act environment arbitrary objective question act current knowledge might analogous motivate human behaviour formulate multi game opponent act markovian environment stage begin stage opponent determine agent agent act payoff opponent select problem reward process first stage want payoff stage second adversarial uncertain environment resource must spend important effort towards future may lead maker explain necessity contribution analyse opponent nature unknown stage gain exploration approximation adversarial confidence well greedy policy however nature greedy perform payoff force explore next introduce environment opponent nature adversarial sec act reinforcement confidence bound conclude link problem theory opponent payoff reveal regret utility maximum utility stage total require strategy stage remain environment perform stage stage act throughout stage markov tuple index shall use class text represent sequence kernel opponent reveal begin stage opponent encode task go rl map rl agent acting process mdp equip reward discount utility discount map ts apply payoff function experimental rl payoff maker choose policy use interact environment jointly select action
column select selection algorithm run nk nk reason call advantage avoid load whole memory none three step svd algorithm memory expensive fast algorithm maintain require whole section several run natural bag natural uci dataset biology bag dataset vocabulary million sift matrix upon large truncate svd experiment matrix dataset source image image matlab implicit subspace error bad conduct ghz much ratio pt pt experimental match run become efficient fast small row randomize problem fast scalable art e algorithm require row achieve subspace require subspace nk nk ok moreover enjoy loading make demonstrate effectiveness efficiency low proved problem work sake call algorithm nk x compute weight j kk I guarantee definite efficiently eigenvalue eigenvalue eigenvalue comparison moreover overall equivalently column convention adaptive give r ai ii bc ac ip c take symbol bold matrix distinguish vector ji accord lie r w w complete large singular column f c lie span singular along lemma nj first r r jj r prove lemma propose line lemma algorithm row give r prove randomize adaptive near target column r cost line total mr mn space thus assumption time algorithm stage nk nk nk cn li china important nystr om approximation approximate term advantage possess tight low complexity avoid maintain memory several improvement scale computation matrix stock web video bring modern effort understand factorization method informative facilitate principled truncate find svd little concrete difficult uncorrelated people claim interesting basis insight actual decomposition technique construct stage row selection implement widely recent work later sufficiently particularly ratio p divide method parallel require art subspace exactly relative cost svd numerical unstable impractical paper art bind list notation review section mainly section compare art defer let th row column frobenius norm svd write eq top u z discuss development algorithms additive section column error ratio hold aforementioned stage stage relative seek stage stage solve relative ratio select achieve ratio algorithm inefficient step svd exist column q expectation right input since time consume employ approximation speedup via approximate randomize svd target factorization optimal dual spectral nc ac optimal sampling present column define ai nc expectation r procedure give theoretical work three theorem
probable dense linkage linkage observer knowledge order convention black observer white zero unknown thin edge study formulate observer star topology utilize show facebook separation facebook number typical topology rapidly count heuristic neighbor far exactly study first contact topology obtain direct reliable raw detection community within dense linkage community linkage implement mathematically partition detection become maximize quality modularity adjacency edge modularity essence result reader lot clean amenable reason people start detection large truth validate quality recent year node associate school company person reality name put result ground truth reasonable observer local topology recommendation mean level application automate categorization cast label predict confusion show fig notation way label depth cc analyze reasonably kind possibility variation informative section computation conventional multiplication classifier framework assign score node score reflect one strict feature heuristic metric eq cut feature train probably pilot potential investigation observer compute adjacency matrix degree degree n inverse get also walk pr interpret distribution step pick walk node successful web axiom fit biased proximity visit node high proximity observer likely version informative however proximity observer score evaluate normalize biased instead stationary know put high walk w x node initialize portion pass neighbor use positive community connect connection rest node put capable distinguish numerically investigate matrix eigen provide irrelevant stationary bounded matrix multiplication separately total complexity computer asynchronous manner algorithm arbitrary portion share direct e call termination imply walk j since portion neighbor complexity multiplication complexity real also tight benefit structure graph currently user view list direct perspective consider dedicated friend friend observer eight run class rely user actually collect eight eight follow quantitative one observer denote manual review large community community interested ability approach name node default observer regard regard preprocessing cause grind impossible name list experiment observer friend positive practical look framework calculate rank graph roc approximately line distinguish informative human instead positive choose expect dominate curve everywhere experimentally effective run simulation plot curve partly focus compare light dark get get node return know statistic point typical user university threshold node reach drop study graph cut drop difference whether applicable current heuristic simply intersection observer intuition two likely intuitive bias reason information node connect contrary low community heuristic exactly many beyond node observer basic approach pr fig g high pure bad heuristic actually analysis user manually positive improvement already considerable bad concept put high observer vary alternative node fill positive put repeat round randomness scatter improve little I improvement besides manual labeling cause find cost degree vary scatter obvious plot heuristic perform application suggest v put node benefit friend recommendation already observer meaningful automate categorization randomness evaluate next evaluate heuristic plot increase node observer mostly l curve reach degree prohibitive degree heuristic note labeling node degree node recognize induce roc evaluation promising application realization multiplication observer section behavior use high two ghz cpu matrix multiplication section norm implementation stability denote choose fig exponentially second algorithm enough practical part fig also indicate approximation suppose require make upper converge proximity score use detect sharp detection leave detect constrain formulation work tackle locally author topology discover propose solve combine globally propagation community node combine locally detect community work go auc l roc extension current direct friend protocol help trust overlap community lot proposal available detection essence reveal seed observer seed reveal detecting seem people topology natural check friend list allocation observer manually degree directly however degree know label random node regardless paper community limited topology topology cluster traditional community collect establish find rank adapt justify applicability evaluation datum topology small discover commonly manually label boost fig service communication presence operating characteristic roc curve rate axiom department success service centralize lead serious concern operational robustness service party possess entire social service like friend community tackle detection impose page rank justify detection topology automate collect practice demonstrate adapt pr heuristic manually friend vector boost relative auc scale facebook part people daily life share platform role friend discovery community formation interest success service serious concern operational knowledge profile activity user obvious seek control overcome drawback decentralize implement party possess entire server super user even restrictive network book friend still effort design building adapt p design trust concern bt enough people within scope privacy monitoring attack difficult support critical discovery recommendation architecture many datum availability global require discover bag part observe acquire scenario community centralize one require localize centralized setting worth researcher tackle label lp switch
loss previous first relax mixed optimize relaxation utilize design input prediction classifier subset intermediate classifier determine tree distinguish circle black classifier child parent full base validation test node produce prediction two path deep handle threshold lead hard therefore input iv I sigmoid child exp aa reach child j node parent child naturally node risk probability likely risk classifier serve cumulative illustrate highlight input follow root denote along exactly analogous except weak classifier zero exactly weak feature weak classifier free reach terminal along marginal random reach l v use norm norm final naturally encourage combine penalty split cyclic v classifier traversal reach terminal depend overcome term relaxation prove minimize function show introduce alternate auxiliary perform conjugate latter follow guarantee decrease respect classifier jointly lack tree optimize help validation compute node removal decrease performance case even ndcg pruning relaxation term dimension mix classifier correct classifier optimize classifier terminal make final prediction final tree weight carefully specialize large yahoo rank set category feature feature correspond synthetic feature learner usage design identify perfect feature per test input along identify expensive mean task public yahoo challenge extraction weak query indicate match ndcg level relevance unless node hour tune classifier minute yahoo set coefficient classifier ndcg versus cost weak derivation insensitive fine early stop validation evaluate six sensitive curve simply compete improve evaluation reduce overall propose node discount correspond significantly improve accuracy curve limited evaluation compare beneficial ndcg long budget reduce achieve ndcg oppose tune ndcg yahoo input pass branch branch correctly rank low branch away share apart investigate fraction extract yahoo observe depth expensive deeply input precisely obtain high ndcg introduce novel time cpu principle fashion partition identify cost feature region allow accuracy fraction tree relax minimize classifier unnecessary consumption application far specialized case incorporate demand author thank suggestion world email spam filter cpu challenge balance test accuracy principled dominate drastically tree input individual optimize specific space tree match learn email spam test reduce unnecessary classifier imagine introduce email spam email web service daily day filter compute prohibitive introduce time classifier address principled manner reduce cascade incorporate dynamically path traversal expect cost tree single loss direct expect lead expensive necessary make framework relax mixed relaxation derive optimization synthetic effectively classifier search significantly current art sensitive web regularization feature extend cascade mostly classification cascade several classifier order center input predict cost cascade enforce feature reject later cope particularly effective skewed imbalance majority contain face often haar suit different significantly reduce never learn dynamically select feature average detection case fundamentally mind input terminal path prediction recent work speed vs evaluation differently feature extraction algorithmic mutual feature extraction tree possibly similar learn direct different adaptively learn motivation algorithmic different regard introduce formalize sensitive consist categorical limitation focus throughout linear decision trick particular first depth
fill f f v f label label v n v x b semantic assign factor indicate determine fix type template describe manual f gm size gm ex model gm gm minimizer minimizer size library allow structure prototype modularity tool file format potential exchange ex template define discrete wide art restriction impose factor factor handle efficiently function differently several parametric e metric dense interface graphical file format line tool modular graphical optimization equally standard tool rigorously c base definition operation include summation marginalization conjunction variety fig deal occur repeatedly store impose graph format extension automatically template elementary library property order grid contrast support drawback table prohibitive model factor focus impose restriction contrast library set publish cost slow optimize mrf twice fast modular corner height ex bp cut wang decomposition branch expansion swap subgradient bundle branch semantic graphical w operation determine semantic
truth original translation affinity cosine similarity two soft ml enforce help unable soft spectral translate surprising version enforce translate use translate original merely tradeoff integrate partition document word fmri fmri scan person consist scan node absolute correlation brain activate indicate certain instability scan subject thing scan result spectral spectral cluster fmri different min cut little insight activity subject transfer scan constrained c cut agree fig far california center cognitive individual cognitive scan rbf kernel enforce constraints fmri scan constrain cut similar cut great disagreement original vice constrain interpret fmri scan clearly study people disease width principled spectral cluster soft constraint flexibility framework side information metric transfer use dataset exist technique validate acknowledge automated discovery explanation event environment nsf grant nsf enhance usa university california usa cluster hierarchical constraint algorithmic setting one many develop contrast effort implicitly encode link modify laplacian principled explicitly encode optimization encode degree constraint constraint unconstrained remain empirical artificial encoding property extensively process community superior traditional deterministic polynomial ability shape cluster cut capture cluster fail fig validate spectral originally propose unlabeled information laplacian become insufficient toy cluster undesirable side large expert explicitly assign must link short solid line cl dash line spectral successfully recover neither complete demonstrate greatly improve result infer example document language context transfer laplacian domain side information transform pairwise ml cl constraint side belong graph cluster area constrain category incorporate exist mean study prune intractable build partition constrain spectral promise assign cluster inconsistent spectral develop category enforce directly graph modify graph second constraint category several limitation design binary real real belief yes many lead example could reasonable ignore small portion exchange natural interpretation either encode enforcing incorporate flexible address limitation benefit cl accommodate relaxed moreover constraint addition handling allow new entire alternative distance metric consider framework amount soft enforce threshold amount ignore exchange side raw consider objective explicitly create optimization unconstraine special turn produce interpret perspective embed standard benchmark transfer section apply namely fmri comprehensive task briefly survey previous constrain spectral formulation empirically conclude constrained extend form convert incorporate spectral group th affinity laplacian encode walk cl penalty affinity mean propagate affinity regularizer principled decide constraint guarantee subspace indicator accommodate inconsistent partial grouping enforce make sensitive extend propose spectral combine use incorporate pairwise constraint cut stress clustering well establish spectral incorporate spectral compare technique follow model spectral laplacian self reader familiar material skip formulation rest list symbol mean affinity degree unnormalize normalize constraint graph undirected vertex set matrix call l call unnormalized assume let one equal show eigenvector cut graph constraint principal cluster solely decide affinity laplacian extension incorporate side information reflect graph spectral pairwise soft new formulation constrain show generalize eigenvalue system encode traditionally link indicator belong ij increase sign soft believe belong magnitude belief value believe different assign similarly belong assignment constraint rather constraint individually substitute constraint unconstrained constrain graph constraint optimize objective solution unconstraine cover special eq encode trivially solve constrain tucker find set candidate satisfy condition choose force feasible lagrange accord kkt take constraint check solution complementary imply either trivial second eliminate unconstrained cluster focus hold kkt condition assign variable solve explicitly produce infinite feasible solution relaxed color side satisfy sign contribute positively sign color assign side cut case formulation joint numerical possible cut coordinate correspond axis line visualize unconstrained cut cut numerical randomly horizontal indicate solution cut threshold constrain show way unconstrained affinity relaxed assignment indicator practice partition side encode soft inconsistent algorithm labeling constraint always consistent algorithm par spectral big solution normally naturally way usually instead feasible preserve eigenvector associate eigenvalue mean embed specifically encode scheme eigenvector feasible routine discretization relaxed matrix due orthogonality step help moreover treat ideal cut cost column respective cost remove rest constrain scenario source edge target derive form soft affinity carry structural knowledge graph cluster knowledge target correspond trivial second large eigenvalue guarantee feasible eigenvector begin experiment soft fashion metric translate transfer fmri analysis study incorporate meaningful truth partition constraint technique list labeling metric knowledge arrange accordingly implement available segmentation pairwise outperform interpret human meaningful berkeley benchmark image image compress use segmentation set segmentation segmentation segment save irrelevant segment pixel close segment blue black belong white visualize reconstruct indicator unconstraine spectral partition pixel ground fail background correct introduce block bound corner block ml pixel cl pair pixel gradually help supervision successfully blue red note try aim unconstrained block fig right two part one red human spectral fail background fig spatial continuity block background intend assign background side cluster thus face achieve image alternatively block rest examine underlie truth claim extension cluster question ask converge cluster spectral draw sample truth constraint ground fig dense point unconstraine could ground ground truth adjust rand ari ari truth partition truth exactly constraint record ari different consistently ground constraint insufficient well lead perturbation constraint provide quickly robustness create double background although cluster correctly background uci constraint choose six breast cancer diagnostic optimal
axis time age period subject segment age entry may death study excellent extensive diagram disease axe age direction three system axis life line plane life triple age duration certain pointing illustrate line subject birth disease go direction disease henceforth life line death without whole life disease person record people affect person incidence person year often take usually divide subject spend event subject group risk spend group equation become diagram clear age product whose rectangle risk sum risk spend dimensional rectangular time risk volume subject spend voxel graphic field seminal present terminology rectangular volume call six face voxel adjacent plane voxel parallel union voxel face play voxel grid life voxel age rectangular voxel depth calculate subject starting point intersection voxel voxel arrange regular equivalent union voxel intersection us intersection plane operator formula define set occur intersection happen voxel order subject sort division voxel summing times yield period person year calculate intersection voxel faces million form implementation tune implementation ghz patient next method article patient population age assume study assume death censor aim transform age calculate f year show person
necessity adaptation infinitely eq index thus contradiction infinitely many combine eq thus necessity robustness establish metric almost attention distance form otherwise x value nonnegative lastly prove robustness geometric robustness say robust p subset equal subset z z z p z framework indeed example partition ensure framework pay convenient prove frobenius frobenius norm f tx tx tx tx x x special order derive however stability ability recent approach stability frobenius arbitrary use norm induce row c example proof version norm function cover kb b bb derive metric formulation use technique completeness consider bilinear mahalanobi simple bilinear triplet robustness generalization sparsity induce applicability framework advantage flexibility robustness tackle metric setting method adaptation promise robustness property input minimize maximum belonging concept optimization deal z n l l z z inequality proposition robustness bound p weakly robust n n p p p sample consist equality apply thus contradiction p contradiction value exist constant fx x approach proof satisfie z z tx tx x x u x product continuous w take frobenius solution subset b term training get consider obtain tx tx j solution triplet optimality derivation partition tx tx tx tx tx tx tx tx x c first follow cm attract lot interest decade robustness introduce xu metric illustrate result include automatically adjust similarity function training tailor improvement attract interest decade see survey rely resp metric generally mahalanobis parameterize psd seen find linear well function without psd result distance use performance despite practical success unseen datum metric independent identically iid indeed give extract consider near example diversity still assess convert result online generalization rely al uniform stability regularize induce propose pair constraint metric robustness xu allow generalization bound closeness input result etc triplet work assume build weak necessary generalize well fundamental illustrate deriving argument work vast regularizer without example simple organize algorithmic metric necessity illustrate applicability framework derive bound metric discuss related work norm similarity training build z label generalization loss bounding notion algorithmic xu classic learn base deviation two close formally subset every belong prove true latter ensure obtain notion space say convex quantity finite cover consider partition partitioned belong adaptation learn pair pair fall metric say formalize np jk quantify robustness robustness sample relaxed robustness easily extend triplet base iid triplet share interpretation z k z eq generalization bind present concentration help iid multinomial random huber generalization metric iid iid parameter z ic z ic c z j z z b n I bn bound lastly equation previous robustness give adapt straightforwardly triplet output theorem robustness property relax robustness z ks z associate robustness come iid draw metric
classification svm community room discuss technology specific separate convex optimal separation ht nonlinear disease disease goal support machine dimensional hyper plane separate space interaction certain gene roughly piece create technology achieve feed try ht gene linear polynomial sigmoid ht svm fold cross rbf sigmoid sample decrease svm literature transform kernel include spline histogram student etc modification hundred thousand kernel paper would new actually separate disease possibly case interaction meet challenge effort generate true modern statistic complicate huge able national maintain claim claim replicate condition finding medical survey journal come trial far examine claim american medical journal journal national cancer institute come observational likely false claim researcher relevant blind control field gene identification microarray category observational scientific gene gene cause flip coin consequently adjustment fdr gene show efficient detection taylor situation tool machine randomization technique laboratory point justify logistic logit versus link side simply boost adjust often single gene imagine gene solely responsible disease protein gene primary gene gene fortunately identification technique gene study differ discover induce tumor formation powerful feasible identify gene cause genome pool reasonably sized could examine rigorous approach microarray first identify disease study stochastic gradient identify pool gene interaction interaction situation possible cancer correlation pick technique important fdr least neural network decision problematic crucial distinction gene irrelevant false non discovery gene subsequent biological false gene lose exploration classification positive false specificity roc curve roc precision recall etc commonly produce accuracy select parameter currently growth tackle hard datum create thousand select variable know many process trying find black cat dark house investigation college student project field gene literature technology sample need select case pls disease moderate search platform beneficial top height pt researcher thousand tool variable responsible trait include education microarray brain imaging name focus investigation limitation variable et al produce stand artificial disease reliability tool widely variable responsible pre screen gene dimensional modern statistical international competition analysis bin predictor cancer research cancer predictor gene expression image processing snp area user predictive user furthermore probably involve satisfactory ask explain predict different la explanatory statistical extension find field microarray identification tool every cell express express cell cause trait compare cell allow comparative express cause formation hundred hundred point microarray need statistical size attract amount survey reveal whose group two category include correction discovery bayes mid platform parametric reference bar et etc vector machine neural near diagonal linear discriminant I bayes near rough pattern base fisher discriminant mahalanobis latent approximated microarray pathway neighborhood fuzzy mutual numerous reference instance et al wang bar et al lee et al reference category list variation instance analysis al eight representative test structural plus non lasso elastic nine toolbox generate vector least modification hundred suit hand cart variation literature include index genetic fill wrong path come neural variation cart radial basis rbf network architecture software package nine kind architecture function kind lot google neural marks day situation statistical et gene equally whether translate wang tool achieve false gene identification microarray vast reliable article learn analysis microarray generate simulated microarray employ identify differentially gene classify dataset result dataset motivated gene cancer patient outperform accuracy top may scenario datum may analyze microarray relationship significantly tool explore microarray large analysis performance model publish classify purpose fewer publish analysis subsequent cancer note classify gene rate ht n svm svm al lee bar analyze microarray leave validation leave one model run base train pls nine table always classify microarray pls variable square available dimensionality reliably thousand subsequent pls achieve always classify cancer elimination gene lowest eliminate without gene cancer however cut top gene slightly pls pls utilize cancer neutral networks nn pls neural analysis split validation cancer display regression pls neural pls leave pls regression conduct place pls gene select split compare neural pls notice pls gene desirable classify instance plausible use different classification rate explore gene method statistical three appear seven statistical pls statistical ht gene gene gene gene gene gene gene gene gene gene gene check ensure partition different seed split find place gene statistical take appear encourage pls error seven small standard corresponding chi square thus conclusion draw separation maximum run seed fix logistic parameter pls ht traditional cutoff significance standardize estimate outside fail normally distribute cutoff value estimate cutoff reliability estimate neural ht lc top cutoff elimination forward contain behave pls table say conclusion pls complete separation regression know pls machine model next provide patient analyze team medical institute institute technology imagine conduct microarray magnitude cost alternative scenario budget collect compare recent article van interaction reality expect build base purpose instead dataset generate microarray need mathematical equation select gene dataset design state state responsible disease complex nonlinear predictor eight simulate normal disease linearly contribute disease state gene gene normal gene disease disease disease disease dominant breast cancer attribute two breast cancer breast cancer genetic lot involve gene move cutoff function cutoff normal heavily skewed display disease disease remain disease disease present exact cutoff value disease simply cutoff disease different gene gene interaction gene contribute disease account represent scenario disease gene similar disease except nonlinear combination gene individually combination q gene interact single breaking et near neighbor obtain maintain patient outcome decision making regard current option rational patient high clinical pick disease summarize rate x boost pls x nn none x disease exception excellent job method contribute classify gene disease much disease disease level influential boost pick subsequently desirable identify biological method x nonlinear interaction important pls irrelevant gene none irrelevant gene irrelevant x turn disease identify table several statistical statistical correctly interaction gene column give pls lasso irrelevant raise achieve gene determine state compatible finding article et markers important use angle regression fold run encourage fortunately boost correctly though unable minor clean cut tree pick leave cross stand popular network irrelevant gene decision boost related representative basis center center create histogram cancer panel patient panel histogram unlikely happen disease cause popular accuracy believe disease ultimately statistical disease cancer patient balance reality try handle technique cancer patient equal patient work well give cancer patient work model interaction effect gene recall important disease gene interaction disease moderate accuracy show include nonlinear x nd rd rd interaction semi boost boost disease patient patient pick relevant minor summarize gradient find al compatible accuracy accuracy fdr gene fdr x gene gene fdr gene gradient boost procedure able pick microarray experiment year may many consequently handle highly turn gene gene lose recover nevertheless gene screen fdr fdr cut gene number fdr gradient well able nonlinear phenomenon
write also expression proposal r walk notice e early rejection actually albeit negligible factor density suppose prior early simplify verify early never take summary importance whereas essential development build close achieve determination automatic exploit classic hold loss choose therefore abc output e thank regression approach linear square optimality order measurement dy jj nj particularly summary addition lasso lasso regression coefficient return software prediction abc methodology multidimensional process previously introduce methodology handle general time vector artificial affected identification mean previously methodology knowledge abc pilot identify posterior mass suggest schedule automatic pilot abc region posterior use set parameter datum iii simulated artificial statistic iv statistic density knowledge informative prior prior region abc question prior determine distribution markov mix reasonably distribution course produce satisfactory enough augment prior mechanism iteration mcmc allow generation large avoid mind set truncate iteration metropolis increment attain long draw discard next section transition proposal density density easier simulate approximately sample exact posterior corresponding small acceptance posterior ease methodology package although find otherwise two inferential carlo strategy particle method introduction smc example simple necessary regard comparison tolerance sequential carlo particle fair abc drug literature devote longitudinal modelling follow although mixed drug concentration drug receive subject elimination rate drug intensity stochastic subject nine min min drug e reasonable sde min practice logarithm cl readily available therefore datum generation k cl perturb obtain however inferential algorithm typical sde euler divide abc multivariate regression exact inference denote start must obtain artificial separately abc truncate impose determine posteriori abc walk reasonable ability explore surface approximate rate acceptance rate discard burn th equal correlation subsequent save computer memory spirit plot bandwidth vs plot outside final draw panel limited draw mix well discuss large biased select allow accurate balance filtering give encourage ess part intel core ghz gb ram rejection produce acceleration true lin ess abc ess ess ess result carlo smc likelihood hasting allow inference ultimately call proved approximation algorithm particle filter resample particle randomization consider prior abc run particle acceptance case carefully code trajectory suit explain poor nine surprising due setup density regression order ess length chain consider obtain lasso statistic train use abc via report bandwidth retain outperform ess value mean fail often literature handle conclusion least methodology remarkable approximation sde limit sde fully measurement error define represent simplified auto protein code specific reaction represent via specie specie reaction negative integer specie value presence law degeneracy therefore redundant prior copy genome reasonable replace occurrence follow assume associated reaction occur continuous dna dna c rna rna I could elegant solve root definite trajectory simulate guarantee stay absolute square disadvantage preserve dimensionality increment unnecessary redundancy setup rna represent mathematical reformulate system partially observe system design define via rate time denote denote exposition dimension sampling equality hold inferential trajectory numerically euler stepsize different coordinate set setup ii perturbed vector ii denote vector object prior abc execute million equal correspond discard purpose report ccccc sd vs bandwidth evident range allow interval theoretically could inference practice safe imply plot sharp increase decrease result associate mixing region draw long mean look vary ultimately draw posterior notice excellent ratio ess eight ess use draw implementation affect measurement variability return estimate partially unobserve total available perturb independent consider variability uncertainty placed generate long draw acceptance retain result despite increase uncertainty setup initial residual variation compute ess ess ess partially error propose reduce experiment practical application methodology largely automate thank informative summary encouraging monte carlo multidimensional demand flexible limited modelling stochastic latent via multidimensional sde mcmc alternative monte notably excellent inferential possibility avoid one device difficulty try implementation smc hundred thousand parallelization deal multidimensional rather error available see paper sde model incorporate science grant tool author acknowledge comment associate quality work considerably model equation relevance grow already financial growth multidimensional challenge theoretically either prohibitive time sde limited package implement abc provide keyword rejection space equation chemical reaction likelihood popularity practical due would computationally considered review dynamical whose unobserved dynamic solution equation allow fluctuation behavior area model mathematic population dynamic model chemical inference work sde noisy moment issue bayesian multidimensional sde abc tackle difficulty stochastic represent state wish sde introduce scenario system unknown know corrupted generic reasonable conditionally independence illustration consider I development assumption infer complicated consider consideration correlate wish form carlo mcmc algorithm fail reason include explore chain surface difficulty sde trajectory mcmc proposal locate difficult sde transition underlie diffusion inferential sde available review sometimes prevent situation many presence considerably sde exploit recent computationally algorithm exploit abc computations stochastic chemical sde already potentially sde common compare simulated statistic role bandwidth consideration closeness trajectory block abc lf generalization additionally case via fact markov section initialization simulate start start start sim law conditional sim sim u sim sim sim recall marginal
scalable feature employ learn factor object inner lie ease excellent performance representative multiplicative receiver factor factor also representation focus capturing discover interaction relational useful absence capture interaction effect object redundant incorporate structure relational observational structural discover intrinsic property object precisely assign object block cluster contribute coupling factor model discover well local paper contribution novel function latent factor propose mm latent extensive conduct synthetic predictive introduce algorithm section describe experiment real provide work give represent miss use mean observe cluster datum denote latent membership object assignment ik membership relation predict among consider relational factor generative interaction hybrid object base relational bernoulli function define follow characterize variable encoding attribute user introduce latent matrix latent link exist object within denote object cluster within receiver inner object cluster actually model discover latent use type relational cluster fix also fix mean structure way adapt structure kind bias generally side information model hence interaction q represent object denote associate pre could assign observe benefit correspond block prediction interpretable blockmodel factorization style traditional process selection accurate latent preference certain successful generalization capable dense interpretable cluster assignment link pair cluster hence integration multiple generalization prediction impose latent factor follow description summarize build section latent maximize monte adopt cost expensive latent follow factor globally algorithm learn fix update instance learn feature one equation feature iterative rule newton logistic complexity respect latent cubic generalized function commonly style concave function step maximization mm factor derive aid auxiliary increase update q estimation maximize low learn feature fix auxiliary eq proof low optimize maximize transfer iteration need optimization function assignment relational latent log assignment value relation directly map term kronecker product matrix learn generalized model optimize obtain latent fix update latent derive update rule latent similarly latent factor employ kronecker rule function monotonically combine learning update em factor information learn latent latent require possibility handling demonstrate term nmf indicate nonnegative semantic membership blockmodel use relation latent latent factor latent relational task task prediction relational relation consider random train relational receiver operate characteristic curve check difference discover assignment well relational normalized way examine represent specifically two inter connect connected object cluster check relational propose different reveal structure nmf indicate model fair comparison wide selection number work check model repeat average roc link task well propose indicate efficiently cluster e g nmf due discover special integrate block conduct assignment label ground measure obtain directly latent assignment factor model feature label score observe well feature prove relational compare propose dataset task also latent vary latent increase however two link online social user consist social link structural choose relation entry time number model outperform model latent lead compare effect take dataset prove factorization respect model indicate effective specifically propose rather bernoulli may prediction moreover vary wide range high performance fix social effect vary real true label object evaluate latent assignment look auc respectively simultaneously construct paper citation experiment case genetic rule theory paper indicate truth g nmf dimension note dimension latent discover future evaluate table achieve reveal simultaneous latent adjust feature citation contain cluster content paper model dense one structure cluster feature relational feature varie extent experiment c c classified distribute prediction inner product lie ease continuous excellent performance representative latent generalize model capture equivalence latent structure dyadic block network example relational group membership dirichlet
variant evaluation corpus ng performance keep ng three compare asymmetric lda five burn optimize versus df df total ni corpus tend pattern many topic ng decrease investigation explain phenomenon table ni proportion ni ng compare characteristic word corpus document corpus tf ng examine word ng note systematic pattern ni alone sufficient class quantify df df show scatter versus word ni circle corner locate tend ni suggest one ni level show pattern word category ni ni windows ni file ni ni ni df category ni ni ni learn ni recurrent feature ni basic ni ng order ng neither order systematic addition stop topic corpus ni posteriori word likelihood result study lda hold likelihood higher comparable ni topic comparable word ni form contribute test stop informative likelihood likelihood combination preprocesse cccc cccc ccccc corpora comparable ni word comparable accuracy ng outperform five document linear one classification fold cross validation accuracy reduce exclude ng ng symmetric divergence capture several major inconsistent output token pose topic assignment token vi topic assignment token vi thereby minor propose metric maximum minimize kl generate consistent topic sum find topic attribute minor topic regular ni exclude clear pattern initialization mcmc run ni computing coefficient degree divide ni run represent run lda select well hoc preprocessing reduce explicitly non informative exclude vocabulary simultaneously topic combine advantage symmetric lda asymmetric prior apply size get vocabulary monotonically control effective vocabulary segmentation collaborative find topic show incorporate selection improve improve latent allocation multinomial distribution entire vocabulary contain topic adopt variable modeling lda excluding corpus robust discriminative prior document lda classification well asymmetric consistency comparison allocation corpus finite distribution vocabulary unique word frequency preprocesse contribute corpus systematic corpus relax must topic subset vocabulary study depth lda model achieve short identifie generate semantic topic topic specific aspect probably globally form topic word token word necessary form use preprocesse also selection dimension inference carlo demonstrate correctness model world compare lda show find topic robustness document classification lda occur frequently corpus contribute discover priori latent however exclude truly include lda selection conduct simultaneously latent combine search lda topic vocabulary preprocesse topic vocabulary topic mutually exclusive respectively prior per document obtain sample make selector observe informative informative token collapse update metropolis rely di word token depend assignment token update vocabulary fix varie di rely carlo integration q done burn correctness dataset synthetic corpus start ten topic topic probability word topic five word ten informative topic generate document random proportion draw token topic draw informative synthetic
define invariant preserve emphasize solver hmc generate eq equivalently express semi implicit interpretation carry size hmc hmc sample acceptance table moment express still term equation solution preserve would paper validity directly equation path computer finite resort hamiltonian justify attain acceptance set q hilbert chain chain rule third I calculation prove denote projection definition last corresponding modulus preserving formally preserve volume transition invariant assume next recall uniform bound need stress expectation integral subscript notice velocity validity hmc calculation forward volume jacobian analogue sense jacobian transform hilbert rwm mala hmc define advanced rwm move algorithm langevin sde drift calculation drift brownian increment sde interesting nonlinearity sde derive euler increment mala euler advanced mala advance mala hmc rwm derive term rwm metropolis hasting rwm advance rwm three sampler together h algorithm hmc mala x v x vx vx coincide mala section hmc appear mala derivative synthesis deterministic hmc avoid behaviour providing illustrate target hmc gain mala rwm mala gain complexity rwm derive bridge target general bridge complexity sampler length bridge analytical appear chain vanish acceptance grow connect proposition derivation thesis similar scaling decide completeness make modification follow derivation bridge expansion correspond bridge orthonormal corresponding eigenfunction co eigen specify expansion iid advanced mala rwm algorithms target probability current position stationarity hmc scale control step write equivalently express effect co power I modulus one power unstable require stability scheme hamiltonian derive probability acceptance constant arbitrarily long informally connect acceptance advance rwm mala hmc step give ignore nonlinear map synthesis effect offset proposal position arbitrarily advanced rwm show proposal distance position fix sub mix mala sde mala carry along continuous dynamic require propagate path rigorous beyond paper accommodate wide unknown conditionally mcmc sampler drive naturally rise due probabilistic brownian motion inferential issue relate applicability advance succeed parameter cover transform offer extension irreducible utilize transformation framework irreducible require ease exposition univariate however advanced hmc also section sde model treat continuously sde transform q brownian motion conditionally eq generic verify advanced density arise carry calculation assume time determine datum application sequel target integrable path reference motion boundary brownian bridge recall specification covariance space brownian motion bridge definition abuse proposition term integral practice hilbert space path specification correspond derivative eq appear specification continuous piece linear turn respectively necessarily lie hilbert space weak continuity hmc ess additional present e sde construct euler diffusion skeleton set relative value advance mala think hmc step indicate much step overall hmc perform ess particular remain level note substantial advanced mala hmc mala improvement rwm sampler rate drop relative rwm mala hmc n hmc relative rwm mala hmc rwm mala stochastic volatility analysis record daily frequency volatility hmc consecutive write poorly mala rwm nevertheless enough hmc reach advanced nearly time rwm mala acceptance mala version rwm mala hmc step hmc step numerical illustration simulated survival event drive hazard figure despite fluctuation hazard decrease could case follow trajectory draw exercise sample sde think behaviour keep ability shape hazard table calculation experiment associate extremely acceptance rate rwm poorly acceptance move mala well advanced hmc specifically hmc fast rwm mention figure diffusion process determine hazard display credible indicate shape hazard capture rwm mala briefly case sde drive one far omit parameter context equation sde coefficient indeed existence v ji k transform suggest wide scope advanced mcmc absolutely brownian motion relate briefly verify transform onto drive wiener sde simply write mapping driving imply sde remain calculate involve w driving noise see appear integral expression functional process path check differentiable condition advance develop apply driven diffusion validity establish relax diffusion scientific contain functional integral extend diffusion handle drive wiener direction algorithm facilitate updating either potentially sized block boost important posterior correlation path critical currently investigate extension advance hmc diffusion successfully fact support pseudo mcmc formulation diffusion see choice hamiltonian prior exploit mass location specific mass boost efficiency memory diffusion cover sampler block time lack beneficial hmc control try grateful comment greatly improve thank dr lot ep rwm mala relevant working result suffice point replace second high crucially grow provide acceptance analytically inequality grow slow retrieve rwm analytical inspection term clearly last vanish term analytical rescaling write bridge thus mala calculation indeed term eq come last h vanish use analytical expansion measure bridge get thus exploit recall calculation one operator eigen stand sequence expansion eigen examine eigenvalue verify calculation rest henceforth recall x eq prove illustrate proof sufficient proof calculation notation equation derivative mapping dirac delta immediately expression expression statement stochastic practice involve take much derivative denote consecutive instance q calculate term summation unless last overall operator covariance standard easily combine give line bridge immediately appear line consider difference construct covariance motion follow recursion derivative recursion calculation derivative notice approximation look equation k calculation get vector term q multiply obtain calculation assumption persistent advance available intensive advance familiar chain change law arise context drive wiener constitute emphasis advanced hybrid carlo make drive move analytical computational advantage regime latent model advanced version mix refinement inherently path finite one apply gaussian mixing stochastic carlo distribution application rapidly practitioner ideally suggest flexibility direction propose study advanced version performance advanced distribution change focus drive differential concrete brownian motion advance computationally demanding reproduce direct indirect ability realize drive mechanism importance behavior sde variable within sampler fast critical feasibility advanced method development take target measure hilbert exploit evolve advanced time projection use inherently sde difference commonly approximate infinite dimensional path mesh look advanced rwm adjust langevin mala hybrid mala hmc dynamic whereas rwm use proposal emphasis hmc employ hamiltonian sde computational advantage experimentally hilbert value rwm mala hmc require strict derivative paper simpler minimal wide range practical inherently specification concrete mathematical verification well infinite mesh mix practical run dimensional projection difference another contribution order avoid walk behaviour greatly outperform flexible phenomena biology molecular survival see application large class specify diffusion coefficient respectively form sometimes imply physical law langevin due quantity skeleton path study extensively observation regime appear pd bioinformatics stochastic observe error financial application model diffusion event involve observe unified consist diffusion complete conditionally diffusion proceed develop hmc inferential whereby sampler path augmentation mcmc address arise characteristic diffusion diffusion diffusion e conditionally dirac cause identify noting unless instance consider computer currently available sequel proposal path overall diffusion process typically brownian motion path candidate path case perform poorly many target one advance connect issue achieve consideration involve brownian bridge approach gaussian method advanced sampler issue advance hmc sampler provide powerful latent blind update mix material regard hilbert advanced mcmc sampler diffusion advantage family general coefficient regime section generalization collect material separable presentation later coincides preserve mathematically diffusion upon square integrable reference brownian boundary covariance gaussian operator bridge q brownian motion brownian specify brownian motion brownian involve eigen eigenfunction constitute orthonormal hilbert particular iid dynamic accept force target amongst alternative hmc synthesis reject test version hmc advanced target appear
achieve high forest due overlap always total combine fista combine tv include compare superiority compressive sense multi method cs validate forest conduct reconstruct mr however achieve good reconstruction fista measurement fista fista respectively contrast feasible way datum channel number fouri significantly fraction measurement bottleneck call literature final contain visual measurement cs sense configuration fig image similar forest compressive sense reconstruction recover fouri channel clinical diagnosis image discuss forest sparse fista solve recover run converge ratio snr fista fista cccc snr fista fista comprehensive algorithm increase improvement reconstruction fista performance sparsity gained combine tv however fista easily color capture optical camera represent blue three color color human observe color channel highly joint regularization gain snr regularization wavelet reasonably htbp color fista fista fista recover penalty different color hyperspectral band spectral utilize sense cost huge imaging compressive sense acquisition image reduce band scene band forest band hyperspectral image site show band band sparsity band joint sparse convenience wavelet band modeling achieve high propose compressive family numerous field sparse benefit theoretically validate compressive fast forest problem practical superiority forest sparsity tree sparsity computational convenience root prove paper dependent accordingly combination sparsity number first rip diagonal matrix tm n x diagonal probability indicate q choose prove c complete l b forest less tm li email investigate compressive channel hierarchical channel follow forest intra correlation family compressive sensing theory forest channel far sparsity theory multiple measurement share validate experiment demonstrate propose sparsity structure sense compressive tree technique become popular compressive sensing recover shannon sampling suppose matrix wavelet coefficient perfectly satisfy isometry property rip large find hard impractical basis pursuit prove cs lot efficient sparse classified group beyond sparsity structure come compressive standard sparsity zero datum arise cognitive arrival estimation compressive medical common joint joint solver bayesian contribute estimation structure already utilize image nature signal image approximately relationship tree zero channel sparsity hard implement exploit subtree hierarchical structure rely parent tradeoff often approximate parent assign joint study combination far compressive problem datum across channel channel differ fully guarantee practical researcher give intra ignore inter propose call bridge connect trees forest compressive forest measurement much tree sparsity structure forest imaging image parallel well forest application organize exist work tree review forest conduct compressive signal compression integrate sparse measurement measurement measurement restrict isometry rip subspace cs gaussian graph rip isometry isometry robust require measurement quantify rip lk rip probability intuitively subspace coincide intuition result prior utilize cs datum basis wavelet subtree kx subtree sparsity wavelet coefficient subspace number reduce proof rip recover measurement classical partially channel learn intra result sparsity sparsity sparsity especially exploit sparsity tree forest sparsity gaussian however data compressive follow rather dense system zero dense less analyze exist structured sparsity concentrate random extend block bind tm result dense entry depend sub energy concentrate e channel vary depend measure block diagonal matrix forest still experiment focus sparsity article structure group although sparsity model forest parent parent pair channel becomes overlap approximate assign example encourage zero encourage forest efficient iterative shrinkage thresholding fista fista convenient formulation composite forest sparsity article fista accelerate minimize object function smooth nonsmooth transpose n x closed form soft sparsity second close fista fista overlap introduce contain else coefficient row use constrain utilize alternate direction alternate formulation positive subproblem ta tb order fista fista fista benefit clearly ax whole b keep
achieved maintain u column update z jj parameter l update element outer hyperparameter maximize ml negative nlp cumulative gaussian finally distribution unseen q u effective vector site optimization f exp loss function stagewise basis add coefficient optimize coefficient exp al likelihood design stagewise additive show site select basis loss leave side inequality note monotonically scale training firstly unlike combination write separability function successive weak distribution keep previous optimize desirable log strict sense approximation secondly gp large get sparse along figure without reference value basis forward function notion stage iteration equivalent closely select updating site update simplify tt j jj j add basis j x u k site parameter approximation site previously add site fix relate predictive nothing training basis parameter stage coefficient summarize exclude see associate optimize next viewpoint propose select basis work min select vector cost useful achieve work adaptive accord add normalizing computed sampling along correctly predictive vector relatively misclassifie find violate difficult technique dataset move desire movement enough nlp improve wrong help getting alternatively get generalization minimize condition summary experiment benchmark test dataset large pick test add use reduction pick dataset conjugate specify outer optimize hyperparameter keep track good nlp loss evaluate nlp three constraint result site equivalently dataset figure minor difference effective estimate site certain aspect variance numerical due nature method loop optimize become slightly computation restrict evaluation therefore experiment change vector algorithm goal basis demonstrate effectiveness conduct experiment nlp loss panel sampling consistently perform dataset nlp make predictive distribution performance reduce working size see sampling nlp generalization see column gain basis ensure selection conduct performance partition compare nlp obtain partition observation nlp difficult relatively nlp r higher perform well base well gain w test error well method six dataset entropy gain conduct partition set nlp result hoc reveal significant difference measure gain nlp low overall p performance base measurement show take second dataset gb ram find fast matlab believe operation significantly expect speed improvement estimator viewpoint vector site selection complexity selection storage complexity several relatively set nlp loss dashed sampling dataset vector order yahoo com department institute designing generalize construct additive model boost effective stage new site vector basis achieve conjunction site complexity hyperparameter dataset aim full input memory basis far develop classifier propose sparse gp base informative particularly inspire filter hyperparameter site matching hyperparameter estimate optimize marginal predictive nlp gain validation experimental benchmark dataset showed particularly require vector achieve compare validation generalize expensive need efficient construction additive like boost introduce basis loss site coefficient effective cost reduction use selection experimental give entropy information gain section present design cover experimental generalize well represent latent
alternative detect value rejection zero remark relative rejection nominal large investigate instantaneous relationship fundamental proportional relationship price refer fair taylor concern link make study view macro u instantaneous causal period stock price us department bank series non series autoregressive test adapt time outcome residual implement study iteration instantaneous causality reject nonparametric nan period c c c see adjusted var deviation display c c c display two break point trade balance balance service see economic health country u balance service similarly account service service instantaneous causality relation analysis department frequency available site bank difference var adjust check suggest statistic display test quite view variance study value test correspond display instantaneous causality unconditional unconditional investigate kind instantaneous causality important unconditional vary propose test outcome set bootstrap framework structural proof introduce q give pd du tt v martingale central limit expression obtain appropriate replace e di theorem follow sake autoregressive conditionally process increment pointwise convergence proof hold similar pass economic survey directional influence reveal causality bootstrap guide chapter identification quantification causality investigate heterogeneous splitting bootstrapping box bootstrapping unit root regression heterogeneity causal stock interest university economic model linear causality stable multivariate autoregressive work I change multivariate regression volatility transaction work efficient estimation presence causal system unconditional variance financial work les shift series autoregressive variance cm corollary abstract instantaneous causality unconditional investigate test avoid precisely white autocorrelation consistent correction suffer severe test investigate instantaneous keyword var causality define financial lee others network et domain causality relationship account past improve information case causality relation var process test restriction innovation tool statistic correct covariance nonlinear pass model account unconditional autocorrelation correction unconditional structure van u investigate unconditional highlighted production sale break variance variance constant unconditional lead instantaneous causality unconditional numerous zhang test detect unconditional rao non unconditional variance xu reference unconditional variance stock constant restriction structure highlight standard instantaneous causality implement use provide suitable correction statistic severe situation precisely detect alternative periodic sign change note previous intend unconditional instantaneous unconditional variance distribution statistic variance asymptotic widely unconditional dependence restriction constant establish modify preferable test base constant unconditional plan variance causality var property test unconditional variance avoid take consideration test investigate finding autoregressive var triangular form subscript rescale approach measurable matrix assume gaussian retrieve tool develop suffer drawback test instantaneous causality piecewise smooth change allow unconditional variance xu reference therein propose use commonly instance structure view dynamic testing restriction variance follow h usual kronecker ols establish denote alternatively ol residual autoregressive important length instantaneous causality lag structure hand autoregressive propose fit particularly ol lemma may error unconditional constant drive markov pass detect unconditional weight statistic z choose statistic involve autoregressive order adjust uncorrelated surprising exclude statistic investigate asymptotic standard establish eigenvalue addition also standard hereafter consist instantaneous appear reject hypothesis causality resp test turn study situation converge strictly follow grow detect causality enough consider stand alternative relative st h c st st every error assume iid structure constant compare ability test instantaneous causality centrality w assumption suffer severe power consequence test intend account consider bivariate case r spurious unconditional variance avoid summary correction contrary addition non unconditional well test prefer time power bootstrap introduce ts consider h ts ct non fourth causality literature investigate var reader therein resample draw take iid give ts b generate motivated fact restriction test residual equivalent bootstrap replicate pattern variance weak
sensitive learn insensitive medical diagnosis business costly unbalanced situation cost rule insensitive insensitive highly make important develop extension technique current understanding extension machine architecture theoretic many extension svm insensitive hinge sensitive group category address adopt majority class due clear effect informative approach involve base transformation modify svm due cost unclear modify naive boundary movement bm shift adjust theory would svms accurately area aid sensitive calibration transformation receiver operate roc curve performance boundary movement separable sensitive optimality modification separate plane penalty bp svm introduce slack training transform bias obvious limited slack slack standard assigning svms fundamentally sensitive specify risk function classifier sensitive risk enable svm sensitive rule approximate sensitive bayes loss classic avoid method produce sensitive result cost closely appear change upper consider sensitive learn moreover sensitive minimum sensitive metric connection roc robust reflect tolerance briefly review generalize connection cs present cost examine classifier sensitive present demonstrate classification map class draw function negative assign minimize loss risk minimize risk predictor provide bayes decision minimize loss omit notational simplicity risk decision prove finally bayes boost hinge svms loss margin loss assign penalty positive encouraging loss enforce margin conditional minimize generative formula derivation consistent rely comparable goal find maximize maximal equality theorem hold appropriate margin invertible symmetry eq q design bayes invertible satisfy equation novel bayes consistent previous check derivative extend connection minimization false minimized predictor include associated classifier implement sensitive sensitive cost minimum risk associate sensitive minimum rate sensitive error generality equal highlight fundamental conditional maximum write intersect trivial thus prove note property concave extend paradigm predictor minimum conditional insensitive certain use g generative invertible ensure bayes optimality necessary lead apply bayes rule classifier approximate sensitive bayes cost bayes form arbitrarily choose close conditional loss equality say determine assign require enforce symmetry property satisfie property risk bayes speak sensitive classification algorithm predictor concave risk measure simple conditional reduce obtain I derive illustrate application cost I start recall adaboost cost sensitive easily satisfy note sensitive symmetry property eq boost extend bottom cost minimize hinge result sign sign replace insensitive cost counterpart sensitive conditional hinge loss condition close zero conditional criterion degree hinge hinge slope margin slope positive weight heavily leading select component result sensitive svm minimal insensitive svm class small margin slope assign higher enforce risk quadratic insensitive soft control parameter responsible sensitivity type impose example margin assign separable obviously define justify guarantee implement decision ii approximation cost risk guarantee heuristic solution intuition problem linearly separate green line arbitrary maximize margin equally distant near class blue irrelevant also maximum positive margin specify increase increase positive enable control limited level undesirable general regardless preferable attempt error bp necessarily cs allow choice positive independent unlike cs margin class clarity formulation svm bm regardless separability slack write dual norm vector svm dual dual solver act coefficient insensitive ci scale svm major ci dual relaxed modification nontrivial connect sensitivity sensitive ci dual allow extra subsequently cccc f effect norm dual consider positive classification class usually lead different intensity majority range training example linearly example imbalance solve apparent implie take low imbalance ratio decision class boundary margin ideal result cause value bm bp result implementation margin classification bm svm dataset imbalance non sparse highly sparse sparse induce norm regularizer lead balanced use choice zero result figure highly boundary choose regularization algorithm choice uci dataset ratio sparsity deal implicitly prevent movement discriminant class enforce margin class cost cs svm class equivalently extra make asymmetric favor cost sensitive k kk corresponding show cs dual coefficient regularization hyperparameter space minimizer dual depict problem formulation cs svm dual substitute appendix ig retrieve problem reveal duality primal decision vice regard expansion bp extra different cost sensitive number basis contribute part highly discriminant mostly dependent kernel basis majority training pursuit add majority cost sensitive implement computational medical diagnosis retrieval business decision make rise concept dependent ed main ed consider accord accord train example probability resample method suffer cause implement bp svm call ed hinge ed hinge dependent ed hinge ed benefit include asymmetric margin ed experimental sensitive svm base ed ed bp hinge svm evaluation prior respectively measure simplify call insensitive risk zero classifier produce induce component optimal theory find follow optimization equal cost sensitive zero one choose optimal risk minimum know classifier well cost roc robust performance measure auc evaluate area within true negative region tp tn roc true region respectively auc tn auc demonstrate cs ability sensitivity specificity bm bp svm group namely dependent sensitive class cost learn example dataset experiment far explain section create cs svm show specification experiment dataset point multi different imbalance ratio http www edu tw target type bad heart presence breast diagnostic breast cancer diabetes cancer rbf hyper perform auc fitting grid cross evaluate appear bold hyper train bm cs generality perform grid actually hyper grid implicitly cs svm asymmetric margin advantage cost http www finally tn auc know problem readily modify hyper test determined phase
function network transform function toeplitz unit bias constitute finally feature see patch descriptor scene weight share network share scale extract scale justify remove sharing predict pixel easier consider grouping neighborhood suit pixel formulate search adapt neighborhood minimize unique assume component minimal formally th partition explore subset methodology confidence good partition define pixel component let set pixel wish component explain cost set lattice overall component classical simply explore depth finding figure aim component kind leaf root minimal component optimal subtle class good root weighted two neighboring function region span region construction quasi linear naturally produce merge component horizontal hierarchy equivalent thresholding map equivalent cut hierarchy various one neither propose hierarchy produce produce classification dendrogram dissimilarity two ultrametric map contour contour constant thresholding map yield contour show simple dendrogram confidence component class available predict simply way achieve illustrate feature part mask component produce vector background representation max grid invariant relation attribute nice shape object handle dominant combine pool solid function recognize underlie object grid fall aggregated pixel fall cell descriptor encode relation experiment define number class paper predict distribution matrix note represent interested find cut optimize power edge qp energy equivalent approximately foreground fold sift flow liu compose split training image obtain semantic label test test range scene type network channel bank map pool dimension map map produce filter follow pool layer dimension feature component combination filter locally laplacian pyramid pyramid rescale output network produce feature map parallel find decay expand irrelevant include rotation paper hierarchy cover volume complete removal informative segment segment optimal include bad stanford system multiscale alone describe report complete example dataset show baseline multiscale network multiscale predictor classification aspect clearly suffer consistency hierarchy network input vector unit stanford sift sift dataset sampling class balance equal network balanced frequency discrimination recognition view balance give balance work poorly dataset large amount example extremely overfitte sift show demonstrate remarkably stanford server c multiscale net multiscale per accuracy report author modern core intel competition c liu multiscale net per pixel average per frequency multiscale cover multiscale per accuracy class accuracy multiscale balanced framework use operate raw mid convolutional supervise scene parse rely segmentation tree image segment contain instead cut cover extract consistent except production state art stanford pixel per compete single construct contain segmentation optimal improvement learn learn segmentation produce institute mathematical york new york ny universit paris si paris france parse consist belong involve parse segment dissimilarity simultaneously dense encode size multiscale convolutional raw segment cover node aggregated fed produce contain segment tree class distribution segment accuracy stanford class sift class order magnitude compete approach produce scene require recognition contextual integration simultaneously label pixel come distant category pixel indicate human nearby grey part road building cloud depict rely scale extraction multi scale dense vector around context convolutional apply multi laplacian pyramid convolutional fed raw tree graph pixel pixel connect near measure two segmentation merge span final segmentation subset wise feature segment encode grid aggregated grid component max feature center fall cell produce classifier apply aggregated feature classifier histogram cover segment define attempt segmentation pixel complexity linear produce full second adjustment threshold three key multi convolutional region class decide segment oppose object part object procedure optimize scene parse variety year rely mrfs account rely pre segment candidate extract individual segment neighboring segment consistent segment aggregate train scoring individual like train feature question scene parse make decision histogram look fine somewhat densely pyramid coarse hence scale encode decrease author candidate aggregate segment tree allow rely base cut finding cover system densely multiscale pyramid convnet feed raw mid big dense efficiently detection image segmentation particularly publish scene parse network feed scene parse base align label produce convnet boundary take segmentation representation pooling paper point seek image typically problem rarely contain object properly poor observation predict integrate grid finite enforce usually achieve
high x take optimistic costly proposal incorporate gram efficiently weight represent proposal corresponding take acoustic word max thus efficiently dynamic viterbi backward forward weight either max suppose produce edge draw unlikely thus reject produce account realistic trial experience sequence refine extend account stop process observe threshold refine stop process maximum evaluate retrieval possible word convert mobile phone assume english million language sentence remain length number retrieval experiment sampling latent token decode top plot viterbi fix sentence decode gram iteration reduce gram gram hmm gram gram c gram hmm token refine reach acceptance ar reflect gram input take iteration successful note trade variable resolve batch step note time find good iteration ar length move average trial os approach graphical loop function potential undirected unary function potential determine minimal bind span intuition gap outside improve upper correspond partitioning configuration write kx k potential unary potential eliminate loop particular equation neighbor replace condition different remove proposal refinement piecewise iteratively partition grain graph cardinality grow exact past decade rate refinement could refine proposal sample experiment show help stop refinement acceptance sample binary unary normal certain ise time elegant properly exactly target rely assumption typically attractive coupling strength make hard use policy take proposal conditioning pair policy addition use reject sample two reject refine contain conditioning variable reject strategy experiment select probable iv greedy policy total maintain queue triple condition ii refinement policy policy reject policy iteration conclude iv computation take trial estimate accept refinement unbiased current rate estimator decide expect trial distribution spend plot acceptance figure start decrease regime acceptance rate refinement see iv despite policy look optimal ii apply confirm reject adaptively region refine compute propose viewpoint rejection functional upper sampling refine refined decrease depend class max bound gram probability give decode hmms problem simple illustrate decode graphical reject refinement computational attractive model speedup motivate development extension two simple typical agreement free syntactic low level gram structure language improve limited refinement improve element partition example sampling far refine conditioning value region refine incorporation high n algorithm asymptotically produce os approach combine adaptive sampling refinement present inference hmms discrete elementary rejection produce sample produce propose os exact rejection space upper proposal dynamic rejection refine complex way implicit constraint obtain relevant acceptance never activate never explore region become explicitly gram latent hmm unlikely present never rarely explore treat consist proposal assess dynamic select interesting decoding construct os hmm attempt generalize sampling improve curve author propose curve rate technique continuous line convexity exploit tight tight piecewise rejection consider graphical introduce sampler increase variable recursively precede graphical similarity cascade sampler configuration dynamically proposal os unify algorithm sampling move background notation l px px unnormalize normalize define able distribution easily give rs follow unnormalized precisely ii dominate ratio accept rate measurable area illustrate illustration attempt say acceptance attempt space measure value q think roughly abuse function generalization avoid confusion say sense normalize dx f find informally sampling tend tend formal note connection mix mcmc hasting sense os almost measure unified os describe follow goal os sample refer cover density space os efficiently limit one refinement small natural control candidate optimization small costly one prefer simply good either certain ratio threshold
objective default inference interval match calibrate reference prior despite effort satisfactory inference goal seed plant formalize extend association unknown model clear equivalent I attempt accurately predict conditioning focus fully alone insufficient accurate therefore call amount association auxiliary set probabilistic start association auxiliary produce post probabilistic yet I three pair collection set valid combine set modeling probability without inferential true calibration dependence I probabilistic meaningful provide I well description calculation I output posterior belief poisson provide summary belief posterior belief consequence calibration property output control fundamental present give I solution marginalization nonetheless illustrate I conclude section website consist value depend induce variable equip unit cube lebesgue model observable observable scientific investigation formal describe produce realization single population familiar simulate satisfied interpret link variable surprising association many association triplet equal association help resolve uniqueness I sample practical prefer association prefer evident view moderate section role characterize observe unobserved quantity course never inference solve solution continuous could well mean continuous identify fix determine integration part reveal invert inference sense true inside information highlight accurately employ simple general description map measurable predictive draw good predict unobserved predictive design variate choice perform interest default predictive random set predictive subset description assume predictive satisfactory predicting available combine intuition set consistent unobserved random satisfactory predict capture assertion assertion act summarize subset association dependence singleton emphasis validity separate make remark first consider nan simplification induce ignore I modify fails discuss follow assess conclusion extreme convenient call singleton summarize report predictive singleton belief plausibility symmetry apparent plausible statistical singleton assertion function plausibility neighborhood poisson plausibility function singleton assertion treat observe argue observe plausibility similar predictive plausibility poisson belief function dependent familiar fact trivial drop measure recommend calculation precisely represent knowledge gain claim auxiliary familiar association primary determine auxiliary predictive certain aspect I clear hope accurately ordinary random association proceed space reasonable natural characterize summary evidence favor assertion predictive evaluate random section desirable assertion question argue meaningful indicate meaningful calibrate amount mean appropriately refer calibration definition predictive random ideally variable function precise predict unobserve stochastically q proportion possible mapping check related validity easily nest valid nest measurable predictive support definition outside set realization iff rich special random validity nest claim I I validity predictive essentially prove correspond I belief refer suppose assertion I belief I state plausibility I suppose I stochastically small turn hand complete proof key predictive I particular predictive valid whenever change validity parametrization include transformation forward correspond I addition provide plausibility frequentist I base frequentist valid type rejection rule therefore type error singleton counterpart frequentist eq plausibility plausibility coverage plausibility plausibility inequality classical plausibility display plausibility interval well plausibility certain exact interval show fisher I singleton sense correct singleton large inefficient validity shall assume ignore conditioning belief function predictive light shall predictive sense ab b u b u eq exceed call I base ratio unity denominator follow theorem theorem valid kind consideration assume predictive construct serve clearly nest bt ta iff monotonicity ax ax omit general reduction technique reduce I conceptual problem since idea prefer presentation sided assertion assertion one assertion I stochastically I stochastically predictive random optimality property light notion optimality assertion one side possible sided assertion suppose define resp optimal predictive nest case similar eq consequently stochastically optimality previously treat predictive leave sided assertion versus I x classical pearson strictly connection test example omit side difficult sided classical I assume assertion unity random set property corollary follow roughly speak stochastically loose test goal thing continuous parametrization argue certain property nest subset shall balance corresponding q take predictive set set measurable score last follow sense least must score balanced choose reasonable balanced random side side optimal balanced proving set satisfy start depend implicitly shall decrease function distribution satisfy analytically typically require condition verify b argument xt family natural condition score describe simple sketch idea balance around make small make score balance determine integral definition hence optimal absolutely assume relaxed thing although illustrate phenomenon thin gray density line convexity figure interval black case integral break take part make gray sense fails transform score score symmetric correspond support x minus plausibility conclude consistent intuition match discuss identify plausibility function show optimal balanced gray default plausibility interval horizontal line much shorter I one might consider nominal transformation short plausibility interval coverage set score balanced default gray vertical mark balance predictive well sake refer reader elsewhere suffice plausibility balanced I require need compute indicate show plot two plausibility balanced although shape vary balanced plausibility plausibility plausibility x black gray suppose inference standardize reduction sufficient sample mean variance justification though step auxiliary student degree non centrality dependent marginal slice space work q consideration thing central develop evaluate use entire integrating set plausibility plausibility interval invert usual frequentist interval mean also agree frequentist plug style marginalization whereas ignore I marginalization available discuss dimensional consist assertion simplify presentation emphasize I produce well product auxiliary vector value simplex association characterize case remark sort easy alternative perform sort component ignore partly partly assertion datum association belief vector greatly simplify plausibility illustration ratio plausibility predictive neighborhood power three test equal ratio old test power connection divergence two substantial due assertion new I default method entirely fair assertion drastically first experience knowledge fundamental science combine respective post probabilistic inference calibrate plausibility frequentist bayesian goal simultaneously I surely benefit I framework logical meaningful frequency calibrate probabilistic without latter property something inferential able final I depend user association particularly effort predictive particularly multi make ambiguity random frequentist choice neither argue uncertainty association therefore prefer framework feature attack depend assertion lead drastically output slight predictive construct variable relevant reduction connection theory statistic parameter developed bayesian theoretical I promise reference expect inferential framework frequency acknowledgment grateful effort partially foundation grant dms implicitly balanced express predictive let text balance b tb sign neighborhood iff balanced remainder vanishe term hence
broad community student arm serious imagine diagram familiar experimental property encounter broad scientific mostly irrelevant category winner anneal backtracking anneal perhaps familiar markov chain employ projection von feasible hyperplane version intersection family backtrack mathematic backtracking determine exist despite computationally problem backtrack alone examine alternative arrange fill cell follow three rule grid rectangle rectangle rectangle rectangle appear complexity incomplete belong exponential nonetheless exhaustive force matter execute simply option anneal alternate projection yield good approximate discussion student computational backtracking grow become partial begin backtrack block potential discard en backtrack construct cell cell order cell possess cardinality come cell list cardinality come cell character string alphabet cell limit treat backtrack grow grow element take backtrack string stre first another backtrack back string discard replace another grow backtrack depth search string string dictionary constitute tree traversal systematically move branch prune pruning branch implement backtrack backtrack virtue exist mechanism node infeasible grow parent draw child grow parent none child parent parent draw grow edge child node child infeasible child child node node attempt solution beyond partial eliminate consideration anneal attack combinatorial define state function quantify ideal sensible constraint zero annealing propose choose markovian current process history decrease accept early high stage temperature physics anneal broadly local integer integer appear nine anneal start feasible proposal projection projection fix theory intersection original complicated know projection projection rule namely projection onto ap simplex I fortunately purpose either alternate advantageous simplex whole intersection alternate project onto several smooth tucker involve multiplier solve search subset contrary small substitute function q denote th entry equal inequality dominate sort describe alternate projection relaxed imagine generating constitute solution reasonable constraint feasible tensor heuristic generating integer put probable integer unknown easy construct imputation relax solve specify constraint projection amount constraint requirement digit appear average constraint requirement digit amount contribute digit take pair projection cycle slightly code sake omit test imply medium hard allow project origin projection backtrack medium hard th projection sim anneal backtrack l medium l heuristic successfully method category computation ghz intel processor ram anneal backtrack implementation c show backtrack vast probably go hard simulate annealing find alternate turn hard rectangle rectangle anneal inspection reveal admit constraint show typical simulated anneal something cpu solution versus backtrack point line solve alternate tends backtrack student mathematical science combinatorial neutral starting stress poorly solve hard combinatorial certainly electrical engineering coding assume dominant science phenomenon computing continue execution drop second tackle large third certain notably communication imaging amount create modern dna sequencing still origin adopt
recovery choose cs pursuit denoise formulate bounding standard l q semidefinite sdp software etc similarly frequency recover fouri transform reformulate signal sparse multiple helpful operator balance transform sdp electrical produce long usually require wireless device signal domain signal quantization random multi balance sparsity l solve sdp solve l norm relaxed square l recovery signal recovery sparse signal static reconstruct signal distribution ls optimization newly propose l reconstruct signal balance measurement quantify root eq th simulation normalize simulation simulation choose fig three person patient patient time signal recovery performance length reach perfect constraint price noiseless available rmse fig reconstruction sub ratio profile optimization fig well bad optimization reason l compressive degree uncertainty l second second long optimization signal newly multi encourage multiple enhance develop hybrid method decrease bregman accelerate multi compressive linear sub investigation signal sparse domain improve multi programming formulate multiple linear constraint improve performance example result newly signal compressive sense cs attract employ preserve projection rather compress new recovery technique cs original measurement classical exploit zero analogue analogue digital operate cs sub obtain time aic convenience formulate matrix use
backpropagation enable perceptron know mlp time whenever simple linear value mlp learn series order computation question question model immediately network prune train large mlp remove connection heuristic introduce criterion prove surely bic layer result establish series generalize iid bic suppose suppose hypothesis likelihood converge toward square index class score establish ratio consistency bic application find suppose admit practical perceptron determination unit one unit keep mlp section mlp cite iid stationary model involve hereafter let markov finite space hybrid hmm e weight iid data peak financial mlp obvious reason compute shift inner see pointed give space generalization common object evolve function mlp neuron property mlp approximation statistical see also alternative mlp hierarchical mlp map som artificial som visualization map low grid string neuron codebook prototype prototype method som prototype point point vice versa away assign neuron prototype consequence visualization capability illustrate instance som design vector survey series build upon recursive version som tree cluster version som temporal code advance include symbol string som som component write represent slow correspond month year correspond month etc series slow year know month suppose series dual lead value view give vector future term would one ahead forecast however generally forecast nonlinear explored forecasting separately mean prediction profile som normalize transform profile hc profiles som lead prototype prototype assign predict standard forecasting scale match neuron instance day week predict profile average rescaled scale become naturally som naturally context normalization one calculation shape mainly another som segmentation technique represent instance som advanced norm visualization capability display globally survey categorical adaptation som way multiple analysis categorical bt contingency table contain multiple bt else bt previously transform individual som use bt som train provide categorical two som world economic field extension neural previous sense construct specific hand strength new technique present general som dissimilarity existence vector format frequent directly belong dissimilarity map negative minimal point dissimilarity readily set string string distance hypothesis minimal rule base choose lead version proceed observation dissimilarity minimizer distortion neuron modification know early version choose associate neuron som modify implementation som concern school transition identify generation sample people market nine category include education stream appear path lead stable situation west however north south region north region contract red contract west transition service education path north initial contract become south east exclude light dark color nine extension som dissimilarity propose avoid constrain deterministic relational number neuron dissimilarity function positivity kernel inner follow convenient come elementary operation datum rely batch som quite assignment distance translate solely kernel
would give calculation term first order induce correlation integral arise parameter residual deconvolution reconstruct contain problem statistical mechanic cell type sample turn ever optimally dataset base experimental issue leave acknowledge discussion research expression level typically type mixture proportion sample principle possible underlie result deconvolution complex support cell two dna set cell decade cell arise cancer proportion proportion concentration gene concentration gene call residual sample specific cell take concentration gene combination contain different mixture product constant cell type variation across sample fluctuation proportion basis reconstruct mix deconvolution reconstruct product mix sample mix fluctuation proportion induce positively fluctuation fraction vary outside molecular outcome example context face recognition analysis question paper fluctuation need algebra formulate deconvolution statistical physics accuracy trivial model suggest algorithm formulate deconvolution optimisation demonstrate drawback derive constraint algebra mn denote gene unknown right side equation gene level hundred thousand condition residual fluctuation type cell induce bold symbol measurement bayes probability marginal keep calculation tractable mean different choose contribution constraint mix proportion delta hamiltonian deviation third proportion pattern cell measurement physics proportion hamiltonian describe focused part phase set mechanic x aa aa aa aa aa tt numerically similarity cell average system thus replica saddle numerically case cell interested difference reconstruct mixture number normalize reconstruction surprising bring induce gene cell reconstruct finite sample reconstruction non value carlo reconstruct average target hamiltonian agreement analytical fig boltzmann sample deconvolution measurement reconstruction weight arbitrary configuration generate within positivity proportion new configuration accept acc configuration enter algorithm create accord mixture need distribution datum allow space high mixture mean configuration visit also deviation deviation quantify reconstruction limit serve reconstruction target deconvolution deconvolution start guess frobenius improved distance solution never guarantee reach artificial outperform accuracy algorithm right plot estimate solution reconstruction bayesian algorithm clearly reconstruction serve uncertainty
round cost size number round round stop meet cost exceed communication cost number k e party party protocol real dataset six party party gaussian carefully partition contain positive lie protocol real uci dataset separable repository protocol work perfectly separable limitation linear protocol design noiseless work communication claim noiseless perfectly entire present well bold require low cost compare time case frequently far desire specify circumstance well per player perfectly player separable synthetic aforementioned protocol perform classifier produce despite rather accumulate become algorithm exchange point c acc acc acc protocol dataset acc acc acc cost protocol dataset case yield classifier party show communication cost variation number finish increase multiply express word htbp party protocol table present party protocol party protocol party simulate player simulate second input work work storage back complete pass continue player arbitrary first completion pass work stream fix dimensional mean low point algorithm communication streaming look deep streaming room turn streaming pass need pass result solve distribute linear communication linear programming constraint streaming stream multiplicative weight allow programming general lp form value multiplicative solution hard constraint call solution approximation would achieve protocol search soft typically feasibility parameter I I xt xt describe party distribute protocol asymmetric value maintain construct feasible consist original feasible use update weight round merely compute soft program apply base algorithm minimization etc protocol hyperplane arbitrary framework distribute applicable wide distribute pt pt goal learning distribute multiple expensive prior propose separable distribute party classification protocol considerably demonstrate distribute across problem convex point streaming adapt multiplicative update set extend range major challenge learn minimize overhead communication carefully point node classifier extension player distinguish linear separability make node paper substantial classification present rely construction exploit multiplicative weight specifically desirable theoretical communication moreover player demonstrate stream weight convex problem task minimal party background protocol party present study section distribute focus infer accurate sub classifier individually improve protocol like admit partition dataset depend extract batch accumulate regret bound recently improve margin apply specific subgradient communication learn resource use demonstrate classification greatly communication major bottleneck many percentage minimize round motivate aspect algorithm consider distribute total agnostic agnostic draw total communication bound decision list proper extend preserve distributional resource process key primitive focus way general weight streaming distribute addition party possess party may negative example correctly assume classifier misclassifie pass pair communication word extend communication issue require count bit instance player cost goal protocol allow get dataset family vc random long exist probability party draw identically party henceforth protocol restrict size construct classifier well threshold axis align hyperplane one bit distribute hereafter way protocol back protocol use communication rely pruning whereby either acceptable classifier classify correctly contribution base communication sampling party player report sample party select union hypothesis vc exist way classification way communication general known word hypothesis vc clarity label protocol round construct weighting end set point perfect deterministic presentation bad two party mention deterministic version weight send initially weight multiplicative propose party misclassifie point correctly intuitively ensures misclassifie eventually weight future round replace line weight deterministic see protocol support point compare ab htbp randomly analysis multiplicative update resemble first key use collect ds although run time vc hold error could fraction mis sufficient specifically yield follow party succeed construction perspective slightly party round round weight construct misclassifie c c misclassifie end misclassifie point point relate inequality hence word communication suggest set lemma weight probability protocol need misclassification set apply failure protocol cost formalize party protocol constant round jointly use communication party protocol obtain classifier arbitrary say protocol player jointly player theorem parameter report fraction player failure round union lemma union party require communication round party terminate round communication correctness round theorem aside party party collect claim randomize achieve construction failure round party misclassifie result protocol linear communication classifier party party scenario amongst naive voting
predictor threshold z stage behave criterion satisfy collecting start decrease gradually maintain close cell analytic table benchmark cell generate build benchmark production maximize optimize medium contour x c ji constant zero prior gaussian kernel kx x function directly corollary theorems z e corollary entail select first locate center inside hyper sphere whose hyper sphere point inside value fc speedup speedup match cl benchmark denote experimental benchmark dimensional benchmark benchmark iteration dimensional benchmark note typical encounter recall theoretical suggest estimation expectation confirm estimation estimation observe good output experiment ei max ei algorithm expect improve experiment value min effectiveness matching approach batch set batch pure cell sample finish step experiment output constant add experiment repeat several way include demonstrate particular function justification theoretical indicate experiment selection meet approach benchmark sequential ei experiment setup table perform well exist practical less matching show constant competitive bad hybrid batch suggest stop criterion fact identify condition stop batch performance degradation ei investigate batch goal policy problem clarity focus maximize concave contribution firstly systematic limit universal bias simulation outcome estimate provide bound relate good experiment secondly propose behave optimally add dynamically early behave gradually move toward batch policy synthetic speedup theoretical light approach ei recall introduce conclude proof theorem conclude proof optimality get ei taylor expansion finally observation exactly addition quadratic root try introduce proposition aim optimize evaluate either sequentially evaluate multiple mode generally evaluation number iteration systematically hybrid dynamically batch size justify eight benchmark result achieve substantial compare try costly evaluate point optima black characterize example motivate enhance micro efficiency surface property problem run expensive consuming focus maximization consist function via select cycle repeat meet stop experiment policy take capability run experiment parallel motivate one select heuristic select experiment improvement ei select ei select add repeat batch speedup experiment sequential policy especially total experiment motivate dynamically batch paper simulate batch purpose select outcome pick batch policy pick upper bound proportional naturally rise hybrid sequential ei next outcome gp choose process go experiment small batch large appeal behave experiment stage prior sequentially evolve manner experimental speedup performance increase experiment speedup increase model probabilistic experiment ei criterion ei gp build outcome unknown new gp variable kx kx kx estimate output consider x location e variance new set characterize change variance z k ba b point enable previous scheme fast complexity actual expect heavily available td tell proportional parameter intuitively know return observation tell batch remark trivial counter intuitive gp optimistic practical bind would focus error oppose simply take corollary capture remark entail computable us criterion algorithm current estimation want accurate selection require suppose capability run speed without sequential
low permit tensor decomposition unified express observable sum rank parameter extract decomposition symmetric solve point variational orthogonal decomposable tensor perturbation variant decomposition subtle unlike decomposition robust use random decomposition tractable manner analysis I would accumulation avoid previously complexity finally address execute dimension require number entry combination robustness decomposition overall framework tensor decomposition model long history across decomposition idea gain numerous thorough application independent analysis decomposition separation statistically mixed observed ultimately projection contrast excess fourth method tensor particularly rigorously analyze bound decomposition estimator relate algebraic simultaneous markov via pair triple probability table later hmms topic lda context explicit estimator identifiability argument source develop tensor develop overcomplete source tensor long recognize algebraic work moment work concerned basic question identifiability structure work symmetric orthogonal tensor combination tensor tensor property decomposition although eigenvalue counterpart computationally symmetric orthogonal decomposition power tensor analyze fact imply general obtaining focus machine statistic far heuristic maximization require practitioner employ computer science machine address issue discussion hardness range mild degeneracy condition tensor observable notably observable propose successfully hmms subsequently generalize extend class low state update estimate emission organize basic definition model structure estimation extract certain decomposition tensor power analogue arise deal tensor euclidean restrict use array pi pa basis multilinear map follow matrix I identity observe basis tensor consider sometimes invariant permutation array I I check order small notion reveal singular decomposition similarly rank one term notion considerably instance clear tensor addition removal approximation norm operator denote tensor first exchangeable invariant exchangeable view conditionally identical simplified model document assume distinct document draw discrete variable draw independently document moment joint word estimate document encode vector conditional moment eq follow x structure reveal certain decomposition exchangeability triple resp form section yield tensor desire various moment mixture spherical covariance start covariance probabilistic mean choose component exchangeable vector spherical covariance differ draw topic document x I spherical covariance variance small eigenvalue eigenvector eigenvalue I common ica mix denote x fourth tensor derivative tensor x correspond excess furthermore increasingly simultaneously correspond mixture oppose dirichlet positive document hx independently specify topic iff word pseudo characterize concentration entry extreme behave interested independent mainly comprise x know assume distribution parameter style inner sep hide inner hide observed name name observe h observe inner sep minimum cm inner sep name hide observed h view call class observe conditionally independent similar require identical markov certain certain variable conditionally importantly may even mix moment k possess roughly speak recover easily handle value linearly independent define three observation concern model mixture product align require certain role incoherence explain distribution partition three possibly view achievable instance align incoherence entry assume rank large span hadamard basis cardinality incoherence condition degeneracy component isolate least partition dimension independently pick put entry put asymmetric component lead mean dimension slightly great base multi extract structure create three apply rotation three x dd invariance spherical check view mean column rank nd surely long imply choose kt ta e jj th large analogous rotation causes fraction large suffice discrete state observation initial chain whose markov give rank iii find decomposition discuss show decomposition estimate decomposition matrix v exist view two scaling fix linear u view eigenvalue uniqueness q maximize zero maximizer subject orthogonal eq maximizer efficient multilinear u tensor unique fortunately permit decomposition mild condition attention minor collection vector correspond scalar focus odd add convention follow generality since analogy two way view fix approximation generalization linear eigenvector satisfie assume eigenvector replace decomposable tensor subtle arise linearity multiple status eq proportional eigenvector spurious say unit repeat start map unit euclidean robust tensor orthogonal imply orthogonal order uniqueness sign orthogonal zero appendix readily orthogonality consideration robust whereas eigenvector large eigenvalue since sign robust fix map third orthogonal isolated maximizer local ica use order adaptive isolated implication two decomposable I function u tu u ii reliably identify iteration analogue moreover tensor model reduce orthogonal single k general form decomposable recover see discussion simultaneous degeneracy linearly scalar imply condition generally degeneracy tensor rigorously formulate smoothed first transformation matrix orthonormal define w identify eigenvector upon eigenvalue everything robust eigenvalue eigenvalue theorem discussion robust eigenvector take convention eigenvector eigenvector correctly along relative latent role applicability tensor characterization provide robust eigenvector latent maximizer objective interpret along tensor decomposition decomposable decomposable moment tensor unlike matrix extract approximate decomposition detail employ establishe convergence iteration decomposition determine convergent point subtract term element repeat power eigenvector ti k c perturbation analysis per preferable estimate eigenvalue tensor power update start symmetric perturbation symmetric empirical decomposable derive population moment throughout reduction whiten theorem return perturbation provide kt tensor orthogonal orthonormal iteratively time call tensor return estimate eigenvalue return least appendix one algorithm specific algorithm perturb decomposition use intuition eventually contraction eigenvector dominate bind note determine converge frobenius euclidean expect tensor perturbation stop initial tensor exhibit effectively tensor build address future research consideration may model topic distribution document primarily initialization document eigenvector application orient tensor variable practical concern arise tensor bag moment document document moment order triple length implicitly document contribution tensor check quantity triple contribution moment allow multiplication document corpus document matrix serious form multidimensional array value typically symmetry store third memory prohibitive dimensionality tensor procedure avoid linear single algebra computation efficiently span use compute whiten matrix product implicit access need explicitly form whiten third w implicitly via core computation perform finally order step non entry computational tensor power decomposable operation random eigenvector extract dimension k reveal value pure note recover eigenvalue unstable gap appendix worth differ roughly gap remove initialization work crucially population algebraic desire perturbation moment exploit convergence directly practical error observe perturbation power example discuss remain previous work accuracy orthogonal tensor method relative cite improve depend requirement contribute complexity far extract need suggest number perspective simultaneous consideration model robust guarantee experience tensor method exchangeable model misspecification indeed affect result reasonable nevertheless robustness advantageous suggest complete microsoft partly mt aa supported award award award theorem completeness decomposition converge repeat set lebesgue measure v exist lemma convergence claim claim first sufficiently thus eigenvector eigenvector point variational eigenvector isolate maximization unit stationary eigenvector pair characterize isolate enough local maximizer must point tu ti nu isolate orthogonal depend cardinality q u isolate maximizer note open pick small maximizer stationary maximizer suggesting constraint decomposition constraint isolate fix relax relaxed power projection analyze iteration complicated tensor initial rate soon dominate due appendix subtle due consider number implicit probability least random vector absolute integer least uniformly distribute sphere l unit separate k comprise normalization specifically least cumulative cdf j argument display inequality hold replace eq define tensor next lemma power method throughout define tv combining read overlap case first case split r sub claim last turn universal I consider three phase use ii iii remain phase iterate hence also hold iteration moreover I rate r claim rate iteration count satisfy imply induction quadratic therefore hold lemma replace check iteration argument
numerically calculation weight event histogram weight determine square candidate kernel could continuous central position practical usually significantly bandwidth subset provide description time find forward take determined minimize expression measure linear degree center evenly spaced start forward step fit consist try give large reduction find procedure stop otherwise include satisfy stop include fit fit positive take increase certain threshold backward step iterate reduce remove try forward threshold opinion stop band around bin integrate interval element singular bin histogram define physical compact error unfold result shape singular bin comparison span case expand unweighted fit indeed underlie unfold degree constitute theoretical properly ignore practical sure good vary function discuss scale quality residual estimate normalize residual quantile position kernel consider significantly width avoid number least bin leave unfold scale parameter function general lead bad fit observe narrow feature favor statistical fluctuation therefore try range overfitte region value satisfactory fit literature leave good method account uncertainty relate also experimentally resolution bias gaussian resolution function fig show measure simulate event distribution show determination monte weight event proportional position analysis scale normalize residual represented weight seven kernel list quality superposition kernel residual plot histogram measure histogram weight unfold illustrate unfold unfold true distribution range vary bad fit also significant quantile small range confirm simple bin measure calculating predict residual content calculate exclude bin unfold determination select unfold parameter minimal statistical error overfitte sum interval band p cm produce statistically independent histogram calculate unfold bin bin make bin independent bin bin q bin bin bin eq unfold total q root characteristic unfold bin gaussian kernel error unfold agree actual finding bin bin unfold result one observe bias distribution value become illustrate complementary grow tend rmse cm cm cm cm bar deviation histogram bin bin show strong region determination unfold datum unfold ill solve solution type smooth act regularization cross integral unfold source solution unfold criterion discuss unfold unfold example property validate execution unfold ghz spectra multidimensional handle properly determination grateful discussion careful reading manuscript ng physics box due resolution detector unfold know ill example smoothness positivity kernel act cross determine parameter numerical simulation statistical unfold problem pdf characteristic
guarantee kalman define unbiased positive partial order define define sampling period order otherwise force dynamic trajectory input solve everywhere time differentiable differentiable derivative mean twice qualitative mean modify filter scheme trajectory make propose modify scheme input scheme scheme piecewise take input piecewise regression improvement identically distribute bound indicate suppose measurement correspond even right parameter entry similarly diagonal side lastly ready unit zero everywhere else leave side filter advance first leave vary computed state estimate give sure try exceed filter pointwise polynomial r strictly speak result apply filter imply perform much discuss intuition measure filter preserve property state bound assumption filter weighted implementation computing choose filter empirical give compact replace leave sided derivative time derivative consume filter implementation compute filter coefficient compute filter provide know dynamic trajectory se mode exist nx nc triangle inequality first term give assumption preserve composition subsequent issue domain occur lie define use pr showing probability convenience far show give kernel c q simply term act ensure lx lx lx v n continuity whenever proceed analogously immediate noiseless intuition correlate instrumental nx pr variable k corollary pr another u pr nx I I uniformly converge convergence series setup often term define deterministic evolve time know se ergodicity hard general se ball center satisfie assume se know true approximate result noiseless se approximate arbitrarily nx g estimator nx iw I triangle setup match interior point cover consistent control second govern would always system explores formalize always exist construction limit subsequence visit limit exist correspond trajectory theorem control oracle converge se appropriate estimator note drop base upon research laboratory agreement air office research agreement fa provide identify rich model system technical proof reason measurement give proof concern simultaneous state dynamical ordinary ode different model prove nonparametric design deterministic numerical implementation distinct part section concern state ode concern control technique assume vector dynamic ode matrix appropriate describe dynamic control input generate piecewise appropriately design
construct confidence plausibility attention complement assertion optimal I something need proposition use desirable set accomplish iteratively intersection idea order equality collection nest equip admissible predictive since eq corresponding plausibility I event treat want break validity follow impose respect vanish include expectation rank suitably balanced remark accomplish numerical parametrization natural representation e respect substitute order choose high rank fact absolute unfortunately satisfy formal recursively permutation achieve optimal describe substitute expression negative contain element algorithm implement stop side positive experience occur tolerance initialize r e k r r r approximately hold close plot indeed realize fluctuation seem zero maximize fluctuation row plausibility belief plausibility classical testing versus size plausibility procedure somewhat less iff plausibility treat random look plot exceed demonstrate I plausibility function criterion thing look stochastic plausibility stochastically former panel I tend outperform approximation plausibility dominate gray line plausibility I plausibility plot plausibility frequentist plausibility various general normal plausibility peak describe plausibility plausibility confidence interval I plausibility unlike I plausibility coverage see confidence interval sort mis model xx solid gray plausibility solid black I plausibility variable challenging consider review introduce I frequentist develop count comprise independent establish mean fact ignore constraint positive I observe must lie consider intend case large mass c constraint belief case constraint validity elastic formulate detail boundary plausibility point imply move predictive recursively refer method balance plausibility black line gray plausibility interval short variety confidence adjustment ms plausibility interval function column winner nominal interval describe plausibility wide must frequentist procedure say remarkable plausibility htbp arise develop theoretical computational contribution construction approximately predictive develop handle challenging high herein part challenge problem binomial handle optimal focused primarily compare frequentist performance plausibility derive tool develop procedure probabilistic summary inferential meaningful singleton hypothese exist general satisfactory probabilistic point help topic identify speak dependent large predictive belief validity bias make I plausibility value difficulty interpretation however meaningful I interpret plausibility claim acknowledgment national science dms dms dms poisson fundamentally problem application physics challenge even large approach construct approximately poisson sort background belief predictive random score validity discrete datum fundamentally example problem physics datum frequentist method reference bayesian popular conceptual also suffer prior inferential challenging handling primary statistical knowledge consensus reach desirable inferential measure uncertainty observe logical way bayesian function plausible hand frequentist hypothesis procedure justify ii error pre assertion majority mathematical definition frequentist interpretability frequentist fail satisfy frequentist application take property carefully reference guarantee fisher theory calibration ii I build inferential reflect inferential ii continue regard strategy develop I review reasoning framework moreover output I poisson naive analysis brief introduction I mean namely side develop I construction recursive construction I side translate directly estimate poisson directly distinguish comparison compare frequentist mind tool procedure fisher build framework monte plausibility note predictive set optimal recommend predictive section particular belief function assertion I stochastically validity property description property value validity whenever predictive consequence I versus testing rule control frequentist plausibility unknown invert plausibility nominal plausibility frequentist plausibility meaningful free probabilistic claim sharp interpretation herein valid want validity belief subset agree assertion probability datum show admissible bound happen nest admissible bind say I result assertion exist admissible
dynamically mainly item frequency advanced enumeration cube monotonicity anti exploitation solver monotonicity property constraint significant previous relation frequency generalization enumeration option equal frequent mapping problem return maximal start frequent reach short maximal eq adopt expressive enumeration strategy anti maximal valid undesirable consequence search optimum potentially need particularly critical database maximal frequent maximal order relax frequent early type frequent search frequent happen clause find frequent repeat search next cm benefit guarantee maximal iteratively length low restriction maximal add form however solver support addition removal pseudo boolean add simple initially maximal frequent fine enumeration option encoding option different boolean encoding support removal clause additional deal cover encoding description additional clause vector search clause satisfied search item belong assumption solver solver removal closely maintain refer incremental clause encode although clause clause add clause grow encoding clause part reasoning outside solver clause maintain solver finds learn clause need satisfy clause strategy storage reasoning iteratively binary clause option item assumption control variable search option distinguish search search encode boolean solver removal clause clause implementation need adapt turning solver solver interface removal clause allow interface solver deal depict option introduce either adapt encoding result medium clause I adapt solver suggestion find set maximal fine suggestion variable wide variety factor dynamically attribute scope sensitive manner decay variable activity activity dynamically tune finally solver resolution adapt sensitive transaction item focus improvement must support addition flexible constraint evaluation option cp property observation retrieve support encode expressive subset cover enumeration option anti imply growth advanced search support introduce solver option one previous flexible interface efficiently solver synchronization parse low frequency solver comprise solver open cover option source simple iteration support assumption clause although solver try learn clause turn impractical adopt solver belong solution support uci repository transaction divide suffer scalability nature therefore intel ghz use operate implementation propose transition low high threshold fig fine solver natural option suggestion decay propagation remain locally maintain remove value domain never satisfy constraint solver deal summation summation constraint deal constraint weight summation discover early whether summation boolean target occur update item possible activate support simply summation branch frequency clause clause justify implication impact solution retrieve dataset nominal attribute label attribute threshold depend near case frequency exceed enumeration aim verify frequency formulate tune base enumeration strategy compact suggestion transaction orient adopt search strategy accord search adopt suggestion good frequent among transaction suggestion guarantee discover major obtain large frequent value frequent otherwise prefer although maximal translate generic translate implication distinguish cp type constraint stream apart simple illustrated ii pi p r r solution implementation large need transaction partition use integration frequent find voting constraint gain measure formulation technique stream pm cp user define novel combine constraint axis predefine expressive constraint variation briefly contribution expressive transaction involve build purpose pattern pattern direction account pattern produce pattern importance pattern devote adopt cp highlight come association due task cp encourage heuristic value extend adapt representative model frequent pattern use principle reduce redundancy maximal frequent pattern pattern variety concept conceptual cluster frequent I attribute stream formulate ip research boolean rough building classification verification accord solver example task multiple representation interval pair interval exist b database arbitrary frequency expression additionally probabilistic use mining expressive formulae option enumeration encoding important adopt association perform efficiency problem high fact clause understand tune solution extension evaluate impact efficiency discriminative exploitation limit intensive rule tune nature density item item transaction transaction exploitation adopt enumeration search level expressive constraint besides anti monotonicity extension dataset require discretization beyond vector boolean reasoning frequent structured new assessment necessarily rely base additional knowledge certain cluster wish relationship mining tree exploit cp verification monotonicity rule advanced bound rule prune speed turn cp option partition stream stream decade mining programming combine manner state formulate boolean frequent mining multiple task drive enumeration option although majority scenario appear non competitive cp cross pm large cp programming wherein variable state greedy pm artificial intelligence develop highly tailor towards specific cp generic high modeling language solver rather pm domain pm count shape minimum efficiency bottleneck cp solver deal option frequency though cp solution scalable pm expressive user drive search nature lead improvement wide diversity cp solver solver boolean evaluate although commonly adopt pm pm discover pm need efficient cp solver extensive cp behave pm section introduce enumeration search option describe implication review finally conclude remark research item call transaction transaction simplify language ti e p item product easily convert database pair item discretization convert row map transaction occur coverage cm consider database pm database frequent frequent place database fix core task pattern classification propose et pm powerful across variety application adapt traditional accommodate easy change support selection drive cluster background knowledge spurious interesting mining drive mining represent support variety expressive impose relaxation pattern bioinformatics cp like already support expressive cp cp task variable transaction assignment transaction transaction define set transaction cover frequent restrict assignment neither fix result different tuple database transaction mine coverage frequency constraint provide essence modeling language current art formulation rely adopt approach dedicate solve boolean scalable solver decade solve thousand clause potential formula clause handle solver encode enumeration alternative option cover map cp clause constraint adopt transaction encoding formula derive equation equivalence mm cm consider transaction binary clause clause cm encode extend clause cardinality need adapt focused transaction transaction encoding formula cardinality ti mapping formula boolean sequential diagram sort
output moreover avoid computation joint drawback formulation kde input overcome generalize allow kde input value step kde operator value rather scalar input exploit describe kde choose kernel recently deal link kde since step rkh space provide structured work base structure evaluate kde paper value joint space output kde data mapping structure output value learn input kde kernel kde formulations ridge learn drawback take dependency datum feature mapping retain output propose value kernel build output encode relationship output component output kernel allow kde kde take feature contrary formulation operator kernel kernel detail table summarize value kernel negative x j adjoint kx value kx value call reproduce input space scalar operator value technique function kde formulations kernel ridge substitute analytic block substitute step structured explicit quantity readily computable output compute input follow correspond operator inherent difficulty problem unknown kernel trick solve image problem thus trick trick implicit mapping material trick pre th express note form scalar formulation kde building suitable output correlation operator kde order advantage kde formulation dependency objective learning kernel functional functional consider value discrete continuous act scalar value operators rkhs receive considerable simple dependency successfully play kde empirical operator last scalar value operator value operate encode specify belong separable beyond separable value address propose conditional define operator kernel correlation effect supplementary material pre suitable tensor jx value product interesting work reduction work tackle operator main view believe reconstruction optical value kde second problem kde lines digit apply kde compare rbf input ridge try yielded performance perform digit select randomly handwritten pixel digit fold testing induce parameter kde rbf kde kde covariance kde conditional kde method value handwritten value promising kde explain fact kde kde kde formulation formulation dependencie contrary numerical output optical subset handwritten mit language character binary pixel representation consist word base handwritten character table experiment measure correctly recognize cross character polynomial third character propose feature mc vc vc py maximum length except easily separately character cm c kde kde experiment operator operate output output consider input allow study output learn dependency kde formulation value kde recovered kernel kde take interaction output input address issue introduce effect structure problem state art well observe conditional covariance covariance kernel another future base structured generalize l reproduce straightforward reproduce implicit feature adjoint symmetric sort increase almost eigenfunction orthogonal dot delta orthonormal basis detail kernel form compute e eq adjoint value yy kx yy yy define variable rkh l ii idea eq take form abc come conclude gram pre image computing compute large operation matrix incomplete reduce acknowledgment thank helpful suggestion de op situation mapping object characterize structure statistical synthesis algorithm structural becoming increasingly extend refined sophisticated need handle output difficulty arise increase depend class value euclidean structure date develop output output output deal scientific community little output recently effort
simple sequence density hasting alternative draw sample pdfs metropolis adaptive however though proposal adaptively proposal converge propose adaptive mh analytical whereas shape become finally eventually produce quickly draw pdfs technique available expense organize section alternative show section letter e letter denote unnormalized normalize proposal pdf whereas unnormalized proposal unnormalize general unnormalized pdfs adaptive rejection construction tailor appeal reject improve mean acceptance unfortunately pdfs unimodal improvement propose pdf normalization procedure time grow index piecewise construct possible construct tangent involve straight pass suitable linear density also piece functional straightforward well since piecewise easy draw reject e approximation one construct table build accept sort stop else back include indicate proposal discrepancy approach ratio close ensure cost htb tangent point line every reject hull construct well approximation close expect acceptance denote respect pdf define distance curve pdfs acceptance w tx vx px everywhere disadvantage density concave pdfs extension deal pdfs log concave metropolis mh rejection arm discard improve standard however sample accept another statistical guarantee accept distribute sample arm arm describe whereas pdfs step rs add point improve sample initially rs happen determine whether accept notice arm reject accept support rigorous mh omit see auxiliary observe link construct pdfs fact issue study procedure suitable markov build discard otherwise px k tm stop support straight line arm introduce piecewise form q htb important new point inside affect adjacent region therefore subset proposal practice pdf slow general target concave form drawing proposal require know straight see might suggest entail point contiguous straight construction without form piece use quadratic order proposal otherwise build well limitation arm technique univariate pdfs satisfied proposal stay target much procedure improve everywhere indeed classical pdf neighboring interval building allow improve even inside proposal interval drawback point eliminate show example remark guarantee except special structural limitation region inside x px clearly illustrate problem arms pdf multi modal show point tangent construction within c support contiguous vary within pass incorporate arbitrarily figure within figure b sequel construction depend straight pass pass look apparent point add inside contiguous interval c limit incorporate close locate line adjacent interval tangent show minimum tangent line straight tangent moreover tangent stay pass meaning even therefore standard arm entire pdf example discrepancy target trivial drawback arm unfortunately problem unbounded major pdf reason standard arm ensure keep cost low strategy improve arm two variant w arm proposal instead clarity technique idea possibility follow inside decrease evolve grow stop adaptation convenient simple draw sample tx tu tm tm tm tt remark point arm accept rs acceptance always accept never incorporate point reduce happen code implement unchanged calculate control step proposal pdfs difficult chain adaptation observe coincide issue chain current balance could good proposal remove final return follow fact technique firstly provide incorporate tends effectively vanish new proposal close close pdf arbitrarily algorithm support unfortunately guarantee although good simulation arm adaptive arm indicate build pdf use possibly account support convenient procedure e one draw tx tm kx tm tm lie proposal independent mh technique markov invariant procedure already calculate density improve arm density also subject restriction arm good support add computational cost easier inspire straight go extend straight line towards minus plus mathematically although look eqs simpler involve calculation one moreover procedure construct piecewise line mathematically proposal e uniform figure construction describe concave log since guarantee except first modify tx vx log target mode upper location depict obtain produce concave finally build structure completely arm domain pdf basic straight point piece tail unlike area indeed follow simple calculate pass tail construction use draw pdfs interval htb pdf domain procedure computational arm type remark could inefficient arm without due explain exception lead combination probably stay depend possible initial procedure describe avoid speed convergence avoid tail proposal initially construct attain proper proposal pdf add tail algorithm tail second arm incorporation support know kind tail reduce burn period proposal build instead straight arm build pdf arms eqs build pdf mixture q mean estimate distance proposal iteration form inside proper therefore inside note adaptive construction proposal slope tail scheme never table show chain build correlation consecutive always discrepancy pdfs std avg avg avg estimation second std estimate avg avg rs std avg avg support rs control table scheme decrease minus plotted mean propose arm perform h b figures eqs figure figure notice iteration two variant arm propose standard arm greatly improvement expense support moderate increase bind slight notice building strategy propose original third quickly procedure support great incorporate region piece pdfs curve shape procedure slowly second always support fourth arguably provide density efficiency proposal relationship approach proposal pdf see discrepancy speed convergence automatic pdfs good approximation issue complicated target approach building rejection arm important arm introduce allow draw concave metropolis log pdfs multivariate within univariate distribution gibbs highlight approach build limitation arm inside prevent converge alternative arm scheme target statistical direct inside adaptation one adaptive mh previous sample ensure measure computation pdfs numerical approach arm accurate consecutive pdf issue two guarantee discrepancy target pdf quickly imply proposal close mh parameter pdf ensure r increase due incorporation support inside moderate remain effort implement remain control arm evaluation proposal pdfs finally notice technique reduce construction slight test arm important fact dedicate introduce procedure piece plus pdfs tail standard arm efficient draw directly piecewise approximation pdf among partly author thank di iii de discussion
identify event put together event observe trace equality present eq keep use gradient generalize number neighbor I rare zero element similarly element correspond scan usually furthermore arrival enable arrival estimate employ last assign fig word reconstruct spectrum notation fire stop criterion meet receive electrical stanford work statistical signal message pass bs university physics theoretical post de th de sup sciences research institute berkeley usa centre de la stanford associate electrical award receive national science award research receive university electrical experience modification conventional pseudo trace overlap little currently employ hardware highly trace efficient massive computational expect domain mass mass ms family chemical application technology protein protein biological interest biological sciences medical development identify analyze protein chemical protein throughput include measure exploration technique configuration mass utility application consist three module possibly advance phase open analyze mass module common force broadly effectively stream toward mass particular stable mass act scan wide band module detect detector surface motion benefit alternative technique essentially mass rate advance range basic acceleration drift detector electrical acceleration effectively fire region drift proportional acceleration hold mean therefore reach detector proportional technique accelerate drift impact detector continuous sample discrete call noisy scan repeat hundred thousand many consecutive chemical reaction analyze precede automate fed automatically metric describe mass mass sensitivity speed mass specie specie mass refer detect acquisition per trade among metric speed accuracy sensitivity acquisition consecutive e scan detector fast acquisition eq drift region acquisition far apart increase factor detector characteristic speed reach mass remain improve mass length drift region obvious requirement sensitivity could restriction monitor chemical reaction use mass significant implication hundred thousand improve speed keeping improve automate drug simultaneous speed fundamental conventional choice mainly simplicity particular sophisticated technique resource generate mass power trying call subsequently accelerate drift detect hadamard reach detector frequency ideal shot output signal detector shift index long spectrum spectrum deconvolution implement efficiently hadamard inversion ignore require substantial modification hardware accelerate mass improvement hardware simulation real improvement scheme detector scan abuse scan process average obtain accurate let scan infinitely multiple overlap scan find vector transpose matter convention refer spectrum bin represent detector generate detector refer short rate response one bin describe without minor modification present simulate use algorithm single scan electrical spectrum expensive perform drift scan repetition n incorporate powerful scheme f fig require collection scan overlap subsequent trace avoid overlap consecutive relax random superposition q superposition overlap total firing consider adjacency matrix bipartite row bin scan bin scan trace refer version row case measurement observation bin reveal adjacency ls reason lie approximate shoot signal similar arise limited shoot limited regularize likelihood stochastic optimize regularize bin span algorithm log log attempt ff presence chemical iteration let cost equation right part negative calculate gradient follow firing constant estimate adjacency repeat criterion step worth note mention bin spectrum fig spectrum thought event cause problem terminate slow confident generalize instrumental spectrum chemical scan length interval average ten evaluation fire prescribe pass marker level indicate start marked event corrupt henceforth trace unless mention across electrical additive electrical detector mark high select potential potential impact event event exceed peak height event trace support valid preprocesse trace event whereby trace scale amplitude purpose identify enable evaluation constant event match negative match exist match say match negative rate tp tp rate fp fp ill false metric width pose state ratio acceleration four fast satisfactory accuracy false factor ten false proceed negative rate value discovery establish iteration fdr design firstly require discover truth rarely consider costly mistake miss line type fdr converge ground reconstruct inspection match vs fdr bottom admit peak peak ground truth difficult detect right plot list peak pick practice publicly available peak magnitude spectrum less tp peak pick estimate peak value slight abuse peak match within four curve acceleration acceleration acquisition acquisition pick truth spectrum list pick pick amplitude choose pick run pick peak ground truth significant peak peak ground estimate top peak achieve perfect reconstruction acceptable inferior acquisition outperform fast achieve decrease peak match fig top peak fdr since detect performance remain unchanged dash solid acquisition peak pick factor cdf intensity ratio peak peak understand achieve sample confidence show vs false acceleration ten construct nine ground interval standard obtain mapping spectrum adjacency simple trace equally cause curve conventional e obtain perform acquisition time long k
thank anonymous remark weak assumption fourth constant consider analogous b apply fix kl n provide high q sufficient schmidt calculation assumption conclude markov derive previous apply except show kl lead define r center x v functional translate design eq conclude proposition adapt regression supremum nr x inequality conclude bind derive cn cn markov derive lemma shall combine assumption derive k investigate take expectation upper derive observe derive n n first eq integer k g note freedom kx x x kx kx derive desire result finish denote freedom k kk k proof get apply since k eigenvalue france universit real combine test analysis interestingly drive knowledge slope level assess slope range numerical linear constant denote intercept represent slope function center random variable lack effect test widely literature generic procedure support address dimensional belong henceforth endow sake clarity thus center independent unknown vector stand essence testing general originally briefly review result minimization square penalize plausibility spline estimator thresholding hilbert functional principal span previous estimator reference therein property estimator rate convergence aforementione recently prescribe basis e spline estimation rely tuning quantity variance fact smoothness operator unknown literature test functional statistic functional drawback set arguably issue apply bootstrap asymptotically paper power viewpoint non test correspond projection power section mild additional sharp art principal rely perturbation theory result rely section power power close problem testing small distance powerful minimax achieve separation distance formalize minimax control functional wide class optimal suitably unknown choice quantity regularity regularity lead poor performance issue minimax simultaneously regularity testing viewpoint combine test test technique spirit require level power analyze viewpoint section test minimax distance involve procedure illustrate simulation discuss involve perturbation give technical refer hilbert euclidean product covariance trace schmidt basis increase eigenvalue correspond sequence eigenfunction sequel line expansion pca tool functional expansion uncorrelate exist center uncorrelated principal amount eigenfunction eigenvalue unknown empirical covariance sequel functional usually technique appeal variance core procedure seminal convergence assess asymptotic tuning besides plug create usually introduce dependence sequel hypothesis dimension expansion introduce define vector intuitively proxy reason stand orthogonal main classical projection kl j nj orthogonal instead behave statistic least prove numerator correspond norm statistic statistic et al consideration transform naturally unchanged replace crucial synthetic functional white exactly normally distribute admit fourth moment error noise fourth constant jj assumption since operator come behavior fourth moment truly mild brownian sequence exponential j restrictive restriction dimension projection classical eigenfunction dimension eigenfunction become difficult dimension exist constant proof show degree argument rely normal assume belong advance span test extension belong span capture onto probability regularity look precisely consequently convergence regularity power trade term assess optimality procedure assume regularity ellipsoid regular positive denote supremum ellipsoid number correspond minimal hypothesis ar tt separation infimum quantity minimax separation ellipsoid notion separation distance boundary ellipsoid exist process ellipsoid upper upper bind positive define minimax consequence separation multiplicative constant interestingly separation distance behavior increase regularity regularity quantity sequence separation test exponential decay achieve reliably estimate eigenfunction lead restriction art condition conclusion achieve optimal regularity take zero lead large good correspond term procedure achieve trade prior regularity dyadic priori discuss kl test accord one refer correspond hoc test approach vector independent conditionally conditionally simulate conditionally work carlo choice obtain appendix require take size small let admit n counterpart multiple counterpart constant proposition gaussian test power minimax optimality rejection pay replace term appendix us sequel stand integer logarithm constant direct minimax satisfying term ellipsoid minimax collection nest constant infinite ellipsoid consequence price pay simultaneously consider collection adaptation framework compare low decay separation proposition decay separation distance interestingly decay experiment brownian eigenfunction operator brownian use form evenly spaced testing experiment compute carlo quantile smoothness stand explicit fix smoothness dirac center trial rejection ie confidence procedure implement ghz intel processor cache gb cc c c cm pay correction furthermore become smooth perform third table correspond sequence become dirac test evaluation quantile computation second confidence cc c cm cm c cm cm cm h cc cc cm cm cm cc c cm cm cc cc cm cm cm two testing completely drive assess address focus extend testing v nj kl kl th behave p degree situation kl ph slight monte state describe nonparametric unknown well prescribe fourier decrease slowly project necessarily suit prescribed spline eigenfunction procedure fact randomness design prescribe also unknown combine procedure emphasize perturbation integer jj orthogonal projection onto span translate analysis I nk j correspond sx class operator numerator statistic square estimator generate n proposition adapt proposition conditionally size conditionally small respect quantity prove appendix operator replace definition use replace three behave distribution rely theorem tell prove appendix b kx hold statistic three n union rely condition enough appendix b deviation turn order conclude derive term assumption
alpha old split problem need monte carlo alg propose transform mdp policy dealing representation must map choice combine let sub original action binary mdps mdps denote define mdp knowing mdp vice versa original mdp environment binary operate mdps mdps binary light training try binary policy manner describe sub problem mdp increase problem complexity sub f except instead action sampling see choose action call action state decision different sense mdp equal noise correct split study present respective define spend multiclass one spend current spend monte cost classifier come multiclass cf cost reduce policy mdps since possible learn binary cf result number show addition illustrate complexity computation really assessment turn day computation car ht number x axis first line second rl vary discrete action experiment obtain set range performance reward episode length quickly second problem grid correspond agent mdp construct action set define contrary car notion consequence agent reward avoid reward small action intractable action learn particularly bad state problem number trajectory hinge perceptron iteration code random procedure alpha reward three figure action average reward first similarly strategy involve binary classifier performance indeed learn baseline intractable non trivial policy approach intractable car action consider explain performance set require last figure number r cccc car sim algorithm rl mdps monte phase costly couple lead theoretically assess efficiency method reinforcement generalize leverage deal space additionally behave impose binary search space mdp thus place solution value rely solely although dictionary key rely code actual hierarchy output mdps learn obtain classical implementation optimal policy work depend automatic adapt particularly theoretical plan relation fact deal thousand study international conference reinforcement learn scenario rl hundred sometimes rl obtain multiclass output code set reduce first classifier classifier dictionary mdp far set tractable finish experimentally demonstrate rl mdp rl learner environment action though understand rl face many dealing generalize large learning explore difficult regularity smoothness action great go planning problem action regularity knowledge regard consequence action sub draw idea multiclass assigning bit know real world problem propose computational leverage idea learn allow optimal mdps first reduce even brief overview rl section using explain mdp factorize accelerate analysis relate work cover key classification code tuple transition function choose expect immediate immediate article adaptation formalism multiclass integer binary code multiclass classifier predict transform supervised infer pass hamming look tree contribution part building begin show integrate around assign action one action binary action code code combine policy original want code action
co co et co co stage find propose approach fitting coefficient select tool refine improved efficiency description dependence application real interesting dynamic ar select fit bic discard ar explanation increase sufficiently inclusion similarly good bic illustrate construct coefficient dimensional var eq iid show stage fit summary define likely procedure use truly conduct correspond parameter execute modify ratio refinement truly exclude inclusion substantially discard pre specify replication refer modify bic refer bic panel compare compare procedure zero coefficient original one forecast bic fit include oracle bic raise implement var lasso var column stack ar matrix usually constraint matrix specify ar rank constraint ar var illustrate consider var express equal non ar estimator ar equation product l know coefficient parameter estimation therefore iteratively ar lasso fitting var ss var compact product l iid minus var ignore additive constant penalize choice ss log tuning parameter control var estimate respectively sum residual minus log likelihood lead method var derivative ss lasso function ar entry matrix lasso lasso var model ss since lasso fit efficiently angle lar descent ss var complicated involve iterative var lasso cast q eigenvalue decomposition define root target residual variable explanatory target var pt iterative var th ty fitting penalize var choose tuning cross use determine explanatory order determine drive manner take fit ss lasso var lasso var apply coordinate descent minimize aforementioned iterative ss lasso model parameter give minimum average ten give minimum average either lasso var ss acknowledgement would thank air science dms part support nsf grant google award section autoregressive var widely dependence series unstable overcome stage var many ar ar partial spectral together quantify far illustrate datum google trend levels air autoregressive sparsity coherence autoregressive model temporal multivariate series consist serial dependence series suit var ar moderate prediction interpret drawback sparse interpretable description popular penalize problem determination use lasso penalty tailor purpose lasso advantage selection estimation small setting tendency var formulate regression series treat variable explanatory theoretical temporal dependence response explanatory fitting stage screen conditionally conjunction information contain spurious far refine fit stage screen strategy ratio organize var section procedure var model stage causal lasso stage trend air contain supplementary pp interpret far pa without generality mean var matrix var want ar need dimension large prediction interpret description also sparse preferable fit sparse var ar develop fitting stage select ar coefficient correlate free correlation frequency specifically coherence introduce correlation effect removal effect result linear k opt another marginal correlation residual distinct conditionally uncorrelated lag lag residual cross density reflect correlation marginal spectral coherence distinct marginal scale residual series show computed spectral process via invert dimensional marginal challenging simultaneously f spectral coherence compute one spectral density encode pairwise conditional series selection sample e concern relationship zero distinct conditionally dimension entry replication spectral density remain conditionally use inverse pair series conditionally uncorrelated pair ar lag fit ar possibility use refine indicate correspond conditionally stage pairwise var particular ar belong lag rather individual connection ar approach achieve return zero identically correspond conditionally uncorrelated word need separate depend quantity summarize different modulus summary statistic supremum indicates conditionally correlate therefore rank summary high low way add base var select var select ar control ar coefficient use bic simultaneously bic eq maximize estimation var detail find appendix restrict specify respectively stage summarize stage distinct marginal inverting density construct distinct marginal ranking select sequence choose give shrinkage adopt obtain ar coefficient much fully parametrize var first stage execute group model examine employ determine introduce g examine multivariate question ar causality series causal relationship shrink e propose group purpose conjunction bic discover ar coefficient reader refer correlate marginal bic however spurious zero ar stage model pair group refine eliminate spurious ar rank absolute non zero constrain estimator column stack coefficient matrix estimator stage triplet absolute ratio high ar retain complexity ar retain bic contain pt zero coefficient triplet high low consider select coefficient top triplet execute appendix correspond bic bic numerical stage approach stage compete stage google trend concentration air lasso var formulate select ar ar loss choice square residual minus affect ss coefficient estimate mse mse ar ar estimation panel figure replication ar ar ar color shows correspond ar see panel belong series stage contain spurious ar coefficient position I circle panel c stage refinement implication first circle stage select true coefficient probability circle panel approach implication show stage correctly ar coefficient panel ar coefficient lasso ss medium circle circle ar unbiased legend circle approximate truly var spurious consequently interpret tendency select ar consistent bt true ar coefficient panel circle percent replication ar compare impact display ar coefficient approach well remain variability stage matrix therefore adjust change lasso var interesting change variability lasso lasso change panel increase circle first suggest increase ss increasingly temporal spurious method influence change variability ss ss result lasso address lasso benefit lasso ss bt bt stage method respectively horizontal ar stage past ideally coefficient stage lasso much ss increase estimator change bias significant estimator nearly vary systematic lasso datum view notice many researcher internet activity see researcher top google select produce google trend google trend center surveillance visit google data surveillance first trend report possibility forecast google google google trends fine report google trend throughout surveillance national fit trend week region north south due simplicity apply range stage parametrize bic stage coefficient far select ar var ten cross var non coefficient lasso x refer ar coefficient retain curve bic varie panel dash minimum compare structure discover ss ar interpretation region surveillance estimate diagonal reasonable activity previous week week region diagonal var signal var hard interpret lag dominant ar ss since description structure model discover activity ct nh indicate panel within moderately lag cross co mt la nm ok general lasso zero large absolute ar lag var three week week quantity ahead root square h forecast summarize forecast
site global vary specify conceptually specify comparison distribution option explore note record match status purpose consider count agreement individual sum within alternate specify choose repeat distribution draw posterior draw independently specify stop converge distribution decide record link review leave record record cut assignment group examine cutoff value potentially record equal small file size divide structure draw cutoff factor transform sample scale pair large match simply one draw multiply specify distribution latent class one beta could transform scale proportion approach implicitly exchangeable match specify field vary independently probability across block without strength block parameter separately insufficient matching agreement version allow correlation could restriction number across surely poor block match indicator simulate sampling conditional hyperparameter matrix distribution assign match initialization option randomly represent match per assign cutoff cycle draw converge target indicator value enforce constraint eq indicator comparison block hasting hasting hyperparameter draw hyperparameter hasting bernoulli stop converge converge suggestion make metropolis hasting step use hyperparameter calculate sm sm hyperparameter value follow step outline index replace parameter hasting outline index need tuning parameter accept approximately adapt tuning second tuning replication perform two condition across block block link together correctly estimate file record organize yield record match seven probability match increment agreement among information depend acceptance hasting step performance complete linkage reflect simulation area work interesting census health automate apply file would regard specification record linkage available linkage site site way decide site discount analyze site file challenge block study cycle gibbs sensitivity specification one indicator restrict future metropolis hasting new instead acknowledgment work contain report partially support grant selective author necessarily national security university else zhang linkage author north hill national center health census present record linkage w record linkage international exposition office management budget j base estimation record variation linkage factorial survey methodology match record linkage journal american statistical monitoring iterative simulation incomplete via journal american association b record linkage model record linkage bayesian survey method f calculate marginal association simulation h sequences stochastic gibbs image transaction intelligence hasting monte record linkage york science business media advance record methodology apply matching census journal american linkage health file link american association imputation record survey record match census record census comment american york city l new york hierarchical linkage department record linkage national death journal iterative record linkage american statistical association latent linkage link american research york discrimination dependency linkage model record linkage american statistical association research linkage census conference advance linkage american method match record linkage ed cox p new york pp record linkage rl file matching identify entity file include count survey study file identification file name find score status come comparison prior class rl key metropolis linkage rl name activity individual one rl file record correctly associate two survey frame survey longitudinal study file rl activity review subject file identification file error rl trivial unique one name address record one file comparison comparison record initial file requirement outside block much evidence favor pair could imagine take determine statistical supervise analysis logistic fit score status unsupervise model group initial rl hierarchical rl impact distribution current study vary across block hierarchical use inter heterogeneity development study rl error rate record linkage section report file file correspond file contain identification variable file similarity record pair agreement file base birth name house name age birth file name standardize accord address standardize separate day month pair consider field record equal agreement disagreement many typical informative location suggest match record determine evidence census file group record well reduce
estimate three ad quantile model high rmse consistent high percentage regard copula constrain quantile ht htbp quantile cccc cccc copula ht ht c ad patient drug blind clinical period laboratory raw data transformation heterogeneity dataset apply log consequently post adjust median quantile equivalent laplace zero scale find indicate association baseline whereas baseline model combination value low e baseline simulated baseline sample transform box cox relate modelling firstly implement univariate describe variable stem observe high post period trial pre period tail simulation incorporate result univariate find effect estimate quantile unconstraine proceed joint simulate post baseline population fit dependence assessment estimate via regard order make justified significance hypothesis conditional exceed threshold output nan order strong approximately order level quantile appear quantile figure order maximum majority order change quantile conditioning place state take survival early simulate post baseline subsequently constrain equation replace likelihood estimate produce simulate simulated survival uncertainty variable induce change survival ht behaviour tail order quantile standard laplace dash vertical correspond statistic row mm third column conditional quantile jx yx solid dash grey estimate model identify assess however insufficient duration clinical trial tail laboratory relationship attribute strong dependence methodology formally assess view behaviour test dependence baseline adjust formulation build extend order trial bound additional mainly asymptotically evidence tail behaviour predict high modelling attribute use survival indicate order constrain modelling primarily baseline quantile propose order effect come remain longitudinal order consecutive acknowledgment program ep mathematics discussion constructive comment constraint dot constrain explain find profile surface conditional small quasi highlight joint affect quantile area constraint join mostly quantile near line show bottom side join boundary constraint induce small quantile area induce quantile constraint section order large tail complement estimator composite additional maximum generalise distribution stage simulation necessary approximate derivation variable analogously outline conditional quantile limit copula second comparison independent coefficient model tail dependence constrain refer ht stochastic performance assess denote monte conditional
versa denote dags arc either dag arc lead variable variable arc arc model arc model network model choice class integrate smoothly arc bernoulli marginal among simultaneous multivariate specific specific find know expectation immediate covariance interesting basic minimum follow close q construction multivariate bernoulli multinomial effort property cover contingency analysis variable tt uniquely identify eq moment two bound prove either constrain segment axis maximum deviation equals reach maximum bound eigenvalue lemma family dag let appendix component probability associate derive distinguish correspond among mass possible configuration identify non informative posterior set useful identify clearly dags useful reference determine arc probability one configuration edge arising property display informative behaviour value derive section consider arise behaviour explain tendency represent relationship underlie almost surely exclude edge consider good closeness multivariate intuitive consider simplex point boundary origin intuitive entropy however model point comprise edge three edge brevity covariance position show entropy due increase difference eigenvalue fact nearly strong close axis distribution correlation clear multivariate similar bernoulli many reason acyclic arc dag direction influence arc even close form make distribution obtain reverse direction arc another every acyclic immediate include arc dag include arc dags direction arc arc arc arc arcs q expression arc arc arc belong arc additional replace undirected combination belief arcs direction uninformative case expect covariance arc converge arc equivalence direction associate way order arc drastically reduce free arc numerical arcs dag dag arc maximum quality examine possible dag value theorem compute dag size generate probability clearly figure estimate limitation easily via enumeration quantity evident absolute edge absolute interpret arc ij marginal arc remarkably graph dag node arcs approximated probability graph motivate case outcome contribution presence direction imply arc direction dag decision presence arc direction reach maximum covariance tight decomposition dependent form equation joint distribution dag arcs q appear tight light correlation compute figure bound dag loose value covariance dag dags non correlation vary dag modulus node represent apparent appendix dag size enumeration arc uncorrelated property dag support sparse proportion arcs converge assume conjecture proportion eq furthermore arc common strongly arcs modulus arc take value modulus value modulus dag conjecture two concerned range structure display little substantial eigenvalue graphical representation variability illustrate usually multivariate normality generalise value property bernoulli total easy bound eigenvalue generalise hadamard negative matrix bernoulli respective maxima generalise variance strictly equal zero convenient rank present option behaviour square frobenius arise case high unstable unique none correspond make interpretation seem arise let unique global maximum correspond matrix approximate normalise follow normalised vary associate display variability vary interpret goodness one total rewrite provide complex different present assumption algorithm datum result rewrite dependence edge arc distance use efficient furthermore possible commonly use literature rely variation ham knowledge possible include example restriction degree denote tuning measure learning algorithm increasingly hc towards never coherent lastly world attention reason parameter assign support drawback firstly effect presence arc conditionally secondly uniform assign discriminate network sparse preferred example arc order parent topological clearly variability prior challenge undirected acyclic graph make problematic propose alternative structure focus respective smoothly extend frequentist present literature behaviour individual reduce super exponential inferential task network moment easy interpret algebraic thank college comment suggestion author real real prof inequality inequality eigenvalue trace proof uniquely multivariate definition would assume theorem direction direct graph arc eq arcs arcs family hoeffding
delta quantity construct gradient procedure begin optimization parameterize kernel mean variance property normal boundedness satisfy importance section cumulative distribution normal interval iteration dimensional density multi gaussian kernel rr truncate dimensional multiplying correspond multiple pair learn cf demonstrate deal parametrize space learn constant multiple diagonal key issue datum square feature offset normalize run validation average run method maximum objective divide average examine speed method coordinate one per polynomial method sampling algorithm kernel parameter machine repository dataset htbp l consistently stochastic coefficient per difference naive slow algorithm representative mkl examine mkl generate regression dimension input choose free separate datum method learn decide cost linearly make keep exhibit consist build recall dd specifie add constant polynomial degree possible method training distinct fed keep gb run increase depend linearly kernel experiment variable large space learn predictive performance start kernel consider suited dimension among close aim mkl new algorithm kernel also product interaction feed kernel six uci method mkl kernel degree specification set overall non interaction linear bad believe observe kernel easily prevent overfitte assigning select purpose repeat confirm nd nd nd finite mkl time experiment fast almost since learning counterpart recall mkl multi slow dimensional infinite considerably fast dimensional since perform tb nd nd mkl finite breast diabetes heart exponentially predictor randomize mirror descent derive optimize variant trick careful unbiased keep combinatorial practically representative literature optimal solution kernel satisfy strong apply applicable beyond study combine combination gaussian remain acknowledgement support proof rate proximal proposition carry argument solve lemma strongly I hold one express divergence thus plug identity introduce part define convexity notice side linearize appear standard proximal learning convergence algorithm give analyze adapt martingale difference sequence satisfied boundary optimizer optimization lemma term yield q develop bind expect measurable together combine result rule bring thus k adapt martingale strong furthermore k old book sure bind sum place k seem strongly case substitute hoeffding martingale obtain union write primal lagrangian n multiplier readily primal equality constraint clearly feasible condition maximizer expression readily mild derivation quite argument find e specialize loss thank implicit provide case derivative manner combine implicit continuously differentiable list far shall readily verify cm ari science ab consider linearly combine predictor number large method whose scale intractable propose descent computationally variance propose base ideal proportional magnitude correspond surprising result coefficient combinatorial structure kernel allow sampling look computational predictor kernel mkl interested base come view nest convex first optimize weight minimize empirical risk randomization mention paper update argue keep variance actually low kernel make potentially use bound prescribe linearly mirror similar prediction considerable attention challenge interest lead overfitte infeasible avoid combine important induce follow approach call use range encourage group actual reason convenience trick see exhaustive survey reference multiple popular decide priori map appropriate task priori choice fed huge increase bring even infinitely kernel soon bottleneck linearly large mention infinitely lack guarantee consistency practically kernel new present specialized version give experiment formal index indexing define input x w consider solve optimization eq specify later nf nx penalize empirical know favorable pass sake simplicity abuse define mention mention call group introduce encourage sparsity handle come however reason convenience reason feature thus kernel penalty form jointly reproduce hilbert rkhs learning rkhs particular page equivalent lf see risk instance mkl therein allow run memory exploit finding slightly notation symbol convexity minimizer unique exploiting convexity use randomness propose randomized coordinate thus alternate point separately design proposal would alternate minimization investigation definition let nonempty barrier gradient correspond bregman kk mirror descent assumption note since infinity tb size subroutine k k g slightly form literature analysis respect norm time deterministic finally hold q q theorem strongly convex implementation project choice use two subsection choice make algorithm start first verify way expression calculate stop iteration include sake section ix nn I iw base f compute proper differentiable derivative write vector define sampling gradient index case whenever derive stay define k belong compact note latter condition empirical k k see efficiently may also store infeasible project project show simple result base calculation follow implement importance sampling tb randomly use algorithm presentation necessary drop gradient drop dominate integral countable back formulation counting measure form parametrize parameter come notation learn product base shall learn kernel combination set ix index permutation define statistical model interaction kernel cardinality depend write rest emphasize without keep write index draw description denote matrix high product hadamard overall would subsample empirical bring leave future exponential algebraic efficiently hard bound independently tb
suggest user web search cascade consider present select list reciprocal cascade observe click family straightforward via triple user choose nothing estimator probability define member order conditional theorem proposition reciprocal representation preference acyclic loss loss recover recall pairwise rewrite discussion consistent special recall structure ij derivative fisher inconsistent exhibit inconsistent employ consistent precisely one coefficient inconsistent simultaneously associate loss edge assumption consistency preference acyclic strategy solve convex procedure broad description presentation specific difficult grow fortunately leverage minimize minimize function argument zero idea et specialized composite objective maintain live iteration score secondary concern verify method insensitive aggregation estimate denote minimize loss display risk reference different pair broadly plot match consistently plot appear sufficient loss aggregation risk salient moderate large effect interestingly significant improvement consequence sampling attain risk asymptotic score display risk suggest minimization empirical risk feasibility design consistent consistency ranking ranking base inconsistent rank surrogate partial preference behaved empirical benefit surrogate experiment thus toward formulate rate rank risk interesting grow infinity least solution asymptotic normality robustness risk explore formulation supervise acknowledgment thank anonymous associate helpful comment valuable proposition national engineering research laboratory office contract query procedure consistency provide rank rarely preference pairwise comparison rank aim partial preference however raise serious many commonly loss comparison yield surprisingly motivation present rank base aggregation preference risk show parallel complement procedure task class year significant notably strong theoretical quantify rate qualitative aspect extension increasingly understand overall satisfactory world label classify real value rather list supervise complete provide treatment example retrieval goal user object probable disease recommendation system aim order accordance preference population document search engine must page statistical fall understand aim characterize statistical inference generate mechanism theoretic preference draw consist candidate item nature preference set retrieval natural preference select return discover provide specific ordering index retrieval need natural language assign score adopt theoretic perspective risk intractable machine select fail ranking reasonable generating mechanism remainder failure strategy aggregation begin loss propose literature discount ndcg ndcg require preference relevance preference makes asymptotically minimize ndcg loss preference collect trust biological evaluating involved click feedback even consideration collect complete highlight human difficulty relevance lead researcher suitable arise world patient treatment match pair store moreover show human reference predict item associate preference prefer associate zero loss intractable convex fisher link fisher consistency consistency hope ranking guarantee hope generally computationally fisher favorable surrogate consistency fail practical strategy satisfactory justification approach difficulty rank preference aggregate theory statistic preference aggregate theoretical establish discussion aggregation conclusion proof defer begin several way preference arise aggregation base match comparison may naturally generate practice preference preference direct preference order example response query engine order record user store track customer provide preference candidate example preference specify partial candidate candidate order preference item wish treatment aggregation us aggregation appropriate case pair relevance weight adjacency item entry nonzero aggregation strategy query represent user preference query adjacency matrix capture mean preference thereby item partial preference framework formalize hereafter specific existence series ks pairwise loss eq adjacency hereafter procedure sequence conditional law aside aggregation addition assumption loss ms ml assumption asymptotic place guide pointwise conditional risk predict closure subset query motivate minimize preference go follow face difficulty minimization nonsmooth typically minimization precise formulation demonstrate commonly use pairwise aggregation tractable procedure losse second consistent surrogate statistical procedure statistic law sufficient query countable pointwise bayes risk equality countable infimum qr f throughout exist consistency achieve risk draw uniform measure assertion rf completely condition fisher general analyze fisher consistent main surrogate fisher fisher pointwise fisher begin define measure classification surrogate suboptimal risk suboptimal conclude completeness provide material pointwise fisher consistency hold proposition clear consistency strong consistency situation connect weak surrogate fisher pointwise connection indeed space finitely multiclass classification setting assumption depend partitioning local often continuous q satisfy intuitively property surrogate score surrogate loss surrogate coincide fisher state give supplementary material hold fisher loss definition structure conditional theorem consistency uniform loss consistency coincide assumption restrictive practice sufficient condition hold weak satisfies fisher structure surrogate risk finite preliminary focused set preference demonstrate popular surrogate inconsistent pairwise loss preference work explore connection focus loss place function aggregation let direct dag direct impose separate distinguish avoid preference generalize disagreement describe et use number equivalently return bounded finite preference assumption let polynomial widely proposition loss convex function minimizing minimize fisher convex inconsistent pairwise show surprisingly surrogate inconsistent noise minimize preliminary loss surrogate function penalty surrogate preference aggregation risk dy surrogate convenient let edge low condition self reinforcement adjacency reverse equal definition graph global preference reasonable consistent nevertheless surrogate inconsistent low see form even noise margin encode margin preference base loss inconsistent noise weak theorem preliminary work noise set inherent development loss achievable review fisher preference typically section combine strategy procedure discount gain ndcg loss loss like score et general ndcg monotonically increase argument inspection standard ndcg criterion form measure relevance item form generalize precision assume surrogate ndcg family since permutation induce fisher ndcg recover result al surrogate preserve order integrate tend infimum satisfy zhang present example loss note differentiable contain example deep ndcg loss extend uniform set loss invariant shift dependency among order select satisfactory position let denote indicate empirically user rank product indeed value relevance independence regardless supplementary let permutation minimize preserve property fisher ndcg see variant corollary suitable condition conclude section space ndcg consist value think simply order direct acyclic transformation score place advantage finite exist give loss readily pairwise consistent surrogate fisher tractable aggregation partial datum list score make obvious procedure estimator law develop analogue population define belong query count develop surrogate expect risk converge uniformly analysis complete require knowledge surrogate develop broadly instead surrogate trading limit aggregated draw assumption loss loss might appear high datum substitution large final grant perturbation moreover choose obtain broad surrogate structure surrogate appearance query size expectation surrogate risk target suitable asymptotic procedure hereafter sequence scoring query convergence occur representative bind empirical risk argument hold proof material loss true probability query tail reasonable receive entirely day finite main concern behavior contain norm loss lipschitz continuous constant satisfied contain interior cover number assumption place give enable condition interaction aggregation datum point grow state shorthand exist sequence expect consider cover condition arbitrarily one relate growth directly sequence satisfy additionally satisfy eq function fix similarly argument n l n consist functional represent ball mean condition familiar necessity empirical loss condition
fair code regularization research topic comparison interior method group variation proximal employ publicly matlab implement coordinate descent refer implementation fista group fista instead refer observe coincide tolerance thus tune stop level range value approximation prox ip support prox concern label corrupt otherwise additive rescaling coincide variable sure working solution determine correspond variable small return nan solution build geometric order robust time ip prox ip prox c ip prox report entire regularization prox ip reason apply store il time know step fast benchmark fista familiar prox fista fista different projection project newton fista refer former latter protocol describe first correspond relevant index ten vary group space amount take thought correct loose iteration necessary fista cc c cc number iteration respectively computational fista fista comparable behavior generate coefficient subsection differently subsection problem cyclic cyclic yield guarantee comparison compute cyclic cp latter stop compute two repetition deviation computing plot cp much show thank apply overlap deal preprocesse selection present analyze gene pathway cross select fista two loop fold validation cross validation split entire fista comprise loading gene validation improve probably absence unstable overall computing overlap problem group advantage wise expand build belong scheme use optimization enable use selection high throughput acknowledgment absence thank carefully read paper appendix project need concave convex ht f usually employ k x px f di intelligence mit usa universit di mit consider sparsity induce allow overlap nonsmooth optimize version proximal nest accelerate proximity exploiting devise strategy thank pre empirically toy keyword regularization popular way arise process broad sense possibility write towards model example sparse minimize motivated explore regularize exploit pattern solution example partition group priori estimate know latter group euclidean achieve consider potentially goal union group common bioinformatics throughput datum gene mass characterize sufficient encode online database bioinformatics penalty admissible pattern arise apply expand paper direction coordinate proximal variable implementation overlap lasso generalize group loop base accelerated version name fista require deal large group exploit operator minus convex active group active active allow projection via cyclic accelerate reduce solve correspond via project newton dimensional extend conference contain presentation proof organized highlight short cast group overlap modify penalty compare extend describe scheme precisely recall proximal subsection proximity present projection active use projection norm perform performance study propose scheme run art conclude real improved preprocesse improve newton norm pp propose type group continuously differentiable simplify exposition penalty canonical identify belong adjoint lead support I e entry union subset see group group penalty study involve penalty complement complementary many algorithm study hand p overlap call allow case strategy potentially goal propose variable practical hand norm valid element belong sparsity instance empirical risk take form hold infinite dimensional kernel hilbert space readily due smoothness trivial moreover high use possible discard satisfied eq projection give proof rely allow proximity proximity conjugate homogeneous obtain minus convex derive convolution compose projection conjugate I indicator unitary ball contain convolution conjugate conjugate unitary ball rewrite indicator function basic projection restrict denote group solution number denote I g contradiction brevity thank hypothesis definition q every precisely cyclic therein propose exploit practice prove fast projection tolerance outer onto intersection cyclic projection modification cyclic ht trivial c lemma ball soft depend recall sort newton onto amount minimization problem dual advantageous much lemma theorem solve onto f lagrangian duality primal efficiently project newton report basic simplicity project newton thereby avoid overhead mention next prove recent provide proximal tolerance x I follow thank equivalent thus l equation use cp happen basic backward long guarantee unless strongly convergent exist enjoy algorithm minimum rule internal admissible projection report guarantee choice would ensure strong see proposition adapt comment subsection complete employ propose
know measurement variable estimate sense measurement question describe know question want optimal even example cm stand expectation estimator mathematical consider define mmse estimation world bias admit bias order mean random world appear aim measurement density property label example expectation mean find tell wiener filtering estimation even rough kind come specific straightforward signal simplify suppose assume gaussian absolutely information additive high frequency zero estimate low frequency gaussian hypothesis simple although asymptotically unbiased consider measurement independent unknown calculation pass certainly repeat however restrict case unlike two give explanation give let obeys freedom estimator value coefficient variable follow show distribution negligible consequence huge induce huge huge negligible divergence consequence give realization negligible huge express respect almost expectation value though measurement figure great non negligible mmse logarithmic estimator solution would course confidence obey freedom mean high confidence order determine suggest replace call fisher location pdf I transform characterize density imply constant probability density hand direct yet constant obtain biased world measurement lead linear unit u mean extend estimation mean definition log unbiased euler make origin event unbiased wrong induce try unbiased return light consideration variance ensure measurement available difference situation mmse estimator propose also construct minimum mmse mse infer product lead gaussian complex form case adequate verify efficient estimator measurement available irreducible biased direction give location ensure three estimator university e wiener filtering datum coefficient mean square error hypothesis noise cross vanish determination error filter replace lead mmse posteriori square coefficient relate biased q normalize obey monotonic give immediately measurement lead pure view calculate gray france email france
take reduction describe condition sometimes convenient plausibility space assume plausibility evaluate important I belief meaningful validity I proportion outlier hold singleton fact predictive plausibility familiar stochastically dominate set need measurable subset valid predictive contain measurable k set hold I however association statistic exist ta mild separability trivially nan case hold association hypothesis admissible plausibility generality reduce baseline I subset set follow p support consequence I notational consistency proceed plausibility evaluate justification right hand complete singleton point plausibility decompose piece plausibility general prior often focus probability value p I posterior interpret perhaps surprising give bayes view attractive approximate light argument concern value fail satisfy value explain respect posterior different prior plausibility describe situation arise mean I predictive set empty moreover sense I one literature uniformity variance goal quite plausible plot plausibility horizontal characterize plausibility keep plausibility develop p hypothesis I plausibility value light parameter clear modification I plausibility via method numerous part author recommend value odd interpretation strong suggest adjustment ever alternative familiar valuable offer connection plausibility p I I plausibility understand evidence plausibility useful I p correspondence probability I unified grateful suggestion associate two national grant dms dms dms demonstrate value plausibility nice ease use issue comes keep brief detailed description idea method find present statistic far trivial present I side albeit plausibility frequentist test goal predictive precise shall p stochastically stochastically value function binomial mass propose scheme carry binomial brevity paper show illustration plausibility function several determine plausibility case plausibility shorter similar hold hold statistic propose p plausibility inferential practical inferential plausibility notion plausibility practitioner use nan connection p trivial parameter inferential plausibility function statistic sort medium simple avoid reason frequent inconsistent people sense goal plausibility plausibility define within I build upon principle inference global mild choice statistic valid I plausibility understood plausibility highly plausibility fit way practitioner interpret p plausibility calculation avoid logic use interpretation summary prove beneficial organized notation p incorrect interpretation plausibility corresponding I bayes problem trivial two example binomial conclude remark observable sampling problem hypothesis mathematically subset write nan alternative whether thing occur relative chance occur false put suggest simplify tx intuitively compare outli conclude conversely sense acceptable reality adopt arguably interpretation go something observable actually sort program condition potentially incorrectly value support interval fairly medical effort confidence interval value free difficulty bayes factor simple valuable contribution inferential I exact prior assertion unknown accomplish explicit auxiliary random predict obtain random I exist confidence reference prior develop sort parameter invert target uncertainty probabilistic inferential assertion special
mathematical text section subsection paragraph eq number eps abstract
look opt gradient warm describe analysis algorithm return terminate solution always direction whenever check easy opt satisfied great therefore similar control practical I implement homotopy homotopy iv explicitly exploit linearity path verify modification code inner cm dataset full approximate path stop ill check experimentally verify correctness approximate duality gap check present complexity interestingly look example path full exhibit complexity also significantly path homotopy worst formally optimistic require number path approximate homotopy acknowledgment nsf grant dms dms regularization path popular optimistic always obtain every path duality analysis practical approximate either observation small problem induce penalty prove quality purpose formulation require value induce control regularization sequel follow classical optimality uniqueness rank subgradient optimality say otherwise uniqueness necessarily w j also reduce w hessian reduce strictly rank formally full uniqueness w sparsity consequence linear patterns path moreover limit easy always present homotopy initialization trivial compute path x record cm maintain optimality correct reasonable make work real become ill may typically truncate assume algorithm single jj enter even problematic view segment homotopy present issue show exponential see state segment therefore w w convex rank remark w optimality increase obtain contradiction proposition build regularization lasso path extra increase multiplicative adversarial design adversarial let span small bound two zero possible optimality inequality verify last last definition sparsity pattern characterization last mainly continuous continuity argument appear segment characterize easily show continuity argument path lasso linear lead segment recursively segment matlab numerical us small segment example quickly lead close publicly optimistic regularization path another refined path quite strong analysis tailor gap lagrangian minimize primal formulation function dual call optimality say solution build approximate perturbation condition condition satisfy dual duality cm w q satisfied path simple opt dual piecewise control machine upper contrast cm obtain guarantee
electrical row think additional v simply give give unit precise unit flow note identical message close message approximate ji ji lemma induction height message easy compare achieve ij I ij iv trace submatrix apply transformation ji step take precision ij ij message height p li il il l optimal argument let transformation ji ij ij induction correspond message satisfy ki ki satisfy induction hypothesis ki li li ji ji fact term complete approximate message ideal state set subtree ji obtain q ji ji induction height base leaf consider leave define subtree simply leaf happen notice ji q r constraint ki ik li il ij ij ik ki kt li il hypothesis height satisfy compare application claim compare eq ji ji along addition operation recurrence go message induction remain ji finally account trace eq introduce round tree unbalanced simple exponentially always balanced decomposition allow polynomial balanced theorem exist decomposition requirement construction diameter decomposition decomposition height decomposition element precision dynamic programming tree follow run ij term require tree produce transformation net leaf extract mi I mi zero matrix lemma produce pass support net passing claim construction edge scheme mrfs width semidefinite element mrfs whose arbitrary instead net net polynomially consequence algorithm mrfs precision impose net wise round precision description describe round mrfs mrf lemma corollary theorem pair observe usual w orient vertex vertex sequence child w I iw lx j tw j scale w positive diagonal dominant new connect obtain submatrix size row index let observe describe run ideal message pass construction net bound give integer support reason way factor ensure rank message ideal small iff round full q observation well eigenvalue within factor semidefinite eq follow imply next net rounding pass decomposition svd see round orthogonal net ingredient net set orthogonal obtain rotation second ingredient define ready net precision v v diagonal set clearly small precision eigenvalue message pass fine net next formally input orthogonal transformation satisfy p orthogonal statement round near e v computation u v p require unit sketch proof triangle u z triangle side schmidt orthogonal column apply schmidt round gram schmidt onto span along give gram schmidt apply triangle give net round message big ideal message error set observation approximate message big approximate message ji ij ij ij ij program subtree almost lemma analysis element round section due svd round error unchanged compare error wise rounding exponentially height whereas svd scale linearly hence provide brief omit depth identical message height ki ik li il ij ij ji il il ij ki li respectively successively application increase rounding inequality analysis induction term equation ik ki il remain obtain scale message message main mrf input mean decomposition requirement page use give net respectively empty extract approximate mi I mi height time construct pass per pass thm observation exercise remark corollary thm thm conjecture example random field observe total variable np supervise greedy graph joint markov gaussian mrfs give budget minimize particularly provable error main mrfs tree widely inference see class mrfs among computer state describe relate research denote denote row non occur matrix symmetric chapter order positive semidefinite positive matrix use eigenvalue rank large let index subscript clear fully scheme mrf matrix rank matrix markov clique assume mrf know w origin center gaussian e minimize I expect error equal conditional actual light express n gaussian decreasing follow state given find version exist small gaussian free chapter mrf fix q set density select laplacian define precision treat whose precision thought variable main finding np hard give greedy graph base message difficult formulate mrfs extend width require theorem programming mrfs graph expect error early process krige see unobserve krige variance solve krige I krige also extensive literature explore criterion entropy area quite similar problem e g overview objective aim provable goal average unobserved variable equivalent trace submatrix matrix experimental optimality minimize trace conditional experiment objective differ want prediction unobserve require unobserved budget formulate expression semidefinite nystr om nystr om approximation process first subset width mrfs exploit precision matrix mrfs strategy om greedy heuristic see provable bound moreover rather error tree mrfs motivation study exact extensive graphical mrfs chapter mrfs involve inversion subset hard show field hard g maximum framework second free mrf thought analog ise computer e discrete mrf uncertain adaptively rather ising prove solution feasible regression within special hardness regular graph nod regular entry q expression step harmonic iff iff budget hard regular graph node np bound set iff question average error since I e notion rather return b fa fs cover compute know minimization problem result know electrical see consider elimination decomposition survey decomposition tree width small yield assume width cluster give decomposition one trick like add empty exactly presentation edge split consist eq set separate variance depend variable mrfs happen joint precision intuitively induce prior markov property allow use message pass density matrix moreover v elimination cholesky row ji lower desire upper eigenvalue elimination cholesky take representation two transformation marginal transform marginal precision q transformation precision think operation first take principal inverse integrating intuition precise application small later discuss message follow fact principal generally matrix decrease invert submatrix invert submatrix large decrease submatrix small set lemma belong subtree note belong subtree define abuse actual need notation subtree subtree subtree q error go algorithm version message message produce approximate message split ji budget allocation mrf come gaussian mrf allow justified message proceed round round leaf receive exclude message otherwise express later notion maximum message message send happen leaf height argument il must observation describe compose cluster consider receive argument compose follow infimum easy describe
practically number psd section effort decomposition mainly influence calculation considerable reduction computational achieve psd paper di wind term simulate variate reasoning eq tm q approximate latter read store novel digital field develop spectral fractional transfer white time valid alternative case psd di extend use spectral two consist perform spectral decomposition psd eigenvectors multidimensional uncorrelated white unitary wave engineering design di calculus journal physics conference series fractional calculus probabilistic characterization pp g di use calculus di representation density fractional pp di r exact stationary colored fractional pp ed pp di digital wind field velocity pp di digital generation wind pp perspective record pp mathematic science multidimensional pp york modelling pp p wave moment di di ed universit di universit digital wind velocity field fractional calculus taylor digital velocity field fractional show construct coefficient fractional moment target superposition fractional white process simulation taylor practical digital wind velocity field need wind vast refer overview different attention simulation wind velocity field digital wind I superposition digital filtering commonly ar al et method wind field contrast filter equation fractional appear wind method slowly become engineering brevity sake fractional refer topic publish novel fractional differential extend investigate digital assign wind wind velocity psd organize multivariate wind velocity due assumption second characterize velocity field specification moment variate along remark straightforward theoretical concept wind reader comprehensive reference wind velocity think around stationary value locate velocity represent vector spectral psd read q spectrum velocity spectrum calculate flat neutral consider cite paper psd velocity longitudinal coefficient eq coefficient determine accordingly next section attention psd construct need reference sake wind field let matrix element hold density eq di di wind form normal increment bar complex conjugate kronecker decompose coherent elementary q eigenvalue form point di frequency pay consider combined start variate next extended gaussian stationary variate assume wind assign psd propose follow firstly psd term fractional spectral output psd show fractional moment linear transfer filter taylor contexts di al generalize taylor di variate topic z store finite use process solid ground simulate fractional stationary psd suffice ideal system white noise process zero power characterize impulse response namely transfer relation read characterized calculate fractional impulse transfer notation last integral eq transform fractional operator relevant psd reader must carry truncation discretization discretized counterpart superposition fractional integral white continue example simulate digital mean calculate form reasonable achieve function good firstly calculate return implementation require step integral gaussian white firstly fractional integral gr discrete fractional approximated become true also time partition amplitude realization gaussian gr approach efficiently transform p calculate decrease many suffice word calculate realization follow show target correlation implement see mean w strength component output impulse integral output linearity mean characterization linear output course linearity system next represent fractional transfer turn sum spectral assign fractional converge analogy eqs impulse transfer absolute eqs approximation truncation interval integral eqs
lead weight logistic g specification posterior generic coincide let independent model yield maximum likelihood result let fully data logarithm detail hypothesis indeed local coincide remark previous matter marginal density parameter likelihood analyze mix weight log nest fact consider depend empirical simulated equal proposition reduce step log function require ng calculate variable depend density respectively step likelihood maximization equivalent generalize contribute log stationary derivative algebra maximization reweighte ml maximization estimate density give closed g rest specify value whether sufficiently acceleration l kl log analysis algorithm stop introduce paper different classified type accord datum arise know numerical study consider criterion ml estimate rand rand rand comparison pairwise perfect make small difficult interpret ari chance possibility classify observation ari rand partitions original ari measure goodness fit linear detail goodness special pearson remove impact sample square goodness fit glm difference achievable dispersion form binomial call generalized q form divide dispersion respectively illustrate artificial aim analysis herein write environment package state nest show numerical concern accord poisson regression group group case three gaussian conditional accord random fit follow discrepancy q respectively poisson reveal bivariate group generation binomial gaussian estimate fit model scatter model display different joint binomial misclassification error model gaussian approach binomial mr group fit binomial binomial bivariate randomly binomial give ht give estimate display scatter list misclassification datum modeling like binomial outperform possible see ari poisson artificial size respectively poisson specify round fit poisson scatter plot ht fit ari obtain model poisson underlie ht ari datum number price run unconstrained poisson group large unconstrained result poisson outperform result value directly two class constrain poisson attain measure goodness substantially parameter confirm prove coefficient cccc poisson library consist application million application associate bic unconstrained poisson outperform result respectively outperform result attain value goodness group ht ht cccc concern length stay related display moreover b base plot scatter ari poisson clearly perfect separation contrary able evident poisson carry patient poisson outperform yield perfect separation class result ht fit study effectiveness comparison mixture regression result theoretical investigate reveal poisson regardless strongly outperform give gave outperform outperform outperform give comparable slightly outperform give comparable outperform show covariate figure highlight performance clear highlight well covariate range confirm obtain contrary covariate figure paper generalize categorical depend numerical covariate come mixture show mixture linear artificial comparison cite paper distribution extension student student estimate e currently extension issue concern first behaviour em suffer confirm g initial marginal distribution maxima finally come discrete specifie coincide get recognize generic specify sufficient complete since posterior reduce neither attain proof sufficient logarithm algebra estimate like prop prop prop remark di di via weighted modeling come heterogeneous family model exponential family link suitable mixture nest
last independence poisson process markov ergodic interested steady equilibrium probability neuron equilibrium satisfy supplementary neuron word neuron ergodic product marginal neuron use unknown variable external arrival neuron weight appropriate spike combinatorial system recurrent use application characteristic connection circuit connection cyclic cause history store internal state recurrent tool forecasting efficient gain form neuron receive recurrent linear produce output connection reservoir reservoir neurons reservoir reservoir circuit possibly reservoir expansion separability approach observation assumption linear often sufficient task know model input connection reservoir possibly integrate neuron activation layer allow connection input concentrate reservoir connection reservoir row contain update output vertical concatenation dynamical input produce series evolve accord spectral radius role reservoir two suggest intrinsic backpropagation etc simultaneously temporal weight compute regression ridge widely literature use research empirical reservoir respectively depend start performance preprocesse rescale offline ridge adjust autoregressive bt bt bt st traffic service european minute training neuron map configuration suggest neural topology trait traffic united education research internet traffic triple compose study validation sample ci last use reservoir zero use radius reservoir good performance improve neuron reservoir use criterion reservoir unit version present slight empty european significant performance accuracy obtain version largely reservoir important reservoir reservoir linearity necessity illustrate estimation reservoir interval begin day set figure instance type reservoir neural successfully temporal learn good relation reservoir performance significant property simplicity reservoir counter software aspect example reservoir reservoir decade computational name recurrent process recurrence neural circuit occur rapidly success neural drawback propose come design consist reservoir recurrent position example performance largely tool powerful complicated engineering many design machine come random mathematical network literature actually interpretation mathematical neuron queue spike among space internal neuron fire spike potential neuron increase supervise descent easy hardware implementation introduction nevertheless suffer feedforward topology refer rnn avoid use rnn network concern circuit topology recognize powerful machine traditional limitation come difficulty implement main drawback guarantee algorithmic sometimes train require learn relatively reservoir overcome drawback topology ten neural adapt connection output approach achieve begin section iii discuss also idea inspire experimental end conclusion discussion regard network propose concept neural context node neuron neuron receive one negative neuron neuron receive spike neuron receive spike decrease equal far strictly
whose zero formalize produce function respective source common expert similarity score strong recall take unless state shall empirically goal source attribute score interpret entity match magnitude achieve depict typical merge thresholded produce feature scoring processing matching produce clean cast release employ pair entity hash movie entity stop rare matching scoring runtime absolute difference cast entity goal formalize amount desire confidence recent community task complexity perform present approach transfer problem inter overview intuition transfer obtain learn set task net wish learn statistically separate structure share subspace depict task jointly take model difficult achieve transfer element transfer learn appropriately reflect task euclidean ball arise set source pair handle source approach function source treat task would source accurately score pair derive er produce appropriate allow fit available degree freedom amount consideration specify er problem model perform serve purpose store large statistical feature interaction capture act weight place score split half finally pair multi state baseline section allow result separate pair lead quadratic training could motivate transfer pair effectively example successfully use either learn separately great labeling expense represent extreme form none limited information characteristic adapt introduce capture movie generally weight account induce also program pairwise develop technique experimental proceeding recall index furthermore denote pair model solve convex estimate take moment encourage feature vector match pairwise function build alternative option term machine extensively encourage closely exist avoid overfitte weight due desirable encourage procedure act non parameter encourage capture part source represent assumption appear decomposition act sparsity allow away nominal source avoid overfitte tune complexity control select extend result popular number risk functional work practically linear learner svms proceed derive result derive tucker condition condition eq half difference subsequent choice remainder simple require characteristic common characteristic across solver source dimensionality must rely solve minimization proceed optimize algorithm estimate current soft whose maximum intuitively vector direction decrease case term truncation operation second solve problem dimension gradient smooth objective geometrically next art present gain world synthetic dataset scale source pairwise pool argue machine er assume pair impose example expensive consider share transfer model hence order example heterogeneous source denote function learner arbitrary vary source source identity pair encode flexibility individually implicitly take machine svm widely flexible mapping also kind success adaptively balance label svms regard er adapt however typically moreover may learn classification svm build general art implementation fair comparison experiment standard computational package library implement employ fold validation grid netflix investigate algorithm match vary produce potential label evaluate precision recall q recall positive negative employ six movie internet movie movie title alternate release attribute perform string stop cast absolute runtime year human source pair experimental movie learn scoring source precision order specific available pair movie conditioning subtract dividing place feature raw true attribute entity interval entity source level score attribute entity incorrect score inequality property er synthetic stress test approach pair movie compare pr deep understanding transfer method require labeling achieve four apply unobserved curve vary three pr curve figure summarize nine fix match vertical expense achieve total little training inferior characteristic behave manner available datum become adaptively dominate baseline recall trace endow structure significantly shift pr endow per source per trend observe figure apparent dominate recall row two assign another apparent width band poor movie focus learner vary transfer experiment take deep look source fine generating explore take individual difference flexible though freedom learn increment source increase intractable labeling compare observe far improve perform quadratic note require significant unable adapt increase method able example source quickly achieve test baseline method fast inferior test increase example prior vary consider er vary first highlight labeling barrier er multiple source transfer paradigm transfer numerous explore serious varying traditionally researcher pool vary level propose matching require evidence similar effect heterogeneous matching motivation paper via real weight grain transfer source simply learner comparison balance transfer task investigate selection heterogeneous acknowledge tailor experimental google quality digital library approach label combine thorough evaluation human effectiveness er work
clearly tradeoff z proposition lemma n plan online planning focus exploratory planning assess regret algorithms plan good effort regret introduce new carlo search exponential scheme part dedicated exploratory objective superior improve exponential bind exhibit attractive process model uncertainty agent action reward problem assume agent flexibility generative expressive type simulate execution mdp current act maximize accumulate reward use accumulate predefine handle mdps exponential lead researcher mdps plan mdp focus consist planning schedule external follow recommend action state action environment repeat action recommend step go assess sub closely capture take optimal policy follow begin recent mdps plan mdps sparse ng offer action discount time exponential discount sparse offer guarantee really setup mdps day mdps successfully partially adversarial relative planning ahead comes expect plan mdps barrier contribution carlo search guarantee choice space dedicate compete exploratory objective objective grow current reduction suitable reinforcement mdp benchmark art variant result factor upper attractive carlo search planning depict construction root issue store node phase recommend compatibility prior nod mdps distinguish subset action c ss actions monte concrete ground later describe core illustrate tree end state counter update action propose cumulative multi armed bandit mab select still sample transition sample popular appear expand dag expand node expand tree counter rs never adopt rule time select protocol root cumulative regret incur explore forecaster possible logarithmic cumulative seem planning reward bandit minimize cumulative simple objective polynomial rate reduction reward simple attempt recently plan general optimize online attempt rather barrier bad case reduction reduction regret plan achievable introduce state guarantee zero achieve exponential multi armed bandit stress bind cumulative minimization mab exploratory recommendation sampling mab action exponential reduction exploration exploitation situation mdps mdp answer exploratory mdps special intuition mdps complicated go act simple induce applicable optimal exception provide action pseudo need information alternative root objective identify optimal actual exploratory compete actually priori devoted confidence improve act reinforcement plan bound hyper exponential separate aforementioned exploratory investigation state mdp outcome horizon get separation exploratory concern mab mdps arm flat act state flat pair action specify act mdp step arm action encounter arm mab flat policy loop update obtain therefore arm mab recall mab actually compound policy policy outcome policy consistent note sampling systematically update stochastic uniform iteration arm sample iteration consistent arm hand q turn bound trivial h note well hyper regret rate transition help require reasoning concern planning introduce refer allow large extent family vary switch update node pair associate initialize value initialize mdp sample end either state reach phase policy accord policy expand update accord action follows call generation exploratory planning aim improve current candidate specific estimation recommendation tailor switching exploration choose sample accumulate accumulate reward depend subsequent overcome novel modification seminal hoeffding select exponent refer reader discussion mdp go applicable induce exploration along action reward horizon applicable lemma horizon hoeffde around serve induction horizon go action applicable online planning accumulate estimate select increase accordingly differently stage decrease put could course weight reasonable rate reduction regret perspective empirical differ point addition action reward responsible accord mdp reward similarly provide explicit expression constant recursive bring potential benefit horizon step line fact hoeffding inequality modify capture hoeffde sample bind q exponent multiplied pose tradeoff decreasing reduce term leaf go tend grow perspective formal guarantee appeal nevertheless try optimize optimize lead obviously would rough optimization valuable theory experience similar bias mdp mc plan original evaluation domain environment wind reach location neighbor move duration direction move diagonal take straight move wind wind fast tail wind assume mdp space action position wind mdp length terminal size path two consider area recommendation orient end state term problem grid branching depth different grid size show base modification replace policy motivation behind show slight modification updating switch identical four algorithm within suggest recent work parameter result respective original assess choose consistently outperform margin grid large relatively short planning allow attractive guarantee effective likewise practical parameter game person sum game go winner decide reward terminal assign terminal move tree move max act depend payoff aim possible objective min many player payoff branch tree domain convergence configuration average appear encourage quickly carlo online planning mdps guarantee interest guarantee goal remain future discount mdps infinite employ use account additive action value horizon infinite horizon mdps set inspection unlikely sophisticated sample another speed action good action towards mdps hope focus root recommendation planning sampling employ scheme previous difference play formal unclear reduction acknowledgement supported carry microsoft center partially support air scientific fa rely correctness well claim diagram depict overall central claim depict rectangular concentration binomial bernoulli equivalent choose state via action sequence apply action exploration end go function accord q pair consecutive iteration consecutive finish exploration action put consecutive exploration phase apply phase iteration finish apply
read tree variable binomial distribution derive parameterized tree range slight preference tree preference detail hyperparameter hyperparameter affect affect hyperparameter range range read nine per read binomial frequency figure structure likelihood although prefer vary pearson range setting range structure range place choice setting although read baseline slightly x due merging whose nearby distinguished note infer pearson recover read frequency consistent goal establish integrate order remove bias correctly sample simulated read frequency reliably recover representative good high dataset run combination simulation single report sequencing sequencing suggest single cell confirm provide truth show major single cell confirm frequency copy read count plot order partial via high parent sort parent ground clarity represent cluster place frequency clustering interpret plot indicate reference figure include single plot representation indicate largely consistent indeed posterior file reference list major estimate single appearance tree switch frequency two table variant allele count allele ci ci b ci ci ci ci ci ci ci ci systematic bias note support tumor cell agree e early root secondary uncertainty rest tree table branching lot allele suggest k cm x allele read count read depth allele ci ci ci ci b infer show figure stop five posterior relatively probability probability probability file ground legend tree largely perfectly reconstruct cell explain systematic deep sequencing nonetheless evolution sequence frequency quantify different point span patient identify sequence tumor label frequency reconstruct evolutionary cancer semi manual automatically allele allele patient reconstruct describe sample cancer share evolutionary history frequency read likely normal case appear sample tumor little tumor good tree agreement frequency figure tumor structure structure likelihood allele tumor sample indicate cluster dot plot form structure multiple frequency reduce evolutionary tumor reconstruct produce semi manual call tree visualization plot represent uniquely reconstruct profile tumor enforce structural population frequency able relationship tumor case detect tumor sample highly constrain drive semi define frequency estimate sequencing put read enough relax hard constraint large possibly phenotype clear ground gold cell sequence heterogeneous cell ensure potential magnitude manner minima represent determine reader extend recent whole genome copy attempt profile genome sequence site use extend full population change evaluate area innovation modeling allele replace binomial allow variability observe allele object follow place weight parameter draw parameterized dirichlet datum fix component stick provide point center e dp stick breaking view break stick location stick break large let stick break stick break mixture component parameter draw result eq every object take stick stick produce extended stick rooted agglomerative internal exploit sequence positive use index tree let denote zero indicate therefore denote child node stick break allocate respectively whereas determine sequence construction tree note concentration parameter model throughput sequence genetic population variant allele copy let allele denote whose probability population dirichlet variant position cell fraction cell population position count posterior appear inside summation compound single draw rewrite gamma dirichlet count useful population nonparametric prior avoid group evolutionary root tree stick breaking count tree structured process count stick indicate hyperparameter indicate latent include frequency population distribution root node evolutionary section crucial stick break infer chain monte gibbs involve subsampling sampling stick length stick break hyperparameter contribution frequency way proceed evolutionary subsampling procedure sampling frequency population node equal enforce constraint satisfy result use denote auxiliary singleton root node node denote parent population child node population appear parent tree current tree weight sample h root create queue q frequency weight non leaf hasting auxiliary frequency asymmetric dirichlet proposal markov chain converge sampling iteration posterior root initialize density extension cf multiple allow tree structure read match variant allele position let fraction population tumor probabilistic share stick break hyperparameter multiple frequency evolutionary constraint separately tumor move base tumor plot summarize visualize tree cluster direct fraction mcmc place aggregate information mcmc summary possibly quite uncertainty assignment input co difference number number assign cluster cluster cluster assign burn fix metropolis hasting dirichlet mcmc sampler initialization pick auto package sampler complete trace autocorrelation plot burn period figure file dataset input experiment observation additional file acknowledgement science award publication software publish manual high sequence quantification frequency single nucleotide variant heterogeneous tumor evolutionary population tumor reconstruct however reconstruction available condition reconstruction describe evolutionary history uniquely frequency tumor nonparametric prior major joint tree identify frequency high generate multiple consistent frequency tumor order real comprise tumor demonstrate branching inference agreement ground apply frequency mutation small partial order supplementary material background disease often characteristic genetic substantial effort genetic identify tumor contain reconstruct history tumor population quantify whole sequence tumor deep sequence tumor associate nucleotide frequency reconstruct evolutionary history multiple tumor sequence frequency methodology evolutionary infinite site tumor frequency reconstruct major tumor automatically perform reconstruction topological triplet consistent great branching contain population must sum
generative pd px conditional state proceed usual sampling indirect possible use auxiliary variable similar sampler contribution theory backward posterior treat duration single notation standard model message filter backward condition message transition modelling assumption unfortunately duration support lie backward fail duration code length propose behave like truncation define duration well sample result incorrectly admit entire sequence infinite hide markov truncation countable message filter sampler sampler state duration distribution space slightly modify application sequence generalise markov transition state duration speech speech synthesis suitable dirichlet hdp hmm allow address transition experiment three state distribution rate pd uninformative duration distribution inverse iw burn learn observation transition visit comparison per would forward computationally impractical second except separate reveal duration show indicate sampler hmm sequences truncation work explicit hmm available letter cardinality nonparametric variant develop black box hmms consist indicator direct duration duration truncation require perform hidden hmms tool variant formulation duration time specification duration priori geometrically hide semi allow explicit parameterization forward backward though short hard code run representation tractable lie within outside incorrect duration reflect allow pre duration hard duration exact incorrect develop hmm key hmms countable state duration countable box specifically estimate demonstrating technique finite duration duration review inference capture state duration observation consist distribution observation duration observation time segment markov chain
closed restriction resolution restriction inclusion axiom discuss straightforward fix deriving appear derive rule cut cut eliminate clause cut resolution restriction clause indicate dag resolution paragraph dag component original derivation node correspond node restriction formula subgraph long resolution use first explore associate clause add allow clause restriction instead simply require utilize clause appear demand utilize clause appear never clause clause derivation equivalent add restriction space remain restriction closed restriction close resolution assignment correspond construct derive clause derive subsequence establish carry construct subsequence step immediate add originally derive likewise derivation construction include therefore thus resolution proof appear elsewhere history alg completeness formula return otherwise either variable naturally convert return correspond limited kb partial draw process assignment valid note previously give find proof theorem proof size bind find reject accept clause precisely refer appear width naturally width proof exclude clause originally resolution generalize resolution e present general straightforward omit iff resolution run resolution resolution restriction close resolution assignment resolution proof take node new dag note step derivation derive dag latter therefore width obtain implicit learning resolution kb target either exist partial else valid respect run assignment length resolution naturally close proof utilize learn section search resolution restriction background briefly proof resolution formula clause formula introduce detail recall conjunction derive essentially formula infer given elimination simplicity derive elimination power include restriction axiom naturally hypothesis otherwise four since otherwise conjunction appear likewise elimination must evaluate otherwise elimination introduction new one introduction derived derive introduction original suppose apply first first rule apply possesse bound alg say proof width reject efficient iff derivation main conversely width run width take derivation formula input tuple derivation via check checking appear width first checking formula common derivation collect elimination width width length formula conjunction formula linear checking width formula note syntactic close let appear derivation formula simplify decision implicit kb target suppose either n assignment calculus simulate prove heuristic could computer algebra gr basis fulfil due heuristic practice nevertheless represent potentially furthermore calculus arbitrary present hypothesis calculus enable solution original correspondence boolean serve polynomial boolean formula product degree variable axiom infer multiplication derive derivation encode modify allow axiom course step calculus combination also note generality restrict great boolean axiom could multiply axiom formula original replace repeat trick time trick appear rest refer formula nothing lose look ahead focus restriction calculus formula system polynomial restriction restriction representation since vector express conjunction sc verify restriction polynomial arise apply effect equation evaluate kind nevertheless system equation np np interested solution calculus encode encoding undesirable recall correspondence require calculus simulate polynomial calculus extension polynomial calculus resolution introduce related axiom cut capture coefficient multiplication purpose formula assign whenever calculus calculus step restriction restriction axiom easily assign otherwise latter boolean axiom calculus axiom axiom simplify axiom inclusion axiom inference preserve say multiplication axiom clause consider formula representation formula degree polynomial fix multilinear ordering nonzero coefficient polynomial calculus appear used proof resolution simulate calculus calculus degree calculus construct lie f polynomial derivation otherwise bp decision pc theorem solve calculus resp run calculus work reader gr bad run paper case alg calculus need restriction system restriction easily polynomial calculus proof degree note restriction calculus valid calculus resp recall polynomial calculus restriction degree decrease restriction learn calculus list either polynomial least calculus resp else q integer interested cutting integer satisfied integer fractional current cutting cast et system cut natural furthermore cut formula encode cut plane syntactic analogue enable although cut connection work cut integer attention integer program boolean value axiom allow ix b multiply integer key division positive integer common crucially due integer round cut inequality name cut technical much note trivially derive cut allow ix ix n find convenient restriction evaluation intuitively end broken stage assignment formula induction else either induction evaluate false case immediate formula note construction matter formula must cut restriction close axiom restriction hypothesis likewise axiom assign simplify axiom assertion axiom remain rule evaluate derive true sum evaluate therefore partial addition simple ix ix division develop syntactic cutting decision main use appear formula sum naturally every appear also restriction cut norm cut appear remark resolution cut sparse bound intuitively cut norm easily contribute assume cutting generalize restriction know special case cut restriction size nevertheless cut sparse cut plane restriction let cut plane partial assignment proof restrict cut encoding bound sparse norm appear appear assume cut else reject accept analogue decision alg programming sparse bound cutting cutting solve cut run constant sparse algorithm width guarantee conversely step note could repeat axiom need otherwise table eventually formula remain integer nonzero choice cut plane formula consider formula formula check checking formula take arithmetic claim apply implicit cut sparse cutting far list cutting assignment process probability bounded cut derivation else run unit operation assignment introduction primarily motivate weak issue concern form problem include axiom informally normally nothing take aside problem assertion impossible reason fail change world axiom etc fully toy domain show algorithm plan early explicitly modal criterion planning planning approach solve system reasonable approximation may fail likewise fail circumstance discrete time markovian might reasonably expect axiom valid small course solution probabilistic plan rather valid pac semantics rule pac plan broad involve reasoning semantic limited decision classical semantic concrete suggestion result perform resolution proof satisfy work al certain choice resolution modern largely et explore work speak algorithm variable satisfy decision add search different variable rule consequence notably propagation false except improve semantic one might learn learn extend partial resolution depend ask actually decision actually arbitrary e number iteration feasible decision match nevertheless obvious tuning illustrate might world modify serve solver highly optimize easier sensible exist solver implementation simply assignment loop remain count bind validity crucially clause common solver expect sharing solver black box formula assignment another sophisticated reasoning pac semantics involve pursuit reasoning might possibility hide since mask independently underlie enable useful extend produce formula satisfied find resolution perfectly although goal one consider et learn even acknowledgement heavily influence appendix show implicit efficient broadly speak axiom formula axiom satisfy say assignment probability valid hypothesis validity proof algorithm whenever argument axiom collection distribution whether polynomial efficient serve formally implicit learn thm partial axiom system query hypothese assignment formula polynomial pair hold assignment distinguish case polynomial know likewise guarantee recognize assignment point precisely behavior pac simulate must decide first hold part checking time decide formula distinguish case usual axiom since check coincide formula formula derivation often optimal two space root induction must worth attain axiom formula root formula continue subtree label carry space subtree utilize space derivation derive clause label root much derivation subtree derivation order derivation root empty resolution step empty still assume derivation either include contain clause occur subtree overall derivation space subtree remain configuration derivation subtree clause clause appear actually et refer number correspond parameter knowledge old although heart recurrence proof exist convert normal resolution dag cut direct clause label contain along variable source cut space normal general initial eliminate introduce along encounter cut past towards finally replace lead proof cut node set cut internal label clause source cut step leave third eliminate subtree root close source replace subtree subtree label subtree mention still child subtree great subtree increase describe theorem find exist return none derivation clause step find path root choose labeling clause remain recursive call describe recurrence verify induction note check bound find proceed child subtree tree induction hypothesis carry derive root mit mit edu theorem lemma consider answer formula base represent partially weak semantic semantic version space calculus explicitly formula crucially explicit knowledge example implicitly consequence distribution essentially agnostic pac logical reasoning hand background axiom may perform logical collection semantic typical know unless visit seem infer explicit knowledge whether broadly pac attribute sentence introduce describe pac query background axiom collection partial cope target roughly efficiently use validity preserve introduction cope rule quite impose class assumption partial assignment restriction impose applicability remarkable approach discover completely would agnostic al et note agnostic distribution pac pac also theory hypothesis barrier less perfect seem useful perspective ai problem typically frame axiom nothing axiom useful therefore desirable learn utilize furth fundamental application approach propose know graphical logic programming inductive programming address kind task distinction datum demand require simply answer naturally work framework task efficiently much svms boost predict reason reasoning couple task state could distinction broadly speak prove aspect incorporate cope handle clause mention able probability agnostic inductive inherently kind valid intuitively quality pac semantics quality pac learn logic precisely target attribute attribute formula pac learn every binding agree valid may particular formula threshold partial guarantee evaluate infer conjunction sequence valid polynomially turn property semantic special sound sound pac semantics sense degree validity merely note without derivation al actually return detail mention sake interested syntactic restriction restriction wish contrast rule syntactic restriction follow sense restriction close say formula hypothese assignment phenomenon restrictive impose artificial difficulty source integrate extract answering
algorithm develop section integrate elastic net induce symmetric vision box fall box constrain adopt surprising box non absolute keep instead problem box easy prox bind dual thank closed function check cut consistent obtain piecewise high interested preserving learn manifold aim neighbor minimize distance nearest neighbor low preserve manifold regularize tradeoff regularize convex solve h adopt optimize objective equivalently eq closed piecewise solution trivial overcome initialize wherein minimum write presentation part second totally unique slope convenience slope slope piecewise strongly convex unique point change suppose wherein proof update one nesterov since nmf explicitly nmf incorporate inherently structured item belong pattern feature greatly improve sparse representation successfully e group objective follow index group sparsity usually wherein group modify smoothing g gs optimize index group overlap eq gs convex slightly project project onto define sign therefore solve project ball projection complete time nesterov adopt simultaneously eq optimizing neither slightly eq solved solve sequence wherein counter construct optimize approximation decrease efficiently solve fortunately answer rewrite proximity dual problem back problem normalization equivalent solve construct optimize combination historical point smooth proximity efficiently replace sentence modify leave proof work due limit converge multi effectiveness negative sparse group nonzero select introduce elastic effect elastic coefficient select correlate take sparse part group negative term extension induce euclidean smooth nesterov optimize solve update net induce negative similarly h convex wherein replace adopt trade role part spectral cluster laplacian laplacian specify graph data symmetric inspire cluster q neither convex smooth involve fortunately equivalently rewrite solved successively update variable discuss normalize cut ht section compare residual nesterov optimize synthetic effectiveness compare conduct illumination video sequence challenge face image compare robust analysis include stop consecutive stopping first decrease dramatically ht iteration nesterov practical compare synthetic study conduct start identical dense high obtain ten versus round cpu obtain cost fewer rapidly confirm scalable also real world I face face pixel seven matrix value cpu second observation summary suggest relatively nesterov capacity challenge face dataset traditional algorithm include seven individual eliminate effectiveness deviation train jx face obtain test project wherein pseudo inverse neighbor test efficient implement bring element aforementione suggest method representation sample learn cosine nmf percentage dx x jx five outlier laplace conduct clean contaminate evaluate robustness setting encourage ht extended image pose illumination manually align image contaminate type pca use baseline outperform robust cause study robustness figure outperform contaminate laplace heavy tailed show noise contaminate successfully effectiveness base different algorithm reduce successfully laplace contaminate interesting noise confirm observation heavy tailed extent ht face database contain individual illumination position simply illumination give accuracy deviation contaminated figure almost contaminate image mainly contaminate illumination successfully low reason contaminate poisson ht select dimensionality observation laplace noise laplace noise image contaminate e face image obtain capacity compare dataset wherein reconstruct reduce show face contaminate laplace successfully conduct simple type experiment evaluate representation conduct construct wherein evaluate ac mi selection repeat report ac figure even contaminate perform presence serious illumination outlier cut successfully image eigen image second eigenvector recursively partition pixel equivalent e g row correspond berkeley wherein edge node product feature node parameter partition laplacian wherein normalize similarity label segment reduce dimensionality segment figure ht correspond successfully separate object mostly g low find three successfully find last help mean segment perform simply set aim segmentation deduce illumination sparse capability compare video surveillance background challenge foreground video reveal background foreground evaluate capability surveillance video whose select frame low background foreground video figure successfully separate detail recognition illumination reduce dataset illumination estimate rank face capability illumination removal conduct extended dataset matrix keep consistent illumination removal result four remove illumination confirm obtain figure computer set inherently many type gradient view vision retrieval annotation beneficial learn multiple factorize structured sparsity learn space view share compose view learn dictionary group contrast respectively achieve great robust heavy noise sift multi group sparsity wherein keep ht row compare image collect object feature construct half image remain image set latent experiment aim compare construct classifier mean cc map reduce versus present basis second zero view different reduce learn pattern different view imply learn private view view table map outperform present heavy tailed laplacian low much account low recover comparable robust component illumination suited negativity keep objective neither propose include nesterov successively update fast make propose nesterov approximation nesterov smoothing improve develop elastic nesterov optimize inspire develop experimental several show comparable taylor taylor nmf approximate two extension kullback leibler present nmf tail decomposition robust estimate part thus contaminate extend various develop box constrain elastic induce symmetric major fast extension residual nesterov approximate row row develop iteratively solution flexible extension extension nesterov smoothing recursively set smoothing iteratively extension conduct experiment synthetic surveillance nesterov variant nmf non factorization nesterov smoothing rank decomposition nmf popular approach approximate negative rank factor account particularly video negative negativity factorization negativity constraint part support success video environmental propose greatly lee nmf achieve great nmf enhance computer discriminant fisher regularize nmf recently nmf distortion several liu mathematical viewpoint nmf variant minimize kullback distance poisson call short optimize multiplicative tailed image sift contain noise traditional nmf tail lie show distant illumination nine component space presence outlier knowledge illumination video traditional outlier knowledge sparse model matrix capability laplace tailed capability perform contaminate outlier extend develop box follow integrate grouping sparse develop elastic matrix try model nmf program slow scalable fast residual nesterov smoothing approximate outer iteratively scalable flexible objective another synthetic world scene surveillance dataset efficiency nesterov smoothing optimize face illumination residual optimize nesterov extension solve nesterov smoothing conduct efficiency nesterov variants vi fw optimal non smooth fortunately nesterov show approximate primal dual strongly prox project space feasible dual exist w however solve prox prox function prox control smoother bad lagrange multipli corresponding constraint median back form smoothed continuously differentiable gradient continuous w define use smoothed smooth naturally motivate optimize nesterov optimal construct auxiliary historical sequence minimize solution determine constant round historical achieve optimize point respectively combine check precision exceed
f universit scheme achieve comprehensive wireless enhance wireless device effectiveness current focused detect primary tv wireless wireless business system resource primary cognitive exploit tv band service challenge secondary device context desire sense device identify secondary wireless band group decision rule first base signal psd cyclic cp take classify psd band signal study detail cp pilot network identification signature signal scheme signature wireless system exist scheme uncertainty problem receiver sense receiver validate combine group algorithm parallel unique feature utilize detection detection utilize list spectrum receive signal false alarm detect tv signal practical pilot period symbol detection metric compose adjacent autocorrelation analyze noise pilot mode pre align dft dft length cp length cp dft length cp ds par ds psd utilize slide name cp autocorrelation lag dft total cp cyclic cp cp length cp symbol slot symbol accord frame frame probability form threshold mac behave structure detection compose summation probability peak ratio par estimate psd tv dft step segment rectangular achieve resolution par psd calculate since high par psd experience selective call par classify logic threshold receive process metric decide channel ds different unified algorithm algorithms mode classify long receiver frequency normally simple exclude psd ds cp sum autocorrelation autocorrelation performance band stop eliminate sensing performance frequency psd psd sense par ds notice threshold cp sum noise practically nu become inaccurate mis original metric require denominator dimension receive require call inherently nu configuration spectrum signal specification utilize classifier cp dft td dl mode mode td dl dl cp cycle fm evaluate white signal capture combine snr receive noise directly sense test use ability scale cp nu nu model limit calculate fig show algorithm notably nu detection become nu close ideal nu ms tv channel center acquire transmission
noise also consist norm marginal marginal compute fit optimization use good report error fit empirically weight norm norm compare collaborative movie dataset netflix rating c movie netflix max test max trace protocol root netflix norm reliably select evaluate max give rmse smoothed weighted netflix local max smoothed weighted give achieve compare smoothed empirically norm give known norm netflix achieve rmse improve trace introduce max norm generalizes examine family max different option weight max norm lie improved matrix reconstruction simulate movie rating suggest improve material prove sign generate independently complexity first xx x k smoothed use complexity smooth start prove u write l apply hull norm x apply result last norm norm ball finally return rademacher prove u u u want need eq mark fact sufficient sdp formulation sdp ba ba bx see quantity inside square minimize infimum attain minimize subject norm semidefinite program main norm special element bound see note sdp lemma equality calculate write show cc ga cholesky since therefore see q theorem exercise title title institute statistic introduce norm norm generalize exist max norm weight norm extensively unweighted conservative simulate netflix norm theoretical matrix reconstruct accurately many modern application collaborative netflix video norm low rank rank describe introduce generalize exist trace norm include lie trace max norm give improvement exist norm allow scale defer material take n analyze accord similarly column common reconstruction trace denote number column unbounde choose emphasize trace location observe guarantee recovery factorize definition intuitive q rescaled rescale norm still regularizer error row marginal improve place penalization represent weighted column use weight trace poor suggest smoothing avoid marginal smoothed trace smoothed column empirically weight yield vector uniform trace empirically supremum generalization norm norm max large standard trace empirically smoothed trace smoothed amount marginal smoothed line simplex weight smoothed trace norm simplex row th compare max minimal decomposable trace et balancing learn adversarial arbitrary affect supplementary material penalty convex determine depend impose low simplex intermediate max norm correspond rely equal rescaled norm take smooth two familiar situation want flexibility trace norm generalization correspond singleton exponent smoothed yield e hence column loose upper trace sdp easily supplementary material sdp write semi include mention sdp approach norm max factorize version norm dimensionality max applicable wise respectively special writing simplify use trace norm factorization last material prove look unobserved approximately ball gain norm think root square whenever ball hull extension trace statement hold trace convex corollary sketch u see convex version supplementary material weight dual inclusion max supplementary materials bound max smoothed norm
per hold fair nb also poisson likelihood th document link also nb base appropriate dirichlet dispersion list table sharing mechanism various algorithm lda nb nb process nonparametric automatically learn active fair comparison consider sampling process document topic weight tb sharing mechanism evident nb infer nb dispersion transition smoother order order last iteration corpus along axis either convenient document specific red per hold list lda refer lda concentration parameter optimize base crf hdp dataset vary nb lda nb hdp nb beta crf hdp gamma active percentage setting train partition table rank bottom proportion usage small nb alone direct generally zero well explain mark nb process mark combination would allowed responsible could indicate frequently document indicate heavily percentage confirm mark nb negative count naturally apply disjoint mixture model nb process completely dirichlet borel connection discover unique augmentation marginalization nb process able fundamental property theoretical structural advantage distinct sharing property nb make discrete model importance nb dispersion ratio acknowledgment anonymous constructive comment manuscript pmf joint count f lm p rf ml l sm pmf express let pmf l f pf p follow proceed thm theorem computer nb process employ rate normalization whose marginalization lead count model nb distinct reveal bivariate connect chinese derive augmentation gamma advantage nb variety nb beta mark gamma sharing construct apply make exist poisson factor importance infer dispersion chinese restaurant completely measure binomial process measure analysis predict claim disease document corpus count geometric originally model membership word member assign topic appear latent count variable mixture random assign count model multinomial dirichlet dirichlet conjugacy dirichlet mixture enjoy popularity group hierarchical share hdp conjugacy solve alternative construction chinese break distinct atom extra expressive tractable paper count count assign nb perform modeling nb process bivariate connect nb chinese restaurant table develop augmentation marginalization nb compound representation efficient bayesian nb employ rate normalization necessarily process mixture model marginalization since nb directly beta nb conjugacy group hierarchical employ share nb probability measure wide variety model provide efficient flexible construction part appear material conference paper provide logarithmic connect chinese table restaurant number nb gamma nb process process nb construction wide nb process nb thorough investigation property relationship key modeling nb dispersion empirically paper review commonly prior study nb process gamma nb discuss nb modeling example let dp aa ap k assign infinitely disjoint borel sign evy random measure l evy even poisson intensity nonnegative random machine continuous base evy poisson valid measure gamma ab evy cr analysis evy cp c base measure draw beta therefore invariant normalize process process base dirichlet completely correlate recover g predictive also chinese restaurant number nonempty chinese restaurant customer random chinese restaurant pmf variance equal due heterogeneity difference individual usually variance poisson restrictive gamma pmf dispersion nb coefficient term show generate compound poisson logarithmic pmf mr nb widely investigate numerous nb parameter nb conjugacy dispersion challenge use provide incorporate robustness bayesian able incorporate information bivariate distribution total number customer representation binomial restaurant nb various distribution poisson bivariate distribution pmf chinese nb joint logarithmic provide describe count equivalent circumstance customer customer represent augment rr augment connection various show corollary key paper allow derive posterior understand fundamental hdp nb connection process gamma reduce group completely random measure subset yy illustrate united count datum measurable disjoint define poisson share completely let equivalent let measure place gamma measure atom lead e evy nb derive draw consist finite almost surely nb critical distinct atom atom number work commonly use mixture model chinese restaurant customer count augment compound impose gamma conjugacy finite lx lk atom g kx ji ji rule chinese restaurant describe multinomial poisson equivalently without variable poisson dirichlet use mixture inference augmentation rigorous discrete process augment regardless whether base continuous analytic gamma dispersion modeling group nb process restrictive proportion group count distinct poisson model count group poisson compound poisson representation bivariate jointly count customer nb nb suit derive analytic posterior joint count group modeling sharing dispersion express express jk nb process augment jk normalization group gamma nb process distribution closely construction fig graphical negative gamma poisson compound binomial chinese restaurant construction construction corollary allow form x gamma exchangeable chinese crf process variable become thus augment hdp nb theoretically process random borel count normalization hdp marginally practically corollary conjugacy achieve analytic usually alternative construction crf stick break representation concentration nontrivial infer simply one may augmentation method practical become base may estimate especially share analytically update play role hdp readily mass analytic nb variation gamma model construction treat hdp group process log gaussian analytic posterior provide option purpose mixture compare beta priors gaussian nb share nb dispersion nb explore group nb dispersion group atom gamma nb hdp normalization follow beta member beta nb construction find beta conjugate apparent lack exploitation beta nb let express dispersion parameter note nb empirically beta nb beta dispersion parameter nb consider hdp motivate construction nb paper beta nb measure count normalization modeling hdp suitable mark nb nb dispersion beta process independent mark conjugacy tractable nb mark beta gamma process point mark govern nb nb draw marked draw construction focus normalization fully tractable link ibp beta spike various nb modeling corpus group document constitute group exchangeable index simply word nb modeling nb express express hierarchical equivalently fully exchangeable drawing insight model denote place dirichlet topic nb nb process since nb nb nb mark beta nb process nb word without document time variable count produce topic loading atom encoding relative importance importance atom tn n n kn jk would exchangeable topic unify connect model differ function kullback leibler kl divergence dirichlet dirichlet topic mixture algorithm nb gamma lda hdp hdp geometric geometric nb nb crf hdp nb nb count sum factor gamma nb parameterized construct variety nb topic count imply setting eight differently nb modeling improve model gamma close conditional posterior require nb lda tuning topic replace nb topic share statistical strength hdp nb hdp fix e comparable construct process process model replace draw bernoulli use explicitly count gamma comparable focus ibp compound nb infer fit gamma explore nb dispersion hdp
advantage generalization exhibit power nature nature tail kernel generalization far introduce similar laplacian kernel hilbert kernel demonstrate apply machine regression svms indicate counterpart kernel entropy generalize event logarithm case shannon functional additive nature kullback minimum discrimination theory shannon moment entropy existence second obtain normalize constant maximization lead doubly translate version write retrieve special multi dimensional gaussian laplacian gaussian inside define kernel l write kernel similarity decrease kernel increase rate decrease parameter similarity laplacian see decay rapid slow satisfy theory al exist space definite point positive first present require counterpart x kernel n lead positive laplacian certain range law behavior positive definite obtain retrieve kernel rational quadratic obtain laplacian define retrieve map higher dimensional reproduce kernel hilbert rkhs definite significance rkhs rkhs exist explicit gaussian taylor rkh continuous non rkhs fourier transform integrable lebesgue must note negativity corollary describe determine kernel q transform follow constant derive fourier transform similarly fourier expand exponential term component odd expression infinite series substitute gamma half integer observe summation observe result agree radial radial present correspond laplacian fourier provide kernel gaussian set perform breast mass heart auto quality red machine class svms similarity variant various separate hyperplane lead formulate c c laplacian illustrate hyperplane various kernel boundary say value compare svms multiclass polynomial kernel dx accuracy obtain use kernel varied type laplacian mark good kernel case require normalization set laplacian well counterpart justify flexibility achieved low law distinguish sometimes kernel rarely upon function datum fix linear optimization wave uniformly space sample c data c parameter laplacian gaussian various fold cross power law relatively behavior depend polynomial paper laplacian distribution retain law kernel counterpart kernel kernel positive rkhs propose law recognize law community far introduce lead theorem motivated importance
million hundred text provide classical problem explain variance pca less researcher hoc rotation technique thresholding include svd generalize converge local semidefinite hoc solving indicate compute principal pca achieve hard cardinality formulation describe safe method block ascent propose observe real datum allow reduction safe elimination pca become pca review base formulation highlight safe elimination pre pseudo extend semidefinite pca encourage sparsity loss generality cardinality cardinality write convex leverage safe elimination pattern correspond optimum relaxation variance feature less elimination look safe elimination huge third practice bring heuristic rigorous feature rate algorithm converge coordinate ascent well converge row row context sparse estimation precisely apply problem impose update imply element element entirely diagonal directly general lagrangian technique augment add objective optimize exploit come optimal apply strictly convex address solution problem column consider problem fix translate problem inverse problem relation valid fx ts ty z hz hz x relationship eq ts ts tv ts tu u stage first qp exploit reduce solve polynomial equation follow summarize denote remove column iterate constrained coordinate solve polynomial j apply ascent converge optimizer coordinate solve product fix size compare htb fig much sparse eigenvector publicly available news repository text collection record contain file gb give gb memory difficult technique present ascent pca variance drastically show fig elimination computation text hope principal corpora cardinality end equal cc fourth education mind repository label document level article website htb component pc nd pc rd pre processing search range implementation ghz processor gb top principal st nd pc words rd pc word pc th pc surprising safe elimination
learn course possible block system block gradient magnitude hessian bind second noise relative figure move recent memory component gradient wise adaptive learning tb gradient hessian want increase increment decay quickly step allow eliminate otherwise update correctly local curvature invariant obtaining estimate hessian estimate gauss diagonal exponential move curvature close infinity instability hessian necessary regularization average sample start algorithm keep exponential average robust near value initialization detail sensitivity empirically simple algorithm sgd whether good estimate block specific estimation tb use variance term operate architecture similar adopt connect regard fact overhead even trivially follow reconstruction variant regression understand behavior sgd function quadratic figure effect fix rate rate htp htp realistic curvature example rate schedule optimum cc ce ce cr mse adaptive memory appropriately initially quickly gradually gradient allow algorithm increase variance circumstance curvature vanish clear htb ccc cr obtain epoch ten initialization mark bold statistically significantly sgd observe full element almost significantly tune tune cr training average ten marked bold differ tune benchmark line total error set balanced statistically main outlier element lead overfitte compare training sgd network training sgd detail good hyper low average line show dataset digit small namely reconstruction use four feed forward softmax regression multi parameter mnist case second denote c layer perceptron cross denote use mnist fully connect layer perceptron hide hide couple reconstruction reconstruction tb formally processes sequentially apply affine transform element output feed loss delta loss mean number hide layer initialize cifar add term avoid numerical instability diagonal never htp circle drastically benchmark table table across per sgd main outlier lead overfitte training table hyper sgd mnist benchmark scatter obtain well tune cifar perspective learn initialization long learning optimize expect update expectation norm linear experimental completely need manual systematic make relate adaptive automatically adjust distribution period system fine dynamic change landscape drastically circumstance appear deep heterogeneous careful input structural classical hope enable truly box want cl helpful discussion title part grant descent quadratic classical base lyapunov stability positive super martingale since sum everywhere tb h give I global case detail learn hyper auxiliary initialization long surprising accurate average interesting quantity compute unstable safe benchmark careful rgb rgb rgb stochastic gradient descent learn decrease adjust error one variation make task sgd approach systematic effectively remove tuning scale linearly dataset recent interest besides fast sgd sometimes well get sgd require manual initial anneal schedule particularly stationary contribution novel rate possibly learn different minimize estimate loss every separable capture estimate practice common local lead variation none manual tuning variety convex rate systematic signal recent review practical sgd machine learn quantity gradient decrease loss without require anneal change schedule quadratic preserve convergence describe couple possible index datum contribute
come difficulty directly outside mistake simply vector hypothesis induce hypothesis equivalence perspective indistinguishable picking change perhaps weak simple tree leave mistake learner representative hypothesis adaboost classify scheme w different mistake row mistake arbitrary open tie break formally also important set closure often consider mistake effective important zero indeed imply adaboost update would arbitrary mistake certainly adaboost mistake adaboost initial pick point w calculate adaboost fact trajectory understand secondary converge study adaboost natural essential establish invariant section establish existence preserve decompose segment define w w w similarly suppose w clean inverse simplex space regardless decompose union w w w express secondary initial average round converge classifier goal adaboost carefully certainly converge round adaboost must grow classifier tx change prove formally discuss theoretically section converge condition kind something adaboost intuitively adaboost converge crucial understanding discuss state give apply secondary center notion able need reader detail existence context great ergodic capture say mean f two proposition state show ready measurable function adaboost exist like subset limit adaboost abuse define tw typical dynamical property adaboost set turn point mistake motivate theorem besides adaboost w w j w recall equation optimal adaboost eventually away adaboost condition instrumental prove measure satisfy formalize speak condition say long weight tie hold world dataset try practice conclude remark optimal eventually pair either condition become roughly condition mistake tie tie dimensional subspace implication exist thus adaboost weight positive converge example evidence condition variety high world behave equivalently remove mistake remove create dimensional removing reveal dominate mistake mistake remove mistake removal become empty update must least positive mistake every presentation avoid clutter repeat phrase explicitly technical mention state theorem condition proposition ergodic formal measure capture paper cover second quantity adaboost combine hold borel dynamical borel preserve way extend construction assign whether sense discuss give practical perspective next w adaboost w simplify denominator inside term w lebesgue measurable implicit lemma appendix technical average adaboost borel function adaboost tw statement tw recall form yield convergence adaboost almost statement repetition clutter presentation phrase state implicit secondary useful delay next let measure follow w obtain example state theorem notation mistake directly representative hypothesis unlike hypothesis need scale appropriately exact select confusion function adaboost build converge optimal adaboost let impose deduce whereby exist x tx take c converge theorem condition use low theorem tx tx tx row difficulty extend input correspond mistake mistake hx I equivalence perspective indistinguishable picking trajectory however learner hypothesis pick simple simplicity number regardless equivalence representative briefly whenever learner finite number selection candidate construct pruning remove never repeat mistake representative reflect bias tree simple complex mistake mistake mistake remove dominate mistake equivalently row set mistake sometimes well mistake mistake must correspond representative weak final adaboost sample extend theorem section prove extend instance whole outside mistake hypothesis mistake produce differently weak learner use put classifier representative one demonstrate adaboost build combine converge representative hypothesis tx tx f tx similarly arrive closely follow full replace yielding exist intuitively boundary hard think degenerate situation know borel almost correspond subspace think unless something special ensemble weak output adaboost likely borel measure adaboost converge borel reasonable relax discuss appendix assume phrase proof classifier implication adaboost subset adaboost exist tx tx f converging limit express dp space sense place exist dominate hx say h dp dp dynamic adaboost away impose outline proposition establish weight represent satisfy tool aspect converge training generalization probability devote fundamental dynamical tell invariant couple deal system formally call induced topological induce treat w directly proof open set convergent motivation additionally continuous map admit limit convergent second convergent must let sequence subsequence g g tie sequence subsequence contain compact subsequence w g close contain tie impossible tie equation clear q continuity clear equation construct whereby compact convergent converge satisfie convergent compact understanding continuity mention tell away set mistake inverse mistake adaboost away w w w w adaboost rearrange update use algebra w I iw weak suffice element case depend w manner exist hypothesis equivalently mistake example optimal adaboost round start initialization weight interior already assume make w continue template claim know w particular mathematical construction contradiction conclude reach finally compact space follow admit invariant borel state slightly tailor follow borel continuity condition follow state whereby mistake round decision heart cancer behavior depict plot suggest text look mistake ignore mistake adaboost draw cc margin sign round margin round evidence sign sign round bin positive adaboost train logarithmic sign margin example histogram sign zero discussion depict margin round boost cancer margin objective empirical adaboost favor tie condition result one adaboost begin potentially theoretical pac generalization adaboost empirical adaboost context author student supervision perform preliminary test attribute define implementation implication simplicity effectiveness arguably adaboost practice decision relatively small c mistake mistake inverse false show simple adaboost display mistake differently optimal adaboost suggest rule ever reduction repeat whole decision induce dominate lead constant realization good plot repeat perform split realization dominate hence include lead size realization density fit provide effective decision table number adaboost world publicly available uci repository note c classifiers train total dominate breast decision use percent classifier column total dominate respectively number validation test table like time opposite rest refer reader thorough cycle adaboost cycle minimum maximize margin boost previously interest find also inconsistent synthetic adaboost open problem select adaboost adaboost meta use learner adaboost classifier decision classifier adaboost round see plot important growth suggest adaboost ever repeat classifier logarithmic learner mistake finite unique representative base base representative logarithmic growth figure continue long run breast number adaboost dimensional suppose course adaboost open effective imply fact adaboost illustrate take weak respect adaboost tight course assume world uci dimension hypothesis dataset abuse notation style convergence account hypothesis tt adaboost h adaboost representative determine also respect exploit empirically also potentially draw application adaboost use commonly representative follow respect recall practice part plot expect round round reduce considerable certainly trivial dependent uniform error general curve convergence even still insufficient increase experimental require pair tie tie weight limit provide evidence evidence dataset representative purpose demonstrate validity available world many researcher adaboost decision heart breast cancer tracking good mistake optimal mistake mistake ignore mistake adaboost initial draw simplex traditionally weight mistake tend happen go minimal refer support term index example output always index negative mistake representative round cause row essentially part mistake tie ignore mistake cause jump round support stop appear tie zero suggest decision optimal adaboost boost cancer figure round little plot margin essentially view converge predict adaboost adaboost tree move increase stop sequence still question interface algorithmic behavior discussion statement contribution work present thorough enough stand technical establish thus reach prove validity tie future address current common ml seminal publish ml adaboost adaboost produce margin begin whether cycle open empirical carefully synthetic impose yet contribution title adaboost cyclic margin reasonable implication seminal partly prove tie result largely use assume synthetic example provide evidence favor tie real world significant establish important relationship tie condition employ adaboost suggest imply cycle observe tie evidence figure immediate tie adaboost generally imply reasonably logical evidence none inconsistent practice repeat far lack tie adaboost tie condition imply adaboost cycle similarly emphasize considerable empirical whether cycle adaboost world interested adaboost cycle evidence cyclic observation informally steady imply view cycle come broad main average class generalization evidence know explain adaboost converge evidence tie condition word favor sensible natural reasonable involve potential serious work quantity object leave proof adaboost condition obviously considerable favor improvement perhaps adaboost lie simple adaboost mapping state introduction optimal vary slow hand strong may generalization adaboost tree leave formal tie create conjecture seem believe result likely involve weak generating amount relative characteristic induce currently interact hence careful statement output minimum weight among respect weight believe extend tie case purpose avoid analysis set beyond non adaboost weak output may weight hypothesis dataset extend version adaboost tie tie would serve avoid weight adapt analysis require careful significant effort beyond scope regardless appeal tie relate phase fundamental implication condition impose adaboost adaboost risk bias deterministic classifier via function perspective intend converge amount something converge time logarithmic tight generalization see selecting seem distribute dataset use benchmark roughly drift state consist weight outline corollary adaboost secondary quantity main derive margin converge learner result limit would conclude adaboost boundary measure probability draw boundary converge manuscript conference except thesis science author part work technical report appear national early award cycle round average almost ergodic tie thus adaboost initial suppose cycle n f cycle denote without n f result ergodic theorem perspective alternative shorter thought equally simple appropriate adaboost tw convergence adaboost cycle paper leave formal statement cycle invariant measure skip suppose adaboost cycle borel cycles cycles brevity measure borel invariant measure give proof cycle adaboost cycle cycle adaboost w nc define collection disjoint pair algebra derivation order well transformation invariant singleton sake singleton contradict element cycle yield element hence either singleton empty present brief formal definition depth subject interior axiom call order pair topology implicit call contain pair satisfies negative triangle whenever call topological topological neighborhood center bx topological topological inherently perspective every topological limit general mathematical topological concept space denote set exist algebra set non set countable countable collection pairwise order pair another inverse e triple event law axiom measure measure note measure satisfy measurable transformation measurable pre measurable preserving measurable decision round adaboost un dominate attribute generalization split round boost vc exactly class split size parallel split induce axis vc dimension threshold apply node ensemble mm constitute consideration probability draw probability ready prove sketch idea specific normalize formal pt set integer substitute eq union obtain turn particular substitution ready provide follow space size specific weight constant simplicity k k k apply bind expression eq turn particular substitution another dependent bind classifier probabilistic account logarithmic output adaboost achieve select representative function hold draw theorem dataset I basic induced hierarchy base classifier hierarchy note normalize formal suppose k positive substituting expression turn particular substitution similarly half decision dependence decrease corollary practice considerably considerably small f measure stability depend h q follow take side equation dp dp h immediate infinite w equality lebesgue dominate third function lemma theorem conjecture condition theorem department science university ny usa department stanford university usa usa department computer ny usa stanford university usa department science usa adaboost learn practitioner mathematically elegant theoretically sound practice test researcher decade machine establish classifier margin round function weight adaboost dynamical provide condition employ tool theory previous work cycle actually evidence hold weak condition evidence real world formally analyses optimize burden adaboost popular implement practitioner mathematically elegant theoretically sound researcher point characteristic adaboost exhibit formal understanding well characteristic converge adaboost refer intuitive description reach call adaboost term adaboost learn weak base call begin boost output good hypothesis base classifier class relative argue subject stability round world decade intuition performance adaboost increase round previous frame adaboost dynamical sufficient existence invariant briefly implication community adaboost cycle employ tool ergodic adaboost impose call rather weak condition tie consider condition impose margin paper optimal adaboost always meet tie world try create big lead hundred perhaps thousand classification available differ bag bagging use learner bias adaboost use learner form evidence bias variance converge gradually hundred add work I five speech hold want broad impact adaboost perspective form present paper establish demonstrate implementation applicability success reduce theoretical hold artificial intelligence experimental clear last decade award year still widely theoretical essence popularity still state consider adaboost perform input training mt ti w ti tx f tx adaboost stand adaptive boost algorithm round sequentially fig algorithmic description output ti overview remain part remain introduction adaboost attempt explain view subject contribution significance work adaboost section round adaboost margin error mild substantial formal proof mathematical tool adaboost learner dataset substantial evidence favor tie summary discussion open problem adaboost add weak hypothesis intuition suggest combination increase long meanwhile performance ensemble improve stationary iteration go general accumulate appear adaboost generalization round appear reader evidence max theory put disease dataset heart disease dataset axis letter note time unlike previous canonical adaboost seem overfitte behavior vc dimension adaboost seem attempt context bound really explain generally inconsistent fundamental graph remarkably generalize combination complexity long meanwhile remain work adaboost raise previously adaboost try misclassification rate dimension margin th minus vote adaboost produce publish margin adaboost less accurate apparent drive force behind far generalization training independent provably reasonably tend margin effective adaboost generalization margin loose precise couple question different overfitte take appear question ensemble notation question I bayes asymptotic predictor inconsistent year long time thought answer yes claim behave like select adaboost strong law number certain learner adaboost version article mostly risk adaboost put put try indicate go generalization adaboost set adaboost instead behavior resemble produce behavior understand informally follow conjecture later conjecture ergodicity finite misclassification low weight increase complement decrease top understand infinity combine produce see evidence favor growth pg addition although optimal make ergodic ergodic dynamical paragraph turns establish adaboost suggest adaboost classifier condition turn characteristic learner adaboost exhibit tendency overfitte dataset heart disease plot pg overfitte datum stability stability help adaboost overfitte adaboost stable analog gauss minimize enough terminal part population learn paper similar sense tool real g invariant requirement tool e convergence round weight example margin show converge learner favor hold high ultimately emphasize ergodic cycle turn need reader discussion last technical adaboost real world dataset delay condition concept round tie weak representative hypothesis one minimum significantly express mathematically provide validity converge answer question go happen believe contribution area reasonably positive fundamental question state wide behavior reasonable generalization cyclic non cyclic mild extend remarkable adaboost explain converge us community evidence trivial trivial know exist dataset implication adaboost round come perspective emphasize space ideally goal classifier something mostly version forget nothing property adaboost boost effective establish convergence generalization state formal present paper need recognize usefulness important well understanding iterative size least decade asymptotic variant non similarly behavior convergence view despite know update interest ml fundamental area without work generalization condition empirical provide convergence rate deviation bound performance adaboost asymptotic result convergence among thing mild informally paragraph quite reasonable mainly adaboost tie without formal adaboost provide considerable evidence hold uci repository find practical adaboost believe concern ml article work publish adaboost quick clear address reason work actually discuss form variant understand go discuss believe relevant previous concern manuscript try put present work believe concern concern form aspect adaboost minimize type demonstrate adaboost iteratively exponential weak context section minimization understand exponentially fast show adaboost minimize provide prove enjoy achieve explore primal adaboost deal term attention complete version manuscript repository concern several perhaps meanwhile interest basic adaboost classifier average relate mostly within consider round adaboost let variant consistent stop example inherently concept distinct consistency adaboost error training say differently concern generalization important note adaboost machine complete version manuscript complete mostly deal convergence put close enough dynamical demonstrating cycle case prove strong asymptotic fully understand margin paper understand cyclic remove condition cyclic whether adaboost cycle date present relate version adaboost version I course like start regard art prove regard art paper follow behavior adaboost cycle adaboost support margin train typical mistake equal incorrectly element transpose mistake value mistake matrix hypothesis incorrectly semantic difference syntactic prove simplify adaboost cycle indexing mistake adaboost vector margin submatrix involve adaboost produce theorem start discussion previous paper something line several adaboost margin interest state adaboost turn good approach margin lead maximum margin support cycle also prove cycle tie tie adaboost cycle fact adaboost easy formal ergodic straightforward approach extend relatively via large typical select weak hypothesis relate condition selector characteristic adaboost convergence state state experience high real dataset condition formal formally speak adaboost face hypothesis w ti concrete selector great fundamental involve aspect get work begin kind adaboost classifier generalization minimize exponential iteratively loss procedure understand quick later enjoy rate achieve similar implicit meanwhile adaboost averaged margin concern adaboost zhang recently adaboost distinct consistent error limit set error iteration sample early adaboost early adaboost case application ergodic cover technical adaboost high technical formally roughly adaboost tie weak ti delay formal later concrete prove behavior cyclic case cyclic behavior higher dimensional always remain date arbitrary weight measurable remarkable explain converge
skewed database statistically plausible goodness fail event via imply law indistinguishable fact give treat event treatment theoretical single primarily drive perspective statistical physics population individual limitation apparent law remarkably robust vary somewhat year change international independent attack economic development country size age experience attack illustrate ci confidence interval observe sized tail log ratio ratio neither significant indicate law plausible alternative ambiguity illustrate difficulty identify tail likelihood rare event imply slight visual deviation empirical tail three past severe automatically minimize kolmogorov statistic tail truncate empirical provide thorough motivation ks law discrete tail appropriate count finally use yield city empirical leave large derive analyze covariate versus international country bootstrap accounting tail event bootstrap parameter agree historical probability interval event global historical record unknown jointly estimate difference concentrate produce tail cause slight curvature empirical mid arise aggregate event single curvature automatically wide ensemble correspondingly density confidence confidence law estimate robustness especially alternative different estimate form alternative x restrict appendix exponential heavy yield heavy decay asymptotically fast law parameter choice marginally traditionally negligible event comparison tail alternative yield ci ci chance power law consistently away outli bootstrap well free law none place failure law attribute roughly track different covariate illustrative factor covariate instance international different rand database exhibit appendix ci figure covariate exclude international event develop important tail specific power law exhibit heavy tail fewer significantly handle categorical covariate produce estimate examine hazard large contribution follow political generate principled statistical forecast event make stationarity estimating event rand calculate number past trend increase exclude significantly maxima severe remain latter trend decade expert new decade lead unless replace potentially specific prediction optimistic attack year return ii status remain refine forecast production use quantitative forecast estimation historical generating estimate facilitate alternative model table summarize forecast scenario status chance decade optimistic event forecast chance strongly popularity great likelihood progress move toward optimistic event place range statistically unlikely illustrate say calculation refine incorporate address treatment event several investigation stationary generation unlikely term technology exhibit event use identify subtle trend impact implicitly actually occur interpret condition historical effort extent policy impact accurate achievable policy actor similarly actor responsible incorporate capability say grained make country incorporate attack likely unclear estimate scale appear play perhaps independence scale central averaging birth rate population contingency unclear although event study control appearance small scientific true event whether classified historical development forecast provide principled naturally incorporate source uncertainty tendency skewed produce reference properly historical rare occur six time could probability observe sized world year choice much anomaly six large statistically unlikely historical record event conclusion frequency degree economic development country international type large international chemical rare use oppose datum historical hazard event change consider forecast next year comparable course treat phenomenon hold example micro extremely long deviation overall process true statistically justify global spatial long temporal scale apparent scale sized global event fundamentally event unclear likely generate event question global event expectation aim event incidence dramatically tail frequency severe suggest possibly political social effectively describe detailed trade benefit preference actor investigate global pattern offer complementary rational actor framework implication normalization tail likelihood attack integer I incomplete equation straightforward past show provide discrete take moderate value experiment percent automatic goodness ks method fall select fit value event several improve statistical analysis large alternative tail modify significantly easy present tail quantify regime analytically draw ensure generate fluctuation power law whose appear generative error mae ratio observation fairly error decay decay add outli small absolute mask example reveal mistake panel probability contaminate fairly sample method correctly relative probability present parametric model ii iii covariate refine historical large process I random distribution iv version use department report event historical sized slightly normalize constant value repeat event process approximate continuous event observe one simply value complementary cdf substitute rand database substitute one period calculation decay slowly power law model severe restrict least accurate repeating chance event event rough total main alternative global global database count evenly record fractional yield analyze rand law fit estimate binomial large inclusion highlight broader good etc round report figure drop cdf round power law number thus year ci repeat along law notably heavy rand center roughly agreement rand furthermore estimate dramatically rand support rand mainly international country international international character provide robustness versus international ii change decade pre event database heavy tailed international international ensemble visual empirical international rand event dash ci dash international large seem confidence fundamentally cause tailed international include international exhibit analyze condition estimate event economic operation development cover rand event tail decay slowly chance year period slightly unlikely event result fit place weight yield ci attack generalize algorithm covariate case event classify chemical biological include device iv sharp vi empirically least generalize carlo categorical original jointly fit covariate least zero apply ks technique tail univariate hazard hazard repeat bootstrap tracking probability drawing set bootstrap estimate cutoff size unlikely calculate event type great type chemical object estimate hazard value indicate dash historical frequency historical rand future exhibit relative frequency attack unknown exceed trial category w online discuss skewed political economic network six large event life modern accurately estimate upper tail present generic make combine parametric historical observing sized result condition global variation economic international datum drive next decade attack event modern history people attack sized decade answer trend global political rare determining generate event common would long expectation planning effort chemical nuclear event pose quantitative mechanism historical record contain event estimate mechanism large alone big fluctuation poor tail misspecification severe financial model event aggregate covariate example resort effort qualitative human actor attack quantitative adversarial treat event like outlier say little hazard algorithm event social particular make broad rare combine likelihood model quantitative uncertainty
variational guarantee define family ensure alternative motivate provide optima view proof college science bt united optima discrete unified description circumstance concave particular method sparse maximization vector differentiable prefer alternative relaxation wise popular discrete simple expectation distribution define space adjust lower tight enough mass restriction bind objective smoothness deviation objective dispersion increase give sufficient concave purpose way construct alternative make q bring differential lebesgue integrable ii integrable non discrete differentiable normally sufficient prevent form even quasi concave advantage concavity kullback leibler variational generalize linear distribution concave pf concave mean factor use triangular element component concave top figure lasso optimal lasso objective away closely match objective result optima differentiable section lasso objective bind problem choose approximation optimize bind implement surprisingly sophisticated main interest svm l bfgs alternative square encourage objective origin optimization variational gaussian expectation affine objective variational q cholesky cumulative hessian straightforward differentiable use alone z split term give maximal objective hence assume optimum find find optimum within first normal iy ny mean dominate solution sparsity iteration approximation inverting gradient whose numerically closely experience optimize set time reduce absolute current cm sd sd iterate sigmoid c large sd iterated ridge sigmoid test solver schmidt matlab relative since optima measure algorithm give mean deviation indicate capable optimum scale well smoothed approximate use sigmoid lasso subsequently effective optima also introduce component converge gradient vector underlie element subsequent create clean label label run rise impact vector package competitive state package nesterov method backtrack line shrink convergence relative cpu time algorithm implement support machine find hyperplane separate hyperplane dimensional separation possible map space separable machine add slack misclassifie datum cost soft solve lagrange multiplier convex refer preferred convenient deal primal remove primal problem objective minimize slack possible yield minimized slack loss misclassifie hinge whole hinge loss convex convex across hyperplane allow reproduce hilbert without without objective kernel k nm amenable choose gradient variational difference difference maximize condition tolerance hinge derive newton huber form huber tighter way give hessian dimension whether quadratic hinge loss small huber solution point lie portion offset except hessian show definite cubic inversion newton construct generating choose completely separate kernel positively label multivariate normal cost coefficient finding value matlab training set normal modification shrink huber dual bfgs package treat huber similarly analytically component iterate toolbox set matlab scale quadratic highly small finding fewer extremely fast surprising shrink roughly gradient opposite evaluation huber take inherent shrink schedule advantage shrink impact huber
code sd sd vc vc element proper appendix thus sd sd sd equivalently great sd j subset fr bound e p deviation close clearly ix code theorem point coarse x least theorem difference loss double deviation loss original loss deviation loss least eq consequently sufficient handle range eq proposition union stage theorem appendix lemma elementary sd p l remain prior across prior choice assign infinite dependence dictionary cardinality bad cover good good overcome require device justify unlabele yet switch rademacher average predictive level coding measure unlabeled simplicity guarantee bad case special disjoint conditional rademacher exploit proof lemma trivial non continue important role overcomplete set deterministic previous low deterministic immediate probability bad event occur last line treat subsection choice hence large deviation code denote mostly random sign definition q routine sign fix rademacher bounding rademacher average dictionary factorize u dimensional dictionary choice encoder gaussian behave standard normal fix proof use show difference perturbation solution linearly constrain program dependence index lemma supremum expectation supremum depend k rademacher dictionary factorize isometry rademacher complexity partition index index occurrence precise index note minimum taking intersection approximate disjoint union union arbitrary choose disjoint stability approximation disjoint useful rademacher complexity loss rademacher combine independent draw proposition result prove theorem bound elementary choose nearly bind fix unlabele eq sparsity margin label unlabeled sample digit versus atom level zero per dot code train digit versus atom sparsity code dot margin away separate employ task digit predictive coding per gradient normalize unit apparent sparsity indicate colored axis non trivial margin margin factor bound stability consequently infinite unclear encoder important open rely strong require turn true greatly would constant work upper sparse thing remain unclear one fundamental importance dependent provide small encourage code margin follow lemma key sparsity establishe perturb exploit old flow sparse code review optimality imply code operator likewise let observation symmetric argument second norm optimal close q consider dual definition follow optimal claim hence eq convenient roughly stability optimality combine eq rhs obeys newly bind rhs implying perturb sufficiently consequently q objective equivalent formulation z plug expansion incoherence result xx sufficient proceed characterize reformulate satisfying must consistency check optimality satisfied clearly sign check hold sign consistency satisfie replace side chebyshev loss assumption follow expand replace choose yield equivalent new related solve eq q choice assign q recall measure sample entire equivalently level u x x k apply program constraint denote equivalent optimal z z add yield cc zero become reduce operator z select similarly coordinate cover number radius space dictionary property one must careful cover proper cover proper cover cover simply subset ambient banach space banach cover cover number cover banach cover proper cover number name ball radius hypothesis name dictionary name name joint description name name return name description singular incoherence sd sm description second label description description w w description description average x ss theorem ex corollary coding learn hypothesis code recently variety establish bound predictive cover overcomplete exceed dimensionality setting stability sparse encoder measure fundamental stability lasso characterize stability code dictionary overcomplete decay infinite respect theory dependent dictionary architecture predictor theoretic many learn architecture base representation representation learn represent k z predictive loss coding seek dictionary evidence abstraction higher predictive intuitively atom generalize theoretical knowledge code previously generalization set extend introduce certain difficulty control perturbation predictive perturbation stability hard realize assumption interaction learn dictionary assumption hold newly provide ambient dimension infinite setting dimension free hold certain mild perturbation overcomplete incoherence system induce column bind absolutely stability dictionary perturbation collect measure product unit ball equal consider iid combination atom dictionary xx solve risk erm objective k space overcomplete handle overcomplete produce fast linear hypothesis predictive class hypothesis measure code objective formally code regularizer analyze erm objective know return certain sparsity stability holding present potentially additionally influence stability learn well apply define encoder induce regularizer design encoder perspective theoretical behave lack strict image unclear parameterized map drastically small perturbation hence perturbation central statement throughout sd value among guarantee code sense introduce property encoder point exploit sample property give point sample importance margin property flow intuitively inactive second sample hold label learn hypothesis highlight stability sparse dependent relate overcomplete minimal learn root unclear set introduction free dependence incoherence exhibit appear significant setting let match rate fraction open question represent dictionary reflect code perturbation quantify sparse stability theorem enough primarily extent extent dimensional previously factor unclear appear price encoder encoder stability error lasso stability stability readily use ensure result learn incoherence sparse perturbation since reach theorem pass stability drastically need avoid aspect margin hold margin away zero inactive atom margin integer small learn predictive learn incoherence trivial low mixture elastic fall guarantee scenario induce regularizer simple datum overcomplete bind infinite dimensional simple attain label iid drawn marginal specific cover cover dictionary metric essentially due main allow depend label completeness prove overcomplete deviation empirical deviation risk overcomplete lemma specialize
combinatorial graph perfect approximate generation generate approximately bipartite combinatorial function mapping relation arrive approximately satisfy assignment feasible uniform generation element speak reverse precisely sort object string uniformly output r randomized output previous paragraph input perfect assignment boolean uniform assignment satisfy assignment boolean hypercube sampler generate assignment proceed briefly statement definition stronger support put element uniform thought suggest impossible efficiently achieve even problem generation satisfying differ thus impossible meet another generation require object definition would approximate generation attempt intuitively appeal schema algorithm accuracy object interesting efficient inverse generation efficiently solve hardness version object define boolean term etc relation sample satisfy assignment high aware unsupervised specifically sort uniform satisfy assignment member know sort similarity since difference goal goal output significant demand hypercube hx set constant acceptable hypothesis function variation distance indeed inverse uniform additive aware boolean error discussion essentially directly yield boolean assignment element uniform access learn additive less algorithm draw failure positive uniformly finally reconstruction subsection inverse problem inverse hard inverse uniform reconstruct mrfs much make past decade concrete problem seem mrf reconstruction reconstruct need impose uniqueness random goal high case underlie give approximate generation note deal satisfying assignment assignment negative present present obtain approximate uniform technique count statistical call section roughly dense generating approximately possible uniform count need reason detail section technique function class construct online point know uniform generation counting generator random satisfying assignment carry stage obtain result check statement problem uniformly second study class full size formula boolean variable run output formula challenge formula time around view output hypothesis term observation since uniform distribution algorithm improve art learn open use query formula carry technique main class either reconstruct reconstruct efficient inverse generation approach hardness problem near computationally signature scheme public roughly speak simplification see full precise statement parsimonious construction signature inverse uniformly generate satisfying hardness positive lie quite boundary generation statement construction signature scheme time uniform problem either follow formula intersection signature hardness parsimonious type hardness approximate class hardness result generation particular obvious produce instance assume seminal np oracle approximately interesting ask require adaptive np whether hardness understand polynomially assignment satisfy assignment formula access simple elimination formula access assignment formula signature hardness hardness sometimes barrier along line statement technical version construction circuit verification result construction construction easy plausible assumption generation construction generation computationally uniform version plausible uniform hard uniform general organization section approximate inverse suggest probability point ds definition total ds dx dd function usually threshold function variable formula stress view take input dimension denote conditional distribution f familiar counting version precise notion approximate count boolean time output generation function efficient f let class variable randomize uniform u uniform dependence dd tv inverse need circuit bit bit clarity emphasize accord formally boolean approximate access test one thus need select pool candidate hypothesis c finite exist sampler evaluation oracle output call oracle perform arithmetic operation output certain crucial difference proposition explicitly carefully kind essentially detail full competition e algorithm failure describe analyze competition subroutine proposition accuracy confidence parameter kk om j could seem thing bit simple approximation thing since approximate totally competition iw variation purpose ideally oracle approximate turn similar ij iw particular proceed crucially multiplicative exact show eq similarly establish establish iw yield second inequality hence choose oracle provide subroutine oracle make draw perform arithmetic operation p j subroutine amount consequence confidence use ix empirically consider h I subroutine subroutine j ij follow decide membership give jx random sample query total claim perform competition set estimate ij jj return competition competition output sample arithmetic straightforwardly verify remain correctness union correctness hypothesis immediate follow competition winner intuitively likely winner competition winner winner r z ij r mutually follow chernoff j beyond stop winner competition ii winner otherwise paragraph competition winner correctness follow compute read sample read bit operation encoding set hence read need bit operation belong call oracle run dominate oracle call competition distinct exist output sample access oracle draw n dd j exist otherwise sample running bound choose hence never competition output ii j competition never variation probability failure assume element ii conditional condition easy proposition extend sampler invoke repeatedly event repetition ever failure generation conceptual heart speak algorithm construct essentially state roughly speak approximate uniform generation iii counting query algorithm suffice uniform generation statistical recall natural pac learning allow estimate access value v sometimes write state accuracy follow boolean randomized input oracle query maximum tolerance parameter ever say bound succeed independent say learnable independently sometimes write accuracy throughout independent pac label robustness useful nf minimum value tolerance ever execution complexity access simulate hypothesis confidence time introduce boolean give unknown output subset much smaller particular enough moderately dense take variable boolean input fx g g state conceptual follow maximum query minimum ever course inverse function analysis expression long uninformative inverse approximate generation work three main conceptual g generation support variation step hypothesis third notice seem introduction reconstruct object process standard consider fraction would kind run obtain good variation imply learn motivate possible accurately query positive draw technical learner issue conceptual seem precede motivation issue arise implement step concern algorithm close need able easy obstacle see version efficiently query claim issue handle statistical previous paragraph sufficiently accurate small provide query oracle succeed multiplicative full accurate function approximate counting accurate estimate value cover obtain distribution guarantee true variation hypothesis proposition find use distribution distribution generic count version theorem ready proof need query positive example sample algorithm efficiently valid e one oracle simple whenever simulate early access sample number computable tolerance use take range additive use empirical ix ix refer empirically lemma identity simple estimate empirically confidence way subroutine sample query fx f expectation dd need total subroutine run bind bind latter triangle simulate performance evaluate minimum provide course execution exist access efficiently accuracy dd om fx procedure query describe algorithm output fx hx fx simulate f obtain moreover confidence time certainly event query additive guarantee algorithm complete tell query access draw estimate sample even exactly sampler sufficiently distribution bound dx desire imply subroutine additive accurate estimate proposition distribution efficiently sample evaluation output subroutine circuit compute om precise subroutine simulate u failure gx hence get access count computable u take confidence bias run input return counter return claim argument assumption gx fx recall present succeed satisfy algorithm dd output outline subsection fx run n learner generator simulate run sampler rejection accord approximate generator hx ix output let exclude code approximate generator error fx hold sampler polynomial correctness step succeed confidence assume throughout definition satisfy property u fx event distribution definition generator probability call union bind failure call successful obtain goal call subroutine time give sample obtain finally obtain conditioning b successful fx hx union analysis hx observe hx x u hx ii remain item run establish I close total attempt draw output sample output default fx tv consider produce immediate triangle everything first probability code call approximate generator call fail least thus probability draw hypothesis fx hx imply therefore conclude desire big probability satisfy assignment failure tv happen lie happen draw identically tv claim u fact writing equivalent construct bind eq rhs proceed finite term rhs term zero property third property tv rhs turn bounding case guarantee learn numerator eq imply conclusion bound f g f h g imply third bound complete finish establish run verification run run regard algorithm claim function parameter sample union turn make draw approximate oracle subsection sampler close satisfy fx going set use need approach follow target available subsection construct detail randomize check high pass pass check give desire approximate evaluation g mx return fix consider run ng ig follow eq ng ig ic guarantee q draw I md probability none go forward hx multiplicative chernoff bind failure failure circuit value output construct simulate evaluate either evaluate step definition generator string behave consequently take lie combine recall desire string alg ready theorem complete nn right element negative ok correct iteration reach mt say mistake learn end section general give problem class boolean think uniform parameter run fast term n significant learning theorem provide count main technical contribution give fourth ingredient present analyze give algorithm exactly generation n approximate counting count tn fast know literature formula run boolean rather formula close intuitively provide target actually know learn accuracy explain e theorems class length boolean run evaluate query hypothesis moreover polynomial property let boolean x fx kk fx note pass learn target function intuition behind oracle hence target tolerance learn tolerance learn boolean regard theorem sc u gx sc c time abuse terminology function theorem step algorithm execution simulate algorithm theorem requirement succeed bound describe straightforwardly verify overall theorem target quite reasonably consecutive repeat high pool reasonably straightforwardly algorithm f output initialize ss repeat string claim start precise jx st term draw independent sample iteration least ss ready prove claim candidate iteration union every jx jx sf take term union term define gx item fx gx f bind least satisfy pm x approach term conjunction inverse generation class run kf kk counting formula formula boolean sum disjoint define express tree special first value get work satisfied assignment add say include high consider satisfy assignment certainly position exact choose satisfying assignment include repeat union hence boolean proving whenever hence clause prune pruning sample uniformly random statistical thing notice deal exact counting oppose chain one distance run via count really point expand space consist simply write trivial writing variable sign make I imply hence I whenever variable assume leave depth thing make since repeat round going want variable tree sum learn much let begin initialize repeat nf satisfied assignment say c ai nc ix assignments independent choose uniformly conclude assignment small remove less hardness generation class boolean learning theory base hardness hardness study hardness learn introduction potential inverse generation reconstruct sampling assignment object generation one constitute proof generation efficient specific step provably well effort get hardness come two signature provably hard hardness assumption assignment hardness code easy satisfy standard subsection hardness detail prove relate hardness speaking say signature hard generation intersection function begin public scheme simplicity suffice verification algorithm triple take message signature verification every signature first fix signature public valid sign message possible sign message signature range potential signature message g special sign sign message sm dd dm message similarly algorithm next notion attack signature holds choose need define notion hardness approximate uniform tn inverse algorithm generation invertible relation reduce binary hold furthermore class binary relation whenever say invertible corresponding relation relate inverse scheme boolean tn efficient generation invertible signature verification public sign point point generate translate sign generate new sign signature scheme formal contradiction generation close target key pair adversary signature v polynomial circuit invertible adversary adversary signature I xx follow claim random let correspond distribution statistical marginal coordinate special image apply likewise apply instance run time produce whose output definition probability adversary succeed produce adversary time adversary contradict security scheme literature requirement similar signature different generality state follow slight variant appear probabilistic eq mention state theory plausible sake hold hardness go hardness et signature use say signature signature message every message signature construct signature variant go back get special signature state special obvious special sake signature signature scheme function invertible generation constant depend hardness generation invertible conjunction clause standard invertible exist inverse corollary formula learnable uniform generation formula variable fact invertible reduction polynomial invertible say problem formula variable occur hereafter constant approximate invertible reduction time reduction follow show boolean inverse uniform intersection approximate integer np complete invertible outline boolean assignment show easily verify invertible see polynomial observe instance intersection hardness result directly prove np extension hardness uniform generation formula conjunction clause trivially begin one need follow hold instance circuit efficiently computable follow helpful invertible one let distribution obtain uniformly uniformly choose fact uniform extension assumption corollary assume inverse uniform close target signature security v v meaning adversary receive message signature choose satisfying assignment adversary denote output call close hence imply succeed signature pair ensure probability success run probability reduction reduction technique invertible reduction note invertible invertible reduction many virtue time invertible construct introduce every put vertex graph assignment every cover computable invertible assignment vertex cover assignment vertex vertex assignment create every vertex cloud polynomial formula map cover vertex cover cloud include else include vertex cover vertex size construct formula correspondence satisfy assignment cover far vertex map vertex cover hand assignment cover assignment map least satisfy assignment satisfy cover gx map conclude true hard approximate uniform absolute multilinear class variable degree generation absolute every monotone formula degree true hardness intuitively correspond function efficient generation reduction would np invertible make thus prove class algorithm code mac hardness generation class begin definition see triple property randomness verification message purpose hardness result special hardness specify generality mac say message exist likewise code say potential tag message tag r tm r tm r cardinality tag security attack message eq see hardness inverse uniform mac message inverse towards contradiction generation run output statistical security mac tag independently mac special uniformly tag message satisfy set taking compare definition output sampler satisfy suitable choice recall contradict mac unlike signature scheme cf hardness belong special construction intermediate verification construction hardness tend hardness approximate uniform intermediate year hardness noise version return independent sample study intensive fast take assumption let let henceforth assumption conjecture seem closely assumption whether formally perfect investigate run plausible strong dx vector input output state say henceforth similar consider show imply minor change security along line minor change proof subset problem hard proof lemma key assumption imply assumption get ready generation random string randomly verification tag operation mac theorem suitable mac assuming describe mac observe mac describe mac mac assumption provide description mac mac mac except exact come uniform remain verification generation easy towards class boolean n multiplication verification r say uniform generation algorithm crucial cardinality return randomly let draw ir procedure simply run distance give efficient approximate give quasi polynomial uniform crucially answer answer fact hard detail versus assignment show randomize polynomial formula assignment immediate even f tn get every element particular inverse approximate uniform generation draw trivial argument generation uniform hard inverse question function uniform generation polynomial approximate combinatorial object assignment let undirected symmetric generation relation vertex graph receive n easy permutation understand define graph unlikely decade effort fast strong claim establish uniform uniform relation randomize
asymptotic normality establish sufficient formulae literature asymptotic direct state checking avoid transformation model l semidefinite ccc nonzero assume contradiction vector uncorrelated follow since converge surely express converge weakly random uniformly uniformly bound surely infinity surely show investigation state model space except ergodicity satisfy short basic equivalence model mainly discrete estimator main time provide auxiliary result follow state discussion space condition develop finally introduce technical need derive process randomness increment almost evy evy mt gaussian l evy purpose enough evy l evy moment use time process l evy process differential dimension drive evy true general l process differentiable interpret consider suit type equation output multivariate causal easy negative part unique strictly second eq b immediate drive evy representation analytical allowing replace apply complex spectral density h relation prove tu satisfy ready q fact spectral density last h complete converse identifiability proof find rational f uniquely determine transformation orthogonal sample n nh partly space zero tu rational less notion explain origin terminology rational triple call realization algebraic realization convenient let realization realization important realization notion play definition mn realization n investigate realization minimal matrix tu assertion straightforward generalization triple differentiable show characteristic q side second infinitely distribution th evy analogue parametrization family continuous strongly normally strong consistency consequence four hold space identifiable multivariate satisfy logarithmic moment mix theoretical applicability practical research go notion confirm study bivariate present smoothness drive evy satisfie parametrize consistently normally distribute additionally evy hold unable verify analytically explicit check computational effort description literature canonical decomposition rational rational determinant polynomial satisfy polynomial root triple degree denominator polynomial kronecker index k small kronecker transfer process process unique structure block th vary concentrate kronecker rational invariance inherent restrict parametrization non description integer rational kronecker realization parametrize matrix th matrix order normalization one special h ij b h preserve multi structure matrix prescribe method inverse example cc cc ccc cc cccc cccc n cc cc cc z z z parameter present bivariate index drive evy gaussian stock return increment determine choose skewed covariance simulate apply eul stochastic making value realization compute numerical routine conjunction trust mean deviation report moreover entry asymptotic display c std est std driven horizon second report fourth deviation obtain one bias small accordance large deviation construction confidence uncertainty deviation desirable overall simulation study acknowledgement support international science universit grateful grant author acknowledge financial study process theorem theorem theorem eqs eqs process general space move result linear model many decade applicability detailed discrete precise formulation investigate second quasi maximum surprising mathematical case eq high noise sequence space aggregate asymptotic normality general satisfie restrictive satisfied sequence absolutely process recent deal weak make mixing model analogue familiar move introduce drive motion evy heavy tailed occurrence path characteristic many series finance multivariate give possibility time series interpret expression apparent process describe detail approach investigate reference therein typically digital observation become recent year variant maximum investigate second estimation multivariate second base spaced discrete coefficient restriction drive evy however autoregressive drive evy consider consider autoregressive letting infinity autoregressive move drive evy parameter frequency time horizon equivalence aim directly therefore condition induce turn paper sample finite moment mix ensure absolutely continuous component need appear rather univariate variance look behaviour function high organization develop general moment result strongly distribute infinity section establish multivariate grid relation continuous identifiability able main result consistency asymptotic sample final canonical demonstrate applicability stand transpose image identity rational indicator stand ccc ambiguity use respectively constant change space time consistency wide system result apply property estimator strong section linear characterize strictly stationary cc mean eq state output simplify considerably eigenvalue unity class aspect filter theoretic series output importance eq closure x immediately imply process uncorrelated absolute unity discrete steady kalman eq write process representation allow representation eq together collection stationary variance tr n logarithm always assume state fail difficulty history therefore approximation steady pseudo recursion one additionally kalman recursively advantageous burden kalman asymptotic deal paper kalman converge steady eq n main state strongly surely prove impose guarantee absolute unity matrix non hold suppose follow positive exist assertion consequence entry claim noise usual ergodicity chapter process via move consequence pseudo identifiability assumption parametrization state true estimate consistently quasi asymptotically term recall process f hold describe far parametric accord impose lie impose condition mapping continuously differentiable imply moment space sequence ergodic alone amount independence result sense impose condition strong mix strong mixing always satisfied restriction appear autoregressive equation exponential output impose condition whose distribution possess asymptotic normality parametrization derivative gaussian vector stack top asymptotic initial lf parametric state l covariance covariance deterministic actually fisher alternative task detailed scope work sense need concept limit estimating framework applicable l rely filter likelihood also hessian achieve kalman equation result pass nz r l ns n l z matrix regression claim il consistently estimate describe deviation technique exist bootstrapping technique extend considerably strong strong estimator sometimes work estimate turn moment reason gaussian know ergodic strongly consistent despite step kalman filter kalman theory obtain true pseudo observation function unique true divide surely function evaluate stationary pseudo kalman filter surely cn n exponentially positive positive c uniform iterate last almost claim observe infinite k complete imply moment one claim thus hold sequence view approximation result assertion ergodic consequence sequence compact analogous sake j surely j tc j j minimum difference element span definition orthogonal expectation dm dm dm remain argue inequality inequality strict alternative inequality equality converge l minimize observation imply complete suffice show neighbourhood lie every neighbourhood define equal normality assertion l idea say estimator central translate asymptotic estimator technical extend pseudo steady kalman derivative obtain certain covariance scalar different strongly bound use gradient quasi uniformly pseudo extended derivative step allow show asymptotically determine rescaled exist invertible taylor function divide number true combine strong consistency third term derivative control normality collect kalman
cut balance effect unbalanced balanced partition cut value lead balance term partition balance toward thm sec provide flexibility tradeoff constrain avoid cut size sometimes constrain draw separate hyperplane two volume volume contain exactly range matter scale regularity proper choice unweighted graph vary cut area notice minimum attain area technical connect point close unweighted compare graph small mode nearly near penalize unbalanced unbalanced way statistic sec rate compose shape cluster match sc fail reason sc cut balanced position cluster recognize region spurious along curve big sc outlier singleton recognize enable robust choice improvement focus unbalanced match rbf include graph sec rbf choose partition near parameter suggest nn meaningful point tb vs dim digit set total vary proportion fig graph adapt c vs vs vs rbf rbf rbf error vs vs rbf rbf matching rbf rbf rbf rbf several database dim letter dim recognition handwritten digits dim order even optimize illustrate network small detection small gaussian show cut average cut cluster leave cut fig cluster away decrease phenomenon curve correspond cut value minimize happen since represent impose flat attain deeply already point find say leave method relatively insensitive choice construction belong low encounter namely rarely point context anomaly detection complexity statistic construct spectral semi supervise unbalanced systematic neighborhood effectively robustness outlier degree framework cut detect multiple meaningful synthetic ability detect indicate utilize crucial step cluster rbf poor unbalanced spectral put balance cut unbalanced adaptively neighborhood tend neighborhood vary naturally cluster justify idea limit unsupervise semi set tool represent graph base learn identify critical unbalanced proximal unbalanced arise graph construction refer construction include nn graph link node graph rbf nn close versa recommend robustness propose suppose however unbalanced graph appear poorly conventional unbalanced proximal put importance balance size lead cut meaningful reason outline whereby objective proximal unbalanced section edge near explore section synthetic show significant construction investigate reason sc construction unbalanced justify let draw iid surface edge spectral denote variant cut unbalanced fundamental drawback unbalanced dataset illustrative draw iid proximal unbalanced density examine construction full rbf nn parameterized sc depict choice balance cut axis line axis sc seek achieve cut cut relatively proximal unbalanced relatively control balanced cut possible parameter account increase understand acceptable threshold outlier globally large small cut ratio discussion clear adaptively neighborhood nod plausible region adapt vary degree proximity comparison construction method learn calculate choose base distance dimensional anomaly detection sec range extreme construction close neighbor neighbor point follow optimize control minimum difficult uniform regardless degree around remain parameter third cluster algorithm alternate approach plus reader final step small cluster size optimize admissible small
sparse sparse component gauss sparsity variance component parameter treat unknown appendix se simplify output match rely principle match oracle em adaptation illustrate signal performance optimal form mse measurement completeness compute se recursion outperform adaptive reconstruction result characterize nonlinear poisson cascade response early pathway visual cascade field neuron amp combine estimation propose connectivity model gauss measurement pass vector q measurement channel nonlinearity also characterize ml section vector imply initially simplify iteration correct induction adaptive continue enough algorithm oracle know vector output asymptotic short mse nearly oracle know em adaptive essentially computationally scalar problem applicable provable learning learn possible extension implement estimation additive interpret product value scalar transform reduce update rule simplify relate need review vector element regard component pseudo lipschitz nature abuse notation q topological write write say topology follow along adaptation argument generate adaptive line z tp tp z precisely set prove non proving limit hold usual proving limit induction need argument hold thus derivation induction argument note limit respectively suppose since mean ab apply induction apply prove first end b h old k since satisfy limit also induction output conclude limit continuity prove argument proceed continuity continuity limit induction make equivalence p converge finite obtain equation similar prove limit two application assumption verify adaptation function fix scalar eq wish suffice convergent subsequence z z lipschitz show maxima maxima unique convergent limit condition satisfy analogous continuity manner immediately theorem part proof since already need need adaptation se adaptation definition hypothesis theorem false ed ed possibly non cascade model transform probabilistic possibly measurement call adaptive enable joint learning channel algorithm recently em posterior approximate message pass methodology apply prior identification nonlinear cascade dynamical system predict scalar perform method asymptotically consistent oracle know correct remarkably arbitrary systematic nonlinear provable random assume identically pass address distribution transform bayesian apply cascade dynamical response reasonably call pass belief propagation receive attention compress sense survey approximation relate exploit couple amp large performance condition optimality consider amp extend however although formulation attractive distribution limitation learn performance amp generalization algorithm general domain adaptive case amp adaptive scalar se converge coincide performance oracle know value remarkably result apply essentially unknown enable evaluation prior compress nonlinear cascade simulation method simple exact performance provable consistency em generic term mixture gm e step approximately parameter gm appear output simulation remarkably distribution sense argument present equation confirm numerically special particular choice contribution rigorous justification analysis however methodology way spatially open extended well alternate present source output call iteratively combine amp future work apply simultaneous unknown appeal conceptually strict alternate problem belong class maximal minimax approach recovery minimax may achieve oracle know em method due provably organize review non adaptive characterize demonstrate consistency demonstrating conclude conference appear paper proof description additional describe algorithm input output parametric see method along adaptation procedure case output set estimate describe two choice two variant bp approximations mmse vector function quadratic sum bp map equivalent mmse estimation scalar version mmse map generate random se empirically moment hold surely variable analyse remarkably arbitrary distribution function nonlinear letter compute similar physics I tp ti z j tr n tr standard algorithm consider adaptive show key modification two correspond output know datum understand via observe identically attempt empirical eq right attempt maximum ml depend role similarly joint component density ml estimate covariance briefly ml propose bayesian amp em sum amp output provide amp correctly distribution give implement via distribution every converge distribution give perform procedure particular choice adaptation rigorous point update simple family optimization search see similarly develop consistency expectation random variable output random ml adaptation convergence satisfy function weakly pseudo detail adaptation regard scalar evolution define similarly see weakly lipschitz functional uniformly open eq lipschitz continuously assumption c technical mild class functional function appendix continuity average q pseudo continuous continuous similar functional ml functional satisfy assumption vector output se equation empirically
code ccc combinatorial submodular middle disk axis row extreme middle fa consist list p mm mm equal trivial non explain cover fractional allow interpretation interpretation define small induce fractional optimal probably fractional weighted cover value minimum weighted cover combinatorial convex combinatorial positively ask exist large h wise extend combinatorial positively homogeneous coordinate decrease envelope minimal cover inequality homogeneity wise since prove second coordinate monotonicity homogeneity fw relaxation norm collection clearly submodular cone submodular cover norm integer note rather cover overlap group overlap w sum term sum situation subtle perhaps distinguished general sparsity behave correspond fractional weighted associate latter different still latent cover number one statement discuss support form core set allow intersection set actually must norm define weight submodular cover advantage need assess empirically remove instance three norm naturally relevant sparsity direct graph chain grid sparsity acyclic nice count worth present penalty calculation submodular thus envelope prior combinatorial submodular overlap natural function I norm include consider standard exclusive example formulation also much exception vector coefficient index combinatorial exclusive lasso sparsity impose group w otherwise explicitly pc j g j norm relaxation last would see result formulation w fa fa w easy verify fa exchange maximization eliminate dual actually norm norm particularly interesting since section operator form focus fa monotonicity thus example partition empty mention section group perspective study special norm submodular present submodular derive g extension fw extension extension vector set turn minimize e minimize sa submodular canonical submodular I q recognize extension simply negative function play crucial norm submodular particular norm say augment set strictly cardinality stable allow sparsity separable submodular soon set stable set terminology word sa fa core derive concentration inequality cardinality previously notion cover combinatorial show submodular extension submodular low combinatorial envelope converse envelope submodular retrieve minimal weighted cover context structured efficient gradient convex principle solve solution proximal proximal efficiently another submodular literature minimize separable submodular polytope dual fa sequence submodular w proximal problem maximize concave submodular divide take involve j ax ct ax ax see appendix allow support rate norm require weak submodular local small result sort reverse triangular involve gap complement separable consider response define study induce determine assume extend recovery condition proposition retrieve proposition extend support recovery normal jj extend condition let restrict eq concentration normal control via result diagonal stable either k overlap overlap surely specifically interval induce rectangular support p illustrate submodular group blue green red interval rectangular pattern lead possible combinatorial natural regardless support unfortunately envelope norm lose relaxation tight candidate axis make candidate interval design matrix norm elastic choose overlap w notation since unweighted example precede discussion regularizer recovery square permit optimal path hamming figure assess incidence generate support vary cosine report hamming figure outperform counterpart reasonably ham weight overlap although dominate hamming outperform neither well achieve vary lot indeed relaxation interpret prior unit relaxation display edge corner priori generally amplitude little vary encode regularization visible term similarly error relaxation likely corner value improve note relaxation slightly square first encode sub coefficient support grid constant combinatorial I cg distribution noise amplitude propose relaxation allow recover principle induce establish code combinatorial submodular establish priori question priori yield performance oracle specify priori acknowledge european project like thank provide fa p bb definition duality indicator function formulation show exclusive computation low envelope function g relaxation note illustrate derive let w positively remark fw opposite bound norm sum theoretical norm imply immediately therefore dual fa fa norm resp would three necessary sufficient condition characterize subsequent let minimizer set singleton component strictly w solution latter unique consequence convexity decrease respect equal contradiction order level j partition component eq eq stable statement fa fa p value j thus extension hence irrelevant formulation equal value I reasoning apply corresponding decomposition algorithm indeed denote I w proximal operator amount w last simplex decomposition minimizer q using obtain directly respectively proximal operator proximal section apply decomposition submodular section vector follow x mean covariance r r c jj covariance large invertible q jj jj jj jj p unique global simply show jj r c r j j result consistency let property decomposition condition thus z derive q z z concentration lipschitz bind expect minus plus minus pt plus pt team sup paris france proposition remark theorem axiom induce simultaneously vector norm obtain function moreover link representation lasso field identify express encoding support regularizers generalization lasso particular group formulation submodular convex reader overview relate detailed presentation given vector find appropriate combine enter convex control motivation stem regularizers penalty restriction unit model relaxation assume coefficient unit ball seem desirable assume continuously alternative paper combine penalty appropriate relaxation combinatorial function preserve capture envelope introduce relate convex group lasso exclusive discuss case analysis experiment section motivation follow part part parametrize encode approximation vector motivate surrogate code length naturally appropriate indexing denote shorthand element norm let would natural define cover denote penalty regularization positive since relaxation natural convex non arguably compute requirement regularizer ask formulation rescale instrumental give positively constant pointwise maxima call da h w hence positively homogeneous penalization gets obtain p w introduce support structure encode combinatorial envelope actually able capture intuition many number formalize next lemma minimal subset q prove redundant non redundant remove redundant recursively redundant still remove stop imply motivate upper combinatorial envelope compact capture canonical
mean natural derive embed simplicity assume underlie belong reproduce hilbert value operator exist h cauchy schwarz norm circumstance wish measure estimate embed sparse lasso rkh start multiplication closure pre hilbert pre take closure hilbert space reproduce know isometry idea I countable countable arbitrary exist k x v kx v kx first absolute kx kx norm continuity continuity operator fourth like show v I think I replace term x divide also operator direction one spectrum h independent value problem sure embedding rkhs kernel either non compact certainly new subtle first thing definition assumption measure etc normalize rkhs need hold xx iy I xx g ix show sup extend check regularity need measure reproduce problem reproduce assumption assume integrable x g g g contradiction together reproduce problem reproduce assume f iy switch mean eq fulfil nuclear sequence nuclear state element extend uniquely potentially kernel arbitrary forget lebesgue another incorporate guess could rkh actually want guess learn rkh learn powerful universal I constant rkh hold dim bandwidth control rkh bandwidth rkh allow old minima compare old paragraph couple assume hold pick fulfil issue cost contrary bad occur solution likely cost suggest balance surrogate tell roughly speak cost advantageous sg think people real see link easier need sg scale element scale use opinion make less theorem theorem claim example cs ac uk ac uk com com cs ac equivalence hilbert rkhs embedding distribution vector value regressor introduce function rkhs intuitive justification furthermore regression derive sparse version consider result value minimax state art valid assumption minimax upper rate rate reinforcement cholesky year framework reproduce space introduction one representation conditional rkhs appear naturally expectation ease rkh embedding application generalize expectation multivariate establish norm embedding behave hope obtain conditional expectation valuable incomplete prevent technique like resemble dimension access rich vector value apply result rkh embedding demonstrate connection provide embedding practical theoretical theoretical side establish novel significant improvement also show embedding resemble regressor year embed increasingly operator expectation naturally machine expectation advantage embedding integration reinforcement independence motivation space generalize expectation conditional embedding rkh norm conditional embedding behave would expectation feature condition valuable characterization since optimizer cross validation value loss value demonstrate provide embedding important side embedding give due measure require intuitive rate analysis side embedding resemble parameter demonstrate embed regression loss resemble apply embedding derive version embedding resemble apply minimax class low derive validation demonstrate conditional regressor connection embedding provide solid justification strict suggest regression specialize efficiently burden compute embedding somewhat choice place embedding connect establish idea investigate obtain version embedding introduction rkh reader give yx l map element thus apparent expand problem embedding define k h kx kx chosen suggest embed underlie section present terminology mean empirical ridge rearrange yet establish natural value draw value convergence derive sparse practical kind derivation estimate embedding rate alternative address key mean embedding value ordinary problem go back observe would regressor form label version vector establish regression draw empty measure one function vector take value analogy vx continuous reproduce exist evaluation mapping value relation reproduce property hold v unique sense unique isometry reproduce rkh limit isometry closure importantly perform replace estimate restrict rkh r thus arrive regularize q q hold fx tell prop value least estimate depend sample convergence th upper drawback fulfil embedding function rich restrict embedding conditional regressor hx complicated optimisation like restrict pick rkhs rkhs h v l operator kernel note function directly risk relate relation replace take loss add pose problem prevent hilbert section value rkh function satisfie embedding rescale embedding song attempt sec analyze immediate validation usual hold subsample j grid choose achieve good fold improve embedding present l ii hold separable vi fulfil examine risk objective x lp objective case r minimizer bad expectation subtle closely expectation originally function element acting function minimizer convention x couple worth hold choice approximation rkh decrease fulfil condition apart obvious word important thm divergence optimum suggest risk tell balance roughly speak allow function solution advantageous last comment estimate predict expectation value apply study rate considerably current state intuitive assumption obvious investigate detail section situation approximate sparse large want look norm objective equivalent optimum computation equivalent solve fista reach per ij zero otherwise q employ replace norm norm n ij encourage approximation output penalty ik link develop useful make use label embedding incomplete cholesky distribution reinforcement discrete done start direct angular policy useful dimension cosine angle angular cosine angle angular velocity approximation cholesky approximations generalization report value derive cross establish well low number problem valuable employ embed hilbert schmidt infinite schmidt result technique deeply embed sparsity equip certainly regularizer demonstrate fulfil thing separable space operator etc space infinite space compact compare element decay something thank ep union fp integrable particular fulfil fulfil x ii exist h x furthermore right q assume show measurable norm x uniqueness way h p e mx h x reproduce lem thm get eq reproduce proof like x follow q functional hence integrable consist furthermore borel algebra fix h nz h bn subspace thm iii sub converge weakly th closed iii weak hence x lx integrable function approximate simultaneously l compact pick product let ia lem represent integrable construction wise measurable measurable measurable gx gx value hence supremum p general weak certainly general measure borel topology finally similar x induce construction space algebra borel space exist lem space cb integrable fulfil group category summarize fulfil verify fulfil separable completely separable able certainly dimensional countable h closely separable finite respect rkh separable vx I dimensional schmidt simple borel algebra continuous continuous borel couple generating discuss detail want infimum h term rkh norm converge infimum infimum attain iff rkhs norm intuition fulfil appendix integrable h generating formally measure transformation induce concern generate fulfil follow appendix guarantee fulfil need lebesgue fulfil concentrated element integrable l furthermore integrable v h integrable integral v sequence converge infimum infimum intuition fulfil need sense optimize integrable norm definition find converge infimum infimum attain rkh rkhs hence weakly iii furthermore close weak thm move suitable equality dimension
rkh recognize explicit trick symmetric k finally kernel taylor expansion reveal correspond rkhs extend allow variety kernel necessarily common choice option variate express dimensional obtain restriction model mean I array note wise eq array implicitly level separable dimension exclude fix vector explanatory form independent mixed stage exclude array density z km variate b z multivariate obtain estimator explanatory model b k km j variate b kb test univariate kb l kb statistic lambda hypothesis available generalization curve array response place observation take design array mx x zero hessian maximize linear estimator attention know array along mixed estimate efficiently likelihood decomposition maximize kronecker product eigenvalue eigenvalue decomposition along estimate using criterion eigenvalue covariance real section show change proportion array array variate randomly cell implement cell proportion calculate independent replication improve htbp american combination trait protein date incomplete array datum selecting cell cell addition snp marker along dimension assume trait dimension trait observe estimate replication generate combined environment mapping trait height yield year condition marker fashion accuracy array entry trait correlation miss corresponding replication setting datum involve model sample increase stand array variate correlation response miss estimate know covariance htbp formulate array variate develop estimation possibly application way imputation estimate array part array value reason poor might dimension decrease extension model another array array suitable array variate response dimension allow make prediction combination separate section remark challenge deal high arrange array propose method variate partially observe develop imputation mixed effect model multi define algorithm spectral recommend simulation real life involve effect trait array variate array variate repeat array stacking array etc variate dimension array delta array kronecker array accomplish array multiply arrange form include video spatial measure array kronecker parsimonious variate model sample array array many main array development effect useful effect incorporate along mean variate model calculate likelihood method explanatory genetic marker model updating define semi array variate model study follow density delta give order matrix generalize multiplication multiplication array element wise tucker mode operation unfold element th stack dimensional level operator vector stacking array j jj j follow useful property array normal variable kronecker delta covariance remain assume root decomposition definite put overall matrix th evident context stand zero dimension miss em usually utilize go back adopt partition represent miss eq zeros eq vector lr rl covariance calculate assume write stacking therefore treat model variate flip prove attain parameter variate normal flip variate incomplete variate kronecker delta imputation modification flip repeat likelihood datum last update calculate
compare descent correction correction stop last correct coordinate descent simulation newton type correction descent correction invert big condition scenario coefficient solution make explicit find notice involve spurious terminate extreme predefined terminate pre application upper reason common restriction transaction cost put maximum select stock limit stop rule always point experience decrease set solution towards kkt hold otherwise coordinate check two depend choice mcp usually accomplish cross argument problematic glm bic add extra top bic subset pre estimator mcp concavity regularization mcp glm propose lasso convexity mcp penalty main glm condition hold mild regularity manifold dominate concavity convexity therefore small kkt valid global condition eq kkt long stay decrease mcp exist package mcp exploit active method appear later method mcp decrease stay activate lie estimator sign step experience point satisfy kkt later long convex path stable accordance size active active derivative respect intercept penalize sign enough order mcp hard enjoy adaptive rescaling similar replace diagonal coordinate algorithm update coordinate rescale together new need u kn k avoid mcp although long derivative correction avoid compute implicit derivative likelihood turn formulation coordinate stop lasso activate mcp inactive activate b mcp lasso lasso discuss mcp possess stay value interval tuning parameter conduct study package penalty highlight first use enough fast package handle penalty penalty different adaptation method logistic popular lasso mcp compare package report include cv fp tp mcp model sparsity level setting covariate zero four setting consider table repetition perform dot path mcp lasso path dot mcp respectively dash line dot mcp solid line dot correction panel line small get select correlation selection criterion tp method newton unstable quickly recommend hybrid newton path switch fp tp result table however provide mcp find overall compare job mcp provide increase mcp correction correction exceed square miss lasso conjecture modification subset active variable job mcp penalize particularly penalty entire example different take time dimension zero repetition solid line dot mcp dotted line solid line dash dot line path line line becomes select get near zero positive fp report repetition standard error c package fp tp cv cv mcp mcp cv mcp cv mcp cv cv mcp cv cv mcp mcp cv fp repetition error l c package tp lasso cv cv mcp cv cv mcp cv cv mcp cv mcp logistic lasso mcp flat level mcp square root jump lasso path cause correction path concavity continuous get smooth tend yield perform much well cv case mcp job fp logistic regression main reason penalization another interesting behavior mcp present mcp parameter previous work warm start find covariate change quadratic path warm affect repetition cpu l previously come microarray project gene profile free survival patient diagnosis subject available negative gene negative year small regularize regression suitable impose mcp accuracy yield package split set time test simplicity use tune mcp turn high mcp path little misclassifie subject notice test mcp test well mcp l error propose calculate penalize likelihood estimator real evidence update correction vector concavity mcp maintain stability newton public website x regression condition give mcp penalize kkt define rescale derivative eq iid poisson regression kkt active mcp penalize poisson kkt q thank associate constructive comment author thank na stroke york ny superior approximate penalize penalty concave mcp hybrid modify method coordinate simulation statistical include modeling miss first many promising bioinformatic finance throughput easy make handle mathematical attempt aic addition also unstable dimensional computationally prohibitive numerous attempt modify burden regression equivalently pursuit penalty convenient regardless penalty well mcp also linear glm dimensionality view interested dimensional likelihood estimator lasso regularization parameter vary penalize least lar homotopy entire path linear mcp local approximation yield optimize lar unbiased linear penalize quadratic spline penalty coordinate descent considerable attention include include linear derivative glm regularization calculate estimator glm small active lar also regularization generalization net exploit coordinate glm likelihood ordinary differential base quasi lar straightforwardly coordinate mcp scad glm quadratic approximation descent estimator mcp penalty adjust concavity penalty find minimizer less change glm rescaling range relate paper include glm different difficult explicit glm feasibility previous warm convex inspire rescale concavity adaptation mcp glm algorithms concave glm mainly mcp extended quadratic penalty detect
acc image simulate namely fig truth separate evaluate recognition contain common sub image randomly image image sub sub image simulate pair thin spline example test point perturb gaussian independently perturbation detector collect sift descriptor use correspondence true among score traditional statistical acc affinity acc gs tune ground eliminate outli elimination acc utilize information outli elimination vary outlier acc gs task challenging method design however outperform find acc gs ab ab c construction omit construct store initial cluster affinity pair cluster require complexity initialize cost find affinity cluster affinity cluster compute complexity matrix complexity loose algorithm reduce maintain store near cluster updating affinity ab ab ab ab ab neighbor affinity affinity neighbor neighbor therefore measure ce ce define overall mapping small ce result c link mnist region inspire difference connectivity sort consecutive score treat outlier acc gs method outlier thm com engineering university advanced technology chinese china simple agglomerative explore different role concept cluster average characterize around insight define affinity average affinity fundamental vision outperform computer involve cluster mean determine cluster agglomerative begin select affinity merge agglomerative conceptually produce informative agglomerative limitation computer different shape manifold conventional agglomerative usually directly pairwise manifold cluster sensitive noise tackle agglomerative various machine rarely agglomerative build nearest nn show lie cccc average linkage c linkage linkage use fundamental concept theory characterize affinity vertex dimensional vertex cluster reflect near density cluster density effect successfully wide web social show result vertex define via affinity inter vertex two truth synthetic affinity naturally affinity vertex advantage noisy multiscale cluster comparison linkage linkage affinity propagation ap sc direct cluster multiscale multiple greatly automatically extensive correspondence demonstrate superiority art fundamental matching suggest many affinity express product external library extensively employ cluster finally acceleration especially scale dedicated agglomerative linkage affinity distance capture global datum sensitive linkage method propose mining community satisfactory fail tackle dimensional sophisticated affinity observation several agglomerative min cluster toy suffer affinity min describe structure affinity inverse affinity slow sec segmentation besides agglomerative mean use mean density shape spectral handle greatly existence eigenvector laplacian affinity propagation explore intrinsic message perform manually spectral cluster graph task affinity keep algorithm utilize robust set vertex vertex edge manifold high space weight nk degree begin initial iteratively affinity merge nn graph vertex construct initial undirected subgraph path component undirecte direct undirected edge present detail initial v ccc cm affinity square fair close keep affinity affinity keep affinity agglomerative affinity affinity vertex vertex connectivity quantify concept vertex average ji w sec characterize nn size normalize favor cluster instead merge small connection normalize well unnormalized degree vertex merge cluster strongly connect mathematically vertex cluster strong otherwise intuition statistic affinity robust asymmetric summing q symmetric affinity affinity efficiently affinity index correspond easy lemma theorem use average linkage three conventional linkage although find linkage nn pairwise linkage linkage simply direct base linkage measure b experimental sec superiority linkage implementation accelerate time merge ab ab ab use ab store asymmetric formula ab row material formula present material maintain pair cluster set affinity cluster neighbor nearest cluster merge update neighbor probably among near create neighbor nearest b summarize material please time section demonstrate experiment run matlab ghz carry six publicly benchmark database write digit mnist database testing dataset image intensity feature euclidean pattern linkage link cut multiclass normalize cut sc spectral use handle distance distance fairly fix set dataset algorithm cluster performance quantify
irrespective deal deal outperform dependence clearly parameter depend run accordingly increase almost inefficient ylabel white north blue thick cd option solid dash bar chernoff inequality use easy obtain e start start end nk da ks kt ks end end bn k start start parameter output u theorem theorem axiom novel attribute time evaluation sparse graph sampling kronecker neither actual far enable rational design algorithm concern scalability become issue especially become million twitter regard kronecker graph attractive traditional latent scale thousand scale million recently realistic practice surprising clearly parametrize million order expressive recently name multiplicative argue matter attractive generalization ask part sampling address na expect I extend theoretically outperform knowledge address behind certain match classic accept reject address proposal compactly guarantee integer direct order target direct graph straightforwardly convenient adjacency exist I call edge allowed call either graph kronecker multiplication matrix define real kronecker product np kronecker product parametrize additional parameter kronecker call observe edge adjacency note matrix definition adopt convenience array expect view associate node digit verify edge write may resp denote absence assume attribute obtain bit node representation node need bernoulli variable additional j sampling entry individually computation graph edge normal chapter convert position ball drop coordinate solve employ divide graphical appendix code rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle edge thick rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle node scale rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle thick thick cell imply observe edge position divide weight fourth choose recursively location place pair generate multi poisson independent instead poisson good g poisson parameter taylor interested graph poisson modeling consequently generate sample analysis bernoulli non negativity enforce parameter bit entry efficiently reveal difference th map bit integer sampling matrix rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle scale rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle color easy verify chapter adjacency matrix sampling cc ba eq filter poisson adjust step remain summarize first generate convert generation pseudo reject target first generate like find remainder section devote behind construct proposal generate definition hold investigate complexity generate edge expectation overall complexity speak irrespective resolve suggest heuristic instead careful color color color behind var mean thus behave high rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle node rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift cm rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle yshift cm xshift rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle rectangle concentrate highly probable relatively construct eq bb cover hand b cover frequent color notation follow prove discuss validity pseudo code overall generate calculate expect find many illustration legend pos south pt marker xlabel ylabel expect color black x marker thick dash blue thick dash km legend pos width height pt marker xlabel ylabel black marker marker dash blue marker thick I b overall k km e simplify equivalent least attain guarantee estimate select value empirically scalability design number edge algorithm write ghz title expect ylabel cycle legend pos north blue thick bar explicit error mark options solid error bar cd explicit mark scale title xlabel edge ylabel cycle black white legend pos west blue cd x mark thick dash cd xlabel ylabel name white legend pos north west bar cd explicit error table color red mark solid thick title xlabel edge ylabel legend pos blue mark bar cd explicit x mark option solid dash x xlabel ylabel black legend pos north bar red option solid cd x cluster scale xlabel run cycle white legend west blue error
mass pt component length put assumption hence put mass portion away support length portion maximize two together long k cover ball gx gx dx dp direct p b supplement p p follow effectiveness semi establish density estimator density appropriately sensitivity two show adapt cross validation excess ef rf f tv notational denote expectation everything ef nu inequality fx mm ef ef ef therefore cb inequality define rearrange use concentration probability respect validation follow sufficient achieve introduce series set use slightly roll show sample size repeatedly sample computed estimator size range supplement repetition error bar indicate size increase decrease beneficial provably outperform mainly distribution concentrate family control supervise semi supervised depend hence advantage extend density diffusion estimate density sensitive work broad two exist besides estimator report air support air force fa use label datum together invoke marginal ad hoc work example foundation analyzing method include mostly unlabele together label know motivate select huge much toy example unlabele challenge boundary label datum reasonable link unlabele estimate ss cluster call distribution develop always explicit outperform supervise precise improve inference paper improve manifold assumption parameter control assumption strong add deal outcome formalize define depend consistently supervise condition predictive depend control define denote datum unlabele estimator supervise conclude estimation label extension discuss condition discuss useful cover therein statistic include paper method strength classification covariate upper could smooth dimension free create boundary rate approach may version idea follow metric sensitive minimax discussion technical article let sample let denote work area region put mass precise pair curve unit everywhere curve eq supplement correspond euclidean definition formalize define scale space lf lx lp z n p ball cover cover function estimator mention showing concentrate small covering say eq short path connect regular covering cover cover ball regularity thus diameter use estimator detail supplement event depend unlabeled suppose every n since condition distance eq b h I simple page result np nm characterize much suffice intuitively
impose must family maximize natural parameter variational variational update gradient term independent variational connect therefore summation term term use h put together initialize pt update ie low q update precisely variational pass message conjugate factor unlike kullback problem consider might term univariate simplified case derivation calculus technique suppose denote write eq dd da ad express product simplify detailed manuscript cycle n ts f ty I I q I q negligible pt pt variational belong approximate posterior normality indicate perform discussion inverse message ic qx r j v say element update obtain formulae parameter conjugate vb update compute ne q simplify update connect evaluation gradient vector poisson evaluated analytically handle construct effect variance negative use quadrature compute gradient reduce high integral univariate give appendix run additional update experiment bind trend instability begin converge change initialization increase monitor change bound lower maximize variational tight useful traditionally computation bayes role prior odd favor q ratio marginal likelihood favor comparison likelihood review low place log obtain equally probable verify comparison cross selection mixture note bic straightforward degree criterion account estimation uncertainty default address issue see performance consider set initialize update cycle significant penalize quasi assess gold via use interface implement specify parametrization specify produce well use penalize quasi three discard chain apply deviation time update effect code dual processor windows pc ghz study poisson intercept logistic intercept intercept logistic intercept model set generate problem may matrix fix effect penalize initialization serve guess estimate penalize quasi algorithm different include l algorithm mcmc gold root c c penalize fix update partially update posterior effect center quite close deviation effect shorter lower attain parametrization update take converge produce parametrization update logistic centering center posterior mean deviation partially parametrization center well mean center take partially parametrization fast provide penalize quasi difficulty mcmc logistic posterior deviation fit mcmc second magnitude example produce fit close parametrization took attain high easy center perform big parametrization tuning although use partially update dash posterior estimate partially dash line figure mcmc could parametrization effect quasi fit previous real center partial centering centering tend consider variable status child time age intercept slope estimate different penalize quasi likelihood initialization method center short deviation improve upon improved deviation fast fast example hand straightforward aic analyze whether intensity sample call parent extra take treatment parent parent per model stage pt final th offset ij ij ij ie correlation intercept henceforth determine interaction retain model ij ij ij ij preferred model arrival drop ij ij ij prefer drop drop treatment turn drop e ij ij ij u indicate none time well effect add arrival c time e conclusion parametrization thus sufficient taken fit short likely cross deviation solid parametrization dashed center fit partially parametrization centering tune slightly close mcmc density good tuning performance pass recently focus poisson logistic longitudinal quadrature intractable integral parametrization center partial automatically toward center case partially parametrization upon well center fit close partially parametrization rapid center particularly slow partially parametrization center give tight parametrization variance improvement could parametrization offer fast alternative mcmc compare bind selection without x pd r bind p q poisson response I qx I logit link function tv ij evaluate quadrature see appendix p nr expression r bx nj ij univariate ij r bx b vector say q I gauss quadrature gauss quadrature integral correspond quadrature sample range recommend gauss quadrature ij sample appropriate deviation mode ij rx ij significant implement gauss package quadrature approximate integral quadrature example l partially support reservoir research thank constructive suggestion improve thank available simplified multivariate pass reading width effect computational center partial accelerate bayes vb implication examine partially model four variational message partially determine parametrization third accelerate accurate approximation center chain greatly choice parametrization improve partially hierarchical bayes deterministic interest vb intractable factorize factorize implementation algorithm message passing examine partially vb model four variational passing partially able adapt quantity center parametrization show partial approximation center produce generalize inclusion account observation applicability numerical quadrature mcmc integral intensive various likelihood approximation consider integrate nest focus logistic mixed model datum literature parametrization partial technique inspire van hierarchical center mixed slow old center complementary role neither partially parametrization lie center parametrization gibbs yu boost mcmc center expand vb method update speed vb parameter seek hierarchical show parametrization property gibbs sampler similarly center vb hierarchical motivation partial vb algorithm td w iw x iw x ty tx iw linear effect prior know let tuning parametrization interest leibler divergence qp py py py leibler minimization leibl maximization lead py expectation vb pt iterative iteration tune rapid convergence center true posterior recover partially suggest partially parametrization outperform vb rest organize section parametrization passing discuss briefly consider simulate conclude cluster denote th distribute canonical dispersion specific assume depend effect predictor q vector fix effect allow
inequality establish close v sa connection induce exactly incur encode close close quantization wasserstein past highlight link hold dx key show support close close combine behavior measure problem quantization quantization codebook situation depict draw intersection nan ambient absolutely jx decomposition atom would count atom double counting imply correctly quantization codebook clearly quantization cost measure absolutely part hold support unit constant note strong know convergence number formal manifold effectively force assumption absolutely describe several widely interpret output free pca performance quantity minimize set see reference connection problem respect average distance minimize datum behave question characterize dx performance quantization measure absolutely continuous inequality almost surely strict x dx k characterize measure vary choose widely deep include characterize empirical population fundamental speed law number mean measure propose diagram show decomposition prove use decomposition arrive optimal whereas label simple ht blue color triangle depict upper optimal quantization manifold technical bind optimize side probabilistic bind convergence absolutely measure intermediate optimal appropriate small output good bottom figure give absolutely let hold stem bind term c output k optimal suboptimal estimate introduce currently match low mean automatically institute technology mit edu problem metric support establish learn theory bound classic probability derive course probabilistic convergence unlike study random error probability typically analyze density closeness two see reference therein low particular estimator manifold pointwise convergence topic riemannian manifold wasserstein distribution approximate measure respect deep connection field quantization sec sequel widely unsupervised mean estimate wasserstein novel work field technical summarize estimate measure measure convergence number unlike probabilistic probabilistic bound measure end formulation well relate dimensional manifold inner product let measure th moment wasserstein law respectively guarantee metric space consider probability distance sequence iff evy point excellent wasserstein difficult combine wasserstein bind strong wasserstein distance use phenomenon algorithm unsupervise extend sec sec induce population measure useful h old take strong establish population convergence sense arbitrarily fast convergence upper absolutely recently propose prove optimal relate distance permutation matching
dx frequentist jeffreys normal arise formulae lead contradict full frequentist use really lie interval determined factor full determined formula jeffreys depend construction dx bayesian approach coincide frequentist frequentist determination confidence frequentist coincide jeffreys prior support investigate relation bayes frequentist determination coincide always coincide jeffreys value probability frequentist investigate frequentist frequentist x dx frequentist determination coincide frequentist coincide jeffreys ref relation credible interval find probability density probability random lie unique limit interval inside big outside determination confidence general depend determination level consider dx correspondingly eqs find frequentist
completely turn observation highlight conceptually simple equally many part detector split comprise asymmetric introduction result significant gain happen increase translate aspect fail provide nonetheless example object pose car g illustrate scheme try look encode homogeneity visual homogeneity within simplify lead well semantic heuristic directly appearance cluster insight detector refer involve annotation object heuristic work learn complex several computer viewpoint annotation associate separate right cluster group video simple instance cluster annotation space category significant cognitive seminal rely member closely relate train use belong positive classifier although promise overfitte emphasis place example explore intermediate spectrum detail detector analysis label bounding box positive instance wherein training model latent binary task hinge training parameter control separate hyperplane indicate latent using mention early step initialization initialization experiment find algorithm euclidean difficulty merge phase calibrate appropriately address transform output sigmoid yield score comparable possible specifically directly reliable one calibrate learn logistic overlap score box indicate predict box help detection experiment car cat table tv result part turn twice resolution challenge protocol row baseline row show visual detector baseline improve detector job cat class supplementary material category aspect ratio leave heuristic ratio rather part template twice relative template amongst high template detector pyramid fine pyramid visual par result align see supplementary thus model discriminative detector train visual detector seem visual actually suggest relatively visual part issue detector root resolution template eight part template detector part detector latent round detector preferable difficulty analyze tv plot variation gradually increase around key requirement success latent appearance initialization compare aspect aspect base lead performance drop notice initialization discriminative help mistake initialization first intra scene category exhibit visual viewpoint scene category visual expect aid scene scene scene database collection organize exhaustive scene well fine grained scene arrange leaf category node hierarchy original experimental human arrive e choice versus level classifier category level half training half classifier belong except serve negative instance positive ignore care tackle intra p feature representation descriptor bin contain patch filter acknowledge unlikely multiple performance use descriptor choose simple analysis focus generic see material category evident take close discover discover correspond level contain front car fine grain category category subsequently deep assign category category seek analyze benefit annotate run initialize initialization unsupervised interesting supervision create grain category may wherein imagenet basic category fine grain need annotation effort contrary belief part contribution need benefit performance simpler interpretable benefit scene human supervision grain contribution part detector orient part secondary vision include ordering
observable term physical physical model value model evaluate uncertainty variance model rather specific demonstrated example commonly illustration dimension indicate correlation use effort calibrate denote pdfs represent conditional note denote probability bayesian acyclic form conditional probability parent representation uncertain well herein relationship write construct chain direct need pdf experimental begin normally normal probability discrepancy process index deterministic deterministic conditionally deterministic variable exist conditionally variable derive observation set correspond similarly eq represent experimental input calibration base point practical series technique quantity available bring actual interval likelihood quantify likelihood develop normal half multiple interval available observe thus lie hypercube hence derive mixture value type challenge particular prediction system replicate may take characteristic g tt time discrepancy k likelihood series measurement time input choose equation complex pdf need make may need time theory matrix instead series replicate e repeat another attribute directly repeat assume variance repeat series series negligible treat condition consider become simpler simplify become real operation prediction kalman extend particle calibration etc implement interest determine identifiable infinite bayesian sign pdfs pdfs flat infinite non identifiability practical non identifiability structural identifiability redundant identifiable level identifiability insufficient bias noise due successful help redundancy detect identifiability overcome develop identifiability analytical explicit state solution analytical various analytical non whereas address second identifiability necessary identifiable rigorous identifiability analytical function theoretical profile likelihood effective detect identifiability preferable non identifiability become since evaluation require likelihood taylor identifiability model analytical detect identifiability insufficient likelihood thus demand detect taylor practical identifiability physics calibrate new form discrepancy measurement mean random expectation accord expansion show use number e experimentally observe input setting uncertainty system system infinite satisfy vector identifiable assume quality cause identifiability former suggest model either insufficient continue infer non identifiable order redundancy detect identifiable may eq retrieve checking dependency since column column column dependent identifiable use set identifiable derivative identifiable parameter remove th column remove model see physical corresponding identifiable analytical expression derivative forward give model rank whereas identifiable identifiable also identifiable calibration due reason function consume actual sparse may along simplified discrepancy another sequential step pdf represent probabilistic repeat mostly surrogate much surrogate krige gaussian polynomial svm machine surrogate uncertainty uncertainty random various source uncertainty natural variability datum input likelihood calibration even surrogate case approximate actual parameter surrogate instead function radial integration pdfs parameter efficiently quadrature widely computational effort conduct unnormalized mcmc slice etc multi physics individual analysis ideally calibrate experimental model method discuss employ variable versus versus share network model individually existence two connect full network option calibration present option represent except e posterior pdfs option expensive dimensional option calibration follow calibration present calibrate parameter combine option pdf pdfs base computational effort option option option flow use calibration pdfs whereas base bayesian calibrate option calibration existence basic bayesian method issue application calibration identifiability difficulty multiple physics model available parameter relate physics numerical implement multi physics rf device identifiability perform use first taylor series base identify rf physics unknown temperature six barrier height fp attempt frequency mass conduct nm mm combination repeat four current second available series observation replicate noise series time term gaussian example least square without consider difference prediction square correspond experimental versus rough appear exponentially significant varie adopt input exponential discrepancy addition variant trend covariance identifiability select since addition identifiability unknown corresponding suggest identifiable taylor form likelihood formulate observation construction require determinant inverse rise around determinant difficulty subset reflect closely precise convenience advance sparse account posterior use range kde posterior pdfs length corresponding pdf significant expect calibrate predict validate calibrate model map unknown fig density calibrate without calibrate model blue prediction observe fit prediction square standard probability cover prediction account take full pdfs demand square expect second later reflect wider early indicate discrepancy density device switch apply contact exceed threshold device close reliability device euler model simulate device take apply air input loading pressure air gap correspond force device geometry material boundary condition device long term loading calculate value cause contact due limitation currently measurement early collect device keep switch form study type device device relative type device construct hour discrepancy construct relate give common third option present option identifiability calibration check expansion method device directly half bayesian obtain rank e identifiable point also examine identifiability hour identifiable since measurement option bayesian prior pdfs pdfs posterior pdfs black fig note pdfs example except calibration pdf calibration calibrate posterior pdf fig marginal pdfs corresponding statistic calibrate pdfs statistic fig h posterior prior common pdfs calibrate grid computation fig option pdfs statistic calibration draw fig due pdfs first two pdfs calibration use give difference posterior pdfs insufficient reduce addition table third two calibration option give calibrate bayesian integrate calibration computational source available discuss calibration interval
write inference make intractable tractable fully factorize posterior nm n nm component turn factorize cluster instance unity note depend therefore bind maximize bind r keep update unity however part perform step inference estimation without class cluster visualize arrange distribute scenario arise target different location instance denote classification refine without share site careful look equation reveal server server site dataset locate site site cluster split multiple place always performance transmission one another summary updating update variational server helpful conceptual sharing need server place server computation carry illustrate fig respectively framework case class site avoid due space already precisely datum assess capability e store location semi benchmark create supervise ensemble target label remove logistic regression discriminant ensemble privacy preserving present accuracy post hoc show significant among accuracy motivate soft transfer acknowledgment support university usa privacy combine supervise site privacy aware computation instance target accuracy share extract knowledge centralize file database due variety recently emphasis source via simultaneously mining approach preserve mining technique restriction inference database ii record iii technique party party communication meanwhile notion privacy expand substantially focus privacy recent approach differential privacy notion impact large mining association also limit partition partition typically privacy true early datum algorithm site privacy aware bayesian combine effective address combination classifier prove cluster ensemble improve combine ensemble classifier cluster nice unsupervised classifying thereby capability motivation mind combine instance idea classification issue provide label base detail employ label instance target datum apply input consider produce consider comprise align label refined target label instance assign
temporal nonparametric predictive greatly hard nonparametric em like yet mixture pose identifiability suffer identifiability identifiable spatio dynamic simulate new one condition check simulate prediction unobserve principle intensive much plan imaging analyze forecast indexing variance delta become finitely gaussians conditional weight make forecasting hard parsimonious sufficient unique forecasting forecast point forecast view highlight benefit us past light cone configuration say mixed optimize h note theorem conclusion conjecture definition exercise lemma remark summary department nonlinear high spatio simulation extend hard non asymptotic greatly forecast limit implement available recently introduce reconstruction spatio temporal every spatio temporal associate prediction future single agglomerative state minimal capable soft often predict well robust allow introduce soft version optimize like naturally interpretation state em automatic corresponding prediction predict hard fix notation spatio process random tn dd optimally future u conditional x propagate therefore spatio neighborhood possibly past analogously event could influence dimensionality light figure illustration plausible find still spatio temporal pattern predictive x refer configuration predictive minimal invariance light cone regard relation among process oppose realization encode sm thus simplify estimate prevent use joint pdf satisfie completely distribution restrict applicability cone forecast alone spatio implicitly specify predictive I exactly correspondence mapping equivalence class predictive state formally keep distinction predictive hide soft mapping interpretation minimal state introduce component randomize version optimal constraint tell get useful fix give solve evaluate candidate moreover nonparametric direct nonparametric likelihood kernel approximate variable play role turn variable current likelihood come brevity x soft log update n matrix kde th get hard bandwidth r estimation normalize ij forest currently slow conditional simulation adequate start difficult bandwidth even increase feasible either would search cf split x train obtain predict iterate pairwise test k sample mse mix state variable since none estimate like weight update step although simulation henceforth usual stop em algorithm density test metric merge datum drive fitting model start reach optimum iid thus cv integrate mixture datum weight difference q combination mode simulation predictive mix practical evolve accord evolve observable integer word say state sample near site include become kx k start state nonparametric estimate top plot log realization vertical right discard trace alternate blue patch observe clearly residual fit residual mixed initial first half estimate drop mse reach optimum merging forecasting return optima prediction fig show mixed practically compare visually indistinguishable residual obvious forecast independent realization train future outline state low initialize initialization remain nine mse field mse mix
accord default comparison simulation candidate estimator value eq computed option estimator evaluate empirical square compute average number positive fp repetition explanatory control large three follow z g z g model give design table show performance minimize understand ideal minimize redundant large pl perform well pl nonzero additive correctly pl bad redundant component performance estimator candidate make pl cccc sl pl fp refer estimator sl estimator pl penalize refer cccc fp ideal select variable fp refer square fp refer select variable fp standard analysis restriction model denote positive restriction smoothness preliminary theorem concentration around variable primitive condition concentration see discussion component exclude fractional preliminary argument concern restriction grow section second although lasso result selection state j analogue infinite dictionary independent condition large case obtain reasonable believe long stochastically procedure theorem condition c depend eq possible exposition despite contrast characterize variable reflect ii reflects reflect behavior term collect property issue comment proposition lemma go bring subtle technical issue proposition lemma obtain growth dm ns dm allow allow slightly condition exist g c assume magnitude situation unfortunately cover inspection condition exist cover observe replace union proof state connection model additive scope review control doubly penalize general hilbert establish split double remove shrinkage cause simultaneously smoothness two proposal estimator modification replace use group lasso jk adaptive lasso post adaptive g besides notable group smoothness estimator notice analysis additive separate sense argue introduction adaptive oracle strict sense directly comparable investigate estimation dimensional additive especially generic overall help bias double penalization explore suggestion practical use c depend index argument define go event may c j q fact similarly invoke restriction ij jj n bind q c substituting wish j noting use p desire share dr author economics mit greatly grant aid lemma section proposition remark regression nonparametric potentially additive step implement penalize square penalty reasonably despite intuitive theoretical selection random generally contain redundant additive component derive implement group lasso additive identically situation larger sparse let denote without class positive q identification interest additive interest conditional despite redundant make sparse contain ii possesse nonparametric novel term event thereby enforce sparsity result control thereby overfitte result see progress theoretical show e penalty reproduce apply estimator regularity boundedness achieve risk minimax point double penalization selection bring correct choose tuning optimize shrinkage sparsity recognize parametric square select lasso remove observation step additive implement penalize devoted two generic situation variable apply speak stochastically redundant significant interpretation rate could know second term effect select redundant significant one plausible guarantee perfect hard group adaptive lasso need general situation perfect separate considerably fact side achievable make meaningful comparison exist estimator performance perfect first despite smooth primarily first selection hence refine additive recent theoretical apply estimation additive idea deal e approximate class complexity namely deal complexity regression expand procedure regularity estimator meaning achieve rate fail happen perfect enjoy adapt adapt believe result finite two alternative estimating work estimation post analysis extend note build paper important nonparametric ii step smoothness provide proof study rely omit case sequence notation exist independent unit sphere integer use supremum symmetric positive symmetric square root index agree describe appropriate second penalty result give j theoretical make show give lasso function except consider subject tuning control estimator give show give choice let convenient concentrate ig ij ig ig ig ig ig ig ig j dm g jk jk n euclidean know present estimate group lasso
simulation variable exact learn exact elimination feasible structure learn heuristic pc importance bn model evolution gene interval instead depend use distributional present important limitation global normal expression continuous skewed unless cox able power multinomial sequence ignore subsequent inference aware trend particular allele comprise outperform gene complicate case molecular least process perform impossible pathway network prior know selection present drawback separately accept task parallel intensive backtrack information speed possible increase positive false negative include markov merging gene proof correctness hundred observation raise problem correctly molecular extend model flexibility incorporate knowledge modern application field diverse bioinformatics customer survey weather forecast system biology among dimension raise curse challenge affect model associate used express graph principle separation two independence commonly undirected acyclic choice datum aim focus mostly associate conditional table consist conditional ideally coincide dependence structure identify difference theoretical approach conditional base fit al al al structure operate correspondence graph must dependence definition observable really unlikely structure implement rarely far common graphical define practical reason local distribution involve small apply modelling problem single regardless maximal non mass clique function accord chain rule clique property identify graph share child markov make system biology employ gene aim molecular mechanism commonly available gene presence code protein protein result nucleotide biological reason mostly vary nucleotide individual possible call interested grouping determine gene node indirect influence unobserve property completely molecular unable measurement direction causal pathway reflect pathway flow molecular consideration protein interaction limit cell interested one human production etc capture indirect genetic effect unless trait gene genome identifying strongly bayesian problematic care gene trait association expression association may hand gene locate likely together configuration occur less expect phenomenon strictly gene link undirected direct trait compose measure rna pattern probe intensity rna measure several intensity result study include meta difficult practice use furthermore within abundance systematically biased chemical collect grey control involve molecular nature gene investigate pearson robust gene relevance network construct correlation correlation order correlation rather presence determined corresponding genomic stein gene undirected scale direct graph microarray review low mean usually unable equally behave produce feature protein expression protein cell time involve perspective apply datum important note protein sometimes gene study network fundamentally gene expression datum genome rely indirect provide individual limitation gene state individual snps snp differ single pair possible variant combination aa model easier independent aa aa individual allele aa individual model perspective model snp discrete option multinomial receive literature approach variable model additive population structure often implementation regression mostly framework statistic genomic et al graphical systematic extend classic equation panel contribute trait significant lasso context model include bottom panel justify computational interaction snps even able completely hand picture capturing trait I implement analyse present crucially vary substantially depend
potentially dimensional yu segmentation euclidean ultrametric journal pages f journal theory practitioner l scientific wu grid classification international computing mind http f ultrametric model mind ii text content http lee statistical record van van sparse face xu clustering comprise value attribute imply ultrametric topology great powerful hierarchy proximity match address address th thus report grow decision planning include type text structure storing scale capability intensive support exploit datum activity big worth flow topology greatly involve challenge issue directly discovery well logarithmic constant dimensional exploit sparsity neighbor triangular metric space triplet ultrametric triplet ultrametric van range example triangle form triplet side represent among possibility semantic accomplish ultrametric convenient ultrametric dendrogram dendrogram root label precisely term ultrametric set ambient dimensionality define cardinality traversal terminal van terminal traversal ultrametric simple tree ultrametric distance computational need dendrogram terminal path integer let call traversal dendrogram dendrogram day construct map topology ultrametric dendrogram near write dendrogram terminal traversal dendrogram terminal dendrogram traversal dendrogram find ultrametric two terminal twice traversal dendrogram informally potentially common parent traversal dendrogram root dendrogram bad dendrogram structure integer agglomerative dendrogram favorable well agglomerative ultrametric carry one candidate check dendrogram neighbor find firstly low terminal follow cluster tend ultrametric far carry indeed endow ultrametric note carry computational structure ultrametric build balanced dendrogram logarithmic review ultrametric distances neighbor constructive computationally dimensional space retrieval form consider human computationally carry ultrametric retrieval massive require cope volume furthermore ultrametric induce hierarchy support arise directly distance ultrametric cluster store ultrametric tree retrieval easy member enhance inherent ultrametric relationship onto alternatively map onto express generalized ultrametric pairwise relationship partially lattice algorithm hierarchical direct reading scan discuss simultaneously ultrametric hashing bin precision hierarchical ultrametric consist common long sequence et al precision example precision length digits generality order cardinality measure attribute ultrametric boolean work number store effectively many theoretic algorithmic allow scope distance set proximity domain include demand typically domain agglomerative dissimilarity imply time number agglomerative reciprocal near cluster quadratic agglomerative neighbor quadratic agglomerative linear agglomerative algorithm characteristic algorithm often develop handle idea separate group within grid partition overlap cell calculate sort accord traversal neighbor grid follow
discuss dependent factorization method lead prediction time validation several choice misclassification misclassification rate stable minimize misclassification bin emphasize exhaustive lead improvement subsample validation average split train curve test set challenge one view paper mention respect challenge figure pair split bar split cc misclassification cc cc cc cc cc cc profile month period direct classify time namely rating appear useful distinguish member exhibit rate movie day week movie rate provide incorporate day week well rating cc consider week rating importantly pattern member well separated show frequency movie different day week label member tend movie day week mostly movie week repeat quantify empirical rating day define average variation p q tv pd rate movie day week possibly figure phenomenon bc tc tc bc week bc bc tc bc tc tc present three predictor member movie third exploit week good suggest predictor rate movie predictor assume movie assume independent everything subscript fix time movie per month bin occur everything else conditional occur user rate movie everything day week third take rating present rating account give generative movie user around prediction make low approximation user movie time center bin show residual movie well normal user agree overall b motivate user time normal distribution rating condition day rating previous tuple movie rule classifier generative evaluate section low particular ccc derive term misclassification correspond rating second variance residual error probability bin unconditional marginally come day week unconditional incorporate decrease misclassification rate condition week variance misclassification rate report detail provide statistically respectively finally excellent roc area rate product rate define average use original challenge study previous yield excellent improve contextual rating provide rating separation raise scalable incorporate multinomial multinomial vast reduce classification binary fix hereafter whenever characterize denote assume logit whereby feature fit assume attribute user maximize rating implement software standardize solver feature vector day week rating indicator hour implement feature vector week suggest adopt vector week day month adopt rely outperform unified framework describe bin length binary indicator rating scale shift reach test logistic assign regularization feature improve achieve misclassification estimate subsampling rating people fold validation vector challenge note bold proposition corollary conjecture remark usa department electrical engineering stanford stanford track aware movie comprise provide movie user comprises identify find rating significantly useful achieve preference know misclassification incorporation contextual play ever availability source information social pool view investigate relation social behavior recommendation result summarize table remainder describe challenge explain performance overview c size consist rating rating user movie provide movie rating throughout first user movie rating training composition belong tuple comprise movie rating user movie track rating train vast majority form user form consequence slightly misclassification size provide roc true rate false entry respectively entry assign minus false positive roc way total per consider user roc positive rate positive union class incorporate increase amount contextual effective tool movie hand latent large infer rate select movie vector generalize temporal variability movie latent first rating miss factor take temporal rating aforementioned rating provide misclassification roc curve throughout ti matrix rating column movie column regularize n kk ki j f j excellent performance performance prove suitable relaxation alternate update minimizing stop iteration quadratic inversion convenient instance ij e I define matrix read I proceed analogously rank prediction bin duration rate movie rating tensor represent rating predict miss section matrix
q provide follow subsection back research focused efficient estimator valid prediction set thorough efficiency lebesgue measure set level one lebesgue upper level prediction band normal center gray observe band quantile band optimal area validity require band refer joint type validity illustrate example validity band band q satisfying asymptotic efficiency band satisfy fact atom euclidean radius conditionally valid finite validity thus trivial validity shall instead band asymptotic finite asymptotically supremum validity consequence asymptotic efficiency marginal validity notion naturally validity partition band locally valid q sample valid prediction available thought become validity validity conditional validity strong validity whose elementary omit marginally valid validity one technical band mild regularity conditionally valid marginally valid prediction band straightforward event case marginal increasingly close therefore simplify construction kernel prediction develop sample suppose set q p statistic agree augment principle fit exchangeable p invert hypothesis reject density bandwidth mean high avoid augmentation validity order increasingly show appropriately characterization level version state construct slice marginally band measure marginally valid region band fix slice joint marginally let define band marginally band local band also simplicity presentation support side kernel bandwidth augment check band finite define finite locally marginally fix another independent conditioning optimize optimize effort interval approximation analogous finite local long satisfied density little inside validity must subset smooth approximate prediction argument choice cubic histogram bandwidth argument density counterpart definition solution existence guarantee contour smoothly estimate density roughly oracle first approximated set conditioning rank plug regularity quantify put conditional density old correspondingly kernel concept also old enable regularity xt cx notion exponent condition cut value set mention empty simply put optimal level difference validity know smooth give asymptotic efficiency prediction band satisfy validity rate minimax risk valid infimum follow result low error distribution fix hence proof use somewhat band note bin marginal consider allow bandwidth bin validity small measure procedure detailed preserve marginal validity drive split sample subsample bandwidth subsample prediction band split sized construct cx c work locally marginally bandwidth construction efficiency excess risk scope section centered performance band valid band equal sized whereas marginally band although locally valid coverage marginally band cover bandwidth panel car original car per acceleration use book linear linear per per power figure transformation sense intuition inverse per power panel prediction reasonably band wide narrow large panel band measure enhance smoothness construct ensure output without tuning procedure conventional band truly valid coverage sample construct free validity efficient achieve completely believe first band property rigorous bandwidth sketch argument exces study stability plug show excess band construct band exploit supplementary technical give old text book old class h old uniformly approximated polynomial integrable old class kernel convolution lemma total atom fix total mass uniform imply hence q q conditioning old consider assumption constant q consider fix theory universal constant q l xt fact fix suppose satisfy estimate satisfy pl c moreover modify argument assumption l l pl pl pl pl part second result eq lt direct measure estimate omit dependence set level obtain I easy inequality conclusion bernstein union ignore tuning generalize symmetric conditional density support take verify hold verify py x py note let uniform verify histogram cube pairwise separation density band l
weight eq section weight expression derive statistic reader test longitudinal explicit location subject multi reader roc large variance get marginal longitudinal also asymptotically take q estimator simulation estimator finite statistic simulate roc longitudinal reader importantly power expect optimal equal weight longitudinal repeatedly repeat measure subject study coefficient simulate statistic auc use show bias square square coverage close asymptotic show indicate violate well method surprising disease status curve base score roc monotonic new score empirical roc power equal let simulated dataset estimate weighted define simulated power weight datum take normal x choose ik ik define give within simulated auc square mean square coverage correlate nonparametric study diagnosis medical cell cavity see challenge exist include diagnostic history diagnosis outcome address issue investigate whether sequentially aid disease reference participant participant digital conduct conduct modality set participant correspond digital report question clinical diagnostic diagnosis original participant gold standard four weight weight estimate weight set indicate precise diagnosis report method apply compare diagnostic marker longitudinal illustrate study propose method conduct study finite set complex subject extend constructive suggestion award american act national security part american institute child human solely author view national institute health derivative finite partial set derivative normality marker therefore element element expansion side asymptotic normality combine device e htbp rmse rmse norm weight htb bias rmse bias coverage corollary conjecture condition department statistics usa division research national child health human development usa college public health human sciences medical imaging modality test marker longitudinal roc method weight area roc maximize detect imaging modality asymptotic power apply diagnosis newly image modality discriminate subject imaging modality distinguish diagnostic marker identify exist specificity correctly truly diagnostic characteristic roc tool compare marker curve accuracy marker apart threshold marker nonparametric measure auc distribution normal marker method perform assumption roc nonparametric robust compare empirical auc roc intersect roc curve marker interested range acceptable early cancer part nonparametric specificity classify non subject may thus desire marker range rest follow multi reader test roc longitudinal roc report performance roc apply example diagnosis notation suppose marker location marker normal subject total pairwise function marginal similarly define q iy roc obtain compare evaluate marker roc longitudinal roc effect among estimator covariance structure generalize estimating derive large estimator valid work reader subject imaging device one rating location subject denote reader approach compare image modality device reader special auc reader combination propose weight possibly empirically modalitie reader roc instance homogeneous rating equal weight datum introduce reader experience vary greatly biased auc estimate difference consistent detect modality calculate weight explicit expression estimate eq marker come longitudinal marker several time roc
directly comparable contour mae contour line dependency axis analogous rmse similar trend mae display indicate count density remain general example pmf sensitive count sparsity long valid level multivariate univariate prediction contour curve contour horizontal imply depend item item contour show count last summarize dependency trend absolute baseline insensitive name highly l variable user average dependent user w count item default item nmf pmf dependent pmf pmf strongly dependent cf count mae rank indicate dependency function density good varies non linearly count especially svd pmf perform analogous mae rmse function asymmetric undesirable word true rating specifically star belief difference recommend important issue recommendation error recommend preferable item high penalty predict bad high opposite recommendation loss mae rank rank recommend present consideration slope well good conclusion identify cf user average item base regularize matrix ii nmf individually slope cf display accuracy baseline svd slope pmf fair fair poor good good poor poor asymmetric poor fair ndcg poor fair fair poor fair good fair fair fair slow fair slow slow memory consumption high high well little short asymmetric asymmetric long use adjust large dense long repeat major conclusion svd pmf variation perform mae rmse except nmf asymmetric evaluation one simplicity vary accuracy user vary relationship dependence dependency appear influential trade factor low consumption accurate algorithm experimental help resolve specific hand efficiency important slope source software reproduce experiment cf conclusion describe practitioner implement recommendation examine novel art rapidly propose conduct study several collaborative filtering art variety context control criterion conclusion collaborative rapidly classic include method base around factorization include factorization despite considerable clarity central clarity method differ substantially user filter method perform setting collaborative compare investigate mention concentrate classic recent art control user comparative follow generally speak factorization nevertheless distinct accuracy depend user number relationship memory consumption crucial describe implementation study describe introduce recommendation collaborative broadly speak software recommender broad seen consider recommendation base recommendation information main system filter former explicit concern information gender location item recommendation category filter cf use user exception rating rating hold rating column typically example five star netflix recommendation rating implicitly activity web interpret positive rating extremely item hybrid content combine user content comparative study serious collaborative system task experimental content tie likely transfer rate different collaborative filter usually model recommendation rating rating recommendation predict rating user whose rating item item motivate user item rating estimate desire rating cf average rating vary considerably example averaging include neighborhood vote inverse recent neighborhood construct kernel estimator ranking predict model rating recommendation include slope successful factorization regularize profile miss nmf variation pmf pmf maximum nonlinear mae root rmse predict observe item range aside purpose test f argue metric conclusion concern relative cf however paper motivate criterion write survey cf extension concentrate hybrid art include netflix recent recommender system explore additional issue couple memory model cf focus comment computational issue mae netflix competition competition improve accuracy rmse netflix recommendation win dynamic achieve win linearly combine netflix competition use competition evaluate cf item represent entry old standard dataset rating item rating describe concern study conduct experiment facilitate implement list public update state art research elementary baseline average constant prediction prediction item rating use factorization category user memory default default regularize pmf pmf pmf slope rank cf experiment netflix benchmark cf literature recent benchmark measure sort rating matrix row realize specific pattern select subsampling require sparsity sort corner sort contain top item word subsample prescribe purpose time train conduct processor ghz gb memory investigate dependency user rate variability dependency determine start prediction quantity user multivariate accuracy variable row mae panel focus cf baseline item count rmse evaluation trend omit voting user cf variant work compare algorithm quantity top row slope intercept appear intercept algorithm mae slope indicate decay mae table method user sufficiently svd little difference simple item svd pmf sensitive baseline nmf insensitive sensitivity item cf extremely effective user dependency user count bad memory base pmf pmf pmf rank cf represent user count mae prediction loss analogy fix respectively middle mae category coefficient get overall svd factorization method simple
n show lemma regular estimator converge let matrix regular w nn n way already mathematically conduct c std firstly selection reduce denote dimensional normal matrix true determined element take sample generate sampling trial normal exchange metropolis eq six model criterion compare standard define set true experiment secondly use experiment reduce theoretical reduce case average c c c std applicable firstly let study qx px minimization leibl generalization define minimization important learning criterion well aic bic regular onto singular statistical v n p e w statistical prove regular model coincide pn singular study remark set singular respectively small model model property viewpoint simultaneously mean weak selection main purpose jeffreys recommend jeffreys however jeffreys theory model selection minimization secondly numerically free firstly asymptotic large expectation disadvantage huge cost present secondly sampling approximate arbitrary last whose analytically numerically evaluate strongly depend true statistical nan hypothesis procedure recursively nan asymptotic contain statistical table model determine training energy conventional recommend asymptotic importance lastly relation algebraic paper statistical prove affect negative value secondly set give statistical set conjecture relation lastly hold odd likelihood widely true expansion bayes threshold partially science aid scientific keyword fisher positive otherwise statistical minus asymptotically approximated schwarz energy invariant real parameter several discover methodology true present bayesian log mathematically expansion energy numerically information bic onto statistical otherwise singular model neural mixture reduce regression network markov layer rule singular word hide random phenomenon naturally normal neither issue theory denote independently loss minus log free energy understand minus logarithm marginal play evaluation bayes equivalent minimization bayes energy asymptotically likelihood schwarz bayesian normal energy approximate bic minimize kullback rational real firstly singular algebraic geometry behavior free algebraic fact machine artificial neural mixture regression mixture boltzmann hide statistical indicate quantitative singular depend know apply evaluation present estimate widely show inverse purpose mathematical theorem firstly temperature temperature satisfie secondly even eq asymptotic singular lastly regular regular statistical cost generalize version true section notation singular representation theorem corollary mathematically main section result name free loss posterior optimal log loss kullback leibler real threshold table variable name paper average loss true leibler eq exist minimize large unique arbitrary function set element equivalent eq define characterize purpose present prove definition say say hessian strictly regular mm mm fundamental open kernel define several open condition hessian definite satisfy value invariant statistical indicate present conjecture analytic function leibl prior nonzero compact local resolution local c c jacobian determinant b essential respectively even satisfied regular statistical canonical threshold theorem generalize true unknown case make even present temperature unique temperature therefore behavior mathematical fundamental log threshold law mm main theorem three important corollary model let canonical mm bic singular regular bic small constant regular general asymptotically distribution singular hence replacement appropriate corollary converge law nk r zero nk strictly respectively maximum w u ij j inequality determinant schwarz inequality mean complete proof evaluate integral integrable generality consequently coordinate note arbitrary understand random process fundamental integral asymptotic two loss essential denote dirac hence behavior value expansion measure
pdf bad estimate improve large method work far cycle mail estimate density estimation large local suited heuristic optimisation entropy anneal everywhere describe computationally form j pp j order maximum principle cc kf kf value estimate shannon entropy condition ease end divide smoothness give locality small condition neighbourhood anneal noise introduce pdf eliminated move final result determine experimentally reproduce well support assumption theoretical consider function center pdf neighbourhood replace perturb average assumption restrictive since taylor pdf total compute approximately distribute large number variable normal average est dx dx put taylor h x f want put numerically nevertheless relaxed term p third equation exactly c solution otherwise equation label solution another put increase vertical allow pdf moving carry dx important
iterate full rank imply establish set verify satisfying requirement broad undirected practitioner look heterogeneity model provide give dataset sufficiently empirically sparsity conservative practice compare literature first show exist force contrast deterministic focus sufficient existence failure polytope sequence nature sparsity avoid polytope evaluate nine size tie college division costly expect theorem variant degree correct blockmodel may establish national health research office office lemma early derivative bound change lemma omit three c p p ij f r r approximation x kl l kl kk l kl approximate verify hypothesis newton inequality l lemma bind lemma consequence corollary aim hypothesis simplification corollary obtain finally simple logistic implicit datum heterogeneity broad property rise likelihood regime practitioner choose context fit variety dataset implication approximate long recognize call assess goodness g facilitate g exploratory outli context take edge appear extent heterogeneity commonly practice node heterogeneity node direct therein take edge implicit depend model residual community associate linear since canonical analytical convenient take outer practitioner lack issue sparse adjacency regime wherein typically purpose essentially parameter log member broad irrespective node degree binary symmetric zero correspond graph node diagonal family probabilistic nan family let specify model introduction parameterize function complementary link logit undirected version exponential note degree case intuition equally likely log alternate parametrization degree loop see appear commonly generalize placing degree model high lead expect whenever log introduction approximate technique suffice gain claim likelihood intuitively behave mean rough bernoulli treat poisson poisson every correspondingly respective maximum include furthermore log likelihood evaluate pair absolute specification arise assumption component log smooth satisfy assumption edge result log evaluate analogously notably corollary total misspecification theorem specifically error ease reference existence likelihood newton
forward pass despite necessarily computational nature especially abc remark smc abc estimate normalize functional hmm scenario need already abc smc scenario density smc quantitie whether associate generate smc abuse x incremental smc incremental instead herein denote add subtract nx one follow grow conversely nx remove smc hmm dx dd dd state facilitate investigation obtain approximation answer smc numerical abc consider forward study elsewhere abc improvements hmm utility high investigation batch static worth smoothing estimate simulated parameter implementation forward particle approximate smc abc label abc dynamically drop smc abc approximation hidden accuracy abc abc discuss calculate nx distance smc hmm corresponding similarly average error smc abc allow understand impact describe preliminary become conclude numerical study smc abc smoothing respectively mean error display expect fall smc control outperform illustrate error small vast interestingly abc accurate smc counterpart smc stability therein smc run ne ne smoothing perform abc error distinction subscript display note abc comparable even marginal estimation mainly hmm smooth quite biased substantial h basis specify prefer perform would appear series forward necessarily functional improvement accuracy produce parameter bias smoothed link method quantity dependent see trend parameter investigate smoothing static hmms likelihood construct abc work batch estimation practical arrive online investigate methodology online consider article abc procedure acknowledgement author support grant acknowledge acknowledge platform self explicit consistent notation rely hx gx avoid notational independence additive assume expectation w abc n ng nx hx quantity consider induction one conclude deal dx p dx qx ng h qx particle convention filter exist proof ng n ng p add definition sum separately r clearly q second subtract function two bound n q similarly deduce ng n theorem follow therein latter depend upon repeatedly omit description abc auxiliary smc work nx dx backward remainder adopt decomposition use p depend depend elementary exist abc decomposition denominator apply use uniformity us algorithm remark school business south mail department apply national mail engineering cb pz uk mail bs tw mail mathematic college mail hmms density involve value precision quantify abc expectation regularity adopt form quantitative treat simulate static particle batch markov sequential unobserved dominate conditionally dominate depend functional approximate hmm expectation functional score see rarely smc rely ability pointwise conditional evaluation avoid evaluation simply close secondly evaluation technique interpret instance abc recent context hmms abc see abc note fix lack abc smc error provide objective theoretically empirically perform static hmms hmm incorporate value auxiliary approximation turn admit technique scheme control sample note adopt filter backward smc smoothed functional estimate expectation similarly smc expectation hmm study smc expectation hmm error control particle mcmc order hmm particle use smc generate state hmm particle hasting place context smc adopt adopt backward pass summary contribution nx henceforth quantify henceforth refer illustrate finding abc characterize result combine smc error numerical section article proof negative addition banach endow norm tv tv pa via advanced computational tool assume throughout associated completely idea facilitate compact exposition hmm bayesian even advanced tool deal tolerance follow represent statistic note data accurate focus return density probability particular abc maintain markovian help abc smoothing resort smc mcmc consistency grow intrinsic bias bias noisy article address potential hmms smoothed investigate concentrate particular facilitate smc abc minor modification adopt gx fx functional rather lie however typically employ smc perhaps discuss verify demand assumption establishe fast estimate grow linearly overall parameter necessarily dominate abc abc context smc recursively particle resample smc terminate sample whether stop else return choose implement smc resample perform increase commonly refer resample particle replacement I alternative degeneracy drop go former case degeneracy degeneracy bottleneck static latent central path degeneracy hmm particle variable smc back sequence whose element smc degeneracy effect want forward filter backward algorithm simulation eliminate smc nx px p note nx dx eq lead idea particle write perform eq apparent recursion v px assumption asymptotic linearly
available analytically consider control chain recursively family recursion rewrite recursion eq traditionally noisy sequence define I h closely relate effect negligible ergodicity sequence start various quantity crucial analysis assume traditionally modification recursion indeed major difficulty dynamic stability property rely good ergodicity vanish instability exist ergodicity ensure result aware numerous scenario follow significantly different developed dividing prove boundedness simpler use section sequence transition uniformly ergodicity trajectory accurately visit eventually provide deterministic stable establish sequence analysis ergodicity study reasonable state boundedness lyapunov away norm although symbol section lyapunov subset continuous compact introduce need describe ergodicity precisely stepsize account denote crucial j establish follow infinitely q eventually prove equation scenario soon ergodicity example focus condition combination exist conclude mcmc let convenience successive separate deduce imply l induction obtain x borel condition notice construction briefly size stepsize form robustness tracking application adaptive know impact convergence vanishe consist adaptively sequence due far give modification x h behind expect probability mean stepsize recurrent aforementioned neighborhood essence homogeneous recurrent scope establish section straightforwardly chain mcmc aim optimally chain invariant specifically walk increment algorithm concern situation cone definite circumstance implement algorithm theorem establishe section conclusion eventually compact boundedness already show sophisticated robust version tail situation stepsize sequence multivariate scenario research project symmetric density increment acceptance aim optimize result display broad tailed situation sequence case contrast scenario scale prove new stability prove scenario improvement thank early initialization increment stable x scaling rely establish establish establish algorithm n contour proposal standardize follow quantify ergodicity vanish eigenvalue become matrix choose statement apply assumption bind weak assumption vanish density previous walk notational piece instead piece assumption increment distribution away twice define r tail x increment proposal symmetric condition condition tail already argument presentation handle crucial difficult establishe scalar increment b vx rv rv vx r proposition vx vx vx vx vx b x vx vx inequality detail straightforwardly lemma z z z eq treat fashion state prove establishe imply differentiable z c exist j convenient x x q z x term bound address x thank know recurrence soon establish satisfied time homogeneous control resp w proof establish lyapunov result depend w inequality x denote x c b w yield w deduce jensen constant appropriate vx n vx b vx naturally let possible lemma theorem imply subsection proceed decay tail target cover stepsize stepsize q exist r infinitely proof subsection differentiable density moreover absolute first notice assume practically relevant rwm proof notational simplicity function establishe consider control valid obtain expectation yield assumption cv deduce algorithm u cx one lead notice I lyapunov exist q p x w cv cv w cv combine intermediate yield x c notice choice implie since pt introduce stop proceed claim stop would page care omit notation I I monotone thank obtain conclude result pt see sequence nonnegative number sequence change part cx cx negative setting constant exist consequently xt consequently deduce monotone statement deduce eq imply existence conclude xt x conclude vx vx q dy x almost sure acceptance pt thank sufficiently take therefore z z z z b expansion integral z x first indeed ensure vanishe obtain term bind choice conclude therefore case aim combination line establish x equivalently pt pt choose conclude let sufficiently ensure conclude pt z x bind taylor enough proof cover imply hand z hand x xx exist chebyshev deduce p lebesgue measure integral x z deduce statement cf r mathematics tw united capital support project letter al recurrence set control markov chain area lyapunov parametrize family transition combine lyapunov lead approach process crucial practical recurrence call adaptively statistic control numerous encountered control markov transition control markov consist define family mapping concern stability recurrence process relevant tool process depend process parameter particular interest open despite relevance discuss follow one invariant ergodicity lose organized recurrence drift stability characterize respectively drift characterize recurrence worth dynamic exhibit separation main behind recurrence clarity remain abstract practice main show familiar dependent drift characterize condition characterize abstract focus practically update cover subsequent find noisy aim section highlight central numerical early adaptive chain type optimize specifically result apply approach adopt throughout drift commonly potential lyapunov function drift lead follow together transition explicit exposition cover statistic lead dependent exist aa notice hereafter convenient mapping cb ec I neighborhood appear abstract motivated concrete situation simultaneous
derivation write zero view claim conclude particular recall decrease resolve second application formula back integer define turn supplement proof proposition provide concern like lemma state zero sub obeys nj square gaussian exponential inequality sub gaussian q absolute denote maximum sub unit eq entry concentration gram matrix lemma union matrix mean shall make use sequel entry population gram becomes accordingly combine lemma exist variance position lemma combine sn equip auxiliary let first restricted rely recent therein variable say parameter essentially simplify sufficient row pr eigenvalue set parameter satisfy eigenvalue c sub self paper consequently virtue exist claim event depend row suffice write transpose transpose decomposition unit vector equality eigenvector cauchy schwarz estimate moment third analogously proof scale expand square treat minimize separately read derive decompose remain aside constraint unconstrained minimizer coincide rule eventually read comment meet fact case condition shall comment provide zero minimizer substitute back turn obtain collect apply consequently depend make use yield exist suppose fulfil hold conditional depend ij x e two table report table l nn nn nn l l theorem conjecture lemma sketch support multimodal number paradigm regard necessity paradigm negativity regression property meet non regard hence support view noise vector throughout concerned number even hope recover sparsity constraint simplest support pixel intensity time histogram count rate negativity numerous deconvolution diverse field image reference excellent seem simplicity solve minimizer solid appear reference decade author permit reliable sparse signal study body negativity alone may suffice recovery unclear continue hold realistic consideration apart intuition specifically dimensional paradigm prevent adaptation noise enforce main contribution characterize certain tailor negativity thereby apparent understand empirical success non establish design achieve regard support recovery combine survey overview appeal decade computational popularity paper empirical limited performance arise responsible sub optimal rate burden grid since small literature recently level free extend conference publication set set self may certain equipped resemble available develop use bound section provide sup datum section look empirical deconvolution appendix contain apart supplement denote denote obtain correspond denote matrix matrix row likewise identity vary symbol respectively scalar simplex set positive differ line line positive symbol denote asymptotic understand w array x n state say position follow submatrix column result excess resemble contrast correspond certain extra design necessary self establish slow self decomposition square modify quadratic role coefficient may negativity square reformulate least square ill square contain fail hold many minimizer position condition interior cone negativity yield perfect fit light constitute pure concern overfitte quantify turn study estimator expression mixture zero conclude nx strong obviously half constraint turn design still overfitte satisfy order quantify define fulfil duality convex hull scale origin sec separate origin argue overfitte fulfil orthonormal fact comment proposition qualitative main illustrative purpose entry zero onto entry n fit cf bind consequence far rather mild additional tail noise comment explicit correspondence level monotonically function understand desirable estimation rise general prediction error correctly problem excess nearly constitute persistent spirit notion persistence distinguish consistency bind recommend roughly computational lasso design regard improvement slow lasso require parameter non context noise continue section elaborate property admit establish optimal guarantee relate selector similar support state condition share selector different I sub least follow assertion analyze condition regard estimation j constant weak several comprehensive random statement line addition eigenvalue lasso sparsity attain apart factor achieved support lasso less multiplicative combination property eigenvalue satisfied resolve apparent contradiction satisfy restrictive isometry rip condition constraint discuss result linear bind rate partly restrictive term compatibility place compatibility sufficient bound distinguish additionally thresholde geometry subsequently study thresholded sequel two purpose need give projection subspace span orthogonal context lemma contain let minimizer negative problem proof separately establish minimizer next require quantity nothing else introduce respect term hyperplane accordingly q duality highlight connection assumption submatrix invertible appearance quantity upper set summarize sufficient amount separation effective provide entry broad roughly os may imply level relative empirical discuss performance well understand light reasoning ordinary cf subsequent discussion necessary generating arbitrary least fulfil scope application remain design bind value scale ss singular value detail class call even imply may thresholde might recovery basis regression specification purpose predictor give rise ranking arrange easier equally order hold denote whose rank rather moderate setup need rank top speak operational replace variance dimensional continue advance minor efficiently qr decomposition removal accurate coefficient elaborate similarity regard result succeed argue general attain provide specific finally benefit negative negative condition negativity coefficient condition employ e support necessary point necessary recover absence condition require highlight conceptual connection note distinguish scheme non recovery low minimum coefficient suppose playing role considerably literature fulfil impose regard c need support entail appendix issue provide represent bad sup e mutual incoherence require fairly restrictive bound two list non thereby formally lasso already restrict specific design slow spirit design paragraph attain estimate sup room leave present design empirically regard estimation recovery crucially succeed recovery without little familiar set directly specify tune close look basic design uniform distribution distribution generally multinomial integer sequel instance theorem bound counterpart well recall self condition turn design light statement restrict x center entry eigenvalue counterpart random scale c center entry q np contour plot solid dash dotted display sample size complementary experiment fix gram matrix decomposition interior vary correlation mass small ij ij reason reader refer supplement brief confirm replication relative bound away zero dramatically short compress cs recover sparse vector adaptive setup linear contaminate additive fall model attribute rare disease strategy affect form testing discard repeat adaptive aggregate group retrieve assignment need proper fact number ensure linearly otherwise prior zero light paragraph measurement sparse recovery approach base discuss set adjacency distinguish since constraint therefore self characterize aspect relie provide sketch paragraph several range component replication consider vertical bar empirical distribution bottom surface quantile summarize panel configuration stay set sparsity shift toward display reason quantile top case consider surface plot form fix roughly hand performance sparse spike deconvolution appear location spike goal spike potentially signal demonstrate deconvolution candidate position densely noise eq think place densely mean may spike process fast theorem result adequate eigenvalue lr part vector return bar represent dot median respective position replication square prediction spike evaluation gaussian construction gaussians evaluate result different parameter ii choose validation tune cross validation spike panel replication inferior roughly substantially ridge remarkably concentrate near spike present investigate lasso drawing independently rescale gram matrix numerical figure set gram setup lead design regard deconvolution sparse fails presence place densely concern component order consider localize function center center draw uniform uniformly sufficient amount separation choose scale uniform turn yield perform parameter control magnitude design aspect fraction vary fix configuration across thresholded nn negative matching omp regard recovery additionally nn subsequent regard table contain supplement thresholded regard recovery I permit support recovery estimate determine manner fix slight advantage equal estimation negative lar obtain solution hold solution base constitute second advantage recover support usefulness thresholding nn obtain non check recover omp serve success h consideration thresholded version excellent recovery superior thresholded lasso difficult parameter regime strength high experiment reveal may competitive recovery select correct pointed thresholding stress regard lasso entail design thresholded threshold adaptively without ii occur apart competitive behind partially keep succeed situation omp success consequence cf success omp error accordance sparse close nn arise extreme l l nn l nn l nn nn thank helpful thank read provide valuable suggestion early comment gaussian parameter generate obeys tail combine fact use satisfie condition vector setting square decompose eq property claim presence h establish expand minimizer lower bind self property j obtain assertion would immediately already feasibility strong substitute add particular adopt analog divide side assume c trivially event apply latter assertion rest invoke along omit auxiliary kkt exist imply solution least equality constraint separate substitute estimator value must observation second minimize negative cover tie think considerably mean parameter first analog hold minimizer observed equip bind
unless clarity exposition regularizer penalization every regularizer virtue prove coordinate multi perform without generality permutation reformulate complement respect imply complement positive iterative suffice g positive e z descent learn convex subproblem section e k still still k subdifferential nk n k reduce remove hold depend sample equivalent regularize separable permutation k u reformulate connection structure solve sort time I quadratic quadratic q relate lagrangian partial solution feasible separable belong prove difference equivalent solution replace inequality equality boundary lagrangian tucker optimal finding sort straightforwardly search regularize remark quadratic trust solve extensively study mathematical optimization trust region arise behind approximation original quadratic inside circular trust usually eq therefore involve provide separable trust eq lagrangian case assignment unconstrained solution inside trust region relate hold region become prove solution theorem optimal block method detail careful time experiment polynomial infinite accomplish implement accomplished newton trust initialize perform newton nk formula iterate split sorting use regularize separable region section penalization unclear penalization lasso contrast penalized penalize reduce penalization solution namely precision task eq sequence regularize logarithmic subproblem z block definite precision additional learning reformulate let b enforce complement k k strictly constraint initialize element logarithmic separable one related lagrangian duality strong positive matrix change sort newton method continuous tucker optimality additionally inverse inverse produce similarly order permutation index b b b tc initialize newton case problem separable solve duality initialize iteration initialization lead k initialization synthetic test truth repetition topology undirected weight ensure closeness truth measure leibler minus fraction exclude specificity minus purpose lasso also show kullback leibler recover edge remarkably comparison bind produce kullback observe significant difference kullback leibl divergence versus diagonal kullback roc curve choose penalization task recover ground remarkably well task upper rule graphical regularization tr kullback leibler always method regularization real world brain subject condition subject acquire second spatial template voxel group please see region span entire effect side six fold subject figure task observe penalization bad therefore multi bind graphical selection tr well multi second idea graphical fmri remain testing procedure six method stable perform poorly multi method multi upper b value significance previously log likelihood technique report minus compete method multi different r tr well r r r mi method mark statistically method statistically subgraph learn structure subject interaction negative interaction red subject method c dataset world publicly state brain outside brain reference template space smoothing perform extract matter group please appendix span effect region side subject r r b site site task one third third validation remain six repetition make subject take turn validation report likelihood scale visualization log comparison moreover well upper bind result penalization well difference statistically mi di multi method task cs tr bind penalization produce subgraph learn site structure produce consistent collection site covariance blue task replace regularization provably show multi trust region recover topology truth cross high compete fmri lead ground experimentally well penalization believe negative way extend practice analysis sample hope trust problem fmri comment grant da leave leave inferior leave leave red xx edu multi graphical boundedness lead provably sequence minimization subproblem efficient experiment well dataset learn aim algorithmic challenge super measure model non inverse equivalent measure technique enforce precision sequence regression encourage structure allow imaging region interaction become setting fmri available consider generalize learn graphical replace regularization norm prior sparse contribution coordinate convergent multi learning learn solve penalization experimentally relate short conference paper general assume assume experimentally recover ground truth always discuss mrfs pairwise
permutation popularity avoid adversarial total priori packing generalization pack goal coming generalize choice present notice ratio despite pack lp recently obtain value state obtain competitive notice guarantee competitive actually various option choose option online learn classification explore see make suitably topic seem pac weak allocation column replacement replacement use independence substantially improve requirement low model behavior lp algorithm permutation still understand requirement column handle column keyword turn engine furthermore guarantee column understand case fundamentally latter difficulty call obtain competitive packing state depend bound connection online lp pac classify correspond consider family classification learn obtain goal whole classification refine bind covering consider group classification budget lp lie subspace large mutually bad classification column classification much small indicate capture lp column approximate lie induce classification classification subspace embed establish useful possibility face set pack lp column lie lp section follow section pricing find dual infeasible classify otherwise classify column homogeneous hyperplane motivation select column reduce solve lagrangian multiplier obtain lp appropriately scale column index output feasible quality solution ultimately lead dim contain column b index use completely determined simplify scenario dim contain x feasible least denote linear classification column property work decision make analyze complementary budget ix ti scenario ii reduce cost concentration inequality argue although chernoff type obtain learnable scenario either classification look sample put depend skewed compare usually equation unfortunately overlap scenario skew skewed introduce right expression classifications b ix ix bx significantly skew direction equivalence event serve skewed additionally reasonably concentration inequality ability term I similarly contain discussion set put chernoff bad size collection unfortunately enough allow give flexibility give unfortunately contain set find directly case inclusion lie subspace single take minimal lie suppose subspace contain assume hypothesis equivalently norm span sphere namely close definition homogeneity norm lemma think robust phase lp column fashion guarantee dim theorem fix solution dependence arrive improved generalization see combination lp denote aim lp describe find lp bs try partition bad classification index let u dim subspace column fix integer number tx bi order description integer think acting phase robust phase union column error notice online robust routine formally lp whether introduce work generalize allocation difficulty classification linear conjunction subspace seem strong enough classification geometric course dependence side competitive possibility permutation real interval random every give duration least modify eq lagrangian lp x sx optimum follow latter linearity fix duration proof let lp complementary loose complementary complementary definition since feasibility get complementary get sx sx simple helpful bounding event occur take iw iw iw iw proof alternatively column jt c analogous claim canonical point whenever limit negativity third inequality observation thus conclude suffice family follow budget set inequality v intersection hyperplane hyperplane l face cover empty face hyperplane inequality mm face page conclusion paragraph non empty let lp denote column hand inequality triangle lp eq lp suffice optimum optimum use argument easy feasible conclude analysis change tp ss slight possibly infeasible mean coordinate dim integer sa tx bi say sx sx ba sx ba appropriate prove scenario definition budget tell ix condition ix classification unlikely infeasible classification total budget follow eq event similarly replace expression event classification take let contain contain reasoning event mx construct set follow lemma j equivalently concatenation union check I element impose denote good scenario definition last combine lp hand side lp return solution jx jx verify return column dim linearity n employ summation bound pt packing lp arrive variable maximize lp
representation gene node interaction plot consider interaction ie isolated gene appear split cutoff significant per genetic pathway modify already know knowledge often aside believe nuisance role interaction example seem independent highly correlated ignore age correlate adapt deal nuisance resolve issue correlation assume potential compare remainder usual replace mean center give motivate appendix section show condition asymptotically interaction fdr rapidly result asymptotically base mean variance realization infinite covariate order furthermore r j sufficiently little bit straightforward assumption fairly trivial must correlation difference may reader might bound independent necessary tail share one relaxed tail converge consistently probability statistic converge thus fdr condition fdr proportion remain otherwise algebra become clarity carry scale little procedure nice effect give rescale permutation like clarity originally observation class class ie minor statistic large converge estimate technical denote mean let integer correlation covariate every k tn q notation particular one pn theorem may bound fix cutoff use n also note straightforward give permutation original statement already relaxed consistent discover permutation discuss permutation base method backward efficacy simulate convergence rate plan replace similar reason hypothesis group implicitly nan nuisance computationally project onto nuisance runtime permutation dominate nuisance regression nuisance runtime mle grow nuisance much technical theorem manuscript technical lemma imagine consistently correlation fisher change formalize order r j ki one rate begin value ki pairwise inner formalize variance distribution write eq first note complete rate random known correlation distribution small subgaussian bound variable lemma wide light tail agree lemma begin hoeffding type inequality j I tn quite inequality triangle jx moment generate know subgaussian random sub triangle sufficiently take far apply union sufficiently combine cutoff ie find begin slightly discussion th letting split range change simplify size simplify straightforward lemma theorem sup distance bind sufficiently sup correlation begin reasonably sufficiently correlation close matrix integer average pre class class probability symmetry begin explicitly quantity contribution eq similar large great lemma make tight bound lemma along fdr converge satisfying satisfied great lemma combine probability get violate complete helpful comment corollary interaction dimension issue model assumption nominal issue permutation testing search differ method simulated find significant false discovery especially presence effect real genomic although gold tell restrictive assumption logistic discovery rate many modern massive amount science throughput vast accumulation technique classical error situation label covariate vector class class mean instance patient belong class develop disease biology detect consider covariate fashion main approach many logistic regression post hoc nominal nice summary subtle effect fdr regression call contrast backward propose attack potentially continuous show straightforward insight fdr asymptotic go specify generative row nonzero index pair furthermore test generative standard past pairwise logistic normality calculate standard estimate fdr problem approach even misspecification cause anti fdr term feature try add move avoid permutation method joint nan main interest difficulty resolve main effect nice adjustment heavily issue back logistic correspondence generative model mention equality toward class simplify logistic term traditional interaction correspond particularly satisfy marginally permutation test logistic forward logistic include interaction match conditional interaction marginal logistic interaction nonzero diagonal entry standard interaction thing happen interaction characteristic single toward end nan interaction pair test property interaction necessarily find forward omit interesting approximate describe x n mean matrix argue test interaction correlation coefficient variance version variance q compare nan work interested difference high calculate calculate assess significance long distribution instead directly discovery fdr reject truly procedure cutoff short know hypothesis nan numerator nan scale denote calculate nan permutation standardized permutation class label estimate number cutoff significant cutoff choose significant statistic interaction correlation summarize center calculate correlation matrix class fisher u kt execute standardized use new class fdr interaction rank testing interaction variable restrict consider necessary first variance correlation backward permutation scenario serious similarly real usual poor job find interesting attempt simulate version biological protein gene act biological diagonal block correlate interpret
complexity pattern variable check pattern mind l h j j elementary pattern set extract census mainly old adjust datum education capital capital week country response indicate set contain ht age age service state pay work education college school th status service sale op house armed force relationship individual pac capital gain individual capital capital capital week work week less country individual country united states south china france pair elementary event intersection elementary data age age education three say pattern pattern disjoint covariate implication form meaning exceed threshold probability occurrence condition event support frequent simultaneously correlation event name association next subsection pair bernoulli multivariate categorical marginal random pattern binary characterize metric sensitivity specificity true predictive u target proportion three classifier since true increase value high attention decrease turn procedure follow highlight binary base binary liu definition else definition easily association sequel binary association statistical discretization numerical suggest choose sort continuous variable expand choose improve learn naive recursively find partitioning step use association widely prune exponentially growth follow rule satisfy condition pattern redundant practice add exact maximum redundant frequent redundant indicator sensitivity specificity predictive etc pattern look step set define rule pattern classify pattern among focus satisfy contain redundant profile profile low prune redundant moreover respect well classification classifier redundant final constitute contribute rule bring basic principle pruning generate end subset nest h u sort opposite pattern nest one worth generate high good criterion help redundant algorithm huge redundant proposition cn redundancy remove pattern j tell performance prefer short hypothesis discard j follow u l corollary h mention l statement equality pattern small ratio high misclassification associate weak hypothesis consider opposite discard accept l u x u nest nest generate short predictive small ratio point li anti monotonic property hypothesis opposite discard generate nest nan accept equality support profile propose stochastic sample randomized u try nest pattern whether hypothesis vs nest try test pattern vs define try equality false x nest observation pair random independent deduce binomial take uniformly distribute stochastic reject n kx kx use test summarize step pattern test positive nest pattern nest frequent frequent generally less pruning value reduce pruning eliminate nested pattern select relevant nest pattern nest general predictive logarithm positive patterns yu yu u pattern pattern asymptotic normality ratio bernoulli indicator identically distribute accord suppose central multivariate delta method nan hypothesis following select relevant select pattern quantile validation redundant n diag nc method three subset machine learn size number select technique generate draw empirical depend write whose summary replace observe replace unknown distribution original let sample denote g test hypothesis proceed simulate replacement denote value following select pattern select pattern shorter frequent bootstrap g diag b diag b datum come uci repository environment nominal h ccccc nominal breast cancer diabete unbalance conduct sup database unbalanced assess statistical start choose rare class select observation rare observation combination classifier specificity etc performance proceed roc fitting conditional discriminant network law raise issue select sensitivity specificity etc roc generally binary forest classifier compare step adjust word whose measure well design create example replacement sample segment class near perform near paper generation artificial smoothed bootstrap approach combine sampling estimate fitting logistic generate principle show specificity auc specificity boost cart boost glm specificity specificity auc boost cart boost error auc specificity boost cart glm specificity specificity auc boost cart boost specificity specificity auc boost cart boost glm specificity auc specificity random boost cart boost glm specificity auc auc boost glm specificity auc random boost glm cart c specificity auc specificity auc boost glm n auc specificity boost boost cart association well mine discover interesting relationship variable near missing automatically interaction building predictive classification critical learning interaction search combination deal
enable well discriminative embed call simplicity formulate equivalent regularization original domain kernel solution xy denote gram matrix yy xx yy way optimization generalize eigenvalue yy yy projection basis help kernel yy degenerate either invertible formulation general yy yy xx xx yy yy another laplacian tend estimate collection denote empirical laplacian yy yy xx xx yy xx yy yy extension locally non linear find embed nearby high exploit graph sample affinity sample gram laplacian minimize embed find nearby build preserve relation manifold gram x x column zero solve calculate next find n therefore extension cluster method sc normalize cut nc concern admit detail precede integrate extension self self pca formulate localize scatter matrix localize matrix scatter laplacian lb weighted laplacian gram matrix generalize pairwise provide unify easy methodology cover multivariate paper design new general method formulate scatter augment generic multivariate scatter gram matrix generalize expression compact highly include also several localization generalize eigenvalue extension methodology design desire multivariate methodology adopt template appropriately methodology specific massive text article video difficulty find intrinsic trend nature analysis traditional hide embed actually report annotation fisher multivariate cca least square pls formulate generalized eigenvalue scatter augment scatter tackle number small deal non cca formulate augment instead scatter need overfitte fit smoothly outlier locality fisher discriminant reduction lot major formulate introduce et al show generalized square mild la square design researcher exist seem good address develop tailor specifically view discussion multivariate analysis methodology exist template new method combine template appropriately characteristic yet knowledge calculate combine enable supervise one label unlabeled follow fundamental method section review analysis viewpoint review template design desire precede section various linear extension help dimensionality several clustering analysis method concern set dimension follow brevity suppose center co occur sample resp resp occur sample I first concentrate pair many scatter matrix quantity denominator ambiguity convert lagrange multiplier confirm transformation unitary implie embed uniquely determine arbitrary practically useful heuristic eigenvector eigenvalue deal convenient close xy scatter xy extension scatter let expression graph laplacian theory definite matrix independent yx yy cca vector cca mistake equivalence cca sample case projection direct viewpoint objective follow calculate derivative square independent direction minimize maximize xy xx suppose equation objective cca cca part already reveal property discussion use arise exploratory linear variable often variance whose top eigenvalue minimize substitute x obtain n description subsection least pls pls try find direction observable predict pls space pls formulate interpretability original formulated follow mean every cf discriminant analysis da marker regularization popular technique optimization include area statistic machine sometimes call popular utilize ridge norm regularization square xx xy equation objective ridge derive yx xx ridge addition several include form detail previous major technique preserving seek embed nearby space reduce affinity heuristic n th neighbor choice minimize derivation convert fisher supervise discriminant overcome original similarity obtain affinity matrix lb lb lb n local manner preserve contribute near neighbor search follow supervised discriminant
without appeal eq corollary q exactly prove simulation entire good general purpose decentralize design provably approximately publish main grateful pointing tracking decentralize learn measurement distribute protocol noisy measurement make distribute cope vary inter communication noisy connect agent system mobile physical environment network reach full uncertain environment require development protocol distribute persistent environment goal change environment subject failure hope contribute development environment protocol capable study protocol evaluate structure tracking chemical mobile sensor circle sensor initially sensor sensor sensor neighbor challenge protocol adapt neighbor node converge challenge far wherein lie mobile subject inter agent connectivity noisy speed protocol especially concerned identify salient inter communication edge exchange message neighbor convention scale connect connectivity condition graph maintain node noise use corrupt offset neighbor mean random update wireless relative measurement frame alternatively may measure measurement measurement vector assume node incorporate node making namely pass successive precisely let measurement positive motivate collection mobile sensor direction take protocol describe social interaction collective cost associate majority individual rely near update describe remainder neighbor node intuitively seeks align neighboring node node align motivated number advance neighboring metropolis within lyapunov stepsize recent agent stepsize crucial intuitively avoid response accomplish ensure decay stepsize analyze collective collective motion vary stepsize similar reduce social target converge estimate almost surely stepsize nonnegative initial remark proposition may early conclusion noise spirit early protocol think protocol possible variation quantitative rate take place particularly adopt introduce metropolis walk move whenever neighbor walk walk start node number small least time node decay variance eq general expect convergence alignment believe vary connected measurement possibly communication somewhat time decay decay depend limit large solely possible nearly decay examine every node analogy consensus convergence bound counter high connectivity explanation mathematical phenomenon low update asymptotic choice decay g stepsize good effect maximum various topology mention graph time considerably time inverse exception multiplicative factor time concrete possibly vary grid node fall scale dimension learn noise constant choose protocol moreover slowly decay period rigorous result network observation incorporation communication first agent indeed learn classic attract couple decade due part application multi system paper work game distribute attract attention reader mention survey much link failure reader representative spirit recent vary node consensus iteration certain lyapunov certain closely relate number recent paper detection protocol like mention consensus consensus idea literature phenomena estimation contribution tight choose limit static unknown compare speed setting outline comprises broken piece step essentially basic fact devote solely bound analyze proving conclude prove several subsection begin begin whose positive entry direct edge undirected standard th basis use th argument simply subsection lemma useful immediate corollary provide multiplication statement sum side row side suffice entry side th may use change multiplication kl square introduce connectivity nonnegative stochastic denote metropolis aware make name geometric picture entry hold keep measure much conjunction decrease multiplication symmetric require bind feature nonnegative denote small positive graph weakly diameter evident definition necessarily orientation immediately without loss generality problem index absolute path make else assumption absolute applying schwarz continue preliminary prove proceed inequality increase positive deriving current devoted analyze consequence corollary lemma prove product moreover logarithm definition use last bound prof yet inequality continue bind eq rewrite equality bind whenever integer indeed consequence lemma claim suggest provide suppose b rearrange need argue lemma occur q simplicity henceforth notation place turn main decay special lemma rely previously subsection suppose nonnegative adopt shorthand break j piece piece term quantity piece piece upper eq piece sum straightforwardly put turn extension later proceed lemma suppose nonnegative corollary subsection place protocol begin proof theorem bound step convenient omit bt symmetric measurement time dominant matrix measurement measurement variable else variable put together use kt kt every q observe satisfie involve sum nonnegative row sum square lt kt since rt inequality second consequence definition lemma main assumption apply component assume remainder convergence strategy repeatedly sure exist observe three fact large consequence four remains apply iterate expectation obtain assertion infimum undirected communication graph satisfy connectivity k lm k km k km lm v km complete last strictly argue infimum undirected communication connectivity speed nonzero determine expression infimum scaling conclude measure achieved give set infimum nonempty nonempty connectivity measurement former latter expression expression infimum function imply fact infimum since finitely graph measure length proof proposition split proof recall two begin analyze eigenvalue connect one time walk probabilistic namely transform node every observe introduce node way stochastic within initial transition probability q necessarily eigenvector prove proceed prove theorem namely apply time use lemma expectation obtain use eq algebra exactly let assume q far assume proof portion imply claim associate argue big association introduce abuse notation contain else say measurement measurement happen uniform nevertheless assumption measurement say
terminology discussion course take let identically selector perfectly intuitively least one perfectly calibrate tend average well proof notation bag removal bag sometimes bag take selector randomness equality conditioning bag reflect notation average weak predictor suffice selector perfect calibration argument replace left expectation element decide bag observation side become arithmetic become equivalence equivalence easy care probabilistic ignoring prediction predictive make object overfitte validity regularity small irrelevant imply property complicated map say order invariant predictor function change natural invariant iid invariant selector whenever iid perfectly calibrate belong selector satisfy calibrate definition rational bag stand matter size bag bag hold bag predictor requirement essential depend predictor perfectly calibrate impossible indeed perfectly calibrate union set test bag obtain test selector probabilistic length probability case calibration train fed test fix threshold alternatively apply attempt score fix scoring calibration follow compute training observation score unique easily algorithm predict direct find close tie method tie never happen experiment refer overfitte give score classifier predictor object scoring predictor unless bt behind scoring label different different scoring function score flexible apart matter score validity predictor show corresponding give proposition arithmetic reproduce distinct would like increase place recursive partition consist formally perhaps element ratio notation proof division cell leave partition assign repeat decrease terminate one lemma recursive general predictor computationally inefficient large training scoring q pre spirit inductive avoid inductive pre train predictor section follow proposition pre predictor corollary bag predictor predictor order nice demonstrating calibrate fold average fold compare loss pair probability extract simple preliminary describe function loss predict whereas fundamental loss equal predictor assign mode independent online cumulative predict true main e course outcome regret minimax intuition interpret unnormalized normalize get case equal version towards neutral typical never ever predictor remain interior produce average section discuss score omit scoring tool nz decision j tree bag bag lr na nb network calibrate function svm standard however inaccurate score input see role comparison publicly purpose nine label uci repository breast cancer diabetes voting vote vary well order practice calculate test mle infinite incorrect confident compare dataset experiment remain drawback score test label object score training scoring simplify set predictor therefore never suffer hand proposition predictor cf simplify observation marginal somewhat score high violate predictor however proof artificial hope nearly valid experiment w computational split set w loss accord mle table probability direct bold classifier short name name svm especially sigmoid method redundant table come logistic sometimes output close hardware improve bagging bag decision tree bag involve set instability bag log make datum calibrate bagging rmse also often prediction whereas square produce obvious suffer infinite prediction mention arbitrary calibration simplify worst indicate table notice four calibration combination classifier would breast become mle algorithm early poor interesting interesting equal number dataset rmse despite theoretical validity similar confirm study preferable computational efficiency perform mle well mle set introduce predictor thereby applicability experimental suggest potentially well calibrate probabilistic log yield improve calibration probabilistic predictor theoretical guarantee share interesting multiclass solve multiclass predictor predictor simplicity rest ask set value fix function yy ad yy ad thank helpful proposition
dimensional think sparsity concentration many plausible remove single practical aspect realistic software available many go implement imputation readily available software approximate well test concentration hide propose variable nuclear norm penalization hide achieve apply likelihood variable moment matrix joint q except matrix iteratively replace finds guarantee penalize stage save converge minimizer generality identity determine write statistic involve miss overlap result clique vs six left imputation representative graph case isolate display unstable bootstrap simulation edge result figure edge stability propose correlation identify ed alternative paper effectively replace nuclear penalization paper easier choose reasonable penalty trace could combine
expert eq key step satisfy maximize complete third inequality fourth inequality clearly end shall ucb restrictive derive qualitative statement easy uniform finite support ix assumption strategy reason appear later uniform I request generality investigate go way expert good ucb oracle strategy limit particularly interested limit section shall derive limit section request expert go infinity surely interesting replace denote index draw discover expert interesting expert step request expert step step interesting successively suffice show surely elementary enough take fourth borel permit almost surely infinity accord conclude study sampling cycle uniform request proposition make precise extent exact proof omit go ii asymptotic optimality good ucb assumption converge almost eq item exist q end proof ucb proportion display draw intuitively discover possibility correspondence quantity draw converge furthermore ucb algorithm denote denote geometric permit comparison ucb uniform unseen interesting item arithmetic ucb proportion unseen small arithmetic mean expect get large proportion item expert unbalanced item find time ucb solid scheme dot present size remove interesting chose plot former easy visualize number item also section clearly ucb moderate ucb outperform sampling simulation conservative round run actual miss course long remain illustrate good artificial probabilistic expert geometrically expectation interesting item number loop good display figure ucb perform sample preferable fact proportion tight concentration inequality mass acknowledgement especially anonymous suggest section proceed obvious playing policy respectively deterministic sake clarity everything go randomized choice item round observe round imply sequence independent one miss large I conclude observe f discover interesting item run history history expert also expert request define consider item unchanged force hand behavior play moreover prove conditionally obtain everything obtain proof hand imply case sum final loop oracle policy hypothese ii open request advance make interesting request go infinity almost almost define eq find good allocation convex solution hence eq instance obtain denote give conversely eq limit proportion item oracle converge proposition ex financial engineering edu department electrical engineering science ac universit fr power expert optimistic paradigm estimator prove expert restrictive optimality provide finding request finite probabilistic request distribution discover request issue amount identify indeed security system lead consequence country electrical identify occur note usually security every contingency identify unfortunately credible credible contingency usually available time security therefore frame propose address rapidly credible significant simulation point new probability strongly sometimes choice possible within therefore engineering build raise try answer able contingency instant result security application web finite set denote assume independent describe pick index observe horizon item request element choose accord time strategy interesting small expert strategy expert non support restrictive support cardinality description generic term good ucb rely lead ucb confidence armed bandit rely analysis ucb large capture property contrary armed bandit bound understand make future policy key grow analyze first paper ucb close precise emphasize bound explicit ucb probability interesting occur sample fraction emphasize denote modify impact least armed reference rely index ucb miss confidence ucb rely tp estimate accordingly ucb without assumption expert shall make assumption mass request significantly difficult hand assumption miss mass reference arm form ucb meet shall good ucb hereafter satisfied loop denote request expert next expectation number find crucially rely yield consist choose expert prove ucb arbitrary policy horizon grow problem indeed thank discuss alternative ucb obtain obtain obtain eq jensen cumulative well bound armed bandit regret completely
average third model average three component merge three component ii average I ari lead great average ari compare ari averaging probability ht average bank window measurement originally collect consist available data mixture bic bank either ignore model I merge within component component classification good ii inferior species ii result slightly inferior ari probability average window bic ii property species world r mixture use within window perform averaging average respectively breast case establish status package observation percent composition respect eight nine model mixture lie within choose component breast end accordingly model inside window equivalent report classification bic three scenario generate via generate scenario generate perfect datum consensus cluster outperform consider ht case bank propose paradigm dominate window probability merge ari take window approach one think use model model average probability average probability carry probability efficacy merge herein real performance issue three instead choose three seven seven slightly comparable approach component average perhaps great notably probability avoid component performance great may argue compete window preferable end user want model term carry usual base carry fact irrelevant average limit clustering consider penalize future herein perhaps begin corpus family cluster family illustrate non family window work explore lead weight focus herein also analogous fashion acknowledgement grateful member cluster comment suggestion computational award research innovation clustering model report good circumstance criterion often multiple thereby cluster result weight average component membership determine closeness paradigm merge method merge effectiveness model use mixture datum approach mechanism accounting model compete approach successfully cox regression comprehensive previously application try least square indicate voting algorithm similarity partitioning arise merge mixture often back model cluster social network expression cluster report well small case criterion component relevant argue criterion approach away necessarily matter reporting value paradigm within report average idea cluster mark significant accept consider herein average produce interpretable directly average probability remainder outline section describe average merging component conclude discussion suggestion work recently mixture parameter total introduce component impose covariance way quadratic spherical volume impose constraint eigen covariance g proportional eigenvalue impose parsimonious provide member member family carry detail model ht orientation na na axis axis pp pp pp range model associate classification report popular criterion maximize bic component wide nonetheless small necessarily well predict classification gaussian cluster might model merged different cluster hierarchical entropy fitting subset procedure criterion merge minimize misclassification procedure maximize adjust rand model recent work sharp although discuss merge fitting mixture skew skew merging give inferior flexible fitting mixture herein arise care would need mixture skew inference family criterion proceed without ignore difficulty practically impossible decide frequentist model take consideration average posterior model prior give performance implementation probability compute overcome choose fit long discard analogy average integral bic q bic equal herein compute describe merge window introduce suppose component model want merge produce component mixture another representation original mixture could combination gaussian case gives merge merge merging outcome ari rand rand compare data partition agreement divide correction ari perform ari agreement ari random ari herein merge component purpose merge reference case window reference component merge perform give number ii equivalent assume bic take regard merging illustrate inside four component partition correspond underlie merging call reference avoid confusion denote seven merging represent nd component go component go go component new component possibility put merging reference partition arise merge calculate row ari reference partition merging store merge combination choose correspond ari base cluster application g via classification either probability otherwise allocation merge merge simply merge within drop case component weight cf probability classification addition model time window estimate use predict averaging probability focus merge classification classification average come parameterize interpretable average careful might equivalent related problem switch mixture strategy match minimum mean generally naturally objective averaging approach situation model average reference model discard merge carry tend cluster averaging averaging merge averaging may reason component say averaging single inside mixture cluster arise average produce average real simulate ii successfully window component
copy journal library service sometimes refer time day copy volume include c data science institute scientific science citation social citation processing year text format format restrict quantity journal keyword exclude publish latter medical algorithmic exclude started note rapidly storage medium sense production additionally allow previous book cd explain retrieval thereby domain content support wide area system retrieval standard early come year either book cd term service supplement journal european north american branch largely branch north classification historical web site journal classification volume always date journal last cd due plan web completely access service www edu volume explain carry service journal appear service reader journal paper criterion employ paper classification journal obtain institute information service paper sciences citation social sciences citation index database service book profile contribution journal paper procedure paper profile service contain profile suggestion improve composition profile collect volume record number year year example cm author another publication publication book publish work business school field multidimensional application cover fu hill huber rand wishart wolfe increase see figure net nonetheless couple grow research production iv give follow economic start replace jointly uk university key table name pick appearance publication title title biology physics mathematics engineering literature economic service follow explanation change service cumulative user interface van books journal process long successively observable attribute observable firstly eigenvalue factor semantic term relationship project post salient hence profile source text label therefore analysis occurrence plane issue content axis dominate connect around north west factor year little north regard management typical later year regard look rather dot figure crowd project cite come follow euclidean correspondence full year min project passive hoc subsection project minimum euclidean distance aggregation hierarchical initially figure relate clear branch dendrogram mathematic management include cluster discuss association order home pattern label fisher hand rand fu wolfe hill cover huber cluster classical characterize dominant period mathematic certainly cite period come modern see management characterize cluster broadly pattern cluster tune role certainly characterize statistic would projection open search show repeatedly e author web wide set appear three record practically return display record display grey display imply e access lead shift role subsequently management trend massive proportion service continue challenge greatly reference http www dl survey special publication pp analysis link computer science engineering journal visualization open platform http security profile author journal book title recognition l tm pa pattern bs j ann fu ks syntactic pa fu ks syntactic pattern k discrimination hill huber ann maps multidimensional principle discriminant infer b rand multivariate string ed ss rr principle numerical j annealing manual wolfe ct automate search service citation service term produce decade distribute journal increase production post quantity couple approximately major analysis time mathematic recent time centre management different
filtering node weight assign set item rate edge indicate chance walk rating assignment depend rating graph I nod item content profile social assign walk weight chosen eq indicate walk move graph node vector walk back follow interpret node path recommendation directly rate likely user similar item rating recommendation prediction equation iteratively rank node recommendation rank represent separate category tag sort exclude user recommendation collaborative filtering similarity item rank rating item q rank weight predict rating rating item rating evolve recommendation change incremental estimate every scope recommendation similarly recommendation prediction recommendation system sparsity namely effectively recommendation na recommendation study social tend similar availability network offer extra social node recommendation recommendation user user connect recommend item user order experiment benchmark website post long review randomly item trust datum set consist movie profile gender rating rating user evaluation recommendation percentile size high percentile rank st recommendation consist item percentile prediction set rate relevant recommendation information art collaborative describe recall compare baseline warm start percentile warm start note considerably start h cf recommendation item content recommendation target improve start rating perform collaborative filtering support science science foundation technology grant research office grant nf dr wang valuable mr experiments department electrical engineering university nj filter build recommendation hybrid model walk system consist item content user profile social setting graph recommendation prediction algorithm evaluate start system hybrid last decade early system achieve recommendation serve component demand service amazon netflix recommendation intensive work improve recommendation assume user recommendation explicit rating item challenge recommendation preference information especially helpful give recommendation user little rate good integrate user social collaborative preference user user collaborative database prediction net dissimilarity bipartite link movie movie average build rate node random recommendation give rating edge user rating score unit user social make rate contribution hybrid collaborative incorporate user item user history recommendation construction edge assignment application recommendation multiple list list rate rating netflix implicit click user movie date item correspondingly profile gender denote user contain social
often jointly conditional sequentially chain initial note variable da one q jacobian determinant auto px da generic call markov leave sample z n n n da da px da slightly great case practical increase computational negligible benefit sampler quite simulation family sequel direct result px da term intermediate essentially implement kernel reversible marginal imply density chain generate law note auto correlation generate da algorithm trivial da remain mcmc scheme improper shall correct remain unchanged concern px da improper although translate almost everywhere markov transition density reversible integrable expectation expression markov initial distribution da px da address haar da transformation haar px da scale transformation haar px reader scale variance da proper haar px da indicate generate subscript proper haar da haar small asymptotic variance measure note augmentation counterpart subject proposition augmentation stress step kernel result suggest modification px da perform da eq translate specify optimal variable regard regard b conjugate gibbs meet follow conditional follow set v unchanged fix multiplicative important make hasting describe briefly mdp way multivariate probit set zero g however notation benefit exploit da arise joint abuse density admit gaussian come discussion put aside px e da conditional augment px da z z ig gamma appear da incorporate connection description generic px da abuse jacobian variable px da step hasting kernel proposal quite percent improper corresponding improper prior sample careful improper able proper may reward px da model iteration result truncate call step hasting joint verify z could implementation one structure detailed far impractical state setting may value onto discrete see mapping assume ix easily controller generation likelihood invariant scalar even px da transform application dependent study think identity noise mdp computer game henceforth henceforth mdp consist component block take angle move combination necessarily boundarie overlap state state occur map configuration configuration allow reach square bottom remove row irrespective termination reach subsequent influence state consist configuration total outline section control independently detail assume reward observe q case likelihood algorithm scaling function capture height square column square difference column automate system emphasis make prediction setup amount post burn da prediction basis end assess number draw subset prediction assess subset assessment action situation resort heuristic explore recover purpose qualitatively typical lead style region rarely second column third generation game immediately termination termination occur step full concatenation three assessment value px incorporate run run burn hasting relatively uninformative prior value improper improvement post histogram marginal significant autocorrelation da da indicate px da standard da observation observation increase qualitative characteristic prediction show block sequence state indicate predict action qualitatively obtain lastly move play inference observation termination step play well termination occur step aim unknown play subsequent assessment da experiment rate trace histogram density action use result even three information pressure allocate aim validate statistical record inference player action preference amount record player force play appearance player second record allow fall translation approximate posterior marginal pressure value indicate allocate preference panel term mode make make decision allocate difference computation action situation arrive gradually posterior monte carlo amenable kind sequential computation possible model paper microsoft indicate da predicting rank microsoft support choice da move comprise operation definition may routine integration integrable change line establish state procedure omit scalar decompose w denote cumulative several algorithm sample truncate marginal metropolis hasting proposal current issue approximation one latter quickly first multiply discard refine newton curvature mode acceptance program request implement un un problem controller statistical markov adopt unified framework new include essential good illustration apply controller observe action variety economic aim mechanism case assume arise specification state controller choose receive instantaneous specified state evolve action receive reward controller horizon controller noisy mdp involve specify reward aside difficulty specify heuristic observe adjust controller repeat simpler desire behavior human generating know controller intelligence biology economic aim purely statistical tractable behaviour statistical advantageous principled properly uncertainty controller mistake jointly future action model specific functional parametric maximize likelihood observe respect perspective augmentation make contribution adopt action mdp inspire statistical gibbs subsequent enhance new devise sampling infer expand da px da improve upon da simulation involve moving augment extra computationally da da px algorithm propose movement augment translation scaling also implement metropolis proposal augment illustrative controller apply game literature treat give player perform posterior player solely specify reward learning demonstrate model organization detail inference da px da assume discuss practical issue numerical various px da px present capital case realization short density mass random two jointly density marginal omit indicate correspond transpose comprise denote omit similarly otherwise mathematical denote concentrated point mdp comprise markov control process additional optimality control control state action probability evolution determine mapping value call reward consider criterion discount accumulate horizon discount ensure lead expectation finite set discount
work flat clutter object visually give consider symbolic object al consider place recently successful scene understand et propose dp learning object present idea specific mixture consist mix proportion mixture model finite model several independent model choose parameterized component product model cast define new combination total would prohibitive method draw component obtain observation effectively computation challenge optimize model jointly distribution mixture model informally present section refer model stick chinese restaurant process select everything assign exclude infinite model dp breaking process except sample number exclude conditional replace however assumption hold general control tune concentration construct smooth single two auxiliary distribution multiply l fx differ depend conjugate prior long prior multinomial sampling hasting present concrete section differ prior hard conjugate case lda employ hierarchical describe proportion symmetric multinomial whole vocabulary size lda allow share world type define fig assume probability model maximize integrate document inference follow classic q difference quantify goal provide supplementary iteratively infer document estimation challenge variational since lda term hard close expression convert eq value gradually violate bfgs quasi penalty one type separately pose assume therefore mixture assignment turn give use sample first behave document baseline hybrid object share infinite pixel represent group ground block diagonal draw I two draw wishart synthetic term density estimation pc z contour successfully identify correct specie specie average fm figure four corpus section appear http ac uk include remove word word article take article result fold learn topic article vs vision article one document leave obtain lda number significantly across lda trend present ability hierarchy sub tree topic proportion fact topic change number domain training dataset would right average different largely depend vs significant baseline consistent topic less multi parameter topic fig model obtains demonstrate different topic dimension contain keyword digit sp bottom topic set affect topic heterogeneity try asymmetric set worse lda hdp lda fm topic cccc vs cn vs force digits statistical realize algorithm cell rank list dimension different popular change object room location room scenario place test room place room fold location difference human pose object predict truth place object together due object share pose multidimensional membership mixture mixture multidimensional infinite membership finite build two model increase robustness sharing situation challenge sampling ability achieve model application pt pt minus pt cs membership membership generate jointly share unique variance nine require parameter fully describe hybrid mixture dirichlet mixture allocation combination topic well mixture ability complicate simple model vocabulary document build document analog word type let fig different five parsimonious scenario work multidimensional draw mixture covariance result parsimonious structure take topic combine specific control topic paper unnecessary identify structure multidimensional sharing parameter different mixture dimension effectively parameter different topic model hierarchical dp branching need accordingly hybrid finite employ name area topic development corpus latent variable statistical mixture indexing later propose corpus document share
observe slightly value large ise mrf mix close due dependency size number accuracy indeed set experiment show component compute size mrfs experiment ise mrf regular lattice sample require minute ghz intel run critical phase dependency mrf fig logarithmic logarithm logarithm field component significantly behavior statistically component decrease state indicate ising compute estimator small performance explain gibbs sampler explain degradation ml variable estimator accordance compute mrf chain generation minute ml similarly exchange close degradation letter depend intractable expectation mrf monte carlo frequently bound turn efficiency method process extension mrfs currently investigation future include derivation bayesian rao bound assign proposition letter consider rao field exact interest likelihood intractable derivative bind successfully ise monte carlo intractable estimation intractable receive recent computational statistic signal estimate markov mrf research unbiased mc letter addresses rao mrf knowing point view limit information carries specifically define practical perspective mean unbiased mrf likelihood letter address statistical propose express mc specific mrf widely namely ise remainder letter class work propose original method application methodology ise present conclusion whose eq mc normalize integration appropriate model important comprise laplace multivariate distribution frequently application intractable inherent evaluating integral establish lower existence weak regularity inverse q q rarely address letter propose exploit expectation approximated integration relate substitute whose element previously known physical property knowledge inference expression replace perspective alternative fundamentally expectation intractable efficiently approximated mc integration mc approximations suit number mc integration simulating chain carlo mrfs output algorithm markov summarize ergodicity sampler guarantee increase please invertible enough stable regardless simulate letter mrf use used z simulate specific ising wang ising mrf completeness discrete value mrf q
parameterize vertical positive isotropic hyperparameter align k km hyperparameter variant predict might resolution along little resolution parameter implement automatic relevance determination ard individual scaling adapt datum value variable important variable less variation coordinate zero coordinate could rely state benefit remove irrelevant decrease stay variant identify combination variable rotation coordinate system direction advance along axis tb calculated ard convenient establish identity matrix one compute derivative h ij x partial computational derivative completely automate set unnecessary state aspect particularly reason see consider formulate solve fit transition episode align ard ii good implementation ghz conjugate gradient give adequate visit trajectory differ start predict slightly known substantially state selection iii note iii choose dominant take look vary whereas along opposite diagonal ii along benefit hyperparameter likelihood predictive circle train insight gain look iii decrease fast consequence complexity capability help well indicate effective efficient small important employ sr online learn count keeping size essential good approximate sr ii iii center select approximation f subset detail reward except episode leave leave transition ii result without see however ii irrelevant unable weight consequence ii additional gain look cf likelihood moreover completely set rapidly even ignore task effective sr base approximation automatic feature generation primarily principle rl ultimately theoretical guarantee due less state action product policy optimal approximate policy iteration world batch setting online gain selection improve research artificial intelligence laboratory grant nsf fa class science edu rl value rl world application associate hyperparameter benefit solve evaluation problem fully allow turn enable relevant subspace eliminate variable substantial approximate value function approximation arise horizon rl problem satisfy function manually important require would could automatically adapt without resource factor easy example relevant input lie rl regularization promise instead specify individual specify key demonstrate selection rl sample transition automate hyperparameter likelihood loo minimization gp policy method benefit base search manual sophisticated concentrate effort part really matter improve generally space generalization despite rl explore variant stochastic hyperparameter use score online agent standard gps employ since approximation structured follow contain summarize benefit large problem sr approximation selection derive associate illustrate representation parameterized rl dimension directly target function guide either bellman reward unsupervised graph connectivity conceptually relate approach hyperparameter rbf basis location either descent bellman basis adapt individually overfitte place problem point use complex x r produce x value regard share gps traditional machine require inversion dense would regressor initially choose approximate take denote k mm mm motivated example nystr submatrix nm ij approximation nm nm plug eq b b similarly gain storage every prediction additionally evaluate sr produce degenerate predictive variance near zero far novel project approximation expression add term combinatorial find subset summarize forward step element well specific criterion general unsupervised target cholesky aim reduce incur equivalent schmidt every active span residual
exist latent scale regularize section scenario code writing intel ghz processor either dataset dimension computation initialize stop objective careful inspection reveal newton find quite apply generate tuning newton event safe stability panel display coordinate coordinate model non search dataset objective indicate objective panel figure plotted well low dense panel however small point suggest spike htbp three bt dotted cd plot different bt coordinate cd line initialize nonconvex guarantee global recommend initial algorithm experiment extreme case start avoid exact pick represent three run recorded point objective report thing value diagonal case try initialize bt get adjust tune safe careful initial difference diagonal initial surprising sparsity start column reach regardless coordinate good zero diag diag diag bt cd sparse bt bt sparse bt cd bt cd bt bt cd dense bt cd bt bt cpu cpu run element bt maximization cd much easy significantly numerically software package new author plus ex plus minus wang statistics sc edu apply lasso element marginal independence generate propose solve graphical coordinate attractive discuss descent basic covariance minimize matrix shrinkage norm graphical shrinkage element exposition method difficulty exactly zero point zeros encode marginal component concentration graphical concentration associate conditional objective impose challenge minimize complicated minimize solve current alternative investigate computational fast numerically test coordinate model variant framework find posteriori estimation regularize graphical map estimator know representation suggest consequently graphical expected inverse criterion remove term depend focus partition transformation drop fix rewrite conditional give conditional maximum every total cm step column algorithm partition repeat
exchange datum presence exchange drive scenario convergence cost pick stochastic run exchange assume regressor still covariance eq repeat validity either statement sufficiently justify ignore depend power square satisfied neighborhood covariance moreover rate two mean mean adaptive diffusion link introduce block quantity define perfect relate follow expression limit latter analogously devise adjust noisy exchange illustrate construction strategy weight update since simplify problem separate associate variance incorporate information exchange continue leave combination need motivate recursion node view manner similar steady satisfy recall r fourth factor ignore size moreover product comparison multiply instantaneous enable variance coefficient one q converge close replace equation adaptive construction several diffusion variation nod temporal strategy state highlight select network share combination add processing example solely current neighbor store estimate say smoothed enhance mean especially motivate strategy smoothing mechanism continue satisfy model sec continue scalar weight instant coefficient solution minimize still argument arrive version strategy incorporate intermediate estimate nonnegative need simple three filter smoothing illustrate letter label step sequence diffusion adaptation spatial combination temporal adaptation result variation filtering reduce variant processing reduce diffusion strategy extend argument sec doubly temporal processing adaptation spatial six variant satisfy note within operation moreover perform reduce reduce conclusion early outperform diffusion I u iw k u iw diffusion diffusion k j iw k u f relate reference start exchange add useful spatial occurring follow use project raw refer act project onto vector vector projection projection temporal plot table base use past current ki ki ki k p temporal processing tend across square enable least design distribute column length use instant vector model snapshot capture collect likewise history collect q estimate square represent hermitian represent hermitian common choice square time instant error occur heavily occur close diffusion develop rely solely compute node recursive square follow every matrix time node scalar nonnegative algorithm instant node incremental intermediate diffusion incremental step neighbor intermediate quantity matrix substantial resource matrix entry diffusion square start diffusion recursive equation stage introduce follow choice strategy variable q correspond illustrate special relate equation q approximation diffusion lead enhanced consensus update space smoothing briefly diffusion kalman filter evolve accord linearly objective track system interaction observe vector measurement measurement zero zero mean covariance matrix uncorrelated solely evaluate approximate filter denote block matrix unitary unitary appropriate next useful bound norm useful size argument nonnegative note equality induce establish bind attain left norm obviously q therefore establish dimensional equivalent norm exist nuclear define sum hermitian definite block norm absolute matrix specifically introduce block matrix matrix property maximum obvious vector vector establish maximum sum entry sign moreover combine next establishes block norm combination right matrix entry introduce matrix establish property maximum evolution error network hermitian hermitian hermitian spectral radius argument agree large radius block radius readily establish reverse entry large hermitian induce radius explain diagonal left stochastic block diagonal hermitian block matrix never exceed conclude matrix conclusion corollary stable lie unit propertie lemma conclusion stable possible appear lemma lead part stochastic prove converse choice must measurement possibly evaluate require processor node centralize operation suffer communication central processor problem fusion critical failure central processor fail need recursion elegant whereby interact locally assign node assume combination entry combination satisfy describe operate repeatedly neighbor iteratively state collect recursion express entry collect entry convergence condition tend initial consensus doubly condition state doubly fact verify induction validity likewise identity matrix iterate kronecker eigenvalue fact coefficient stable necessity must doubly doubly condition q rate since doubly know us term magnitude equal follow magnitude radius arrive condition successive iterate consensus strategy determine large worth doubly also satisfy condition combination doubly regular satisfie therefore ensure generate towards list connect consensus condition doubly satisfy condition connect associated nonzero combination weight include therefore towards namely multiplication entry power connected value path corollary consensus converge comparison strategy strategy employ quantity construction keep iterate converge consensus algorithm strategy quantity side process exchange update filter diffusion collect instant inherently keep streaming diffusion incorporate instant diffusion manner optimization iteration cf cf turn influence evolution interesting advantageous manner consensus strategy initially information exchange ease derive body optimize intermediate combine likewise neighborhood combine subsequently update also motivated consensus suggest e early instantaneous side equality diffusion strategy consensus influence help incorporate reflect likewise strategy rely desirable evolve recursion aggregate diffusion set matrix mean sensitive establish matrix size satisfy diffusion regardless whenever regardless contrast consensus individual stable diffusion relation consensus development application diffusion adaptation insight contribution student student laboratory http www student tu chen yu lee author also yu early chapter r work nsf grant author electrical engineering california usa email edu electrical university california suit decentralize various self behavior nature adaptive consist link connection topology solve inference relation streaming condition adaptation agent provide overview diffusion strategy adaptation network divide section motivation square distribute strategy descent diffusion diffusion noisy extension consideration appendix kronecker b laplacian block reference consider say q agent agent become neighbor enable optimum decentralize spatially rely solely continuous sharing neighbor localize article connect network separate disjoint subgraph network arbitrary connect node pass intermediate figure provide node able sharing directional particular node degree higher denote situation relation node flow nonnegative neighboring node receive reliability assign use subscript subscript denote send different exchange zero mean though node reach node likewise whereas situation use fig draw information control turn also reverse htb denote send datum send exchange illustrate neighbor leave admit interpretation example entry location location chemical localization identify band share communication medium agent albeit frequent among beneficial general important lead solve strategy overview area adaptation illustrative diffusion strategy enable adaptive useful consensus distribute nevertheless explain diffusion consensus strategy body present treatment distinguish quantity use random example letter quantity font letter need distinguish capital letter small letter scalar letter refer scalar variance distinguish vector scalar letter scalar thus refer time presentation letter row convenience symbol complex conjugate entry stack denote argument top vector maximum absolute among eigenvalue drop use article cl font capital letter letter scalar small letter scalar letter scalar entry diagonal entry top argument weight induced identity drop reader development optimization process square advanced skip read start cost utility algorithm nevertheless convex cost minimum choice dependent purpose concept underlie diffusion focus mse network beneficial argument extend mse cost individual cost already summary non solution agent would operate determine interaction derivation reveal converge tracking effectively relate noisy region observe ar scalar agent spatially nod eq far assume allow noise vary figure consist node spread space figure neighborhood parameter equation equivalent relation arise context process jointly sense order case noise profile allow depend index vary spatial definite convenience shall definite hence result follow recover multiply take obtain linear moment square mmse quantity mmse appear find quadratic respect find see therefore solve mmse coincide explain justify deal linear besides mmse noise level expression result square error attain find minimum recover expression attain mse location substitute simplify find expression mmse determine information successive become necessary devise measurement adaptive alternative node observation evaluate successive pass adaptive operate compute instant many accurate yet structure note discussion article structure process iterate instant compute exist appear constant positive discuss article size replace filter normalize satisfy slowly virtue size turn size environment reason shall ensure continuous learning equation quantitie deterministic interested nature step especially filter useful actually variable kind stationary change estimator tend towards virtue mean ideal variance filter loss adaptation gradient cause actual steady amount offset size smaller easy introduce priori measure assume instant variance priori mean excess quantifie offset adaptive far steady small step small well performance adaptation regression accordance variation noise perform however node model among mean neighboring share estimate network starting motivate node carry adaptation manner go achieve develop algorithm manner objective add individual note every verify global optimization desire scheme ahead square relate move finite impulse agent interest communication biology agent probe additive situation illustrate operation highlight inside diagram assume mean process assume white spatially scalar agent seek identify collect column input express likewise parameter equation continue expression therefore mmse attain location alternatively adaptive show limit perform worse small since observe underlie among achieve global cost adaptive manner ahead third relate source chemical localization application agent allow adaptive biological behave towards away e g tracking motivate situation target node agent spread assume aware point agent express inner product ok access noisy practice distance direction towards denote assume perturb continue either perturbation model deviation occur direction along direction perturb relative along amount spatially assume compare contribution assume independent find measurement since assume linear model q difference eq vector moment continue side compute however equation justify linearly estimation attain mse information iteratively adaptive implementation early performance therefore large distance direction towards location beneficial going way develop algorithm ahead role application help highlight advantage adaptation track change mobile close agent become dependent actual distance target vary well mobile time target close expect remark hold eq lead repeat lead variable eq nevertheless adaptive work directly reflect change statistical track non stationarity slow fourth relate sense cognitive type primary secondary avoid interference cognitive device detect band condition carry aggregated primary band communication environment consist primary user secondary spectrum facilitate secondary function say scalar expansion angular measure spectrum vertical therefore interval basis possible illustration purpose divide location control illustrate basis large class spectra h gaussian collect primary column collect basis express alternative coefficient secondary primary user spectrum determine stage involve repeat accommodate albeit primary secondary secondary highlight path location similar path primary primary power secondary user receiver collection coefficient primary vector contain primary user every power spectrum discrete frequency mean variance assume estimate every instant discuss ar localization implication vector scalar continue collect quantity eq subscript expression early difference scalar represent continue application secondary aggregate spectrum secondary amount vector secondary reconstruct spectrum kronecker verify moment secondary determine solve follow minimum error knowledge similar lead verify attain performance level level location alternatively similar secondary noise perform secondary secondary user observe arise natural expect among beneficial going performance develop mse tool frequently application sum assume article nevertheless focus column assume clear get square compactly view covariance eq covariance definite sensing focus derivation argument come beyond sec covariance ensure seek couple even determined describe lead estimate realization adaptive rely instantaneous powerful behavior change change reflect observed behavior construction strategy enable track act individually limited power perform well node process across consider development series approximate amenable distribute alternative motivate start introduce neighborhood mean relate row translate add denote say right use associate fig also eq weighting eq amount taylor series side consist quadratic cm along edge weight node local substitute second expression minimize follow relate equivalent newly side local cost correct cost minimizer value likewise neighbor suggest alternative cost lead powerful solution neighbor function node expression latter optimize weight matrix knowledge however non arise whose nonnegative coefficient node substitution property ensure constant un weight positive use hermitian characterization eigenvalue approximation theory newton former reveal select scalar set adjust far ahead replace argument modify global rewrite exception solely neighborhood soon fact node minimize evaluate start positive denote evaluate size parameter vary size sequence tend shall case develop return arrive iteration correction form implement step add correction update intermediate gradient actually try stand examine minimizer value actually perform step minimizer help throughout neighbor neighbor incorporate information approach widely item replace observe coefficient relate like treat weighting theorem collect translate one nonnegative satisfy intermediate scalar correspond weighting coefficient coefficient use first scale explain column row stand example illustration hand case correspond coefficient connect instant strategy combine update exist intermediate node perform exist estimate neighbor second aggregation combine neighbor step reason name adapt combine second adaptation follow reason combination real influence information exchange aggregation strategy reduce step locally node observe pass exchange gradient local strategy q significance applicable cost quadratic detailed involve neighbor label evaluate neighbor subsequently dot represent diffusion involve exchange exchange represent return reasoning step arrive nonnegative coefficient strategy step combine estimate intermediate perform aggregate estimate neighbor exchange node information intermediate simultaneously name adapt first involve step strategy combination reason network special exchange rely locally node rewrite cost follow involve first exchange structure fundamentally lie update combination adaptation step ease reference list derive follow early use dependent require reach consensus agreement likewise scheme exchange aggregation appear subsequently mean decaying adopt towards function l k du w k w strategy weight word set derivation comparison common strategy doubly row reveal converge common consensus strategy agreement end adaptation flexibility process tend square deviation manner require index vanish size property adaptation function adaptation combination topology mobile solve knowledge agent expectation become arrive adaptive coefficient adaptive usually start convenient clearly view approximation successive iterate iterate nevertheless shall continue use ease implementation data realization implementation practice approximation introduce update interpret involve cost affect diffusion close list htp mse individual adaptive l l w k w l operation diffusion time instant step first receive neighbor information use node combine intermediate neighbor node step exchange aggregation instant strategy step combine estimate aggregate information exchange node receive neighbor information intermediate estimate exchange exchange aggregation node run kronecker delta situation study implementation noise necessary sec rather useful descent general scalar set correspond implementation correspond become furthermore correspond reduce stand alone recursion exchange examine fast converge measure subtract relation relation compactly collect collect node vector represent likewise entry neighborhood denote neighborhood combination q reduce appear block likewise kronecker replace matrix entry matrix ease reference list sequel variable nm return error end purpose non node evolve special evolution vector distribute implementation examine convergence quantity block block extensively exposition reader review stated lemma establish descent establish end converge solution node optimize pick run diffusion estimate solution positive step nm stable meaning eigenvalue lie ti nm verify ensure matrix long weight matter influence nod actual classical stand alone bi directional stochastic combination stochastic employ non q row add early satisfied weight share strategy enhance step size optimize doubly satisfying consider distribute diffusion mode individually say condition decay establish satisfy stability verify also covariance e eq hermitian matrix coincide large jensen hermitian matrix nonnegative scalar choose radius r jensen decay zero ti nm likewise determine radius matrix note ti nm ki u k r nm individual spectral radius handle size assume covariance enhance set size stability condition case meet hold magnitude diffusion decay rapidly diffusion doubly stochastic ti nm strategy information exchange convergence enhance ci consider optimize stochastic situation mode algorithm operate descent satisfy meet decay zero ti nm r nm nm previous theorem highlight important fact role combination convergence diffusion influence stability matrix radius ti nm r dynamic error move adaptive implementation gradient noise steady perturbation average reason mean special resort measurement order become update adaptive scalar coefficient respectively matrix choice correspond adaptive correspond likewise correspond share become operation run classical stand alone performance diffusion exploit indicate related likewise analyze strategy network present early shall satisfy form white spatially far network turn result expression well simulation small independence stand filter whether estimate adaptive diffusion implementation towards introduce desire measure measure would term implementation capture priori second component variable confirm always least priori order quantify particular excess steady quantify offset square square indicate stand filter non section examine among node influence since operation network require nevertheless say expression performance interesting subtract relation recursion replace matrix instantaneous I compactly quantitie error likewise individual diagonal say matrix contrast r simplify combination eq time reduce column vector zero eq covariance relation end evolve operate individually non across would evolve appear replace independent early noise error recursion replace I would say asymptotically follow statement theorem analysis pick right network measure describe converge optimal k combination combination influence neighborhood lemma become classical result stand estimator node case converge bi diffusion choice convergence optimize pick network adaptive left theorems stochastic enhance satisfying network datum condition consider diffusion node operate case meet select decay rapidly enhanced uniform say independent far employ meet diffusion decay rapidly consider establish strategy enhance enhanced ci cost pick satisfy situation measure operation network run diffusion operate individually case choose require magnitude diffusion decay word diffusion highlight follow important matrix influence mean stability influence matrix matrix stability influence convergence since radius ti nm converge fluctuation examine evolve steady node evolution mean steady quantity datum square thus square evaluating quantity hermitian weighting say reveal energy block hermitian nonnegative stacking column write sequel convenient start compactly short note constant denote side q side evaluate expectation expectation compatible nonnegative define evaluation order continue sufficient exposition focus sufficiently term power size ignore let topology statistical profile expression transform follow compatible side coefficient matrix reasonable replace expand plus square whose effect term notation square convenience notation exploit simple rewrite appear relation independence impose sufficiently size power step ignore leave stochastic hermitian nonnegative definite actual recursion side equality relation transform recursion expand convenient exposition except ensure stability fact establish relation remains bound converge steady radius steady value adaptive square lie strictly unit satisfied satisfy define moreover rate determine compatible form note stability ensure also stability second order would stability require size stability stability node steady term error I n diffusion strategy satisfy hermitian expression average therefore order select q matrix early repeat substitute arrive diagonal obtain likewise diagonal except index whose network measure obviously implementation perform series express alternative filter expand desire simplify regression employ combination matrix doubly refer uniform variance noise across network sequel sec expression kronecker identity compatible likewise introduce noise simplify additional representation two separate dependent compare use steady iterating compare expression recursion relate measure evolve clear stability term grow unbounded obtain average node evolution select substituting arrive correspond follow recursion network table ahead summarize derive section doubly combination eq row auxiliary verify doubly part outperform information profile consider exchange scenario follow description expression derivation get performance exchange perform exchange measurement case would doubly stochastic eq without exchange exchange case end effect exchange correspond note follow difference performance tr u c network
bayes neural network propose neural network layer well go thus algorithm natural part role learn representation share neural outperform hierarchical n sentiment al answer wang name entity chen al extraction chen et parsing know still investigation recently learn work seem however instance small vector highly whether sentence encode segmentation module module stop word text come intuition word topic occur target word w word type encode position word order feature cross module preprocesse experiment window yield well use speech tag part speech tag word nn noun phrase vb dt feature encode word sentence follow seven na I neighbor logistic label na I bayes nb conditionally independent nb na module al neighbor choose close learning near take vote nn extremely use near neighbor principal pca eigenvalue problem covariance magnitude basis small et nn nonlinearity introduce trick map reproduce rkhs convenient nonlinearity implicit mapping compute use q center equation eigenvalue like pca basis square eigenvalue double dimension use polynomial regression technique probabilistic prediction prediction softmax et sgd rate sgd perceptron mlp feedforward artificial mlp layer layer view input try output classifier layer mlp layer stochastic gradient hide architecture perceptron find hyperplane programming control tradeoff width extend form optimization nonlinear rbf deep graphical hierarchical latent start learn feature layer field restrict rbm recursively parameter set fine backpropagation discriminative second adjust architecture deep belief sgd function hold cross validation tune iteration rate hide layer restrict boltzmann rbm another connect different make rbm h restrict boltzmann machine make visible unbiased rbm try reconstruct function bias visible layer respectively activation rbm achieve descent start take input rbm try h restrict boltzmann machine lot divergence cd introduce perform gibb gibbs come various relation rbm combination stack rbms complex label current backpropagation bp smooth possibility optimum backpropagation slight model task instance datum grain measure performance micro recall side frequent sense word task nn perform improve polynomial kernel rbf kernel principal component machine comparable among work good logistic perceptron mlp make nonlinearity work well outperform baseline learn algorithm rbf nb mlp rbf micro p lower nn well na nb strong bag binary bag sentence work mlp good feature outperform mlp nn na I nb feature among work mlp bad feature score score outperform logistic regression mlp outperform learn feature feature nb feature speech svm score higher improve little worse still outperform baseline much improvement alone basic make fairly belief word three compare art superiority art support machine outperform reduction pca I however regression perceptron significantly speech local much feature need help extract engineering deep nonlinearity instance lot cross show instance rate direction firstly representation helpful improve per secondly source one incorporate department cp ac intelligence laboratory institute technology ac intelligence laboratory institute pi ac rbm
important issue machine learning show bayes boost machine classification fundamental comprehensive boost svms prove label previous consistency pairwise surrogate class difference conditional consider expect distribution minimize challenge consistency surrogate loss formulate base new consistency whereas yield calibration necessary necessary insufficient hinge loss calibrate inconsistent consistency ranking quite setting instance consist accord yet label instance exponential loss hinge loss hinge inconsistent auc consistency surrogate logistic surrogate loss surrogate generalize calibration necessary yet insufficient consistency surrogate loss present exponential well general surrogate loss equivalence loss auc present denote underlying distribution identically otherwise maximize auc fx f risk infimum take measurable get hard practice optimize auc hinge infimum instance define define follow auc show sufficient error auc calibrate auc eqn commonly hinge hinge square expect pair instance different consistency instance minimize minimize study focus conditional risk hinge contradiction consideration loss simplicity hinge loss contrary prove calibration necessary auc consistency lemma surrogate loss partly motivated converse direction inconsistent auc hinge surrogate inconsistent detailed defer addition hinge absolute prove inconsistent auc loss surrogate f inconsistent auc hinge loss absolute convex calibrate lemma auc long lemma insufficient auc parallel binary classification error auc consistency whole present detailed defer differentiable present simple lee hinge provide condition bound later theorem surrogate prove consistent exponential surrogate x f f consistent consistent auc reformulate follow consistency weight loss x consistency hinge convex increase derive variant hinge consistent example consistency norm hinge loss hinge surrogate x immediate auc hinge general hinge eqn f auc hinge loss inconsistent auc loss loss hinge loss bound corollary consistent detailed logistic fix rf instance minimize norm hinge regret bound exponential focus fr partly theorem logistic hold detailed corollary rank bound q negative instance e x f hinge norm hinge loss hinge auc corollaries regret whereas provide regret analyze auc regret bound learn improve accuracy infimum take measurable popular formulation loss logistic etc risk f f infimum take measurable begin bind eq detailed theorem classifier optimize accuracy optimize auc classifier easy smooth score rank exponential x f section learn optimize exponential proper theorems rf f rf r pr f asymptotic f auc consistent threshold consequence auc infinite sample adaboost training consider show equivalence auc provide equivalence calibrate exist auc differentiable auc conditional assume convex eq case obvious g exist similarly eqn instance eqn give f f hinge construct eqn yet complete begin crucial differentiable increase definition contradiction similar introduce far derive give differentiable function therefore f eqn contrary eqn immediately complete e jensen c f expectation f c x f f get therefore x p inequality apply exponential logistic function immediately therefore proof eqn hold q easy yield surrogate x r r x pairwise loss auc e x x f complete area roc diverse convexity auc algorithms surrogate calibration yet insufficient auc e hinge calibrate inconsistent auc base asymptotic consistency approach surrogate loss least square etc prove derive obtain many surrogate finally bound equivalence accuracy straightforward adaboost limit infinite design pairwise propose require scan auc superior auc hinge theorem area roc criterion diverse imbalance optimize convexity pairwise etc
notation z q fix simplify hereafter ease gram lemma simplify q density starting matter appear I third calculation put full see sum case claim simplify combine row eq fix index actor let th actor domain operator abuse word row identically markov proof leave reader survey proof force start proposition fix fix treat recall fix claim empty f convergent subsequence consider arbitrary convergent subsequence observe algebraic real convergent subsequence real orthogonal real algebra fact take subsequence distinct diagonal generally see eigenvector hence algebraic eigenvalue orthogonal implicitly moreover eq fact convergent share common subsequence convergent activity doubly three develop infer actor latent stream actor process latent actor movement model govern dependence population actor position online actor value numerical scaling doubly classification g present ever challenge wide inferential contain message short message infer community asset actor word income summarize retrieve past transform detect mathematically streaming time actor represent actor message suitable deal notable cox hazard doubly also know although self sometimes cox hazard reference information transform actor multivariate take suitably statistical hazard activity actor model bernoulli family multiplicative actor time mild consistently participant explicit actor model intensity model spatio information series multi model among actor pairwise overlap interval actor mention thought capture underlie social empirical dramatically useful information doubly process drive latent process interaction actor proximity position close actor configuration message model actor approach representation formulation population position convention communication activity undirected mean know actor receiver probability probability proportional set k wise vector order product give column th entry slight vector entry number actor fold bold letter bold letter denote actor identity column one abuse notation indicator confusion actor big observe actor longitudinal actor feature frequency pairwise activitie illustration far position detailed diagram message generate actor community population member distribution proximity actor message distribution path eq value vector covariance positive smooth degenerate smooth degenerate diffusion taking th coordinate finite dx modeling example yet co exist dimensional motion stationary relative center ht b order actor population define begin give taylor value actor actor actor roughly speak influence actor space actor actor latent discussion motivation appendix actor deterministic path assume latent position assume let column vector transpose compute symmetric xt xt b message send actor actor message actor experiment actor possibility actor actor time interval jt ht paper continuous x formula update posterior present formula actor generalization case actor notational complexity nn ni jx ji ij dm ni ij hereafter develop actor dirac delta generalize formal adjoint u update projection latent systematic cf cf idea approximate weight discussion generality dissimilarity arbitrary solution mapping run dissimilarity relationship g read influence last section fix note away subspace lebesgue algorithm existence simulate property fact poisson ij u u ij ij ij jx x interval practically mean normal experiment hope detect possible position even though information estimate experiment environment end actor far away mode affect actor small eight actor illustrate eight actor able adapt five actor axis jump actor update discretized unit mixture initial actor sample distribution drop low simulate actor value near show behavior vary intensity produce simulation observe amongst actor change unit interval amount eight actor message equivalently actor hand eight actor tr vertical latent eight actor position eight actor experiment dash filter position actor position eq sufficiently interval height recall close connection model paper general setup experiment experiment comparable generally roughly amongst actor translate actor rough stochastic run q move average fix haar modify straightforward reader latent instead get large medium showing compare clarity suffer embed dissimilarity color clear lag accuracy clarity compare figure misclassifie indeed figure embed position true unobserved position bit intel core cpu ghz machine gb ram actor take red cluster gb monte replicate take slot replicate take ari maintain ari maintain x nearly prior show embed row position embed embed illustration actor activity complete cluster posterior present level level posterior online study often address stop illustrate simplify mild task structure crucial maker whether act confidence infer measure establish take explore robustness
exploit suboptimal example adaptive adaptive incorporate localization check play suboptimal nature abstract level subroutine return round sub particular often procedure exposition block block learner stay start start localize history move last learner suboptimal incorporate localize parameter guarantee relaxation x else round x admissible relaxation regret localize meta block block pp take advantage optimality situation literature conceptual online setup strong adversary meta either say next separate minimizer round localization update round next gradient recover follow style gradient specify admissible suppose play strongly specify turn study well unnormalized remain winner end localization q erm minimizer say minimizer non minimizer behind strategy next expert round gap suffer relaxation algorithm exist eq loss localization consider expert become winner round winner throughout cumulative throughout game suffer heavy dominate depend choice investigate full minimizer future history erm erm consider online lipschitz loss localization localization strong norm small eigenvalue exp loss dependent localization gradient pay adversary play adapting sequence play happen play sub adversary rate already adaptive localization remainder try introduction notion relaxation burden development like stress thin air rademacher obtain computationally relaxation something one would follow omit explicit sake constructive instance response relate protocol proper learner select simultaneously suffer improper learner side information pick nature simultaneously improper version easy may equivalently protocol learner necessarily mostly focus improper distinction difference improper write eq side achieve equal result specialized section relaxation extend term consider prediction complexity define contraction style future definition latter future term sequential rademacher attempt step rademacher condition conditional relaxation obtain relax finite proceed weight rise spirit attractive function function sign relaxation achieve yet lead admissible alternative method regret guarantee predict label correspondence proposition combinatorial analogue alternative remain trivial algorithm technique consider relaxation sequential rademacher involve coin computation couple expensive generally realize rather strategy draw sequence solve optimization estimate utility move random solid furthermore randomized strategy informally idea trace form mix term drawing verify make infimum expression inside outside example randomize tree variation regret walk related idea interestingly many turn rademacher complexity constant factor supremum draw erm rademacher complexity relate finite natural method perturb previously kind randomization rademacher new case style later trace norm completion expert one rademacher complexity complexity later verify consider x partially average randomized method relaxation forecaster see sequential rademacher equivalently bad classical rademacher ensure draw rademach argument regret rademacher particular setting expert specify game need appeal case distribution algorithm pick look closed randomized perturbed simplex claim especially attractive draw provide perturbation unit draw rademacher draw independently pick enjoy eq weight perturb uniform surface follow show draw likely produce direction ball unit previous address ball sphere consider randomize round draw pick enjoy importantly depend follow perturb adversary simplification perturbation begin game method also outcome play adversary contrast version side present pick observe exist x assumption relaxation assumption admissible loss case assumption rademacher complexity instead box access return bad depth round expect regret expectation whenever strategy g convex scenario static make outcome suffer lipschitz loss sequence turn expert sequential classical rademacher rademacher reason sequential rademacher general conditional classical rademacher average admissible absolute forecaster static expert plug value thus variant forecaster relaxation prediction supremum potentially preferable idea start note convexity solve online alternative game pick choice pick adversary indeed convex first game eq linearize relaxation admissible specify regret algorithm compute linearize note need evaluate expectation enough draw estimate close randomized draw one sequence show linearize already randomize idea expectation relaxation strategy specify expect employ alternative novel completion predict unknown collaborative pick shall predict relevant round entry know advance value algorithm regret bound depend order entry mild condition function regret entry game two alternative problem subsection alternative variant small trace regret guarantee moreover variant also computationally present equation hand spectral round matrix ask notice round efficiently subsection round second second loss equation trace position elsewhere ask fill inner respect entry first value problem yield involve trace constrain per randomized specify use situation constrain treat adversary illustration constrain within mention step selector define algorithmic step supremum satisfied expression feasible rademacher complexity relaxation upper complexity relaxation mirror step size problem mirror universal log factor exist whose conjugate q regularizer descent sequential relaxation arise relaxation adversary let relaxation whenever relaxation form mirror bound remarkable universal mirror algorithm algebra th admissible opponent repeat obtain remark leave side process recursively immediate consequence minimax q infimum expression sign supremum arrive equality effectively add swap follow admissible exponential infimum weight improve function rademacher relaxation show weight relaxation arise rademacher future tree last represent worst remove argument see general mirror admissible relaxation optimal one attain infimum optimal relaxation yy f g tf increase term hence concave rise eq point clear force fact value obtain value familiar update plug algorithm admissible derivation smoothness remove eq establish relaxation recurrence gd block block get gd interesting two block relaxation turn start initialize block length trick round round strategy pick apply also note case summation block continue become small block trick entire round regret block regret initial block sub gap minimizer round already play round block suffer playing rest thus suffer let think choice two whose sum introduce arrive bound justify gd zero bad depth let consider optimization eq write side rademacher substituting complete part finally relaxation sequential sign sign class take possible denote sequence projection relaxation q I factorize derivation randomized strategy propose match randomness arise eq theorem compact last draw draw multiply instance convexity minimax swap infimum supremum need non randomization extra throughout observe remain q coordinate j remain consider eq hand symmetric sign coordinate expectation argue ball ball rf hand ball h conclude round play randomize define randomized view view satisfied lemma show randomize specify admissible conclude statement randomize admissible additive hence sphere equivalently euclidean observe use omit symmetric prove geometric argue uniform decompose sphere align htbp direction vector length plane span angle deal dimensional span align coordinate read prove side mind four square cross square root completes yield assumption dropped expect specify choose start however calculation identical thus conclude need strategy propose rademacher tx convex first minimax eq minimax term prove analogously pick definition notation step minimax bound term expression write pass supremum upper shall start show admissible game pick adversary directly pick achieve equal last theorem pass case choice achieve relaxation convexity convex interval admissible bound w randomize jensen convexity pass variable pass strategy plug back q inside convex interval hence conclude yield increase I yield supremum two equally divide yield term distribution irrespective strategy round draw hence almost proof mirror let pick expression instead square derivative force conclude tree rademacher smooth case update specify follow proposition yield small pass must loss potential minimizer block admissible eq acknowledge nsf grant dms minimax bound constructive know derive one perturb forecaster emphasize rademacher complexity algorithm allow fast learn localize include randomize perturb method completion trace problem learn player observe distributional assumption past decade refer comprehensive sequential learning average number measure classical supremum rademacher concerned supremum martingale dyadic though complexity tool study value algorithm constructive literature study naturally certain sequential constructing corresponding method improve view constructive upper value surprisingly method perturb recent method show whenever sequential rademacher randomize certain development norm perturb prediction expert throughout paper appear air sequential rademacher understand inherent one aspect development average key also perform hand relaxation localize arrive complexity complexity lead fast equally localize present adaptive take move suboptimal study localize develop
categorization optical character recognition spam bioinformatics prediction carry standard scenario popular sequential baseline label propagation intend way global vs perceptron sequentially node test diameter imply affect prediction common underlie similarly intuitive fast algorithm majority predict time requirement need read approach sense node affect adjacent introduce method graph scalability comparative way span edge run give generation span span create via visit visit probability vertex adjacent visit yet span fast generate approximate spanning minimize sum tree short twice diameter short span root node random diameter check carry edge neighbor rule edge span near neighbor combine unweighted span nn take weight unitary span actual run span also classification span run one vote experiment choose world preprocesse normalization throughput protein dataset combination edge inter create exploit weight spam train split training remain unlabele create follow experimental euclidean j compute generate frequent category connect belong class functional depth classification database order associate task binary multiclass binary task combine try proportion size particular set macro average binary result contain table dataset resource run moreover inferior span dataset decide go refined tree table report seven span six split per tree well span purpose level achieve operate span see contain report span show algorithm span tree c rate take across use dataset span result relative among algorithm systematically run storage quadratic see near dense large situation real perform span tend original graph approximation actually practice fast way generate span inferior take take implementation actually rely span simple improvement reach introduce analyze algorithm term ignore linearization phase algorithm provably evaluation previously online predictor moreover combine span batch label scale analysis reveal loose tree slack second express low object well reflect tend graph generality prefer bias sense robust perturbation introduce ask robust number mistake weight question nice active classification support google ec publication reflect view reference main prove cluster sub start terminal cluster terminal empty close ii node adjacent node near mistake make iterate argument subsequent mistake mistake q define summing expression step jensen k obtain split predict split sub connect almost cost extra mistake edge suppose split node mistake make respectively pdf semi drop light increase mistake removal drop make coincide split sub cluster number mistake bound one mistake due let sometimes cluster overlap node fashion mistake argument mistake make conclude edge let linearization mapping edge pair start define visit sake visit terminate backtrack node path visit contain distinct construction pair create single traversal mapping remove edge eliminate transform l prove definition transform spurious create spurious edge create spurious free create spurious edge sum spurious elimination edge linearization let part immediately remain description pair gets replace edge establishe label free run line number mistake exhibit mapping edge union pre establish partition partition turn r last recall spurious edge correspond exploiting finally conclude proof lemma free hence associate giving turn lemma linearization span tree mistake equality lemma jensen concave corollary grow polynomially sake contradiction assume r n free edge give free contradict theorem proof part case adjacent flip cause terminal total sum weight weight twice therefore hence mistake node inverse polynomially mistake proof span label span span polynomially bind via summarize combination test macro set tree span perform thorough report four c split error split error class h c predictor average train split c measure l c c error rate train c f f theorem lemma universit di universit di universit di sequentially predict label arbitrary logarithmic random induce adversarial characterization achieve mistake polynomially random span tree time world compare perceptron propagation approach represent node item weight quantify similarity datum code input datum web spam categorization edge model relevant classifying order learner must predict label observe world application involve I role scale online edge advance start look kind graph unweighted online prediction mistake induce adversarial I mistake obviously lead span whose edge minus span mistake predict previous span maximally span tree predictive recall online suggests span prevent span edge adversarial assignment yield mistake unweighted term span tree unweighted prediction uncertain paper bound fill gap lead bound hardness weight weighted graph online must span exhibit mistake weight total weight spanning predict tree extremely like stress central issue involve context slow consider impractical operate near neighbor besides run linearization bring benefit turn perturbation label clearly practical empirical evidence compare literature graph prediction perceptron kernel version propagation baseline perceptron propagation method carry sized world dataset test span tree span tree aggregate majority vote comparison predictor baseline recall basic section survey literature relate mistake span tree mistake restrict match connect quantify provide constant devote let undirected labeling ng protocol predict possibly randomize adversary parameterized choose labeling node adaptively choose associated mistake occur learner total note depend randomization labeling learner use randomization adversarial choice requirement fact randomize increase mistake regularity regularity follow label free quantity I e weight define fix denote maintain tree q expectation choice rescaling probability thereby irrespective measure weight large dense unweighted large give big happen I disjoint path span see example ball graph edge contribution must contribution large bridge one independent path edge imply clique survey literature closely relate end learner perceptron embed canonical basis perceptron gd expect bind dense order investigate span mistake bind perceptron suggest algorithm span diameter reason behind span hand guarantee concentrate remain introduce show unweighted reduce span linearize first visit give line graph extent diameter inductive nn small ball diameter take cover recover unlike perceptron hold trick unweighted perceptron tree help diameter number perceptron like nn mistake speak way consider graph clique unweighte much scale cover clique hand large tend stay constant get close diameter become include tree effective strength flow square edge diameter clique sometimes yet fair algorithm logarithmic propose refined exploit paper application weight diameter diameter version speak select dual obtain logarithmic diameter unweighted select appropriately comparison present one paper scenario problem graph label energy minimization instance subject span draw long history g unweighted span tree sample graph reduce technique take due hardness reach walk connect light edge experiment test approximation building span tree finally generate conclude span tree reduce g result currently show node label deterministic mistake make pdf adversarial strategy graph consideration node node set contain small adversary node unique label previously use weight edge belong span edge node label matter mistake yield illustrative less label labeling connect mistake mistake unweighte set bound drop total mistake q mistake line free contain terminal terminal proof cluster split terminal node bind mistake near mistake must occur iterate get mistake k k jensen follow omit drop mistake mistake refine mistake solely let edge include eventually terminal terminal summing mistake bound mistake I w last mistake occur obtain number mistake minus contribution include bound mistake predict generate span span way mistake span tends reduce light theorem span mistake scale affect rescale w mistake bind every graph node substantially connect would uniform rescale follow polynomially connect fraction overall pick polynomially connect ratio total weight span corollary satisfied example nn contradict assumption connectivity need fact drop mistake extra mistake property bind elaborate compare algorithm perceptron section span tree distance depend weight reciprocal covering case unweighted whereas norm mistake approach namely diameter bind unweighted generally cover ball seem allow parameter diameter typical unweighted clique algorithm mistake corollary yet fair deterministic compute laplacian cubic quadratic tree interpolation initialization whose quantify comment drop run trial memory space constant linearize line initially terminal traversal make calculate constant top complete binary construct leave simplicity assume node build th maintain I subsequence leave reveal marked figure reveal dark grey link depict mark traversal operation predict traversal stop soon mark traversal begin go right determine look close locate start mark traversal mark path order find close mark go child node keep via distance determine trial visit trial place gets mark visit occur subsequent visit guarantee subsequent visit stop operation preprocesse linearization calculation distance trial show label graph significantly small running polynomially extend span tree labeling perturbation label explain beginning operate graph linearization whereas principle difference always couple edge unweighted moreover node label linearization step generate consequence linearization graph sign item hence star graph linearization tree free label regular labeling see quantifie mistake
nearest express interpret rule instead still neighborhood svm optimization finding recommendation author national national science foundation thank yahoo yahoo near amongst compare efficacy three none metric lead particularly improvement rbf introduce combine mahalanobis rbf capability nine benchmark difficulty outperform accuracy establish serious metric selection apply sake binary scenario restrict reason svms popular perfect furthermore maximum margin reliably generalization perhaps importantly svms highly boundary overhead trick map dimensional infeasible explicitly svms completely inner computed efficiently infeasible ij k svm entirely sake brevity interested reader description offset separate hyperplane hard margin ensure label hyperplane minimizing error solve equivalent separate hyperplane slack optimization simplify section svms well positive basis rbf dissimilarity must scale
dramatically variant demonstrate significance estimate vote actual opinion crowd take majority member collect vote crowd overall gold refer gold entire crowd initialization figure initialization gold gold accuracy gold improve initialize perfect expense budget gold performance substantially agree crowd gold standard reward vote yield crowd receive preferred crowd vote gold illustrate believe handle note achieve binomial early rather parameter trade would achieve illustrate point curve course increase help exploitation parameter example shrinkage uncertainty ask vote correctly highly seen correctly label make stable consider label mistake iteration mistake constitute vote choose recover cause mistake early one purpose reduce almost estimate approximately early stage majority vote prevent chance uncertainty vote explore quality gradually vote exploitation quality majority vote quality exploratory scheme high accuracy collect vote quality overall remove collect vote well approximate crowd crowd member vote discuss specifically exploration estimate exploitation crowd majority vote modular outline pool majority vote overall modular outline determine overall label entire baseline exploration exploitation idea behind crowd challenge approximate vote aim complicated distribution accuracy truth consider independence sub crowd crowd majority calculation crowd majority vote majority seem essence yet pool try like thank helpful project mit intelligence approximate crowd majority opinion crowd crowdsource present online dynamically exploration vote crowd opinion crowdsource useful vote quickly vote every member crowd may nature crowdsource vote crowd budget sampling comprise capability sample align opinion effectively crowd determine crowd crowd arrive require explore vote want vote likely decision vote align crowd limitation pay lot vote estimate align crowd majority decision pay vote suffer decision clearly exploitation tradeoff exploit mainly agreement crowd main modular template crowd include exploitation choice arise choice template dynamically determine vote make request uncertain iteratively quality estimate keep exploration exploitation iteration rate datum sufficiently experimental demonstrate identify member crowd approximate discussion representative crowd whose vote crowd vote hence vote informative approximate crowd vote voting wherein vote differential weight vote contrast vote wherein vote multiply scheme place emphasis vote high rest organize within modular approximate crowd baseline baseline specific choice conclude remark crowdsource increasingly lead resource amazon collect multiple generating label collection crowdsource clearly effective little disagreement approach develop reliability example crowdsource label treat demonstrate label preferable single label cost negligible pruning label less improve vote collect discard lower viewpoint effect researcher explore trust majority vote characteristic true sometimes difficulty effort prediction item quality however seek correct opposed turn know task precede effort collection simultaneously stream estimate exception reasoning label data accuracy classifier vote active labeling example pay price per vote reflect crowd majority vote sure repeatedly vote example look check whether potentially majority vote vote vote could bring criterion add candidate get follow quality add assign majority l c l l l conduct separate model crowd separate recommendation movie vote subset rate rate subset original rating vote boundary chemical document track define competition develop retrieve overlap list citation dataset article appear select document category divide test format half develop learn adaboost na I bayes svm decision several decision specific total combine half total baseline crowd uci repository collect census add noise add diversity strongly diversity issue aim varied prediction prevent due baseline accuracy half combine quality assessment vote build interval estimation ie refer item reward metric critical student sample vector majority vote select size subset exhibit criterion ask vote cause strict hand control use cost adjust spend compare demonstrate ability accurately vote modular view impact module
turn brownian distance distinguish marginal euclidean namely distinct independence measure informally speak independence easy problem homogeneity testing hypothesis borel pz w recover characterize statistic trace trace hold case diameter respect square integral operator xy operator apply induce distance characteristic briefly interpretation however interest completeness characteristic characteristic norm weight aspect kernel let translation kernel borel weight function integrable translation translation embedding one attention embedding eq discrepancy type ensure distance need proposition coincide whenever generate n z z inequality convex clear induction reverse triangle jensen z z imply z k k nn give natural interpretation w embed generate half addition finite also must mmd impose condition underlie type mmd energy conversely kernel small c gauss ii type type I type type extend analogy moment xy xy also kernel em x p yy xy schmidt yx x translation assess various multivariate microarray tumor event small type report trial high vary thm thm thm remark independence hand energy covariance literature embedding kernel establish energy negative kernel embedding rkhs family independence one member family statistical machine community space sample accomplish consistent random finite dimension moment community introduction discussion advantage certain expectation distance lead notion metric negative machine reproduce kernel test embedding associate schmidt criterion rkh far graph despite similarity base kernel link context statistic straightforward confirm integrable question rkh dependence formal extension brownian distance family induce kernel consequence large get sample bind third relation characteristic hold quite unlike characteristic kernel via structure definition rkhs theory relation rkhs distance relate quantity mmd schmidt respectively give quantity distinguish empirical test describe investigate variety strength family introduce concept reproduce kernel rkhs review relation space reproduce rkh associated rkh reproduce map say need triangle empty say iii say terminology type hilbert euclidean space use hilbert relation theory hilbert space exploit negative symmetric definite summarize nonempty set let definite call brevity hereafter simply abuse terminology work distance scale equivalently obtains express canonical rkh z z z imply every proposition clear positive kernel combine valid negative zero induce zero e z energy covariance measure demonstrate discrepancy measure criterion sign energy distance use characterize scalar coincide covariance imply negativity every mmd introduce measure dependence characteristic choice weight expectation pairwise euclidean recently establish replace type domain embedding kf embed obviously embedding measure discrepancy mmd characteristic characteristic characteristic characteristic interpretation thus immediate application distinguishing characterize satisfy additional property extend r finite p kernel embed well space say first q finite directly type measure finite moment check whether characteristic state although rkh hypothesis sample biased induce pairwise result k center statistic sample replace distance require nan chi yield bootstrap spectrum operator study context compute gram aggregated entry concatenation sample center nan estimate converge spectrum covariance respectively chi square analogous design attempt estimate distribution interpretation test independent variable conservative compute associate kernel approach section experiment kind synthetic multivariate second compare gaussians differ univariate second univariate frequency frequency plot design indistinguishable outperform bind conservative checking quadratic significantly test differ similarly differ kernel advantage similar cc cc mmd distance mmd various sample addition investigate value case perturbation kernel value yield high frequency result kernel advantageous distribution sensitive distribution fine differ high frequency appear real rotation rotation right use propose ica dependence fill pass random dependent uncorrelated dimension plot vary obtain wide range performance dataset case small rotation
forecast rule expectation scoring encourage careful probabilistic scoring definite nonnegative scoring proper yy score example class rank probability score score calculation cover apply score calculation dirac future observation incur bayes pattern theory break whenever metric continuous definite family member interesting open question equal cone generate asymmetric function near neighbor acknowledgement author grateful von foundation thank reference j harmonic protein bioinformatic I neighbor rule comment probabilistic pattern new york netflix challenge statistical learning introduction journal statistical statistic forecast journal american strictly rule prediction american association ari advance american association learn statistics j graphical journal definite space definite transaction american physics remark cover universit observation well perhaps surprisingly large refer cone metric negative slight explain performance measure success reasoning analogy loss metric definite prediction predict observation real value structured forecast population sample predictive performance mean loss forecast future observation information approximate cover misclassification risk twice elegant thought cover paper seek remarkable section satisfy consider prediction forecast predictive performance evaluate proper analogue dirac discussion relate neighbor let hausdorff space borel algebra measurable argument consist yy function cover contain predict past bayes risk belong class slight extension assume bayes measurable cover inequality desire argument play role harmonic arise structure increment ari example reference negative hold finite dd space mm qp dy qp mm special case summarize kernel apply reach generalization standard
sim w nn false w pick col nf w nf col col col nf col nf col w x mr r matrix exceed sim sim nu nu alpha n sim cn nu sim home sim home sim c nu sim home nu sim mm doubly matrix deviation stochastic projection doubly sum doubly scale non contain component say scalable number doubly zero scalable permutation eq thus approximate reduce corresponding subject generic computable since carlo sort computable formula doubly stochastic satisfied identically check scale connect independent lie additive row col maximum equal variability doubly gamma hold scale doubly sense close maximum apparent appear matrix accomplish linear proportional prefer provide moderate matrix doubly attain equal permutation extreme doubly stochastic easier doubly doubly stochastic permutation weight average helpful norm extreme doubly one helpful hilbert deviation large scale convex fix moderate deviation purpose deviation consequence though affect improvement approximation apply matrix obtain doubly value exact nevertheless decrease agreement unclear sequence moderate decrease block block associate doubly stochastic table error evidence suggest error moderate deviation n row doubly deviation value weakly strictly doubly associate assume moderate degree explicitly summation calculation nominal also least sense sum circumstance similar conclusion sum example restriction expansion odd term hold scalar bound degree restrict restricted expression r r r r r constraint value row index x express reduce form restrict sum arise speak sum sum row column index hold constant elsewhere one block element relate lattice write value may put together scalar restriction block reciprocal factorial partition q neither partition least upper less either three term symbol block pair series convergent spectral assumption expansion group degree six less express far n compute adequate gamma freedom e odd scalar chinese projection eigenvalue size matrix moderate occur moderate threshold comprise odd exist arbitrarily indicator corner approximation compute panel magnitude standardized residual log middle panel residual plot plot panel plot tend occurrence leave panel residual odd type constant moderate mean square approximately gaussian happen inequality rate structure matrix block structure slowly decrease course must depend generated positive deviation expect tend large doubly pr w adjustment sample table approximate polynomial journal approximate n matrix theory new york generalize relationship doubly ann alpha alpha return cyclic product else alpha alpha need paper else alpha alpha else alpha weight diagonal iterative fitting make col sum alpha alpha du n alpha sum alpha cycle u true log strictly positive rarely occur return alpha cat fail alpha alpha alpha alpha rarely alpha exp exp x x pi x mu mu else x nu nu
score shift unit orthogonal selecting index h aic rank usually unimodal dependence integral copula exponential estimator goodness fit test role understanding comprehensive speed issue ask give h scatter em section thm thm example modeling dependence united science many department statistic college set big size consider first value orthonormal mx x comparison enable expectation lp representation tail normal height keyword phrase comparison density lp co moment mid function orthonormal nonlinear correlation parametric modeling modeling quantile theoretic fisher algorithmic discuss researcher many include research quantile theoretic modeling seek comparison measure call tail decade extend discuss theorem describe routine specific pose involve method box intensive come applicable traditional plot exploratory polynomial score function discrete separately mid dependence continuous density divide fy discrete concept contingency ccc discrete give density denote connection major apply density innovation mid observe copula comparison comparison special iii score estimate regression quantile true construct relation x construct shift polynomial cosine regard mid observable construct schmidt power constant shape polynomial take copula influential product score function selection increase increase chi freedom drive chi discrete contingency dimensional moment traditional sample sum copula function integrable influential score maximize dependence sum influential exact combination compute vi display interpret fact lp integer satisfy x high tail polynomial lp score extension extensively develop skewness concept regression expectation apply naive regression quantile scatter notation compare display lp rx rx united aim bayes theorem fx fx interpreted formula probability unconditional marginal formula heart discrete f copula density continuous x express theorem odd du x x x express theorem equivalently linear logistic density exponential smoothing estimator
case approach show optimally incorporate even consistently robust nevertheless contrary exploit improvement proximity obtain gain match hmm access side side wrong naturally recover hmm partial turn state achievable moreover training expect support sequence hide access partial hide state derive accordingly considerably improve performance baseline training application speech processing bioinformatics language process stochastic generate unobserved state certain regard hmm particularly concentrate hmm alphabet sequence form chain observation hmm completely emission maximize expectation local equation ordinary observe sequence instant partial state sequence ordinary hmm along span observation ordinary generalize might ever say circumstance mathematical derivation em incorporate derivation state incorrect provide define simulation confidence match quality side ab fall category partially supervise label model suitable unlabele contain noisy may occur label date study relative frequency initial model feed ordinary hmm rigorously incorporate ordinary framework likelihood estimator know theoretically analyze mle maximization special set state derivation provide explicitly hide sequence furthermore state brief hmm derive iterative equation incorporate partial noisy iv remark briefly hmm sake notational finite derivation readily extend come outcome discrete time hmm alphabet hide unobserved set generate py py pz ib hmm characterize hmm without equation give maximization procedure recursion observe give q observe transition ease drop sum observation model incorporate proposition explicitly backward h marginalization use definition update markov reach q side update forward update backward find reflect ideally name accord unknown bring immediate set confidence accurate guess present confidence low confidence limit iv transition transition state relate forward backward definition q x j j n otherwise proof wherein markov condition observation side summation addition information likelihood maximize via auxiliary em provide incorporate definition convergent least let carry maximization split log division probability obtain maximization involve side derivation derivation indicator numerator denominator probability outer summation sequence observe divide estimate new incorporate possibly corrupt information next new scenario simulation test along relatively high emphasize exact hence side confidence side I match sensitivity side viterbi use average efficacy method compare state ordinary
cauchy also stable detail iid eq need act simultaneously include subspace challenge although correct behave classical affect distance histogram quite tail behave median distance preserve cauchy near subspace query obstacle qualitatively increase much heavy ii well behave tail projection increase distance distance good mis detect low sharp guarantee choose correctly significantly close need argument rigorously key sec need establish theorem argument proof routine technical lemma close unique long close p p lemma purpose appendix numerical fix cauchy bad subspace task projection come something strong denote hold bad subspace use unit sphere argument p associate close follow tail fix vector exponent unfortunately give power detail defer sharp w bounding cope heavy tail insufficient elegant argument rough basis analogue orthonormal preserve preserve cauchy appropriate value detailed calculation tie instead ask become fix nearest one rest near rest relative distance control become subspace compare target state theoretical demonstrate relevance success failure sample new project practice maintain subspace pool economic yield perform subspace search consider case employ augment alm numerical solver affect extend otherwise solver project handle solver subspace iid pool take randomly subspace induce reasonable simulate error divide magnitude add fraction growth corruption retrieve necessarily subspace pool report note distance small enjoy chance preserve near gap entail dimension htbp visual varied induce gap thing simple pick pool repetition refine fraction low half find corruption vary illumination well nine phenomenon physical cause low formulate recognition subspace norm face recognition recognition happen prefer save detailed discussion line nevertheless popular recognition extend face align face image acquisition randomly image moderately camera angle great angle face suppose closed world present evolution image stay ns achieve perfect ambient take nine experiment achieve imply search represent evolution success gap theorem need achieve already repetition significant extremely first htbp increase weak exponent become experimental result confirm specifically take high extreme achieve accuracy back refined description dimension recognition way htbp vary evident order recognition preserve recognition subset testing find corrupt original pixel resolution size location accordance gap take case effect gap gap gap drop rapidly corruption suggest accord significant corruption level level demonstrate norm ns variant htbp performance exhibit less beyond pay working dimension efficiency bad corruption corruption consistently applicability proposal recognition multi library library comprise toy pose take illumination direction take although nonconvex shape nine recognition corruption percentage corruption c corruption ns method tolerance variant report performance scheme recognition illumination choose datum perfectly corruption level beyond scheme remain gradually beyond burden expand rapidly associate predict lower large obvious run largely determine rt concrete task object instance straightforward exhaustive dimension cost search boost practically generate iid corruption fraction take htbp run bite matlab ram turn regression plot vs logarithm turn matlab scale see recognition run significant random experiment confirm projection figure dimension search empirically tie part provide essential fact rv stable rv constant strictly e distribution characteristic stable characteristic rv stable call stable characteristic distribution stable also stable exist virtue stable iid rv symmetric sequence complete distribution standard cauchy hc remarkable aspect standard cauchy invert characteristic scale control fact subsequent addition follow half cauchy rv fact iid sake page rv weakly converge f z g ax q half rv determine appear half cauchy expand series center plot convention subscript subscript figure htbp iid cauchy corresponding cauchy interest behavior sequence ds ax stable eq strictly eq note assume application k derive claim kk k obtain constant proceed condition let column independent sr rd existence subspace existence basis existence condition lemma well write last well condition whenever finish side relax apply markov take loss rv arrive claim expectation complete net q least choose bind notational eq intersection j side triangle obtain intersection remain ensure satisfied choice ensure choose eq tc simplify bounded constant ensure solve sparse record stable sparse c query similarly matrix theorem tm ok subset detect near subspace span canonical basis e projection parameter tuple subspace gap identify support assume partition I I subspace enough guarantee constant identify corresponding obeys put construction hence hypothesis proposition kt ic require start rr q sr p nontrivial probability tie submatrix rank foundation partially university zhang object collection linear high ambient determine naive exhaustive large burden significantly query dimensional distance linear program independent get back space exhaustive preserve nontrivial order multiply logarithm subspace dimensionality hence vision application investigate empirically subspace cauchy projection recognition w zhang space dimensional intrinsic structure central computer vision impact denoise object view match associate image say illumination recognition nearest put image prescribe paradigm face development sufficient achieve appropriate model depend encounter variation illumination capture linear model whereas variation alignment highly metric set appropriate observation perturb gaussian induce norm datum error due alternative model input would resource ambient application quantity could dimensional subspace well justified illumination variation build induce p z p x x p turn empirically count robustness current analysis likely justify image moderate cast unique convexity robustness subspace query determine objective interior method thing careful discussion expense guarantee achievable even aforementioned fit aforementioned tuning prohibitive object useful problem recent modeling error correction norm provable corrupted imply even nonzero recover present theory recovery strong recognition subspace precisely distinction cardinality hard find small agnostic solve purpose approximate problem concern individual regression heavily sampling e leverage score
word replace exist real world document collapse magnitude provable practical recovery machine estimation robust probabilistic latent allocation topic multinomial vocabulary topic distribution lda normal document begin document position assignment topic document matrix unknown stochastically never expect task topic case dirichlet lda maximum estimation distribution np result typically optimize markov provably parameter provide word topic separability word anchor word occur partially could document let topic topic learn document q learn additive unfortunately solve inversion make tolerance word topic two step identify recover anchor co supplementary material high learning follow recovery anchor purely combinatorial find anchor word recovery column anchor notation refer document outline set anchor tr inversion simplex original poorly likelihood recover anchor whereas besides ignore co estimate inaccurate probabilistic describe notation document assignment topic role anchor step word matrix normalize row store normalization anchor ar ta index anchor index special hull row anchor word fact pz k w negative hull mix easy co occurrence algorithm recover objective reason measure understand particular maximize observe count problem constrain gradient write k prior recover recall constrain least pseudo learn hyperparameter bayes calculate pz pz pz specify recovered find grid perform range material also polynomially word anchor anchor infinitely document hull simplex anchor number perturbation hull define simplex randomly subspace origin notation denote compute complement give anchor word solve whether anchor span find hope exactly anchor choice succeed anchor guarantee recall robust convex hull rest reasonable define polytope lda let hull write cover follow hull vertex appear perturbation cover vertex provide help infer topic separability material find argument volume simplex determinant vertex algorithm prevent exponentially proof run alg vertice phase point vertex simplex anchor word question variety also assumption hyperspectral clean whose prove infinite able give provable guarantee perturb g sample non negative factorization separability application topic application document provable guarantee even programming slow comparable three gibb average iteration burn train set evaluate performance correct document document dimensionality sparsity generate corpus generate document guarantee separable large york times article vocabulary length corpora ny document document topic corpora reconstruction true word matrix well align topic learn intractable reliable topic interpretable semantic metric coherence coherence use avoid occur well quality perfectly co frequently redundancy inter topic probable overlap expect ambiguity recover heavily linear size second second corpus second corpora semi corpora synthetic corpus corpus synthetic recover algorithm document bar show poorly noiseless infinite corpora recovery low variance reduce mcmc corpora recover corpora match comparable ny times corpus poorly corpora much recover zero error even noiseless lack separability semi synthetic corpus test separability add anchor word unique topic assign probable cause great reach exactly problem even correlate corpora ny article symmetric add block result fig recover bad case dirichlet corpus especially robust consistently corpora uncorrelated zero separability produce quantitative qualitative probability hold document top topic panel ny times split fail small ny recover produce worse hold token algorithm hold pair within range document method recover unique consistent sized corpora supplementary material ny topic shown observe tend specific gibbs file read internet email site www algorithms yet provable
sense hadamard means exist solution unique stable article organize review ode solution examine introduce prove consequence discuss relation incidence three state figure go transition density people consideration incidence disease individual number time assume homogeneity duration path figure henceforth homogeneity assume close birth furthermore continuous age member population number normal respectively q linear structure incidence combination normal age allow application ode unknown ode interestingly ode follow depend type ode ode ode analytical analytical equation depend type ode last par examine dimensional ode e system however obvious proven solution fulfil subsequent application derivation age equivalent costly ode solve information age relatively study application effect cause incidence interpret problem oppose infer incidence e cause ode ill pose sense sufficiently pose differential equip na pa pa ill pose study relation incidence link dimensional ode article show meaningful ode change type implication analytical exist important ode derivation specific proof ill distortion incidence disease consequence unclear ode ode incidence medical trend appropriate formulate partial subsequent disease disease diabetes diabetes duration u shape duration unlikely expect term I
reconstruct reconstruct explicit easily convert exact entry entry uniquely many position sufficient uniquely entry relation observation rank pattern special allow minor specific combinatorial position completion property completion algorithm separate relevant estimate miss entry allow estimation estimator completion whether term encoding position necessary theorem sharp apply determine minor minor experiment movie predict underlying matrix completion key idea set rank additionally fundamental algebraic observation question unobserve specific structural show numerical fundamental aspect algebraic set characterize exactly polynomial relation polynomial connect major ingredient independence independent row finite characterize perspective enable completion broadly speak two spectral j n rank generality follow variety r n algebraic vanishing vanish vanish view position entry vertice mask position bipartite tb every unobserved informally go interaction structure structure jacobian smooth jacobian j ji mn denote kronecker row entry order satisfy let n part nan parametrize see nan jacobian corresponding position row submatrix correspond zero entry purely combinatorial structure make boolean irreducible variety hausdorff kind call generic generic statement concerned proposition tell mean polynomial imply open topology surely respect show generic jacobian generic first compose critical attain maximum critical point prove uniquely reach first generic intersection devote mild answer position relate row jacobian show separate observe either real start precisely define mean imply observe position finitely fix infinity whole finitely whether position care issue nm kk finitely rank position position entry position finitely jacobian generic implication subset combinatorial independence generic language say finitely closure equal closure rank perspective prove combinatorial condition relate finite describe closure testing entry correctness continuous closure closure compute jacobian decomposition great residual return routine svd ram high rank detect minor uniformly elimination factor project neighborhood fm rank smoothness position finitely generic finally statement directly closure preserve generic finite every argument smoothness essential identifiability statement phenomenon explore imply similar jacobian show use parameterization another section show whether mild analogue well exactly one practical relevance mean analogue theorem finite entry uniquely true matrix analogue constant theorem proper overcome geometry defer statement state position position call uniquely nr maximal index uniquely finite check position analogue jacobian characterize general gr computationally impractical jacobian finite jacobian entry characterize space intuitively concept mathematically jacobian rank kernel stress call denote remove jacobian allow unique remove matrix maximal stress depend entry rational proof generic define generic stress uniquely stress theorem stress unique generic observe stress jj remain correspond position real number instead field rational substituting thus compute evaluate left correctness one imply similar consideration analogue remark algebraic irreducible cardinality depend whether section nm correct size stress hold keep dimension keep development finite define remove alternative probably statement pre image let stress inverse neighborhood e df leave definition stress prove determine row observe uniquely row recover column span connect reconstruct miss entry cover one call associate address reconstruction version wise actually answer general arbitrary one concept set circuit connection reconstruct circuit unique let circuit mask correspond scalar circuit rank fulfil circuit polynomial determinant generalization algebra matrix vanish call circuit scalar make circuit interpret let circuit circuit entry entry circuit observation respect property completion circuit position unobserve entry circuit polynomial entry general exponential circuit circuit correspond circuit minor closure mask neighbor denote intuitively iterate edge terminate edge add bipartite closure ki ni neighbor k jk e terminate say terminate e reconstruct minor uniquely crucial return subgraph efficient many vertex employ would one entry general strategy position observe fix construct position cause grow span tell position terminology dependent already spread sparse section basis maximally basis edge consist basis unless circuit I independent subgraph linear closure closure notion subgraph v subgraph position finitely basis part corollary rank set row column vertice q statement statement first sparse generality empty time edge induce treat symmetry situation need n maximize product n er hand rank sufficient independence bipartite mask sparse amount basis together along preserve infinitely circuit circuit jacobian vertex circuit imply core show rank section proof generic circuit stress circuit power see circuit circuit stress support column index zero stress hold concept core useful position circuit concept let denote subgraph minimum degree core circuit circuit follow need subgraph lie inside note thing closure circuit circuit rank circuit rank c circuit tangent stress uniquely identically remove would none coefficient addition polynomial vanish open stress mask equivalent bipartite random value interested bipartite graph enyi bipartite bipartite row random bipartite vertex vertex regular regular mask p n cn isolated vertex connectivity suppose depend p subgraph core graph span smoothly low completion rank generic incoherent perturbation show generic sampling powerful experimentally rest heuristic generic call conjecture threshold circuit circuit subgraph core edge value smoothness imply threshold thing together circuit inside core threshold reach give structural circuit core circuit yield threshold nonetheless moreover threshold establish rank span nonetheless completion explore experimentally also conjecture regular provide incoherent case regular mask dense enough start seem quite practical favorable entry add edge closure conjecture investigate finitely position phase transition case matrix quantitative investigate influence entry particular completed complete slowly entry occur entry various condition consider uniform sampling edge order edge matlab sequentially preserve property estimate success plot successful solve rx degree b chance success minor pass around closure nearly show minimum b exceeds require minor start minimization tb phase mask mask almost regular mask edge independently list list become th order way take check mask average make phase transition regularity experiment mask varied phase different plot regular plot bottom row regular transition sharp grow rather slowly likely column tb devote publish entry follow convention movie row grow core standard entry complete core size core grow vast majority entry majority column attain rank row core exponentially speed rank rapidly start empty core finitely identify check whether minor closure core theorem check process figure number determine way thing point correspond transition core start exponentially simultaneously experiment result reconstruction prediction one matrix choose entry independent figure three implementation competitive outperform low compare change actual predict priori actual error independently algorithm entry quantitative bin qualitative figure come measure calculate predict mask mask multiplicative entry priori variance mask affect known entry less come successfully entry structure mask pattern mask structure structure adjacent square mask depict red blue entry half implementation blue increase increment completed predict square code respective color error variance x axis complete entry logarithmic error
form instance observe situation ii x case substitute formulae situation forward x x formulae previously multiplication hmms maximize issue formulae multiplication operator common forward backward geometrically machine I architecture suggest solution keep track parameter store expression recommend initial become example u arithmetic em repeat iteratively formulae consider segment emission emission normal backward obtain heterogeneous ib ix segment emission segment transition identify transition hmm regardless occur observation equality follow base allow specific marginal regular heterogeneous chain change point cp contiguous homogeneous formulae occur plot estimate posterior mining set dot horizontal line state dash line dash r dot horizontal segment posterior probability solid line segment segment mining hmm emission segment greedy change state poisson initial transition change occur display probability change I segment contiguous state segment hmms almost historical perspective look literature year occur end though meanwhile coincide define end world posterior dot lrr horizontal dot line cp estimate breast cancer dot lrr line posterior marginal segment segment dot line segment point r line bt log reference genomic segment genomic hmm simultaneous point multiple simultaneously identify outcome implementation deal resolution current snp wide amount bioinformatic typically find contiguous extension fast rule smoothing technique model comparison find two modelling address computational issue allow hmms bayesian model variable dependency direct acyclic dag code edge dag factorization evidence obtain fashion bp sum product product message belief secondary shape structure call quantity distribution term typical criterion schwarz integrate complete assess exact bp forward hmms kind evidence bn framework permit extension realistic accounting variable straightforward aforementioned segment model segment segment specification two segment common segmentation advantage segmentation toy distribution probability state lambda transition pi pi reference change location pi lambda backward recursion probability evidence check ok b sum pi lambda posterior col blue col col j col col need large segment c parameter segment ref lambda lambda lambda lambda k lambda evidence consistency ok change cp matrix cp lambda lambda c segment col col col k change col blue col cell bt breast cancer lrr display probabilitie distribution change probability mean example transition segment r figure approach point begin end contiguous ir draw minimum proposition lemma heterogeneous finance biology characterize number change follow real value ideal non interval change emission observe segment index position segment two refer refer simplicity clear commonly use hmms bioinformatic widely financial series music brain image security hmm observation switch likely occur hmm inferential state quantity classical forward backward hmm change accomplished maximization method reversible jump mcmc recursive sampling characterize hmm mcmc summary inferential involve find chapter estimating include frequentist modify forward backward state hmm investigate penalization information adjust previously location present framework hmm change summary procedure hmm discussion property current efficient may change hmm permit use hmm variable simple dependency variable height text depth var node var var var edge thick edge thick thick thick thick thick hmms prior knowledge aspect hide markov introduce notion set outcome evidence evidence know unconstrained case contain hmm allow base level underlie appropriate share segment hmm transition observation evidence usually state homogeneous often segment overlap state segment transition increment segment segment particular effect arbitrary computation forward backward problem technical detail simple hmm find apply evidence evidence accomplish sum joint variable perform highly small cache avoid generating time tool factorization order factor factor depend place eliminate accord eliminate cache eliminate recursive backward message thus naive elimination consider elimination
sbm class let straightforward q space regularization free free maximizer within within diagonal close eq mle exactly mle substitute profile high set cluster estimate class incorrect assignment use divergence dimensional diagonal asymptotically decay additionally denote block tight high proportion population assume block define theorem require two expect high slowly specifically every equation identifiability class merged concerned ensure assumption make implicit block affect large mle block memberships blockmodel scenario particularly heterogeneous computing mle computationally intractable pseudo likelihood fit replace element without modification fit adjust remove block estimate meanwhile hundred remove mle return reduce nearly mle reduction optima initialization make seed follow motivate connect call neighborhood combine node pseudo spectral cluster laplacian eigenvector diagonal whose element grow everything else simulation sensitivity algorithm heterogeneous diagonal htbp keep population grow ten right panel block separate asymptotic expected connect eight noisy ten axis examine deviation model diagonal sbm maximize probability asymptotic identifiability node correspondingly demonstrate mle represent involve propose estimator incorporate insight gain straight giving definition outline expectation maximizer maximizer first step difference lemma building block establish union regularize likelihood bias tradeoff bias necessary concept refinement idea refinement connect choice similar make ab maximum aa argument take aa aa j concentration inequality asymptotic degree true om nc equality r pm leave maximize tractable concept refinement connect refinement regularize refinement log regularize refinement regularize refinement define partition index low triangular define notice assignment induce correspond straightforward refinement partition triple partition assignment large together process terminate pair choose routine connect refinement contain satisfie far essential refinement partition later previous subset result refinement analysis refinement partition remove define set block set group refinement refinement refinement refinement partition correspond refinement increase set taylor grant research dms pt mail edu mail edu fan mail edu pc pc pt theorem theorem sbm thank associate comment grant grant minus abstract blockmodel network maximum neither grow grow allow fast additional motivate empirical physical develop maximum likelihood estimator paper introduce regularization parametric word phrase consistency pt recent advance technology produce complex community highly actor identifying help question field depend interest interact element people computer cell communication network page provide community discussion topic network might contain activity relationship essence observe ability correctly understand lead blockmodel community actor belong connection node add rigorous likelihood blockmodel clustering blockmodel line author plant stochastic blockmodel spectral recover plant consistency plant paper blockmodel spectral high first statistical criterion though finding various brain find size brain human roughly brain suggest human maintain community roughly people refer similar human blockmodel asymptotically previous even result grow enough paper size slowly high ignore fast eventually contain roughly subgraph remain estimate stochastic blockmodel setting unified parameter set restrict space amount potentially technique lasso restrict space restriction diagnosis graph statistical propose restrict
ds overview come processing remarkably result history go name variable model develop fairly however representation issue namely mention argue solution algorithm near discussion bridge particular develop weighting lead somewhat aforementioned reason second wish argue general necessarily hold possibility datum design isometry coherence circumstance efficacy appropriate sparse update modify orthogonal pursuit omp stagewise f lasso selector view homotopy method take path direction forward stagewise fs prototype analysis note lar omp bp solution uncertainty homogeneity design pursuit argue uniform preferable growth argument selector lar ds validate repeat variance uncertainty lead prediction reduce lar formulate challenge pose importantly derive example illustrate sparse sometimes less motivate response imply interpret uncertainty arise source express variance absence admit application discuss repeat remainder variance challenge true solve ol expect bias towards depend signal noise new exact know explicitly error bring uncertainty residual access structure variance v system light great expect error hence become prominent track high briefly greedy sparse forward fs particular particularly understand main algebraic interpretation implication selector fs small initialize identify correlated update stop otherwise qualitatively find coordinate high minimal like implicitly solves often express attain increasingly main accumulate design equation unless scale uncertainty scale uniform contribution slight abuse modify noting solve specifically norm different comparable term say identical scaling associate uncertainty obvious correlation fs recall transform penalize scale lasso lasso penalty represent uncertainty recall penalization uncertainty uniformity uncertainty notation path geometrically uncertainty homogeneity uncertainty lead constant growth growth course random deterministic random carry expectation rather specific path scope brief moment connection selector tuning selector distinguish pursuit algorithm explicitly linear respect residual seem would property variable notice feasible result residual correlation expect explicit quantity fit panel identically lar scale line however associate less panel f right residual lar lar apply characterization datum without highlight challenge datum efficacy scale motivated candidate natural chemical different predictor molecular measure twice detector ratio spectrum peak measure allow pre processing peak may ratio relative control fraction infer nm visible light linear calibrate measurement sample biological run fraction proxy chemical composition mass sparse brief approach sparse correlation peak break redundancy cell hundred peak incorporate peak ensure noisy peak little consideration sample signature figure mass spectrum show height cross every peak among especially estimate variance normalization indicate proportional marker radius clearly peak variance attribute model nest fold outer validate choice lar lar chapter predictor another fitting lar peak validate matrix estimate detail improve validate lar ds lar figure consistency ds axis scale accurate associated plot figure noise scale number indicate lar solid line region uncertainty scale uncertainty slowly scale input almost provide graphical scale red peak indicate circle number sample increase accurate scaling remarkably lar ds function remarkably lar somewhat surprising expect selection utilize zero datum either neutral selection lar ds scale agreement lar peak ds alternatively total distinct peak lar ds combine common state front make formulate say reduction cv increase well agreement lar ds still practically ideal circumstance scale lar recognize interest peak less particle furthermore unknown peak small far unknown peak chemical mechanism argue uncertainty challenge knowledge focus term context uncertainty regardless enforce uncertainty selector lead residual reflect characterization application uncertainty improve lar peak addition one scaling
child denote way dag restrict find subtract way connect arc leave reconstruct drawing possibility leave incoming arc old result old receive arc reject arc term arc old fall drop draw arc old arc accept rejection highlight possibility constructive preferable draw parent invert arc number child enumeration recursively remove reach restrict avoid directly dags partition node partition dag easily restrict replace expression ratio section dags dag triangular influence bernoulli act uniform space dag underlie preferable practice dag rather dag arc basically recursive enumeration receive link link term binomial possibility arc exclude could theoretically arcs dag arcs possibility even add dag rational store integer limit reduce towards dag large term include model average parent necessary number mcmc treat dag analogously dag express split two reduce hybrid dags weight calculate perfect rational sampling mcmc offer remove bias operate wrong advantage adopt triangular adjacency produce markov avoid efficient perfectly uniform dag vertex compare dag chain dag recursive enumeration mean practical limit behaviour base dag belong alternative cost check approximately small markov adapt dag include restriction parent node idea could integrated scheme procedure combinatorial go add resample new connection move reversible one amongst alternatively underlie combinatorial sampling act space matrix dag partition produce avoid act structure sense combinatorial score order satisfactory possibility seem dag several set identify dag probability know markov equivalence dag although formula run dag correct essential implementation compare dag essential fraction move essential suggest dag little exact essential graph also dag importantly number essential graph dags dag corresponding essential actual might sampling uniform dag possibly essential sampler real label responsible background depend background satisfie result inequality frobenius square must exponentially fast constant maximum bound return irreducible uniformity eigenvalue one unity difference uniform discuss upper comparison enumeration study low weight evenly dag overall would weight amongst dag handle focus arrange empty dag still spectrum dag arcs transition q assume cover dag dag intuitively dag encode far store looking fill matrix evenly amongst dag high suggest matrix require move irreducible integer binary treat integer follow bit store grow multiply done perform multiply copy operation sum final multiplication add multiply calculate recursively without overhead recursively multiply calculated computing affect repeat bind complete step limited decay give simplified length representation numerically single dags step integer subtract number binary sum discuss sum sampling time seem complexity choose discuss complexity lead code precision integer package recursively store number sa n sa kb binomial readily recursively loop computational overhead r sample store value recursively generate loop km sn ki ki l ni l j randomly draw sample dag correspondingly column row define power bring replace since rgb rgb universit acyclic graph representation underlie network represent practical reverse engineering gene great require analyse acyclic uniformly enumeration consideration discuss enumeration provides actually acyclic us large idea enumeration various restriction include restriction enumeration hybrid markov chain graph acyclic acyclic graphs dag collection represent relationship short property independence apply relationship seminal paper tool infer structure model particularly important light drive dag class reconstruction review development dag discuss simulation study aim assess reconstruct graph datum crucial space relate remove currently rely chain limit dag carlo context graph uniform dag require neighbourhood expense extend dag recently pose negligible triangular adjacency matrix ensemble matrix provide dags point hill slowly uniformity small increase uniform sample essential evaluate certain count dag insight dag landscape assign structural therefore strategy base recursive enumeration dag combinatorial uniform dag enumeration note early report think impractical evaluation complexity show enumeration expensive establish practical enumeration exploit dag fast minimal triangular dag uniformly devise share easily restriction method parent certain perform triangular hybrid mcmc offer avoid bias vertex dag admit label dags asymptotic dag find dag entry vertex remain acyclic low triangular node eigenvalue dag easily sample triangular graph dag several permutation example dag arcs arcs non uniformity entirely start arcs proceed repeatedly pair direction otherwise long remain acyclic arc operations arc cycle stay exclude sampling reduce marginally chain path dags arc empty reversible ensure approximately long chain elegant dag apart remain acyclic order edge cycle applicability dag arc irreducible uniformity element moving consider go dag arc dag none remove arc add arc arrive arcs transition path move empty path order last move actually require probability dag uniformity derive triangular uniformity markov markov needs say accurately background restriction twice often uniform uniformity fine additional one twice entry ensure useful much express term importantly uniformity element probably element satisfied logarithm consideration dag easily find respectively remove arc improve verify uniformly dag away first step chain also discard like may check graph convergence start dag randomly uniform enumeration label dag incoming arc far classify label dag dag figure dag allow possibility arc old give way give dags arc turn inclusion dag finally order small integer multiply store many dag arc receive bold arc allow remain triangular bold case arc nod node arrive sample dag dag sequentially marginally well chain integer perfectly uniformly sample dag significantly chain require subsequent uniform dag rather quick integer still obtain run core intel processor gb ram method per machine dag perfectly recursive become irreducible recall take reasonably uniform chain hundred million thousand time enumeration hour give complexity thing bad enumeration account store memory computer several fact see next access memory reach space exact dags growth lie behind atom observable universe move dag extremely multiply exclude lr dag uniformity sampler sample limit dependent weight shift towards meaningful store small iterate long return enumeration sequence limit behaviour reconstruct order step dag sampling upper complexity therefore dag build dags procedure remove alternative dag sample vary triangular zero rare dag limit nearly uniformity return exact enumeration indistinguishable enumeration variation enumeration markov enumeration sequence order partition introduce partition accord dags dag account connect edge empty order partition integer represent correspond size large previous partition treating build split adjacent ensure th draw exclude step become irreducible metropolis hasting partition dag scheme dag digit single flip directly modification affect edge hence partition simplify split ratio convenience evaluate calculate weight take account partition allow dags preference move calculation chance move note suffice complexity completeness acceptance involve clearly large repeatedly draw soon unlikely event draw total binomial bernoulli effectively soon average move chain overhead
another implicit population crowdsource offer expert quality concern estimate expert labeling yet high base label ir direct considerable toward effectiveness relatively blind label via crowdsource regard pseudo gold label aggregate gold label build develop aggregate set classifier evaluation reliability tradeoff investigate crowdsource direct evaluation supervision method generality test establish rank correlation estimate figure depict conduct classifier crowdsource track investigate reliably classifier expert much crowd blind correlation evaluation result score see significant score benefit outperform blind complicated label yield though blind crowd likely common accurate broad concern human labeling relevance ensure label consistency decade system human evaluation system evaluation positively human system document vs retrieve wu exploit popularity frequency document retrieve fusion blind label al labels et pseudo generate evaluate demonstrate label exhaustive expert pseudo aggregate classifier consensus pseudo gold spam filter predefine applicable label effort effect direct performance label blind crowdsource class et evaluate pseudo score combine research integrate redundant produce area crowdsource aggregate human I method supervise consider majority vote expectation em variety calibrate naive glm adaboost also label expert evaluate classifier notation classifier example label example turn three human level gold estimation sampling arbitrarily reduce majority vote pick vote round decision q round vote tie break generally varied specificity definition relevant maximization em probability classify membership eq indicator receive iff unknown em classifier every independent probabilistic maximum summation become maximize miss crowdsource though rather way method naive nb support generalize glm adaboost class training expert set detect section unsupervise metric consider default nb supervision infer gold likelihood compute glm label glm fit example return response binomial learn plane decision learn support class adopt identifie classify dot meta iteratively weak respect adopt well adaboost method classifier begin describe dataset study use score difference score discuss test correlation achieve label per ex crowd train test crowdsource track comprise task collect crowd involve aggregate crowd task yield classifier output come distinct expert university final balanced track crowd label collapse single label majority across range classification positive negative metric fair perform operational pair difference compare preference induce ranking order hard tie question rank classifier preference minus pairwise correlation statistically significant vs vs obtain largely correlation score score rank adopt measure ranking swap pair closely tie measure reflect score outlier worst receive low whether estimate concerned determine ranking correlate nan ranking classifier rank ranking correlation exist tell directly detect one achieve involve triangle significance ranking expert vs familiar ir letting correlation score ranking hypothesis denote reference expert estimate ranking compute statistic freedom triple order evidence evaluate classifier crowd estimate achieve correlation expert begin section show produce classifier early establish diversity label score expert compare score supervise supervise present section place evaluation c c c c begin measure relevant diversity overlap coefficient intersection size union show standard study crucially classifier exhibit extremely low vs classifier bad suggest likely widely agreement represent fair chance exclude remain indicate moderate acc pre rank c tie rank statistical testing ht acc show tie rank significance confidence measure classifier accord actual table four tie tail pair indicate vs merely classifier classifier specificity actual test significance confidence result insight summary reliably crowd enable high pearson metric crowd significant improvement rank crowd low condition answer method correlation alternative testing label axis line acc metric typically correlation correlation four blind follow rr next supervise nb calibrate axis acc plot line detail achieve acc specificity lower significant see crowd blind evaluation correlation achieve similar simplify presentation correlation correlation improvement crowd observe crowd method wider across metric crowd blind range c score correlation swap blind acc acc acc rr glm measure ranking refer classifier achieve achieve statistically tie rank indicate reflect confidence rank tie achieve expert validation raw value reflect rank achieve statistically well tie rank indicate crowd direct correlation direct select evaluate cccc c cccc c c c swap type acc pre acc acc acc sampling nb alternative rank data blind crowd combine approach direct directly crowd significance difference prediction indicate confidence inform validation top gold blind evaluation correlation em absence correlation method consistent early crowd significantly rank offer potentially tackle aggregation gold gold well concern however pseudo gold labeling error derivative score rank quality secondary estimation four acc pre secondary primary combine swap negative metric hence correlate indicate c acc pre glm method predict score ranking measure pearson coefficient pseudo validate intuition correlate effectively predict axis axis indicate predict pseudo gold strength pearson expectation pseudo achieve coefficient correlation gold accuracy critical pseudo gold label evaluate classifier absence truth label effectively predict rank acc difference label correlation difference reasonably large metric swap acc pseudo evaluation pseudo gold gold gold vs expert grouping value metric label vs correlation value shown swap well difficulty uniqueness pool acc blind analyze classifier classifier test effectiveness worst achieve representative evaluation select predict rank score present row well rank
package code age f na na na na na na na na easily person site gender analyze key format omit item display ht option specifie contain expected one select fourth estimate option reflect typical item total moreover display fairly monotone behavior cover exploratory parametric currently analyse quite check reason apart software computer smooth prominent software year run within user perform several package handle believe enable pose plausible option extend package smoothing well serious statistical currently nonparametric program health international center california kernel smoothing implement option curve add subject choice mail alternative category must reflect accordance degree option option order naturally set item hereafter sake option often response refer comprise set ji jx assume select response reference mathematical underlie trait measure measure refer choice item typically proportion subject quantify discrimination account option characteristic mean universal among literature survey name aim analogy classic parametric structure estimation idea model vector interest item difficulty discrimination assume accordingly might approach unless although experience confirm communication appropriate display justify year recognize paradigm give item characteristic basic set statistical require package perform propose approach nonparametric smoothly keep ij consequence preferable weight amount trade continuity non argument far largely nevertheless common choice assume vary highlight subscript important different use replace nk consideration estimation process begin transform common summary alternatively come quantile ability denominator avoid infinity f standard equally value span ordinal ability consecutive start group n bandwidth plug former kernel generate estimator formulate framework deviation induce consider validation considerably high simple widely context let kernel nj remove estimate ordinal ability omit statistic nk vector denote suitable visual inspection graphical estimate pointwise point relevant extent consider moreover useful error argument ignore substitute complicated approach interval estimate several quantity compute take jx j score coincide refer option straightforward define kernel jx jx jx jx jt really respectively analogy single function call counterpart preferred substitution people display axis interpretation method fails affect extreme generic relative say differently summary addition characteristic describe tend step time step iterative refinement really reason consider package reference generic see simplex dimensional nonnegative coordinate vary move curve analysis simplex location convenient triangle unit length side opposite vertex unitary sum one triangle regular item option nevertheless three four option main create option item illustration capability number alternatively list weight item use simply correctly option preliminary item necessary vector option option scalar item analyze subject rank nominal item rank item vector equal complicated weighting specify help section default alternatively point default value quadratic select global computed user input numerical opt cross validation specify implement treat option add multinomial subject one class function return brief description variety select option evaluation point observe subject score mm subject quantile subject score quantile probability overall mm km option correspond weight matrix contain error mm mm ht method allow exploratory return correlation mm observe vector ii nj evaluate alternative return subject expect list package return return object return ht mm simplex plot high option display multiple plot group include contain response student item option analyze create key format produce smoothing default gaussian kernel measure performance correlation value create become available overall display produce display display blue red use dash fall item figure plot figure different item apart item item monotone problematic trait level option trait select measuring trait contrary item subject trait level item display power near subject great subject roughly select regardless total figure top student recognize option incorrect select incorrectly expect total constitute probable aside since expect select correct option consequently incorrect code display figure weight blue correct pointwise interval dash red illustrate via argument remove change specify wide total high due datum less precision plot average group include triangle simplex illustrate plot item naturally normalize experience people tend eliminate option characterize option item generates display figure ht curve medium red green blue value break three ordering level possible make consideration format correct term requirement toward important individual trait figure fairly item curve concentrate close correct student slight tendency wrong speed bad student item triangle figure see share lie option perform way simultaneously show define x paradigm consideration rank ks compute pca pca produce graphical principal represent item place component axis difficulty ability extremely principal discrimination item high slope case top respect component user identify discrimination possess strong clearly conclude principal plot tend summary composition dominant subject red line
formalize outli region locate region outlier cell outli poisson region poisson variable ax indicator region cell count name notion outli easily estimator exchangeable robust preferred procedure contingency region respect outlier look ensure full space call submatrix subset consider definition pattern cell full strictly coincide add minimal cell yield non singular pattern outli identifiable rule identifiable develop notation model correspond cell take yield indice detection omp define identify set large majority minimal ml x w w idea outli detection cell pattern outli region set minimal notice minimum identify base knowledge omp outli cell identify identify number minimal cut discriminate discuss ml j nj jx w enumeration minimal minimal potential outli free ml procedure open choice estimator call also derive categorical detect outlier adapt pearson least chi residual create subset minimize chi tuning generation subset code j ml j ml adjust perform difference outli detection base analysis estimator large table analyze pattern contingency design eq indicator design latter parametrization usual parametrization implement former proof many formula become handle cell table min case configuration submatrix hand produce strictly minimal first singular complete among case different algebraic investigate notion describe problem example nevertheless structure cycle cell exactly submatrix definition submatrix cycle case cycle ingredient cell cell independence cycle cell cell row design coefficient nan minimal pattern conversely submatrix nan submatrix indicator row belong coefficient cell negative therefore column coefficient cell choose cell positive otherwise iterate reasoning another cell certain column two definition corollary following produce strictly pattern table uniformly tuple cell cell produce cycle cell strictly select uniformly statement table empty cell row remain cycle must empty cell row column exclude remain minimal cell cell force constitute loss form cell cell strictly choose minimal need pattern stop two pattern section present outlier discuss situation conduct exclude omp poisson contingency table value contingency cell hence upper bind expect small bind corresponding contingency place row outli moderate outli contingency table analyze extensively create work three two type cell notably scenario time replace contingency affect simulation table similar finish generation contingency minimal constrain outlier type outlier outlier outlier identify analyze comment satisfactory table outlier outlier notably hence position within major placing classify outlier reduce considerably evident case outlier rise parameter estimate almost respect seem notably small scenario well scenario valid large though less regard outli rate outlier discuss method motivate cutoff consider case contingency nan plus outli cell table size proportion detect outli motivate compute proportion display good trade outli first discover outlier treat independence model become apparent detect social network go applicability general h ccccc within mi mi mi point omp outlier e outli look stay cell study cell sure outli cell result category contingency class yield contingency table omp identifies identify surprising hand model wrong seem plausible alternative offer satisfy final present behavior seven work class concern friend daily week week friend cover category whether daily identification classify outlier outlier yield eight definition pattern yield time omp one outli cell eight yield solution produce result detect outli time cell time detect rest cell time cell interested outlier cell outlier detect method omp contingency regard outlier get see co friend omp good contingency detect simulation real summarize consider provide detect table use increase outli simultaneously detection see sized big table procedure suggest omp outlier scenario detection however expect dimension scenario identify outlier perform
ranking return return importantly set depend concern social person loss application respectively incoming training neither differentiable development end consider set would edge latter optimize conventional hinge simply equation denote relation beneficial differentiable function edge knowledge forward relation restriction product mapping express pairwise kronecker feature pairwise dual choice rbf rkhs closely function compact bound rkhs dense universal define universal base interesting optimize universal apply kronecker pairwise expect low amount universal consistency kronecker consider valid relation approximate closely whenever expressive illustrate detail domain knowledge relation similarity relation relation domain object need learn symmetric relation binary call symmetric real analogously relation relation edge become undirected domain metric similarity symmetry incorporate framework modification kronecker symmetric kronecker use predict arbitrarily type relation symmetric relation rkh knowledge learn unnecessary expressive still let type relation call hold relation place reciprocal edge graph induce edge appropriate application arise bioinformatic represent preference win real high interpret interpret set instead incorporate kernel symmetric reciprocal also rkh reciprocal kronecker arbitrarily reciprocal let continuous relation dense kernel model reciprocal relation give detailed algorithm rank system equation relation powerful tool matrix correspond reciprocal matrix vector use omit consideration follow matrix skew armed definition corresponding reciprocal symmetric note cover encounter encountered consideration involve node set reciprocal kronecker ordinary kronecker immediate consequence kronecker claim pay entry evaluation edge entry contain skew straightforwardly contain evaluation kronecker p kronecker kernel represent follow node edge node enable drop present case application graph belong classification bioinformatic e feasible experimental imputation edge notation matrix prediction parameter whose represent dual solution contain per occur regularization multiplication thus minimize becomes set system kernel interpret simplified solution enough rule solve kronecker product let matrix matrix shift kronecker solve actual rule concern kronecker introduce namely eq q ss eq side leave write well multiplication time kernel definite nonnegative strictly therefore exist proposition continue reciprocal ordinary kronecker identity certain property furthermore prove multiply inverse result vanish orthogonality inversion identity calculus analogously inversion identity reciprocal shift ordinary kernel advantageous computational cut provide ensure inversion shift kronecker accelerate computation reciprocal inversion identity symmetric reciprocal ridge regression reciprocal kronecker ordinary kronecker learn symmetry encode kronecker kernel equivalent use analogous kronecker label ordinary kronecker equality fact couple suppose regressor determine vector kronecker result reciprocal show way reciprocal kronecker corresponding kronecker kernel use represent representation matrix entry center multiply entry quasi start contain node exactly form provide order compatible entry arrange analogously regression kronecker time ordinary kronecker write equation p invertible rewrite product nonnegative g involve semi definite call empirical e kronecker consideration reciprocal kronecker kernel similar close form corollary concern regression drop fortunately design take advantage closed form solution proceed entry node edge possible edge several edge adjacent write ordinary kronecker analogously reciprocal variable store occur sum solve become identity matrix system linear product cubic nonzero quasi diagonal multiply gradient show early learning process together regularization insight back ordinary learn model wise center correctly edge rather common algorithm relation utility value center help achieve utility form ordinary rank diagonal center make indicate regression eq inversion identity solution lemma provide block center conditional therein indicate regularize prediction minimization kernel level center conditional wise behaviour give compare pay attention corresponding construction rkh expressive wise able express help kernel next focus term wise latter promising predict value ordinary regression pairwise challenge consider task graph use prediction application retrieval preference induce document machine optimize closely rank regularize least square previously minimize equivalent article edge training graph enough train however exist limited training node furthermore mean edge learn need construct store less kronecker use node see assume wise solve avoid complexity present special product close regularize ranking generality first experiment reciprocal task win condition experiment consider retrieval document document summarize goal bipartite specie rank species capability generalize observe conditional minimize convex approximation minimize regression also train specie possible due memory requirement cost enforce reciprocal experiment kernel experiment thus measure variety approach characteristic task solver pairwise square loss experiment train software whose generation detail consist well pair play player player first win player differ move game player another meaningful sensible rank player variation set strategy play move one rank game adapt exist game represent start winner start thus product edge generate kronecker three algorithm directly score minimize edge pairwise hinge preference node preliminary notice regularization reach zero appear run zero rank present successful easier beneficial set pairwise rank approach yield rank surprisingly experiment reciprocal kernel experiment cc comprehensive plot performance repetition regularization behaviour rank range reach start increase implementation reach good even narrow suitable approach parameter highly model successfully enough exist use reciprocal kronecker regression approach pairwise rank c learn rank document document publicly consist document contain document feature similar message document document rank highest next last rank undirected graph relation conditional ranking document grow node nod experiment already million paper experiment training rank directly graph long experimental static rather document already simulate form contain message hardware window message graphic os ms windows mac thus form consist node second training test rank greatly outperform value relation investigate enforce final experiment compatible symmetric increase computational node task effect early discuss contain see quite outperform regression enforce symmetry increase loss rate curve monotonically decrease due scheme sometimes solution demonstrate show training consist million edge generality available improvement base enforce knowledge type kernel potential rank classification problem huge class happen dataset class correct prediction occur capable introduce article unknown rank normally define identify specie profile task profile profile profile scenario profile relational consist edge connect single bipartite detail version divide different form point large class randomly divide small entirely combine profile result also run kronecker experiment result rank rank close method point result specie test rank regression similarity query serve relationship r biological newly discover think try machine kernel similarity ec ec level detail ranking ec rank ground query count successive left start digit ec soon example ec query ec structure bind ec annotate purpose ec lead ec comprise member heat consider art label superposition generation equal part cross validation addition cross validation loop implement recommend hyperparameter power whole hyperparameter benchmark bioinformatic construct compute query appear end represent potential similarity rank query principle methodology cb fp global well supervised outperform kernel put traditional benefit supervise method account training phase retrieval ec conversely characterization ec supervise preserve ec support summarize set different correspond well correspondence truth different distinguish third improvement exploitation direct method indirect approach interpret indirect kernel entry noisy case similar kernel loss cb loss slight experiment prove trained kernel runtime conjugate early stop close rank train edge file software intel gb memory parameter limit conjugate efficiency solver sample fully vary number vary number consider scale behaviour cubic time edge thus observe beyond apply worse iterative cg allow efficient training share behaviour applicable solver kronecker product necessary edge stop cg prove perform experiment algorithm linear solver sample uci repository label point belong solver product feature even competitive e less feature low efficient inefficient infeasible long also beyond scalability large dense graph kernel solver million matter minute solver ranking condition rank loss function generalization motivate rank symmetric reciprocal treat important confirm rank optimize loss beneficial instead moreover boost performance discuss follow computationally present conjugate early matrix structure recommend become form solution proposition recommend method solve magnitude large rank exist acknowledgment thank comment grant research foundation european conference corollary definition domain one task goal consist rank particular object ranking type relational ranking object efficient theoretically regression prove symmetry relation efficiently illustrate symmetry improve present review conditional suppose play computer people player unfortunately game game game tend sense player great mathematically speak cyclic relationship rank supervise biological consist interaction highly confident interaction similarly conditional context predict rank protein likely conditional task differ ranking compute reciprocal relation ranking arise relational relation model social database object information retrieval interaction
ranking dataset os enyi rank yahoo movies division yahoo movie user consist represent movie difference movie user pairwise dataset construct comparison fisher randomly impact base schedule pairwise incomplete team b team team strength schedule among collect rank elimination expect team remain game possible collection interpret design schedule ranking context study schedule games informative spectral connect division argue improve scheduling conference continue construct synthetic estimate graph laplacian establish used section collection reduction synthesis experiment employ turn learn rank statistic schedule conference include extend comparison os enyi connectivity fisher additional study optimally feedback arc rank consider ordinal pairwise preference label specify work preference represent quantitative optimal alternative geometric ranking easily survey literature find design pose inverse e image optimal imaging tradeoff measurement reason cost collect often place acceptable reconstruction construct desirable informative equivalent variation schedule static scheduling schedule throughout scheduling elimination team next round elimination world dynamic scheduling gain view ranking past process dynamic game thus disadvantage scheduling method scheduling focus elimination agree rank particular team objective assume paper investigate round team schedule synthesis maximal study many algebraic review review optimality robustness network failure highly algebraic process determine fire game game connectivity load balance consensus mention rotation measurement algebraic edge proportional quality briefly eigenvalue algebraic extensive give recall incidence direct graph terminology arc direct arc tail arc define edge weight refer un laplacian interpret triple direct arc connect vertex degree define degree laplacian interval eigenvalue nonzero connect second eigenvalue characterize subject wise connect graph u w connectivity small tight incomplete set connectivity adjacent removal algebraic connectivity vertex algebraic bound inequality also perturbation weight increase indicator use obtain indicator respectively weight however unconstraine round greedy add maximize algebraic augment work refer greedy heuristic finding edge large second w alternative complete let arc incidence comparison denote comparison alternative comparison gauss state unbiased equation finding agree square sense square ranking prove empirically rank present datum laplacian square estimator since unbiased fisher collection definite ordering criterion function prop square bi eigenvalue laplacian expression prop criterion proposition datum collection criterion problem reduce synthesis add algebraic interpretation theorem tree condition effective circuit construct identify return comment ranking proposition structured graph laplacian graph compare algebraic os r enyi yahoo movie rating correspond division schedule construct via active accurate cycle mm fill dot dot edge dot edge dot thick style draw fill thick thick thick thick dot fill thick dot dot dot thick thick inner dot style blue fill blue dot dot dot thick dot dot edge thick alg diameter connectivity represent informative ranking computable graph node algebraic connectivity connectivity vertex connect unlabele algebraic connectivity number edge algebraic connectivity vary connectivity algebraic large node equal degree graph add algebraic connectivity study fig nontrivial maximally algebraic algebraic achieve connectivity panel connectivity remains describe distinct graph nod additional connectivity graph prop ranking review review movie yahoo yahoo movie movie movie define pair user movie write note expression arc yahoo movie user rating movie graph pairwise comparison l l movie day friend degree represent yahoo comparison histogram residual top rank number comparison black pairwise give increase movie relative estimator prop compare comparison experimental develop approximate matlab initialize use eigenvector lead eigenvalue laplacian maximum increase rate increase pairwise randomly movie tf yahoo database movie legend comparison color bottom collection square color graph visualization via pairwise top plot yahoo movie comparison improve system enhance generate spectral algorithm generate reasonable position movie movie use full dataset graph plot use figure top panel primary connect weakly advantageous designing schedule division schedule rating division divide decompose division college member opponent member member division game static ratio example american team primarily division within average strength among schedule consider team game per year team play ranking mathematically expert opinion aggregate ranking none none relatively schedule via game vertex indicate conference represent reveal conference indicate text use visualization method division play conference cluster conference algebraic cluster community toolbox algorithm optimal cluster centroid division division vertex team color conference membership bottom cluster interaction conference implication algebraic connectivity os enyi nearly representation schedule game edge game conference use primarily conference introduction note scalar three compare division os r let vertex graph laplacian comparable e harmonic rather logarithm determinant define remark division division record also fig division division table fig enyi take approximately time threshold connectivity connectivity similar scatter indicate graph nearly evaluate solid obtain finally blue table division os enyi graphs sense division division reveal primarily conference conference imply algebraic cut dash fig division schedule connectivity edge consist conference os enyi top primarily reduce ranking schedule flexible contain permutation previously optimal scheduling future game performance schedule completely begin play schedule os enyi division schedule solid line represent dot os enyi circle os enyi optimal os enyi division division os enyi comparison propose division truth rating distribute truth according ranking ranking ranking right algebraic connectivity dataset enhance dataset choose game comparison greedy selection collect initial collect plot rank determine collection strategy error collection thick minus strategy plot vs algebraic comparison red connectivity graph greedy random algebraic produce fidelity experiment design informative statistical ranking heart framework unbiased rank outer identify information square outer inner reduce optimality fisher weight find algebraic division optimality scalar strongly strategy remark optimality furthermore use via improved collection rank yahoo user rating consider show pairwise add contrast datum collection comparison add movie review review review review review improve ranking potential constraint collect extension employ accommodate incorporate penalization compute constraint handle weight least rank orthogonal free cycle harmonic cycle harmonic inherently inconsistent ranking lie graph give first improve remove connectivity alternative estimator method I mark nsf dms grant support nsf dms thm thm question vertice alternative rank agree develop collect least square experimental view problem maximize
eq final arise potentially statistic apply point whenever statistic impact allow issue value produce change detector shift detecting shift difference return find conditional analytically instead compute carlo thresholds stream find successively proportion exceed discard equal successively expensive hour needs perform computed store table overhead give seem reasonable since threshold bernoulli equal conservative table actual empirical variety take seem suitable stream far towards implementation world scenario resource spend calculate equivalent package optimize level correlation statistic sequence failure failure observation compute algebraic eq recursive formulation recall q use though hence smoothed still time far retain memory use efficiency store discard length recently window memory discard calculate whole contain computational naive old discard detector discard point entirely statistic maintain sum old time outside current carry lose give notably threshold post symmetry bernoulli c detector parameter run delay recall degree smoothing c comment occur suffer detect bad detect increase smoothing cause jump statistic partially detection performance demonstrate usefulness worse read relatively poor latter bernoulli everything far design complete change minimax parameter misspecification c detector realistic compares parameter misspecification likely happen early monitoring since insufficient allow estimate design base value correspond misspecification control assume post change design change misspecification occur substantially bad every appropriate detector exact sequence pre unknown device favorable chart change occur relatively stream comparable chart knowledge change bernoulli stream method poorly data optimal realistic investigate perform degree misspecification see drastically tool situation methodology available http website co uk research agent jointly engineer physical grant ep detector c c c detector eps widely approach violate observation occur approximation detection computationally suit sequential monitoring empirically datum find optimal assume detect change spaced datum assume realization integer constitute arise receive finance monitoring increase work hence terminology bernoulli change soon change sequential detect encounter find monitoring change begin reduce successive multiple simultaneously avoid single stream performance detector expect false positive delay detect denote denote false positive say occur play type important sequential assess reason literature bernoulli detection suffer several make monitoring computationally inefficient context require store memory although fix monitoring asymptotically assume substantially work come literature size treat binomial random chart tool chart individually commonly inspection detection treat failure approximately standard tool detect parameter sequence approach transformation chart also pre parameter know however situation sample always misspecification pre paper task parameter regard pre post computable monitoring point highly hence receive stream detect practice trivially side change detection tool design monitoring change extend distributional shift bernoulli implement available section end proceed begin bernoulli deal length develop sequential monitoring discussion follow performance detecting point length neither change assess occur hypothesis observation identically several test exist exactly rely like change detector number change point idea observation break nan sample identically number failure observation reason failure follow binomial coefficient fundamental probability dependency value hypothesis make test know introduction failure first probability failure observation observation identically convenience occur reject appropriately choose threshold change locate hypothesis
literature model autoregressive comparative baseline engineering order show return note make day volatility forecast evaluate mse model series historical windows contiguous historical constitute consistent volatility measure result observe significant improvement term employ metric evaluation return historical volatility horizon step c evaluation horizon step step evaluation c return historical volatility step evaluate covariance asset return repeat experimental copula employ similar input observation comprise available sample interval one comparative art volatility specifically ccc use asset result yield scenario yield one order magnitude yield comparable employ return average prediction ccc evaluation square ccc nonparametric financial model separate gaussian drive law capture model addition covariance asset return build top efficacy several asset observe improvement consider successful model financial return alternative widely field machine specifically novel nonparametric series impose component well capture model distribution tail skewness asset model gaussian mixture conditional volatility asset require tendency asymmetric generation financial market index tail asymmetric autoregressive address dependent volatility finance market approach commonly model exhibit time volatility follow period excellent define field last decade represent squared return variance computation prediction field machine function significant application domain gps several gp difficulty deal multi modal approach ensemble space issue work volatility financial approach effectively volatility introduction novel distribution constitute gp variance process gps model way novel approach nature asset procedure conditional bayesian technique especially process become popular year nonparametric realization seen give shape large highly indeed dirichlet prior selection value turn frequent dominate entire inspire advance switch mechanism prior derive examine efficacy consider financial remainder organize provide presentation present process prior summary regression propose copula jointly model mean conduct application deal model financial final summarize result dirichlet dp characterize referred innovation parameter dp dp subsequently draw joint exhibit sample draw allocation proportional previous allocation concentrate dp scheme innovation parameter role determine parameter induce tendency draw new base get contrary randomly discount innovation parameter yield rich rich assign draw likely effect rarely allow scale note case unconditional distribution stick break construction consider two random beta draw stick representation draw comprise unbounded component density proportion process completely specify define covariance real eq take concern large employ depend application eventually identically phenomenon scalar predict target information contain consist observable express superposition regard normality target observation yield identity conditioning training expression yield q optimization employ conduct type maximum maximization likelihood easy regression deal corpora model comprise derivation infinite innovation infinite reason break variational equal note full prior truncation impose tractable truncation prior impose hyperparameter function variational stand posterior kl nonnegative strict bind tight implicitly consider take variational read ignore constant term derivation posterior involve hold manner consecutive updating guarantee monotonically maximally denote posterior stick break innovation eq posterior regard process semi definite diagonal variational parameter q posterior element element estimate comprise hyperparameter prior maximization free resort bfgs model output derive derivation furth variable eq base variance one component largely essence model jointly various output return correlate give input approach gp aforementione exist learning entail optimization estimate hyperparameter kernel get bad local optima work issue financial devise novel way model tool copula seminal standard restriction example hypercube cdf exist variate copula cdf marginal variate copula unique conversely univariate hold inverse cdf model easy show copula widely assume cdf parameter copula model copula several frank enable capture dependence coupling copula copula model able skewness tail dependence copula rigorous effort motivate excellent discussion copula c covariance specifically vector cdf define cdf correspond give use copula value
make lemma perspective problem low rank similar low cx decomposition regression constrain multiple regression place elementary regression optimal thus multiple build improve previous well nd multiple except instead note simple equality approximation optimal optimality constrain b minimum point dimension randomize output subspace run section connect round follow fast round quickly basis generalize general easy start eq clear number conditioning pa relationship quantitie q recall da easy algorithm naturally connect round symmetric respect connection round deterministic compute rounding separation due convex round ellipsoid ellipsoid find problem algorithmic suppose describe oracle provide ellipsoid round result aware particular use construction find theorem round convex origin center subgradient matrix qr decomposition maintain conditioning column take construction well condition extend construction round leverage next improve run slightly condition run application find condition full compute pa storage work depend well norm condition leverage norm row norm estimate sample row together complexity due reduce modify obtain random construct specify diagonal solution section evaluation implement evaluate cauchy transform transform ct transform ideally evaluate distortion evaluate non convexity seem accurately transform basis approximate describe column well basis transform metric scale invariant condition fail bad input comparable much conditioning ct lead regime algorithm worse base increase perform cc third draw dash correspond difference inferior explore sampling fail completely fail sample ct instead next transform implement section compute norm instead error thus permit indicate leverage similar side entry independently simulate corruption regression regression conditioning instead determine tolerance accept size third correspond failure remove error fail completely fail slightly test certainly favorable error theory fact leverage approximated capability scale corrupt generate standard canonical measurement number entry million corrupt simulate noisy corrupt regression estimate regression certainly robust alternative separable response cluster core condition ct un time fast ct particular moreover store storage spend running implement ct ct reveal inherent ct condition sampling row solution simultaneously check measure error ccc ct randomize error ct well follow ct large third without conditioning corrupt normal treat overall entry draw wise clearly condition sampling ct uniformly entry wise ct maintain base ct measurement adjust towards summarize condition promise digit pass twice cauchy transform analog hadamard base demonstrate fast base improved running extension provide first evaluation evaluation rectangular precision solution acceptable base setting connection theoretically exploit scale david fast problem rescaled row polynomial improve previous fast round well condition basis round time conditioning provide algorithm compare thing asymptotic guide practical performance construction condition fast cauchy transform approximately preserve norm distortion constant cauchy transform coherence embed prove development dimensional importantly sense implement distribution mapping subspace embedding geometric quickly variant hadamard thing construct regression lead problem rank fast give q interested case although least absolute regression programming problem convex relevant work solve sampling basis embedding thereby improve cauchy analog variable slow problem subspace problem construction fast subspace interest call cauchy provide detailed essentially analog represent sense depend preserve distortion moreover actually relate construction rescale fast analog improve result construct well condition regression row probability row regression reduce regression consist rescale independent use black run exploit improvement well basis fast guarantee run improve run condition algorithm improve extension improve run fast computing basis problem condition basis round general conditioning alternate way particular round round much round time present basis round algorithm competitive develop evaluation construction slow cauchy transform core matrix evaluate latter nearly infeasible clearly paper brief use present basis leverage approximation include condition large conclusion presentation assume find minimize mostly set th row th frobenius norm ij p pp vector identity generic may trial due immediate first I cauchy stable distribute cauchy heavily sum note prove cauchy random variable dependent conditioning magnitude application lemma completeness tail necessarily mt logarithmic dependence cauchy assumption random substantially improve cauchy independent latter scale tail bernstein inequality suppose define cauchy transform analog actually present coherence rescale cauchy random hence main theorem quality construction first matrix cauchy random finally combination parameter failure uniformly algorithm fail suitably diagonal independently cauchy sg sn integer reader matrix hadamard may recursively hadamard transform hereafter spread comprise cauchy scale finally variable theorem find cauchy hold far constant since establish factor variant hadamard additional plus evaluation use tail concentration independently bound construction generic change parameter construct random informally slow transform multiply dimensionality slightly construction inequality dy fast mapping give distortion arise random whereas use requirement restriction proof originally appear strong prove correspondingly improve second easily well construction basis approximation least absolute algorithm basis condition adapt basis range u exist basis generally condition norm norm condition summarize run step product condition lead multiplication let projection construct construction section step factorization next theorem corollary combine proof condition construction condition fix construction result require rank case modification small sampling preserve discuss subsequent basically either subspace approximation motivate score let element name random regression rank regression ultimately follow define invariant basis score depend polynomial orthogonal span give lead leverage degree analysis goal vector sophisticated yield run prior fast permit arbitrary detail summarize well probability sample row quickly norm row scale leverage input carefully sample rescale problem theorem summarize improve n exponent multiplication purpose norm row norm condition cauchy construct diagonal solution minimize weight regression dimension construct remark preserve norm mention minor change natural right approximation subspace efficiency solve simple thereby replace expense size improve essentially accommodate detail
effort constant analysis recursion recursion f ff ts difficult q equally q plugging estimate form become integration calculation plug obtain hold essentially thm imply decay average polynomial decay scheme convex choose dataset run gradient extend mirror question individual iterate ht sgd iteration return common heuristic last well rate polynomial average last indicate return last iterate far whether especially implication probability last variability unclear whether iterate behaves comment proposition definition zhang edu nj usa descent simple popular stochastic theoretically decade usually smoothness machine paper investigate smoothness average convert sgd iterate optimization framework prove round iterate non non smooth first averaging averaging attain propose implement simple descent sgd unbiased learning minimize sgd simple highly scalable making scale proceed round describe line round estimate subgradient iterate suitably analysis decade instance reference therein perhaps gap understand look rate budget year attention devote e classical non continuity use solve machine standard smooth objective non non whenever smooth function smoothness smoothness carry see definition error general provably suboptimal averaging iterate average rate result leave significant averaging return iterate often say optimization iterate convergence aware well simple gradient know exist theoretically practical memory one know advance trick natural compare iterate rate average require contribution prove individual general convex iterate much bad pose knowledge sgd rely iterate iterate implicitly somewhat improve error average call enjoy new scheme easily compute study discuss emphasize algorithm convergence strongly focus paper widely bold letter convex strongly subgradient hold instead see wish loss close choose I subgradient individual subgradient error oracle bounds note without much generality size take strong function substitute subgradient assume optimize diameter begin consider individual iterate attempt constant tt convexity extract sum rearrange get convexity trick pick instead plug back iterate imply dividing repeatedly sum remain iterate analyze yield simplify also individual constant explicit convex c begin time instead substitute substitute simplify kf verify eq proof iterate inequality repeatedly sum remain term side use upper bounding plug simplify rate average rather individual iterate rate mainly since iterate first examine iterate assume optimization show tight flexible hold identical thm tf tf get tt obtain bound estimate theorem eq eq general
gradient iterate function sparse simplex constraint author true constraint problem project jointly consider regularize minimization rely low cardinality constraint easy plain letter represent scalar resp vector estimate p c dimension apparent subset without sort onto simplex onto large operator keep rest operation onto simplex euclidean projection onto isometry property rip k k rip assumption project invariant stepsize obtain via similarly rip sub entry ok case approximation guarantee statement task support support unique em w I definition threshold small introduce meet em c em delta delta obvious greedy project remarkably selector complexity sort w unfortunately obvious select grow mean adjust lambda selector overall symbol break tie hermitian semi almost construction call nuclear singular leverage algorithmic affine tractable amenable provably fail constraint relaxation obtain solution normalization meet constraint ingredient rip rigorous analysis unitary f apply reduction similarly eq ability rank jointly generate add follow snr measurement noiseless rank value freedom hence recover noiseless vary know experience estimate convex parameter rank normalize need exploit trace proximal sophisticated nesterov acceleration projection project stepsize acceleration algorithm try initialization x poorly noiseless theoretic high approach substantially convex solution solution neither advantage fewer perfect measurement convex highlight partial eigenvalue decomposition convex overall complicated factor dense solver approach iterative solver large discrepancy leave middle approach middle bottom right kernel relative gaussian additional nonzero tend small weight I x minimize result estimate negativity induce avoid overfitte interpretable one might sparsity context extend cardinality consider follow quadratic kernel priori find width depict individual mixture center around unfortunately program middle window interpretable cardinality enhance constrain order convex useful see profile pdf along position weight enforcing concentrate fully frequency illustrate pdf inaccurate mean portfolio return construct realistic adjust exist portfolio cost transaction naturally introduce cardinality mathematical portfolio selection seek optimization highlight synthetic solution rip goal refine art solver accommodate pursuit criterion solve use equality constraint matrix solver
advance fisher discriminant wireless take protocol extend propose capture relationship classification combine communication network interaction effort put tradeoff stage accuracy overfitte wireless surveillance environmental monitoring age application presence absence physical supervise result take length limited operational operational dominate communication wireless sensor sense popular operate wireless sensor part wireless sensor communication medium successive scenario measurement classification perform collect varie network supervise discriminant analysis algorithm train operational operational fundamentally character detection characterize operational classification accuracy training approximation network proportional throughput operational operational training two network rate reciprocal spend node likely consumption discuss operational back exhibit one regime quite usually encounter literature concern detection function sensor tend aspect model communication therein classification consider application mac focus case likelihood performance consider wireless measurement simplify wireless detection much reality algorithm estimate covariance training encode later adaptive back far less detailed network interference offer cover broad topology remainder organize sensor perspective operational accuracy balance act idea consist take measurement combine classification classification unseen focus fisher discriminant analysis covariance discriminant plug ratio covariance mean operational sensor classify performance specifically generalize unseen operational datum first specify statistical sensor spatial likelihood probability exposition one neighbor near sensor sensor graph element near neighbor graph encode remain element sensor highly accuracy rule known mahalanobis insufficient accurate simplifie decide wireless protocol represent node network set edge simultaneously ease presentation neighbor graph mechanism inactive transmission time unit neighbor inactive try back correspond incidence transition make measure refer throughput may consequently expect operational fraction dedicate operational throughout disjoint first node network simultaneously correspond interference empty simplify particular node state back notational active activity equal throughput operational consider prevent node correlate state use eq throughput choose shorter back position network throughput operational operational parameter spend interference communication node produce activity thus matter expression incomplete activity address include elaborate cost additional computation communication separately learn classifier operational setup associate compute overall accuracy generalization accuracy pattern set disjoint stationary throughput sample pattern sum sensor mahalanobis distance substitute expression four mahalanobis mahalanobis state state weight accuracy derive operational operational accuracy wireless sensor communication spend quantity comparison bayes optimal detector regime regime qualitatively plot operational function fix devoted training cc operational back rate mostly probable drop always cc sensor monotonically operational state negligible acquire state initially sensor approach examine function plot cc monotonically expect decrease local give phenomenon overfitte demonstrate decrease sensor number wireless network effect sensor change initial increase overfitte sensor fix fig examine accuracy comparison devote represent correspond range different part close corner plot extreme maximized increase contribute behavior turn three correlate sensor similar gd gd distance correlation independent fig
innovation whereas datum bias innovation extreme situation parameter decrease shown summarize iii u accept fail innovation iii average reason indistinguishable inaccurate integration error example innovation iv limit iv illustrate respectively realization solution linearization compute thin simulation interval period series I space innovation estimator u compute figure show histogram limit h innovation innovation negligible thin estimator provide biased innovation unable illustrate usefulness order innovation estimator estimator happen one exact innovation insufficient accept step acceptable first accept estimator accept ten estimator well r moment noise simple efficient moment covariance journal molecular nonlinear world scientific simplify differential http b http http continuous volatility market scheme stochastic equation additive bit local linearization comparative innovation diffusion series equation model equation level estimation differential overview review model finance prediction measurement mathematical processing third linearization filter application ed conference innovation j continuous st conference decision usa bold gaussian signal inf comparative likelihood nonlinear dynamical maximum versus kalman expansion graphical eeg model compute interval period interval length period interval compute c period period time length example period interval period sampling theorem conclusion theorem theorem exercise theorem lemma theorem la mail innovation method introduce two filter exact filter increase decrease conventional one enhance intend recurrent situation stochastic identify distant describe subject intensive simple joint unknown form diffusion contaminate noise observe extra arise typically reformulate continuous sde term discrete observation analytical simulate approximation decade see et deal noisy maximize function associate underlie state class inexact approximate like local linearization kalman et filter discretization mean numerical et innovation estimator way actual financial molecular et al et et al feature innovation exact approximate alternative innovation orient bias innovation minimum finite decrease increase normal bias decrease mention innovation local linearization algorithm simulation simulation innovation estimator enhance test reduce distant basic estimator innovation convergent linearization filter present section illustrate various differential differentiable compact q gaussian time increase sequence diffusion kt sde time specifically innovation give observation innovation k denote recursively filter ss first specify prediction normality conditional equation formula innovation approximation early paper like linearization filter extend filter hand filter discretization equation error innovation limitation estimator nevertheless innovation negligible actual variety financial molecular mention cox approximate innovation al similar comparative denote differentiable derivative n n discretization equation initial weak weak definite follow solution give instant z solution approximate innovation naturally continuous state ss innovation filter discretization kind converge approximate innovation euler linearization order might derive reduce conventional filter situation innovation innovation consider linearization scheme filter measure give consecutive state go include approximate weak innovation state decrease therefore convergence innovation innovation innovation innovation estimator q respect realization h identity h k k deal particular prediction filter k k finite imply imply assertion hand depend realization new expectation measure realization mh conclude assertion innovation zero weak criterion clear innovation exact filter decrease assertion error mh deal average estimator state partition innovation innovation give probability mh e underlie probability generating realization assertion conventional innovation section property theorems innovation restriction partition specific application innovation variety reduce close innovation account specification inference automatically concern simulation section approximate innovation relation minimum define generating remark depend index good predictor average loss regularity identifiability stationary ergodic diffusion definite definite ensure ss derivative let innovation mh theorem identity h r e k k er deal particular prediction state positive whereas account k moment addition imply finally together increase innovation decrease go zero exact innovation approximate theorem go innovation estimator order estimator neither restriction partition discretization end previous definition apply define observation twice differentiable transform high definition innovation far depend innovation deal particular denote reduce estimation innovation approximate innovation estimator estimator theorem concern property approximate converge innovation estimator viewpoint linearization reason prediction formula additive noise adequate well performance innovation estimator filter satisfy nz k h recursive computation prediction give prediction variance filter gain go innovation go zero innovation average reduce innovation innovation emphasize whereas evaluate conventional innovation estimator linear equation additive achieve prescribed relative tolerance useful performance four innovation estimator compute conventional order innovation histogram time period distinct observation moment filter formula exact state additive noise observation conditional filter van pool addition van pool observation value multiplicative two innovation four state previously table give example realization equation linearization equation compute thin precise time take state model compare sampling period exact conventional innovation
time trading share day news per record company record word yield start rough aggregate flow news topic individual stock order network flow news apply impact trading order iii peak trading explain news decompose news piece lda global corpus word document simple sparsity multinomial learn easier already remove stop news record set stock analyze record stock significantly fast volume appear indicator function day present time evolution news volume four topic topic support lda topic general every word specific though mixture word draw estimate topic highlight follow negative click top simply reflect repeat phrase repeat phrase corpu difficult would huge topic employ focus top eliminate phrase day symbol number short eliminate topic describe e stock market news suppose topic leave relative trading stock regression trading volume trade median trading move boundary regularize provide presence trading activity contribute determination ten fold perform divide entire ten measure ten cross stability estimate focus peak percentile day day pay attention study period divide overall month window time window peak day fraction explain attention peak day fraction volume article useful sensible correctly fig compare trading trading term yahoo discrepancy quantify quality explanatory peak day define define success trading peak day subtract day among peak volume successfully sense fraction peak explain peak day explain peak day yahoo peak day explain peak day exercise swap extract topic yahoo news yahoo explain trading fig modify decrease considerably substantial explanatory find confirm regression pruning record e g always report stock explain trade rarely stock index news news record stock stock find recall logic topic visualize similarity construct topic link broad list tend emphasis report important piece information influence financial market million news thompson trading landscape news affect automatically simple trading news piece piece trade able impact difficult way visualize whole landscape stock utilize network visualization technique topic careful reading news include confirm piece stock finding power news stock trading activity provide insight news market stock large volume trade explain news trading news novel provide reason success require effect sophisticated probably news source specifically news collect confirm tweet compare extraction activity financial market summarize major external influence financial market news trading activity explain pricing extended universe topic acknowledgment author grateful helpful discussion partially support student service organization support aid scientific trading red plot aggregate volume trading volume volume index news record company latent allocation different topic implement lasso step impact topic peak news methodology text lda associate volume production black trading volume stock dot day select method period line actual trading yahoo bp number yahoo bp comparison actual trading volume topic explain yahoo trading b yahoo try explain trading volume regression exactly peak explain news stock news record period blue circle restrict distribution manual read company course red triangle quantify degree associate quantify word six six topic link topic quantify company top frequent quantify topic proportional explain topic link topic sale drug panel business national budget summarize frequent support financial production percent price percent car car company ht create manual record extent column show stock analyze stock classification stock sale top business accounting trading trading sec recall stock energy bp medical east business plan environmental education business east cm cm management technology economics institute finance university school information school economic mail yahoo co abstract mutual relationship flow activity financial key regard understanding quantify news social financial economic affect trading pricing organize stock seek issue news provide thompson trade stock stock landscape stock price automatically summarize regression activity news decompose modeling use quantify news trading represent landscape stock representative topic able extract piece information stock namely trading explain news trading restrict time news relevant information financial price instantaneous change reflect news news type environmental social financial economic expectation flow project translate demand price imbalance demand towards fundamental present flow view market without feedback occur solely paradigm relate price news relate frequency failed recognize price volume flow concept past present create feedback loop dynamic issue understand news really incorporate price priori nature
claim na quite model practice compare time time slow greedy procedure slow general submodular quite slow perform give feature nb model accuracy result svm nb significantly outperform nb correspondingly case also selection submodular partition modular vs cost associate feature modular cost objective submodular bad approximately strongly evident provide justification heuristic make procedure selection submodular feature procedure moreover various combinatorial constraint may great yet concave extension easy rest submodular material support science foundation google microsoft award electrical university usa electrical usa section definition proposition function submodular algorithm guarantee monotonically reduce theoretically efficiently difference hardness difference submodular function bound minima show submodular function show validity area ever grow problem show submodular function submodular minimization np complete constant submodular submodular modular function wherein element big economic submodular define machine function problem possible model alternatively maximize mutual fix independent submodular submodular subject location property discount sensor may place sensor location g use mutual want suggest initial optimize produce submodular structure generative feature find submodular moreover example particular choose remain may group strategy fast transform appropriate model cost source feature ix hx submodular typical probabilistic desirable minimize probable factor tree bundle form solve programming hold approximate inference define vx width inference vision also submodular cut minimized efficiently address difference decomposition compute minimize submodular way probabilistic class rich potential stand possibly I say submodular contain potentially rich function particularly vision also minimize two inspire address equation minimize modular express submodular function minima question first describe tight modular submodular include describe submodular constructive procedure construction np certain class set find decomposition two monotonically converge note algorithm order moreover give exist guarantee heuristic however additive polynomial computable upper optima mutual near taylor series particular taylor series function q strict convexity tight polytope sub point ny fs fs f parameterized observe parameterize submodular characterize function submodular modular tight set modular submodular submodular review minimize henceforth ds suitable new combinatorial convex provide submodular express define submodular vx vx submodular complexity decomposition low difference gain low tb g every converge minima monotonically every iteration minima checking permutation describe iteration minimizer since modular improvement permutation minima fs fs minimization large reach costly algorithm practice ds monotonically objective minima briefly iteratively minimize modular every submodular tx step submodular maximization complete admit modular bound run procedure mean alternatively modular upper first maximize expression modular replace modular bound tx tx xt every modular minimum non two variant moreover decrease every converge monotonically increase permutation adjacent modular bound early observe different fs modular bound correspondingly choice g von permutation modular correspondingly modular local iteration experimentally quality gx might great h achieve permutation order gain former progress much heuristic indeed address question selection minimize subject minimize submodular cardinality np hard approximate since seem unclear constraint np hard admit constant correspondingly cardinality easily introduce negative function constraint utilize minimum span tree min polynomial non difference submodular function modular directly minimize negative modular subject constraint analyze approximation assume normalize correspondingly section theoretically hardness think justification case inspire algorithm procedure size application list general hence np hard surprising large randomized perform poorly easy np positive computable opt p give positive choose unknown define observe n notice guarantee solve contradiction exist hardness function optimal define g lemma issue unbalanced let clearly use unbalanced query unbalanced query balance regardless actual thus never achieve approximation say easily randomize polynomial factor theorem factor global optimum hardness hold even submodular monotone monotone monotone submodular function eq simple decomposition decompose modular totally modular add decreasing prove monotone decompose monotone decrease modular totally repeatedly low obtain guarantee correspondingly open minimizer ds function generalize local max cut trivially generalize cut polynomial monotone decrease guarantee optima local optima
cover run intersection edge clique constraints lagrangian lagrange derive lagrangian clique therefore dual use cardinality evaluate due sort edge htb entropy size output clique iteration initialize obtain solve c c dual optimization define iteration evaluate respectively correspond primal allow constrained step proportional dual moreover visit primal approximately feasible go feasibility illustrate violate violate observe reduce clique clique e element feasible I convex relaxation equivalent solve dual optimal ic hc clique define span polytope liu trees solution htb infeasible sequence clique feasible initialize sort fractional infeasible add adjacency elimination maximal perfect elimination connect graph average solution edge select edge clique clique run algorithm synthetic distribution several decomposable uniform identity value value normalize definite decomposable decomposable ensure relationship adjacency entropy covariance factor row column therefore multivariate decomposable graphical independent graph structure star decomposable matrix correlation algorithm decomposable generate ten represent multiply dual represent replace negative constraint key obtain tight relaxation primal represent fractional fourth primal represent project optimal fractional corresponding primal dual greedy add mutual empirically correlation simple greedy htb ccc dual htb primal primal learn propose produce order combinatorial liu liu variations pac pac pac jt traffic alarm performance alarm approximate decomposable graph traffic dataset traffic month california location discretize bin learn approximate decomposable graph entropy generate likelihood structure learn traffic dataset high likelihood figure illustrate gain early relaxation decomposable graph empirically currently explore design heuristic speed acknowledge european project dataset thank project true minus plus pt pt sup paris france l sup paris france theorem proposition undirected likelihood np consider involve search problem variable synthetic benchmark set tool collection define probability many domain vision language bioinformatic naturally problem situation might rich discovery relationship tractable often variable simple ensure graph however rich work simply possibility fit tractable number learn design approximate high indeed tractable variational one convex design fine trade apart learn direct constraint type definition independence maximize likelihood factorize context learn latter hard lead various algorithm base search independence test heuristic least incremental al maintain graph kl et convex decomposable give kl iterative cut et base perform independence derive decomposable programming use weak conditional tree use information criterion recursively small paper decomposable datum combinatorial section loop complexity round state method synthetic dataset gain ccc decomposable clique dot set maximal represent decomposable cast maximum combinatorial assume clique clique differentiable potential within clique say maximal clique I connect clique vertex clique contain decomposable time merely problem sufficient decomposable convex zero vertex clique contain number clique contain clique loop blue loop acyclic encode combinatorial minimize eq represent imply clique blue decomposable red clique decomposable graph red decomposable color provide convex relaxation combinatorial number clique intersection property already clique I e clique least select incidence hull thus new constraint crucial maximize linear polytope maximize weight span order select weight form long add restriction see polytope define polytope clique otherwise ideally hypergraph similar subgraph hypergraph hull incidence acyclic may applicable notion number contain trivially select graph forest hypergraph sub
goal motivated help goal exploration allow agent arise easily relate relevant capacity subsequent regard universal define utility find fully specify transition reward need option heuristic agent conversely default mode equip quickly target position middle regard automatically human identify even environment present continuous utility priori specification reward typically try goal choose wrong nothing would death natural balance discretized case learn initially instead interact environment gradient state agent balance wide condition would study priori contribution vector initially state approximately monte underlying imply estimate triplet interact iterated forecasting structure definition illustrate know finite domain main technical portion formal high integral model rather matter go beyond description reinforcement natural coincide way incorporate action loop action high state guide part discover explore without although formal definition motivate example informally logarithm effective number induce action measure extent influence general show left location apart agent want location r car state usually rl factor explicitly salient flat elementary action move indicate direction chance movement result movement agent pick correct car pick drop meet environment place state transition effective step action elementary state slice thing apparent location move want location corner car away situation episode agent stand heuristic agent task specify goal compute alone essentially freedom create salient environment imagine high sensible blind quickly salient state environment remainder typical design establish subset black section define give dynamic system interact space simplicity give interpret natural realize hardware denote go make system interaction action induce notational generality form exhaustive enumeration step every consider evolve step action regard abstraction allow treat simplify drop index instead irrespective loop model variable shannon channel respect possible length entropy conditional strictly speak entropy differential entropy density e entropy finite limited resolution n ip eqs proceed definition look simplify exposition integration replace summation environment label fully describe right transition entry want calculate aa ad da dd x logarithm x aa da dd natural probability among section mi take maximize various illustration mi mi em column illustrate full action maximal action action horizon make stay matter whereas stay important reason horizon increase step large action agent contribute mi zero dominate indistinguishable action action outcome outcome correctly let agent transition system system zero state action distinct outcome briefly discuss mutual channel comment consider controller action main one maximize achieve action fail resolve property distribution obvious action mutual action different state every outcome state mutual still action whereas distinct action highlight degree attain capacity transition action algorithm em denote th achieve since domain notation start obtain normalization ensure x give carry stop look regard assume simple gaussian consider approximated depend state compute input calculate transition fully np I ni p k k k z achieving associate n n n requirement computational neither readily triplet result infer proceeding iteratively predict accomplished regression gps mathematically elegant offer considerable gps exactly furthermore hence certain polynomial instead gps analytically close bayesian gps automate adjust predict perform gps action desire output difference variable independent gps univariate gps deal subset approximation subset greedy aim incur incomplete cholesky decomposition avoid project px th equation mean note set hyperparameter independently selection transition see multiple gps predict individual train corresponding turn sequence integrate unfortunately solve closed approximation numerically integral alternatively laplace prediction iteratively step instead step produce repeat horizon step np slightly different every produce ignore uncertainty precede prediction indicate intuitively appeal salient study continuous reinforcement latter learn need optimization use result behavior actually external subtle detail effect heuristic fundamentally oppose informed need horizon desire extent lead external act heuristic guide agent incorporate intuitively state lead high quantity far reduce consider start know nothing environment combine exploration choose batch gets predict accurate reinforcement optimize physical system describe usually control goal optimize determined drive reach aim drive operate characteristic system alone turn close enforce external intrinsic exactly wide action stay achieve stay certain problem one end depict force plane equation force equilibrium position enough back create difficult nonlinear control space angular velocity force depict model simplify move surface keep continue dynamic appendix roll angle roll angular inherently horizontal turn neutral position deal u propose bar depict around bar force nonlinear control informally degree body system balance dynamically stable task first link bar significantly difficult first link reach close unstable state system appendix equilibrium link deal continuous alone sufficient choose therefore primitive control appendix balance action actually balance choose assume state transition loop current possible describe use deterministic corresponding meaningful usually increase ahead single one action exhaustive enumeration agent number computation perform informally number step grow large cost prohibitive action action hold amount extend horizon go locally inform sufficient treat scenario show start follow action noise demonstrate alone happen straight without future current system controller determine optimal accumulate minimize behavior remarkable unlike value tie operate characteristic alone goal keep go side angle great happen stop matter step behavior run trial simulation show keep stable wide condition dot successfully keep second indicate impossible velocity pointing column angle outli keep stable ht successfully initial vertical bar state long avoid deeply thus demonstrate equilibrium makes actually translate bottom increase reach position vertical bar balance would sufficient phase control reveal high action cm discuss potential applicability space explore case state probability interact challenging space make avoid sample agent agent learner x step action selector action value state would evaluate approximately simplification prediction predict adjust face exploratory assign uncertainty assign uncertainty depend gps implement observe accordingly performance step loop initialize loop observe step use prediction provide adjust nc hyperparameter action easy compute episode start action selector compute learner update train ard automatic unlike programming find agent optimize want goal transition former gps region flat whole predict high thus agent create therefore discretization transition prediction grid attractive reach explore figure base exploration various grid division cell every cost function episode act steady act optimally minimization graph thing fine optimally grid reach optimal episode grid need episode episode episode behavior somewhat bad versus consider optimize performance explicitly agent fast explore model transition visit compare visit treat plot accurately transition go gp confident mean exploratory action know agent behave plot happen first visit cell much even goal desire behavior carry behaviour show hand create fail imagine instance state reader task attain nevertheless task arbitrary nothing specify balancing seem unbiased course outcome naturally arise arbitrary select actual dynamic approximate theoretic maximization place minimal guess agent place goal seem consider thing survival scenario stay move possible related free particularly concern system property discrete display somewhat scenario optima visible dominate state maximization task require picture open question system property seem continuous environment degree globally interact environment otherwise majority pick dropping degree freedom increase box due option horizon algorithm suggest
inductive value define predictor parameter score fold current proper calibration rank example fold separate eq fold give fold popular set encode encode output odd program program default help tree default value insufficient size fold region validity cross interesting package partially foundation discuss approach fisher valid statistic denote chi cross predictor define rather analogue obvious computed fold may theory testing generate testing kind naive tb mm cross predictor ac uk ac uk inductive validity empirically method unconditional confidence inductive inductive predictor typically produce compare predictor validation explores mainly consist ask element number ask predict classification always example unknown label label object label check corresponding level inductive split part value check calibration prediction proper training produce calibration calibrate calibration obtain inductive predictor much use proper develop use calibration prediction fold inductive set combine way combine fisher produce heavily main lead study involve datum predictor predictor measurable sequence stand list training non part measurable interested example choice rule label allow method appendix logit label q letter respectively proposition obvious proposition predictor exchangeability note
minute minute minute minute always dense shrinkage small need smooth preserve ht c efficacy datum one gene denote drug variance use select high variance compute table r dim cpu sp e optimal u kkt hold condition subproblem respect update e second summing obtain eq large equality combine complete algorithm start converge lemma get monotonically non thus subsequence get note imply satisfy kkt unique thus existence limit use identity hence convergence gray condition unobserve aim optimization direction alternate multiplier exploit take thus solve problem numerical synthetic expression datum two five time algorithm latent direction theoretical convergence compute variable graphical selection covariance dimensional joint translate gaussian capture system variable dimensional rapid advance high throughput microarray imaging regime develop penalization receive neighborhood regression scheme aggregation intersection propose jointly neighborhood regression selector constrain minimization call establish decompose graphical establish frobenius rate spectral absolute normal normal norm normal usually define convention graphical set realistic scenario suppose jointly multivariate concentration sparse marginal observe matrix base hold naturally convex relaxation introduce regularize eq theory concern recovery problem determinant cg inefficient special note rewrite dual max dual lasso programming problem interior requirement high thus impractical large problem develop block descent project variant nesterov recently show propose solve graphical selection multiplier solve problem problem proximal apply method block alternate subproblem take advantage problem different type include second completeness cg significantly cpu times paper organize proximal problem alternate direction multiplier gradient alternate method draw conclusion rewrite reduce function drop implicitly function involve operator split monotone operator study due success structure convex arise compressed machine e alternate easy proximal note proximal proximal shrinkage mapping discussion follow natural solve lagrangian multipli subproblem correspond proximal mapping convergence three block error require strong consensus problem respectively describe lagrange multipli subproblem solve subproblem mapping partitioning block second subproblem multipli thus eq substitute equality get admm solve block consensus discuss literature g another alternate direction type reduce block result group new admm multipli proximal mapping subproblem couple quadratic overcome difficulty second proximal note subproblem q proximal solve follow quadratic part proximal mapping r gs k incorporate alternate vary technique low optimization alternate direction method seek completeness include global convergence group implement grouping one find alternative yield practical absolute penalty basically modify q present data method admm matlab intel ghz gb ram admm set dynamically change multiply iteration solve numerical step proximal choose test use nonzero p kp matrix function inexact admm run admm objective run achieve admm iteration objective cc c admm e e e e e admm
extension centrality centrality subsection quantify pair therefore nod probability path bag short path visit indeed indicator equal path least otherwise path find interested give start end assume reach posteriori probability posteriori pair q node network derive formula node ik k z denominator ij j ik jk appear ji ik ij ij ik ik path q substituting element simply ensure normalize result quantity put form transpose numerator read assume normalize simply matrix path normalize numerator derivation form group see quantity neighbourhood within high node belong neighbor call cluster assumption derive numerator belong element element transpose observe express numerator want classify class find algorithm unlabele weight contain contain usually inverse binary indicator index mutually exclusive c hadamard product max supervise refer multiple introduce goal classify unlabeled medium partially graph medium compute network first describe subsection experimental third set class set classifier nine set label average laplacian rl regularize normalised walk discriminative walks walk comparative report follow nine fig set result significance labeling one side determine read indicate significantly well column frequency summarize equivalent labeling rating eight mean classification seven rating consider set total always competitive range set labeling come kernel achieve well suggest labeling rate notice largely notice walk labeling label score percentage eight position since nr drop significantly label surprising remain cm total take table five set seven ng variant ng ng ng ng label class since ng ten ng ng class ng class set record run run report competitive subsection two rl less class last slow rl competitive augmentation increase strongly come contrary system equation compute graph pre cross virtual bag novel describe sum posteriori draw accord sample boltzmann gradually probability path walk set outperform label competitive time include interesting provide feature feature assess experimentally label protein protein could protein path framework thank classification path bag tackle graph nod topology label node assign boltzmann distribution possible path pick path posteriori node path receive appear path connect network invert number group constrain node classify show outperform test state benefit mining centrality kernel semi supervise automatically predefine traditional pattern amount difficult effort reduce supervised mixture learn actually semi label accuracy semi classification receive grow recent year describe input numerous link categorization protein facebook internet dna readily available gene database large unlabele supervise infer effort label work capture path assume weight direct graph cost arc consider boltzmann cost low bag pick sampling start closed inversion quantify extent naturally posteriori intermediate lie n receive path connect define class unlabele show group within framework bag framework compare performance base technique classifier path paper mainly path bag path work semi bag us work discussion finally far background notation classification bag finally introduce direct vertex class class class since membership class otherwise zero node unlabele denote transition cost contain element transition proportional immediate apart building section instead virtual bag connect include graph path path low path also walk transition loop allow exploration pick detail derive relative path graph probability short large probability pick walk towards short path short one bag probability thank author show computed product entry ij define q bag behind bag path appear word node allow paths derivation path model show interpret reward agent along path path expect sub probability base subject intensive wide approach spectral method regularization framework svm alignment regularize rl generalize show large similarity alignment kernel laplacian previous normalize laplacian seem sensitive prior rl alignment sum similarity previous method start step comparative harmonic closely framework rl structural interpretation electrical potential rely walk markov precisely stationary distribution label unlabele interest discriminative walk walk see
take derivative product variable parameterize discrete imply ica show rescale respective distribution cone affine distribute coordinate iid invertible imply iid measurable linearity independent imply result elsewhere independence ica proportional nonnegative join product everything th power rule involve fall positive strictly power interior determinant jacobian row reduce determinant nonzero support gamma compute analog density vector give joint distribution part early separate permutation permutation sign ica sample gaussian introduce ambiguity respect rotation ambiguity ball symmetric different problem solve efficiently ica show satisfy efficiently obtain many question remain come particular iid sample v simplex tu algorithm rd within distance equally near sample pick canonical equally different equally likely algorithm output true process run randomness output optimize analysis simplex v mean pi pi invoke algorithm tu theorems orientation preserve nr discussion whitening mod ica simplex et distribution format choice algorithm mistake thing isotropic subroutine engineering microsoft research science university microsoft computer science engineering conjecture acknowledgment problem give simplex simplex arithmetic polynomial answer bound fourth moment show direct connection problem randomize base representation dimensional uniformly need approximately seem intuitively clear reconstruct require theoretic consideration turn approximate theoretic running know full fact body know affine study intersection half suppose space advantage proper mean output half perhaps approach mind volume contain simplex volume estimator applie situation hard hardness directly find idea ica ica recover give requirement full independence rigorous special generalization rigorous ask convex give simple rigorous popular algorithm use fourth briefly fourth variable input moment sample distribution minima function find maxima minima ica historical remark projection pursuit mid ica problem directly among kind dependency among appear learn intersection intersection space open sufficiently g intersection half pac access membership random effort simple therein towards goal intersection fall seem sample uniform intersection one work learn intersection half space reconstruct direction moment order require recent closely apply variable latent gaussian latent local optima closely showing algorithmic technique arithmetic division root distance access randomness vector simplex run polynomial mention early idea ica moment fourth version ica moment cube simplex useful fourth moment understand moment maxima simplex get ica dependence discussion take succeed error chernoff type simplex distance additive simplex simplex distance within thus algorithm ica substantial learn ica problem ica simplex independent representation ball cone measure representation problem affine ball ica estimation obvious ica implementation ica result one put position recover simplex simplex map full invertible affine translation simplex well efficiently choice simplex hull canonical call begin invertible affine map isotropic bring sample error final result rotation leave denote rotation keep fix recover rotation rotation simplex characterization origin orthogonal normalize result consider third along orthogonal descent sphere find highly maxima isotropic sample come simplex leave closeness depend accuracy mean simplex close total subroutine find subroutine succeed probability definition total variation subroutine succeed simplify case isotropic transformation plane kind need handle successful algorithm exact rd moment arithmetic gradient rd moment simplex rd real algorithm canonical isotropic access perturbation canonical nearly rd precision perturbation canonical limit third moment sec individual vertex position subroutine sec characterize maxima third learning ball ica hull affine hyperplane convenient work simplex canonical simplex origin euclidean ball clear isotropic distance express imply two set nd tv particular trial union p denote x distribution distribute distribute cone measure cone moreover variation distribution th moment derive formula integration newton identity direct computation divide standard simplex moment copy rotation invariant approximately simplex algorithm coordinate simplex simplex algorithm note introduction algorithm also idea rule maxima use first think fast state get solve coordinate simplex simplex rotation thus independently coordinate work u tu tu obvious way moment unbiased give bit coordinate extremely fast equation subroutine version convergence fast show growth overall subroutine copy notational iteration copy rotation leave output approximation pick normalize divide subroutine rr simplex pick algorithm likely vertex equally randomness multiple sample vertex different likelihood outline iteration except inequality quite loose gradient show handle coordinate simplex analysis clearly randomly coordinate value similar argument question happen coordinate happen angle volume happen vertex drop exactly choose evaluation least end chernoff omit thus true least starting coordinate great loss get mind let get last coordinate continue factor iteration r c c omit easy detail succeed overall claim lemma subroutine algorithm use map map isotropic simplex simplex orthonormal basis everything else qr definition let except set simplex distance vertex origin affine map isotropic tx cholesky decomposition rt simplex let invoke subroutine get hyperplane length simplex simplify pick isotropic core first put case carry practical prefer select loop vertex bound choice gradient variation equally likely least simplex polynomial input simplex invertible affine apply transformation simplex output equally transform remain ambiguity rotation remove end total affine transformation isotropic easy application chernoff choice moment imply st far ambiguity precisely moment subroutine fix point like iteration successive subroutine vertex euclidean
flow fix transition enter leave equilibrium detailed balance instead directly sum allow transition equation flip form require pixel intensity case integration momentum corruption sampler image return see technique present rapid dynamic style explore slowly otherwise would due momentum rejection cause back walk slow momentum accomplish maintain exchange probability opposite reduce exchange direction momentum normalization constant partition flip f momentum momentum x fp hamiltonian hamiltonian step stepsize property preserve perform r n draw uniform additionally depend
act like metric notably learn position rather implicit position position single position strong learn explicit metric capture evaluate differ distance image database type etc contain representation position form mahalanobis method detail paper plan future use diverse new mit set grateful part grant nsf nf mind w nf ap finding author reflect view ex minus ex plus ex minus ex ex minus ex minus ex plus minus ex ex plus minus ex make plausible label date base single mahalanobi handle accuracy metric single implicitly metric accomplish random forest base incorporate pairwise representation implement test type consistently fast bad euclidean convenient underlying application retrieval face verification present angle learn random forest emphasis capability space metric per vary space achieve function limitation illustrative elaborate dominate mahalanobis metric representative brief constraint sampling collect supervise group large find appropriate satisfy positively link link point separate expensive component learn mahalanobis explore minimize constraint quadratic develop global learn although single efficient capture mahalanobis point base need store divide subset generate possibility require metric specificity transform binary forest incorporate efficient pair encode absolute demonstrate importance incorporate underlie representation several reason forest yes vote forest vote correspond positively constrain constrain number yes vote relative forest scale powerful handle specific multi independent input fast state multi third forest inherently follow thorough forest inspire several advance learn learning learn adopt remove instance geometric consider interpretation throughout metric vary adapt instance multi pairwise link constraint set I ij specific minimize flexibility f j input function train ij mapping function implicitly mahalanobis term metric j denote mahalanobis encode general position concatenation represent take enforce symmetry metric contain two map region adapt heterogeneous reason metric incorporate absolute position incorporate add alternate example raise order yield arbitrary meaningful usefulness largely include particularly difficult significantly l dataset dim dim diabetes image handwritten digits feature handwritten interested reader forest tree leaf similar forest essentially q output dimensionality compare inherently nonlinear scalable flexible mahalanobis incorporation describe adapt different word select split value sub subsequent refine emphasis localize although negative sometimes construct appear problematic necessity vary feature triangle triangle triangle extensive uci image take position uci neighbor task mahalanobis position dependent former latter rely publish author mahalanobi follow finding section overall ten range table comparable art specific nine category benchmark metric theoretic near feature include position well position result heuristic metric simply obtain medium uci see positive select generally constraint tree test use neighbor task vary vary able gain global set identify neighbor enable linear perform globally nonlinear take linearity linearity become global strong implication specific nn highly retrieval figure show plot uci without position set act difficult diabetes dimensionality forest low ten value absolute position mean metric must address learn increase forest past certain yield possibly
group extensive usefulness drawback notion recognize like level level set parameterize content great probability proportion parametrization classify commonly exp right impossible univariate phenomenon group identifiable set three several coverage orient goal cluster graphic show cluster description methodology set arise independently notice reach maximum fairly maximizer provide restrict sensible detail make empirical level paper paper level note plug excess issue aforementione reference corresponding estimator surely induce modal admit illustrate thorough agree constitute contribute cluster introduce precise population rely answer process mutually positive content population ideal clustering reflect high density begin formalize illustrative exp exp node node right node node first split cm exp node node node partition cluster tree one set methodology identify depict correspond consist reach split branch two constitute partition part refer assign obtain divide assign either lead equivalent left branch branch core tree splitting cluster observe immediate observe precisely minima equivalent consist define minus attain solid circle involve constitute color sigma sigma exp pi exp grid major sample thick gray cs axis axis axis axis idea generalize proportion mixture distribution identity intuitive black line identifiable density function precise theory branch differential manifold critical differentiable reference book useful range represent goal indicate water flow extremely smooth call degenerate critical degeneracy number negative fairly neighborhood critical minus point figure minimum saddle maximum index white color blue function false grid domain sample title view major title width view sample title suggest unstable manifold explain next define minus satisfy integral minus vector descent curve properly speak trajectory water subject unstable critical analogously whose integral curve xt firstly manifold critical manifold unstable define population unstable gradient flow maxima water flow region peak describe look index unstable manifold manifold manifold rw w clusters nan definition univariate unstable manifold local maxima manifold unstable respective point minima moreover focus flow integral curve become respect xt notion domain mode follow ascent gradient flow cluster goal distinguish concept procedure however data procedure whole population immediately objective usually interest minimize natural due involve redundancy disjoint every twice account avoid redundancy measure cluster many add partition partition interpret penalization probability mass partition cycle thick node two show differ low probability cluster denote match depend less component match indicate match achieved match match note replace analogue call clustering explore differ correspond transfer distance minimal partition alternative clustering hausdorff population hausdorff ab two nan two clustering view hausdorff distance express every hausdorff regard distance set standard theory eq demand mean partition really close hausdorff distance check previous hausdorff clear picture wise hausdorff wise minima take copy distance compute add hausdorff instead solely obviously analogue measure lead clustering indicate understand procedure induce precise datum get increase formally mode stochastic clustering sensible plug base clustering naturally arise smooth estimator domain mode candidate kernel extent cluster dimension rule easy nevertheless development scale exp exp plot exp line completely discard step solid line pointwise kernel maxima function really clustering also seem easy clustering estimator surely theoretical consideration previous help compute clustering note know connect induced density use mode surely popular class shift algorithm introduce subsequently highlight engineer every location produce convergent local initial sequence notice multiple approximately ascent since make gradient shift algorithm recognize employ normalize accelerate convergence common point density replace allow produce moreover closed maximum correspond refer convergent j harmonic component definition partition normal mixture density show kernel h nk shift convergent proceed guarantee form explore expand direction nonparametric iterative explicitly denote shift restrict thus reduce per direction research exhaustive include variant mean shift oriented robustness consider shift mean shift previous median near neighbor concept would modify obtain proposal geometric median riemannian application respect shift algorithm third research paper deal subject procedure ideal ideal population notion interested cluster accurately make induced maxima appealing extend meaning point even resort differential mapping cover book hand case key role play subgradient generalize cluster measure distance clustering discuss section aim modal rely pilot density happen method shift surely identify shift locally adjustment suggests perhaps support project de well suggestion concern thought lemma theorem remark example group despite recognize aspect relatively surely classification quite difficult try paper aim nonparametric present branch activity note rigorous methodology author concern formal partitioning problem seem
restaurant restaurant section predict high hand movie recommendation member likely something thus prediction individual recommendation provide preference procedure recommendation likely opinion certain network decision user active node already make explicit statement preference target decide recommend item member influence power modify member social activity disagreement iteratively social influential recommendation individual recommendation take friend make recommendation group opinion future validate real com rating ideal collaborative recommendation setting threshold weight area movie restaurant department electrical university recommendation considerable focus collaborative filter people preference quality recommendation model group respectively individual improve cascade recommendation influence pattern disagreement group member introduce concept rate flexible essential social collaborative filtering study collaborative cf recommendation great success serve important demand amazon netflix recommendation book movie accord preference order recommendation mostly improve recommendation base assume user social people decision datum attack challenge recommendation system social social receive less attention dataset past network provide structure social improve recommendation recommendation traditionally system recommend item individual decision activitie user go restaurant recommendation desired recommendation nontrivial extension individual unlike influence preference individual recommendation way aggregate recommendation separately often mathematical past year pre exist social preference record datum linear threshold effect collaborative provide recommendation recommendation give recommendation reflect setting list user rate preference infer netflix click rate undirected node edge friend case user information organize influential recommendation group follow conclusion future recent paper recommendation matrix propose et network item node aggregate user estimate weight recommendation base network reduce limit target immediate friend consider friend recommendation rating friend rating identical good knowledge consider influence role decision effect result collaborative filtering addition recommendation merging g intersection aggregation particular recommend movie merge result recommendation yu group preference merge individual profile total virtual profile user recommendation opinion formation influence within collaborative system reality crucial people decision amazon choose read particular mind mention book book strongly recommend friend game theory scale rating state inactive respectively influence intuitively trust communication frequency lack neighbor user proceed neighbor must become active either state tendency adopt opinion associate g set rating process operate discrete inactive fraction influential intuition activate either equation user expectation easier opposite newly activate social decide subsequent potentially like activation note assume recommend short local view social social connect payoff c cm c playing game neighbor represent receive represent play neighbor payoff node w choose inactive solution rating active odd active equation threshold valuable minor modification influential user simulate social process binary show initially probability assign influential uniform influence average threshold fig even small portion network majority active newly activate node choose show fig social fast initial influence fig show realistic set node draw various activate half become eventually stay speed simulation result disagreement expect rating newly activate social network majority opinion attribute item recommend ignore inactive recommendation immediate distant friend supplement recommendation user group recommendation differ aggregate prediction merge preference profile couple opinion consideration
experiment potential speech corpus rnn gain source information core training contain select slight disadvantage many core test reduce speech preprocesse apply audio file bank emphasis coefficient compute coefficient hamming window interval create normalised mean rate sequence percentage record rate one record put quantity convert per lstm lstm hide input give counterpart momentum noise stop low network stop lowest weight range decode result lowest aware result recurrent network nonetheless prediction label oppose million performance almost target per much would hope alternatively jointly dataset hmm language corpora acoustic model speech corpora log bit epoch pass epoch rate calculate allow analyse dependency relationship speech recognition directly intermediate heat probability lattice input network binary vector encode character low short vertical common sequence th input bar graph indicate red prediction simultaneously predict correspond prediction output sum give heat sequence show sensitivity heat previous network dependency visible across dark horizontal heat correspond sensitivity space word jacobian conclusion acoustic linguistic speech speech experiment end look wide problem particularly example speech small input label translation due complex alignment acknowledgement suggestion institute research learn transformation sequence sequence speech recognition machine translation secondary prediction text speech name challenge sequential shrink rnn powerful prove capable rnn traditionally require sequence severe alignment many even introduce probabilistic system rnn principle input speech corpus ability crucial human intelligence everything reach everything interact world action important major apply output example transform signal ability identify despite apparent create speak language lexical rnns architecture general expressive conventional particular rnn well store early rnn recent capable state task recognition range perform character delay handwritten character rnn problem advance example rnn frame signal protein output length sequence purpose sequence output length advance prefer distribution output align ideally cover rnn layer long text speech sequence length jointly model similarity graph conditional rnns dependency mark potential close paradigm module often network transformation segmentation recognition define present speech conclude remark give belong input value would receive add omit previous standard add hidden layer feed layer rnns vector whole input rnns extent rnn compute hide tf tt iterating implement network input normalise simplify determine lattice possible correspond therefore similar k lattice output horizontal represent element black bottom node one start product pass red path calculation backward variable recursively initial condition define backward product lattice output sequence forward variable backward speech recognition forward image bottom map variable bottom lattice text sequence way log respect perform diffusion lattice u top q network
independent freedom indicate put everything ec coefficient arrive look cone side functional presence unknown analysis fmri subject receive neutral heat stimulus right rest repeat fmri model stimulus delay second heat fmri region stimulus stimulus delay shift taylor convolution stimulus stimulus regressor eq ingredient structure nature regressor shift plausible restriction specify coefficient illustrate result observation independent dependence iid fields course cone alternative angle intrinsic volume middle appropriately normalize course correlate add regressor neutral heat stimulus cubic scan leave effectively result whiten regressor nuisance calculate depend remarkably brain fix field isotropic volume term make fortunately estimator residual difference lattice take voxel inside course radius ready get p statistic large cone statistic almost identical cone note increase cone getting threshold detect heat stimulus statistic delay activation question powerful test range outside statistic statistic volume statistic cone increase indistinguishable twice f twice statistic volume intrinsic usual half surface area dimensional intrinsic simple intrinsic come formula concave corner lebesgue lebesgue ball boundary curvature volume hausdorff surface sphere intrinsic lebesgue gauss half zero part nsf dms pass field statistic fmri datum delay find euler ec test statistic result ec density formula class begin appear david work together intersection interest david smooth smooth field application last pass cancer author pass david main approach problem formula change preferred euler ec generalization ec volume formula ec densitie back david change concern maxima field field variety test statistic cone package fmri give use ec formula conclude subsection relate fmri purpose define gaussian unknown must mind generality testing unknown shall way likelihood ratio variance maximum cone case normalize span cone field mean embed onto span sided cone statistic field powerful random field outside cone side statistic acceptance concave dominate cone advantage admissible specific always powerful reason would denominator conservative strategy degree cone fmri shall likelihood ratio test pass principle point whole would standard reproducing argument function center statistic region thresholding suitably threshold choose threshold detect control outside something field degree freedom representation locally picture non arbitrary scalar linearly subspace actually compute field solve direct regression fit negative amongst fit fit alternatively separable warm location negative application geometric project could generic span event non negativity restriction satisfied achieve actually face surface region depend dimensionality q probability fit eq correspond depend large effectively generally ignore later ec interpretation limit approximation realization see inf field gray medium patch scalar patch freedom value curve define light field freedom term correlation field theory ec subscript ec paper devote evaluate ever wide field space simple next get ec build gaussian gaussian transformation domain field build discover take volume formula somewhat power ec seek radius rejection denote long omit far generality ec random rejection region ec field field union rejection union arrive ec show ec ec ec lead ec respectively part evaluate maximum generate intersection circular empty ec term intrinsic volume fail convex right ec represent dependence random field ec field ec sketch q face determine negative empty ec determine differentiable define ec differentiable though contain interior critical critical surely critical theoretic computation ec show ec actually sum subset ec field complicated replace decomposition argument complicate prefer complete counting result incorporate work ec elementary geometry remainder reflect origin fail content fail ec ec density field freedom u rejection red rejection region radius rejection cut ec density seek random lead expand appear sided f yield follow expression ec follow ec ec easier closely resemble tackle
segment parameter except hyperparameter vector ensure transition process finite crucially tie hdp limit tie via connection state encourage visit state elimination self generative similar process illustrate self transition define affect global hdp factorial factorial factorial rich duration greatly improve demonstrate outline factorial factorial hdp component eq factorial factorial factorial hdp replace chain semi evolve combination state factorial identifying bayesian uncertainty ambiguity demonstrate use factorial hdp home separation ill condition solve separation uncertainty motivate modeling duration two gibb begin hdp develop limit limit sampler hmm collapse hdp conjugacy hdp develop auxiliary augment recover conjugacy gibbs limit assignment trade integrate transition simplify duration however must sampler hdp resample pass discussion leverage side greatly accelerate perform inference construct duration follow iteratively appropriate respective easily reduce depend prior sample dirichlet diagonal tie resample require care see section introduce independent block distribution employ pass block state must duration message describe begins efficiently sample state conditioning initial draw label sequence limit hdp hmm construct hdp transition motivate fact infinite hdp encourage practically limit transition also represent greatly accelerate circumstance parameter control guarantee grow convenient approximation stick represent probability construct atom limit hdp greatly accelerate sequel long tie share introduce conjugacy hierarchical difficulty clean technique avoid priori sampling technique priori overall easily self link long conjugate resample necessarily easy conjugacy weak summarize generative component deterministic transition transition state extra transition proceed conjugacy general datum augmentation technique describe extend simplify us cycle produce focus namely depend extend row drop parameter convenience write portion generative state represent transition represent transition depict show auxiliary variable conjugacy iterate compare simplified consideration sampler compare sampler asymptotic unity augmentation auxiliary conjugate weak well numerical tp da insufficient marginalization estimating model mention direct assignment hdp hmm transition analytically observation parameter prior represent explicit sequence use simplicity additionally recover hdp conjugacy requirement correctness auxiliary finite immediately infinite dirichlet hdp hmm finite basic hdp hmm da sampling count portion predictive write exchangeability joint likelihood hdp hmm distinction new transition segment case hdp writing segment entire contiguous term segment may merge depend adjacent segment use first score correspond duration duration duration segment thorough affect predictive merged allow must deal conjugacy issue assignment sampler rather segment associate datum serve preserve conjugacy hdp via resample use I I simply self count self self super auxiliary variable way integrate entry auxiliary hdp hmm describe circumstance super may switch many obvious divide though super sampler bottleneck pass step asymptotic dramatically message similarly modify sampling element sequence block label adjacent hdp assignment limit well hdp assignment hdp assignment hdp weak synthetic hmms number hdp unable geometric demonstrate duration distribution geometric hdp hmm little factorial whole home device encode simple device advantageous side experiment matlab compare hdp hdp weak sampler sampler hdp hdp direct apply datum hmm limit direct weak sampler much execute great believe superior whenever choose priori geometric hdp direct assignment hdp hmm compare hdp weak sampler median dash percentile summarize hdp datum dimensional run hmm unable capture non markovian duration statistic state hdp equip poisson duration transition emission reduce hdp concentrate hdp hdp hdp hmm duration distribution geometric distribution one class negative denote analog gamma cover geometric place non well mode hdp hmm priors informative concentrated duration hdp hmm hdp hmm hdp application factorial unsupervise power signal device aggregate whole home consumption efficiency detailed power device significantly solve problem monitor application demonstrate duration speedup rich history variety specification factorial em distinct state mode device sampling model explore additional build factorial dataset frequency period top drawing device identify hour device least sequence rough level duration statistic device home hdp negative binomial power usually hdp accordingly specification duration parameter twice device device perform inference independently factorial hmm bias hmm present device separately signal identifiable jump term device reduce potential able state resample magnitude tp greatly accelerate true device consumption report emission mean run machine core detect second resample duration collect result summarize device fit device parameter incorporation duration hdp detailed comparison finally figure consumption select factorial factorial hdp c house em hmm hdp tp tp tp nonparametric modeling consumption mode user home level priori device provide strong providing duration hdp weak direct assignment sampler duration bayesian nonparametric allow learn markov demonstrate algorithm upon provide tool sequential simple hand free encode device prior mode use mode near specify hyperparameter uniformly parameter setting factorial table observation gaussian denote fix gaussian similarly latent success draw geometric factorial hdp hmm prior parameter encode duration prior extension elaborate hyperparameter experiment emphasize hdp device measure interest hierarchical hdp nonparametric hdp hmm strict undesirable wish encode hdp structure draw parametric set interpretable admit natural introduce explicit dirichlet hide also provide modular gibbs sampler hierarchical tool toolbox demonstrate utility inference synthetic semi markov set unsupervised aim distinguish single audio wish infer characteristic home individual device would knowledge device power sequential build flexible expressive encode appropriate prove datum real state priori infer data issue hmm hdp hmm infer hdp disadvantage markovian lead unnecessary rapid one avoid rapid switching hdp hmm introduce self improvement hdp hmm geometric duration structure duration induce markovian difficult construct sample find efficient important limitation improvement hdp hmm investigation explicit parametric idea hdp allow duration sampling hdp approach hmms difficulty result mix applicability model synthetic remainder duration message build bayesian brief hdp hmm assignment sampler hdp hmm improve demonstrate effectiveness hdp demonstrate quickly hmms unable necessarily require operation chain pass however duration length message express message often duration cause message increase message pass offset sampler experiment hdp treatment employ
recalling explicitly write calculate low gradient paper sgd cg task method gradient element md independence complex jj jj jj cholesky jj notice e mm jj equality jensen yield form optimal observe jj notice process exactly bind goal calculate distribution key prediction predictive gp coupling next calculate via need form fact separately obtain make eq evaluate md yielding calculate calculate j I calculate effect admit gp task sample share gps add variation inference gp hyper parameter model process mutually generative process perform variational previous hidden membership auxiliary induce effect nk j k j jj jj jj learn hyper marginal likelihood low complete data jk represent hidden make simplify derivation variational f q jk jj variational optimize variational em algorithm estimate derivative k variational rewrite thus set zero use obtain complete estimate variational find induce inside extract log term logarithm multivariate integral jk interest follow conditional eq form single write bind calculate derivative optimize base easy complexity key derive kk identity kk km therefore next calculate first calculate sequence operation calculated implementation stochastic coordinate perform coordinate g q index experimental effect white recognize gp combination gps effect newly input task kronecker implementation criterion standardize differ low use sample optimize uniformly task advance maximize mt sd sd single discard sample mt pp hyper parameter mt pp pp experiment kernel inducing obtain optimize likelihood scale equally spaced share whenever center iteration support control three also use gp finally hyperparameter specify initialize spaced randomly assign gradient small subset task time maximum entire experiment algorithm task artificial task interval function eq individual interval sample space draw realization prior function show top qualitative initial variable induce variable clear predictive showing show input recover indicate sufficient time plus propose intel core pc reconstruct profile developed case frequently among task realistic generate explicitly reconstruct profile commonly analyze denote action dirac delta four task previous multivariate normal real gaussian profile interval uniformly testing deal efficiently multi center center arbitrarily number center base significantly outperform method develop classify star star constitute densely sample star sparse effect task star shape inference approach equally dimensionality full h star normalize universal slide center shape variational mixed support evaluation demonstrate interesting online investigate induce feasible make partly nsf paper perform computer usa gp main solution avoid instead desirable group unknown individual problem idea gaussian handle datum validate outperform method outperform art sparse solution multi task many quite task aim predictive advantage successfully area medical diagnosis recommendation screening share parameter multi focus shared represent deviation share hyper portion difficulty number cubic improvement remain example case cardinality approach call induce variable sparse solution multi functional mixed effect variational efficiently hyper sparse sample approximation different point real datum validate approach show full sample gp first first perform organize mixed gp inference gp group section demonstrate use conclude summary mixed grouping model extend grouping give j aim take data effect perform mutually assumption kronecker function concatenation similarly see new extract sharing task lead inference scale though
random walk outcome coin valuable theorem claim coin give relate well use adaptive possible action also outcome first strategy bayesian employ markov problem application bioinformatics medical arm choose receive reward choose rich motivation identify bandit minimize number step regret choose reward arm exploitation exploration perform know address identify good arm bandit fix propose strategy focus reward show bind budget subject quality action address sample pure maximum expect minimizing need pac formulation introduce al et show total identify arm whose reward away arm correctness reward attempt address theoretic outcome exist minimize whose least pac appear theory decade field research decade introduce normally reward sequential variant application achieve arm special know strategy various relaxation independence normality know variance distribution address particular reward arm bernoulli coin adaptively rest chernoff trivial adaptive coin coin empirical coin bias coin yet strategy choose particular coin coin optimality employ tool number coin provably optimal coin probability heavy probability heavy necessarily identify coin heavy goal expect coin history outcome coin outcome coin coin whose strategy say main optimal employ adaptive heavy coin optimal infinite value fix least need general set probability bayesian exploit stage outcome coin p b optimality game game system cost start state cost system token token say pay state system terminate token denote token minimum cost say pure choice token entirely pure playing markov state game incur associated state reach state small play game state system unique pick token minimal straightforward infinite system system satisfy determine walk correctness coin straightforward ratio coin coin coin history coin log likelihood coin begin history likelihood identically zero outcome head outcome coin log coin coin great barrier since coin heavy boundary coin outcome system step equivalent system reach strategy associated dimensional log state real state state random oppose random uniform transition probability locally state coin heavy heavy decision conditioning log consider likelihood pure strategy strategy infinite choose child reach since show mixed label edge onto root leaf let child node adjacent pr xu pr maintain initialize root leaf choose play coin move system coin generate move yu pr yu yu bl xu bs gx use consider leaf leaf node node reach leaf leave incur r relate expect associate non leaf leaf node observe node non leaf leaf node pr xu let incur incur incur root node expect process mr xu g likelihood also coin coin markov random walk
sample exploit number edge induce sample b fundamentally state technique exploit sec accept rw challenge consecutive rw several rw stand technique rw problem issue estimator available sec simulation topology facebook require facebook state paper sample node l art exploit among node exploit edge sample repetition undirecte connect estimate calculate eq every neighbor depend distinguish random replacement collecting would trivial know alternatively rejection currently facebook impractical unless exploit proportional node sample presentation unweighte long collect refer technique approach prominent follow classic population split equal discard repetition discard valuable limit suited population specie biology node separate specie review page likelihood derive define number equation index across entire elegant extension consider node weight I give reduce outperform eq various depend special service broadly applicable ii service iii less prominent technique inefficient often layer widely interpret rough estimator take estimate device area fundamentally describe fig sample knowledge sampling node rarely study illustrate edge count every occurrence probability average facebook basic accept node extend rw graph density interpret choose form node assign weight divide term correct correction finally plug arbitrary possibly repetition replacement cross node consequently reality resemble except repetition lead derivation independently happen employ relatively additional cost list clearly consequently remain discard estimator accept arbitrary node explain category node make discard arbitrary simulation latter perform well especially skewed reason explicitly note discard vs consistently require correction edge rarely correction reason henceforth technique node www connect acyclic rw probability proportional demonstrate indeed arbitrarily say impractical fed directly rw sample poorly consecutive rw consist rw close rw increase rw reduction literature efficient l nn analogous rw take subsample fed sec find easily observe rather different exploit create problematic aggregate denominator avoid perform consistently henceforth yet sample main modify additional transformation consequently easy exclude lie within omit brevity implement note naturally strategy pair come walk result count node neighbor sample consequently number pair make contrast rw dependence sec sec pair node drop node markov rw sample typical happen hundred reduction straightforward implementation estimator lead complexity become sample million facebook node implementation fortunately sum thing especially auxiliary dedicate guarantee rw life topology facebook concrete rw eq eq rw topology ground truth topology come million summarize size grow example estimator sampling graph g email pa efficiency improve topology weight turn give already however primarily berkeley facebook k link k k email k amazon k google pa average highly topology rw gray dark gray perform grey region percentile percentile dotted life fully topology sample corner rw reduction technique medium grey dark grey grey percentile dotted line art rw ten large topology rw topology fix vary parameter margin weak rw systematic topology range acceptable variance effect many well say nature occur interpret contrast regime margin yield value much interpret exception topology rw fail probably lattice diameter consequently rw possibly sample absence rw date performance rw ten long regime rw close fig reveal rw highly node facebook entire topology us sample period rw rw cover user light gray dark gray grey cover percentile median set dot top facebook return facebook correspond translate rarely soon facebook exist continue remain fig rw facebook column estimate impossible contrast apply rw fig concentrated attempt compare entire rw represent grey rw dark grey improve interpret facebook contrast latter rw
q represent approximate value gene equivalence microarray conduct university aim investigate express day equivalence express day priori gene colour long treat ratio gene mean hyperparameter joint appendix equivalence case study approach interest equivalently day log mean distribution flexibility motivate histogram ratio day shape tail compare degree possibility distribution approximate normal therefore eq true mean log gene rearrange give calculation constant adaptation use equivalence region hyperparameter rather approach nine posterior employ em hyperparameter study behaviour observe express variance log value hyperparameter table ratio contain within lie outside increase prior demonstrate equivalence equivalence ratio asymmetric htbp scatter separate hyperparameter appendix htbp nucleotide nm nucleotide gene validate purpose increase equivalence adapt estimate note testing beyond scope conduct assume prior considerably mixture hyperparameter use em algorithm apply full observe unobserved complete calculated substituting algorithm ij give step replace numerical optimisation find I search successive interpolation interval choose hyperparameter proportion em mean consist small observation integer sample sample pair function initial component mixture proportion often method recommend choice algorithm iterate em algorithm iterate estimate achieve value normal dominate normal run algorithm limit initial produce log algorithm compare calculated order show consistent log initial maximum consecutive final monotonicity number b symmetry sufficient case let weighted observe average suppose decrease p odd school sciences sa edu school mathematical sa edu school sciences sa edu testing study define notion strength equivalence wide adjust evidence nan expression formally satisfy consistency requirement strength mean widely gene positive discovery inclusion translate equivalence posterior basic consistency credible equivalence propose analyse experiment keyword false discovery microarray equivalence assess treatment comparable pre interest potentially side treatment costly produce trial level statistical prescribe within provide sound application equivalence microarray bioinformatic study illustrate biological establish expression rank gene profile many profile gene remain part play central role system include disease however population ensure desirable establish gene naturally occur cell baseline rather find evidence appropriate identification stable future variety study population result aim scientific test gene test goal bioinformatics exploratory know rank inferential equivalence mistake test hypothesis hypothesis false dramatically increase adjust value gene discovery discovery adjustment applied incorrectly discover significant controlling example control family alternative derivation increase calculate equivalence describe probability basic strength rank motivate cell microarray conduct equivalently propose ratio equivalence employ algorithm hyperparameter finish brief conclusion credible evidence false outcome perform accord accept nan nan total alternative true suppose hypothesis statistic nest denote return consider unbiased statistic large evidence increase away statistic decrease value eq hypothesis u u se se se z se alternative derivation equivalence pair base significance possible rejection region observe associate recover strength evidence page demonstrate illustrated plot increase decrease equivalent observed confident value margin false fact lack monotonicity increase equal assign incorrect equivalence value highlight
estimation cone sufficiently proximal iterative shrinkage thresholding iterate satisfy contraction denote convergence importantly outside framework estimation either testing framework reader theoretical describe formulate section proof constitute primary mathematical concluding remark excellent exist modern numerous design broadly literature primal method directly primal dual problem optimize block penalize regression solve coordinate reduce lasso regularize iteratively solver graphical lasso implement unknown variant nesterov interest one stem convergence tolerance dual max study similar show sublinear block approach primal take block oppose decrease rate primal hence second locally around rich body numerical provide shrinkage algorithm technique gx hilbert associate differentiable convex semi necessarily often lipschitz gradient low convex operator example fx choice size iterate fx fx optimal say converge sublinear fy within satisfie p I approximate power discuss iterate minimum satisfy fix constant contraction give convex objective prove section constant follow directly lie specify iterate explicitly close convergence property rate I contraction closely relate condition soon smooth become effectively newton advantage bound solution global bring literature advantage element section compare time implement run intel gb ram zero simulate sparse somewhat finally matrix way definite condition percentage nonzero derive show heavily regularization generally numerical optimum duality result sparsity error convergence demonstrate dependence algorithm duality investigate publicly implementation record present far present three processor time time times processor important cholesky advantage dimensional cholesky cpu cpu apply gradient widely algorithm competitive example section competitive condition competitive many national science grant dms national dms dms grant department energy office thm remark statistics stanford stanford stanford stanford tx stanford stanford great community produce sparse inverse estimator gradient method present although numerous proximal attractive theoretical convergence reach tolerance give closely optimal convergence comparison
recovery edge phase transition correct nearly transition newton synthetic blue variable green dark blue plot recovery maximum census consist discrete year state status education level state state com supplement population evaluation set study variable size increasingly need sensible baseline compare multiclass regression condition rest evaluate performance evaluate regression multiclass logistic well objective similar separate regression since figure separate regression outperform benefit test conditional marginal regression education train use figure outperform generative small less size generative outperform conditional estimate computable originally evaluation find train evaluation model well outperform although suggest see specify synthetic acknowledgement helpful lee department engineering fellowship program science research fellowship stanford fellowship dms national health verify jointly check fu j tc sx jx tu compose fu convex iff establish cv cv complement iff inequality entry mrf large model exact difficulty mle gradient difference sufficient discrete computationally involve matrix inversion continuous difficulty method model method propagation reweighte belief surrogate case state discrete estimate inversion joint primary derivative expect use show efficient develop mrf independence q last entire py j indicator let p z st b w theorem conjecture axiom consider model continuous pairwise continuous amenable structure natural generalization line novel undirected regularizer sparsity take path learn value however population click gene expression gender consider continuous previous assume via graphical lasso several efficient find pl discrete markov field py derivative focus connect discrete previously well understand multiclass markov gaussian lead natural generalize graphical lasso block different calibrate fairly discuss conditional mixed view mixed response discuss discuss approach section discuss section calibrate discuss consistency discuss census population dataset synthetic pairwise model pairwise random discrete variable parametrize j rl important regression multiclass logistic regression simplify usual pairwise case critical absence conditionally read iff coefficient capture desirable property distribution property represent continuous eq discrete variable mix discrete pairwise reduce familiar multivariate parametrize discrete reduce pairwise second markov field eq although gaussian differ parameter standard calculation see variable gaussian common mean depend depend know mixed condition model pairwise simplify although exponentially vector homogeneous mixed allow make alternative limit via independence however marginally undirecte decomposable model regression applicable know primary learn parameter exact likelihood decomposable regression knowledge optimization procedure mix maximum obstacle since include special case difficult case well maximum product distribution take familiar variable state multinomial multiclass regression contribute effect take log twice negative appendix separate separate think asymmetric share conditional double expect outperform wise straightforward predict regression confirm separate regression outperform setting parameter dimension even dimension would eq compute factorize q weight uv section edge truth use extended estimator straightforward consistent regularizer adapt calibrate regularizer vector estimate estimate q likelihood edge type difficult result parameter identifiable popular fix issue drop identifiable asymmetric formulation treat differently q group regularizer optimization equation present consistency active inactive inactive orthogonal convexity q lipschitz imply convexity convenience exist convexity ensure uniquely estimate condition active dependent inactive form partition differentiable hessian lipschitz radius suppose give sample select large applie estimator maximum constant true thus determined theorem less useful proximal proximal convex penalty block coordinate descent coordinate close many smooth solve appropriate routine suit proximal accelerate proximal proximal compute consider familiar group overlap proximal simplify soft group soft thresholding accelerate proximal gradient work solve current iterate proximal prox determine line search theoretical property proximal accelerate proximal rate package allow proximal algorithm less proximal next newton nd analog incorporate minimize quadratic center subproblem
extension crp ibp dirichlet dependent crp partition restaurant process necessarily interpretation endowed notion dependent nonparametric exchangeable distribution support atom share across probability share base share atom base result hdp model point share variation population covariate nonparametric process survey number variational method model replace hierarchy variable conjunction dependent base collection atomic eq q simple assume share vary process stochastic make spatial replace create mixture extend incorporate interpret straightforward find beta main random induce dependency markov generalize employ vary across probability construct stick break wherein biased order atom repeatedly break random stick recover dirichlet commonly ibp dependent prior stick break marginal might obtain whose function construction elegant inference go way application related breaking process define measure bound recover stick break prior vary stick break process accordingly vary maintain dirichlet arbitrary result marginally non much easy perform inference hierarchical break appropriate structured jk matrix countable weight construct permutation row matrix process involve break follow construct random atom location permutation g v associate location distance share space covariate variable whose neighborhood e method stick break construction limit dependency multiple dp biased stick ordering size atom tend permutation distance maximum size covariate location tend large use class discuss induce dependency mixing assume I induce dependency covariate restaurant covariate marginally spatial dirichlet extension spatial replace multinomial correlate spatial field field mixture spatial process aim ensure value marginally dirichlet mixture collection field unit yx kx ix surfaces component enforce cluster explore break surface dirichlet break surface select marginally arbitrary stick break similar image author truncate technique employ ibp ibp element jointly element marginal recall conjugacy exist case analytically approach induce predictive induce n observation different exchangeable hierarchical hierarchical model document exchangeable arrive batch create observation crp maintain crp extend approach covariate covariate unlike elsewhere survey share model covariate lack generalize construct leverage combinatorial subsampling specifically assignment customer z leave restaurant scheme proportional say stay leave remain atom g describe sample assignment th crp customer step restaurant restaurant provide recurrent chinese restaurant special customer leave end atom step atom transition measure decay determine alternative crp say customer customer customer interpretation crp define table ie sequence customer distance crp utilize dissimilarity satisfy triangle customer monotonically decay customer assignment interpretation crp concentration customer person function window describe distance customer customer dynamic cluster predictive ibp create covariate customer customer dependent ibp first customer draw say analogously choose occur create dx dependent map poisson evy measure homogeneous obtain rather process distribution use customer covariate try probability popularity dependent segmentation interpolation model evy evy gamma gamma atom covariate contribute measure share atom close vice versa atom appear process window gamma placing allow recover obtain subsample measurable complete atom active center covariate covariate prohibitive create poisson carefully rate vary construct vary kernel kernel l utilize stochastic order use integral perform algorithm utilize metropolis hasting particle filtering infinite straight collection straight line plane pass set proportional set process intersection clearly intersection process marginally process correlation map induce collection poisson poisson covariate origin unit representation describe alternative look dirichlet process describe section covariate covariate restrict large covariate algebra disjoint algebra subset gamma process unnormalize gamma weight gamma gamma process weight normalize bivariate measure dirichlet autoregressive measure step distribute innovation measure marginally distribute marginal distribution depend dynamic hierarchical extend group hdp apply single covariate bayesian combination measure plus innovation eq approach nonparametric traditionally form arbitrary dependent focus dependent base process thorough covariate early community nonparametric process covariate dependent model dependent apply processing interpolation denoise provide goal remove pixel model common pre processing patch frequently patch treat bayesian vector stationary typically accord beta exchangeable well sharing achieve use achieve superior denoise house task sort achieve break achieve segmentation natural analyze song highly time evolution music music allow transition hmm segmentation obviously local correlation suggest use dependent hmm correlation topic model exchangeable collection specific distribution word text document latent finite hdp allow nonparametric topic corpus evolve news volume document exchangeable expect topic time dependent dirichlet create example hdp dp hdp turn use document since active topic appear topic popularity multiple addition vary vocabulary allow evolve recurrent chinese restaurant manner model apply finance volatility price measure volatility versa overview choice stochastic volatility simple return determine positive variance report indicate model fit length phenomena snapshot range nonparametric employ statistic community paper highlight measure hope develop well application hand similarity easier efficient history dependent relatively short modern field drive mostly machine learn community range application dependent analysis also variational introduce often less many meet know marginal certainly put restrictive marginal well rely covariate well sample question still applicable still play loose often lead start emphasis inference dependence appropriate restrictive rigorous analysis nonparametric community develop underlie machine provide development process complementary interest statistic machine dependent extend give collection typically space assumption nonparametric formalize statistic model understand help develop leverage recently paper develop dependent prior bayesian snapshot currently model traditional nonparametric chinese restaurant exchangeable exchangeability series proximal set may exchangeable relatively exchangeability arguably go seminal formally dependent nonparametric process exist prior prior collection member collection similar report nonparametric process range fall sometimes relatively group collection atom dependency introduce stick break locally sequence induce dependency collection locally exchangeable modify exploit collection model difference consider underlie related adapt numerous application point similarity hope aid understand current development new representation first nonparametric particular survey early construction grow rapidly focus process particular dependent stochastic process describe nonparametric prior reader reading model fit specification method satisfy conclusion body challenge currently discuss survey base nonparametric review prior random random measure notation convenience denote unnormalize arbitrary covariate covariate observe random covariate index notation extensively observe spaced observation draw variable useful representation process product let evy evy evy measure correspondence simulate simulate alternative often jump evy poisson
propose deal artificial characteristic movie make recommendation movie technique system improve across deal start rating apply induction hybrid idea behind many recent work exploit history trait social network yu wikipedia article ingredient know advantage content factorization hybrid content achieve improve produce recommendation insight otherwise interpretable recommendation ever desirable act recommendation big netflix diverse distinct even netflix content enhance ensemble proceed review collaborative content describe evaluate useful mention end brief summary review rating user rating form user rating principle indicate certain range indicate level rate highly sparse recall eq rating user item recommender predict would define rate despite often capture fair simple overall user effect obvious please common factorization experiment factorization algorithm factorization use predict factorization use know decompose another user rate user item simply imagine item item nearby long component mathematically factorization frobenius prevent turning view come likelihood penalty come spherical gaussian posteriori estimate bayes computational burden rather substantial result term improvement convenient term subscript bl stand baseline purpose bl convex solve descent move keep vice versa initialize small iteratively later cf factorization outline value netflix review speak perhaps factorization mathematic problem let p p factorization outline confirm netflix work orthogonality degeneracy question avoid practice elaborate matrix negative nmf negativity constraint nmf image reveal recent nmf though structure primary orthogonality svd g nmf sound mathematically somewhat ill focus remain community partly year long netflix stack attribute indicate whether item possess way incorporate slightly extra penalty selective regression constraint share call intuitive notion closeness model mathematically feature preference alignment problem attribute express solve size make alignment penalty size subscript ab biased idea descent apply alignment shrink centroid share page play central role introduce smoothed respective centroid square generalization alignment penalty proportional relation unity alignment smooth monotonic function ht u I become allow share attribute contribute share attribute attribute much enhance create tag exploit tag follow tag baseline optimization user history content apply item content information subscript stand original idea lead mathematical consist particular front multiply alignment respect become selective clearly reveal front centroid shrinkage measure attribute cosine intuitive amount interpretable incorporate information store attribute feature method group conference behave coefficient latent feature attribute feature vector correspond problem become replace regression technique fit sort site latent coordinate site similar likewise specie environment cca latent linear actual environmental site become extremely technique environmental content mf bl ab section table briefly summarize compare ht summary bl ab eq one early factor use allow mind bl recall sum multiply everything calibrate extra include rate least user movie set ht movie attribute lp lower integer indicator whether contain ingredient movie different serve learn rating p truncation movie mae cf mae discrete rating literature increasingly mae dominate netflix share attribute share ab share attribute generally regard ab overall metric mae rmse reasonable fairly opinion think strategy alignment activate certain sensible affect hand item certainly toward little sensible nature result result sensible smoothing behave like ab whereas eliminate alignment main fine provide descent hence initialization reasonably follow first miss regular svd p enough degeneracy somewhat ill pose explicit orthogonality prediction miss remove prediction take initialization would applicable bl extra natural invertible diagonal element practically pose subtle comparison force relative disadvantage explanation fair mixed svd sample initialize bl ab initial yield approximately algorithm geometric explanation initialization force disadvantage good factorization initialization ensure factorization way scale quantity remain fair use list increase contain give large movie e table bl measure could cc reach percent respective pre descent algorithm keep ensure practical finish large differ significantly relatively allow objective iteration figure summarize use content ab bl behind shrinkage error hold set data validation triangle emphasize overall fair section break vertical axis explicitly biased ab recommendation show dimension reason recommendation present recommender distinct e g category plot bl ab use two solution distortion together alignment bias key likewise movie close say child movie three produce alignment bias ab vector item lead bl alignment constrain similarity simple co occurrence merely drive content range similar highly result dimensional one calculate use cosine away meaningful pair example easy people like likewise like movie child favor action movie tell like like insight insight suggest familiar ht vector ingredient ingredient cosine black movie p cosine child action like briefly cf fill miss keep matrix behind minimization behind believe drive factor therefore rating differ explicit zero compress np recovery norm nuclear norm nuclear remarkably restrict rip rip involve minimization norm minimization reason attention limitation example maintain complete sdp matrix completion approach apply although devoted importantly mathematical bring
column ill condition unfold say column factor merge unfold helpful merge matrix highly condition one possible thereby mode reduce matrix method ordinary unfold tensor mode mode merge correctly estimate rao thank special rao product structure column cp high tensor assume knowledge suffice separability linear mixture bss unfold relationship bss respectively cp decomposition bss overcome little general incorporate transpose tensor lie estimation fundamental generally impossible additional corollary likely column rao factor thereby may tensor idea cp bss estimate unique may flexible cp corollary unconstrained ambiguity orthogonal bss employ ambiguity incorporate mild mode reduce perform rao product original rd tensor simple uniqueness mild section interested order tensor rd tensor paper mode involve unique detailed cp unfold via rao projection consider simple frequently mode local solution rd alternate apply order full rank correctly consider factor say column purpose keep unchanged matrix run tn efficient scale mode reduce one large unchanged correctly eq rank always column rank practice efficiency reduce want reduce row reduction technique employ consist lead let lead sometimes maintain essentially adjust sign matter related discussion row equivalent may svd multilinear worth method rd dimensionality often efficiency perform rao formulate matrix product specify apply sequentially original least eq example procedure jk optimal rank apply implementation mode unfold equivalent hence singular uniquely impose follow projection operation constraint element nonnegative repeat alternatively convergence analogy power derive straightforwardly tensor impose repeat achieve actually tensor involve operation extension repeat tensor exploit nature mode tensor convert one reduction algebraic proposition unique theoretically essentially corresponding solution unfold relationship unique reduce order thereby unique word uniqueness condition simplify suggest uniqueness rd important uniqueness point essentially lead important issue investigate uniqueness tensor extend th uniqueness th order unique rank simplicity uniqueness hereafter move without generality hereafter transpose change tensor likely generally increase hand uniqueness order unique th consequently also unique assumption proposition tensor corollary proposition let way th rd maintain assume factor tensor unique word mode possible show ill condition rank factor matrix mode rank way uniqueness relaxed uniqueness propose simplify may feature method may interested play task etc consider extract rao product actually th order consistent order uniqueness reduce mode high theoretically unique exact compression common approximate practice performance provide q obtain rank consequently able give essentially truncate cp proposition fortunately rao product rao conclude actually almost direct hold optimal use initialization different tensor unfold rd etc pi interference unit proper synthetic noiseless propose matlab intel cpu gb windows feature boundary add tensor combine cp include toolbox matlab set construct perform method perform way finally recover use monte run evaluate solution plot run minima give mode analyze unfold important method influence configuration randomly select tensor unfolding bad perform hierarchical alternate constraint replace algorithm perform achieve way final tensor meanwhile improper tensor result global limitation unfold achieve cp cp simulation improve feature technique normal component shift frequency jt configuration neighboring setting performance run fit correctly actually uniqueness monte c c cp real analysis image object angle simplicity object image scale perform influence center achieve cluster typical rao final see method slightly bad mode rao satisfactory able proposition c runtime accuracy iteration bottleneck paper reduction tensor efficiently incorporate unfold respect method overcome bottleneck cause component full loss structural essential uniqueness mode reduce tensor theoretically investigate confirm algebraic rao role tensor intrinsic one determine choice phenomenon develop future statement show statement show also regard note prove p procedure proof factor merge essential uniqueness lemma jt jt author thank anonymous suggestion original manuscript receive ph south china china laboratory advanced brain signal brain institute interest processing processing sc ph dr degrees electrical university institute electrical engineering system electrical technology title spend several electrical engineering von laboratory brain head advanced science technical translate chinese china china ph china technology china laboratory university two scientific paper conference research interest automatic control blind accelerate cp widely tensor square tensor frequently performance bottleneck overcome realize rao unfold mode promise bottleneck high mild convert rd theoretically order tensor essentially analyze presence show easily cp decomposition decomposition alternate tensor gain importance representation practical attempt find informative sparse tensor attractive account latent name extensively study decade blind separation unique mild useful factor cp matrix another form rao product remain specific
preprocesse extraction classification boundary segmentation knn ann classifiers improve state art classification six various promising recognition toy object realistic new progress aim experience recognition action video recognition intra class clutter object static problem handle bag feature support remain generalize experience spatio spatio bag weak approach outperform realistic show eight class include water read automatic automatically label dataset author location human event system section two foreground section solution define detect particular wide classification video condition main location compose four phase follow segment stream video capture candidate key separate background image video define description foreground foreground event result detail characteristic phase segment detect actual camera frame correlation shot sequence frame camera frame shoot detect similarity previously mention two depend camera movement instant movie instant video digital mobile camera shot video capture frame video convert frame convert frame original color frame calculate block mark new shot frame convert q red color separate object video play important decade still object hard adjacent spatially coherent demonstrate combine advanced computational model stream sensitive use initial object equation change drastically object background region color intensity correspond people experience vary color vary red back vary color vary component vary color become increasingly space feature pixel background moving object foreground statistical background color component background detect video summarize capture frame input convert color equation statistical pixel frame background pixel measure current frame pixel otherwise foreground pixel pixel follow q mean detector image isolate connect interesting template descriptor interesting area intensity orientation point detector computation matrix salient region step position extraction computationally expensive describe feature develop sift approximate multiple image limit intensity decision tree event track feature lx pyramid gaussian smoothed derivative q q sift descriptor mostly sparse event instance characterize training use parameter testing determine determine kind near neighbor knn classifier svm propose briefly give subsection neighbor neighbor find close set classified classified classified predefine query find number query point minimum near neighbor neighbor majority near neighbor query dimension advantage nearest robust noisy near neighbor determine neighbor query instance inspire system neuron work solve neuron artificial neuron separate negative class training vector label belong linearly separable geometrically find maximal solve weight call complexity superior capability function extensively past popular radial multi general independently discrete sa site seven see sa mean hold seven order water breast plain important significant gave hill try mark place move remain cc seven east day day fig category video resolution category video format system result record retrieve relevant record define relevant retrieve recall usually equation respectively video shoot shot identify accordingly c sa false precision knn ann svm performance system cc human spirit use web character video sophisticated tool action text discriminative representative however boost huge making applicable motion human action descriptor aggregate optical flow measurement spatio volume center channel optical template match spatio correlation motion energy create moment mahalanobis moment description avoid explicit rank constraint intensity spatio enforce spatio temporal video location several explore control recognition localization recently dataset manual annotation develop
cca retrieve visual database cca space combination tag cca retrieve modal embed tag directly search tag database tag scenario database variant keyword query retrieve set accurately go cca coherent tag decode g scope task similar neighbor tag return five tag tag accord diverse dataset local somewhat task protocol task subsequent tag annotation automatically v refer view baseline tag visual tag supervise semantic keyword unsupervise automatically computed tag I completeness view cca cca give embed model detail compare obtain intermediate visual tag baseline structural formulation tag tag predictor ridge task predict visual validation cca tag unlike cca cross retrieval sift sift h pca cca without learn weight classifier keyword feature modal retrieval stochastic function difference explicit use early sgd iteration pick dimensionality embed achieve good cca overall slow early stop use cifar conduct detailed study include feature tag purpose perform evaluation cifar dataset learn cca imagenet use quantitative imagenet validation split validation search query retrieve fraction imagenet search query tag exclude embed truth query examine percent closely ground example car car ground auto accurate image tag search cluster mean retrieval row initialization method cca image tag image search map nine tag section evaluate dimensionality reduction involve third view cca tag search visual apply slightly less significantly tag retrieval help possibly tag noisy reducing motivate give platform core ghz gb ram take cca versus minute nine minute get scale try likewise thousand completely infeasible platform point address finally report retrieval view cca fairly center image close tag present number tag view truth table method cca approximation tag discuss section compare solely cluster bad performance tag base cluster visual demonstrate view c visual bad cca v baseline tag cluster range nc method cca tends depend use tag joint example image visual coherence show tag though completeness figure still long correspondence frequent entire cluster confirm difficulty alone help explain incorporate cca l cc cca cca cca cca v scale cca v cca v multi view denote euclidean correlation cca nc clusters cca weight tag automatically interactive system user adjust concept query weight image quantitative annotation annotation neighbor imagenet cca space tag ask member research tag suggest mark tag image interface tag present tag correspond hard complete tag human voting tag gets mark three report tag frequency cca v lead cca v vs cca test wide randomly embed image sample query retrieve image use cluster nc end cifar surprising large semantic ten precision since ground keyword per keyword image keyword top I cca cca v structural refer query split around report ground annotation cca cca cca task v find weak model entirely surprising however tag mixed concept fewer well intuitively rich diverse truth annotation unlike cca strong cca especially noise cca flexible suitable tag image search cca c return compound query consist tag compare tag view cca cca tend retrieve compound annotation cca cca cifar three image result confirm tag produce explanation result image specifically either abstract object suggest tag landscape light appear somewhat try suggest specific wide cifar embed solution exploit multi important subject search explain information dataset give view supervise cca image database query query database mark tune nc report tag search dataset diverse absolute accuracy may database image annotate retrieve query quantitative evaluation cca still consistently wide work wide cca well cca tag noise qualitative search finally image tag present approach internet tag semantic start view cca work performance third ground truth search keyword cluster quantitative model cca cca consistently outperform extremely appear second simple concept cifar tag actually capable label even concept tag sensible concept attempt discover structure may connect visual tag latent image semantic view cluster indicator highly nonlinear second expressive cccc cca cca cca quantitative successfully visual consistency diverse flexible retrieval system capable visual semantic discover tag subsequent cca projection content internet like user retrieve query tag keyword manually adjust weight keyword serve basis discuss satisfactory develop sophisticated incorporate consistency name intermediate recognition parse retrieve embed tag retrieval return lead improve parse describe image tag sentence generation may become easy anonymous constructive comment implement manual auto support nsf ap microsoft fellowship website http www edu microsoft view department internet tag image tag tag annotation analysis cca popular latent capture semantic category mutually exclusive way supervise come truth unsupervise automatically tag ensure accuracy learning process multiple visual feature cca produce qualitative view task scale internet dataset keyword image associate internet order enable retrieval scenario search image annotation task must meet requirement big give extremely heterogeneous annotation million flexible cross retrieval tag enable tag without tag promise cca two visual common maximize modal embed vector treat image text retrieval principle handle cca cca cite use cca consider correlation feature improvement two semantic semantic high characterize image concrete figure three semantic view keyword think view visual tag stochastically tag vocabulary semantic may object category correlate etc alternatively semantic object attribute thus image annotate ground color shoot semantic give mention ground three high view cca embed embed three embed confirm derive capable considerably diverse ground truth keyword define ahead annotate realistic tag underlie fortunately clean truth still semantic explicitly informative signal search keyword retrieve tag know search ground category search effective view unsupervise tag inspire information semi reconstruct distinct modern method find necessary multiple nonlinear cca scheme improvement dataset contribution explicitly incorporate semantic function adapt cca retrieval scalable discriminative visual view reduction section ground truth annotation unsupervise tag vector confirm help improve perform comparative tag extensive evaluation image tag tag protocol vision community jointly image area exhaustive several important line work research focus occurrence image tag generative lack annotation image challenging contaminate internet tag tag frequently characteristic localize establish relationship conceptually compare attempt annotation unlike concern symmetric model correlation treat tag assign scalable inference test tag tag nearby cross tag tag recent canonical cca cca image modal retrieval modal approach relative importance word user annotation develop cross image embedding use domain unlike cca retrieval annotation approach add third semantic underlie basic cca pay price mahalanobi relate near optimize optimize neighbor annotation modal retrieval evaluating traditionally content base image tag image retrieval use query tag though directly contaminate purpose collection third interested evaluate automatic traditionally help sophisticated scheme annotation adopt experiment latent tag transfer annotation approach obtain relevance image retrieve unfortunately learn computationally expensive scale annotation consist tag dataset k descent simple image annotation annotation system use optimize dataset million learn feature model distinction tag image annotation dataset image single label hierarchy annotation occurrence different tag retrieve belong category plane tag query incoherent exploit annotation multi limit visual would impose multi decode outside scope paper image classification web baseline tag multi another way bank concept unlike produce embed supervision computationally intensive training view semantic scalable cca assume training image associate visual tag discuss stack number image may entry rest several keyword possibility indicate scenario possibly noisy annotation tag tag point learn rely instead formulate projection embedding project embed vector distance pair eq th result understand term align tag term view cca objective align tag formulation embedding formulation trick cca linear combination one must solve eigenvalue infeasible base reduce mapping use describe trick substitute cca eigenvalue w ij ix jx w iw add constant covariance view matrix space learn view without tag retrieval near neighbor implementation validation aside section range typically fall cca function distance embed dimension latent magnitude project objective reformulate view let view cca whose corresponding experimentally lead high retrieval euclidean section visual text mapping semantic unsupervised third appearance nine use different filter channel base dense corner patch mean build pyramid sift sift texture large codebook spatial pyramid dimensional joint bin feature compute set deviation descriptor adopt wise combine quite put respective mapping dimensionality reduction top approximate combine feature feature vector dimensionality balance efficiency origin construct frequent vocabulary summarize remove stop include etc characteristic etc tag tag tag feature two however tag require visual sparse show actually embed sparse compressed top sophisticated rank tag list early user salient image initially cca indicator straightforwardly
maxima different natural multivariate extreme weak convergence univariate reasonable apply affine transformation margin sequence normalize stable distribution th convergence take know strictly normalize maxima vector henceforth weak distribution margin sequence copulas maxima component give check joint margin maxima copula arise call copula call extreme value copula exist copula extreme copula arise copula copula say extreme extreme copula coincide copulas maxima max stable also conversely extreme stable tend fix limit max summary distribution appropriately maxima margins extreme max copula margin extreme margin copula former tail go infinity tend good copula characterization rise function denote generic one exceed percentile distribution copula originally involve maxima familiar univariate tail copula component exceed percentile simultaneously q govern formula dimension somewhat retrieve margin value copula tail copula even zero depict cc homogeneous eq similarly simplex w homogeneity frequently write union consequence q multiply let extreme must satisfy display attain independence perfect association max property copulas function law form fr reverse yield fr margin notation transformation disadvantage action neighbourhood identically geometric success return time return pareto find map evident dependence function take mode vanish origin intuitively grow profile degenerate equally depict exceed interpretation order statistic fairly max device represent tail measure parametric obvious work measure profile margin condition different fr independent vector calculate copy fix q j rescaling case margin compare identify da comparing give absolutely law profile profile variate distribution put variate stable profile dependence random profile encode dimensional face simplex empty let otherwise measure profile contain simplicity apply order parametric remainder work ensure perfect constant finite profile dependence spectral discrete atom suppose ensure dr w spectral profile discrete magnitude give max spectral call value factor outcome result type event random indicator b pair pair stable dependence asymmetric parameter extension equality constraint yield one indicator switch structure keep law equal empty way structure think logit indicator copula device copula associate model let tail dependence know dirichlet dirichlet model dependence mixture introduce transformation yield generally max stable eq variable count variable expectation exponential polynomial dependence bivariate random margin put stable dependence q bivariate standard put expectation stable function calculate yield author thank journal manuscript centre participant encourage discussion strongly author grant science contract max universit random component mathematical suggest stable univariate multivariate various stable device stable distribution device role yet fully exploit copula measure extreme value concern sample component bring univariate extreme ignore mathematical max
irrespective statistic accept homogeneity even hellinger large refer logarithm negative log likelihood generalization exact unlike one ratio facilitate comparison likelihood display entry homogeneous proportion give underlie rl hellinger frobenius please p value frobenius display entry display assumption table log hellinger please value display homogeneous correct rl log hellinger negative note frobenius distance value display display without optimally display party party b z lrr party divide square root party z treat lack recover another efficacy recover lack efficacy another root lack candidate lrr divide square root treat treat treat reaction reaction prior square corresponding reaction frobenius distance classical homogeneity paper typical similarly important circumstance classical statistic fan wolfe fellowship p foundation section remark homogeneity proportion contingency probabilistic process generate fixing row illustrate frobenius schmidt classical statistic log ratio hellinger distance divergence family hilbert schmidt consistency homogeneity chi fisher exact divergence root sn sn n sn categorical contingency table table provide comprehensive common fluctuation homogeneity proportion displayed consider homogeneity row construction therefore table assume homogeneity generate draw identically draw note rp count equal observe please construction exact confidence observe draw consistent homogeneity see discussion interpretation section value measure canonical possibility hellinger hilbert schmidt q take number member divergence member read advantage illustrate advantage statistic recommend use classical sequel define several section conclusion definition p involve compute probability carlo guaranteed specifically perform generate draw rp draw depend define draw row discrepancy simulate p equal discuss number compute similar procedure bound report present
ibp generating process ibp pre perform vocabulary ibp due restrict sample single atom posterior ibp use sample estimate predictive maximum binomial distribution train document classify ie label ibp uniformly explore way distributional exchangeable allow distribution flexibility future benefit distributional exchangeable column useful datum poorly suit propose exchangeable exist model set ibp relate gamma poisson exchangeable infinitely contain flexible model specify row distribution ibp impose distribution exhibit give ibp marginally appear want flexibility allow non marginal per ibp use markov see nonetheless ibp feature prior similarly wish exclude interesting arise exhibit perhaps team encode per tend parsimonious situation imply ibp well text tend suggest impose tail ibp point exhibit measure feature remove random unchanged far exchangeable important tell exchangeable distribution write probability exchangeable predictive px n measure may motivate exchangeability nonparametric ibp binary exchangeable row infinitely column measure process distribution bernoulli beta process bernoulli assign form parametrization beta commonly use ibp scalar cdf atom distribute accord piecewise countable jump ibp cumulative atom beta give think matrix infinitely binary result entry row row non entry column exhibit rich rich give time appear precede vary distribution degree ibp introduce extra beta limit result ibp feature point disjoint replace beta measure beta process include ibp exhibit feature law exchangeable gamma poisson process correspond column process ibp atom result negative integer matrix exchangeable distribute binomial exist able sharing feature ibp remove ibp power customer try distribute aspect elaborate marginally row ibp conditional distribution location trial sum restriction specify row priori bernoulli exchangeable mixture recover ibp ibp marginals poisson ibp see case construction explicitly condition draw de mix measure infinite construction exchangeable discard outside interest certainly find exchangeable sequence exchangeable restrict infinite process infinite previously assign count restaurant restrict sum chinese restaurant restrict chinese exchangeable sequence restrict predictive exchangeable restrict distribution process consist mass index red represent blue ball pick note color return ball sample color iff exchangeable trivial ibp restrict probability model give example exchangeable introduce normalize restrict previous broken exchangeability obtained directly restrict ibp section restrict ibp appeal exchangeable modify ibp scheme exchangeable refer ibp modify sampler crp distribute binary accord entry per location non th sampling especially hasting entire binomial recursive fourier skewed approximation metropolis hasting wish evaluate ibp unable analytically predictive use ibp predictive n equation describe exchangeable modify experiment appropriate datum design careful latent improve evaluating
find manifold window manifold enough bend local moment matching already match discretized consider density moment mild suffice chain converge convergence chain chain precisely point minus proportional sketch ergodicity convergence counterpart asymptotic distribution interesting note operator equation multivariate see kind specify component like density noisy approximation step small magnitude intuitive point neighborhood appropriate normalize define local mean scale take assume calibrate mean mean obtain chain ball define approximation definition result q mean step mean equal asymptotic moment version target similar observe term produce gaussian version construction make follow vanish desire neighbor moment minimize reconstruction estimator train contraction whole encoder equivalent criterion corruption set completely covariance encoder already set corresponding example go infinity go around taylor expansion neighborhood non infinity neighborhood dirac delta rewrite choose dirac expectation arise covariance get derivation happen force practical see relate asymptotically decrease jacobian parametric infinite perfect much magnitude indicate look increase singular relative direction practical want take non generalization parametrize perfectly generalize learn reconstruction equal even jacobian mathematical link situation control variance hyper optimize inspection introduce many non density window generator mix cover much true density window larger likelihood result auto encoder train generate likely look novel approach density moment auto justified chain local moment auto decade everything capture first present justification auto particular denoise interesting advantage criterion estimate difficult future follow try answer happen learner non consider compute future give rise sampling moment extend auto encoder experimentally give mathematical algorithm derivative dx mse x r q respect trace x obtain j starting x get j x x j c c solve limit note go goes plug recent encoder job unknown datum density contribute mathematical understanding phenomenon help justify auto previous covariance capture jacobian novel density call local matching propose extend auto encoder learn aspect generate learn manifold identifying set point configuration observe plausible attempt recent year feature auto encoder stage encoder decoder sample data feature stack deeply abstract much evidence suggest perform purpose classification generating target manifold concentrate locate manifold variation movement stay density remain concern many feature criterion implicitly learn whole aspect essence density formalize exploit question contribute propose motivate intuition jacobian encoder direction tangent plane direction preserve reconstruction auto reconstruction entropy loss easy success criterion parametrization tie weight force tie singular contraction singular least direction little variation encoder train denoise criterion corruption mostly corruption handle mathematically expansion rx rx x derivation auto contraction parametrization mm couple respectively
edge satisfactory merge intra benchmark four reveal art meanwhile possible intractable exclusive major approach hybrid evaluation major statistic score mutual mutual advantage cost however multiple another drawback usually require observation sake constrain constrain algorithm use conditional ci test reveal dag pc drawback ci variable parent besides orientation perform search scoring super get dag serve hybrid integrate constrain min hill superiority reconstruct skeleton network constrain hill search orientation devote overlap popular clique compute belong exist clique http overlap detection roughly detection subgraph detect overlap locally subgraph furthermore augment combination fuzzy define either node accord similarity besides cluster conduct agglomerative dendrogram partitioning overlap identify copy identification continue sufficiently undirected graph weight speak play unweighte one consist unweighted undirected edge community belong community isolate community could partitioned group hierarchical weighted partition number contain row column support node example tp node set partition weight exceed weight co model propose divide originally intractable task scale sampling section serve crucial drastically reduce run incorporate overlapping may ideally possess possess edge community coverage correct edge fine small build unweighted predefine map weight graph keep weight exceed generate partition htb discretization partition step outline step start unweighted step employ predefine weight set translate original unweighted various predefine mutual mutual entropy square entropy respectively mutual standard value edge coefficient standard correspondingly weight low threshold prune sake prefer community organize meanwhile sake wish maintain generate build partition v result satisfy criterion couple robust learn per community challenge mix eliminate bias irrelevant retrieve markov challenge step outline sub sub sc structure sub resolve find community node precise markov intra make problem localize solve markov parent child parent compose child parent word close topological closeness edge play ci measurement among require identification markov divide variable look node edge weight exceed mb xy w xy connect give justification association phase apply markov add heuristic denote mutual candidate node independence remove combine expand small intra member community expand community conduct reason contribute sampling build precise pick inner start neighbor denote bayesian large keep remove neighbor sub community sub community bayesian different constrain score interested confidence relate topology per expectation estimate classify exceed however dag overall dag average transform due current domain consideration remain markov dag assume prior possible order markov chain sequence approximately estimate sub community structure intra community resolve characteristic find major introduce try triplet however fails eliminate indirect interaction one edge triplet indicate interaction miss weak weight adjacent mutual small two first collect candidate triplet triplet node mutually triplet indirect unweighted edge triplet cluster graph subgraph hand indirect cluster direct interaction indirect eliminate correct group employ triplet create node community remove community merge add pool individually community resolve intuitively strategy large achieve expect edge continuously eliminate idea try strategy community perform greedy strategy combine community community community similarity coefficient community resolve triplet combine community hybrid community put community calculation value community initially calculation sum nk efficiency advance add would nn evaluation five close enough indicate learn loss learn alarm consist arc arc arc average network include arc degree benchmark besides weight mi entropy mutual entropy value normalization network weight transform weight certain bin lot possess ratio bin threshold threshold remain edge edge remain average short diameter dominate example mutual alarm short yet worst always belong never drastically average order path range alarm alarm alarm percent percent percent percent percent mi average l alarm link mi pearson second order denote htb comparison various weight l link mi pearson denote htb c alarm benchmark four learn namely pc greedy hill information extreme popular widely score serve hybrid prove art correctness greedy causal http www format default test greedy pc type choose weight equal greedy person threshold manually tradeoff true positive implementation name alarm five benchmark divide bayesian averaging se would could evaluation fair pc structure target keep everything unchanged regardless influence usage regard binary pair metric performance system edge edge classify negative number miss art pc search alarm table dataset recall reconstruct show versus counterpart avg intractable relevant pc superiority superiority serve harmonic global precision intra structure case even algorithm alarm htb evaluation alarm c recall recall score pc greedy avg na na htb evaluation c precision recall f score f pc avg na na na na evaluation c c global precision recall score avg na na htb dataset c c score greedy avg na na na htb link c global pc avg na na precision state cccc significance four cccc alarm normalize cccc greedy alarm present large large divide partitioning intra individually perform partition strategy verify
location location simulation table work spc knot result well knot especially spc simulation r knot knot spc c spc spc spc interval score mcmc simulate simulation table still work slightly spc suggest model process stationary knot model result eight fix knot world collect resolution south sense monitoring precision far costly scale gamma ray front smooth krige calibration hyperspectral ray krige improve ray spectra focus total integrated count unit count carry exploratory spatial prediction effect assess test region contain remain leave panel count colored dot location circle right mean deviation mean process see text region rectangle color count error tc count identify trend two another r knot knot spc spc spc hour parent spc parent covariance distance interval score design knot result low average square knot improve spc knot result accurate mixing slow knot knot slow simulated time slightly focus parameter assess base fair article spatial combine rank remainder process often encounter extend way parameterize ern parent spatial function second component function shape jump describe detail section indicate covariance mat ern closely indicate seem unlikely simple stationary simulation highly vary acknowledgment research advanced anonymous comment grateful taylor collection universit email modern high resolution measure monitoring analyse dataset spatial statistical technique feasible heterogeneous appropriate challenge combine component flexible via compactly address propose extension parameterize base mat ern smoothly model high measurement improvement covariance massive reversible jump sense environmental ground day large quantification uncertainty feasibility angle composite likelihood low compactly function produce contain computational feasibility appropriate achieve feasibility component spatial r small component model differ discretized convolution convolution author view fix orthogonal wavelet parameterize allow formula make clear general covariance fast computation range dependence inherent component rough I smooth short stein effort stein context process divide remainder component covariance compactly support feasible inference contribution specify mat ern parent smoothly linear combination function extension allow location refer knot allow fix priori treat achieve discretized convolution model spatially shape basis infeasible location assume shape special piecewise location spline contribution article parent truth rather way spatially dependent flexible hence preferable modeling real fairly involve large reversible jump take advantage operation strategy chain section deal extension approach use simulated conclusion denote spatial suppose namely simplicity identifiability assume practice origin process model model standard extensively standard computation likelihood modeling problem parent knot knot strongly prior flat improper point dependence write valid function feasibility compactly function great covariance quickly invertible feasibility give describe medium range range spatial account local zero gaussian covariance flat place z modification knot select location jump acceptance bayes fully automatic flat statistic inference true might include observe q first mcmc convergence q var p n explicitly p sect dense full update large employ formula obtain range nonzero order cholesky operation location indicate decomposition fix regular algorithm aside spend evaluate spatial obtain considerable extremely massive achieve feasibility current large spatial result prediction predictive approach knot mat ern comparison vary knot mat ern covariance spc spc special three study location simulation process together mean credible ci pm e without knot true create measurement examine medium model test miss design interval location location refer simulation observe measurement model mcmc plot pilot element per parameter covariance describe spatial trend intercept proposal knot uniform knot eight evenly space evenly spaced knot take exponential model knot basically pick consideration interval credible credible interval penalty contain goal small error average average group ht
observational estimate individual population length period credible result table run trajectory original interval gibbs augmentation detail discuss epidemic ode proceed euler approximate gibbs algorithm update euler time determine translate mix epidemic suitable differentiable sde standard vector write intractable accurately alternative suggest drive skeleton eq transform euler alternate step particle implement metropolis respectively accept formulation replace algorithm every ode part functional accept small variance epidemic simulate posterior draw draw run hold clearly reliability difference correspond formulation datum inaccurate term essential driven epidemic address problem helpful improvement error bias estimate particle filter particle posterior posterior interval cccc I quantile r shift shift shifted median shift shift median shift shift estimate euler discretization axis illustration contact record green observed incidence green contact panel line dark light interval level noise bottom trajectory estimate contact rate h implication dot line pointwise dark light blue interval respectively panel alternative explore diffusion right panel posterior density bm maximize mcmc particle gibbs statistic school economics political uk ac department centre school epidemic supplementary part illustrate various suggestion appendix kalman analysis give regard continuous implication associate provide present specify determine euler ps vector discretization encounter epidemic trajectory approximate euler ensure value choose quantity sufficiently show datum reasonable point theoretically regardless smooth particle particle smooth mcmc affect acceptance mind need get acceptance perhaps beyond observational decrease assess validity approximation extent avoid approximation month bm realistic epidemic reverse select google contact significantly decrease experiment methodology text incidence correspond figure fig moreover present credible interval estimate seem value credible aim assess robustness brownian motion sigmoid curve perform capture trajectory explanation part volatility h plot child additional alternative time conduct main approach integrate imply smooth differentiable path
formally current future cost current satisfie bellman strategy minimize cumulative contribute heavily prior knowledge environmental dynamic traffic advance therefore approach mdp require knowledge traffic specifically adopt algorithm name component actor illustrate select way transform actor action execute actor update td prefer smaller versa adopt actor generate store select policy useful traffic separately knowledge actor illustrate let begin meanwhile load state controller stochastic balance compete objective bs operation exploration exploitation result controller methodology probability call temperature indicate tendency stage exist possibility serve traffic stage however conventional scheme commonly controller bss hence controller meet load requirement transmission traffic load connect bs modify communication join cost bss determine intuitively simplify I minimization energy consumption simplify location prefer choose join bs transmission traffic load consumption transmission stage traffic bs transform meanwhile cost consequently td would k feed actor way update occurrence stage iv policy td action indicate execute stage one action higher execute infinitely limit greedy infinite converge address methodology exploit classical ac conduct bs switch effective energy end utilize knowledge strategy historical period bs switch basically indicate tendency converge tendency bs switch conduct system optimize policy period within attempt action traffic come might beginning exist come traffic high target hence target turn bss mode avoid decrease choose certain controller experience take consideration overall traffic load choose euclidean b additionally p kp actor initially dominate overall hence contribute optimal transfer continuously decreases take advantage learn also get propose connect bs transmission cost start scheme traffic td policy transfer update classical ac result one start several relate lemma boltzmann exploration thereby eq next policy track ordinary equation ode eq asymptotically track ode algorithm transfer transfer factor rl km km transmission macro bs micro bs w operational macro bs micro height macro bs micro channel interference factor arrival file constant discount transfer negligible turn mode due scheme consumption bss turn scheme bss start reasonable since exactly scheme controller find bs simplify adjust table firstly much static traffic load arrival bss turn homogeneous traffic offer correspond bss homogeneous traffic rate reflect static arrival expect energy decrease arrival bss stay bss utilize continue decrease controller understand traffic know well efficiency scheme traffic inferior scheme simulation feasibility save energy fig depict performance energy consumption delay scenario incur increase obvious put emphasis delay choose tradeoff performance improvement begin contribute fig depict task leibl divergence divergence transfer plot impact transfer generally speak expect traffic profile approximate approximate traffic load arrival scheme ac k detailed sensitivity analyse c various reflect file consumption controller new action controller try controller would action large result energy consumption fortunately certainly undesirable action classical ac demonstrate effect file similar arrival constant consumption proportion cost turning utilize bs make difference save red region bss exhibit scheme different bs switch operation vary decision besides adopt actor method reinforcement learn switch energy exploit traffic transfer actor improve learn period propose provably arise extensive knowledge transfer propose approach apply scenario map less straightforward bs dedicate meaningful yet challenging knowledge author l anonymous kind author suggestion improve paper quality national program china green cb chinese education key r china grant laboratory without stage state occur stage simplicity k tt tt jt tt define depict z z piecewise integral jump z negligible hence convergent see equivalent ode claim chen zhang china email edu cn box chen sup france email fr universit sup de la france validate energy access turn bss switching operation load dynamic still forecast firstly minimize consumption reinforcement learn bs switching actor utilize historical neighboring provably simulation various contribute demonstrate feasibility expense delay communication energy reinforcement actor popularity load demand massive consumption huge emission speak account overall emission besides operator doubly five china mobile meanwhile account significant improve energy consumption access reason behind largely bs peak stay irrespective heavily body load aware bs adaptation validate possibility energy dynamically adjust status predict traffic reliably challenge bs besides find turn bss bs mobile mt moreover subsequent user association traffic load bss bs switch bs switch influence consumption word scheme minimize consumption concern effect bs operation present combine bs association load solve instead predict traffic traffic actor reinforcement rl necessity knowledge traffic bss centralize centralize bs operation controller network generation evolution rather result bs switch controller estimate traffic load variation experience possible consumption bs switching take experience repeat know switch bss traffic load profile bs switch rl convergent hence rl bs switch controller hundred bss fortunately traffic exhibit make load aware bs switch moment region relevant utilize conceptual transfer mostly recognize one previous novel appeal lead research activity learn bs historical neighboring region bs switch operation enhance incorporate actor ac namely transfer actor propose previous work firstly save consumption moreover assume already extend rl thereby contribute field especially ac traffic mdp energy scheme conventional focus incorporation idea rl section evaluate scheme present validity mdp state subscript td error occurrence stage occurrence discrete shift arrival file consumption delay summarize run bss traffic bss bss depict exist bs switch traffic bss correspondingly status bs stage centralize beyond bss meanwhile transmission request location arrive poisson point arrival per location therefore traffic example high arrival file bss turn traffic bs ix ix bs versa otherwise variation bs coverage partition indicator describe subsequently whole region traffic load variation constitute state chain transmission user bs convenience change
approximate overhead subproblem suitable issue return optimization adopt method I k k center radius example consist determine index approximate solver enough rapidly point strictly indice fw originally design solve iterate frank wolfe consist local k fw update iterate stop sufficiently suitable proximity search direction towards vertex feasibility assume prove exhibit tendency intuitively lie boundary often problem get close direction reach face sublinear frank wolfe later improvement stop practice tendency fw lead resource desirable towards aim enhance introduce direction know move vertex ascent follow find fw algorithm minimize fw k k iterate feasible direction stationary correspond face lower reach whenever away stepsize impose constraint stepsize continuity concavity strict problem assumption aforementione problem particular ask concavity force perform happen behave unconstrained optimization linearly objective obtain explicit procedure maximize carry compare argue whole associate I k k immediately follow update component introduction sequence make structure output preserve formula increase criterion bc identify procedure differ bc square value get optimal round line join round triangle pt circle circle pt circle bc fw derivation procedure compute r j k ji k apparent identify small step optimal stepsize determined correspond lie remove ball remove hard perform step whenever step ascent fw fw standard expression easily insight away search htb join round cm cycle pattern color black black color node node linear result algorithm substantially need remove correctly optimum possess eliminate get near originally motivated advance lead efficient train use satisfy impose obtain fw objective concave constraint regard addition strict optimum make normalize kernel add objective normalize apparent need reformulate long fw procedure large similarly explore iterate still propose immediately evident notation explicitly target qp j whose column constant g hard write follow k stopping hand concave eq addition virtue feasibility objective function classification acceptable frank wolfe build considerably small fast software evident classifier time discuss several issue several include scalability increase assess capability frank wolfe study significance analyze penalty experiment capability wider highlight purpose paragraph summarize conclusion detail cover number like dataset original number dimension classifier tuning category adopt one beyond binary case cm cm cm feature brief description classic handwritten united states service capability digit create experimentally collect optical character objective black rectangular pixel display capital letter english file bioinformatic regard concern neural international conference network handwritten come national connection record detect type one categorization document appear document extract census original aim predict individual us year series task appear minimal instance subsection svms rbf reason kernel know choice svm capability fw handle polynomial satisfy small svms use among pattern logarithmic set consist randomly computed determine fine parameter tune significant subsection effect fw adopt method require evaluation quickly possibility scale sampling technique call overcome cardinality major previous generally technique rely equal least choice make adopt lrr c dataset computer core ram implement source code report concern accuracy support size number fw respect training algorithm fw contrast considerably pattern great fw importantly speed monotonically range fast time significant training set obtain dataset create scalability pattern fw confirm propose tend dataset actually fw case reach fw moderate exhibit test accuracy fig correspond accuracy speed svms price slightly fw significant grow fw particular peak show case tend however exhibit around ht l l cm acc acc std acc std e letter e e e baseline acc std repetition experiment protein train std std e e e letter mnist protein time second algorithm baseline correspond deviation std dataset accuracy run use fig pt cc paragraph significance perform reject conduct performance adopt method test safe parametric binary propose apparent advantage accurate fw run test cm fw p cm fw rejection obtain book p sign preferred tie employ default correction suggest cm statistic fw accuracy w sign rank test respectively cm binary fw vs fw fw time fw vs run low significance level low summarize conclusion hypothesis confirm time run section run magnitude well conclude training statistically conclude extension fw reject reject fw reject accuracy test significance use pattern conduct experiment study effect fig time accuracy change confirm fast svm h c solve svms capability propose even satisfy conduct homogeneous squared use meta note however determine optimal consider capability fw wide thus great flexibility building summarize result test accuracy svm select experiment fw demonstrate ht cm cm dataset accuracy std times std fw fw fw fw fw fw fw time statistic repetition experiment detail competitive tendency favor speed sometimes expense fw among single subsection several binary subproblem great subproblem subproblem quite run bc reasonably traditional constitute fw instead capability handle show testing offer fw say fw large order probably lack size solve finally fw advantageous offer considerably run comment understand performance gradually commonly scalability regard appear encouraging fw outperform collection cpu speedup increase reach order fw algorithm accuracy clear fw probably explain considerable dataset evident become advantage start optimal support algorithm try number progress towards remove approximation imply weight vanish drop consider possesse remove directly enjoy base computational geometry svm algorithm base frank wolfe scheme analytical formula learn solution compare present compare testing accuracy confirm conclusion reach wolfe build prove software conclusion statistically assess non parametric preliminary handle wide kernel allow flexibility variation theorem quadratic programming qp become expensive traditional mainly restriction adopt condition within svms core task building embed propose novel method svms frank wolfe increasingly complex step case sometimes price binary classification wide support machine problem improve technique svms optimize optimization usually formulate qp require complexity major effort scale dataset structure traditional svm address small svms essentially training call prominent minimal disadvantage method exhibit slow close get slowly sensitive size active like use avoid scale adapt qp still handle efficiently look new propose small contain adopt impose recent independently point build core demonstrate new method software start solve small subset look point outside define large new current approximate prescribe sequence optimization problem use efficient warm avoid full gram ref employ end variant algorithm formalism need wolfe optimization define simplex solution considerably idea nest linearization exact line linearization consequently external numerical solver discover support look separate hyperplane maximize misclassifie unseen deal noisy example compute margin svms address svms see g degeneracy address
spline output consistent evenly spaced covariate visually credible posterior gibbs sampler plot surface generalise mixed spline give parametric basis degree block parametric effective degree freedom draw way density trace goodness number describe effect schwarz residual autocorrelation autocorrelation remain matrix chain u remain residual integral predictive observe forecast forecast ideal forecast calculate routine autocorrelation beyond used may uniformity choose predictive forecast credible distribute normally credible divide half give performance methodology ability univariate bivariate interaction model significant amount illustrate simulate covariate uniform fit trend evenly spaced interval univariate second spline order bivariate consist product two basis six spline order penalty prior direction intercept weakly ar residual area include planning environment portion particle linearly week measure flexible motivated generalise spline generalise average record record different autocorrelation fit spline motivate estimate capture maxima occur trend represent significance parameter type size covariance nd week year spline wind spline nd penalty wind speed temperature nd penalty spline nd penalty nd penalty fit thin spline wind speed effect order polynomial tensor wind attempt spline wind spline undesirable final fitting table wind wind term replace spline spline fourth daily trend replace trend change year exclude inclusion fourth daily trend fit trend daily trend trend basis mesh copy year residual lag hour hour lag capture hour hour fitting daily variation fit daily year year estimation conduct draw discard burn periodic interaction interval periodic similar uniform covariate htbp capture curvature function trace posterior posterior converge unimodal whose figure trace distribute credible interval distinct parameter maximum posterior occur still approximately long parameter coefficient coefficient autoregressive ar residual final distribute marginally htbp contour fit spline figure spline log wind either remove nearby trend year daily trend vice daily exhibit pm trend day daily daily peak pm show posterior residual exhibit autocorrelation semi regression daily trend temporal variability explicitly residual significantly examine model particle air middle fairly peak dependence peak traffic increase hour dependence wind direction mark direction peak occur around effect trend wind proxy good result present effect temperature wind generally wind effect estimate credible see lag credible interval represent specify regression autoregressive daily trend peak occur pm express specification quite daily trend autoregressive daily trend trend autoregressive contiguous model residual select residual find intercept intercept freedom freedom spline around predictor agree spline sd intercept daily trend wind direction wind wind entire forecast modelling base reality weather forecast estimate model observe forecasting rather combine spline autoregressive method underlie smooth autoregressive b spline small none smooth temporal trend result function exhibit fourier despite basis sampler ensure first sample discard sample stable density spline count basis wind reduce indicate dependent basis flexibility univariate basis basis qualitatively model residual show lag daily trend autocorrelation model capture smooth trend explain trend model autocorrelation residual methodology outline spline rather allow knot vary reversible jump use knot necessary smoothing fit choose covariate ensure spline typically knot model product spline investigation particular functional guess random glm generalise analogy generalise product spline identity stay include desirable come argument setup basis form tensor spline present provide fitting non focus cyclic autocorrelation residual model much remove wish division sciences institute health innovation school sciences science author wish feedback suggestion development comment thick observational exhibit cyclic trend autocorrelation realistic forecast combine spline modelling model auto error simulate efficacy particle autoregressive generalise particle continuously series exhibit temporal dependence covariate motivate able take account specify smooth surface time covariate establish spline extend penalty periodic basis basis generalise additive model spline periodic trend spline coefficient evolve admit error forecast fitting assumption independent identically residual modelling autocorrelation realistic traditional distribute error specification move model lag term series inclusion account remove lag interpolation trend adjust examine temporal forecasting forecast generalise model b despite development spline incorporate approach error cubic spline spline spline function smooth combine spline parametric modelling error propose define autoregressive ar incorporate model outline review generalise spline outline penalty identifiability spline spline autoregressive error discuss metropolis forecasting discuss include section series dimensional covariate show model trend error covariate forecast apply record section review light suggestion make extension model present parameter belief posterior belief mathematically condition posterior parameter specify response identically additive covariate discuss semi methodology matlab fit whose omit generalised treat equivalent generalise spline alternatively augment say glm express spline glm glm spline many equivalent spline non knot linear adjacent knot basis overlap desire covariate grid knot spline constant knot basis create recurrence cubic spline three piecewise knot knot e zero interval flexible response basis spline spline affect speed mcmc knot choice particular spline basis make spline include spline order b spline piecewise piecewise individual grey scheme across thick grey thick black show b spline zero covariate boundary preserve impose fit spline introduce spline frequentist spline coefficient derivative spline element spline equal row operator analogy sum penalty conjugate multivariate coefficient spline parameter small concentrate improper penalty overcome add spline analogue periodic trend spline evolve accord daily treat daily evolve accord day obvious factor continuous sense spline term covariate tensor form product identity size element penalty spline formulation penalty term two covariate matrix block corresponding penalty covariate indexing spline univariate reflect different amount smooth dimensional precision formulation amount direction multiple address evolve recover walk spline penalty parametric polynomial model outline periodic spline briefly choice basis parametric modelling periodic fourier series term ensure spline identifiability already treat e equation fourier weakly informative structure diagonal fourier series trend include function high spurious coefficient zero use periodic spline adopt merely point basis term effect basis covariate element simple assume factor precision knot knot space smooth spline basis hour treat lag describe autocorrelation structure residual multivariate prior censor absolute strictly order prior precision weakly variance autoregressive error conjugate forecasting realistic successive within residual mcmc forecast readily important dependent forecast ignore uncertainty forecast would produce formulation respective weakly informative prior respectively autoregressive precision informative block smoothness spline hyperparameter prior treat similarly consist distribute spline penalty coefficient model htb lag lag include autoregressive similarly contain column residual lag euclidean distribute autoregressive condition vice versa comprehensive acyclic marginals hasting fill thick minimum black thick mu edge beta edge eps eps delta edge lambda draw black corner inner
parallel computation provably converge nested problem complement backpropagation setting serial computation reach cloud become device etc thank availability area computer speech compute mac many choice optimize step nsf award input deep net net simplify ignore bias form follow nonlinearity sigmoid per hidden sometimes notational particular constraint single feasible mac qp loss nest subject qp bound derivative us compute c z f nk gradient suffice look gradient r short construct eq kk problem rather qp define problem minimizer nest give qp satisfie nest qp stationary n satisfy eqs point problem remark since constraint infeasible converse true stationary define eqs hold q mac constrain minimizer cm thm prop thm corollary thm conjecture thm wang california sound nonlinear layer sometimes inspire offer way ever example computer front joint consider numerical optimization execution suboptimal system strategy learn extent replace original involve augment without alternate coordinate mac provable derivative competitive serial providing model year practical applicability ever powerful machine translate natural physical limitation serial suggest scalable thousand processor cloud hierarchical neural originally brain successful face define hierarchical feedforward mapping dataset numerically transform output linear sigmoid net map wide net unit encode net cascade scene understand process classification reduction architecture share ideal nest layer objective probably occur stream visual composition produces inherently usually backpropagation compute gradient sgd feed optimization convergence criterion satisfy suffer layer higher slowly slow researcher practice net around architecture net recently initialization much computer architecture mapping decade remain good strategy architecture net front combinatorial search trial costly effort solution often irrespective cascade lb lb lb lb optimization nest call partly parallelization optimize describe give illustrate advantage mac give input deep net ignore k mapping nonlinearity scalar sigmoid output fully connect share term backpropagation auxiliary per unit equality see intermediate hide problem minimizer complicate involve small ill cause may solve mac quadratic qp mild converge minimum original follow path break unfolding every square involve equally condition gradient gradient qp single unit unit k n nh kk separate coordinate derivative step exist new however proximal operator mac complex couple step auxiliary dataset potential operate variable qp objective function iteration achieve along coordinate device algorithm nested architecture convergent nest architecture elimination architecture need feedforward net coordinate semi net hide auxiliary nest step lagrangian rather depend gradient newton inexact step mac maximization alternate multiplier optimization area illustrate mac deep autoencoder technique need mac qp decrease reach mac serial remarkable minima unnecessary perform inexact qp term common net error validation nest sophisticated take maintain mac qp fast post auxiliary coordinate simply kn kk kn weight except fit one prove value another advantage heterogeneous architecture layer perform specialized quantization recognition layer radial weight layer jointly minimal something easy mac autoencoder encoder decoder code net remain mac qp large decrease iteration advantage mac value architecture architecture traditional architecture architecture criterion picking involve suboptimal hand select mac achieve layer let coordinate e ce function aic km choice mac step separate require testing layer still costly parallel alternate run multiple iteration guarantee decrease near mac reduce exponential complex architecture mac rbf autoencoder try architecture early mac mac formulation full generality equivalent optimize jointly auxiliary quadratic exist several line qp give set principled way net nest unit past early recent sparse activation net go early day neural net backpropagation good representation important deal activation objective activation intend distribute representation nest develop net nearly focus reveal parallel processing aspect truly deep extract overcomplete dictionary one nest work layer backpropagation limit nest recent greedy weight sequentially optimize weight converge problem machine homogeneity dimensionality mapping dimensional mapping minimize nonlinear spline neural net radial nonparametric function drive separate infinitely hoc prevent dimensionality supervise dimensionality mapping rbf see version single layer therefore result nest biased summary work solve representation encourage mac auxiliary purely mathematical construct parallelism good none heterogeneous layer criterion separate mac introduce however term functional negative contrast mac new break able thank introduce subproblem mac characteristic deep autoencoder heterogeneous architecture autoencoder speedup mac alternating square nh gauss approximate hessian search search backtrack practice minimizing separate formally enter objective nearly gauss mac gauss reasonable result intend efficient augment lagrangian combine inexact warm unconstrained alternate explore topic parallel mac qp yet processor nearly parallel toolbox programming effort matlab processor run processor report matlab toolbox parallel model cache dataset handwritten autoencoder architecture map dimensional try reconstruct mac qp introduce auxiliary learn curve machine image handwritten digit training equally digit autoencoder logistic layer weight layer dimension fan initial adjusting attain give gradient random weight mostly give initial mac qp run newton gauss since lead local optimize inexact nest evaluate tolerance term tolerance classical stochastic descent conjugate gradient sgd carefully ensure minibatch learning cg regard cubic interpolation backtrack long minibatch batch mode work plot objective initialization validation closely train iteration mac cg epoch red circle mac processor also show right sgd cg iteration error mac qp beginning reach mac qp serial remarkable linear processor objective runtime mac mac c ex nest marker iteration mac epochs sgd mac qp quadratic indicate red solid processor parallel mac curve processor memory toolbox runtime hour mac opt mac c nest marker mac optimization reconstruction sample initialization embed algorithm mac encoder decoder rather rbf train mac qp introduce benchmark sequence degree thus close manifold pick half cat leave autoencoder bottleneck layer dimensional visualize reduce decoder decoder radial hide image decoder concatenation layer total million weight problem minimize plus quadratic layer weight rbf stage encoder one train center mean fix input obtain preferred fully center achieve net strategy network beyond example vector machine slice wish train deep autoencoder construct rbf center alternate mac approach alternate step mac objective embed decoder slow mac mac problem step encoder decoder separately
primitive infection cascade result np another paper cast albeit good knowledge crucial cascade learn mrfs classic cascade epidemic assume parent child child edge reverse happen initially become seed every try child active correspondingly time node become inactive spread infection infect sir epidemic remain epidemic never reach inactive sample path cascade sample cascade four around square around active inactive respectively inactive time child turn inactive step epidemic stop epidemic cascade observe seed infect infection times cascade cascade refer complexity cascade super prior network direct super node parent course super infection find node cascade cascade cascade thus cascade infected speak graph far seed far incoming edge follow infection time node infected node infect presence provide infected attempt edge activation sample number need graph meaningful interpretation need time infect cascade via theoretic knowledge estimation infection find identity associate infection ml value maximize likelihood crucial particularly nice enable particular j set parameter parent every finally sample infection vector seed concave please sum depend variable algorithmic proposition small subset solve parallelization speedup fast super infection analytically cascade reliably optimality program particular complementary condition program neighborhood formally involve optimizer define prop eq analytical cascade needs reliably node consider super graph strength parent cascade great ml algorithm e enough remark appropriate derive corollary depend entire neighborhood I infect reliably error scale value sample great set need entire scale system parent seed reasonable seed next slot make hard neighborhood counter indeed degree node power learn ease restrict algorithmic cascade specifically parent amount generalization enable implementation formally denote child infect recover less information diagram series q entropy e conditioning inequality apply need low lemma cascade decay corollary ensemble graph information bound set use time infected need order specialized case number sample particular degree subgraph thus parent super allow corresponding must big exact remove ml graph ensemble choose result sir cascade establish connection classic formalism mrfs e g introduction sir direct graph define parent e direction arise comment causality result sir sir infect discrete causal govern parent include encode history neighbor history cascade node neither cascade random mrfs classic formalism enable central notion therein collection every condition purpose graph variable clique direct graph connect either child relationship common child undirecte direct present child infection generate sir epidemic direct graph graph epidemic evolution graph imagine divide assume specific mrfs ise discrete pairwise require knowledge enable free test know typically causality generate apply sample leverage example phenomenon alternative formulation acyclic undirected cycle causality empirical evaluation ml greedy synthetic graph pick mention structure govern number infect set super leave infected grid cascade grid graph vary much presence cascade regular ml scenario regular graph see extent reduction sample complexity fact effect super direct graph twitter tweet feed put direct choose person edge weight parent simulate cascade use greedy super axis neighborhood h scatter number infect ml explain treat graph learn super example linearly linear cycle graph epidemic cascade time get infect require establish theoretic paper epidemic cascade extension infect weak infect cascade much ml result assumption reconstruct entire trajectory weak observation fall class propagation focus one decay cascade reach cascade seed reach would efficient case well interesting performance greedy greedy epidemic give ex pt lemma epidemic cascade infect offline online analytical impossible cascade cascade sir establish cascade require ml greedy reliably graph tree theoretic tight super cascade sir epidemic cascade infection keyword cascade field epidemic successive spread failure phenomenon nod large disease cascade recently example include network understand consumption medium e video news article spread twitter facebook keyword epidemic cascade optimize security reliability epidemic cascade spread complex epidemic protocol file fashion basis popular content streaming network learn vast majority graph small etc affect spread cascade cascade spread infer learn first find influential online network facebook access graph user link priori spread offline social practice seem twitter survey drive cascade structure learn cascade primitive investigate summarize many reliable recovery main characterize characterize observation need
sign impose reduce use allow detection dependency furthermore feature weight interaction simultaneously deal redundant redundant increase implicitly deal issue avoid redundant weighted help execute tend systematically solve perform constraint firstly assume feature procedure start algorithm step search eq middle find update accordingly find reach subset feasible rr time subset find sort k close head one estimation form product eq overlap kernel classification otherwise reduce simply gradient positive onto ball carry practice sophisticated e newton super gradient linearly take general method convergence rate rely per iteration line search find sophisticated solver actually decide efficiency instead every iteration significantly iteration approximately section report experimental result backward replace redundancy relevance state select mutual relevant solve problem obtain program trade low redundancy categorical correlation categorical experiment recommend weight solve varied another discriminate experiment toy binary bernoulli take probability chance bit flip characteristic redundancy dependency binary function characteristic interaction candidate selection candidate adaptively scale median available f adaptively distance convexity time randomly method f report correctly select feature measure none tb dataset lasso pc rank redundancy feature redundant forward necessary simultaneously f unstable instability incorrect heuristic become non inclusion many obviously make b since start feature well detect dependency feature sometimes redundant feature keep one eliminate strategy pairwise feature interaction therefore surprising fail information reveal dependency since feature interaction drawback redundancy consider hence dataset pc consider feature simultaneous feature interaction regularization combination detect dependency correctly choose usage attempt maximize manner flip inclusion feature illustration feature problem evident high inclusion drop extreme table considerably value cccc cccc demonstrate conduct experiment real summarize classification dataset cover wide include speech bioinformatics dataset task class per per class uci repository second international evaluating house repeat time trial low entire dimensionality kernel use availability selection competitive especially class pc multi toy experiment pc take redundancy among exception would problem pc dataset component component redundancy consider redundancy see suggest ignore redundancy many unstable another bold face method pair test significance mark problem large computation perform number time number choose alternative interact non interact explanatory detect interact c segment speech dimensionality help facilitate learn predictive cause difficulty include redundancy feature sparse importance keep allow interact dependency implicitly redundancy obtain usefulness method valuable comment international mm institute technology mm department institute technology feature technique less important feature exist algorithm redundancy select key characteristic world attention attempt interaction propose maximize variant output numerical handling redundancy feature mutual supervise number e learning could lead interpretability useful aim remove unnecessary target many show insight ability removal feature relevant explain redundant likely output redundant another feature two context explanatory characteristic interact consider reveal attention focus redundancy study consideration square information output experimentally art artificial real handling detect interaction formulate describe sect several measure argue list regularization weight account balance consideration mutual information possess desirable argument sect propose refer artificial sect sect formal realization feature attempt identify index cardinality index feature accuracy classifier yield classification interpretation wide applicability practice feature amount challenge np guarantee exhaustive since impractical quality issue sect quality sect strategy feature complexity optimization exhaustive optimization find strategy fast feature redundancy feature give feature form joint operator hilbert schmidt could xy k universal kernel gaussian dependence property criterion heuristic set theory dependency extraction mutual information kullback divergence measure vanishe density difficult avoid information accurate computationally expensive existence another square mutual pearson information log kullback like information go goal close estimation formulate least square minimize since find tractable learn admits find optimal purpose q respect call possesse selection fold nk k hold
consensus step accelerate convergence distribute average differentiable communication substitute induction constant induction compare side constant lemma mit edu distribute method private objective topology objective distinct share favorable suitable computation proximal communication network nesterov acceleration multiple per round distribute superior verify distribute enable storage agent number network objective local private information agent maintain optimum updating iteratively team explore regularize square fit distribute learn sample variation order gradient subgradient function computationally naturally implementation typically agent carry subgradient average exception distribute gradient communication converge additive distribute vary maintain update step take step negative function neighbor consist communication agent estimate current neighbor consensus stage nesterov stage bring estimate agent proximal step allow inexact centralized use centralize establish exploit objective proximal accelerate nesterov acceleration fast cite paper method global problem use parallel computation relate consensus grow optimization decentralize subgradient direction multiplier build seminal paper centralize proximal organize gradient establish rate verify effectiveness finally conclude open scalar centralized write average denote centralized standard product ij stochastic exist number hold concept establish key subsequent give section inexact characterize respect minimization optimization attain map optimality proximal operator basic proximal operator scalar view inexact proximal control iteration inexact centralize characterize inexact attain accelerate inexact proximal iterate eq indicate q sequence k inexact gradient verify inexact multi proposition allow omit sequence th order integer every therefore make q objective private function g ix hx n follow continuously differentiable gradient scalar exist subgradient ix first propose proximal problem initial agent k iy I k stepsize scalar represent vector agent communication agent exchange value value step consensus shall estimate serve method agent update estimate along gradient differentiable part multi consensus proximal respect objective follow nesterov type acceleration stage perform us error inexact centralized method step vary condition double doubly scalar say receive neighbor interval exist part agent equal update estimate current receive pair limit establish let shall decrease geometrically respect communication inexact centralized proximal inexact centralized method satisfy q lipschitz centralize inexact simple expansion inequality hz fact optimizer z hz combine desire bound term control step consensus inexact exhibit optimal section polynomial geometric establishe iterate sequence k x l give recursive let scalar polynomial geometric polynomial geometric proposition therefore polynomial geometric sequence distribute step consensus generate kkt take communication iteration number require execute communication great equivalently state proximal operator clear communication sequence error sequence address step consensus wish small guarantee become result equivalent nonnegative hold since
minimizer last follow specific let datum probability summary generative dataset refer list intended admit low dimensional model totally structure contaminate outlier I ambient dimensional ambient increase outli start chance scale energy dimensional parameter result ratio contain information behavior dimensional stability number draw except appear restriction comprehensive statement constant statistic outlier contain recover perfectly stability sufficient condition outlier seem fix ambient dimension l subspace number increment increase increment specific invariant heat identify outli ambient leave find dot regime capture qualitative determine identify experiment repeat probability fit theoretical bind empirical understand behave control definition statistic plain residual l square subspace exceed claim q noise stable regime stability suggest experiment basic perform linear subspace dimension space increment increment model determine close statistic projection give repeat heat blue heat map begin regime pca heat mean top valuable tool robust semidefinite interior problem favorable numerical reweighte whose weight evolve proceed svd exhibit type directly motivate estimate eq seem plausible minimizer computation water water appear box label correctness dataset solve matrix pt heuristic motivate counter weight solve influence increment box guarantee converge pt observation regularization stop pt satisfie initialize iteration counter initial increment optimal optimal objective fail convergence satisfie q establish schema summarize cost read keep typically point somewhat solve subproblem computation calculation arithmetic operation matrix spectral calculation arithmetic operation achieve calculation direct compute thin approach also iteration accelerate randomize page dimension curve associate choice ratio algorithm sequence detail experiment assumption iterate amount omit analysis indicate empirically experiment ambient outlier mark single unique run tolerance compare iterate package precision measure frobenius seem experiment usually much leverage iteration proof take machine algorithm terminate iterate error practice precise result degree reasonable much database numerical experiment involve formulation solving summarize recommend procedure algorithm pt pt center datum dataset dominant assume center center observation center q incorporate center modify problem omit detail second formulation address point solve reference program denote transform experimental idea pca finally proxy small exceed solve huge proposal limit attention formulation polynomial programming success largely consequence hard way standard pca spherical pca observation sphere pca perform et recommend reliable observation solve consist outlier comparison sparsity decomposition consider effective design several generalize single face database image convert pixel face subtract median apply spherical pca subspace choice experiment display project compute nine center every onto meanwhile face describe pca capture well different database original project tune noiseless xu al underlie ratio dimension model appear standard draw subgaussian method alternate high numerical experiment significantly fast strategy readily different situation observation scenario semidefinite relaxation work zhang minimum obvious equivalent indeed together implicit observation yield claim subspace specific tight relaxation find dimensional point analogous inverse determination dimension eigenvalue present know incorporate require requirement convergence idea broad perspective nonconvex problem establish combinatorial relaxation work relaxation proof maintain page statement proof section compare perturb objective see triangle eq justify third reach residual dominate become apparent restriction lemma insight behind involve side derivative direction confident take directional recalling perturb sum separately term involve l twice outlier page outli therefore q take limit determine produce low directional low bind component bind section alignment substitute expression directional bind finish relation draw norm involve consideration explain argument ingredient simple express comment obtain homogeneity thin value dl normalization unitary euclidean side concave construction extreme reach finally state block equivalence norm inequality equality third coincide trace exact model probabilistic statistic statistic follow detailed fix parameter let dataset accord page satisfie except verify demonstrate simplified collect numerical old reach probability bound alignment bound gaussian develop next except add side obtain second right invoke calculation rademacher sequence draw sum unit rest jensen expectation close dominate centering probability f frobenius complete develop alignment matrix know see whose standard normal cumulative column sphere argument distribute sphere gaussian variable degree freedom jensen normal cdf proposition complementary algebra use simple arrange recall ratio begin invariance imply next statistic independent center probability bound statistic satisfie complete contain argue near optimum solve problem correctness statement range eigenvalue q construct via quantity whose either enforce convention proof satisfie optimizer point simultaneously nonzero among therefore objective perturbation expand particular condition ensure derivative choice side continuity solve index eigenvector argument observe semidefinite negative perturbation optimal phase iterate matrix fact point achieve huber highlight interpretation continuously strict objective generalize auxiliary fact argument prove iterate innovation handle definition collect term next verify relate gm relate b equivalent monotonicity eq motivate stop criterion imply f require information bilinear inner strict imply span impose constraint set argument necessary constrain minimum function convex iterate line however characterize global nearly identification define dx x latter reach respective objective zhang support nsf grant dms dms award anonymous remark improve manuscript rgb rgb theorem theorem thm thm definition thm thm fact thm e computing sciences institute technology mc california ca zhang institute mathematics se mn explain along outlier describe reliably low use relaxation orthogonal formulation document confirm find lie near dimensional describe model huge array highlight bioinformatics pt illumination nine body lie dimension camera involve rank concern topic dimensional snp use correlate model difference substantial outlier principal linear sensitive consequence scientific year researcher alternative principal favorable low approximate convex proxy discussion dimensional formulation convex demonstrate cope outlier optimization seek norm matrix frobenius refer refer trace semidefinite symmetric dd trace range great exceed integer respectively extend transform motivate research begin principal explain residual approximation subspace classical method fit minimize residual elsewhere sum dataset appear pca linear algebra convex right nevertheless analytic datum column arrange svd extract principal imagine contain hope another challenging pca linear model less possibility residual sometimes orthogonal absolute book extensive way mathematical contrast tractable although many none minimum proposal linear involve intractable option analyze rigorous linear optimization propose replace straightforward equal eigenvalue enforce eigenvalue convex set observation dimension attempt tight formulation restrict argument effective rank provide good hull feasible impossible constraint reduce ultimately immediately clear auxiliary q norm solution dominant invariant precisely spectral entry weakly straightforward good fit even outlier solve issue accept original believe solve involve subspace two hull relaxation character robust type theoretical numerical indicate type model manuscript appear apply orthogonality approach suggest relaxation close outline develop describe dataset outlier simple appear section evidence effective setup locate outlier ambient property result gives summarize leave remain lie ambient structure contaminate l dimension space dimensional subspace dataset outlier model locate near hope assumption geometric check find subspace evidence outlier exhibit structure unsupervise justified finding let discussion imagine point must sure summary statistic design requirement direction statistic direction along approximate residual pca outlier major linear exhibit signature
cl bivariate margin explain either cl n frank std deviation create multipli car std cl h cl normal copulas lr lr lr c margin margin multivariate normal variate margin bivariate c lr paragraph copula std deviation create multipli car std prop prop prop prop lemma prop detect distributional change series often lack alternative change sequential copula contrast attempt rank computed respect consequence multiplier resample serial account scheme finite sample involve financial present give alternative involve f multivariate regard capture empirical appear little margin unchanged copula evidence detection particularly change practical change structure good copula obviously copula change test functional copula power aim powerful alternative copula crucial lie computation rank whereas rank rank way accurately copula detect quickly phenomenon empirical copula copula another illustration dependence organize follow asymptotic nan contain description multiplier resample validity result simulation defer rest equip uniform describe statistic difference ij l dependence tie process rank compute continue adopt convention write ns test copula aggregate er von aggregate thought find powerful er von nan value distribution limit multipli reject change joint could concern marginal copula change statistic differ copulas copula rank subsample compute process compare use process one obtain analogue carlo usually powerful copula intuitively copula copula nc empirical bivariate empirical course cite serial write sided stationary margin side link statistic difference eq focus briefly denote mix integer say side empirical define strong continuity absence dependent actually pointwise continue weak actually transform convention ns u aa state asymptotically uniformly turn trajectory probability one consequence continuous argument draw continuous margin satisfy corollary covariance compute resample back suffice resample partial observation spirit resample frequently compare behavior various resample result state later case multiplier seminal multiplier extend multipli bootstrap sequential key idea replace multiplier suitably serial dependence h way sequence appendix copy multiplier define multipli resample rescale rank arithmetic convention subsample x ns ms partial simple vary slightly upon follow canonical form effect ns nan reject estimate significance statistic multipli multipli multipli sequence condition copy multiplier strong copy multipli margin satisfy also strongly automatically appendix margin keep assume break assume change copula frank version obtain device hypothesis single occur copula one copula copula marginal rescale rank rank fast base sample package sake brevity table provide percentage alternative involve change copula copula serial independence change copula value margin occur besides conclusion reasonably serial table minor fluctuation statistic copula copula serial conservative size consideration base powerful alternative involve table power low distinguish copula basis low fact change point make involve detect copula another reading regard robust behave rather illustration apply computed index bivariate multipli explain value provide evidence course early decide basis nd compute daily composite former take package th approximate extreme occur period report sensitivity constant change achieve limit statistic resample sequence reject rather concern margin one change margin intensive rank calculation next speed grateful mark anonymous pointing might center nonlinear project foundation contract de e grant p science integer let margin quantile collect convention quantity replace decomposition supremum small ij complete shall case strongly mix statement map put j du jj continuous second side converge assertion boundedness third converge zero u tu assertion observe without ns ns assumption existence q verify asymptotically finally display desire convention sum zero decomposition definition respectively take different value observe exceed specify stationarity markov large corollary show n ms probability exist set u n sufficiently uniformly exist use fact ns
condition truth goal objective capture match experimental adopt fitness experimental free maximize focus art choose going define follow protein logic compatible logical give hypergraph weight finally combinatorial several condition protein logic experimental compatible want total number output compatible implementation systematically generate comparison logical perform identify compatible middle scale case dataset predict whole consequence see equivalent minimal compute together obtain minimal solve middle minimal second take second equal show score second enumeration take boolean show genetic material optimization hour predict whole dataset evaluate depict perfect depict compatible perfect appear number compatible therefore suboptimal run large genetic minimal phenomenon dataset optimization approach minimal recall find global clear nonetheless percentage axis experimental computation plot evolution experimental I contain maximum time score go minute hour outperform issue test conclusion formal boolean rule paradigm rely powerful free open source software main conclusion reasoning solution moreover outperform order opinion optimal issue robustness shall suboptimal gain whether biological relevance loop hope offer author project b partially support protein protein integer program logical steady biology mathematical logic become approach datum introduce heuristic solve paradigm encode logical program answer solution improvement efficiency guarantee well application cell via effect pathway gene define decade biological pathway exist public pathway sign oriented retrieve sign orient inside event signal molecular network vast concern type starting generate high throughput correlate protein protein include control modification key condition design insight control throughput establish comparison public resource integrate knowledge exist recently introduce implement www genetic logic compatible realistic suffer lack guarantee intrinsic train experimental equally optimal experimental regard explanatory related event reverse engineering assessment www project encodes distribute general public offer boolean constraint technology search logic program modern complex preference optimization global reasoning complete search obtain go towards prediction combinatorial logic represent graphical representation logic give motivate formal formally function logic relationship capture graph two protein encode graph either logic gate logical network represent adopt formalism example reader cite literature compatible logic formula logic sign conjunction green activation green node nod drug specie measure toy b logic circle whereas multiple informally logic good explain criterion quite intuitive appropriate way clear give base convenient protein logic clause three input sake loop mechanism transmission pathway include responsible signal input sign acyclic direct measure moreover subset mutually sign happen extract one database compress experiment correspond force tool experimental experimental function rv rv proteins active inactive great low approach future compare formal dataset appear definition recall domain property define example look hypergraph hypergraph capture evidence property logical positively resp formula redundant minimize model reduce formula redundancy logical conjunction
upper evaluate task audio corpus sub sample atom initialize extension use code algorithm number iteration result optimum spectra train dictionary reference line spectrum figure coherence inactive decrease decrease svd result middle increasingly unable consistently self coherence svd dictionary self enable maximize sparsity code atom efficiently dictionary coherence joint atom atom achieve objective code approximate demonstrate benefit coherence acknowledgment excellent implementation valuable feedback draw code learn dictionary successful denoise separation solve dictionary dictionary iteratively factorization dictionary sparse code dictionary coherence coherence class code self residual code goal high coherence self seek trade depend application dictionary effective self train enable trade sparsity approximate frame coherence particular code dictionary place densely region redundancy similarity dictionary atom angle atom self permit well admissible coherence increase avoid atom degeneracy enable span objective self coherence dictionary support generalization orthonormal contain unit span recover natural signal suitably achieve dictionary atom atom densely feature however redundant long unique need omp orthogonality atom maximum note definition self product magnitude therefore consider sec self dictionary magnitude dictionary formally exist minimum coherence self state sparse code recover norm curve omp al control gram approximate expert appropriate family provide adapt characteristic advance therefore dictionary suited source atom atom another atom lie atom representation atom coherent multiple atom practice dictionary self coherence fall self use independently atom algorithm pair inner bind increase coherence atom pairwise iterate step grow present self atom step sum multiplier possible trade sparsity code maximize dictionary signal parametric necessary furthermore make incoherent empirically task training incoherent improve unseen approximation sparsity jointly propose alternate minimization convergence optimum follow svd atom newly update replace large incoherent atom one atom replace atom therefore self coherence update fall atom well enforce constraint introduce atom optimize self minimize thm lagrange multipli trade
supervise account uncertainty unsupervise consider support national science foundation fa cs model online transition specifically achieve complexity unknown balance exploration exploitation able rapidly past rl allow mainly learner capability require interpolation process learn highly accurate provide determine implement control method domain accumulate maximize one rl learn g obtain physical system interact environment real able quickly amount robot interact environment good optimal learn interested state smooth dynamic typical primary keep complexity rl spirit extend continuous space gp part model dynamic environment transition prediction accurate amount implement non parametric gps give enhanced modeling flexibility automatic determination gps us implement exploration way bellman via concept key estimate bellman behind transition simple transition inclusion often sharp bellman actual bellman point gain efficiency compete advantage transition estimate conceptually closely relate fit interpolation primary contribution gps trade practical unlike could implementation learning approach make always dimensionality space grid grid bellman scale exponentially advanced grid allow somewhat increase curse alternatively nonlinear unclear approach really application curse still smooth state close close interpolation work fundamental gps observation small bellman operator rather learn comparable assumption justify criterion specify something generate sometimes elsewhere make amenable tb transition cm mdps reward justify control discretize assume discretize reward action rx x addition continuity fulfil domain violate e car else despite case still action accumulate maximize decision find bellman yield value bellman operator well number bellman uniform order solution affine function grid write grid cell interpolation invariant perform apply obtain apply action respect grid z vector associate point grid slightly discretize bellman contraction point control approximate posteriori residual grid cell modulus continuity continuous way rl problem agent environment sample observe behavior goal keep possible done use place sketch plan interact action discount planning interact current choose execute observe store evaluating criterion plan step exact determine else explain detail functional module essence theory regression particularly flexibility automatically principle via optimization marginal automatic determination relevant determine guide gps target estimate triplet perform dimensional change state variable independently univariate gps prediction respective subset e lack sketch subset regressor approximation element subset greedy aim minimize incur decomposition avoid degenerate variance gps gps rl also study offline gps functional parameterize tv constitute adapt relevance determination projection whereby variance formula find uncertain stacking together per uncertainty ensure receive scalar maximally maximally uncertain discretization planning stage plug let uncertainty reward solve discretize bellman repeat maximum reach necessary gr une case update empirically make agent space uncertain employ bellman suggest show become uncertainty either explore unknown explore exploitation take process discretization keep final call model stage plan summary model planning discount give see update reached examine various well comparative car hill car powerful hill necessary hill opposite car velocity possible reward hill reach maximal episode discount velocity single link car provide need energy create angle angular discretized remaining discount next consider balance state roll angle roll angle bar angular velocity inherently vertical turn usually experimental setup comparison start recovery impossible discount goal link robot height since problem setup state initial episode episode end state pass discount four negligible offline efficiency grid car balance tolerance full planning second problem min large spend planning module offline transition possible domain step balance total gp tolerance hyperparameter would computationally perform single optimal
share availability secondary result scheme propose algorithm asymmetric planning approach cognitive justify important channel policy primary rely algorithm suffer number channel regret suffer end consider play q refer one channel good channel play upper high index round inversion two sum second equality read slot equal number equality us share index channel play substitute combine previous lead end secondary primary limit every channel secondary select access primary rely algorithm expect number coarse verify least inclusion last inequality ease assertion write implicit km mb write second concentration hoeffding inequality exist km implicit avoid sub channel belong proof event km n mb consequently hoeffding km finally fr cs paper secondary exploit primary secondary environment need learn availability channel sense argue mechanism exploitation formulate mab share integrate interference limit learn ucb resource primary user definition channel spectrum access type primary user secondary user access resource dedicate service pool user resource area refer exploitation network challenge one observation thus substantial communication discard unable activity resource consequently interference quality resource realize secondary address key concept order implement communication user knowledge channel machine secondary among multi mab gain impact explore channel element cognitive scheme enable user quickly resource learn interference among resource interference entity single allocate channel deal uncertain environment first formulate network arm mab derive ucb issue share learn iteration unique per interference rely algorithm modification fair third fundamental result collaborative secondary network scenario yet learn sensor environment simulation rest paper organize work find consider mechanism instance job assignment detail discuss describe mechanism empirically evaluate section conclude strategy resource brief relate extensive allocation present various multiple secondary mab perfect channel acquire suggest channel multi user tackle lead nash equilibrium know since several paper suggest mab modeling tackle author context tackle model framework analyze draw bernoulli contexts modification take frequency approach decentralize error fundamental realistic scenario require sensor converge show hand alarm detection signal band free lead miss hand slow rely author complex empirical estimate instance contribution find user compete communication yet algorithm handle mention environment fact provide heterogeneous worth specific framework rather expect resource consequently mab framework refer mab rely close explicitly error homogeneous latter address detail primary channel appear channel channel suppose tackle time divide channel slot n distribution realization availability availability subsection generic index choose channel trial per see refer channel centralize decentralized detection denote sense ta k refer channel slot sense accuracy characterize usually probability detection observation channel finally policy policy scope depend outcome access access I interference primary small although send transmission interference occur transmission secondary fail channel channel receive numerical feedback transmission fail share reward share interface discuss aim promise resource reward adversarial exploit learn firstly environment resource index rely computation usual form index confidence bias resource q resource iteration design among round rest consider suggest method job mainly matrix opposite depend worker combination mainly invert solution refer discrete sampling time refer weight maker let refer operator division refer maker assign resource consider subsection making say optimization decision context network formalize resource ensure consider refer discrete define weight use row matrix iteration maker need fail moreover scenario worker solely relate usually detection suggest base resource allocation compute section share enable quality theoretically network among rely previously notation throughput slot equal free observe consequently achievable channel equal eq job context usually throughput expect slot achievable expect throughput implement channel slot symmetric symmetry channel slot every combine every common index slot symmetric optimal compose channel round describe fair collaborative symmetric bound symmetric secondary user primary channel assume secondary access suffer notation matter expect state upper one hand refer behavior report suggest verify selection environment optimal fundamental allocation mechanism environment tight exploration parameter notice converge one previous scale heterogeneous environment piece answer information paper communication interface cognitive share channel receiver hand purpose assume without thus purpose adaptation receiver band communication configuration solely receiver outcome receiver transmission message receiver receive slot communication dedicate compute reward consequence fast rely outcome band aim herein mechanism describe consider scenario result secondary analysis numerical value false impact analyze resource tackle user secondary user divide set symmetric user spectrum secondary mechanism channel inefficient learn condition conduct generalized scenario average horizon regret four illustrate collaborative iteration theorem vector illustrate matter base slot collaborative perform individual regret symmetric learning quality enable observe figure handle notice figure quickly
show evaluation multimodal black however black function train big unit without duration cloud make actual unit desirable understand concept cost second parallel multiple core optimization advantage good learning argue fully parameter critical hyperparameter examine default exponential core experiment kind interested find take bayesian evaluation local hessian find evaluation determine evaluation perform easy justify extra computation formalism review discuss contribution two choice make perform must express optimize choose must utility determine evaluate prior gp property induce elegant marginalization conditional impact section draw observation variance induce acquisition denote evaluate proxy depend observation proceed denote good standard intuitive gp analytically alternatively ei current development idea exploit consider construct course acquisition exploitation show behave unlike upper require improvement function several machine unclear consuming vary take core parallelism onto modern solely non degenerate correspond function determination ard ard mat ern differentiable quasi newton require squared behavior bayesian problem interest hyperparameter estimate result input hyperparameter alone compute observation estimate integrate sample acquire slice chain computationally assume sensible ultimately find quickly possible acquisition expect improvement evaluation execution rate improve optimize good quickly cost resource know use independence duration ask generally batch parallelism however evaluate clearly repeat experiment roll acquisition policy appropriately balanced exploitation roll intractable monte different complete yield ideally new acquisition evaluation simply whose covariance hyperparameter straightforward acquisition select would operate note attention monte highly effective compare machine gp ei optimize hyperparameter ei opt ei ei time gp ei ei tree common bayesian define compare classification task hyperparameter gradient batch epoch run error report integrate ei outperform logistic grid dirichlet allocation lda direct document generate learn variational minibatch word minibatch exhaustive search hyperparameter make online wikipedia article article split validation word count document must specify weight follow analysis lda generally take five hour approximately processor compare strategy expensive restrict repeat time pick report figure achieve compare number time evaluate duration optimization run choose optimize expected improvement integrate hyperparameter ei evaluation ei find significantly figure ei well run fraction min svms svms dna sequence hide term svms consume grid search report hyperparameter avoid find computationally however entropy parameterize outperform expense protein tolerance grid possible split figure grid search ei mcmc version constrain term time optimization strategy status mcmc superior ei gp ei learn strict exploring parameter ei least compare ei problem optimization error selection critical infinite function commonly restrictive tuning multi thorough beneficial demonstrate often prohibitive explore numerous hyperparameter remain nine layer network cifar code tune human competitive publish cifar epoch softmax scale layer optimize nine strategy validation five separate run achieve well find ei error expert cifar bayesian hyperparameter
distinct iterative aggregation detail centrality score item comparison graph centrality efficacy popular multinomial logit comparison associate determine probabilistic comparison sample laplacian evaluation synthetic estimator popular aggregation task range contexts recommendation system item wish order item player player book book display big collection book pairwise movie compare preference wish score result engine assign online outcome use score computationally work comparison datum reasonable aware question rating rely interest assumption research recommendation matrix arguably provide individual user inconsistent lead aggregate rating design aggregation challenging item study potentially existence set axiom paper interested outcome item consider want score indicate intensity set imagine ranking permutation item time generate permutation become hard stop scientific practical designing fall work call model logit research design price select winner round winner assign player centrality research global ranking item line research refer survey choice pair lead eigenvector slight accounting multiple spectral propose centrality rank eigenvector form construct markov similar ranking build classical novel showing near popular formally popular open exception axiom satisfied axiom make spirit centrality novel ranking justification work ranking comparison several distribution difficult know ranking provide generalize partial expectation scheme comparison provably variation pairwise might comparison additional margin win team score team lead score score generalize traditional distance comparison whether exist meet criterion count comparison entire perfect pair wise marginal limit author order adaptively pair underlying observe comparison true rank centrality produces propose nice intuitive explanation item player random likely go vertex ever depend lose random give immediate neighbor effect comparison capture walk node wide graph notable include version web page share network assign trust use nice rigorous property establish choose unknown underlie centrality note even produce probability centrality may comparison object comparison arbitrarily os enyi strictly positive high compare option show choice summary centrality always remark analytic study walk chain like scenario relate chain comparison use dirichlet form characterize reversible adjoint operator expansion graph result order position position remaining choose verify ordering rank notation comparison pairwise graph connect comparison pair weight edge object fraction weight set pair walk independent entry non negative valid walk maximum node rescaling ensure sum one finally ensure row exactly add node define reversible reversible ideal I proportional chance item item ideal act markov hence least self unique adjoint behavior set due surrogate start distribution large eigenvector stationary top makes formally assign call centrality rank centrality rank distribution suggest alternative receive high object preferred remain since state fix markov chain irreducible interestingly meaningful complexity derive find solution implie per laplacian identify play ability score special compare os enyi graph bound require scale optimal complexity bound rescale euclidean underlie stationary quantify prefer present simulation ranking day international centrality rao effectively comparison characterize centrality notation node min degree laplacian thus think reversible jump neighbor probability random normalize laplacian real eigenvalue denote shall eq eigenvalue walk symmetric laplacian establish graph outcome produce hold probability enyi logarithmic guarantee specifically comparison centrality score comparison everything choose pair compare comparison produce b constant remark immediately go os number thus centrality range resolve score consider regime order great preferred consideration graph centrality effectively centrality require ignore performance centrality reversible random matrix mix walk end play centrality choose pair result gap play goal compare oppose preference wish n metric ordering order iw indicator natural unweighted loss normalize connect begin e representative depict show
replace jeffreys jensen shannon discriminate density class employ shannon gm divergence generalization jeffreys divergence multiple good maximum discrimination naturally binary comparative text dimensional divergence jensen categorization broadly divide generative conditional class generative classifier include discriminant lda directly without function minimize error term classification discriminative classification example support generative classifier small counterpart require training contrast tend incomplete take care entire incorporate allow close datum generating knowledge incorporate easily lastly understand discriminative counterpart method take divergence minimum proved language processing curse dimensionality maximum entropy model generative obtain early number tends prefer discriminative also allow incorporate classification criterion call entropy basic present study binary extended class several binary among unique classification main jensen divergence divergence well arithmetic study jeffreys study accuracie discriminative portion notation label binary space propose conditional form function individual observation q form I presence maximum moment function statistic mean sample draw unknown iterative like descent iterative scaling version jeffreys simply divergence give context problem law divergence divergence jensen appear relatively recently characteristic divergence combination negative symmetric interpretation connection relate seminal entropy text use data label maximize naive propose maximum entropy feature notational assume notation us surface hyperplane strategy feature scenario select maximize class ideally would address classifier bayes label quantity role hyperplane former hyperplane class using indicate I train separate term get divergence distribution similarly instance class classifier indicates given subset I use particularly number high set et feature thereby reduce divergence especially since naive classifier optimality loss simplification greedy extent divergence class class write set p divergence discrimination step deal classification invoke natural distribution problem suitable interpret multi classification similarity class datum mix mutual prior classification basically deal give device among divergence turn measure discrimination divergence property equivalent case become greedy maximize among information latter complexity feature linear text theoretical need also linearity divergence discrimination discriminative measure drawback motivate form notion average however similar basic divergence replace interpretation discuss weight symmetric observation usefulness divergence stem multi divergence help difficulty extra exploit nice follow easily equivalence replace weighted class form feature I divergence calculate rank divergence decision assign density bayes scenario natural performance discuss c testing rank approach multiclass multiclass approach svm discriminative use I cross moment global popular commonly summary well introduce refer algorithm special computational algorithms complexity various feature vector choose rank svm assume svm implement mention technique affect dimensionality addition quadratic complexity indicate I distribution appear unstable skip algorithm determination involve considerable require numerical calculation worth datum hence estimate twice following thereby make twice estimation use moment variety gene number use divergence obtain fold randomly distribution I cross highlight observe distinguish disease prune strategy away gene validation feature os windows window mac mac category experiment class group way different multiclass remove frequency less much corpus jt number corpus represents contribute document table classification propose use fold main
accelerated newton classic adjust curvature step iteration method store full demand task problem ease burden hessian difference iterate use lead dynamically curvature storage acceleration mm employ comparison acceleration scheme direct reader early review instance broad term unconstrained minimize unconstrained f auxiliary function identify g alternatively unconstraine satisfy satisfy sequential unconstrained generate iterative barrier method backward reference mm unconstraine obvious meet speak closely iteration nonetheless since must iterative derive iterative ultimately convergent surrogate globally algorithm ambiguity algorithm highlight mm fortunately carry broad set finding idea well know projection attribute close simultaneous projection enjoy advantage invoke operator find np p two intersect derivation simultaneous projection bregman divergence generate strictly bregman generate bregman bregman onto bregman projection equivalently function c brief denote hessian reader thorough treatment proximity set consider intersection minimize optimal minimize contraction coincide point stationarity point way euclidean euclidean w suffer instability rate accelerate mm systematic quasi newton acceleration reduce number sake describe dual nesterov dual solve dual dual consist I subject primal reduce I maximize sake clarity novel derivation iterate primal respectively derivation nesterov fista appear projection regard project onto doubly nonnegative numerical project symmetric intersection matrix example entry projection onto set project nonnegative accomplish matrix accomplish decomposition outer generate identically project simulate onto quasi acceleration fista decision switch update track progress algorithm eigenvalue newton path reflect switch point obviously amount vary method record include second frobenius fit equally version acc shape criterion readily amenable projection cone accomplish pool arc component project component fitting pair space geometrically penalty constants newton acceleration next result fit track progress constraint solution rule compare fit improvement accelerate version fast acc square convex predictor vector weight seek residual view convexity preserve define concave impose interpolation accomplish minima maxima reduce semidefinite inequality feasible region nontrivial w jk j jk ip jk c jk admit w w nn sum mm reduce p jk jk nn n display point least geometrically took five quasi acceleration second total objective maximal vector discriminant predictor svm classifiers penalization f constraint pass intercept negativity yield iterate except entry report svm set geometrically sequence iteration maximal consider handle optimization choice include cholesky ij essentially cholesky cost huge trivial cholesky factor fast eigen example distance distance indeed city occur rectangular guarantee short spread city norm treatment arbitrary hard calculate fortunately side component I objective minimize separate side sum underlie projection point euclidean distance broader relax requirement widely robust convex convergence prove strong comment algorithm subproblem mm monotone instrumental proving set compact minimize continuously meet scenario assumption eigenvalue hessian take rule practice know convergence unknown nonetheless determine classic point mm converge unique iterate argue turn entail quadratic expansion inequality g convexity g single minimizer proceed check proposition easy verify map take arbitrary descent f stationarity c continuously differentiable limit g strict property uniqueness continuously cluster sequence fix conclusion iterate converge stationary possesse represent converge require sharp ascent ascent sequence penalization parameter expect subproblem possess minimizer subject constraint one justify converge minimizer boundedness feasible principle distance formula parallelization intersection variant competitive program raise question rough insight encounter formula several nonetheless devise area thorough constraint employ advantageous introduce change qualitative conclusion may improve useful parameter estimate although convexity long theory show relax extend target feedback thank associate constructive suggestion comment unconstraine minimization attention connection mm research partially united states public service grant gm appendix ascent algorithm iterative dual project convex reader reference proof background low relation indicator close convex convex thus function bound zero convex attained lipschitz ensure associated function convex convex denote recover gradient important algorithm class ready consider slightly general convex intersection minimize rise ic iff f therefore since separate proximal denote size simplify p combine identity lipschitz thus actually require convergence convex early n mm fail quadratic bind algorithm employ figure depict global fail algorithm geometrically iterate convex reduce descent step size lipschitz circumstance condition maxima minima maxima remain hence meet helpful
proof generate tend certainly constraint merely conditional constraint method causal relationship iv constraint constraint possibly form understand difficulty contain control effect directly generalize control effect yx let vertex appendix always distribution two directly bound compatibility exclude rv style draw minimum mm rv style xshift edge corollary case joint z kp jk unlike strong condition iv program possible constructive crucial determine test graph specifically give symmetric likely graphical approach dag separation derive fourier elimination size infeasible doubly exponential elimination check theorem search intensive case could highly advantageous intensive search benefit much easier intuitive human user iv take state find practice theorem dag observe dd element condition valid conditional density dag remove edge vertex property say pa dd cp unchanged give compatibility pa give strong conditioning factorization clearly pd expression maximize sum take minimize tight analysis lem lem lem lem lem lem graphical acyclic dag neither adjacent latent parent generalize instrumental inequality effect characterize term separation provide acyclic graph dag commonly intuitive display dag identify conditional possibly bias remain marginalization understand merely term conditional independence causal identify hope intensive spirit use elimination determine construct dag relate terminology new derive instrumental applie order say set parent denote adjacent path towards direct path say direct vertex associate vertex admit convenience variable bold letter refer dag say joint v internal vertex call vertex whenever criterion imply dag interpretation dag extra system intervention edge parent dag implication margin latent observable identifiable underlie impossible presence assumption space latent discrete state obey iv inequality specific clear derivation interpretation see adapt provide value discrete obey iv p z p p z instrumental suppose iv effect break regardless factorization z equivalent satisfying quantity equality give dag give instrumental insufficient elimination become moderately sized rv thick minimum inner sep control possible effect give strength shorthand generalization construct eq py py z include instrumental next provide graphical graphical criterion dag vertex dag disjoint
set assumption class metric limit criterion see low distribution property asymptotic claim specific hypothesis learnable binary hypothesis distribution hypothesis class limit hypothesis distribution tight fully characterize thus tight significant desire chance statistical asymptotic tight depend moderate machine margin achievable error write multi refer element misclassification margin receive return classify object size distribution margin define follow excess distribution specific fix require general provide regardless specific sometimes omit simply separable stand real stand denote unit w homogeneous x x w uniqueness sort fix vector matrix eigenvalue use follow constant stand constant rademacher independent variable rademacher respect hx get desire upper margin appendix classic let function dimension class class pseudo linear classifier bad dimensionality nonetheless neither unbounded dimension arbitrarily depend converse dimension arbitrarily norm bound loose tight direction dimension sum dimension average direction capture let integer sub x margin minimum drop x minimum statistically independent first covariance generate small latter quite former ease carry dimensional hilbert eigenvalue imply unbounde complexity cover space cover covering follow class choice sum require notion hausdorff eq minimal f provide mapping hausdorff provide hausdorff distance lemma let let domain dimension dimension follow subsequent corollary integer value consider therefore generality henceforth write x w projection dimension x like class equal w f hausdorff cover pseudo ff theorem eq q theorem bound equation adjust fact side note bs upper definition conclude second right one want complexity matrix eigenvalue example moreover estimate covariance matrix necessary tight dimension answer need relate bind relate sub gaussian prove family closely evident formulation hypothesis class distribution hypothesis linear sample least prove simplicity otherwise replace set addition point hypothesis misclassification show h whenever return h argument last simply half least hypothesis domain uniform label origin return alone suffice hard homogeneous high present simple characterization condition eigenvalue gram require auxiliary lemma state exact margin lemma provide representation pseudo rw w theorem rescale sufficient necessary invertible invertible necessary origin hold exist invertible ready let matrix generalize linearly probable matrix formulate let dm theorem generalize linearly hope extend margin sample bind say sub variable relative also family distribution quite sub inequality gaussian moment psd random useful sub moment moment random define distribution moment orthonormal relative matrix multivariate corner hyper full rectangle low constant depend regard small gram assumption dimension approach example series matrix finite fourth asymptotic use random fourth moment draw equality control upper separately eigenvalue product proof appendix state dd orthonormal direction define natural w draw let low norm control sub moment equation distribution theorem show upper product bound moment margin dimension fully complexity follow two identical sub gaussian direction let gaussian moment moment projection moment point provide sub relative formally depend result characterize adapted dimension conclusion equation spherical alone logarithmic logarithmic stem margin might logarithmic equation characterize behavior improve gap irrelevant feature regularization regularization bound establish one ng assume use low coordinate bound low sample gap discriminative spherical mean v looking calculate generative spherical classify estimate ensure learn abundance unlabele costly active need decide current without abundance estimate distribution adapt dimension calculate easily unlabeled label bind far would information show true margin adapt characterize comparison characterize complexity matrix great conclude lemma since g g sg exist vector yx ki eq dimension prove induction inductive denote projection z r z r denote moment dy tx x expression ds hold fact ms x x prove squared moment psd b therefore eq method matrix fourth lemma work assume eq column far center psd value let ij I ij ik ik index sub matrix row j substitute brevity constant later since inequality l row column change moment lemma
enkf enkf solution enkf first figure right stage start ensemble forward twice dot enkf look apparent enkf linear tradeoff moderately enkf quick turn prior produce fully inverse wishart spurious specification function give produce update ensemble finally note bootstrap useful accounting run carry describe make use enkf explore enkf determine take suggest involve aid computationally demand perhaps simple g scale universe background dark composition fluctuation universe combine digital survey give map galaxy body evolve matter history begin big year physical produce frame slice map galaxy spatial galaxy predict matter universe require effort set scale carry simulation dark matter background force middle frame simulation spatial physical observation digital digital body right matter datum periodic galaxy censor snapshot structure dark spatial matter scale matter periodic cubic lattice since difficulty geometry bias challenge publish line diagonal deviation frame run array table spectrum draw prior application enkf cccc upper spectral galaxy amplitude dark prescribe spectrum gray line frame physical parameter incidence select correspond physical along ensemble enkf describe section vector frame figure presence skewness fit uncertainty give right figure distribution triangle member circle black dot contour black posterior pointwise credible spectrum density universe light line ensemble enkf produce far elaborate statistical posterior similar estimate gp application come effort measurement well ice ice rectangular location ice distance exchange use approximately depth measurement guarantee configuration minimal determinant multiple anneal enkf community sample dimensional range apparent physical cloud ice cloud particle cloud air consumption mb cloud density warm average relative threshold warm ice air temperature ice heat radius cloud environmental air cloud cloud ice fall cloud cloud ice produce could output explored record two air temperature observation directly observe orthogonal long pilot air temperature show green model dot basis element output combination summarize correspond pilot variation output scale basis actual physical light green prediction dark average compute observe simulated field basis produce long pilot black dot solid black variation compute pilot run dash due discrepancy light sample run dark line update scale standardized air temperature field pilot least variance discrepancy precision discrepancy give dotted line discrepancy air temperature govern prior dimensional model define get plug reduction basis pilot vector air figure ensemble sample uniformly rectangle depict figure ensemble rectangle formulation discrepancy proportional physical result enkf ensemble run finally point enkf collect ten successive synthetic highlight enkf show variety enkf use mapping make easy ensemble often gp well deal output helpful enkf enkf start fairly little depend specification relatively exist literature tail clear analysis simple largely implicitly use enkf see ice compare ability enkf quickly rough ready demanding task problem computer kalman sciences laboratory laboratory sciences group national laboratory sciences laboratory institute physics division laboratory high energy physics division laboratory price laboratory fig ensemble enkf effective quantify estimation assimilation weather successively aid forward next recently enkf prove effective estimating describe reservoir history match challenge model calibration involve well computationally demand forward calibration combine observation explore enkf calibration consider keyword computer validation assimilation quantification process ensemble filter enkf prove quantify weather model assimilation dimensional successively physical aid demand enkf ensemble kalman produce update ensemble affect enkf involve describe reservoir production time history especially since involve number demand forward model assimilation static evolve explore enkf collection constrain go enkf address end strength enkf model inverse denote map physical observation noisy horizontal denote result inverse order inverse inverse forward produce physical normally specify posterior trivial deal range demand forward experience range second monte mcmc remain popular explore vector forward model inspire recent focus effort simplify multiple parameter produce conditioning physical four run exploration process gps computer produce infer forward incorporate model function function typically product require dimension take gp conditioning forward example fix variant enkf run result posterior draw paired simulation ensemble covariance setting draw simulator enkf normal motivate calculation variant enkf calibration leave figure vector eq simple problem vector nd write element simulator output normal rewrite commonly filtering gain computation effectively assume relationship induce enkf kalman enkf depict symbol distribution depict gray marginal density ensemble dot draw second basically enkf time member order member whose match posterior enkf update simulator simulator
terminal either load color package graphic explanation terminal graphics macro ltb lt lt lt lt lt ltb lt lt lt lt bp ltb enyi match right generalised random graph cutoff gp walk show rather subtle correlated limit reach cycle large threshold kernel reach walk typical globally variance complicated manner structure undesirable prior normalise deep understanding apply approximation fail regime outline cavity method approximate average global show vector output match obey component independent identically distribute prediction vector n conjugate case effective vector teacher differently normalise change matrix weighting prediction simulation curve list large diagonal plot generally end region spike vertex figure normalise globally normalise student globally normalise teacher locally normalise student clearly large spike edge dy typical enyi match red globally normalise teacher normalise pt dy cm normalise student normalise teacher kernel similarity space graph counter use variance euclidean undesirable modelling walk normalise variance analyse consequence study curve prediction curve obtain depend belief propagation approximation gaussian process cavity belief propagation field wide attribute mainly nature ease kernel space relationship encode ease gps set gaussian explicitly rule require error trace average error example study variety understand parametric parametric gps euclidean graph connection relation include financial space graph become understand expand curve gps output graph rest walk particular correlate proceed walk gp section looking function local extend accurate asymptotic regime curve core development normalise belief propagation act warm locally normalise compare numerical simulation find agreement globally locally normalise qualitatively normalise gp normalised versa result curve prior arise fundamentally different monotonic discussing complicated everywhere jj fundamentally accurate broad range parameter last surprising continuous like correlation space calculation correlation avoid trick inner kernel machine result develop wide kernel normalise graph vertex generic encode connection vertex connect otherwise exclude denote known matrix construct conjunction load graphic need graphic macro ltb lt lt lt lt ltb lt lt lt r ltb ltb ltb ltb ltb ltb r ltb ltb ltb ltb ltb r ltb ltb ltb kernel graph calculate conclusion kernel value short apart remain limit cycle focus regular qualitative vertex study walk completeness gps gps machine direct posterior input correspond observe choose covariance matrix focus gps output corrupt independent identically noise target teacher noise cx cx cx simple loss preferred correspondingly kernel belief smoothness expect amplitude predict one desire achieve location kernel spatially graph connectivity vertex along among usually undesirable model unless justify return depend dependence general imagine show variance walk globally unity ci gps unity vertice single instance present give poisson distribution figure generalise power generalise graph edge assign one generate superposition p dpp know large enyi locally relatively prior even give specific example spike vertex spike edge subgraph vertex additional increase weight opposite subgraph spike large even step would occur centre star illustrate alone degree constrain vertex end probability theoretical prediction large limit fine peak agree linear rather broad tail power feature variance similar unity spike subgraph slow exhibit significantly large accordingly value two constitute vertex retain spread latter complicated ij ii ij kernel euclidean vertice variation computational unity plot figure gp remainder describe terminal option explanation load graphic see graphic lt lt lt ltb lt lt lt lt lt r degree limit generalised exponent cutoff performance non method study square student teacher prediction give teacher teacher average simplicity assume input vertex graph average still specific consider average ensemble degree specify degree equivalently degree degree mean analysis therefore broad mention regular graph enyi power law generalise typical curve teacher distribution likewise assume teacher noise simplify student variance value note eliminate output average general calculate input enter see situation learn prediction euclidean space case gp predict graph broadly ensemble extend discrete graph see fully belief sketch derivation include treatment cavity mechanic eq input enter count average logarithm partition cavity discuss exist graph cavity fully replica power replica index independently independent poisson average average evaluate eventually equation variational justify exponential truncate hand fluctuation issue prominent order unity fluctuation example nearby posterior vertex posterior fluctuation mathematically proportional close contribute compute normalise walk dot centre generalise right compare show line range noise approximation justify large expect package color conjunction terminal explanation load color package graphic explanation terminal graphics macro ltb lt lt lt ltb lt lt ltb ltb ltb r solid numerically graph dash cavity note visually indistinguishable simulation centre law generalise exponent cutoff move cavity look curve focus large regular trivial large stay important conclude naive bayes unity reduction v away vertex volume get distance proportional large summing grow detailed large proportional characteristic cutoff distance decay decay eigenvalue graph identical regular tree explicitly normalise overall color option explanation explanation terminal graphics macro ltb lt lt lt lt lt ltb lt lt lt lt lt lt ltb ltb ltb ltb ltb r ltb ltb ltb ltb curve master curve slow curve numerically far eigenvalue numerically simulate fluctuation graph detail average indeed cavity rely local nearly indistinguishable cavity curve must function relate vertex create factor arbitrary eliminate cavity begin ij eliminate transform integrate term respect coupling link vertex remain interaction binomial walk variable pc pc sake uniformity instead interaction hide definition additional represent via dirac delta fourier conjugate get adjacency interaction graphical apply belief propagation focus globally rescale interaction adjacency product respect prescribe order eliminate subgraph locally tree root approximately cavity create iteratively subgraph belief propagation cavity message vertex factor cavity self consistently character cavity explicitly correspond matrix rather translate equation ultimately large degree individual cavity edge pick random edge degree vertex pick incoming cavity distribute sample cavity equation vertex label omit index example vertex simply also replica element cavity result allow vertex exact worth learn prediction specific regression gp applicable level briefly isolate vertex careful avoid division end find consequence vertex datum give globally normalise cavity eigenvalue regular enyi generalise greatly outperform confirm cavity already graph vertex cavity globally normalise kernel cavity fraction bayes dynamic predict posterior become variance cavity without prediction distribution go cavity numerically distribution prior cavity distribution cavity provide particular detailed tail agreement fine peak shift rare cycle also cycle cavity cavity normalise walk argue probabilistic define normalise challenging calculate ensemble normalised scenario account block cavity update set bayes trick well behave expression parameter determine way predict really want normalise analogue dropping write graphical iterate equation cavity cavity form define equation specifically update one calculate large outcome cavity cavity covariance cavity normalise couple look joint replica method fix equation globally normalise second covariance marginal covariance subject marginal constraint approximated globally normalise reflect cavity message first delta cavity cavity may enter represent cavity send cavity belief set cavity message reach set apparent reason full cavity case look curve normalise kernel normalise cavity show figure cavity enyi random graph law generalise centre globally normalise cavity accurate even capture aspect curve discuss detail package conjunction terminal explanation either use package graphic graphic ltb lt lt ltb lt lt n ltb ltb ltb r random line simulate curve mostly visually indistinguishable result show single line centre generalise exponent curve globally solid line normalise random right law cavity locally normalise indistinguishable curve simplification drop consistency requirement look cavity prediction value curve package explanation color graphic explanation graphic ltb lt lt lt lt lt lt ltb lt lt lt lt lt r enyi vertice cavity prediction prediction curve gp regression global plausible avoid variability prior trivially relate local justify compare kernel well normalise match case kernel kernel bayes reflect answer kernel error try
lemma q regard difference operator simple derivative refer convenience define p consider quantity taylor dt use u u conclude equation apply taylor equation exist dt v dt dt result u use bind w difference ti jt jt jt jt jt jt jt v jt jt jt jt jt jt jt jt jt v jt v exploiting jt exploit also v curve repeat argument recall know hold particular bernstein let independent eq adaptation use coefficient follow condition independent true conditioning behaviour x k v rt w v fix random variable satisfy constant twice focus fix last line hypothesis continue bound integral recognize incomplete l dt e l dt dt dt e standard piece together draw satisfy also lemma processing consist model signal atom paradigm range support particular rely minima analyze paper admit local study dictionary noisy extend previous limit dictionary key quantity coherence modelling signal field framework signal design dictionary prominent effort dedicate g wavelet representation notably many processing decomposition simply therein although sometimes formulate submodular widely definition coding bring sparse dictionary instance quantify differ non uniform later explore consist characterize minima non formulation recover identify important direction signal possibly corrupt outlier compose atom knowledge carry signal straightforwardly noise point discuss local code presence make contribution within probabilistic derive low local reference understand around reference hope completeness result integer extensively frobenius ij extract conversely n square diagonal vector matrix sphere v set pp atom learn ik aim reconstruct atom introduce signal lasso code level sparsity paper processing however context topic atom simplex characterize minima signal accord draw precisely regularize neighborhood subject signal already blind permutation instead affect issue soon carry dictionary unit turn consider unit column parameterized stand v leave drop dependence correspond tangent frobenius w parametrization scalar proof sec curve onto appendix describe require lipschitz minimization since non moreover however close favorable leverage key denote p property appealing make possible form remark sign q sf nf n ingredient exploit sign least understand reference therein ic machine control support square reasonably reference dictionary quantity eq measure correlation column deterministic coherence dictionary coherence light transfer proof rip weak coherence assumption however face ic use rip coherence rip equip present signal dictionary sparse build generation draw replacement define generation yield support whose I background variable g simplicity case formally eventually signal x model prove admit local size detail main local minimum model k follow hold admit minimum highlight distribution keep main dictionary incoherent signal guarantee ball resolution coherence condition see impose slightly hadamard dirac addition handle regime resolution minimum term rely recovery worth note dictionary coherence follow generative reference orthogonal minimum sufficiently relation previous remove unchanged limit mention obtain noiseless result noiseless analysis noisy work differ linearization equality technique easily noisy building structure proposition control many concentrate high correspond bilinear form concentrate uniformity net collection coincide number large recovery signal get seek well rather appendix cr exact recovery dictionary sufficient w event turn question comes study regularize square problem associate see conclusion turn go away noise depend statement proposition supplementary material previous place v conclude highlight experiment zero coefficient uniformly invariance see auxiliary hand know sign close frobenius assignment absolute solve fix epoch random dictionary begin hadamard dirac fix k noise primarily limit plotted fig hadamard mp speak decompose function residual discuss surrogate function building expect concentrate uniformly bound assume second uniformly x instead event around relax expectation seek rather equal show dt coincide definition x function coincide uniformly zero hand indeed r cr exact sign dictionary v control proposition exact recovery consider v modify handle simplify noiseless recovery perturb noiseless signal satisfie almost dt residual find small signal proposition set recall induce ball local minimum recall sphere positive accord prove hence formalize lemma give vice w v result case column uniquely previous vector handle thank convention define j j reciprocal v j come column product term w j p consequence conclude exist finally observe conclusion general c stand universal make thank sufficient k constant keep hidden success theorem even conceptually noiseless regime follow q noise handle jj function particular imply stems read eq c positive radius tt display surely main consequence simplify residual surrogate coincide almost code radius depend simplify moreover ask lower bind positive p universal explicit choice c jj second c define onto w jt jt j jt c dt dt q jt next norm isometry set matrix z z dt matrix control coherence dictionary k q similarly matrix briefly prove introduce positive definite term I norm obtain stand one four corollary computation normalize finally normalize column pi simply v v p I pp overall approach concentration model lipschitz showing lipschitz control together estimate size signal proposition six denote denote w x observe union bind conclude average w lemma exploit cover p respect background cover reader net euclidean resort sphere second dimension net conclusion metric ct check hence n mp mp mp mp mp mp mp mp recall ct k obtain ct l ct ct l ct observe moreover dt estimate vanish
identify convenience sup define closure sup thereby ensure complete classical fr inferential approach relate empty separability sufficient non empty observe event trivially certain prefer value neither existence sequel sample fr space equip allow space metric identically mode mean sometimes denote linearity lebesgue probability assume integrable key make generalization take metric space need class separable whereby real value every every fx fu say indicate rao p value separable dominate continuous integrable ii exist albeit closure within due fr every fr mean quantity pair fr converge q definition tr inequality tr q number maximum prove proof construction choose mention invoke continuity fr hence boundedness sufficient guarantee assume definition mean belong point locate respect square fr locate interval author highlight fr interest boundedness family find prove restrict fr original argument thereby uniform appropriate weak assumption aforementione theorem remain valid fr mean result variable commonly define consideration group pt remark proposition axiom sample mean support air office scientific research whose grant fellowship uk national institute health center mh south foundation college conduct like useful fr mean idea space fr graph fr necessary sequence sure fr mean value pseudo metric exploit fact metric bound fr order generalize metric square summary analog expect value abstract show specification space interest fr minimize notion abstract physics riemannian manifold sample fr cumulative produce convex metric space negative curvature mean interested simple separability mild topological boundedness metric condition modern statistical likely bound bioinformatics dna naturally family commonly similarly bound space albeit metric lead node fr fr sample pseudo compact metric two publish conference probable particularly proper bound subset riemannian manifold make apply manifold mild fr sample non take riemannian manifold take metric boundedness strong fr mean space indeed space difficulty arise special issue also space metric consider several metric hard restrict fr greatly simply importantly fr show fact consideration limit fr difficulty consideration function power set become end resort sample mean asymptotic main paper necessarily manifold play outer fr constitute extension mean point proper emphasis value study sequence mean show outer adequate consistency fr devote fr font scale circle circle font circle circle circle font circle circle v v v ig may graph say multiple loop edge graph statistically type fr interest hamming simple realization element distance element graph fr graph obtain result somewhat average concentrated mean hamming sequel consider separable bound random special popular choice font node circle circle pt font circle circle circle v v font scale circle circle font v font circle font circle circle hamming space endow assume every measurable particularly construct base remain behaved term value element approach marker thus true commonly fr fr exist sometimes refer fr non set observe space endow inner unique random variable fr version fr attain ambiguity refer sequel respectively denote fr albeit fr close clearly x expectation infimum lead result q argument mean particular note assume underlie fr thereby fr convergence define observe treat subset sense ns outer subset however type fr define abstract fr fill gray corner font west anchor east anchor east anchor north anchor node east node south anchor west anchor east anchor anchor north anchor north anchor east south panel close equip fr inference conduct take median panel coincide arithmetic fr evaluate set fr
topic associate topic correspond make efficient scalable become popular parametric specify grow chinese restaurant crp dirichlet enter restaurant denote cluster assignment customer customer give customer rich get rich table customer high get table crp independently topic vocabulary post topic choose multinomial sequential crp exchangeable chinese restaurant process widely medium since aspect network value relationship accommodate chinese restaurant chinese restaurant point context medium say post relationship imagine hyper element element share nu distribution table assign customer neighbor already look social medium crp wide entity post death capture unique label along relation hyper relationship one factor content preference movie evidence topic empty table capture crp post friend case cardinality nz friend influence influence trend national lot country capture construct location country location profile user cardinality maintain explicit relation statistic time capture medium way crp crp define remain individual world user relationship reality act determine content specific affected differently multi relational chinese restaurant aggregate influence label post associate additional take indicate influence label multinomial iid dirichlet interpret influence imagine given assignment define post equation aggregated influence capture r w reflect topic two word post iid pz create help capture factor medium analyze dependency one relationship set world user relation country create across user contrast distinct friend couple post friend three flow user relationship preference relationship distinct dependency create user couple user medium factor distinguish medium dynamic capture aspect fall count extend aspect evolve old topic b topic wide network preference user topic evolve enter vocabulary go chinese account temporal reality change epoch label take epoch hour week dynamic dependency describe separately start popular preference influential evolve need change epoch popularity dynamic epoch make early equation epoch nr nr u k neighbor epoch exponentially nr k epoch define conditional well may friend early epoch sample iid temporal add u time influence relationship epoch distribution evolve capture dynamic component depend epoch epoch define present posterior influence involve posterior distribution couple solve resort collapse gibbs traditional repeatedly sample size sequential old infeasible parametric model scale expense sub begin medium first describe require online set influence factor condition influence user look additionally vocabulary vocabulary topic baseline discover trend exist algorithm experiment collection million tweet extract publicly profile default setting baseline lda generally assign topic available twitter parametric topic also dynamic unseen ability exp n dd log indicate ability different consideration tweet month facebook evaluate record lda topic topic facebook baseline capture relationship acc model facebook index lda lda assign topic post indicate find vocabulary word topic indicate modeling way compare reasonable gold topic task significant trend qualitatively knowledge post accord assignment gold twitter indicator know indicator post www etc label normalize rand measure model twitter dataset correctly cluster virtue input come knowledge well inspection find label propose movie comparison split grain distinction gold twitter facebook additionally indirect classifier topic medium gold model twitter facebook twitter na predict author twitter facebook last month create twitter facebook positive negative equal random user topic near neighbor topic post user post set twitter facebook number accuracy topic baseline well usefulness topic infer consider temporal lda lda hash tag significantly outperform topic task involve compare label post conjunction user epoch user epoch aggregation subset different segment major dominant break label event wide popularity aggregate major wide event discover include sep google wave google popularity aggregate death actor uk verify wide event top search interval specific page microsoft twitter page summary enable discover trend major aspect indicate post accuracy infer factor aggregate perform rich insight factor influence epoch aggregate factor epoch trend trend anonymous plot aggregate user subset heat epoch color probability world trend wide user specific day period trend heat color color interest observe wide trend largely influence happen expense preference happen news death discard gradually user preference wide sep expense popular google wave usually influence mostly event trend aggregate user specific trend uk china heat trend uk largely influence epoch around uk around strength influence somewhat weak china strength various stay relatively stable character trend variety look trend topic character aggregate user distribution influence topic epoch character move world color wide google wave preference preference wide summary wide influence lead insight beyond capability employ core ram employ child read batch plot micro post version sequential process increase demonstrate scalability twitter facebook twitter facebook analyze final performance addition model attempt study analyze medium crp incorporate influence process dynamic essence million tweet extensive demonstrate trend model superior baseline beyond capability model study influence various factor social challenge various type influence social medium network act influence multi relational chinese account user evolve design extensive twitter extract prediction baseline valuable trend trend capability facebook extremely user share view short understand medium become important identify segment security lead major distinguishing feature influence variety four preference event world factor affect user different influence secondly medium datum inherently trend interest influential sometimes lead popularity slowly individual evolve factor intrinsic analysis social exist fall address isolated interaction scalable parametric medium specifically augmentation chinese restaurant relational restaurant multiple relationship assign relationship new parametric process version evolution topic trend rich capture crucially multi perform dynamic million twitter dataset facebook demonstrate qualitatively goodness topic discover user use baseline importantly discover interesting user mostly influence global network influence preference analyse effectively medium discuss media section discuss light work parametric model medium dirichlet dp infinite mixture dp restaurant provide generative description algorithm permutation probable dependent crp dd
crucial since compact unlike countable countable countable algebra borel one correspondence measure separate interpretation separate interpretation hence set interpretation restrict show interpretation give separate alphabet separate interpretations borel establish note rs first clearly separate closed ir I remark precede paragraph eq proposition separate interpretation intend belief interpretation member separate interpretation alphabet countable separate interpretation algebra let sentence interpretation separate interpretation close term type enumeration r interpretation immediately measurable proposition show proposition countable alphabet concept strongly interpretation correspond probability strongly imply let borel algebra relate iff set interpretation sentence set interpretation sentence also separate interpretation probability separate clearly interpretation algebra probability relate sentence interpretation suppose sentence conversely probability set sentence separate probability discrete countable real call borel algebra probability alphabet countable enumeration model discrete interpretation sum separate interpretation interpretation countable interpretation separate discrete construct separate characterize axiom also sentence assign countable exist probability sentence enumeration countable choose interpretation assign mass define countable define sentence illustrative concern sentence alphabet alphabet interpretation standard close close term separate countable alphabet probability example non separate interpretation strongly choose construct form enumeration sentence put separate sentence correspond strongly alphabet exist sentence separate model put mass separate sentence alphabet logical two map suitable alphabet element logical interpretation everything separate interpretation everything far except separate disjoint alphabet sentence domain separate confirm note choose sentence empty form enumeration sentence put mass alphabet sentence alphabet non logical term separate interpretation standard interpretation continue logical add logic constant interpretation still axiom later induction interpretation modify interpretation expand nn n ni everywhere separate therefore axiom either replace subset choose absence operator cause constant select singleton non number enumeration interpretation avoid axiom hand logic formula replace something construct good section distribution logical axiom enumeration sentence iff construction problem infinitely constraint build measure illustrate model combination zero odd I theorem characterization sentence sequence sentence finite infinite complete left depth depict conjunction sentence root ex bl bl tree let sentence sentence pairwise disjoint state necessary sufficient well yet characterization probability alphabet countable enumeration sentence sentence resp iff separate resp follow condition addition resp part demand separate enumeration type item explanation enumeration satisfy appropriate intuition I require axiom put set hence would branch enumeration sentence nothing probably prefer keep thing keep enumeration branch formally branch proposition immediately strongly strongly trivial direction imply thus case induction complete induction argument prove summary imply great conjunction contain conjunction together condition separate r axiom hence allow neither large sm unfortunately item item e separate something else interpretation enumeration sentence separate sentence extended sentence ask meet idea uninformative consistent unfortunately possible concept kullback sentence necessary extend pairwise valid eq finite sentence valid sentence illustrate sentence htbp result sentence depth sentence let sentence sentence implie imply empty sentence sentence extend alphabet countable extend definition equation trivially equation finally fourth trivially induction suppose depth assume hierarchical sentence depth restrict equation sentence pairwise disjoint see q show htbp simplify sentence depth index depth sentence expand let index sentence path valid full valid sentence equation replace furthermore equation sentence hence sentence depth inside proposition conclude brief reasoning discuss inference logical implication knowledge base observe exception construct usual forecasting index belief word intend logical axiom uniquely satisfy optimisation else solution logical case expect logical important kl intuitively sure separate forced sentence separate evidence even model problem black problem integer sentence th uninformative let logical degree true happen apply bi bi observe approach course weak black bn want sentence range give extended bi bn bm bn bi bn hypothesis absolutely sure bi bn black counter ever conclusion qualitatively free free hypothesis quantify inference like type probability x generalize follow possible separate crucially exploit term type long break could confirm asymptotically determine separate whether sentence separate determine separating calculus satisfy sentence separate finitely sentence increase appropriately tree assign probability procedure inductive choice make every inductive reasoning operation approximation assume endow etc expressive order logic convenient definition achieve maintain consistent knowledge g outline proposition remark agent past kind ever observe e agent need action regularity time formalize type intend black immediate agent black formally whether allow inductive system basis process result inform belief derive ensure limit belief reasonable reasoning difficulty get right game three nothing game select one ask preferred behind reveal constraint open nothing behind either originally switch good switch use three predicate define set sentence important symmetric predicate true branch requirement uncorrelated rather scale style circle mm predicate location swap swap swap swap swap j swap swap generality locate allow right root add predicate predicate predicate branches circle fill k swap node swap node g swap swap n swap node r n swap branch two force player leave hand branch player likely outcome research history go back main far mathematical bernoulli historical idea sentence back contain reference important appear theory find reason artificial intelligence third machine artificial intelligence typical useful technical distinction approach external probability sentence logic incorporate mix appear uncertainty careful logic particular logic alone expressive extend extension simply higher crucial high logic exploit admit take return property modelling concept directly view integrate logic outside researcher logic build early sentence bayesian network constrain interpretation system approach example provide integrate inside outside sentence lie discovery currently partly dp architecture agent department communication digital centre axiom theorem ex ex ex school school ng national university ex reasoning difficulty develop system integration logic expressive language order ideally reasoning structured knowledge rather main true sentence wish consistent procedure logic iv inductive reasoning vi quantify hypothesis sentence wish list criterion construction counter finitely sentence suitable sentence bias brief theory might reasoning consistent satisfactory logic formal insight generally expressive knowledge reasoning suitable logic admit high result probabilistic logic write answer computational sometimes logical answer probabilistic consideration work quite idea prior general concentrate develop sentence logic sentence use high employ make interpretation sentence order introduce interpretation connection connect probability quantify sentence limit separate interpretation accept circle countable justification real experience statement relevance first statement furthermore guide less equally conversely short mean probabilistic coin one event sentence possible satisfy I strong sentence condition something prove false chance intend non belief non inconsistent weak version plain strong construct interpretation sentence model characterization general probability interpretation countable domain example important value sentence sentence necessary determine probability constrain probability sentence coin head fact logical axiom number require interpretation prove specification complete proposition hierarchical sentence determine choose biased relative informative consistent prior proposition brief end discussion broad motivation well outline model logic direction extension straightforward nevertheless useful material proof technical wide attention hope reference logic discussion history give logic mathematics situation description logic foundation mathematic several advantage furthermore logic logical formalism area hardware verification order science logic present minor way omit operator logic omit logic terminology along type definition type function convention logical type logical type alphabet constant term term together mm type term type closed term formula formula logical countable truth quantification axiom logical axiom truth x g indicate indicate axiom axiom axiom axiom axiom reduction substitution replace occurrence occurrence provide occurrence immediately logic basis logic simplify notation omit type always x application frame frame call truth member call individual frame type variable assignment map variable element define pair interpretation variable term type mm mm pair interpretation every interpretation call crucial allow logic formula mm satisfy valid interpretation valid logical consequence consequence theory need separate interpretation term separate separate close alphabet separate close close thus distinct concept separate play completeness sentence complete say closed interpretation provide separate sentence suppose completeness proof sound separate expand alphabet separate expand alphabet separate alphabet interpretation alphabet alphabet consistent expand separate sentence fact development carry logic version logic practical extension extension deep extension include tuple logic probability probability connection conventional interpretation make sentence sentence alphabet negative valid valid sentence probability sentence follow intend degree disjoint sentence sentence valid valid valid probability include since valid iff induction induction assume imply valid conversely imply thus valid I term close let sentence range nx range set term type finite close hold eq suppose alphabet alphabet countable range closed type enumeration alphabet countable set countable hence close sufficiently large appear enumeration enumeration type rh supremum conversely sup include monotone replacement class necessary term countable alphabet alphabet equivalent
label approach determine acquire crowd amazon allow collection retrieve area g multiple svms supervision task include amount label amount notably closely computer shoot single training leverage category different source field adaptation class model success verification vision method gaussian trivially transfer discard letting source constitute background modal rather time sample modality provide overview input variable denote denote probability let seek joint px represent available label source tp u tp n generally assume classification matrix column domain da summarize predict unsupervise da introduction relatively label application domain language computer vision reason categorization weight da utilize source across domain make sec survey method multi modal relaxation da supervise model space empirical da arrive formulation source minimization already really discussion consider relaxation imbalance relax imbalance population drift training sense change landscape assumption account need explore make class relaxation target margin classifier situation problem assume datum regardless covariate become fit minimize dominate since simplify biased metric distribution distance prior sense discriminative three distribution target common source standard logistic parameter specifically find solve adaptation adapt entropy argue suggest ensemble portion fig unlabele adaptation adaptation adapt model successfully also illustration adapt gmm gmm figure gmm background together train right show survey support domain train subsequent source thus sec machine basic decision learn source act regularizer existence model domain let classifier attain svm similar enforce proximity support vector close introduce constraint old classified old target specifically vector sign close survey da perhaps domain adaptation create map finding might equal depend entropy feature likely good alignment take account straight forward way propose remove approximated source minimize de u another towards goal mean discrepancy mmd mean two domain reproduce hilbert rkhs integrate function separate parameterized mixture function relate representation correspondence unlabele behave required noun help noun algorithm maximize correspondence recent vision variation mapping label mahalanobis algorithmic domain state ensure decompose mapping change regularizer frobenius encode mahalanobis new become improvement table compare recently goal source transform combination following relax rank constraint lagrange multiplier alm propose mapping motivate incremental create intermediate representation create representation final stack location cm p cm naive amazon naive train image category domain mark bold sometimes call multi learning da transfer prior da formally think transfer task view feature space augmentation name recognition text classification class share formulation author concept assume assume available goal transformation vice versa common begin work popular method project projection unsupervise agnostic project discrimination find minimize scatter class scatter lda direction give solve note eq eigenvalue cca analysis though cca projection onto mutually maximized sec cca formally find projection solving dimensionality write constrain eigenvalue solve eigenvalue problem detail excellent
minibatch sample rather sum subset condition field benchmark closely detail include train accomplish step consist maximize likelihood rbm consist rbms maximize far include discriminative ability somewhat mean exist closely phase criterion code epoch base last start increase entire mnist training last obtain mnist trained rbm follow variational obtain jointly train correctly optimization result number likely introduce boltzmann jointly learn train machine time deep model call visible unit always train evaluation usually apply represent label code discrete label available contain organized unit neighboring layer allow gibbs entire likewise fast variational q summation fortunately structure every element conditionally visible likelihood intractable overcome layer procedure thus layer simple consisting value repeatedly apply field essentially run beyond admit simultaneous excellent invariant version mnist handwritten digit knowledge allow conjunction train boltzmann machine layer rbm rbms train enable jointly train procedure suboptimal deep general optimal influence deeply training procedure procedure optimistic layer able current boltzmann machine deep architecture pool factor multilinear interaction unit layer use hessian without greedy never view difficult show hessian poor parameterize intractable great expense approximate costly search early mcmc take make markov chain
rapidly occur reflect target elliptical multivariate call random field elliptical advantage mix induce easier apply tuning parameter elliptical take invariance namely draw eq q marginally nevertheless previously proposal latent gaussian elliptical slice sampling rejection elliptical transition consider pass current induce purely gaussian prior choose elliptical slice sampling auxiliary sample sample angle choose slice procedure specify current state satisfy slice intuitively slice compare depict elliptical slice illustrate produce elliptical slice reflect full elliptical slice depict elliptical slice log distribution x elliptical slice arbitrary refer need target sample elliptical slice approach put residual density note correct enable use elliptical slice sampling elliptical mix practice difficulty tail rapidly illustrate get back toward hand low unlikely origin iteration elliptical allow approximation gaussian serve auxiliary residual residual possibility combination point equation view augment fully choice drop component elliptical slice update markov transition operator focus simulate heavy simple convenient location density real value parameter observe conjugate parameter update elliptical define current perform elliptical slice new contour sample red new point make way choose maximum laplace regardless section parallelism dynamically require exploratory create markov chain immediately obvious discuss fail way current chain lot shape operator elliptical slice additionally stationary core generalized elliptical slice desire property begin markov chain use determine multivariate parameter apply x individually update chain introduce detailed balance create joint updating multivariate parallel however factor efficiently problem maintain multivariate denote chain practice simulate leave invariant generate chain two part transition use determine use operator composition idea maintain chain state markov chain chain see description choose appendix subroutine return maximum likelihood subroutine elliptical slice update x chain algorithm choose may converge set estimate describe fit poorly appear well k ad r r fit multivariate construct dimensional space identity avoid degenerate distribution intuition emphasize nature important devise sophisticated idea high merely simple seem work choice least likelihood parameter diagonal covariance parameter nature establish aggregate transition wish preserve x stationary desire chain elliptical slice nonzero nonzero transition chain therefore nonzero transition irreducible ensure invariant distribution multivariate approximation fit iterative empirically parallel chain challenge construct manner cost evaluate density function overhead applicable promise complementary introduce technical software requirement addition population parallelization could apply mix compare scale core number ec algorithms environment parallelism empirically mix mcmc sampler seven mix compare chain effective effective effective result metric sample carlo experiment unless burn iteration five variability omit aggregate diagnostic elliptical several sampling direction slice coordinate wise slice sampling variant slice direction direction align mh gaussian current tuning adjust acceptance ratio optimal hasting chain also sampler carlo hmc combine procedure automatically evaluation term include comparison hmc though differentiable principle access manual complicated subroutine instance require image run sample distribution temperature regular interval target produce often practice geometric view entirely principled choose goal box restriction require describe ten shape normally nine condition remain initialize spherical component gaussian spherical component distribute uniformly hypercube breast cancer explanatory breast cancer consist mean prior initialize chain spherical explanatory repository place prior variance spherical origin stochastic volatility simple fit synthetic burn iteration chain take absolute parameter constrain length scale place burn iteration spherical snp gene eight dimension actual genomic sequence burn initialize spherical center origin take leave five bar rescale effective mix accord metric attribute unlike chain hmc although fail snp arise reflect mostly reason rapid mixing even vast away successive highly multivariate build shape skew transition mix arise result length parameterization frequently approximation greatly parallelism b x wish function core available chain triple make sense core chain section equal triple origin degree use model initialize spherical gaussian second half discard second used estimate vector marginal triple five average squared aggregate chain would set additional enable effectively chain enable sampler effect column fail fail burn however increase core provide square respectively converge one chain mh converge elliptical slice approximate parallelism gaussian shape transition chain particular hundred core mcmc elliptical variety evidence number work reduce overhead share chain speed chain obviously chain run slice sampling perform per update location chain narrow evaluation chain rapid chain leave imagine perhaps chain machine gain gain parallelism mcmc able mix area extend take core magnitude multivariate core give feature location mode use mixture accurately gaussians explore leave explore algorithm detail parameter likelihood q update parameter function probabilistic conceptually find practical effectiveness approximate carlo step rapidly mix leave regard multi speed paper distribution take advantage hundred core information chain elliptical slice create chain mix require curvature computation markov carlo parallelism slice elliptical slice probabilistic fundamental machine coherent formulation representation unobserve estimate typically integral rich rapidly expectation carlo simulate chain equilibrium principle gold inference mild limit simulation often expert method space generalize tune multiple core parallel detail arise world dependencie parameterization directly primary efficient markov reflect structure imagine thin region high necessary length metropolis mh proposal monte carlo behavior introduce momentum method efficient move even approximation target elliptical shape require curvature word encode us information oppose hamiltonian valid preserve guarantee limitation expert difficulty stem tune transition mix metropolis appropriate proposal carlo step probabilistic widely develop black box attempt mcmc theoretically understand narrow transition operator hamiltonian slice local search acceptable avoid potential adaptation significant computational
define estimate percentage high simulate axis vertical difference consecutive consecutive horizontal represent difference show must rule class read well multiclass ccc horizontal kernel namely project pca mahalanobis mahalanobis kernel pca mahalanobis previously mahalanobis correspond extreme simulate construct spectral library hyperspectral spectral noise adjust figure present spectra number train repeat hyperparameter term accuracy report vs classifier intrinsic present estimation try fix value lead drastically configuration class increase decrease estimating important another estimate correct live experiment axis represent value classification propose kernel performance result provide mahalanobis kernel kernel pca mahalanobis confirm poor generalization mahalanobis deal conventional sensitive datum redundant class sample could l mahalanobis mahalanobis average correspond mahalanobis load computed set first program matlab core ghz low eigenvalue demand hyperparameter computation eigenvector fast second demand mahalanobi regard eigenvector demand mahalanobis perform fast clear winner process computational viewpoint less demanding mahalanobi might nevertheless mahalanobis demand pca mahalanobis mahalanobis mahalanobis adapt propose parsimonious mahalanobis base space original space split signal subspace accurately purpose demonstrate case accuracy increase superior simply datum mahalanobis regard computational load mahalanobi demand besides consider tune optimization hyperparameter demand computation strategy concern development model conventional mixture kernel could mahalanobis kernel summation intrinsic author tc university definition article exploit computation mahalanobis inversion condition inversion unstable parsimonious signal noise subspace inverse stable use hyperparameter provide conventional kernel dimensional commonly automatic huge number variable hyperspectral hundred pixel gene thousand gene customer web service associate noise cluster filter literature moderate suit pose need specifically exhibit intuitive statistical property space summarize tendency corner tend concentration difficult analyze unfortunately discriminative also suffer tend distant clear near fail distance assess similarity affect neural locally potentially phenomenon consequence space phenomenon follow bellman reflect difficult separability comparison hyperspectral hundred spectral sense former contain hyperspectral conventional accuracy several dr dr reduce mapping onto space meaningful dr dr algorithms without exploit project accord variance independence improve discriminant analysis dr however maximize class scatter matrix scatter condition limit effectiveness popular dr laplacian one dr purpose map onto global discrimination maximize dr recently specific subspace original processing class normally embed white property without discard model dimensional dimensionality discriminative conventional generative local depend neighbor mostly local kernel choose approach subspace mahalanobis inversion whole associated matrix diagonal purpose covariance sample mahalanobis project apply euclidean space would specific adapt data subspace ensure parsimonious characterization possible explicit parsimonious mahalanobi construct mahalanobis several subspace hyperparameter optimize classifier accuracy subspace select hyperparameter section detail signal simulate dimensional report conclusion space curse statistical estimation I restrict covariance reader find clustered cluster generally follow identical similar intrinsic whereas class last understand decompose noise contain noise ci sample eigenvector correspond eigenvector computation usually opposite dot major drastically covariance estimate need furthermore stability eigenvector mahalanobis rely significant discard dimension theoretical advantage two far space seem discard noise even worst consider ccc represent contour b mahalanobi hyperparameter optimize tuned hyperparameter reason know direction optimal classification maximize variation direction discriminative modify discrimination regularize mahalanobis induce hyperparameter analyze section function map onto space evaluation input live riemannian metric equally task compress right
raw exponentially straightforward ad ad independent observed achieve self ad degeneracy query equally ad tuple ad b calibrate ad self calibrate ad calibrate calibrate prop prop prop prop thm nice thm prop sec prop prop sec prop nice guarantee nice prediction calibration expectation condition selection selection define self calibrate sure state query bid raw previous weak define invariance raw map calibration show ad raw prop prop weak z insufficient property behavior occur mechanism prop prop equivalent prop prop nice bid prop fortunately prop prediction f p calibrate ad element also event selection break ad ad calibrate assumption consider prop query candidate tuple share selection ad must ad prop prop imply prop query two c f prop imply nice four equally ad scale ad whenever prop impact thus map efficiency show half bid query add system normally reasonable query ad seem unlikely grain bid running experiment ad prediction nevertheless current calibration iid assumption question efficiency lose randomize deterministic maximize calibrate lemma calibration prediction system statistic calibration adjust prediction click prediction ad want maximize notion calibration impossible detail impossible calibrate prediction give achievable maximize roughly speak already capture calibration event predictor happen fraction occur event problem prediction make decision engine past decade business role engine select candidate ad query keyword reasonably candidate rank ad bid bid click bid show ad top ad show ad score threshold nice prediction value generate equivalently produce mechanism combine value reflect predict face example volume rather new output order prediction consider informally state efficiency computationally map efficiency formalize question mechanism different mechanism impossible prediction calibrate ad iterative fail since ad rate ad np sufficient fairly system bid query selection condition fact calibration predictor calibrate fairly make example easier calibrate classification calibrate example good predictor excellent method adversary easy classifier prediction calibrate sequence predict interaction calibration care theoretically key begin define unit ad select provide raw concern raw prediction establish study interaction engine fix string like car ad click bid pay click prediction prediction drop sometimes ad tuple query ad query drop ad consider ad ad achieve high pick ad model ad raw position ad theoretically change change ad respect ad candidate ad ad formalize distribution ad proportional ic choosing suppose two query candidate candidate think ad occur ad q break tie thus ad f calibrate ad multiple concern finding prediction q able issue assume exactly addition choice mechanism show take selection bid reflect click justified pricing per cost incur query incur engine cost vary ad would notational burden focus simplest generate transform efficiency prediction calibrate say exist first relate offline problem datum efficiency prediction order ad problem nice calibrate map exist maximize calibrate concerned start prediction serve like train large train prediction ad calibrate ad batch may ad prediction map assume calculate exactly ask ad potentially answer iterative calibration cycle start finitely ad map permutation cycle address general restriction select ad ad purpose counter section prediction candidate bid cumulative ad candidate ad three ad tuple show ad ad ad generate near ad occur generate negative ad exponentially far shorthand pick candidate fact ad figure distinct candidate query consider single query ad observe click self calibrate ad click rate show ad start cycle efficiency example calibrate even efficiency access calculate time sort bid must list exactly ad query show ad compute contrast practice possible accurately simply grain estimate select ad query raw trivial answer become show nice answer show ad bid guarantee z tie break answer q
replacement design design practice design require frame list may complex design design use appropriate distribute natural design entail detailed presentation reference substitution extend functional assigning unity elsewhere namely dirac total mass depend respect sample membership indicator assign weight elsewhere thompson weight reader lie information mention nonparametric treat elsewhere rest functional expression base straight obtain argument implicit thompson well directly formula give thompson variable fr measure jacobian identity two strictly unique eq fr respect respect influence exist tm tm dirac definition functional usual definition robust survey mass unknown order von write tn tm tm use tm finite influence nonlinear linearize artificial variable linearize suggest estimate give linearize role exactly design linearize respect nevertheless put know curve discretize describe employ product quadrature view multidimensional vector kt kt kt kt estimate linearize k linearize volume treat france greatly fact de france million request store analyse challenge huge consume desirable consider median curve design population minute consumption aim curve week consumption record week auxiliary population kt week consumption curve peak peak day decrease time measurement effect figure consider strategy fix distinguish kind auxiliary auxiliary opposite sample realize classical survey update practice simple design put chance sample perform survey draw list median unique equal kt kt kt kt kt basic inclusion median obtain sampling inefficient systematic sample efficiency frame correlate adjacent element frame accord week discretization point simplicity systematic appropriate really frame advance solve eq may substantially compare replacement sampling homogeneous worth improve median linearize indeed linearize asymptotic variance sum estimator within usually know strongly computed week linearize compute linearize consumption week use propose split week distance linearize cluster prop allocation allocation figure week within accounting allocation kind build consumption induce consumption consumption week well allocation prop allocation allocation plot second week correspond consumption linearize week population linearize design efficient proportional auxiliary study information cope suggest discretization consumption week inclusion give thompson distinct element replacement use advantage sum consider one simple improve median partition call describe practical consideration may favor perhaps costly whole frame belong membership impossible perform group know improve estimator obtain thompson auxiliary group stage weight size median remark zero group whole suggest aggregate calibrate estimator auxiliary membership argument remain group agree allocation proportional allocation multivariate g linearize variable drawback reduce inefficient improve efficiency week size thompson study post curve four design design replication study equally space discretized quadrature linearize variable cluster space term divide compare simple result proportional allocation design week consumption surprising report design perform estimating consumption fail median believe encounter small need prop also simulation week consumption table c prop remark surprising construct account consumption variable optimal allocation compute minimize variance mean estimator allocation usually survey consumption linearize design need consumption costly obtain impossible storage contract consistency analyze design quality estimator use estimation sampling give deviation design h median prop dash allocation give observe sampling divide ten compare proportional example optimal survey appeal consequence consumption curve confirm rule lead reduction estimator get well nevertheless rather complex future concern auxiliary information concentrated median thompson complex develop general regression constructive manuscript spatial b b estimator median banach efficient geometric median space average gradient component e asymptotic band estimation outli p geometric notion quantile pp storage complex statistic residual methodology estimator survey principal di ai multivariate center area population journal american association influence role robust population g thompson replacement universe b median banach quantile york nd york zhang multivariate usa von differentiable english france de universit email fr fr widely consumption currently de france profile unfortunately outlier high consumption load median avoid whole median test linearize compare area future highlight thompson substitution france load calculation load use future load customer load load curve profile use consumption customer nevertheless highly sensitive presence customer infinity datum force extreme customer view context join company one leave important amount demand load profile demand customer class synthetic describe situation profile profile besides analyze population median univariate central way univariate advantage survey first motivated mainly census spatial median among definition median spatial study detail zhang company device individual consumption new accurate point work census united concern center population median explicitly drawback transformation goes deal variable isometry require invariance share lie
tuning reason tuning parameter purpose except tune unchanged conditional chapter tell respectively sde diffusion sde brownian motion remark nn homogeneous initial degenerate distribution discretize find replace uniquely see limit governed wiener limit n derivation trace norm sde langevin univariate adaptive mcmc properly behave langevin little metropolis adjust langevin mala langevin mala general lipschitz uniqueness lie b sde ds ds take integral ds ds ds use e cauchy ds ds e ds multiplying integrate side ds ds ds ds u ds u ds du f f du equation bind prove sde comparison discrete call try tail compare mh tune fix proposal cauchy generate discard efficiency perform kolmogorov ks remain asymptotic measuring empirical burn high value great choice well adaptive suggest optimal lie simulation extent naive mh heavy lie interval adaptive although compare solution euler discretization choice euler z euler h sde euler mcmc give target compute kolmogorov value table note part sde move leave modified accordingly well almost peak average dominate differ interest quite heavy address adaptive figure plot value e e compare asymptotic generate adaptive e e e e asymptotic use target cauchy cp cp adaptive cp value non adaptive h ht cauchy pdf diffusion many iteration invariant approximation chain limit target limit metropolis early adaptive attention example however adaptive mcmc early scope mcmc continuous fine detail suggest mathematics valuable comment cm cm markov carlo mcmc proposal chain mc stationary diffusion standard limit trivial diffusion plot comparison adaptive mcmc sde monte carlo mcmc consist markov unique invariant parameter possibly infinite mh measure choose proposal probability form generate accept show chain drawback proposal distribution make stationary dispersion interest turn value mcmc quite main concern definition grows follow algorithm algorithm select move proposal multidimensional mh detail technique mapping continuous limit metric endow algebra variable nh homogeneous probability let almost surely specify drift coefficient e k dispersion
branch replace step similar finally replace constraint hand side start cell outer sep minimum fill gray none draw none none fill none draw none none circle fill circle white fill cell outer sep circle fill gray sep cm sep draw none fill draw none fill fill white draw circle draw fill draw node circle source target record linkage record tuple record linkage computation ratio record pair match record subset match vice versa record subset record possibility record record tuple subset weight record tuple membership record belong denote record tuple assign tuple however regardless decision appendix record tuple membership record record order weight determine cutoff weight multiple procedure maximize assign right subject tuple belong maximum record possibility whether record tuple possible increase indexing subscript record membership j p p finally record tuple configuration belong record decision optimal availability assign wrong keeping subject admissible level assign record subset linkage practice matching evidence actual error rate cutoff linkage expect resource linkage perform three record census three gender outline follow find record triplet suitable triplet every field train divide set within triplet belong order method gender triplet pattern classify remain triplet date age explore option construct three categorical variable date create comparison new construct category category date period use age record gap record new think record assign help identify replace triplet multinomial pt c comparison category table linkage option age date obtain hand thought usual misclassification group control linkage carefully scenario misclassification error indicate multiple record linkage comparison date treat date way agreement three correspond agree unit unit agree small age category misclassifie triplet misclassification indicate use properly naturally performance specific link implement bipartite record third approach assignment nominal level triplet misclassification error try decision record triplet assign decision triplet record linkage record triplet one classify linkage provide decision uncertainty something record practice common variability size dependence record field existence replicate record explore propose method affect model generate categorical record measure give represent value field unobserved model record field distribute field generate generate around value value important linkage inefficient need good adequate linkage product census three accord date measurement triplet record linkage option error compare result method city simulation measure panel b misclassification specific effect close subset measurement true triple create option low explore generate contain common five field contain equal also variable scenario category scenario database field category scenario known quality across four database triplet triplet database linkage correspond exclude made assignment use performance report mean assign mean meaningful linkage unbalanced subset extremely small show misclassification panel solid dash error grey represent method quality extra panel large large amount impact misclassification class finally scenario inclusion low extra integration common modern decision common match tuple bipartite incorporate linkage posterior model belong go beyond dependency naive integration believe linkage census census decade linkage enumeration post enumeration census block source use american survey various file incorporation linkage describe sampling census order ps ps ps logit order define decision record tuple assign record tuple subset keep rule record tuple record tuple admissible rule construction divide wrong minimized linkage linkage identify unique available building upon application record linkage merge post enumeration survey census census health care database name integration use ad linkage decision thus suppose record linkage linkage record file match truly match could occur measurement record record refer another another individual bipartite linkage pair file resolve matching record ad hoc resolve multiple bipartite appear survey implementation accurate census evaluation census adjustment require census evaluation match survey response census dependent joint inclusion heterogeneous survey evaluation process list homogeneity attention multiple linkage exception event whenever capture diversity source coverage assessment record system clear record maintain census de de number record conceptual difference whereas national record information daily census occur evaluation census record system linkage source linkage generalize approach linkage supervise record product link union possible agreement record record kind fitting present decide tuple record study different scenario exposition population denote record exist record generating record member vector record tuple entry correspond describe record tuple information different individual I individual record tuple classify record tuple individual order formally set associate tuple partition group entrie tuple pattern common record tuple agreement certain associate tuple set describe match consideration grouping agree similar idea record tuple us record agree agree agree since match partition equivalence record tuple equal one agreement field vector field represent agreement present represent belong connect need link three agreement fr full triplet field probability kp p p pg j pg note entry tuple complete bipartite linkage take state independence multinomial indicator define obtain arrange k k conditionally independent give tuple membership baseline produce
outperform fact filter ds remarkably close length accurate forecast svm preferable may outperform historical evident experiment ds test initial add five multiply figure value interesting condition look around mse begin consistent repetition phenomenon create repetition parameter converge converge wrong poorly repetition result two green red specify pattern location misspecification forecast misspecification factor lead incomplete knowledge system historical validation procedure stop cross pick dimension well range dimension allow pick determined dimension hyper pair l entire sort experience svms ls dimension ds ds fit filter laplace diagonal choice degeneracy calculation use possibility approximation posterior previously create particle particle degenerate employ addition encourage parameter state perhaps forecast mind state state equation propagate specifically one step dynamic practitioner intuition perfect knowledge yield forecasting system know incorrectly forecast accurate system fail highly system paper forecasting historical preferable exhibit misspecification extensive misspecification examine circumstance historical datum predict traffic flow week nuclear forecasting forecasting however vary drastically historical traffic volume predict flow may physics historical forecast forecasting forecast use perhaps city traffic flow considerable traffic flow united lack world record infer ice core preferred perfect one gain arbitrarily system knowledge historical become limited nonlinear include system forecast become inaccurate particularly prominent field connect science chemical determine strength market physical formalism usually equation system ordinary euler approach base approach forecast rely solely next year management give file rough origin test testing conference one somewhat parametric linear autoregressive parametric vector assumption relationship note combination well exposition wish knowledge method describe form various level misspecification method organization discuss historical datum parametric tool approach remove effect misspecification study simulation conclude future may parametric parametric rely describe datum relationship autoregressive historical comparing build class svms latter specifically begin describe support machine ls end square free parameter kernel uniquely hilbert rkhs e consist map suppose sample l svm find unique solution optimization problem hilbert sample diagonal require operation infeasible find maximize usually discuss early process differential initialize forecast adjust current noisy assimilation gaussian noise choice kalman filter class dynamic system forecaster process process assume write linearly additive noise order frame recursive interested posterior merely often filter density know include forecast able filter analytic covariance simply calculate however dynamic kalman provide alternative method dynamic methodology kalman nonlinear kalman linearization initial estimate wrong incorrectly jacobian kalman filter approximate transform variable instead taylor expansion kalman mean transform recover carlo filter importance distribution particle associate importance unnormalize typically sequentially take markov nature measurement x z recursive weight formula choice proposal simple proposal particle filter simply generate evolve forward proposal calculate weight degeneracy resample step remove multiply high weight filtering concerned dynamic information noisy state space state representation equation iid run state filter parameter state filter many detail volume purpose misspecification broadly include traditionally term firstly complete complexity model misspecification incorrect alternatively might attempt misspecification understand term lack availability complex require considerable amount rough elaborate gain mechanism misspecification cause forecasting method forecast distant forecasting misspecification unknown look svm filtering early study learn may really limit specifically dimensional system forecaster mention later svm consecutive observation forecaster complete dynamical system forecaster dynamic forecaster addition entirely base specifically system explore wide length historical fold increment repeat forecaster repetition historical historical repetition filter laplace specify laplace general setup generate trajectory dynamical historical set svm random describe index sample save index examine historical misspecification fully drastically describe system stochastic observation ds none unknown none ds aspect ds ds contain various attempt ds contain detail forecast appendix ahead forecasting rmse
sphere study index tree greedy strong tree margin investigate success evaluation algorithm population evaluation exhibit assume procedure copula fit exhibit sphere bivariate characteristic bivariate explain well summation optimum fraction sum select reflect include provide appropriate situation point interesting explanation simple interaction combine non copula summation linear summation model multivariate well copula copula similar present copula construction exhibit robust multivariate product level copula simplify construction determine selection study example compare tree aic determine tree summarize evaluation I truncation tree select appropriate tree technique aic find pairwise show crucial copula arise propose benchmark correlation seek copula copula product copula copulas copula construction product copula consequence underlie normal bivariate univariate normal copula function bivariate student copula student univariate student distribution degree derivation bivariate copula dependence bivariate transformation degree bivariate copula control dependence denote expression bivariate copula bivariate copula controlling dependence close form numerically derivation formula bivariate bivariate represent dependence bivariate copula transformation copula copula control bivariate copula institute physics email institute physics email institute mathematics physics email study copula build copula copula study dependence aspect crucial success show copula appropriate distribution explicit probabilistic sampling build solution multivariate derive evidence copula separate statistical structure fit rich univariate model dependence model copula limitation copula strength dependence coefficient parameter characterize pair alternative problem multivariate copula powerful copulas decomposition family literature author behavior problem indeed copula identify model build product copula empirically show well endowed variable notion copula respectively notion terminology conclusion consider random distribution every multivariate distribution function marginal copula separate effect dependence margin provide immediate random copula copula give independence margin support besides base reformulate copula normal margin copula multivariate normal univariate normal normal case dependence preserve margin normal inversion estimator ij matrix positive correction margin normal margin I select population individual margin distribution third fourth limitation available copula bivariate parameter appropriate construction approach result decomposition multivariate copula bivariate copula building construction nest call regular copula construction regular rich bivariate copula dependence bivariate copula set node regular constitute edge tree share instance regular graphical four model specific particular denote scale rgb rgb rgb joint consist density copula construction q copula distribution component remain recursive evaluation expression special univariate reduce joint facilitate function define expression bivariate copulas simulation develop implementation specific copula imply way random heuristic heuristic assume weight appropriate variable maximize node root greedy maximize weight original instance add node edge node heuristic structure completely copula decomposition tree require might step conditionally c yy consist fit tree detect truncation level bic expand value truncate tree copula type goodness copula way repeat copula accomplish way bivariate empirical copula copula significance reject copula combine copula student normal low student copula dependent copula copulas tail copula copula likelihood become almost indistinguishable normal copula distribution method w x j expression although present crucial statistical package know function definition summation maximization optimum ensure fair find algorithm optimum optimum precision evaluation truncation population sample interval value take global locate middle interval interval experiment sphere summation asymmetric interval summation joint asymmetric interval ccccc population c ccccc evaluation well large normal margin interval margin global optima capture
short time sampler st lattice site express take integer graph exhibit phase start update algorithm periodic boundary metropolis heat gibbs iterative sampler alg compare confirm convergence gibbs ergodic fast quantity factor acceleration locally rejection transition gap markov st autocorrelation conventional autocorrelation variance autocorrelation bin autocorrelation metropolis heat iterative gibb investigate exponent temperature exponent gain metropolis see st geometric allocation autocorrelation kind mention introduce gibbs review show set well mention update applicable heat usually inverse cumulative inversion apply candidate choose proposal reject autocorrelation explain reduce rejection general inversion reject ratio canonical mcmc candidate achieve rejection picture simple metropolis generate configuration stochastically detailed keep reference rate state px configuration isotropic bivariate x proposal propose candidate current position account broken proposal avoid configuration current configuration proposal use proposal probability state fig process rejection minimize st indeed proposal get short candidate construct several ergodicity extend efficient future review ergodicity assess conventional applicable efficient need analytically grateful valuable discussion present university code library support grant aid scientific program support markov physics university south wide variety tool chain carlo curse care candidate configuration determination kernel construction monte dimensional wide tool interesting topic physics phase transition physics curse effectively suffer configuration update sampling decrease former quantify variation transition assessment latter decrease autocorrelation observable th independent autocorrelation effective roughly total number simulation central care key ensemble replica method apply protein spin wang overcome growth graph many physical model hamiltonian move candidate momentum third determination mention chain construction rejection transition matrix detail however chain metropolis type apply algebraic transition conventional ergodicity monte balance impose method detailed balance impose simulation balance elementary force actual simulation metropolis algorithm heat sampler performance seminal analytically simple reversible optimization transition finite state transition element follow statement ref suppose irreducible matrix invariant call gibbs usual gibbs accept reject metropolis scheme modify gibbs sampler sampler say iterative version gibbs sampler rate diagonal minimized see transition balance condition invariance target meanwhile modification additional axis version axis apply momentum hybrid partly chain generate history discuss asymmetric sphere see net flow hybrid carlo detailed balance recover break paper chain reversible metropolis however al apply solve algebraic algorithm discrete continuous space probability balance visually allocation optimal solution introduce allocation update huge implicitly construct elementary configuration proportion quantity raw flow average rate metropolis proposal distribution give heat find minimize sense visually allocation entire box keep fig ref describe alg satisfie allocate rejection maximum rejection rate certainly reduce candidate rejection discuss tb geometric propose rejection figure weight candidate choose matter box remain box remainder allocate iii ii continue likewise iv step iii box discuss ergodicity previous irreducible condition ergodicity although update interesting ergodicity color tuple state assume unnormalized proportion state update alg configuration allocation arbitrary fix calculate reduce update irreducible define determined alg transition make alg irreducible consecutive update candidate multiplication certainly cyclic loop end visit b monte update accord lemma similarly choose show space irreducible ergodic rejection exist
possibly necessarily identical I graph separate path separation condition separation mit coupling process mit source process mit solely convolution exist mit parent parent parent mit additionally distribution far simplify parent mit mit proof give remark modification mit autoregressive coupling lag connectivity generalize analytical lag mit mit yy shannon communication couple communication channel mi shannon read bandwidth coupling capacity alone interpret strength mit mit might leave exclude parent parent parent modify mit call stand modify mit stand separate mit hold coupling strength link thus mit coupling coupling strength make mit solely couple filter mit numerical next section couple strength numerically mi te mit investigate process independent white process variance b dependency process dependency appendix functional form neighbor lead bias high variance mit coupling strength link depend mi mit mis entropy mis input mi approach analytical exclude exclude different always equal mit bottom panel fig mit red line te gray dotted link series green dot te much mit compare lead te analytically input apart te link dependent consideration mit show input due estimation mit aim measure distinguish couple property equitability parent expect mi take broadly mit coupling distinguished analytical variance add total plot te suffer negative mi mit solid related conditionally line almost nn seem bias small measure te relative mit theoretic mit thus measure coupling strength use te possibly also mit intuitive well interpretable te coupling ingredient equitability interpret coupling assess mit mit x large mit coupling surface area correspond strong mit large increase surface air apparent spurious coupling mit measure strength reasonable picture interaction preliminary analytically mi te coupling strength overcome step lag link link multivariate theoretic mit lag strength interpretable experimentally set couple strength process coupling attribute suggest modification mit mit practically besides extract causal direct indirect connectivity assess coupling acknowledgment national foundation research education research research environmental thank comment appendix proof mit discussion regard couple strength linearity condition mit mit x stationary series article condition parent hold coupling include ti separation time x mutual parent already path towards chain twice covariance variance auto q kronecker delta else entropy derivation markov entropy simple form top toeplitz e toeplitz toeplitz te toeplitz utilize relate geometric toeplitz wiener absolutely toeplitz te integral contour integration te since block zero everywhere simplified complement block since zero infinite te depend strength rather exploit infer involve mit mit source conditional firstly process correspondingly I secondly arbitrary entropy source generally depend parent invariance simplify arrive since independent past add simplify g therefore extend arrive parent remain parent mit additionally ic mit condition entropy first invariance entropy know know dependency parent communication graphical science operator new york method new york j causality reasoning g meet mm mm conjecture united assess strength meaningful focus strength theoretic demonstrate known instead certain delay mutual mit association coupling computable root mit graphical mit component graphical admit low reliable mutual mit practically mit strength compare interpretable case solely interaction particular mit many able exclude theoretic formalize idea general illustrate mit produce grow abundance hypothesis statistical measure key reflect heuristic notion give dependency idea multivariate reconstruct field propose zero imply causal strength alone remain understand interact bivariate coupling strength influence correspond keep solely measure impact realization regard ingredient demand require somewhat transfer due property reconstruct interaction link attribute delay mechanism give already denote assume stationarity te aggregated lag density te choice truncation lag affect via number process influence te reliability causal couple delay influence coupling dimensionality overcome next transfer derive enable estimate remainder another truncation describe somewhat arbitrary lag avoid drastically still quite simple lag lag non lag dimensionality causality sect te strength property mit past parent series infer introduce supplementary detailed detection positive determination coupling strength detailed causality link mit infer link parent certain key stationary time yield realization uncertainty observation realization non process noise always observe mit entropy diagram attribute mit closely couple show link mit correspondingly due mit mit mit sensitive mit use causality non also parent drop source mit causality graph entropy reach measure situation interested matter hold sect mit already estimation truncation analytical property series graph mit website http mit intend different analytical interpretability motivate couple tractable process series graph parent dimensional determinant evaluate derivation formula te te entropy past easy yield entropy innovation term hand treat process apart infinite toeplitz theorem another block thus
unlikely region obtain ucb complement colored panel denote task shrink search space simply algorithm back lattice lie stage relevant inside practice even exponential course improvement begin show explore location residual specifically hull set hilbert projection interpolation hilbert bound well reproduce kernel hilbert rkhs norm bound tp p tuple minimum norm interpolation consequently satisfy assumption lemma follow x location derivative hull pairwise distance diagonal gaussian process subset standard variance vanish point exponential vanish branch theorem state prove supplementary material gp appear interior twice vanish alternatively vanish lattice decrease exponential main use bind utilize together picture expression ucb finding assume compact convention subset give differentiable continuous fx x xx bx df two q specify branch bind like remark theorem ern hessian co dimension function chance lattice fine interior important decay lattice choose sake rate follow branch algorithm cumulative regret ucb monotonically factor shrink necessarily lead regret point large ucb sample every lattice propose modification ucb address key regret evaluation word speak case far discard piece unlikely compare show additional step improvement branch cumulative obtain noisy unbounded search region less go free achieve identify noisy reflect achievable depend suggest noisy limited exclude branch lie outside perhaps cut would encountered guarantee contextual bandit parameter context v stack functional provide remain standard adjoint arbitrary equality calculate claim prove see project follow correspondingly interpolation tight variance inversion second without generality minimum consider exploit need dimensionality large latter prove rkh lemma derivative vanish bind bound sequence sample covering banach set define ball banach space entirely first show step neighbourhood enough neighbourhood densely envelope around tight eventually neighbourhood please graphical compact continuous unique suppose contrary radius exist bx fx tb theorem densely densely inside reason localize neighbourhood regret exponentially proceed twice densely shrink claim prove number follow carry lattice sample iteration begin f sense combine give x neighbourhood precise form irrelevant maximum argument goes depict tb need govern equation differential separable integrate side integral use integer illustrate somewhat satisfied overall expression x derivative indeed cs process observation ucb gps gaussian great prove vanishe rate complement attack attain much fast regularity function near global maximum sake maxima might free could feed consume algorithm global observe sensor deterministic configuration automatic entire architecture hardware uncertainty characteristic elsewhere mention search impose objective vary require change point point tight near elsewhere relax fast discard bound convexity global force reach search guarantee produce function lipschitz give imply methodology interested result stop ideal regret hessian behave cf therein improvement methodology function possible
underlie among give variable assumption sparse try column use penalty copy block condition induce dependency machine partial tool condition input indicator annotation associate feature graphical tag identify dependency tag interaction ignore impose consequently block order estimate impose assumption sparse affect moreover estimate inefficient unfortunately dimensionality document bag web dimensionality order category usually much small formulation precision estimate one consider matrix contribution graphical model point marginal precision dense expression quantitative trait gene gene take effect gene paper simultaneously matrix drop consequently reformulate efficient statistically connection conditional quite partial graphical different impose lead convex formulation lead summary estimate contrast global minimum directly scalability formulation conditional assume sparsity establish numerous precision recent estimation popular precision matrix guarantee investigate formulation know estimate support nonzero separate neighborhood estimation aggregation hide dimensional dimensional write complement exhibit sparse dimensionality low minimization standard advantage realistic many closely analysis precision likelihood convex solution solution precision rate precision approach precision via significantly procedure statistical model estimation study regard multivariate matrix text entry vector row column ij ij ax j remain graphical analyze multivariate unknown discuss paper distribute e parameterize give vertex order pair normal convex sparsity particular precision formulation aim estimate matrix confusion idea eliminate respect follow allow allow convergence sparsity introduce indicate decompose l proposition behind optimization high penalize define separate formulation sparsity induce penalty respectively induce penalty full model formulation significantly assumption analogous situation conditional field unnecessary relatively compare yy yy yx yx yy yx yy yx concept strong q guarantee present probability proper hold compressed follow sense assumption obtain let frobenius minimizer wise penalty interpret element assume absolute c imply may depend least algorithm subproblem respectively objective procedure global q proximal utilize smooth next problem eq subproblem term subproblem inverse product gradient subproblem product evaluate therefore high per precision dimensional standard significant show formulation noise matrix estimation set sub gaussian discussion maximum noise setup true covariance matrix follow remain algebra write rewrite equivalently express multivariate yy generate condition add generate training goal increase compare recover conditional glasso glasso modify recover support via modify selection method adopt study method precision frobenius evaluate retrieval concept stand positive fp stand positive negative recall practice score metric evaluate f support recovery glasso expect glasso select parameter exclude figure since glasso inferior due penalization figure speedup glasso glasso glasso consistently form norm observation consistent frobenius performance stock price dataset parameter sample summarize describe become standard benchmark image retrieval image annotation contain around manually annotate keyword keyword set visual feature publicly construct evaluation feature tag label joint allow available high score annotate make tag average word per word along extract sample training graphical set contain scheme utilize remove transform set category subset sample frequency purpose vocabulary keyword price stock stock stock daily price trade first adjust return point choose sample goal precision condition size test value sec training glasso glasso glasso experiments glasso glasso precision component formulation skip regularization ground truth evaluate recall definition training since construct two link regard connect category training observation small value glasso precision graph identify glasso glasso stock correlate well approximated construct link identify threshold show graph tend glasso potential link induce tend slightly construct method illustrate detail top present estimate advantage conditional include interpretation term dependency variable without sparsity assumption show rate convergence depend partial competitive current derive normally distribute relaxed normality easily set useful undirected believe extension application follow determinant algebra eq q parametrization pointwise objective convexity proposition hold yy yx yy yy yy imply absolute contain th q combining display therefore complete singular last desire bind q verify q yy yy yy yy yy yy yy yy
rule classify every category binary variable represent say instance th otherwise label category simplify general naive analogy classic category joint along feature true instance might seem basically perform naive show neither intuitive realistic clearly clear jx cm j p jx formula probability instance estimate instance train binary assign last give probability would know assignment shall make approximate notice really variable apply equally approximation remove threshold component classifier compute capable deal input classifier propose content binary capable give instance obtain category category compute score thresholded hard categorization width text corner minimum minimum height text height distance text width block classifier cloud classifier load apart contain say usually part classify binary label fast computation needs summarize see neither issue classifier methodology second heavily suitable content classifier possibilitie classifier exhaustive give aware collection combination different categorization corpus together preprocesse news use split name document test assign test medical every assign mesh category mesh huge often category disease category subtree category discard corpus follow relatively news document final document term name split give remove reduce first english apply mark removed occur less document document preprocesse consider base measure shall along brief measure certain label subset measure hamming compute normalize label hamming label range test subset loss case predict set average measure make case hard categorization assign suitable task recall stand true positive false negative categorization harmonic macro micro average version denote classifier tuning aim make result implement evaluate rank instance partially rank label top list relevant baseline secondly obtain well multinomial naive bayes nn use normalize svm choose recall could perform perform gain max combination selection note implementation use contain except alternative label choose base classifier category correct category real usual dot learn maximize criterion generally select instead proposal reason deal fast work environment accurate regression include approximation accurate vector probabilistic vector value classifier value probability simple svm modification canonical reasonably outperform select classifier effort naive machine refer classifier machine respectively distinguish correspond explain also purpose micro macro baseline least one baseline reach macro respect remarkable present run couple classifier micro micro sign perform macro choose kind nevertheless notice specifically true positive negative positive also macro account number improve sometimes counter intuitive significant macro micro second hypothesis say show table sign resp plus bad baseline sign indicate difference time bad significantly bad ccc ccc nb nb nn nn svm svm nb nb nb nn nn k svm ccc nb nb nn nn nn nn svm svm baseline micro macro methodology follow fact technique improve macro improve baseline micro also especially baseline around classifier bad micro compare bad regardless remove capture base assign probability assume example methodology seem benefit lot macro category heavily whereas micro micro quite baseline sophisticated one table value bad nb nb nb knn knn knn knn knn svm ccc subset error nb nb nb knn knn knn knn knn ccc loss nb nb nb knn knn knn knn svm hamming subset alone pattern behave really collection improve baseline version work naive perhaps improve baseline show present measure great explain method great increment macro micro fact ham false positive negative regardless predict baseline show methodology explicitly take account classifier way assign training content build provide document collection content baseline classifier naive classifier experiment tend known technique question first output threshold question explore relaxed environment categorization natural form acknowledgement support pt la machine unlike choose complexity exponential alternative common use take methodology improve obtain train occurrence exhaustive standard corpus probabilistic base classification text categorization category instead nature corpora article belong collection article multiple document scientific paper associate keyword vocabulary subject internet example furthermore tag sometimes tag instead category scene categorization protein medical diagnosis although solve ignore concentrate obtain I
use attribute test set role business attribute parameter evaluate result business entropy indicate guess assignment user good configuration entropy small determine configuration attribute role dispersion resemble must business attribute uncertainty reason business mutual heuristic score generalization experiment organization unit dash generalization split train mix user role role term compute evidence pz numerator distribution b rewrite beta make must normalize consequence bernoulli derivation substitute back eq analytic evidence assume bernoulli bit bit use outcome noise indicator role negative auxiliary variable li solution I derive update energy introduce derivative derivative first equation rich department mine base control give access mining candidate small minimize paper mining combinatorial need represent derive probabilistic model likely generative reflect configuration additionally assignment configuration assignment experiment real role wide variety popular access assign resource access role role decompose role assign role role conceptually easy assignment user experience well recognize may change within different must aspect develop two kind top role top business security policy exist assignment engineering incur pose security simplify automatic bottom engineering numerous role develop notably unfortunately mining algorithm suffer artificial role undesirable link principle mining goal deviation role deviation goal compression role role configuration call attribute user assign limit among make configuration difficult maintain join business within role fit close possible user assignment attempt make business mining process business relevance role control matrix necessarily role arise role mining reflect setting practice role maximally fit configuration access definition algorithm prediction compression address wrong problem argue mining quality solve contribution mining definition role role likely motivate involve probabilistic configuration likely control configuration role role hold generalization role mining compete experimentally role incorporate business mining thereby improve interpretability role follow inference role problem appropriate evaluation class model user world control show business user mine hybrid role mining algorithm attribute draw conclusion explain underlying mining business relevant account pure without special assess mining rectangle text center height em text minimum rectangle text width em text em node auto input nd assignment assign user role assign fit residual matrix encode mining term user di entity induced p heart mining role say search assignment assume principle organized find access therefore perspective business policy reflect business encode policy enable business knowledge security policy business business attribute principle name reflect mining structure distinguish mining hybrid role might contain unknown perturbation control exception fully system error discriminate exception expert rank procedure substantial manually assumption instance fraction determining constitute role problem mining infer assumption view role differ availability hybrid part influence bottom role case remain solve bottom well hybrid assumption influence pure mining influence availability c dependency involve mining indicate generation grey pure mining role mining compression deviation role alternative problem either contrast quantity involve determine mining algorithm little optimal create closeness fit treat mining require obvious distance metric compare configuration hamming role usually know practice create knowledge evaluate configuration generalization use assess learn dataset generalization unsupervise role conceptually general unclear transfer role hold relationship role employ cost along validation mining configuration neighbor user role near mean create matrix row row correspond neighbor hamming divide ratio assignment behind configuration assignment mining generalization configuration mining fail structure thereby perfect mining inferior adapt compute describe agnostic role mining particular probabilistic posterior user discover tailor employ work method role rule core two role assignment mac role relationship instance particularly relevant deterministic configuration convert probabilistic version observe user matrix hand matrix assignment determine source parameter bernoulli distribution ignore moment treat treat random describe entity mean circle variable circle observable user assignment empty hide statistical dependency entity entity rectangle realization entity exist index assignment explanation text px boolean f term must outcome see go value reflect exploit property probability current deterministic role eliminate model parameter observation appendix px chance assign role user role take px ik eq accord entry conditionally complete user extend core introduce role core model depth level role introduce depth role layer super role generic repeat depth see flat section propose variant level hierarchy motivate consideration assign user column group resource department assign similarity simplicity matrix second group boolean assignment motivated similarity resource grant access object orient group execute alternatively group denote user call business role technical assignment track introduce boolean role hierarchy use rule understand boolean partially make business link assignment logical user assignment user assign path indicate assign multiple path easier express union path I appendix expression assignment business role technical role resemble one role switch condition alternative generic strategy arbitrary treat assignment substantially likelihood derivation avoid probability take constraint assignment group turn already domain arise particular decompose role lack one role assignment convert decomposition freedom fit role flat role layer hierarchy constraint template template specialized environment instance binary variable relevant model instance later generation contain l condition become collapse technical role serve assignment tb even constraint group limit belong belong user partition disjoint business role disjoint decomposition reduce hierarchy user role explain determine model repeatedly algorithm optimize cross validation generalization level hierarchy explicitly role role algorithm role provide role role new plus probability reflect favorable assign exist create enter rich influence role configuration role large likely configuration role model dirichlet number role nonnegative role assignment except event role create role exactly role accord kl account assignment active beta derive equivalent infinite propose hill repeatedly keep track figure assignment calculation hyperparameter htb prior circle denote principle add role applicable could underlie assign necessarily assignment replicate user account bit assignment add bit noise bit generate generation assignment flat generate coin flip let binary indicating bit observe depict explicit threshold denoise way denoise role mining show cutoff htb generative boolean mac indicate noise bit probabilistic model role mining require access em gibbs algorithm update current centroid assignment mac role introduce temperature uniform role set low make expectation assignment sum equal iterate modify negative result minima apparent early minima guarantee effect anneal scheme regime ultimately user role assignment benefit decision row initialize row initialize stop user assignment gibbs assignment keep fix exploit exchangeability cluster involve iterate assignment stop several predefine reach posteriori report book keep score compute average entire configuration restriction practical ultimately choose configuration experimentally investigate model artificial mac dataset compare mac world dataset core differ layer additional encode assignment create mac flat second business role disjoint term user role assignment business role role constraint role assign role difference mac implicitly make encode dirichlet variant suit solve experiment control accord mac assign outcome fair coin flip role distribution repeatedly role connect kind infer mac assumption operate role require additional generalization describe section quality difficulty vary level noise user run generalization hold infer decomposition depict right mac trend type level mac generalize mac even slightly bad mac mac range expect assumption behavior mac mac mac violate though extra mac layer instance assign technical flat configuration role overlap term model variant mac user mac inferior mac algorithm result appear mechanism finding role step gibbs scheme annealing perform beta bernoulli improvement unnecessary extend mac general generalization provide criterion select role assumption world focus explain give input mining assume hide noiseless noisy infer input probabilistic approximately reconstruction assignment investigate question conservative configuration reconstruct user assignment provide question wrong wrong input fraction randomize vary mac constant negative false positive towards repeat old error introduce sum old error mac rate negative false bit htb error distinguish negative positive trend configuration illustrate true value reconstruct learn strong introduce addition configuration result uncertainty practitioner mac factorization technique large matrix business attribute business attribute mine publicly dataset system scenario control server customer access configuration create compare boolean matrix different binary component learn fit learn boolean factorization mac noisy dirichlet capable learning factor role cost solver find boolean matrix decomposition minimize boolean decomposition false differently compare decomposition successively create large role select minimized subsample training user hold user model partitioning mac repeatedly validation select role threshold low error train customer user error run run mac user c min run min mac table number discover median customer generalize equally inferior mac mac winner largely significantly fast time author different criterion impossible fair comparison time mac increase generalization grid le illustrated mac generalize good bit bad tb hybrid business mining remain unchanged assignment business property approach hybrid mining fit configuration assignment business modify set role role assign business cost influence business make likelihood business special whereby optimize minima incorporate business toward minima business attribute input configuration meaningful satisfy business predicate attribute also ideally identical account
solution therefore q net norm ball convex hull unit immediately norm illustrate support u support norm differ sp hold norm show generality yield strict inequality strict efficiently solve composite particular accelerate exhibit proximal proximity map sp lx correctness sign sign hence require order sp fista fista elastic net corresponding complexity generalization guarantee expect huge whether tight support gain experimental mean total way contain validation point net lasso elastic net validation set report table oracle v gain show learn absolute elastic learn use response absence heart predictor training select three method version repository name remove word appear randomly split report accuracy datum regularization give heart mse se se elastic relaxation show elastic exactly light relaxation motivate tight relaxation demonstrate well prediction induce evident unit tradeoff population predictor predictor predictive presence beneficial correlate feature elastic net yield less solution direction yield sense course refined handle within question lemma remark case novel correspond sparsity tight relaxation elastic good elastic net support light justification expect justified envelope zero vector bind vector convex impose accurate magnitude convex convex outer convex hull yield zero empirical inside np scale perhaps good sparsity expect bound assumption broadly aid robustness motivate sample use constraint scale identify support regularization predictor standard get tight convex equivalently elastic net alternative paper relaxation whether tight well possible outer bind associate support norm tight convex convex elastic elastic net elastic two ignore ridge three select elastic correlate large may correspond consider hull clearly nest norm ball dx variational denote subset cardinality imply overlap traditionally application case tractable depict notice elastic cardinality differentiable net vector elastic norm support vector surprisingly entry support invariant derive formula every let unique integer satisfy cardinality combine shrinkage large
adaptive sequence k k tt u v u tu reasoning u k check case even odd obtain contradiction consequence several primal primal proportional case least iterate guarantee alternatively fix candidate thus primal average primal prop mirror note apply logarithmic applie focus primarily iterate interesting distributional worth investigation primal conditional algorithm cut acknowledgement partially european author like mark schmidt lipschitz b convex domain denote minimizer duality relationship x mirror recursion average candidate appropriate similar reasoning condition dx hx hx lead get integration part upper bound gap gap fa ax obtain gap pt plus minus pt plus minus pt pc proposition mirror generalize duality regularize regularizer dual interpretation lead mirror primal dual many problem cast situation simple potentially preferred method interior fewer expensive objective certain decomposable part classical subgradient extension sometimes refer frank subgradient adapt algorithm tailor hard show fact b recover previously c dual iterate review primal dual converge precisely minimization hx make follow assumption constant note conjugate bound situation tend boundedness allow explicit define differentiable gradient include cm quantity compute maximizer duality gap run converge rate convergence rate primal continuous strongly hx hx gradient pair primal candidate dual pair optimality subgradient equivalence mirror descent generalize strongly format original typical often regularizer strongly convex convex generally many barrier fit square although continuous entropy exp e proximal step decay extend general max formulation object vertex hypercube rather lead domain associate relaxation combinatorial linear maximize variation interpret formulation square regularizer lead descent hx proposition consider mirror strongly denote minimizer mirror recursion tt u follow adapt strongly x x x hx hx cm convexity rewrite smooth summing function average size sum weight additional averaging come guarantee close convex strictly include mirror recursion prop classical project classical mirror need equivalence assume still optimality prop review correspond maximization domain equal change conditional follow order towards small correspond linearization later search hx f taylor expansion part maximizer dissimilarity add extra proximal may u reformulate f f mirror equivalence mirror gradient lipschitz b mirror descent start
reduce present complexity rule language think complex learn order apply rule permutation great number begin universe vast reduce block addition improve order rule rule give else block reason linearly order rule beyond need less know great give rule gradually rule rule set first need short need thereby ultimately division complexity example rule step purpose rule however hundred seem child significantly exponentially rule relatively quickly number step time fast start end rule adapt rule give exponential middle rule linearly apply middle else proper new rule hence rule order sequence rule less pick give binary search logarithm number integer counting step different factorial n equation disadvantage impractical give exponential rule rule need less small child day learn rule year learn order
nonparametric estimation nonparametric regression show local procedure develop cubic periodic essentially intrinsic difference key assumed section value fix reproduce certain envelope need simultaneous denote eq strong crucial theorem away infinity existence bias result address show explicit bias specifically asymptotically vanish strong condition uniform boundedness amount choose suboptimal demonstrate validity rely simplicity find bt know simplify remove example consistent therefore coverage increase bandwidth pointwise ci exclude cubic periodic share surprising spline structure z thus imply ci periodic spline cubic calculation rely regression weighted value c value periodic belong th proportional show minimum base meanwhile maintain fourier satisfy minimum bandwidth construction otherwise boundary point make band thin hence likelihood require tune vast dealing nonparametric stand technical smoothing applicable composite hypothesis nan identify chi degree freedom nan limit show minimax well sample smoothing testing hoc thorough either eq q derive remark nan nice remark likelihood satisfy n r nh e cc copy direct reveal asymptotically approach distribute specification global next equivalent condition h reveal scale theory regression possibly change bootstrap reasonable value find testing whether belong space integer nan ny decompose l setup show modify theorem testing specify limit negligible minimax general find supplementary restrictive write pt hold positive achieve exist g ap cr detect local alternative turn minimax see testing establish lebesgue however separation norm minimax whenever equivalently local detect satisfy detect alternative since example model z package see case trivially useful identifying quantity pointwise ci become I smooth l z k dash dash length equally spaced covariate pointwise ci true periodic spline figure coverage cp ci space cp computed proportion replication cp smooth report indicate construct coverage band reasonably meanwhile area cover grow band area remark detail conclude significance setup except test provide local polynomial smoothing proportion correct nonlinear show moderate advantage especially intuitive rapidly vanish example test c cubic repeat procedure document summarize band factor tested significance true nonparametric gamma lead choose conduct similarly logistic binary logistic relationship length simulation straightforward give z find approximate eigenvalue significance numerical eigenvalue value simplify analogous consider true z length space test value power various significance level c plot title spaced validity specifically power cp desire level grow validity grateful discussion careful du author thank co comment lead improvement dms award dms study local spline unify first technical tool call functional traditional parametric equip develop I pointwise confidence ii local likelihood test simultaneous band prove point short arise ratio applicable likelihood carefully discover periodic tool smoothing spline variety literature primarily interest conduct together rigorously derive smoothing nonparametric assume construct interval band interval test local simultaneous confidence global property good little systematic rigorous partly important main smooth function cover aforementione relate tool asymptotic property square equivalent approximate reproduce become extremely complicated smoothing spline effective employ process develop new tool directly broad nonparametric immediate asymptotic normality spline lead construction pointwise confidence theoretically valid point length next testing function nan limit chi constant converge smoothness discover phenomenon nuisance arise nonparametric useful band global literature effort estimate see hilbert applicable validity ever nonparametric demonstrate optimal width achieve assessment aspect inference fan al testing namely generalize call penalize chi freedom locally test complicate periodic smoothing spline mild remark reveal inference mainly study however issue currently spatially peak drive future research reduction quantify need see necessary benefit asymptotic apparent mention extension complicated multivariate smoothing conceptually approach rest notation present local several local inference together discuss give concrete illustrate theory numerical periodic spline suppose copy nonparametric unknown cover assume model instead assume specify conditional value nonparametric second quasi modeling overlap coincide combination fa abuse may also refer homogeneous periodic smoothing use rather simplify existence uniqueness guarantee theorem denote order w tail condition continuously concave open positive z z slow rate impose boundedness information trivially order g achieve linearization technique quadratic device technique hence surprising condition cox assume weak ready present technical tool develop practice construct representation naturally apply hold nh h pt z pointwise asymptotic direct equivalent regression result global hold bias true obtain detail explicit pay obtaining could therefore alternative imply envelope introduce normalize eigenfunction satisfy iii boundary condition follow spline weight involve furthermore remove bias exist literature global mean corollary mainly focus conclude global infer construct pointwise local example corollary delta proposition pointwise ci remove confidence interval corollary asymptotically vanish nh z ci propose asymptotic ci aware rigorously pointwise spline know ci coverage exactly asymptotically furthermore ci uniformity across evaluate point however ci valid even short perform comparison case equivalent version heuristic ci nh proposing show see ci wide meanwhile turn correct ci remove bias inclusion limit ii introduce consistency short consideration furthermore demonstrate superior spline ratio testing chi
black whereas similarity dominant stationary blue motivation subject impossible exist subspace subject g measure table summarize datum improve however subject focus stationarity estimation surprising significantly improve subject shift like toy bottom individually word obtain filter parameter apply method combine statistically pair permutation estimate permutation obtain subject actual value cm ccccc audio competition iii subject std ss analyse stationarity activity investigate gain detail one highly reflect relation experimental mode stimulus mainly area responsible visual test minimize presentation stimulus change increase correspond filter feature little subsection show protocol may subject invariant train crucial training system use consist five subject right calibration experiment second indicate record day scheme band pass day subject mean increase effect lower prominent indicate variability less sub sub prominent change non similarity subspace span one change part question score span measure similarity generate actual lie significantly discriminative datum project discriminative direction method non relate noise reduce need show dominant change discriminant show prominent change experimental condition subject subspaces stationarity interpretable meaningful reduce non robust perturbation subspace weight investigate transfer learn covariate shift imaging modality foundation education project university national foundation education frank mail tu de frank tu de tu department cognitive university subject change computer challenge great importance subject reliably use user construct subject aim common discriminative information conceptual difference subject matrix shrink towards reduce shift paper multi eeg meaningful common spatial stationarity transfer learning process gain brain computer reduce construct classifier approach average order quality especially promising setting transfer spatial pattern subject optimization thus incorporate subject construct note strong namely process stationary satisfied common challenge two signal characteristic method classifier regularize wrong direction careful opposite change induce assumption thus remove unlike reduce shift discriminative subspace advantage characteristic spatial filter regularize towards dimensional assumption true discriminative compare unlikely remove towards discriminative subspace effectively complement subspace different characteristic reduce one stimulus visual test phase system increase increase computing filter lead feature stable pattern extract discriminative user prominent complementary successfully promising transfer novel common organize art introduce analyse toy prominent shift phase popularity domain stationary basically strategy belong second category limit review extraction discriminate state induce band filter eeg pass band power condition stationarity original adapt author trade extract feature filter lead stationary ensure stability consequently compute training incorporate ensure remove stationary preprocessing step neither consider instance learn datum discriminant subject brain utilize collect previous kullback leibler subject largely dimensional write covariance matrix control incorporate user restrictive namely subject author recognize violate subject variability thus subject propose similarity filter different subject goal eq apply learn trade value force opposite force vector filter perform high newton kk extract filter spatial filter restrictive function formulate generalize cluster tackle inter introduce way brain information tackle stationarity like kullback adaptation multi main utilize subject principal subject summarize filter invariant briefly extract invariant utilize datum user prominent capture suboptimal toy example see stationary subspace relatively direction behaviour change explain difference discriminative stationary contrast extract represent real experimental discriminative eeg visual since pattern performance switch different stimulus remove stationary classification subject subject impact classification moving generate five effect subspace sum stationary call contribution source contribution source thus toy normally mutually subspace span source variance discriminative stationary source subject trial point later extract per classifier determine classification leave experiment toy datum repetition increase amount make remain rotation aa add simulate dissimilarity discriminative subspace artificial error dissimilarity plot figure subject namely rate however become affect dissimilarity mix performance stationary remain experiment opposite namely fig stable improvement subject show decrease non stationary subspace drop important transfer goes increase although non become increase average behaviour discriminative stationary subspace course subject experiment advantage meaningful contrary discriminative may regularize stay gain stationarity severe consist calibration without perform two specifically indicate stimulus visually appear calibration locate ms band hz filter training hz densely filter three filter linear discriminant competition iii consist eeg subject right behind randomly move indicate presentation period subject relax eeg signal hz hz trial available subject manually densely cover division coincide competition trial trial run subject band pass hz addition spatial
use ni configuration time estimate run quite time simulate draw estimate panel full panel note curve b ni mostly well less effort ni air region daily pm matter diameter index early calculation air health health range minor environmental air take proper map air daily european human health ec region recently spatio dynamic pm concentration use daily pm daily probability european ec daily year map consider probability produce make limit daily pm measure monitoring web denote spatially uncorrelated latent unobserved air assume covariate spatio field daily maximum daily temperature km km spatio temporal assume autoregressive dynamic spatio mat ern give estimate marginal lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc probability level posterior see avoid result show km section family configuration integration right fourth possibly level expect divide forest basis proportion thus noise area pixel km km lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc trend green model method south region show deviation reasonable semi bottom positive significant trend green conclude period conclusion draw seem spatial control uncertainty region contour stochastic precise definition introduce calculate behind family sequential practice simulate even problem extension fall broad category thresholding compare threshold disadvantage many interesting accuracy interesting contour curve quality contour map present family initial comparison complicate far fairly verify situation family complicate family set contiguous could incorporate matlab see material acknowledgement observe system system datum center pm provide web grateful valuable contour curve uncertainty need thorough walk scale version distribution give determined simplify mcmc still computationally cholesky entire set predefine value several well related find contour sequential accuracy environmental area air region exceed daily european health region early key word contour curve interested area study exceed level typical air level scientific imaging spatio might common science want region common specify location marginal exceed hypothesis act control confident information family hence quantify instead want modify solution marginal value define differ basically category discovery thresholding thresholding euler characteristic threshold latent field focus latent measure mean measurement priori motivate problem example go unlikely base parametric importance parametric extended parametric family region contour curve contour curve estimate computational integral test example application consider air north spatially remark formulate hence one exceeds given quantify uncertainty contour curve bound well area notation contour set contour curve point incorporate continuous contour neighborhood complement union interior negative contour contour interior set follow empty contour respect include occur environmental field essential treat contour region close region exceed certain region process level give similarly negative also formulate large another probability interested idea uncertainty might example simultaneous regression interested significantly avoid set avoid non overlap non reformulate case similarly contour set contour contour set let avoid region contour interpretation important uncertainty region union interval level somewhat find satisfying satisfying requirement report wants know exceed might something probability interpret function contour visual tool answer question function contour function member retrieve function interpretation function location close set value member calculate set practice popular practical application setup covariance parameter far contain location prediction field let main contour uncertainty set contour calculate region probability method shape algorithm calculate use computationally demanding involve propose minimize outline simplest complicated base family routine calculation integration node parameter sequentially soon fall large want candidate na I method large probability run integration retrieve retrieve extending set extend optimally discuss section method calculate full vector considerable effort approximate year approximate mc fortunately number technique integral integration transformation transform unit hyper cube transformation separation sequentially quasi mc replace mc see integration problem integral show technique applicable parametric model note motivating reduce prediction use property main matrix difficult inverting matrix advantage sparsity auto cholesky eq depend element matrix integral iterate integral band efficient particle simulator integral last density sequentially importance sampling first l nh dx dx b importance like conditional target update set select resample meet size technique evaluate computationally optimization family marginal calculate simple one base family quantile q marginal simultaneous non stationary example quantile set set fix dependency therefore set seem reasonable grey posterior mean curve contiguous center smoothed smoothed probability circular radius q general disk smoother equal modification result demand method calculate family family choose set parameter increase sequentially order step probability fall optimization use search uncertainty region avoid parametric level pair family family avoid uncertainty cholesky precision sparse reduce point cholesky improved bound marginal perfectly marginal level obtain correction method improve eq large variable perfectly class lower contain node contain node example add node domain integrate sparsity monte computational practice computation section directly unless laplace good alternative estimate posterior distribution use affect appropriate estimate posterior calculate probability ignore conditionally estimate quantile follow calculate b numerical numerically approximate scale parameter measurement prediction krige location estimate together grey cl cl tc tc ct ct ct ct ct ct tc tc ct ct ct ct ct ct ct bc tc ct ct ct ct ct ct bc tc tc ct ct ct ct ct ct xx correctly curve result contour set show red b grey show dark show grey set grey red panel want correct denote requirement difference calculate twice base mostly monte carlo nothing accuracy correct panel b contour uncertainty krige show black level complement union large contour set field mat ern parameter kind spatial center lattice us calculation expansion set field location noise give posterior estimate krige contour gain definition set parameter special contour two small family arguably computational family although large b lc lc lc
contain power normality empirical level test notice significance small true cn ct cl lr lr lr lr lr mp mp mp mp mp mp mp mp mp mp ex g overall close notice test mp rejection vanish multipli procedure bootstrap investigate specifically rate rejection reasonably nominal scenario simulation fit procedure multivariate five absolutely continuous family state continuous copula copula family three nc multivariate normal normal multivariate copula five parameter copula nine margin copula seven margin copula generation margin n nc expectation dispersion order copula use nc margins copulas f nc extension package unlike multipli carry situation distribution matrix decrease multipli procedure bootstrap powerful multipli dimension rejection vanish mp reveal powerful lr lr lr mp nc nc ex nc look table multipli conservative agreement significance level improve strong nc easier notice finally difficult distinguish table make multipli looking bivariate overall empirical comparison also reveal rejection two suggest difference practical perspective power reach scenario lr nc nc ex ex mention early rely package evaluation finite experiment default indeed quasi carlo random number generation therefore second analytically multivariate approach close significantly nine extremely multipli simulation improve change routine latter carry study multipli datum bivariate fit notice could procedure multivariate would solve lr univariate bivariate experiment suggest computational multiplier analyze consist year daily intel microsoft stock sample univariate goodness intel return simulation execution execution multipli bootstrap order heavily ii multipli numerically bivariate goodness n intel stock bootstrap time multipli procedure minute processor compute nc returns third multiplier even instance execution minute hour parametric mp approach lr mp mp nc ex nc execution expense analytical could horizontal little evidence basis plausible financial would thank associate constructive comment proof random belong brownian follow map p respect equip metric probability follow k similarly proposition converge mapping assumption obtain immediate proposition weakly probability proceeding proof turn supremum proposition consequence center respect except compute jx center dispersion correlation matrix formula multivariate discuss except jx obtain whose otherwise obtain corollary compare cumulative function estimate cumulative employ carry goodness parametric easy implement expensive alternative technique fast empirically outcome generic computationally efficient multipli goodness fit procedure sample power model contain nine gain multipli procedure product fast procedure degree phrase central cumulative integer base computed parametric assumption extensively goodness two statistic er von minor variation important goodness concern computation critical value regularity state weak drift goodness theory free several statistic martingale transform boundary review location er use distribution statistic weight explain generic implement resample carry test fix idea let repeat stand version realization convention rejection ns n procedure generation consume consist reference therein technique recently assess nan large parametric reduce minute aim investigate empirically bootstrap sense determine parametric bootstrap roughly power small powerful acceptable recommended become practice multipli appear extend establish context instance jointly weakly speak copy new approximate hypothesis f explicit sensible tend verification assumption family illustration purpose er von reformulate adopt procedure compute n nz z statistic regularity indeed jointly copy limit compute tend advantage multipli parametric bootstrap introduction roughly speak generation fit generate generation multiplier typically fit respectively fit resp multipli realization indicator concern smoothness discuss instance smooth continuously candidate square integrable relate estimation prove hold
flow treat conditionally imply multivariate mean come different slightly align system action due characteristic robot signal different modality follow methodology delay maximize alignment signal delay manually basically gmm approximate desirable compression affect likelihood hundred multivariate case threshold need spurious one likelihood equation basically covariance weight contribute proportion update mixture inverse learn feed sensor optical optical flow prediction mixture rule map probable identify component depict time show optical v mixture capture information check predict application predictor sensor robot new robot active frame activation component event follow active optical optical flow highly likely around second obstacle experiment core ghz processor ram gpu optical object aim robot action perform human decide decision situation stress acquire knowledge action robot five forward velocity angular threshold correspond distribute look optical flow field angular unstable magnitude nearly zero unless ground sequence absolute normalize na I predictor basically expect horizon predict predict optical predict flow vector choose compatible I derivative optical flow confident prediction predict test could predict account show log ratio na I predictor decide experiment platform information optical axis flow present row big flow black row encode shape forward well condition mostly assumption I ahead present mode usually information valuable segment alignment stream align significant reduction na I error e without use reduce incorporate half almost independently gmm action consideration sensitive h logarithm na I gmm test gmm I learn obstacle later object significant horizon second detect robot obstacle binary optical flow change middle optical flow graph optical flow mobile advantage improve dynamic extract accurately predict application model mechanism build robot flow technique particle work principled extension spurious refine underlie ram department electrical engineering college bt email event agent need make term uncertainty internal external environment acquire mobile learn optical flow image learn optical time horizon reinforcement modal simple one interact environment consequence effect agent sensor acquire capability optical scene move movement body encode scene appearance benefit aware action learn optical action capture task perspective mainly motion effect scene appearance discriminate change optical flow primitive day old behaviour mechanism enable term optical dynamically analyse optical flow optical flow prediction consistency predictor state online task build optical flow mobile robot predictor axis interested term option sensor value decrease unfortunately handle optical flow grid field optical variable robot encode angular extract define angular velocity optical current hypothesis optical na I predictor assume decide analyse sparse backward modality use present indicate region show learn useful period line method propose joint optical flow previous action optical flow example flow environment optical optical
scope article successive time process speed count limited gps user nevertheless traffic huge data position traffic compose day et france et france flow road france flow regular weather weather soft medium traffic high due several gps accuracy wrong incorrect variation individual traffic eliminate outlier flow speed free flow datum record reference link enough record link confident already consistent france treatment main issue build weather build weather understand weather consequence road behavior associate traffic weather datum weather propagate get pair find rule forecast weather velocity indeed build include range road yet provide estimation road weather method forecast scope work road weather depend road road weather go fast speed vx vx tt vx speed call adaptive spline k hinge point suffer lack indeed homogeneous among may weather fit drive bad weather variant well thresholde bad weather traffic critical speed decrease wish traffic weather condition following extract neighborhood neighborhood practice arbitrarily narrow enough time find fact thus decide relax traffic stationarity day existence stationarity belong stationarity want weather e e two represent weather section good link statistical modeling nevertheless turn link task residual selection established present match france day also weather france basically link highlight thresholded classical choose one compare good classical selection datum contain sample remain validate select estimate rmse goodness calculate q obtain rmse thresholded fit well thresholded moreover thresholded constrain model fit model fig mean conclude thresholded far advantage homogeneous detailed h apply correction weather take separately road link weather global justified generalized interest link weather condition structure complex different instance three link another network build kind road matter fact road weather may level behavior one h sum link deal linear thresholded linear thresholded model build pair homogeneous among link show parameter km km highly wish network dependence depend correctly highly normalization fig joint concentrate h remain parameter e compare rmse aggregated significant link nevertheless construction road correct thresholde road easily homogeneous thresholded remain eq interesting interpretable road really illustrate fig speed proportion proportion flow speed represent condition let km flow km km road limit decrease thresholded normalizing speed also need quantity calculate speed speed network obtain loss link compare predict weather version calibrate get basically proportion flow speed threshold value difference speed thus correct information weather learn overcome stationarity mix change weather traffic desire decision extend road quantitative major forecast weather contribute forecast company study weather actually thresholded think weather flow global build weather nothing free speed consider weather thank provide respectively weather road road weather france mail understand weather modify traffic dynamic change change thresholded road road network provide speed term forecast linear thresholded road traffic spatial weather accept weather modify significantly traffic actually bad
controller methodology determine whether quantification tractable many temperature temperature quantification j amount solid default controller mark solid line characteristic controller substantial future direction speak experiment energy wide modality berkeley edu master berkeley edu cs berkeley control improve energy efficiency air system potential economic build wide hybrid update model effect air temperature controller large usage justify present energy per p confidence interval united air systems drive research consumption ensure building respect modality methodology scale design difficult weather design operate efficiently setting require measurement moreover consumption controller reduction usage temperature water outside air energy controller controller air single room warm day day mathematical dynamic experimentally usage weather wide building control along distinguish describe controller able begin procedure identify next describe hybrid adaptive vary differ change furthermore robust nominal measure provide experimentally controller controller methodology use nonparametric energy part berkeley efficiency platform building laboratory seven office building measure protocol interface water air air water air throughout operate design modify operation box modification general default controller loop box keep air temperature control box air minute e denote control air send air dynamic nonlinear simplify typically adjacency eq adjacent additional vary principle make affect bilinear finite value mode I unknown scalar function system week identify hybrid system change fidelity modify modeling approach amount hybrid two hour hour consecutive hour reason pick experiment roughly constant construct discuss modeling load change purpose load short within quick span serves provide additional measurement suppose prior coefficient ib j ic j constrain least mode system ensure fact reflect capability box relatively across lastly window air amount temperature box controller controller controller restrictive portion k temperature amount model fitting point comparison point air eq minimum air flow allow building scatter amount controller largely leave approach several air explicitly implicitly vary n consequently simulate minimize cost optimization fortunately greatly change hour change hour combination far reduce specify mode hour four combination low computer mention early need variable would specifically quadratic program reasonably formulation form piece usage consumption water air usage usually subtle energy energy reason usage physical stress fine usage source cost energy really get simplifie model usage energy energy controller behind system nominal use maintain improve rate minute intuition represent correction provide adaptation inherent controller load due weather control air
structure output space yy hinge surrogate loss apply prox sdca note strongly norm indeed maximum definition therefore objective overall imply prox sdca bind option fact v prox sdca maintain w x iw tt maintain explicitly maintain vector efficiently find sgd structure prox sdca output gap eq problem instead imply corollary replace match main duality loss approximate point coordinate ascent frank wolfe match handle regularizer involve notation prove theorem run sdca careful option choose side choice simplification option iv employ simplification convenience list primal dual formulation key lemma update expectation obtain lemma third strongly n dd duality randomly next rely fix achieve maximal dual suppose lipschitz gradient combine lemma tell imply next inductive suppose turn rearrange choose obtain sufficient condition n imply remark version numerous regularize rate sometimes improve follow generic associate regularization structure svm analyze runtime coordinate solve u z u associate optimize dual dual immediately duality gap regard sub version round optimize analyze sdca lipschitz smooth functions lipschitz differentiable convex define smooth generic prox sdca idea describe maximal allow step let sufficient primal lead w v write gradient method sgd show strong result prove prox sdca loss motivate sdca sdca choose maximize rely directly maximize dual general propose choose step increase objective htbp prox sdca scalar pick option large option u option definition lipschitz non iv return prox sdca expect everywhere superior sgd one numerous interest categorization bag short choose approximate base sdca goal solve linear require strongly calculate therefore iv iw w I hand set prox target accuracy prox sdca runtime obtain theory factor sdca sgd require intermediate iteration relevant shrinkage fista approach batch fista enjoy however fista rather runtime q contrast sdca happen choose need regime prox sdca fista runtime
news manual setting already balanced file document actual description typically section subsection sub subsection paragraph example throughout article environment indicate paragraph character indicate phrase code part subsection care begin character english character three display text formula simply notice equation style look slightly display style display equation set equation environment environment structure somewhat differently enter able article conference book article occur text automatically several citation cite short reference uniquely identify document author title identify item file scope detail form citation format across page top table environment wide table figure page bottom page figure detail figure table environment forget file article logical like axiom corollary proof form distinguish document environment create gb ga environment logarithm e list construct use form construct environment gx fx gx gx contradict complete environment source code primarily intend create camera copy omit problematic paragraph acknowledgment make exception book article nothing location acknowledge file rule discuss indicate start b title include body add file run resolve create file file file comment file helpful comment moderately expert read please usa format lars rv rv rv email email alternate tight design file write
parameterize convolution sub reconstruction form q notational brevity combine convolution sum matrix convert map combination reconstruction part integrate straightforward also feature pool operation direct eqn fix reconstruction eqn eqn impose pooling become employ constraint fourth importantly directly pool employ multi model stack manner channel property solely intermediate weight convolutional operation convolution layer derivative repeatedly version filter key section respectively ensure invariant small change pool otherwise able perfectly pooling would generalize axis mean precision introduce weight sum give unit gaussian location show illustration parameterization brevity neighborhood rewrite several advantage sum pool gaussian select allow variation max change invariance small adjust variance continuous occur pool illustration gradient analytic activation enforce motivated factor notion intensity parameterization weight represent individual negative third negative brain finally negativity reduce flexibility improve utilize enforce constraint optimize sparsity two though latter lower toward descent neighborhood layer variable eqn input problem adapt scheme set element shrinkage projection shrink project onto order employ estimation technique advantage layer significantly account architecture manually good inferior per mini repeat minima proceed epoch overcome set feature epoch perform demonstrate explain stage mini datum cause new map turn reconstruction remain continue get gradient early therefore suffer map epoch reconstruct optimal element continue wish intermediate layer optimize assume variable combine layer update pooling variable reconstruction error chain parameter neighborhood q q coordinate eqn pool however estimate likely pooling coefficient map pooling size epoch reconstruct propagate I loop low eqn cg project length output map filter update term bottom map index update filter plane practice parameter infer project feature pool optimize infer shift layer pooling layer filter initialize work well second jointly take trade filter capture detail filter become dot avoid fix learn layer layer layer nice part optimize learn filter initialize random project begin inference start create feedforward forward pooling use moment extract pooling parameter provide initialization pooling filter initialization evaluation mnist handwritten digit dataset instance per compare interpretation image input digits architecture filter map contain connect randomly amenable gpu processing layer configuration input one motivation make top level activation infer input patch pool analogous sift show allow operate high dimensional classification patch map multiple layer patch hyperparameter mnist classification layer epoch pass dataset epoch map test improve optimize inference group connect layer visualization layer model learn fig demonstrate reconstruction b display raw filter helpful input sensitive search infer input activate fig representative selection feature activation activation inf biased view large feature pooling correspond image activation stage visualization examine shift analyze separate connect understand visualize second project pixel pixel projection activation activation layer alternate pooling feature decomposition pixel reconstruction color depict high assignment pixel color indicate color feature color purpose understand also activation utilize show operation map box one second starting show reconstruct layer benefit smoothly non overlap pooling layer long range structure group common max pooling max smooth transition overlap use convolution pool activation ie maintain adjust confirm break regularization display comparison pool significantly max optimize max despite train smoothly throughout tune property pooling stack max cm test encode fundamental drawback procedure improve performance layer separately map pool directly differentiable pooling variable layer pooling inference need examine depth combination inference find training separate phase work training optimize pooling filter combination filter make allow filter move need updating pooling table optimize optimize joint key differentiable pooling mm infer phase update update use model phase update update respectively mm discovery evaluation classification run also reduce compare iteration classification happen removal high make discriminative change level order reconstruct feature pool suit due summation negative pool enforce positivity gradient negative pooling filter feature fig many rd nd map encourage filter similarly layer due notice nd improve varied help cm cm mm previously show encourage hyper laplacian apply train run regularize training control various sparsity
round care use exchangeable exchangeable ti proceed tell sequence identically apply sequence number partition argument exchangeable partition appear draw rest use generation argument belong apply condition frequency partition contain imply every partition exchangeable partition exchangeability draw induce partition construction think partition convention interval break yield exchangeable consistent case collection follow feature exchangeability allocation belong specify frequency summation partition independent ensure summation unity stick breaking obtain break unit return generate stick proportion stick remain length stick v unity convenient condition independent rapidly simulate stick finite fall stick crp chinese exchangeability value mixing frequency outcome bernoulli find break chinese restaurant divide customer crp rest color former collection gray white color gray begin white customer crp whereby start gray white draw replace gray ball white ball number check define crp gray customer sequence ball ball ball number round exist customer customer equivalently customer crp construction customer color gray customer table color denote therefore play gray customer white customer customer scheme gray ball initial white ball ball thus customer table outline recursively independent customer identically table table v identically frequency process well define since frequency sum crp recover stick length choose contain consider feature p sort encounter ex stick crp gray ball ball feature draw likewise round ibp model point initial gray ball white ball iterate across function partition stick length stick length sampler mix impossible partition block feature frequency stick crp around examine finite crp stick use approximation create truncation stick break size exist technique avoid slice thus far posterior length stick different particular use approximate minimize family potential stick breaking demonstrate number stick length ibp example cover build ibp examine slice stick ibp posterior stick allow underlying stick length exchangeable collection appearance random sequence exchangeability sequence assignment give point sampling exchangeability suggest exchangeability stick length random label assign label block belong feature allocation exchangeable continuous open interval correspond jump form partition stick construction partition label measure unit may sum way increase stochastic one correspondence mean choose distribution weight due summation independent stationary measure follow nonnegative left increment purpose wherein ultimately scale assume range interest stationary increment poisson measure countable subset process characterize disjoint treatment drift jump component result write eq countable process evy eq jump frequency jump may frequency hand side jump interval identical e jump stationary transform evy eq drift nonnegative call drift evy particularly since jump process methodology stick generate membership take bernoulli jump success jump every associate jump eventually jump appearance iteration let choose recursively jump see round particular see early derivation stick stick length ibp beta stick connection zero evy parameter parameter allocation l evy index appearance sequence ibp stick poisson suppose point value point also process define measure eq evy assumption distribute beta without atom prove theorem distribution atom weight stick length check recursion assumption note choose total mass poisson desire cf atom rate desire recursion note remain bernoulli minus measure next generate process stick jump however normalize measure finite mass sufficient must stick length partition jump jump exchangeable laplace derivation distribution jump find normalizing jumps exponent exchangeable partition exponent chinese restaurant start gamma exponent correspond crp concentration jump crp derivative laplace calculate partition generate normalize jump derivative always straightforward calculate note general integration yield substitute equation pn b kn b n n line form beta crp desire final laplace exponent know find partition normalize stick must distribution normalize jump feature appearance jump correspond recursively none jump chinese restaurant jump crp stick example notation sum jump minus j j show lemma evy jump density lebesgue lebesgue dt dt change variable transformation former variable latter derivation term notational clutter end evaluate ft gamma determined e k second read ahead imply assignment object random uniformly partition label special general straightforward gibbs stick ibp illustrate find exchangeable usefulness stick representation crp stick discussion jump unnormalize jump label unit uniqueness exchangeability sequence label uniqueness moreover wish associate point height vary randomly specie likewise collection correspond assignment equation book person appearance block similarly appearance label complete let nz nx ny atom completely disjoint jump us size point intensity point yield tuple intensity tuple completely completely random gamma labeling partition specifically come point process parameter along point axis atom plane dp normalize gamma atom length dirichlet form induced partition partition restaurant process feature block I measure distribution beta choice uniform line segment beta value atom measure bottom accord process induce allocation index model first outline context feature structure generalization beyond option remain section infer underlie infer generative block handle dependency parameter particular depend block dirichlet mcmc provide use beta augmentation simple partition exchangeable stick encoding frequency occurrence block random stick length likelihood focus show successive restaurant process specify induced partition form draw dirichlet completely specify induced form I I draw weight extension idea lie scope extension crp form stick ibp beta explore generally demonstrate alternative partition stick length measure expand suggest structure might provide allow feature permutation research national science foundation material foundation award office contract grant point group block notably formal analogous call integer feature topic combinatorial representation analogous beta thereby bring previously achieve process allow collection often move bayesian paradigm rich express belief thus field dominate dirichlet flexibility arguably aim broad kind useful toolbox bayesian mathematical structure useful specification combinatorial mathematical field focus area version classical combinatorial combinatorial analysis establish speak statistic combination latent compositional modeling essential calculation serves exhibit bayesian combinatorial one well dirichlet expand upon notion stick dirichlet process factor dirichlet statistical idea associate underlying factor interaction rise need counterpart characterize combinatorial model connection stochastic model stick break nonparametric adapt expressive specification essential inferential tractable reason popularity dirichlet stick break yield inference algorithm analogous discuss representation associate beta process consequence inference remainder organize follow introduce model allocation exchangeable know crp correspond ibp choice rich stick frequency associate label crp ibp ex stick ibp marginalization crp ibp mention stochastic study bayes suggestion development intuitive idea constitutes formalize idea proceed underlie combinatorial datum index mutually exclusive exhaustive nonempty partition similarly partition exclusive nonempty example n introduce mutually exclusive exhaustive restriction allocation nonempty belong block belong infinitely coincide index consider array feature represent may picture impossible display infinite pixel allocation block block partition block allocation note converse general continue development special partition allocation feature exchangeability modeling rigorously block allocation apply f nf imagine similarly agree randomness say allocation allocation q consistency feature allocation characterize restriction sequence space allocation sigma algebra sigma index restriction circle customer table middle customer customer useful feature emphasis exchangeable partition exchangeable exchangeability block chinese restaurant restaurant partition increase belong chinese restaurant customer chinese restaurant customer restaurant belong customer table else restaurant recursively restaurant table latter next available exist possibility distribution table customer shown summarize choose n consist customer customer try customer gray sample indicate customer exactly index generative model allocation discover enumeration crp poisson number feature factor associate density let choice k n index restaurant finally process allocation generate multiplying factor f f n h n k quantitie ibp exchangeable recursive exchangeability equation ibp see form specify index yield exchangeable crp example crp ibp refer conversely partition label index partition sequence result call give sequence partition interpret contain label feature cardinality definition allocation unique collection share label similarly collection cardinality form induce feature allocation feature reduce allocation concern label feature appearance assign labels st index block uniformly recursively see label use scheme appropriate partition find appearance know induce completeness conditional z probability underlie z n k chinese chinese restaurant chinese crp find rule imagine must
template extract capture think score model database involve face modality fusion arrange vector performance svm experiment arithmetic give svm give equivalent combine previously work various fusion architecture base score fusion issue fusion automatically fusion solve order allow reproduce database first create available database modal tuple tuple modality present subsection present summary two tuple intra capture tuple tuple score recognition time second database create combine template ar database compose individual illumination difficult database reason involve ask time capture compute inter capture subset produce help correspond score empirically choose g validation main difference benchmark modality performance intra important private tuple intra score instead database score system private template adapt physical access building modal adapt logical access web modal tb intra tuple tuple subsection fusion genetic score need normalize normalize operate indeed score come classifier necessarily evolve normalize equation present normalization score capture user call represent stable impossible paper procedure genetic fusion genetic programming quite genetic population evolve create evaluate program represent mainly take child leave kind case ordinary genetic choose tree replace another computer program function build step repeat termination criterion fitness reach reach number fitness measure program fitness operation new generation operation previously individual program create randomly select program example provide mutation new program randomly provide population winner tb genetic reference field list symbolic economic biology bioinformatics compression seem apply find genetic use genetic meet maximal reach half score tuple first tuple tuple program basic constant fitness case thank library fitness generate global computation roc read genetic root function terminal avoid unimodal set function function child depth set avoid stay mutation individual evolve quantity individual much interesting half validation mutation mutation individual scheme generation first generate keep generate program benchmark rule return minimum return score score return sum genetic fitness genetic engine present configuration genetic fitness search validation performance present performance database system fusion mechanism concern art performance see good operator outperform method fusion c fusion far function far gp present performance operator private fusion point give global really compare system fusion scale quite programming give global curve inferior overfitte well state weight fitness criterion meet always reach generation fitness evolution run database scale easy fitness evolution database fitness appear bad deviation fitness huge explain generation say well automatically return score tradeoff pattern already way really train pattern empirically normalize fusion inherent genetic probable simplify another parameter fitness program interpret present depend complex compare subtree tree modality terminal set genetic fusion give genetic weighted time genetic important two method genetic need ten less propose paper approach automatic generation decide automatically fusion program genetic concern design base inspire return threshold heavily database improvement hope fusion system thank genetic genetic engine performance metric add genetic template fusion thank library problem database region financial suffer drawback general capture extraction modality state svm performance genetic give well performance database score classical validate propose method art score performance new modality example classify even literature biological dna analysis signal individual perform dynamic handwritten computer way drive pattern face recognition user method bad whereas performance instant accept several accepted security less well modality correlate generic different sensor sensor acquisition capture texture recognition modality instance right per video compose previous interested fusion combine score compare need evaluation evaluation realize curve information regard quantify device physical attack service attack fusion depend modality metric represent ratio user overall small globally system
contradict incorrect overcomplete solution solution necessity general lebesgue measure zero subspace one large notice instead objective therefore lemma true optimisation formulation multiple cause fact dictionary indeed make much map r r nk sufficient uniqueness particular optimisation unique contradict standard cause pattern need uniqueness boundedness solution bar mention coefficient generate matrix reduce subspace practical robustness successful distance embed sensitive restrict definition isometry measurement proportional total function propose comparison option choose also sparse position subspace quantify number subspace q concept theory binary l lf depend figure plot bound plotted ratio new sparsity ratio small boost recovery see optimisation technique powerful project admissible norm schmidt frobenius norm relate quadratic onto keep large onto keep absolute letting intersection set provide necessarily projection onto alternate projection projection statement projection element zero simply point statement unique choose projection iteratively update current negative gradient direction onto solve f find select selection objective half optimal descent constrain size thus objective increase parameter initial sparse change gradient algorithm check update dictionary selection type dictionary may greedy sparse technique however reference sample signal stable close set subset accumulation h admissible black dot zero dot horizontal line gray horizontal line plot area zero unit normalise norm vector randomly select distribution magnitude sign descent base superiority coefficient recover panel sparsity admissible figure explain panel show clear set correctly recover keep repeat plot transition joint constraint colour high area exact recovery large add demonstrate new dictionary dictionary times overcomplete want large dictionary difficult solve need keep dictionary joint overcomplete environment ghz take another test select dictionary representation learn joint joint well reconstruct reconstruct reconstruct setup audio signal propose dictionary dictionary learning method record mostly select eight hour record overcomplete two dct dirac transform incorporate audio combine atom subset appearance figure low frequency dct atom atom although delta dirac atom window close atom audio database get plot overcomplete dct reference solid line snr dictionary overcomplete run learn sample select learning similarity fast sparse necessary dictionary use mod time simulation show dot dictionary aim dictionary dictionary ghz fast ns implement datum ns sparse expensive part bottom right new dictionary selection atom signal dictionary reformulate approximation sparse coefficient overcomplete joint generally infinitely solution sparsity active show overcomplete joint approximation problem dictionary gradient approximately synthetic support namely simulation select subset overcomplete dct dirac dictionary handle vast need keep dictionary vector multiplication audio small overcomplete seem extension investigate left investigation exact work paris university langevin paris france ed fr representation describe sparsity processing rely simplify unknown domain knowledge signal paper call reformulate consider representation subspace synthetic realistic sparse successful low application elementary plus noise atom set call overcomplete extra failure redundant main lot redundant already use knowledge example combine r select dictionary fast dictionary product give atom complexity search learn learn avoid atom signal represent reader ask dictionary sparse complexity success sparse structure space property affect dictionary cause atom atom similar similarity indeed atom equip line search monotonic investigate test large formulate dictionary overcomplete introduce iterative show section lk find objective decomposition efficient
already show fact explain expect poor performance single expert unlike active indeed incur active poor performance convex weight cause specialize helpful mostly conclude benchmark rule worth good second compound therefore fix operational consist produce forecast hour forecast expert run rule access instance expert different prediction similarity estimation fair expert form expert expert fair weight weight equal expert regret operational section describe weight usual sum real statement figure basically need run round two share deal expert day ahead period convex n f share update correspond name whenever share update multiple empty extension share aggregation operational forecasting regret statement extension coincide instance weight differ loss one quantification appendix note form interesting fair get performance comparison note sensitive initial allocation well theoretically one rule aggregation indicate summarize turn depend weight fair sequel compound excellent significant performance aggregation rule track expert improvement respect aggregation rule obtain prior fair potential black important operational forecasting side review know expert aggregation rule hoc accommodate specialize exponentially average indicate operational forecast rule datum consumption parameter adaptive rule tune theoretically hand improve accuracy note improve finally term aggregation expert come rule anonymous valuable drastically exposition completing write partially support grant adaptive grant induction define normalization expert round expert entail e w rewrite thus appear get note proof straightforward share note rule implementation choose j prior indeed efficient implementation indicate part sequel sequence expert denote define introduce ti interpretation switch inactive stay current active slightly switch current expert inactive switch expert compound distinguish whether prove immediate bound definition loose second induce replace recall distribution update definition transition function rewrite j ti w last fix compound vector tt sequence index recall trick bound note show plus ratio weight use enough also magnitude state entail paris france sup paris france france sup paris france paris en france specialized expert relevant contribution fix one consumption concern general methodology state parameter demonstrate improvement square behavior large error specialize expert arbitrary round forecast past outcome error aggregation expert interest aggregation rule expert bad sequential ahead forecast consumption take variant basic expert call specialized round expert output scenario specialize expect forecast mostly external consumption specialized work specialized expert rather reference formalize specialized expert one paper context specialized pass another theory many field mention forecasting consumption study aggregation rule already load review framework expert discuss take important theoretical bound online put rule provide france market organize standardized historical result rule tune sequentially section also behavior occur aggregation rule bound predict element number forecast henceforth available one forecast active know expert time assume empty produce forecast weight prediction linearly expert precisely reveal start round loss output thus loss forecast sequential ensure combination expert expert shift possible square percentage instance equal dirac weight average aggregation rely denote choose convex put perform exponentially follow loss denote quantity theoretically uniform advance like trick cope limitation concern see different rule sequel replace depend family simple sequel content next introduce statement regret heavily specific loss hand work square contrast compact solely lemma rule two statement exhibit little learning predict j ty rule convex belong forecast proof theoretically choice lead comment compare follow formally generalize trick convex subgradient product subset gradient replace definition regret figure aggregation imply theorem subgradient uniformly supremum vary sequence expert observation third regret performance expert sequence expert constraint expert specialized formally sequence instance call compound expert put vector compound expert compound weight minus one correspond expert convex compound compound shift might rule index weight control aggregation loss efficient exponentially weight spirit uniform fix share specialized depend adaptation set forecaster mixing rate predict ty ti convention denote fix rule use section sake completeness update bound belong varie vary expert forecast theoretically theorem define desire soon theoretical choice section gradient trick replace loss loss obtain variant denote bind loss subgradient sequence forecast aggregation rule automatic advance rule parameter almost optimal indicate online tune rate exponentially bound thus poorly illustration different set result come poorly remark theoretically satisfactory consideration sake completeness first symmetric tuning rule tune equally performance report table theoretical practical theory tune thin exponentially need far concern aggregation set base rule aggregation report elsewhere family rely possibly value past forecast instance weight aggregation rule denote consider equal member family meta sequel offer guarantee term underlie family need parallel instance together meta rule impossible minimization instead whole seem somewhat second meta article report application prediction cluster choose theoretically optimal sometimes losse see improve slightly datum respect sequential target expert good example article application management tradeoff energy consumption wireless tracking traffic also online e vast average expert set enough method sensitive topic forecast performance latter already standardize outline treatment datum section expert strategie choice operational automatic comment robustness assessment strategy uniform forecast expert strategy convex finally pick good forecast expert forecast good element form prediction expert observation period absolutely merely black heavily hour day large enough hour day ahead arbitrarily hour interval characteristic hour hour consider report error root aggregation consumption characteristic lc number day interval expert cc graphical datum leave expert scatter relate frequency activity pair first choose differ one weight instance compound expert explain fact good predicting instance active compound expert require difficult cope active give partition active expert eq active corresponding choice partition choice even relatively choice versus achieve compound achieve possibly benchmark serve procedure benchmark aggregation compound j element accord expert convex aggregation introduce rule summarize tune outperform oracle type rule choice far preliminary exploit concentrate limit carefully weight remain change converge vector meta get somewhat second grid around grid evenly grid evenly grid choice preserve meta grid choice need summarize table comment performance sequential aggregation rule choice obtain aggregation meta grid middle grid meta grid constant p cc convex cc tuning parameter choose meta select rule cc rule use department importantly length period weather cloud wind pattern summarize characteristic detail divide use expert forecast
schwarz differentiable density build exponential need choose auxiliary laplace transform ensure many distribution multinomial von wishart start retrieve canonical usual canonical include parameterization family parameterize coordinate moment parameterization arise see case strictly function conjugate maximum l strictly differentiable compute functional inverse gradient strictly leibl exponential bregman follow moment often divergence use mixed coordinate geometry system orthogonal since mixture single component distribute report unique maximum hessian consistent asymptotic normal covariance mle report duality bregman square euclidean multinomial leibler divergence divergence wishart use bregman duality prove maximize average sided bregman coincide center follow geometry average likelihood reach shannon x reduce empirical bregman center obtain diversity report finite probability exponential latent variable get otherwise mathematically rewrite family mathematical equivalence average write mixture maximizing eq gives hide since per g parameter convention mean sense bind fix function bregman define conjugate bregman decrease monotonically reach correspond local bregman greedy swap heuristic swap take geometric show assignment reach prescribed reach local denote proportion assign th jensen diversity minimize complete shannon simplex entropy weight complete q summarize identically independently family characterize moment exponential family cluster far later I I depend parameter unless reach weight unless local reach parameterized system assignment weight single mle correspond bregman maximization describe straightforwardly code e hard mixture von log compute function invertible approximately proper initialization assignment tx reach per cluster degenerate em identify convergence em gmm way initialize define partition diagram precisely joint gaussian weight ix gmm xy cell explicitly weight bregman diagram design partition assignment mle em bregman tree mean heuristic greedy optimally global swap mle tx yield von need reciprocal gradient focus von remain problem properly step way soft convenience mle properly initialization bregman guarantee initialization far minimize design far et mle may mle infinity see discussion penalization boundedness mixture complexity exponential future compare model modal family determine appropriate many criterion criterion aic problem exhibit make selection unstable interpret nk mixture simplification polynomial complexity practice entropy acknowledgment early prototype discussion center popular iterative fact swap per cluster additive mass cluster generator eq could exponential family split bregman loss center monotonically decrease terminate finite optimum bregman generator minimal diversity cluster rewrite assignment step close cluster since center iterate inequality since function local optimum I I iy ic I I w ni loss minimize cluster variance bregman quadratic induce generator identical et rewrite centroid cluster center within cluster bregman mathematically generator arbitrary minimizer assignment mean monotonically base global bad exponential plane let dual natural consider parameterized matrix usual cyclic product sufficient auxiliary parameter log symmetric eq moment parameterization leibler leibler decompose bregman write explicit usual number ix initialize seed mahalanobis let ix diagram update unless reach reach mle initialization report seed uniformly probability c c eq add seed initialization soft bregman em write deduce conjugate reciprocal mixture cluster parameterization px soft calculate moment parameterization bregman divergence conjugate generator meet decrease complete weighting iteration initialization build mle seed seed tx kx exponential parameterize coordinate versus log parameterization information matrix conjugate canonical natural natural moment usual usual space density parameterization use parameterization convention shannon mixture component cluster hide sufficient statistic maximum weight parameter partition cluster center cluster proportion bregman divergence generator b jensen diversity fp n n fx e f j respect additive bregman keyword frank computer mail frank family traditionally maximization monotonically increase incomplete expect assign likely update duality bregman divergence local complete mle bregman heuristic greedy swap cross entropy proportion update successively careful possible family bregman mean mean denote mixture summarize mixture often mixture parameterize positive definite mahalanobis positive definite draw variate model variate cholesky variate drawing sampling mixture doubly essence multinomial gmm component learn set sample gmm frequency information image figure gmm image model point color gmm representation retrieval ir paper mixture belong bernoulli multinomial poisson mixture model wishart mixture gmm model depict hard
study system various entropy thank small suffer overfitte insufficient approach overfitte prior response drawback likelihood place center key principle principle idea inference combination framework special introduce inference free parameter make size available day day amount datum variable information theoretic inequality capital letter capital letter variable capital letter denote set bold capital shannon equality therefore preferable many require low entropy many system estimation probability combinatorial event significant role regard know overfitte parameter often suffer overfitte viewpoint avoid bayesian background datum increasingly natural add frequency knowledge avoid generation g usually always distribution interpret prior knowledge probability parameter regard generalize although widely use accept axiom probability improper still inference another keep characteristic ml estimator knowledge regard accordingly insufficient small becomes consider regard regard increase principle optimally method physics method utilize ml modify regard essential principle estimator effective I principle within constraint far principle log contrast I low high preferable case I principle necessary devise entropy I principle balance optimal contrary principle branch analogy select achieve balance minimum energy analogy play temperature represent hyperparameter entirely hyperparameter propose method interpretation estimate nevertheless free include treat fix regard therefore potential free energy extract determine still organize theory joint experiment estimation conclude size utilize entropy concept accordingly energy constructing hereafter multivariate knowledge assume I discrete probability joint entropy quantity accordingly come author aspect mechanism internal kullback leibler kullback leibl distribution manner cross according define way empirical random empirical eq relative denote energy internal leibl estimate ml also divergence hereafter minimizing define define practically ml q indicate data temperature often hereafter regard introduce statistical available size new temperature unit uniform denote I distortion divergence divergence temperature datum temperature random define omit recognize temperature assume see get viewpoint select explain I principle similar relationship internal principle probability estimation formalize manner define energy define empirical replace relative binomial principle estimator minimize solve shannon energy alternative energy express temperature estimate distribution physics determine size estimator denote range note propose I temperature limit low kl cross entropy x formula px x px partition mechanic mechanic statistical mechanic probability shannon assume statistical mechanic feature state denote canonical distribution value right log function single value equation meanwhile estimator true consistency need equation accordingly asymptotic limit approach preferable ml insufficient definition adequate free dominate large dominate new count occurrence event count occurrence regard event state uncertainty consist degree uncertainty principle quality quantity regard obtain optimally denote canonical distribution characteristic property mass temperature canonical canonical follow cross principle prove statistical mechanic fluctuation denote distribution define equation equation usual call fisher canonical fisher similarity mechanic relationship relation obtain equation energy differential capacity temperature incorporate belief bayesian count event sum prior bayesian maximize lead canonical express equation temperature calculate replace I dirichlet jeffreys variety entropy px px px hx px hx I dirichlet jeffreys posterior viewpoint estimation kullback divergence average value identical ml inferior overfitte superiority I opposite ml poor tend entropy example entropy one increase entropy entropy tendency view relative I sample size entropy even improper point
well increase overlap little dependency increase criterion result ht variation mse among b outperform criterion case merging component equal however difference mse merge focus simulation compare simulation bottom dependency bold correspond well calculate b b b estimation value estimation explain fact reliable cluster show seem outperform merging probability estimation term confirm study number merge throughout remainder markovian method estimation try markovian dependency make b value regard beneficial interest accounting stand overlap emission neither accord cluster nest correspond distance see display distance spatial well misclassifie dependency two separate consideration account require array experiment array biological protein protein involve genomic computational time section ht method start cluster bottom grey leave diagonal red contain whereas cluster green density project axis unimodal result biological expectation know element name easily interpretable knowledge flexibility model hmm emission flexible provide hierarchical lead local criterion simulation highlight component real illustrate biological describe approach component brief enough link computation require backward dramatically pruning criterion approximation lead algorithm algorithm easily criteria update step get close optimum group paris france paris cr france cr france hmm account neighborhood emission suppose belong parametric distribution distribution flexibility classical combine likelihood criterion select combine criterion derive carry determine merging selection accuracy hmm constitute longitudinal hmm process neighbourhood dependency unobserved emission state parametric class gaussian poisson lead poor fit distribution effort capable skewed tailed multimodal emission cluster latent great interest mixture emission distribution mixture hierarchical review merge good approach limit hmm achieve component hmm hierarchical consider criterion simulation good criterion eventually classification experiment describe chain transition conditionally emission assume emission mixture denote denote transition specify finally emission collection paper devote parameter hmm existence px stand expectation parameter aim x ps pz ps pz conditional step first markovian emission density complete log get cluster complete like call hard observation entropy xu clearly dedicated purpose uncertain simply bic complete mixture behaviour model couple decompose different entropy measure hide uncertainty among may relevant refers class focus integrate complete derive penalization display mixture remain always hide dimension hmm differ criterion go merge initialization note g l simulation performance aim good selection criterion advantage account markovian design dependency four emission mixture six impact markovian ps aa ps dependency decrease
general precise identical expectation effectively reduce expectation avoid usefulness class map lp constraint operate bind experimental performance nonempty whose union relation fine briefly empty binary operation exist element operation composition name play na permutation group assign composition group define iff group call induce fine equivalence refine give action preserve lead vertex preserve g name denote clear cardinality symmetry node equivalent act let edge also arc problem np program graph subset compactly factorize ki ji must argument track argument factored encode graph pairwise feature discrete family indicator overcomplete overcomplete mean marginal overcomplete representation letting parameter overcomplete family tu x family symmetry exponential permutation equivalently straightforward act action remainder hold x x x x characterize factorize tt permutation group symmetric pa pa consequence lemma permutation action variational quickly depend constant map domain size yy yx yx design simplify call model yield strong two partition fully find six method reference domain size logical case dramatically runtime domain suffer expect graph partition partition virtue work observe domain current implementation curve finally cycle illustrate change cut plane size cut plane point outer variational observe substantially slowly affect much need work network converge early observe grain lead slow notably work constraint improvement optimal solution model present new introduce symmetry construction device cycle demonstrate constraint art obtaining extend approach full approximate condition pick express factorize reduce must argument j k x kt kt satisfy pick kx j hypergraph represent structure exponential family model iff throughout change inner product next integrate lebesgue integration establish result linearly lebesgue integral log hx hx hx hx equality also px hold act essentially partition subset consequence summation sequence since jx jx observe linear consequence write denote define contain main clearly induce proceed step overcomplete via action together x indeed u loss v group elaborate mean depend intuition purely formally indicator give assign index obtain I cycle every cycle sized form first connect arbitrary node use straight forward strong induction exist path must pass constraint constraint pass bi node must pair permutation
ridge collect reservoir reservoir analog reservoir design reservoir weight reservoir retrieve straightforward require optical implementation encode light intensity summation time depict panel figure reservoir state intensity optical reservoir generator provide pass send keep constant intensity input output light intensity multiply reservoir weight output analog proportional output exclude reservoir across discretized follow detailed multiplication negative realize follow balanced state reservoir periodic reservoir pair intensity eq light intensity come reservoir arbitrary wave filter choose appropriately output gain drive multiply coefficient multiplication state intensity weight choose summation achieve summation equation give reservoir minus side figure provide integration eq constant significantly single instant begin encode write time integrate equation time multiply would multiply residual value reservoir measure reference node whole b output balanced panel integration equation indicate dot discretize take analog simple encode reservoir define equation effectively would therefore several magnitude undesirable require precise behavior adapt reservoir state desired keep weight minimum force generate hardware implement analog start experimental depict maximum sampling feed balanced response read national digital acquisition rate experimental end circuit simulate circuit apart circuit every external output behind choice implementation generator issue architecture analog wireless bilinear filtering subsequently capabilitie reservoir task become benchmark reservoir wireless channel presence find reservoir symbol wireless analog reservoir input reservoir measure triangle digital square reservoir ideal circle reservoir analog simulate star experimental amount three quantity reservoir digital triangle ideal equation red square experimental third reservoir analog reservoir analog integration circle performance analog fairly ideal significantly digital reservoir report benchmark handle performance node reason loop time hardware increase leaves noise incorrect discretization network nod concept show simulation bad effect circuit network complicate rule may infer contribution node effectively term large capacity nf reservoir parameter reservoir expect extend dataset far fine tuning amount ridge procedure good one lead architecture quite modular add reservoir whether circuit complicated summation increase analog reservoir computer leverage main characteristic operation indeed need slow challenge field million nonlinear node digital recover reservoir output second setup retrieve symbol analog remove rate symbol order reservoir analog use reservoir input successive technique support office grant channel detailed description use reservoir reconstruct symbol symbol together reach path time create gaussian ratio sequence fed reservoir close value measure rate symbol service universit u f cp universit b reservoir hardware hardware reservoir computer digital implementation reach system bottleneck slow digital design analog reservoir computer work real time build test benchmark performance reservoir room improvement thereby major limitation development reservoir computer reservoir computing machine idea computation train external powerful technique range significantly easy neural network easily hardware implementation comparable implementation almost recurrent nonlinear topology network nonlinear requirement rich dynamic essentially dynamic exploration come input reservoir computer appear optical reservoir particularly optical inherent high parallelism without interaction reservoir construct weight reservoir bias actual conventional recurrent network make unit reservoir time step regularize reservoir computer obstacle design nonlinear unit
possibly constraint model clearly fw tw fw motivate minimize observe stream sequentially model observation online measure regret gap accumulate step cost allow good online achievable rest cost e convex unique example setup svm stream hinge formally type stochastic descent achieve scheme online problem computer computer message undirected serial minimize wide objective incur naturally extend cost distribute decrease rate similar serial hold describe endowed consensus assume doubly generalization stochastic detailed propose update update point processor round main within accumulate variable incorporate neighboring gradient accumulate merge constant reduce half per round dual subgradient maintain accumulate achieve perform exponentially step proper kt I kt kt I kt w kt let suppose doubly produce minimize obeys execute node maintain graph topology consensus fast dependence go worse dependence spectral characterize performance process order point available advance minimization account available observe manner hold doubly evolve derive back recursive algebraic bound accumulate triangle term invoke sequence collect far q step depend node iterate describe towards optimum recall recursive obtain use prove arrive eq bind must control round eq consequence view repeatedly convexity theorem integer update need restriction exposition assume select need algorithm round solve constant conclude present previous section easily variance gradient moreover old jt theorem past q hold collect expectation still room optimization processor illustrate simulate arrange geometric input generate classify relative position minimize fast dual simulated boost yield still also version fast end optimization random expect denominator unable online propose batch convex convex optimal iteration batch serial logarithmic every preliminary investigation perform communication explicitly account time require optimization doubly node eq eigenvalue consensus doubly stochastic moreover frobenius show demand less obtain eq since fact arrive true definition lot effort motivated dataset interest towards present distribute computer converge continuous iteration reveal time batch access start rate corrupt additive form formulation scenario sort point tend infinity rely smoothness trivial mechanism batch optimization set processor project type converge reveal sequentially rate lipschitz reference aim processor similar arrange graph gap e optimal processor
les dans un la les de l dans est dans par une le du dans prototype un le centre des es par le dans la des des la assign object par des des le en dans par les la pour un de le si ce une une les optimisation le maximal dans objective une l de la par la base de http www des des pour si de est des dans des de cluster dans le les r un la du la la distance dans les dans un sup par l de est dans une en sa les la par la pr des dans k pour des les les international conference discovery mining usa method pattern intelligence extension pour de des method overlap international conference recognition page consensus overlap microarray bioinformatics international conference mining usa c international conference intelligence cm les sup de de tn overlapping learning issue mutually exclusive may belongs present method overlap separability overlap overlap le de un dans les es pr de dans en pour la mod la les exp et les classification dans en un la en en th en pour les de classification un en les affect es une les ce en et des des g en des r par des des des th des pour al dans la des des les une les la pour la pour des des la la pour des partition le il des
life reinforcement hamiltonian systems energy balance actor storage successfully electrical circuit key exploit structural endowed hamiltonian ph ph widely geometric differential pde concern simplify pde search stability rely technique supervised solve stochastic control explicitly equation controller receive immediate process maximize cumulative reward actor policy actor reinforcement action space disadvantage progress incorporate partial address ph framework property parameterization particular type energy balance pde split part parameterize actor reinforcement part pde see synthesis seek controller instead online structural bring allow control fashion system simple consider reward else counterpart specify desire hamiltonian bring intrinsic energy consumption offer robustness ph learning control point view incorporate yield controller interpretability solution combine advantage aforementione control trend synthesis ph system root stability aim ph experimental show control input ph system solve base present interesting controller objective aim semi oppose supervise learn network fuzzy specification output instead rl consider alternative fitness analogous reward ph system parameterization actor reinforcement introduce experimental conclude natural represent system energy exchange ph introduce formalize ph eq store matrix assume paper denote system call semi bound satisfy call passive closed loop equilibrium desire control loop passive storage add energy feedback balance hold desire energy rank eq solve pde I passive kx eq reinforcement control environment rl mdp mdp action transition process control state apply apply return scalar maximize cumulative expect reward paper discount reward discount policy factor deal continuous necessary actor actor policy approximate improve actor direction usually beneficial actor ac approximate approximated parameterization actor temporal use trace speed include visit state decay rl exploration visit new find policy towards away lead actor pde split loop hamiltonian simultaneously parameterize loop actor rl actor first pde term close energy eq clear call loop energy assignment energy form desire closed automatically guarantee relation storage basis choose grow unbounded constrain parameterize basis impose obstacle stability demonstrate analysis section knowledge energy hamiltonian prevent total grow unbounded avoid possible set control rewrite stack version mean bilinear oppose find actor actor rl condition observe practice fast fast free rl system never unstable additionally policy model approximate rl balancing suffer obstacle application system even position hamiltonian energy hold split shape widely closed read desire energy update zero long validate control physical fig commonly benchmark motion angular momentum full measurable state model p furthermore htbp mass read potential shape one define hence two actor desire two actor ease incorporate symmetry ability problem fourier topology define basis possible integer frequency contain combination order dimension scale several advantage periodic prevent value small topology learn policy third beneficial periodic value momentum restrict adjust actor table low slow result goal entail critical hamiltonian rl position pointing equilibrium position top able back build momentum equilibrium reward unstable mapping e order basis p rd done randomly distribute white incorporate define take policy zero run system trial approximately begin initial position estimate learning curve simulation give table htbp trial trial sample trace max dots average average minute policy zero initialization value optimistic explore space lot learn state back momentum eventually hamiltonian acquire learn minima desire undesirable shaped try undesirable fig extract negative region initial necessary momentum disadvantage control basis also initial constant initialize otherwise stay set overcome perturbation system less present hence conclude local stability calculate opposite negative x look fig gray equilibrium around stability indicate positive white gray region x dot infer naturally function see stability similar behaviour show fig experiment fig algorithm slightly convergence minute near fig attribute system performance attribute optimistic htbp systematically law extra control make actor reinforcement landscape
draw popularity pair illustrate text ideal popularity adjust vote position determine mixture vector text ideal popularity understand suppose everywhere position issue adjust ideal adjust adjustment capture ideal point algorithm vote scan ahead show posterior vote vote bottom involve issue multiple yet issue political data issue ideal classical ideal equal zero model relative depend text different figure e adjust interpretable multidimensional point yes vote dimension separate subsequent capture kind pattern vote researcher leave right divide classical dimension readily nothing tie concrete issue ideal vote bring issue multidimensional ideal explicitly tie issue determine vote unlike multidimensional vote issue relate deviation voting pattern cccc education attack york serve history attack percent community child national substitute college frequent label lda describe issue language proportion dirichlet associate political issue exhibit treat issue encode e collection collection bag unsupervise topic fit use exist scheme document assign heavily influence give interpretable tag readily fully unsupervised case study political history tag provide service service provide pass code phrase cover may label illustrate top perform frequent mixed draw exhibit proportion posterior tie political modeling portion complete give roll text adjust ideal ideal point turn central problem roll encoding prediction ideal position issue bayesian tractable must variational tend gibbs update next section roll select simplify posterior adjust fully factorize distribution family specify latent endow capture specific marginal conservative variational fit prediction excellent variational proceed fully factorize successively update kl divergence reformulate jensen bind closed optimize ascent coordinate ascent adjust close form effective many algebraic derivation take alternative carlo integration approximation give easy rewrite integral exchange order integration apply rule q monte distribution note replace discuss material estimate find optimize stochastic optimization gradient step alone achieve convergence convergence minimize eliminate improvement briefly mean hidden random vote achieve infer expectation induce behavior adjust roll united house evaluate fitness roll issue responsible qualitative u issue preference demonstrate rich explore political house roll vote hold office attack th mark gain take house roll call vote record one want vote useful roll serve record roll www vote receive roll call summarize ccccc ccccc house separately two period vote house treatment house completely fit represented phrase count phrase bag commonly vocabulary phrase omit vocabulary gram far use phrase section interpretation identification ideal arbitrary multimodal address point adjust ideal traditional explore justify first outline issue adjust traditional report quantitative adjusted ideal table ideal adjust ten show ideal bottom row vote separate vote roll call place cut vote improve vote vote education trust provide program community service provide provide develop house ideal model issue ask differ ideal house ideal panel compare ideal axis bottom ideal parallel compare separate line different treatment traditional ideal surprising able explain fact political alone check discriminant separate party along illustrate discriminant bottom normalize political political party additional dimension extent entire house fit vote contrast analysis fold axis logarithm weight vote issue issue log increase relate vote issue issue political single explain ideal demonstrate number explore information estimate position adjust expect adjusted illustrate issue figure inspection relatively house blue red vote political party th house across denote issue show issue great variation issue preference become ideal understand point issue explicitly control fit effect ideal offset observe residual share offset vote ideal issue moderate ideal adjusted right increase unit leave corrected identify preference nonparametric describe label I vector correct issue adjustment issue find expect extreme absolute counterpart several house house correct issue focus vote full vote house correct issue issue preference replication house motivate analysis issue adjust vote increase place improve adjust improvement restrict significant tend vote vote recognition house frequently house local majority party house largely symbolic topic house business national national effort production symbolic vote evident post symbolic vote yet point alone recognize voting behavior issue explain house way sharp voting vote vote house party species power party party house ensure file member decision vote support systematic systematically issue systematically position adjust ideal finally improve vote vote apart position roll capture vary issue issue large history vote well illustrate exploratory tool several way issue differ vote issue activity outside voting issue area modeling position issue change series provide additional detail begin variational ascent iterate lower specify threshold traditionally symbolic expand solution expand stochastic repeat converge upon variational generality perform variational parameter motivate taylor taylor variational notational convenience estimate describe next paragraph taylor monte variational current estimate variational posterior mean current coefficient straightforward eq increase change instead monte sample decrease space pass cdf variational sample note estimate bias sample method paragraph enough quasi select paragraph markov current marginal factorize gaussian form practical implement happen issue adjust parameter little posterior drop pass drop perform second order describe section definite enough also limited step reason issue regularization share inference considerable expense yield adjustment estimate report effect house hold vote calculate b approximate drop summarize well range represent vote good log drop house accuracy house issue issue e service range five consideration result fairly well political table law east business requirement service health day service local child human finance resource p management care force security resource history name elementary secondary education force education international education house house rule vocabulary convert form part phrase word eliminate phrase phrase result gram processing feature characterize classify whether use vocabulary feature number external appear article feature list phrase negative word figure phrase vocabulary phrase contain wikipedia appear wikipedia containing contain phrase phrase phrase phrase phrase phrase term term phrase term phrase appear expect chance david nj cs david science computer department nj fa grant develop specific voting exploratory language correlate political approximate base across inherently multi key variational voting vote vote capture row roll vote item r roll political ideal binary roll give political call vote two voting favor roll point recover familiar division example vote point incorrect vote adjust service point behind classic black line mark vote incorrectly capture broad illustrate vote act h colored vote classic vote four eight vote model incorrectly sometimes vote incorrectly predict circumstance often vote explain consistent political get closely error differ come policy classical political position political content propose political position voting expect develop capture place spectrum inference encode issue unlike political incorporating vote conservative conservative adjust tell hasting issue vote following describe develop algorithm usual mcmc fast year house vote give ideal exploratory tool analyze item decade political science overview historical perspective political ideal dimension vote yes explain explain party issue careful study roll neighbor former make inference assume different point inference political variety incorporate text text focused issue evidence answer political science method behavior toward text vote receive vote predictor vote individual vote well ideal affinity toward type adjust model conceptually content recommendation article already user web specifically design political work item employ orientation orientation spectrum consideration political roll ideal discuss account specific ideal point quantitative political member overview voting record point place interpretable political historical dimensional characterized discrimination often difficulty popularity yes probit underlie empirical popularity vote popularity vote vote yes ideal standard prior vote ideal
norm sdp solver first optimization trace thresholding available sec problem write provide optimization complex selection publish basis pursuit order nesterov proximity gradually use smoothed also idea linearization iteration fista depend know advance indicator zero inside infinite positive definite matrix smooth play central role carefully present envelope well summarize proper function x left side property lemma lipschitz gradient require reason nesterov smoothing derivation somewhat refer propose diameter convex convex conjugate diameter focus identity I envelope conjugate slightly lemma derive follow nesterov formulation somewhat different relationship operator gradient lipschitz control quality approximation kx l fy k l accelerate method track update formula satisfie iterate possibility give iteration share limitation choice additional converge optimum original smoothed objective optimum elementary b b dx assertion follow constant price pay advance logarithmic factor report convergence rate term adaptive version accelerate backward lemma thm eq thm apply use obtain x lemma change parameter denote use obtain apply fista guarantee nesterov smoothed method important provide explicit lipschitz discuss sec nesterov rely bound fix advance set guarantee unbounded practice motivate optimization advance share domain primal minimize problem translate objective discussion point clear constant different direction multiplier lagrangian associated updating approach satisfied make ill suit problem admm linearization alm alternate projection augment lagrangian applicable decompose fista smoothing share another applicable differ nesterov part affine applicability pursuit similar attain boundedness combine benefit max unbounded domain priori number iteration advance max u max regularizer collaborative nice commonly nuclear recently max commonly lack recently way clear far aware non proximal rely regularize demonstrate entry entry max problem eq entry max operator correspond thresholding movie entry among rank matlab stop less small eigenvalue increase proceed projection produce procedure big speed kind strategy semidefinite programming parameter good consider rough run list highly optimize rely method interior runtime increase run memory could run large try hand iteration increase iteration decrease range first yield runtime hour generic sdp convex fast reach hour regardless able reach minimum objective value attain plotted gap sdp practical require significant parameter produce rough solution c c cost datum given solve choice tr surveillance stack correspond sequence frame matrix represent move foreground frame since implementation alm solve method alm strategy beneficial iteration analogously static strategy video since roughly order reduce hardware result number iteration theorem art alm static alm sparse see therein q give
surprisingly device grow popularity mobile device pc combine security require dedicate sufficiently mostly simple short scheme dedicate explicit security attention analyze operate reliable device contribution framework serve behavioral behavioral feature mobile device task discriminative design decision usage continuous insight operational online characteristic identification verification active category behavioral hand feature behavioral aim activity early behavioral pressure identify base input work average depend datum gain popularity security survey modal mobile enable bind sign verification high signature fusion yet highly read enter scheme explicit user particular literature aim implicit instance scheme monitor user digit particularly behavioral study high instance action click work inspire experimental visible one continuously clicks carry move position reduce system flip resemble line signature verification surface geometric signature subject pressure feature achieve differ signature verification signature sophisticated main kind implicit always signature verification paper probably contribution try historical behavioral classifier support unless carry use author two contribution entry implicit second interact like trajectory goal understand would reliably continuously record enough behavioral perform read coordinate stroke addition phone record time cover available phone various extract system profile profile phase classify challenge action frequent primitive perform system slide usually page slide move email web distinguish horizontal within action across extended complex focus appropriate continuous monitoring exhibit discriminative rely action feature continue converge equilibrium point device device long switch mode device begins phase continuously track make consecutive result precision individual influence precision need choice proportional provide detail affect decision usage scenario suffice reliable likely identify cause get conventional modify security configuration phone hour usage standard mechanism gps device scheme operate feasible framework challenge record interact phone try collect realistic analyze distinguish make week investigation aim decision technical feasibility must compare main motivate user produce ask read question reading inform would subject phone mobile minimal many improve assign participant ask start document ask difference image stop user minute pair participant ask week later one participant collect across different interact protocol enable interact phone various document image allow entry document image primary application wikipedia article see panel free go often new article orientation device user could switch landscape mode record sample variable raw event code absolute ms device orientation pressure orientation orientation provide standard reading usually take minute trial minute participant specification appendix possible concept study outcome generally speak appropriately freedom variance operate device time cause condition issue try avoid know serve analyze behavior reflect interact behave use reveal purpose collect limited artificial limitation intra set aside phone user phone possibly interaction regard moreover round record one part document influence user therefore protocol adaptation user improve ability device behavior relatively amount snapshot study term one give device user use aware play ask experience device user world two long stable phase phone phone ideally must phone phone would record acquire record phone take care phone enable describe record report extraction record begin end trajectory encode n na phone landscape record list believe relevant trajectory notice tend device stroke vary trajectory able lift velocity stay latter gets accelerate even distinct direct stroke consecutive scale straight angle ensemble angular dispersion straight constitute project distinguish end leave important time read detect informative reading discard method keep store near neighbor accelerate training k distance odd interestingly nn hard user decision powerful classifier two divide hyperplane margin svms maximally hyper generalize datum individual outlier within margin control trade minimize class linearly feature involve hyperplane kernel transfer space radial basis parameterize tune feature usage scenario way explain detail normalize rejection require fraction recognize fraction reject classifier quantifie resort temporal user misclassification rate far increase detect security equal rate equal fold cross tune stroke individually user estimation several consecutive classifying instead classify majority project thresholded score depend far near put count resolve individual report carry feasibility experimental analysis reliability classifier output analyze influence several vary number per decision stroke lower increase classification stay another affect affect interact device give attack carry please influence turn device decision slide precede adapt classifier early intervention odd could come make second take decision decision might provide phone home restaurant enough increase might insufficient exclusive security mechanism device implementation trade security accuracy experimentally constitute difficulty different experimental inter week assume train stay pick device train week round acquisition record user day every put phone pick record day device every overall already make device user directly device device detect phone imagine extend minute scenario draw htb datum result outcome color black nn colored blue percentile individually median range usage intra outlier seem user reach reach depend quantifie security reduce rate distance interestingly inter week classifier rate inter interpret round round text find text subject round week short user phone way interaction prefer way hold phone intra class record yet week experiment horizontal scenario user way image comparison distinguish way depend chance correct accepted rate future equal accuracy scheme highlight critical account categorical take value read email write email read music operate scenario probably behave use video heavily temporal instability current serve exclusive device satisfy security trade achievable contiguous extend time minute adaptive false favor gps available network computer finding computer claim would differences computer user believe help content collection collection phone classifier get large user long without might consider introduce degree affect variable bias classifier test principle binary variability user clearly affect influence subject repeat one inter via number ten time subject repetition test equal user fluctuation range apparent demonstrate size influence influence phone phone risk might record necessarily slightly dot convert normalize device signature device aware minimize condition much possible datum experimental protocol try strictly protocol check device carry device carry device total phone phone constrain user record device respectively result illustrate phone user collect unclear signature help device responsible relation phone differ nn equally suggest phone responsible however rate inter phone suggest introduce signature alternative explanation number phone collect phone alone reliably detect play role estimate finding attack attack base try random attack sophisticated try observe stroke imagine pressure etc look successful also attack place application device pattern attack success chance argue device lose attack situation construction investigate question serve behavioral importantly
expect achieve misclassification plot zero block include lexical syntactic review amazon website review review review per identify feature uci repository neural use author report selection separately multinomial show utilize sparse gain efficiency normalization center evident k problem plot estimate rna datum patient various unbalanced normalize lasso datum available validation run trend lasso significantly sparse reasonably outperform investigate primarily interested trend error cross validation trend depend restrict multiclass center low interval distribution typically confirm center determine cross generalization sparse give trend configuration specific one degenerate location zero construct site choose singular primary quantity set interested problem non false false negative true negative rate method non zero entry predictive zero central distribution show area average configuration center configuration center accord model thin configuration error configuration result case choice dense area dash randomly thin configuration lasso sensitive look group low except configuration bad configuration group applicable require twice continuously require around bound requirement ensure range nest loop middle loop coordinate descent outer middle inner develop allow part improving middle provide group separable sense separability primary ensure straightforward subgradient b n nx sufficient proposition define closed proposition denote hold sufficiently jj deal hence equation finding point descent provide descent accord cyclic rule compute eq jx jx line search jt gradient need aim cost compute block sufficient consideration piecewise cyclic compute algorithm beneficial broad cauchy multinomial without three time amazon software package interface multinomial logistic sparse group template library template generic sparse package template library external library template library performance utilize boost library boost review library template library introduction library package library list current set fitting group lasso group amazon k see discussion section penalize negative log multinomial distribution primary multiclass classification implement algorithm template library regression present multiclass classification real example lasso example lasso comparable performance acquisition objective would optimize interpretation theoretical lasso different descent interested unconstraine coordinate naturally search denote projection onto block mi mx function say around rule I scheme make twice differentiable scheme rule differentiable sequence cyclic outline block coordinate coordinate minimizer consequence minimizer differentiable minimizer last scheme additional continuously give optimal guarantee convergent immediate descent use differentiable fp cluster generate stationary point point coordinate outline except minimizer corollary minimizer lemma contradiction point otherwise continuity implication choose ax ax sparse sparse real outperform multinomial classification sparse implementation multinomial package implementation high classification amount package group classification gradient descent lasso regularization lasso descent optimization generalize gradient descent consider logistic cox combine optimization descent coordinate modified application multinomial optimization multinomial group group efficient straight forward completely robust minor treat type optimize penalize approximation raise another upper sufficient whether attain approach enable update hessian reduce provide hessian reduce time set multinomial purpose multiclass lasso rate good multinomial lasso achieve example text amazon review improvement rate sparse template multinomial logistic available lasso require multinomial size twice solution define lasso decompose dimension write n group include infimum computable grouping multinomial multinomial multinomial grouping reflect group multinomial problem purpose set profile type classify primary site profile amazon review classification rna disease classify rna characteristic simulate problem assume categorical use parametrization multinomial intercept together identifiability parametrization penalize group parameter grouping part cancer site expression amazon review disease expression k set lasso two preprocessing entail center scaling center scaling feature design column column technical biological normalization procedure
trivial scheme verify use mcmc static metropolis utilize mixture ergodicity unbounded see metropolis combine augmentation augment copula specify distribution new update chain exist choice proposal covariance choose specify update define history j j theoretical motivation recommend choice next mcmc proposal manifold definite matrix property symmetric definite marginal consequence closure convolution wishart proposal matching detail wishart covariance note also riemannian symmetric positive matrix wishart fix freedom select average variate marginal chain adapting proposal ensure restrict matrix combination remain method present table second hyperparameter claim development incorporate estimation posterior gaussian copula augment iv predictive dependence compare incur versus independence claim payment payment copula include statistic perform metropolis iv adaptive metropolis use proposal heat riemannian manifold metropolis posterior convert matrix convert top left table summary payment per payment statistic provide component year decompose year quantify residual currently development constant year constant per year repeat incur principal variation payment development proportion easily distribution leading year incur complete development weight large eigenvalue posterior summary result distribution present involve contour posterior construct use adaptive development factor payment distribution present posterior copula development triangle payment incur copula model full partial estimate quantile leave measure worth note augmentation specify example parsimonious specification allow whether payment respectively present claim mixture via hierarchical copula full payment find agreement attribute bayesian model structure consider additionally model extend combine information present within payment incur allow prediction incorporate achieve develop hierarchical incorporate wishart conjugacy log form copula approach hierarchical capture potential tail dependence payment incur demonstrate incorporate regard augmentation incorporate challenging methodology inference monte incorporate efficiently marginal dependence advanced challenging claim model extended dependence incorporate recently chain payment incur model partial several different payment introduce discovery dp incur copula std applicable e marginal gaussian copula payment incur independent copula full post hierarchical bayesian prior variance independent bayesian scale payment report development cumulative payment marginal mean variate say variate matrix satisfied variate chapter pm rp tx cp partition matrix sub matrix variate distribution degree multivariate function wishart partition matrix denote follow chapter detail variate inverse variate wishart degree denote generator cv cc dimensions function copula family manuscript lemma function copula dependence tail dependence copula dependence copula dependence frank tail frank admit recursive expression q two variate copula u ic u copula weight secondly show b condition satisfied notation copula definition give q domain box odd consider n k copula expand volume box corollary remark proof article recently claim loss coherent particular hierarchical incur claim combine incorporate dependence augment incur usefulness incorporate payment incur result payment payment iii lag year gaussian payment incorporate transformation augment incur carlo data augmentation copula payment triangle adaptation component deal matrix restrict manifold dependence incur claim augmentation department statistical university college uk email ac uk university mathematics statistics school business predict ratio past come variety contribution develop claim however structure formal extend generate copula incur loss year row article significantly extend dependence problem variate inverse prior chain manifold treat unobserve miss likelihood financial security company predict claim account parameter claim combine claim incur loss available chain aim prediction claim incur datum adjust reduce gap drawback log normal model future claim triangle payment incur major loss quantify model applicable loss extend capture additional structure payment incur payment incur datum commonly exist ratio development impact development development period appeal claim practice payment plus projection incur find structure payment incur loss development period period life fully actual different period payment copula construct specify triangular structure loss permutation restricted copula across development period augmentation methodology involve mixture copula feature incur loss structure development lag loss diagonal comprise contain row previous bayesian hyperparameter development variate preserve conjugacy independence copula conjugate model sampler copula dependence non make develop iterate form class mcmc grow recognize inference interest utilize adaptive mcmc facilitate adopt employ scale metropolis algorithm financial reference therein involve adaptive definite create variate proposal mixture modify copula base challenge intractable arise triangle argue dependence payment incur challenge marginal year give payment incur loss integral copula bayesian augmentation overcome model involve claim payment second incorporate claim amount loss know report claim impose develop construct vertical development claim row claim incur year year incur loss moreover assume claim probability ultimately claim value p j diagonal appear diagonal block corner right corner clear definition iterate define dirac vi stacking create derive dependence framework assumption independent recursion l follow component payment incur component claim incur j note assumption make applie coincide claim incur assume tail beyond incorporate incur base particular result cumulative satisfy parameter summarize payment development year loss year conditionally eq conditional moment chain conditional upon consecutive incur loss see conditional payment incur give aspect extend develop independence incur ratio incorporate subsequently article develop extend convenient conjugacy often lose present generalize static assumption use variate wishart distribution develop statistical relevant distributional payment ratio copula log payment ratio wishart property positive accord variate gaussian possible extension second structure payment loss incur loss prior wishart give inference development consequence dependence payment log incur conditional k distribution q degree freedom matrix eq properties wishart covariance conjugacy payment incur loss convenient marginally result random payment lemma find th year claim payment incur specify unobserved year triangle incur specify consider multivariate covariance define family operator permutation permutation dependence specification admit conjugacy tuple tuple tuple multivariate lemma generalization model case year payment loss year unobserve payment incur loss payment incur loss consider year tuple give transform payment incur I payment incur variable payment incur denote I jj follow jj I jj formulate joint incur claim upon parameter accord normal jj j variate allow j either payment delay loss increment assume positively claim increment however structure especially party status change term medical substantially loss lag lag year always realistic feature enhance correlation maintain parsimonious consider alternative structure motivate practically appeal claim estimation claim recognize claim incur month month portfolio claim period claim benefit lag normally follow subsequent correlation equally change claim huge claim cost period replace stream period mainly capture dependence year propose lag assumption dependent lag normal statistical payment year analogously incur inverse wishart define inverse payment hence payment incur loss datum jj element random payment incur loss permutation index characterize distribution covariance payment loss incur loss prior hyper another give multivariate copula form conjugacy gibbs block ii matrix consider vector initial vector give rotation transformation obtain decomposition accord diagonal follow j u u incur loss j covariance vector detail consequence conjugacy directly scheme full estimation ii conditional posterior give independent posterior component payment loss gaussian specify wishart section form ii inverse q study copula framework copula copula article dependence capability remain framework however introduction enable modify distribution model comprise statistic alternative modelling dependence copula model copula g detail method know statistic augmentation combine allow copula model variation comparable present fundamental member family mixture copula characteristic copula member reference variate particularly copula construction analytic dependence parsimonious generator member family copulas frank copula copula denote copula density parameter frank copulas dependence tail upper tail copula dependence class copula payment log model comprise copula copula distributional variate distributional member mixture copula component admit tractable function denote proof provide augmentation situation bayesian intractable consider generic evaluate point wise setting partition admit augmentation due posterior dimensional augmented detail give notation augmentation invertible transformation consider log incur loss give I decomposition observe log payment loss payment loss incur th year introduction notation make payment triangle incur loss un observe quantity triangular loss auxiliary
require stage study current health classic current status study member period disease case age due contaminate technical reason I trial participant randomize treatment make treatment treatment period intend covariate treatment treatment treatment I measure measurement omit reflect fact measurement case set causal systematic inference account encounter world medical stage conclusion causal observational relationship estimate incomplete directly graph project despite effect carry systematic calculus distinction allow description miss presentation causal graphical specify hand explicit parametric relationship presentation effect identifiable linearity effect implication tie causality new possibility graphical second show key format clarity speed communication analysis scientific scientific knowledge anonymous comment description derive population population become explicit factorize accord distribution variable notation distribution argument unless otherwise integral clinical trial integral unknown variable need nested variable q cm abstract assumption design scientific research design miss structure direct causal calculus node causal diagram time observational incomplete make offer causal clinical trial commonly set typically acyclic graph represent direction effect discuss relationship mathematically object causal systematic carefully design sufficient effect specify objective collect collection take account analysis population pressure measurement design causal effect design need accurate reporting causal estimate describe clinical nest causal provide implication cancer effect unknown causal causal calculation assume assumption variable subscript index close variable describe determined symbol population directly circle instead measure circle belong available design consistently design control selection factor status cancer case status control analysis control figure situation difference causal effect apply calculus design rely causal causal extended reflect inference causal benefit calculus directly question relate selection mechanism probabilistic structural causal triple variable determine factor call model respective domain form probabilistic causal pair causal diagram correspond direct member toward design extension probabilistic causal causal follow attribute value determine selection parent parent parent miss node type observe type open open observational corresponding setup determine know principle lose determine determine unknown later population miss record identify truncation selection conceptual union population differ causal conceptual allow area sampling may differ area member priori unique social security member enter study induce consist individual effect also item causal value individual select value definition univariate extended axis may special indicate miss relationship node variable health study life health causal variable time link causal observational relative node visualization observational stage section causal version unobserved unless kind benchmark measurement subsample correlate addition sample design formally determine make measurement undirecte miss concern design description many general cancer operator intervention observation represent represent node income remove identifiable experimental front criterion use formula automate calculus step must accord factorize model notation refer define respect unless specify ignore response ignore note parent parametrization purely vast topic directly applicable likelihood value mention parametrization causal write link parametrize variable collect case design frequency number assume cancer become summation shorthand maximum carry htb cc illustration sum cancer control select covariate measurement probability table non select population difference third round aim causal essential experimental study illustrative design causal make causal missing describe realistic remove name design study two stage collection graph present applicable design design analysis factorize causal design offer point write likelihood need illustrate project aim classic genetic disease genetic project select underlie certain age typically individual examine factor follow disease genetic risk health status understand pressure baseline variable need classic classic baseline health status baseline conclusion make causal calculus causal disease correspond effect disease health causal classic health baseline gene health status baseline observe effect investigate birth unobserved individual make status whether mechanism eq consequently effect genetic disease accounting tell conditioning baseline situation qualitatively kind status project risk account potentially different participant participant must take classic
mc dx operator I lasso produce hard mc line value choose mc select correct estimated type penalty researcher algorithm entire solution e g algorithm algorithm descent explicit maximization whereas lead formula derive formula penalize utilize maximize maximization quadratic coordinate wise equation loading suppose variance update log old old old old old old old ii parameter maximize express penalty include descent vector update equivalent penalize close variety penalty update maximize likelihood diagonal column zero decrease describe column loading loading compute introduce entire path value give penalty mc path produce improve smooth surface initial loading unique variance might value loading matter choose stationary function say value loading nonzero element factor loading value factor loading estimate ij column loading exist element see stay may nonzero zero nonzero produce column therefore adopt newly value loading approximately lasso large monotonically employ traditional apply mse mse mse pm indicate proportion loading zero obtain follow increase often detect e mc much mc detect lasso penalization produce rotation mc yield mse select dense bic lasso penalty rotation mc mc mc mc lasso illustrate usefulness available evaluate add percentage loading approximately three technique penalize via mc analysis pca reconstruct via posterior mean reconstruction mse without reconstruct digit reconstruct loading right figure factor produce smooth image sparse mc yield loading loading yield mse produce traditional rotation descent entire investigate propose procedure produce handwritten show mc load sufficiently datum hour entire path would propose algorithm freedom lasso however topic apply em common regard miss penalize log likelihood take ni te tr f old old ne old e old ni old old e old old old old old old old old old old il proof contradiction element proposition section remark school engineering science rotation utilize loading even rotation produce introduce loading methodology rotation technique via em along descent path permit wide penalty investigate model strategy example give nonconvex rotation useful covariance observable variable exploratory analysis use maximum use rotation technique utilize loading however well unstable e mention furthermore even estimate rotation produce sufficiently case penalize lasso attention tool sparse generalized support penalization method nonconvex nonconvex method smoothly scad minimax mc elastic several path coordinate researcher discuss procedure penalization obtain sparse loading numerically show penalization traditional procedure yet although ordinary analysis early nonconvex yet penalization via nonconvex penalty penalize view I maximum estimate technique methodology solution estimate loading em descent introduce permit nonconvex include package r implement http package traditional estimation procedure likelihood nonconvex penalize method rotation obtain path conclude remark observable vector g unique factor mm pi unique observable distribute covariance estimate close form likelihood g loading generate rotation meaningful orthogonal criterion minimize loading note construct model loading q loading produce loading possess loss recover
carried processing moment demand impossible exception filter analytic problem transformation nonlinear kalman transformation approximate transformation taylor mild linearization acceptable linearization measure quantify linearization deterministic technique instead covariance nonlinear statistical possible linearization ii mean capture linearization characterize term covariance linearization correspond dirac affine calculate year choose example selection paper briefly simulation sec calculation sigma n factor depend scheme transform cholesky scaling hold root factor solve expression note adaptive linearization well desire demand single global linearization especially weight covariance filter local benefit decrease increase nonlinearity perform locally linearization linearization provide linearization assess linearization apply geometrically trace proportional ellipsoid linearization I linearization affine linearization nonlinear mass avoids split irrelevant together select splitting linearization interpolation linearization focus linearization error consider split calculate analytically component choose gaussian formulate replace gaussian vector large reduce degree concern preserve simplify splitting numerically reduce univariate moment preserve mixture require determine symmetry place hold library preserve univariate preserve constraint lead free equation dynamically linearization throughout simplicity determine component additional capture order like skewness split subsequent linearization split linearization control growth univariate q covariance axis thus eigenvector replace eigenvector choose splitting univariate mixture shift scale multiply transform splitting covariance calculation preserve splitting gaussian formulae eigenvector splitting eigenvector large cause linearization merely eigenvalue idea evaluate deviation transformation linearize along split gaussian deviation subsequent linearization desire split l integral along eigenvector eigenvector choose nonlinear integral efficient mean dirac mixture discretization linearization reduction stop linearization delay right node start delay north pos near line pos delay line pos west north linearization sec filter key idea dynamically increase mixture linearization filtering limit memory follow description operation prediction step perform linearize l linearization nonlinearity cause severe linearization large nonlinear linearize splitting linearization newly linearization component affect splitting gaussians subsequent linearization criterion combine criterion drop grow beyond original remain integral tracking keep avoid linearization stop let split base locally linearize system kalman predictor linearization component grow due subsequent step exploit redundancy component reduce algorithm year see typically much employ computational demand reduction predict filter almost identical prediction linearization filter linearization correspond linearize joint equation apply gaussian locally linearize measurement give rise component predict measurement linearization k k reduction posterior splitting max nonlinear consider recursively mixture threshold estimator linearization numerical integration weight linearization error rather select splitting base splitting perform large table leibl divergence approximation large eigenvalue scheme component account gaussian always since split inferior quality splitting importance splitting quality tb kk angle choose k model unimodal exploit filter strong heavily tail deviation threshold filtering threshold filter employ simple weight criterion consider example linearization fair scaling e split reach since exploit linearization error filter pf residual resample threshold simulation root rmse runtime depict good tracking high runtime conversely fast selects exhibit perform large eigenvalue track inferior less thank split besides direction nonlinearity split analogously filter represent split runtime see consume operation reduction already novel adaptive statistical linearization quantify linearization error linearization covariance splitting direction split linearization criterion reliably strong keep number split level linearization corner draw text circle thick fill fill dash sep anchor corner
rich stochastic production line consider measure server server buffer infinite capacity free keep service service go customer node place buffer service customer buffer service customer random mechanism represent visit customer let matrix refer table table firstly describe consider problem criterion produce representation notation arrival time service may note identity coincide customer service wish performance criterion follow network function every service occur node customer third arrive therefore point obtain representation reason extend node performance consider customer average per number time queue denote performance optimize random operation great necessary may represent indicator provide procedure deterministic note sample expectation network performance possess reliability recursive require proof possibility representation obvious deterministic procedure technical order simplify formulae introduce easy value traditional axiom fulfil axiom minimum small element equal denote first subset obviously table deterministic procedure consider network write examine arrival customer customer leave go node coincide one clearly customer provide customer denote happen join queue node account time I j representation finally representation superposition representation theorem require conclude take represent generality suppose entry introduce additional fulfil induction statement assume operation lemma axiom successively axiom statement optimize formulae estimating obtain gradient consider estimate random realization differ estimate difference formulae computation additional function simulation difficult perturbation method sample simulation dynamic combine calculate gradient network provide derivative one simulation compare require alone concern unbiased estimate mse order short considerable algebraic examine easy interpret practical lipschitz family etc various suitable ordinary change verify condition fulfil next technical operation break provide respectively variable w verify instance examine f operation multiplication w one satisfied firstly possible derivative vice versa neighborhood lemma may occur therefore h inequalitie subset obviously q g hold e condition analogous borel lebesgue measure neighbourhood violate nevertheless every hold p condition corollary continuous hold every neighbourhood remain closeness example show define eq although inequality lebesgue follow see nevertheless case fulfil lemma neighbourhood w neighbourhood consequently fulfil belong lemma fulfil neighborhood reasoning minimum index consequence importance obviously stable addition clear corollary deduce example continuity essential important verify fulfil nevertheless example definition algebra operation generate show example family k satisfy conclude fulfil contrast traditional exponential satisfy random integrable network describe gradient algebraic probability begin may mean completion normally case realization algorithm fix activity stop go step complete correctness reliability apply denote sample initial failure element exclude node able well arc set hold save go step rather deterministic mechanism conclude continuous estimate unbiased total fulfil customer hold follow boundedness indicator identity q arrival service previous server contrary arrival customer completion customer th arrive w expression define algorithm incorporate initial value completion add go iii visit customer change key role calculate gradient include function u initial upon add g determine visit customer server free completion quite note gradient give use value gradient measure optimization acknowledgement grateful discussion axiom example class measure expect analytically function give analytical study estimate unbiased rather meet study network perturbation management biology system help analytical method computer behaviour usually main aim performance performance evaluate sensitivity gradient generally estimate procedure
student institute research statistical institute email control along generalization chart prove robust generalize multivariate univariate chart establish efficacy methodology depth bootstrappe directional statistic mid ease chart prefer high speed production relevant analogue basis chart multivariate analogue robust dispersion drawback shift median univariate interpretability difficult liu rank datum datum rank rank moderate change reflect suggest use distribution central tendency performance chart presence chart clearly absence recommend development section program propose average simplification take deviation statistic robust multivariate mean choose depth accept multivariate observation dispersion observation four depth depth depth liu liu desirable invariance monotonicity background correspond normality line limit chart distribution limit chart chart propose control limit distributional transformation distribution chart follow value consider mean suitable choice chart distributional chart instead plug limit choice l say p ensure stay control time treat besides consume multivariate pp test statistic define depth resort chart extend variate th element simplicity mean mechanism bring procedure rational fix distribution construct phase distribution plot size chart size chart convenient standard preferred percentage take percentile table propose chart chart chart scenario shift shift outlier significance previously depth compare chart obviously simulate bivariate normal get select process chart univariate chart statistic bootstrapping control replacement dispersion dispersion adopt compute ti variable choose percentile control statistic significantly recommend multivariate chart shift shift outlier compare liu rank test comparative efficacy shift table liu shift presence adapt poor shift presence strength obvious shift chart chart recommend chart around fluctuation depth cut lead highly control chart long stay serve distribution point huge function limitation application depth depth reliable bivariate dimension depth performance change value evident circumstance liu bootstrappe surveillance change
repeat characteristic represent make life unstable group connect line rank like play likely opinion novel likely situation analytical refine go alternate view player expert give style mark four use position determine dimension reasonably demonstrate level connection indicate seem merely suggest section nn sec sec reference expert reference rescale comparison pattern pca reduce dimension fold measurement include show column examine classifier part already significantly generate normalizing explore investigate way relationship thorough internal player argue adjust condition color limit opponent might differently game stage pattern carefully move game game strength investigate extract summary pattern information various correspondence summarie player play play style player proposal characteristic remain work demonstrate exist practically information summary application inference direction future newly go theory specific go comment methodology mr acknowledge major paper information go games corpus student physics university also ik student move pattern record game propose extract summary several within correspondence traditional go feature player strength style accurate classifier attribute rank internet study go program theoretical playing approximation computer go focus create play position research game record human play game well player information play black assume basic use computer internet review game computer record enable collection far move position base go feature belong class intermediate go present play devise player information way play strength playing move characteristic player reliably game apart practical investigate style extraction game corpus explain apply finding game finally interpretation research direction large collection record format analyze particular player look effect process play move count encounter globally mapping generate describe need specificity tradeoff description various attribute description game sample difference significant inspire candidate pair position feature move last play move edge distance spatial configuration played maintain symmetry configuration produce go square match contain capture move opponent elsewhere shape move corner far local diverse object pattern thus describe multiple intuitive solution scale default process show fig mostly large diameter game pattern always try modify raw count post processing method beneficial normalization dimension knowledge inspection ultimately classifier conjecture frequently style aspect implement use feature go rating stream produce string per move course move play influence playing analyze several basic technique purely correlation vector component produce represent next significant axis dataset lie little pattern pca preprocesse sensitive component hand produce player project map connection assign pattern output represent infer assessment play strength meta country calibrate knowledge pattern output divide difference approximate trivial approximation representation pca dimension define interpretation play aside classifier vector vector artificial output generalize unknown network pattern relation nn naive infer membership evidence couple analysis together technique well pearson product correlation coefficient measure strength compare data validation randomly divide reduce assume inter dependency vector dimension center produce vector eigenvector eigenvalue eigenvector projection follow describe r n compute eigenvalue decrease eigenvalue mechanism relationship player distance preserve consider dataset reflect estimate quantify scalar probability high therefore preserve visualize quantify order member position subject descent know vector player similarity output average illustrate prove study pn network ability classification iteratively reasonably call neuron get output activity activation previous activation call neuron typical sigmoid feed training w n n exist construct approximate player separately cover sized interval dimension fit value distribution element feed estimate deviation element probable c mining method open source framework available majority implement programming library notably mdp library naive bayes around module playing strength play receive rating scale list scale rank game review game player strength game force discard various represent positive representing rank number represent etc create perfect correspondence measure dimension pearson see satisfy imply extremely trivial group strength confirm correct accurately strength confirm suggestion examine pattern position suggest pattern alone certainly thorough reason play pca strength nn classifier classifier play player perform reduce explore game determine accuracy small pattern compare fold measure standard approximated file select column describe game square classifier obtain albeit network classifier network due network sample small list mainly comparison perform nn use three architecture comprise c train take explore look vector attempt pattern style source game collection subset player choose within player play play frame analysis traditionally expert allow predict similarity pattern discover move ask mark four give style safe style base experience expert terminology go player much room confusion concept extra style accurately reflect diversity dimension research experimentally pattern year play r r year player complete answer reliable though beyond aspect pearson manual consistency game game age reasonably expert decision dataset either se li chen deviation receive three playing sort game go play several distinct diversity analysis outlier high common extreme style prominent balanced careful player regard ten pattern player set three pearson pca dimension correlation year
consider random error satisfy parameter become case error denote element specification norm norm page k combine thus become specification practice popular fold validation little contaminate performance ols pure obtain identify small contaminate pure testing contaminate merge penalize method identify minimize step deviation positive section integer large number infinity covariate estimation penalize lemma still stop provide supplement critical properly case covariate follow specification difference formation change highlight continue conclusion valid analytic property additional cause state theoretical assumption frobenius fourth consistency hold depend fix estimator infinity appropriate integer eigenvalue bound properly scale asymptotic assumption via extension theorem suppose estimator consistent supplement theorem hold asymptotic level estimate x plugging theorem require handle theorem condition q estimator simulation parameter specification p cp estimator know zero least square hard pls hard square pls soft penalize square hard pls pls soft ts oracle iy refer thresholding squared rmse penalize examine pdf realize realize leave rmse horizontal line rmse minimal rmse rmse panel soft parameter rmse symmetry rmse rmse forms rmse around soft cause rmse optimally rmse pls pls hard hard come pls hard soft pls pls perform significantly ol depict minimal rmse contribute rmse ts generate figure rmse similar fix increase bad estimator much ol penalize hard pls vary rmse around pls hard pls grow ols four rmse p set eight vary simulation report regularization step similar rmse step drive pls interval pls hard pls small size rmse drive penalize panel pls pls pls pls tell closely phenotype thousand snps three chinese false discovery proportion discovery discovery population filter deviation regression loading factor snps biased number proportion filter clear large filter filter slightly filter filter optimistic ccccc discovery number consider structural regression model exploit parameter consistent contrary possess partial selection thus structural consistently asymptotically previous show square continue significantly naive least square advantage interval verify genome association testing equipped drive penalize false discovery proportion estimate illustrate regression model high structural fast sparse phenomenon appear penalty impose structural parameter save proof ps ps p ps I ns p proceed theorem notation x xx assumption cauchy schwarz adopt related theorem matrix deterministic bound hold nd proof condition ds dt p n dd proof explicitly eq condition dd next show special q conclusion n condition condition q thus therefore estimator lemma supplement copy denote bound notation sufficient bound lemma p lemma dd n notation next desire result apply eq times v thus q q assumption dd v v p definition ip dt theorem converge denote p note thus nr theorem since notation lemma estimator supplement occur assumption long lemma assumption condition consistent occur desire need theorem specifically result theorem eq converge first n lemma note q lemma dd dd long corollary notation sufficient q next zero eq note lemma dd dd material section true thm thm conjecture rgb question pc partial sparse penalize principle extensively structural structural high dimensional estimate derive consistency possess partial selection consistency interesting partial phenomenon structural consistently estimator improve procedure challenge suitable region dimension slow drive simulation real analyze material paper methodology almost namely hold overview sense contrast another model different depend model arise arbitrary statistic factor statistic z dependence statistic unobserve critical discovery estimation parameter replace model critical possesse reflect instance nonzero measurement response contaminate response outlier several seminal mle presence stop work exploration assume come common elimination nuisance conditioning solve principle economic remain write kk condition parameter detail supplement mainly possess consistency consistently estimate penalize one step estimator estimator asymptotic thus construct efficiency step parameter penalize method rigorously penalize derive two estimator propose regularization extend covariate analyze proof positive random copy variance exist positive mean kind large understood penalty soft lasso hard general penalty function minimizer simplify minimizer soft penalty penalize estimator interpret minimizer subdifferential calculus theorem necessary ordinary integer simplify derivation keep message simplicity statement stop stop every algorithm stop simulation suppose limit corresponding limit estimator solution follow nonlinear soft eq necessary lemma must conceptual confusion minimizer interestingly huber q huber equivalence penalize huber indicate naturally formal least indicate datum huber sparse compare paper dimensional high review robust impose problem obtain estimator notation expansion n index I similarly operation set properly great greatly asymptotic index analytic expression derive property theoretical specification regularization addition result
class dissimilarity happen performance identify way eliminate improve collect identify graph edge connect node root refer graph extend supervise learn et al fundamentally content output node put pressure sign content feature weak address purely work semi solution lead propose mixed ir converge exist even relational many ir vote relational harmonic square use devise although straight forward form similarity dissimilarity hyperparameter two graph losse graph relate approximately call easy usefulness quickly parameter demonstrate et mixed systematically usefulness web citation link page page poor organize extension ir discuss hyperparameter notation use represent associated edge connect belong class either weight correspond label node distribution interest dimensional unlabeled namely scenario edge belong divergence kullback leibler shannon divergence etc shannon smoothed version divergence term information datum match regularization underlie graph regularization fitting regularization node connect term edge violate suffer objective q measurement distribution help regularization constraint find fashion iterative kl regularization ir clean often useful method mix graph way normalize w j w practice pure achieve improved node initialize j z ji jt formulate collective classifier relaxation label rl relaxation labeling component collective keep probability instant relational classifier compute unlabele sum neighbor may sometimes perform mixed modify ir method label k characteristic role key graph characteristic study researcher method method study characteristic class graph coefficient base graph represent weight weight ia ib respectively usefulness use graph negative factor type graph method degree way similar note pure description benchmark experiment binary people cs generate equal document categorie window categorization non link dataset page link page follow construct relational specifically weight knn relational purely derive comprise computer paper relational relationship seven class neural etc convert seven refer movie movie office versus movie share production company edge production two dataset available next dataset comment dataset percentage forming realization one labeling experimental give table vast performance good intermediate ir mixed set dataset small original graph coefficient good range increase move lowest extent positive variation occur somewhat away attribute fact toward fall similar conduct pair significance compare ir dataset significance graph intermediate subsequently graph ir use identify beyond scope occur graph demonstrate application next dotted black performance ir ir cv ir ir cv cv ir ir propose signature page http com refer http www com refer graph edge page signature match range put score signature accurate consider binary first page rest page table since varied evaluate auc ir method graph set cv technique quality grid interval cross validation performance inferior graph see performance instance improvement significant cross validation choice good performance slightly inferior estimate since auc around graph method restrict base perform ir ir cv obviously three competitive tune section computational et al ir fast time also provide competitive mixed get improve al
vice likelihood regard nuisance function regularity obtain property smoothing estimate curve q k I predictor I hand profile log need ii k km k obtain newton involve four ik tm repeat convenient ii alternative see chapter al category specific incorporate account order would need vary application vary increase curse dimensionality structure modelling computationally could profile context integration could h efficient fitting parametric incorporate gender region age flexible model fit irrelevant alternative parametric odd effect gender region age party support surprising significantly relatively category east substantially raise lp party strong represent decrease support support irrelevant conservative decide amongst within argument party obvious perhaps past g computationally implement context nest adequate framework category flexible case value confirm problematic political party say party enter type datum additive wrong conclusion average derive likewise extent peak lp low implies type potentially probability support low increase find level level support group party work across high drop away class confirm party mostly student constitute strong prominent role great background involve student anti nuclear movement surface interpret exponentially display party lp versus thereby scale difference clearly origin drive factor lp recognize group small party notable support lp old typically involve parametric section age function reveal monotonic led number covariate political party lead comprehensive profile model inherent assumption relation response explanatory overcome modelling model separability sophisticated modelling additive modelling design arguably interpretability age model misspecification extend discuss response mathematical property consistency efficiency estimator remain parametric profile logit give detailed subsample particularly case nonparametric approach due precede political composition interaction age highly shape party likelihood notably smoothed field keyword multiple investigate influence possibly categorical response behaviour demand provide mean factor support political great policy designing useful outcome opinion fisher et however conventional adequate complex pattern profile manuscript political party usefulness basic parametric thereby demonstrate superiority identify profile gender nonlinearity covariate interaction different model multidimensional explore thus usefulness flexible explanatory little utility recently nonparametric propose special generalize qualitative cf respect modify bayesian approach manuscript kernel specifically profile estimation spirit extend inter model multinomial response motivate introduce formulate kernel procedure fitting parametric party section benchmark result conclude remark motivate political party aim dominant political multi party collection interpretations broad party historical background party comprise political party form conservative party party party lp green party five consideration govern diverse group green local green nuclear movement movement movement form east unity party although system call party social throughout manuscript extract economic variable political party party commonly literature economic log e logarithm region origin gender region origin whether person
study n array investigate channel array difference baseline interpretable practice describe agglomerative approximation search ht implementation crucial interact gradually merge fig dependent reveal response improve however increase tends increase overfitte sample effect information complexity log size gene merge minimize l q agglomerative initialize univariate potential pair direct singleton merge network newly terminate continue merge distinct mixture neighboring demand task gaussian process variational type co mixture activate condition overall correlation gene profile fluctuation covariance component however heavily complexity leave difference response generative mixture maximum limited gene adapt gene sample per previously use pathway pathway base pathway www implement knowledge pathway tool consider pathway provide pathway set format interaction consider remove normal post gene measurement degradation minimize pass include biological replicate platform finding investigate human expression platform biological replicate available root temporal multiple definition probe genome would provide expression plus array identical compare set technical rely annotation alternative use available probe com validate pathway interaction network sample body compare approach wide response genome interaction network response step response step detect gene expression find connected subgraphs subgraph gene response detect list gene algorithm randomly keep association original assess influence prior gene include finding investigate significance detail detect response expect difference group datum test condition response qualitatively similar pearson detect characterize centroid response characterize response within gene datum consider validation corresponding predict response supplementary report readily applicable modeling wide implication finding investigation coherent response highlight functional outperform comparison finding identify supplementary median distinct response supplementary one distinct significant difference observed validation response qualitatively similar supplementary central fourth level group pathway alternative response may share mechanism may reflect reflect overlap pathway start pathway remarkably pathway pathway association provide supplementary file general signal system network cell growth could six pathway pathway interact pathway diverse vary detection detect expression array individual visualization mean baseline ht selective activation association response response grouping condition accord occurrence expression share suggest highlight functional connectivity share differentially gene response co expect individual overall connectivity signature co occurrence reveal investigation reveal supplementary detect fraction differential median gene response compare alternative detect amount response coherence association mutual information class detect statistically association condition average label supplementary high compare confirm detect describe obtain median detect significantly comparison detect response seem however finding biological gene affect analysis could specialize activate share functional help formulate context define support validity detect large response finding highly establish finding response biological network include pathway protein connect characterized process provide readily interpretable biased study bring prior information increase however collection pathway hundred pathway affect predefine demonstrate account aspect pathway pathway pathway predefine module association multiple pathway description consist module procedure rigorously identify coherent module interact gene response increase account measurement principled interaction factor bind protein base pathway database interaction limitation connect principled collection availability contain valuable condition specific experiment enhance directly exploratory task provide disease genome scale key comparison systematic investigation response unknown motivation subset process model focus interpretability since set potentially disease differential network subset patient many collection update extension datum topic another treatment interval remove criterion response without significance majority response verify datum biological sensitivity could seek constraint impose related cluster agglomerative model base selection connection feature global optimum model exhaustive combinatorial network problematic solution link genome interaction solution achieve compare interaction power agglomerative pair find merge affect considerably mixture run application computer ghz supplementary investigation interaction network group interact gene differ highlight gene measure activation characterize wide quantitative validate identification characterization scale diverse wide cell activation potentially help novel hypothesis gene previously context european research institute school science information technology department computer science box version manuscript biological process interaction associate response reveal unique share yet novel interaction method search local know interaction gene guide human pathway response functional condition context gene availability available contact molecular induce change level produce huge body cell public gene pathway model protein context activate reflect expression although indirect wide availability provide investigate detection response
guarantee worst running provide generalization inspire concept game theory game contribution technical contribution behavioral formally identifiability game concept trivial game game bind analogous vc dimension number equilibria datum game equilibria baseline approximation completeness search game well player show convex produce game player brief delay compare without reader aspect constitute attempt visualization similarity previous model behavioral game community ai nf nf n n n n n nf ce discussion graphical game form game payoff learn equilibria pure nash equilibria ce equilibria equilibria number agent relatively payoff learn game provide time learn game extract temporal dynamic branch thus vary probabilistic vs achieve utility exception assume action noisy set differ focus past behavior model method payoff agent reach vote vote availability lead use conditioning neighbor also validation player researcher payoff explicitly potential payoff e pure degree reader payoff determine assume joint observable non game furthermore validation game graphical datum mention direct cyclic parameter maximum describe although game direct cyclic structure factorization player probabilistic differ nature approach well suit mostly consequence keep mind competition graphical except behavior aim seek ai community game variety include identification popular social computer community excellent area social discuss depth discuss self study diffusion influential individual game type read game either behavioral start article couple year estimate influence consider lead influence g take theoretic approach static game approach strictly theoretic within static game framework make assumption choice assume conclusion player game see g e player game theory rational act independently subject irrespective core concept x prescribe equilibrium notation game theoretic model form direct player arcs payoff player set parent neighbor context graphical game payoff influence weight threshold f ia player define quantify player play graphical main contrast perhaps still influence modern view network compact uncertainty analogous intuitive description interpretation structure game become paper influence influence voting game people present influence unlikely ground influence u ever respect encoding game joint capture property really certainly direct influence type learn depict type proper rigorous scientific structure applicable shot generalize generalization require effort simplicity reference section discuss main later misclassification likelihood result causal take strongly indeed game lowest differently problem priori one find equilibrium would context game maximize low behavioral play cognitive social principle ml tradeoff complexity performance apply joint choose uniformly uniformly complement mixture stable outcome game formally pmf pmf need enforce also enforce mixture game completely mixture induce pmf action identical drop context parameter q q game equilibrium greater trivial game mixture p valid tuple q part definition depend prove contradiction q q contradiction definition game call refer tuple hypothesis consist tuple identifiable similarly game game brevity trivial pmf game hypothesis space trivial identifiable proposition completeness trivial trivial identifiable game identify discuss nash equilibria response equilibrium discuss realistic mathematically tractable bit concept human behavior social book behavioral address nash reason understand illustrated chapter propose nash behavioral theory superior setting payoff g logit interpret temperature available payoff propose payoff indeed technique hard graphical game mle payoff unclear tractable apply sophisticated extension scope paper unclear logistic regression logit variant behavioral model account cognitive reasoning effort usage assume payoff fair human experiment behavioral theory scalability game unclear view vote individual hope take one study concentrate vote reason scientific process sometimes alone concentrate publicly hold decision voting record decision case record information pre final work within model population justification make unclear gain randomization strategy introduce mixed pure realization general game measurable process complex expression thus appendix furth equilibrium could easily natural expense produce generative less amenable form see know priori action equilibria pmf bernoulli parameterize derive equilibria mle probabilistic definition q n prove mixture property kl remark proposition case induce pmf note low non game set mixture give furthermore trivial identifiable game give infer state payoff induce generative exclusive different property joint note game mixed equilibrium state previously work concept equivalent mle game reflect generative hence select equivalent seek attempt create model objective unknown assume truth learn I nature essentially translate ground game illustrate several parameter ability know selection core ml choose ml learn precisely elegant multiple invoke generalization practice pac support standard measure via invoke toward well know exactly prefer reason exploration interpretability everything else equal explanatory measure likelihood preliminary work algorithm heuristic operate among alone would prefer former show generalization maximum upper vc maximum achievable generative mixture e last technical include vc allow expectation denote game game maximum probability least objective b elaborate result clarity presentation generalization number player logarithm pure equilibrium game bound network layer unit input function unit term hide term weight quadratic respectively term hide weight define neural equilibrium oppose output instead dimension mixture section equilibria baseline exhaustive us negative show nash complete general learn player probabilistic loose allow obtain tight bound expense hinge primal program program ij ij regular derivation add slack duality logistic I behind decrease choose latter logistic show bounding generate strictly upper bounding ii generate claim iii logistic identity e strict show exponential function positive prove claim suffice set I simultaneous minimization become bfgs expand onto negative enforce justify minimization structure game absolutely proportion equilibria assume player large connectivity analyze structure present utility restrict assume enyi connectivity define player produce player proportion equilibrium player concentrate ingredient proportion equilibria simultaneous logistic game simultaneous svms degenerate case x lc c l l subdifferential simplify I I svms simplify j e degenerate solution j l c l smooth lc x claim termination b point careful reader subdifferential vanish proof still concentrate ingredient equilibrium show player interestingly ingredient bounding proportion equilibria main absolutely I x condition normalization axiom I b nx I I hypercube cover hyperplane recall note hyperplane zero half important surface I except hence state probability lebesgue still bind depend norm instance arbitrary covariance additionally requirement uniform subset entry allow graph equilibrium absolutely absolutely random draw I I I equilibria furthermore statement let I n f cc use hoeffde simultaneous svm additionally exponential exhaustive baseline proportion equilibrium sigmoid plot ise green class drawing probability component view nature arc every node truth p threshold generative move nature purely data validation respectively respective regularization ise model use describe repetition kl respective theoretic ise considerably purely probabilistic across nature seem learn model purely kl value much purely ise end bad relative probabilistic vs outcome summarize figure help still learn show superiority game ise player gradient ise np use propagation statistically latter value select value report leibl precision minus equilibrium minus exclude equilibrium closeness recover model truth world report synthetic equilibria proportion equilibrium significant avoid bar clarity marker simultaneous svm il sl simultaneous regression kl search equilibria model equilibria equilibria equilibrium equilibrium resemble truth game exhaustive method ground nash equilibria accord set mixture truth repetition lower exhaustive proportion equilibrium ise exhaustive exact likelihood suffer consequently convex simultaneous equilibria equilibria grind truth recall additionally proportion equilibrium resemble mixture ground slightly second synthetic interaction nash equilibrium accord repetition ise logistic low loss equilibrium equilibrium additionally equilibria parameter experiment minimization improve increase slightly repetition real almost move interesting arise concentrate decision course please understand leave diversity influence induce interesting nash equilibria magnitude sign b truth validation convex outperform maximum model synthetic proportion equilibrium remarkably well equilibria ne low kl experiment two approximation remove true equilibria maximum equilibria loss loss player contain repetition require bias number convex outperform maximum proportion equilibrium observe high density obtain low inspection ground truth usually seem equilibrium maximum computationally feasible game learn leave th row influence learn weight value top corner bottom right corner produce c directly influential least regularization voting loss publicly vote vote vote researcher g vote equilibrium np sampling randomly split repetition turn train maximum proportion equilibrium convex simultaneous logistic low kl equilibria equilibria learn game counterpart normalize influential high respectively influential th list aggregate party strong party structure game vote cast interestingly decrease voting adequate argue log action action action single agent work regard game produce order address identification influential refer behavioral individual conditional inherently without produce regression condition heuristic please area manner learn study plus get technical need common player utility despite similar estimate generalization standard also generative theoretic equilibrium well evidence support theoretic long extensive benefit find contribution successfully graphical model game theoretic model research payoff utility assume player play simply condition extend broad class boolean action binary feature player player non version still extension analysis linearity additionally new vc extend parameter regularizer future sophisticated process upper bound hinge account variation influence voting difference term influence bias topic preference grateful improve discussion topic behind suggestion improve presentation motivating causal comment question wider grateful share master thesis reference thank anonymous anonymous reference overview composite likelihood appear ph nsf award end dynamic discuss manuscript certainly prediction knowledge publicly vote vote seem sensible modeling perspective nature vote detail indeed go availability process considerable burden something world motivation view temporal provide wrong often inherently express separate differently treat individual example invoke page intermediate need general temporal dynamic model poor really care stand interest recognize significance social behavioral science economic high abstraction going decision reach growth physical international shift engineering entity goal course computational reasonably end state model extension mixture player probability action parameterize otherwise complement parameterized pmf behaviors q datum likelihood keep constant step hard keep constant combinatorial tractable provable distribution maximize change generative parameterize player action probability pmf behaviors q pmf alternate method step change jensen nature likelihood step yet another changing pmf hard formally surely pick computer university ny usa behavioral class parametric game stable influential social task also cast estimation goodness complexity capture equilibrium control number equilibrium unobserve generalization mle probability world voting record briefly applicability game modeling ai formally study behavior book core solution serve outcome system self interact setting role infer causal outcome therein say computation significant computational game community ai compact game decade game large encounter game graphical ai player determine player neighbor neighboring decade constitute arguably theory considerable progress ne see therein play prominent role ne games example non game theoretic motivate current work influence identification
auc gene intensitie roc foundation european community agreement review manuscript interest u bottom fully scalable fully spike presentation score value full material method abundance solid dash dotted line u classify cell exclude quantify forest validation mm pt title short microarray running department laboratory university european bioinformatics genome uk department material engineering de es mathematics centre european bioinformatics trust european bioinformatics genome uk department box email keyword online robust probabilistic average accumulation standardize collection function scalability current form bottleneck microarray collection constitute major source genome wide scalable measurement calculate training overcome limitation scalable online large microarray include thousand array alternative readily applicable preprocesse probe update batch novel take full microarray collection moreover collection probe effect affect various provide tool control available http www package accumulation house public create microarray standardize overcome inherent bias analyses measurement hundred genome disease size portion microarray collection gene million array database able combine analyse array challenge preprocesse central multi array combine array probe effect performance multi array technique limited requirement million probe bottleneck comprehensive meta collection variant standard preprocessing develop tackle rely probe term restrict microarray applicability microarray probe platform platform independent extract probe collection scalable introduce scalable probe online limitation extend averaging framework estimate online hyperparameter allow rigorous microarray ordinary consecutive batch requirement respect probe collection readily short microarray moreover analysis probe performance collection probe platform provide collection step preprocesse normalization probe probe array preprocesse applicability collection due memory reference extract complete requirement time hyperparameter appropriate microarray quality background probe apply collection step store disk preprocessing step operate quantile basis estimation quantile normalization average probe implement scalable quantile calculate distribution batch jointly normalizing array use parallel calculated averaging hyperparameter approach introduce online consecutive probe hyperparameter update correct normalise transform finally batch initialize equal prior probe batch provide batch final probe level batch ideally expect yield batch probe last probe summarize yield final summarize probe model normalize level gaussian probe probe gaussian affinity ij model additional assumption probe affinity estimate variance analysis randomly affinity posterior give version develop validate full advantage form update prior prior probe hyperparameter posterior hyperparameter interest estimate probe nuisance analytical available marginalization slow point fast approximation iterative optimization newton sample size follow gamma yield readily give equal j information update value final retrieve update hyperparameter next give hyperparameter complete yield probe probe probe probe issue microarray preprocesse assumption formulate linearly hour preprocesse file four processor core preprocesse classify traditional preprocessing multi array preprocesse perform array combine probe accuracy array incorporate preprocesse scalable preprocesse large scale microarray collection standardize database probe array probe however applicability currently limit variant additional incorporate utilize batch available heterogeneous microarray collection requirement procedure contrast fully readily preprocesse reference http edu microarray preprocessing spike quantify focus website design moderately sized scalable however applicable spike probe microarray comparison probe method field array single figure summarize complete available outperform u u set outperform bias fig false difference salient test interestingly although outperform fail spike outperform scalability scope summary preprocesse standard wide scalable collection type probe set tree split exclude comparison outperform pair batch outperform considerably limited scope batch array contain multiple test summary technical data pearson share package correlation batch pair supplementary requirement array applicability sized data probe preprocesse probe reference form bottleneck meta analyse microarray collection take advantage nearly million array array key introduce online individual probe effect large collection scalability limitation current preprocesse sequential hyperparameter batch microarray collection extend probabilistic alternative fully readily probe term hour run file optimize efficient processor notice yield result additionally probe affinity omit speed computation preprocesse analogy allow advantage method preprocesse array provide array diagnostic purpose suggest probe rna degradation snp annotation modeling yield source probe remain poorly model probe scalable comprehensive collection short array bias tool microarray probe single model variance recent probe comparable outperform utilize outperform model microarray collection primary scalable algorithms effect separate scope finally applicability algorithm currently limited plus spike microarray outperform scalability online currently fully scalable preprocessing applicable
strong organization ice event patient stay wind bin california area gene black fix produce moderate support alternative plausible stay city moderate conduct fix yield exponential cutoff law cutoff spatial ultimately distribution surface invariance truncate exhibit large systematic indicate alternative likelihood nest cutoff give ratio support law indicate probably law power moderate power law fit alternative remain plausible alternative complex often suggestion argue identify quantify straight doubly logarithmic plot straight law furthermore problem test extend researcher reliably investigate law hypothesis collect lose properly claim extend objective characterize generate question power whether follow pose largely scientific goal law fundamentally tail law part challenge face make phenomena law interesting reach modern hope statistical present aid contribution share available http www support institute phenomena law accurate law complicate fluctuation empirical tail lose adapt statistically include hypothesis kolmogorov goodness fit compare effectiveness quantify datum heavy tailed law attract broad consequence appearance phenomenon span physics economics sciences quantity law say exhibit invariance empirical presence underlie process network know law generative mechanism facilitate statistical event branch specie unit diameter rise two diameter diameter forest ad leave know test similar tree forest scale consider another interest intensity wind speed knot impact modeling frequency process datum rely upper frequency reliably tail follow existence upper fluctuation information lose convert count tail unable distinguish power heavy tail normal present principled answer statistically power typical u census city united york times extensive law find review obey draw pattern hold law researcher critical start detail statistically law hypothesis discrete goodness fit characterize fit plausibility range exhaustive principled consider law rather narrow adapting popular quantify impact impractical impossible histogram measurement receive due inference tool currently present power testing plausibility compare make assumption logarithmic bin synthetic accurate size amount lose statistical investigate hypothesis bin fit law consider plausible hypothesis power tail alternative model replace principled minimum description length real world exhibit tailed power finally highlight continuous paper material derivation indicate law constant function formulae simple distribution analysis case method count original discard retain range bin count let bins boundary bin convention bin count fall bin density subsequently external source otherwise raw exponent small bin study law first fit imply straight doubly straight seem approach parameter case tend sampling fluctuation pareto th early th researcher na I generate give mistake see fitting law generate na I estimate scaling parameter choice assume provably estimate maximum multinomial sample material section focus elsewhere symbol symbol power sample scheme space must numerically maximize em complicated section bin boundary successive analytic obtain bin hill estimator associate positively bias material role equation tail explore detail demonstrate experiment draw practical priori truly power choose fit model section sec alternatives law variety ten power result bin technique produce ordinary complementary regression bin operate doubly yield significantly value dramatically especially poor linearly f tail count induce significant ordinary complementary call frequency tail f complementary c bar sometimes dramatically accurate tail equal hill mle robust square regression mle quantity tail body aim simple value power discard fit uncertainty prefer conservative avoid maximum always use choose inspection log datum avoid method choose distributional synthetic principled none accept choose high reason require goodness fit options example pearson practice like find ks superior fit bin ks empirical bin reach law ks law poor ks distance high effect illustrate method methodology rt index minimize bin boundary idea sample minimum sample smoothing parameter size law logarithmic variety slope slowly numerical size logarithmic one threshold across second bin boundary characterize vary bin versus estimate bin true figure scheme dash show reference reliably slightly ks moderately rt accurately slight deviation law bias versus logarithmic target bin induce substantial second true slight rt rt work large reduce slight method logarithmic scheme kind number bin tail bins bias power vary sample bin decrease implication researcher must around method generate draw bin bin c logarithmic absolute bias decrease fit say constant perspective theory use asymptotically start dominate visually hill beyond maxima yield ks hypothesis power dominate correctly nontrivial makes inherently fail power law systematic deviation range power follow law quantity sample model imply mechanism law imply care empirical law actually measure describe allow model plausible generating observe bin wide normal power histogram axis despite one law power hypothesis sample dash law notably determine reject end plausible plausible alternative answer likelihood additionally impact power test count count would know plausible give goodness quantitative question turn represent likelihood deviation close attribute fluctuation reject view reject consider model law step law choose remain case empirical statistic bootstrap law distribution call law times fraction count generate law increment bin increment bin proportional count repeat desire ks statistic fit correctly original datum subsequent precisely estimate instead yield bias synthetic answer datum accuracy know true value know generate decide reject relatively conservative power law threshold avoid fact chance power value correctness arise well law test necessary cover number bin shape yield underlie happen goodness hard rule little bin fit analyze tail require frequency statistical power ks effectiveness goodness fit test various sized synthetic log alternative strong wide size bin reasonably law plot size law draw model expect rejection reject note however rejection power law require coarse provide follow law tail produce datum even demonstrate bad argument several compare law validation bayesian construct loss reduces interpret far commonly alternative theoretical mechanism effort application expert constitute reasonable normal power cutoff table bin likelihood pt cutoff pair logarithm natural make distribution indistinguishable event subject fluctuation direction unless require log chance log use fit fit bin law small say sign sign reliable favor technical likelihood supplementary material hypothesis answer datum sign without regard power power cutoff true yield must material draw law normal use boundary power dash performance hypothesis log range general hypothesis favor hypothesis difficult datum also power imply generative implication illustrate second log bin sample compare likelihood supplementary compare law well law law correct figure grow scheme interestingly reliably decision favor law illustrate alternative size already fit power hypothesis supplementary material step scheme quantify scheme power law hypothesis illustrate information pose question scheme logarithmic power achieve limit equal fisher scheme approximate equality supplementary material vary scheme fix show scheme estimation arise variation span source bin arise asymptotic agreement come dashed line log datum power axis make correct step framework axis sufficiently nearly eight option experimental phase fine grain collect maximize accuracy hypothesis log law plausibility computing law log normal reject law experiment scheme e log
follow likely occur likely likely occurrence event likely occurrence fundamental reasoning belief law probable limited knowledge say imply logic binary therefore knowledge possible universal applicability belief cox sentence consider strictly however coherent reasoning identify value act abc formalism determine complement happen form abstract naturally calculus measurable show partial belief represent show order course term proposition reject sentence accept necessarily sentence accept value belief notice value great value satisfy requirement aforementioned value dimension remark bayesian include abc formalism objective belief subset hence fill gap know minimal readily c case knowledge hypothesis experiment namely evidence favor value whenever decision derive loss issue future work minimal feature apply hypothesis note variable value differ drastically maximum likelihood estimate therefore logical reasoning proportional nan hypothesis sx take say induce compare difference value extensive review refer oppose posterior distribution requirement conclude reflect probability long posterior zero sharp directly latter section procedure produce readily classical alternative want specify computed compute respective computational principle main approach likelihood principle violate nan specify propose otherwise interpret aforementione treat undesirable nest hypothesis nest give evidence principle actual great report word distribution among many thing undesirable aspect evaluate critical value decision decision possibility choose critical threshold vary correct critical internal undesirable value incorporate scientific elaborate discuss know statistical scientific significance reader employ however internal undesirable certainly bring problem implement without logical open value next deal article rigorous mathematical treatment likelihood concave compare confidence region carlo require e actual criterion decision acceptance replacement value measure acknowledgement I acknowledge wish suggestion write manuscript attention report motivate student think contribution statistic anonymous helpful suggestion lead improved paper definition conjecture mat email corresponding value regard value scientific ratio measure evidence logical requirement meet e measure study abstract belief calculus formalism use establish end short problem abstract calculus measure base nested science discuss limitation well alternative parameter value alternative propose objective evidence connection calculus status proposal specific observed capital letter denote frequentist paradigm test cox least statistical common informally define small probabilistic specify common set advance threshold reject interpretation feed drive think statistic point serious logical argue highly nan conclusion arise regard implication frequentist detail parameter lie interpretation explain basically attempt possible properly understand statistic avoid decide family p measure restrict namely value family p p happen asymptotically probability way topology complement pearson significance value ratio testing account nan hypothese consider side computed likelihood reader asymptotic mean reason health g genetic study one frequency two know homogeneity frequency homogeneity mild logical requirement claim evidence widely hypothesis nest within logical reasoning datum word compute respectively dimension hypothesis compute metric may happen may evidence reject limiting happen exact consideration identically random identity example suppose ratio statistic hypothesis sample size show evidence ratio value statistic logical contradiction frequentist job define scientific take decision problem conclusion value decision hypothesis present produce surprising conclusion form regression usual non say generalize section plausibility therefore regard plausibility fail hold value short value final great closure inside way least point closure gradually confidence dot dash dash confidence general fashion measure relation region hand specifie specify test confidence readily compose see procedure consistent hope draw community conduct interval fuzzy compatible recommend author confidence level fuzzy whole interval membership goal paper author naturally depend consider connect formulation quantify yield nan prove essential evidence consider metric asymptotic statement confidence region interval level law weight evidence hypothesis definition although paper point relate monotone transformation transformation work version reader extensive review method interior hold theorem hypothesis fix evidence invariance confidence assume prove show ab plus concave increase alternatively ss basically strictly connect proposal author value boundary confidence notice converse typically approximated degree strictly upon
linearly study es function linear es increase invariant basis strictly l situation parent far optimum optimum undesirable situation handle increase search algorithm handle sa es es characteristic section study section variance result search normal zero standard multivariate zero ni tt selection child select f define adapt size path multiply approach easy front distribute path length length update decrease short determine simplification update consider rule analyse throughout linear determined distribute es vector generally non linear path alone would es geometrically log step size change formula immediately sum case constant prove es geometrically recognize thank quantity number investigate es e equal prove lemma number I surely deduce integrable show strong law obtain q sign determine infinity independent see idea normal invariant g gx dx recurrence recurrence vanish three converge size es surely expectation prove sure convergence satisfie expectation c analyse conclusion get remove use development formula equal separate side go c right summing expectation rhs positive increase geometric divergence step geometric mean affine geometrically geometric geometric divergence dimension pressure double future show hand increase increment suggest likely equal furthermore c equal I development prevent ex curve quantile set precisely quantile median figure evolution run variation deviation plot curve bottom deviation divide relative must agreement constant three careful variance relative critical make deviation investigate es affine compose rigorously prove behaviour contrast sa increment deviation increment go mean signal ratio go give stability confirm partially grant research mm cumulative adaptation adapted measure call path
bind noiseless nuclear nuclear spectrum net since parallel sum penalty penalty provide meanwhile roughly correct call calibrate net develop replace scale soft decomposition result level smaller calibrate net perform outperform method calibrate bind use sample extra appear bound appearance sample seem paper describe spectrum net convergence derive calibrate frobenius mapping letter op ie step rank minimization thresholding solve matrix tie need complete equivalent n lead e compute convergence result proof iteration may em calibrate e nonzero projection uv p net proof proper lie proper still infinity calibrate modify provide target imply support consider matrix ratio normalize suppose coherence hold let satisfying imply rd constant noise frobenius magnitude simulation use element equivalently existence spectrum net penalty n c remove recovery noiseless sufficient rd right shall independence requirement aspect symmetry set theorem replace root size requirement remove factor requirement root aspect ratio square block satisfy coherence provide random matrix iid projection noise htbp calibrate lasso modification estimator level penalty solution correspond solution calibrate spectrum always ny training error rank f report three level proportion result average replication net spectrum modify calibrate spectrum net spectrum true spectrum lasso estimator analytic unclear control approximation taylor expansion nuclear control tangent spectrum e omit uv tm p uv uv uv spectrum member differential h uv inequality n uv p uv uv h h p p inequality singular error follow derivation find uv uv uv p f acknowledgment research grant dms dms dms grant slightly provide l converge iii lemma remark department statistic paper concern calibrate frobenius penalty incomplete current guess scale soft thresholding singular result converge cause frobenius coherence condition penalty nearly order unified analysis noisy noiseless completion proposal
rs snp associate significantly manuscript analysis central genetic approach phenotype binary phenotype account serial longitudinal estimation novel center efficiently phenotype selection bayes simulation demonstrate utility application genome wide phenotype relate diabetes keyword carlo sampling occur genetic influence phenotype many genetic study trait diabetes disease study disease common phenotype conceptual phenotype disease phenotype response mutually correlate trait take increase jointly many collection type diabetes exploration sequencing sample longitudinal measure phenotype body mass american diabetes trial longitudinal diabetes longitudinal combine longitudinal study study phenotype measure genetic factor trait phenotype longitudinal family trivial correlation phenotype repeat phenotype serial complexity phenotype discrete b joint effect phenotype difficult serial factor apply conduct quantitative trait equation methodology study correlate phenotype formulation latent rely observed phenotype latent conceptual disease status widely field economic become increasingly attractive genetic trait sub common genetic trait study outcome covariate longitudinal consist part relationship characterize genetic marker account serial direct part longitudinal continuous desirable genetic raise challenge sample inefficient algorithm center expansion computational organize section discuss consequence na ignore structure section mcmc design phenotype selection selection bayesian extensive simulation study apply method genetic section conclude recommendation discussion outcome phenotype measure individual individual repeat measurement outcome u represent conceptual phenotype trait via mixed q covariate direct effect direct effect factor loading phenotype reduce capture mutually binary mixed link probit logit gain known representation probit response recover dimensional marker possibly clinical latent mixed indirect covariate effect phenotype interest effect model genetic significant specific represent subject ci modification nonzero model phenotype avoid assume covariate part individual status practice analytic burden assume independence exist ignore cause incorrect phenotype response individual index sample decompose j ci genetic marker phenotype first simulation longitudinal equation ignore conclusion longitudinal covariate efficiency popular rely modern computational markov dependent paradigm offer principle incorporate genetic contain outcome direct covariate indirect interest algorithm statistical unfortunately mix along sample chain next specification modification da sampling relatively phenotype extend phenotype phenotype gibbs sg due vector mixing improvement center hc hc move move hierarchy overcome px px auxiliary couple load update likely update vice versa hc ad ci original expand parameter transformation gibbs sampler induce distribution hc apply since assign prior prior analysis instance add flexibility improve behaviour suggest use prior loading set conjugate expand model parameter n specify prior hyperparameter degree freedom df prior respectively phenotype selection specify spike hyperparameter notice induce spike spike conjugate expand apply gibbs obtain appendix step involve involve suppose trait first phenotype one concern mcmc mix continuous specific gibb expand notice strong apply layer da scheme leave prior parameterization unchanged call doubly expand center hc summarize gibbs j conditional k j remain continuous part draw illustrate autocorrelation expansion medical interest determine phenotype study truly status loading phenotype phenotype loading conservative assess loading spike prior spike loading prior belief recommend correspond phenotype priori latent small determination phenotype posterior discuss introduce approximated mcmc threshold loading include practical discovery fdr phenotype factor central assume two factor support support calculation challenge dimensional arithmetic path specifically unnormalized via calculate identity expectation density py g py choose estimate grid calculate context bayes factor different remain concerned interested factor loading phenotype remain unchanged binary phenotype obtain calculate show bayes df bayes df another grid size approximation suggest grid big grid due distribution close suggest small grid observe good important inferential genetic study marker covariate component clinical marker interest two compete link calculate practically study method ignore method introduce detail simulation model phenotype specification family allele families segregation serial ar autocorrelation phenotype five time unless specify phenotype run discard first burn five phenotype interest table replication study ac hc px da scheme propose reduce increase efficiency improvement evident standard illustrate square error rmse result rmse response continuous phenotype discrepancy factor however phenotype serial estimate correlation incorrect serial correlation cause quantifie phenotype variable clinical covariate spike prior four phenotype binary phenotype choose phenotype strong association phenotype replicate phenotype disease status replication phenotype moderately associate phenotype specifically consider also phenotype via df phenotype truly bayes correct phenotype truly associate factor criterion correctly nan estimate criterion little identify weakly inferior explore selection purpose indirect genetic assume marker one standardize continuous covariate standardized effect part phenotype compare assume association genetic marker criterion covariate respectively covariate estimate consistently
depend dimension estimate word word may require sample situation near say mass contain frequent translate guarantee svd svd procedure simplified take v u provide fast accurate outline perturbation detail detail sample obtain whiten understand complexity dimension rather depend let focus analysis issue x w column deterministic perhaps surprising unit unit unit notable error skewness consideration dimensional notion sparsity decrease often burden explicit sparsity helpful skewness whiten complexity model skewness tend surprisingly beyond requirement read david zhang thank insight preliminary claim lemma section condition definition example topic model generalization cluster observation latent document increase power come cost challenging unsupervised distribution parameter wide probability contain term fourth via value scalable operation topic rather agreement powerful incorporate broad interest observe occurrence word document document sparse determine occurrence dirichlet dirichlet allocation permit multiple correspond hidden finding estimate either method maximization variational factorization modeling count document specifie topic specify word topic match observe efficiently observed decomposition low moment fourth exchangeability availability hide excess knowledge exchangeable principal pca two decomposition first data svd utilize high order find direction exhibit moment perform latent factor scalable applicable wide model include exchangeable model exchangeable model poisson generate analogous decomposition base order recover third moment suffice document finally multi view draw independently available topic present discrete model recover simple eigenvector presentation focus utilize moment correctness plug plug section matrix efficacy appendix topic document provide guarantee albeit topic essentially overlap condition separation separation condition learn statistically document topic separation utilize multiple document provide provable certain condition separation utilize anchor topic provably use negative procedure problem motivation limited provide result progress algorithm take certain technique know setting idea utilize evolution multinomial extend mixture single key ability multinomial factor method quite general hmms body algebraic problem blind approach decomposition see tailor source separation usually much without algebraic utilize idea column noiseless algorithm insight independently hide exhibit rather solution suffice estimation simplify eigenvector long necessary approach hmms approach exchangeability permit rather additive key technical contribution dirichlet careful gap topic approach exploit work separate variant canonical correlation permit procedure guarantee recovery h k hide th l first analysis four mild identifiability goal particular assume rest example gaussian consider component somewhat classic mixture gaussian permit responsible generate state component responsible impose noise linearity count sequence well understand separation without exchangeable independent factor case skew consider dirichlet present exact decomposition svd high third algorithm extension topic factor model throughout pseudo column allow square moment matrix leave singular q return pairwise project permutation identifiable column form transformation sign central third moment skewness skewness false positive return subset sample sphere consequence canonical assumption uk whiten possible w uniquely determine singular every singular nonzero reconstruction w entry standard positive skewness direction skewness practical freedom multiple find singular easy determine skewness straight estimate skewness suppose singular step skewness singular matrix find matrix u v set singular reconstruct eq subspace fourth moment tensor central fourth denote excess hope order factor subset uniformly sample sphere return canonical skewed note incorrectly skewness provide distinction remainder argument product simplex modification parameter manner prior denote modify second modify dirichlet equal rest extreme central eq expand exchangeability x moment find remark find identity set svd leave reconstruct normalize set pseudo show recover permutation column restrictive tuning also latent allocation positive return column random vector sphere return vector consequence rescale variance normalize algorithm finally claim hold functional form limit skewness eigenvector base show skewness appear skewness modify implication note effective skewness skew lda conjecture fourth moment utilize alternative model dirichlet application natural come different intensity prior
multivariate total identical mutual multivariate extension information information mutual combinatorial conditional mutual information upon mutual information eq similarly entropy mutual mutual information follow introduce lattice integrate statistical introduce notation define set multivariate lattice expansion information joint special lattice variable special give tb derive three basic rule mutual identity view equation equation lemma joint conditional show three rule mutual inclusion evident multivariate hold mutual n h x mutual decompose hx equation chain apply repeatedly x n n entropy variable partial equation total decompose explicitly mutual information dual follow probabilistic variable entropy conditional redundant entropy mean equation derive rule decomposition decomposition conditional entropy residual nh h entropy mutual spatio temporal variable essential let multivariate condition zero otherwise lattice delta transfer come rate free variable come entropy follow prove go flow entropy come lattice difference non go flow go flow inequality minimum lattice depict bivariate information cube cube plane indicate entropy depict I k also depict colored outline outline circle code specifically transfer red outline transfer variable theorem less come sum negative information blue circle outline indicate go go transfer circle residual entropy entropy total correlation x visually node lattice reflect mutual respectively indicate green flow cm cm cm corollary support mathematical proof theorem flow stand lemma mutual information information lemma joint mutual lemma neither stationary depend past state conditional mathematical variant convention cover except
match consistent denoise matching arbitrarily expect want conclusion equivalence equivalent contraction parametrize per force correspond jacobian minus jacobian energy interpretation concentrate near score nearly density tell direction increase ridge direction direction decrease direction manifold return mind derivation particular ridge expect reconstruction move manifold near eigenvalue direction manifold besides derivative property immediate recover integral path stay approximate pick line use hasting reject acceptance compute train capacity process concentrate train exercise along require qx show hyperparameter explore work train learn draw model mind original fall region come would certainly maxima kind lead ratio property metropolis hasting embed plot pair read pair leave concern spurious auto encoder capacity end fail properly desire reconstruction towards density case work region close dot thing outside noise small enough example find right sketch illustrate situation properly reflect accurately stable manifold concentrated conceptually spurious assume case end completely involve corruption capture recover generate denoise form closely estimate property second derivative contradict reconstruction derivative hold simply virtue reconstruction reconstruction alternative maximum likelihood unsupervise need rbms boltzmann machine analogously score matching estimator density minimize confirm mcmc approximately sample metropolis hasting question since heavily notion notion already norm show trace rewrite motivate continuously define expression applicable point fix part leave make notation denote first term implicit lagrange calculus reader either component optimize separately expression px satisfied optimum become hypothesis vanish linear obtain eq substitute side get substitution situation try series like dr kx term show involve get eq eq could would need study could possibly local appeal precise denoise auto capture local function second derivative move justify procedure motivate moment auto encoder continuous function ball product ball pdf ball around stick ball everything rewrite formula first local base associated obtain estimate direction q dx px x dx x dx hx come come probability term transform anti odd x z look front reason px hx remainder main exact expression make number ball origin integer absence absolute put negative theorem corollary drop odd become integration unit get determinant jacobian let radius dot decomposable corollary let show yield integral ball understand integral rewrite h xy h observe multiply lot power term yield odd function power dx dy reasoning write u ji integral function previously unit ball continue conclude translate give match auto auto encoder good job capture manifold regularize reconstruction characterize auto capture derivative contradict interpretation theorem generic depend parametrization encoder encoder would train similar auto encoder training criterion corruption contraction similarly consider criterion maximum likelihood involve show sample confirm aspect many focus region point variable plausible representation discovery feature latent generate base reconstruction auto variant encoder representation argument empirical well train counterpart generate quality restrict unknown concentrated region density core manifold identify question concern error criterion implicitly whole aspect essence target formalize link answer establish implicit define statistic contribute proof link criterion rest contribution auto generate estimate increase correspond hessian second access approximate mcmc achieve hasting approximate auto auto encoder recover distribution compare project visualization order presentation differ order gap name review exposition capture training thank reconstruction map maps back reconstruction made view encoder regularize denoise regularization regularizer basically attempt constant possible away precision make neighbor high value otherwise possible correctly reconstruct derivative direction remain direction manifold case direction around direction auto choose ideally tangent direction encoder linear encoder capture manifold extreme want nearly special obtaining function flat near yield towards density point mm large identity obtain regularizer point towards peak encoder corruption loss actually auto encoder auto encoder encoder whose contraction encoder regularize minimize regularize reconstruction element use easy strongly term tie variation direction fig denoise stochastic source mostly noise corruption easy mathematically many proof serve train consider reconstruction result make reader understand study effect equation convolution involve neighbourhood view noise neighbourhood total quantity include weighted numerator simple additive try pre give equation strongly minimize expected thing expansion go good reconstruction something equation allow asymptotic go family distribution mass connection auto taylor encoder expect corruption x loss take expansion similar encoder contraction impose alone motivate loss instead plus penalty analytic auto encoder contraction denote penalty coefficient involve notion notation see alternative way behavior continuously differentiable differentiable twice minimize derivative expansion understand function expansion wise behavior use calculus allow recover score know nothing minimize score subsection practical loss draw alternatively auto encoder train online gradient update stream auto encoder minimize generate auto encoder auto minimize empirical loss available capacity minimization ideal drop information train sufficient model capable function work round numerically evaluate setup describe pick quantity purpose give example happen model minimize train fashion discretize score get arbitrarily illustrate score ex fit divide evenly discretized loss free thing discrete score htb visual expect predict extend manifold one visualize field parametric encoder numerically along denoise auto corruption noise encoder decoder optimize bfgs use corruption bfgs run bfgs convexity solution corruption also final outcome reasonably may undesirable training select visually outcome find along field towards density close play vector place reconstruction implicit high manifold role function regularize auto small example maxima place whenever path region median region median clearly visible
incur take action loss compare action rest present slot loss pick environment action library marginal additionally convert gain library store action action similarly train idea control little gain sensitive multi environment slot action fill evaluate formalize calculate slot carry formulation advantage control instead loss regressor slot estimate marginal computed regressor slot per feature benefit regressor produce pick target environment establish sensitive slot sequence distribution environment greedy allow algorithm cost sensitive fs nr hypothesis fs define cost sensitive classifier nn thm nm thm misclassification marginal x dependent multi classifier hypothesis minimize additive n budget control take execute make correspond pick additive sequence state fs regret class class square slot reduction action fs classifier reduction regressor fs class sensitive planning recent show hard avoid soft penalty lead trajectory suitable often seed straight environment suffer local provide seed chance minima seed contextual overhead evaluate library negligible rely plan arise research difficult circumstance initialization position distance field seed leave target seed library environment performance train regressor without seed note regressor subsequent regressor also include remain action difference initialization library order order sort regressor sort regressor maximize absolute benefit slot failure execution trajectory execution robot short seed rank performance fail free plus trajectory execution time long computation environment summarize straight unable free result execution single regressor choose primitive provide seed benefit pick trajectory seed fail sequence failure execution second compare initialization seed find environment avoid obstacle enable produce planning environment top slot failure rate fast trend bottom execution column straight heuristic planning mobile library feasible trajectory low traversal portion repeat trajectory place challenge challenge generate tx robot maximum trajectory sequence cluster center compute whole overhead run histogram use online slack sensitive slot report traversal sequence trajectory choose appearance step fall empty around generate trajectory sequence try avoid theorem yu liu university pa edu order library optimization static account within description goal efficient reduction order repeatedly classifier regressor approach formal sequence cost prediction contextual optimize control library mobile web ordering admit interpretation item interpretation real requirement relevance diversity member library context library refer rank choice action trajectory mobile optimizer dimensional trajectory control library collection trajectory mobile demonstrate trajectory robot al expert dynamically trajectorie store control span feasible library specify action library mobile robot task traversal path usually constrain motion model dynamic library robot library every traversal robot goal minimize trajectory sensitive seed bad initialization suboptimal even action trajectory act predict library feature fail initial seed early fail library feasible time find naive closure fail similar principled reduce diversity art either library ad action past success fail g unable back heuristic code plan predict robust action predict label number difficulty selection predictor within expensive explicitly multiple regressor environment performance guarantee selection module sensor system leverage classifier map action action contextual loop et outline good attempt submodular submodular et al contextual section planning contribution near move predict make arise domain search library demonstrate efficacy robot plan mobile robot generate sequence either rate success etc diversity library formally monotone submodular sequence sequence fs concatenation increase monotonicity addition sequence sub modularity library optimization attempt optimize cost well sequence cost environment high cost dependent environment evaluate et monotone
entropy panel version namely stepsize search weight vector value value plot difference plot plot conclusion converge maximum entropy goal align hope well goal ar discussion support european project city paris paris theorem convex optimization conditional minimizing moment discrepancy enable result optimization consider approximate integral reproduce space learn approximate algorithm entropy attractive method field estimate mrf entropy requires learn mrf answer query empirical follow recursion observation statistic empirical frequentist mrfs parameter update motivate perspective mrfs mrfs explore update effective decrease regularity decrease fast collection representative super contribution frank wolfe another explicit minimize yield gradient active algorithm exist infinite algorithm bad sample property satisfy interpret integral hilbert rkhs mapping rkh identify associated kernel update f rx illustration figure constant self cm cm cm perspective corresponding moment match approximate approximate similarly fx schwarz control less rademacher estimate generalizing relate bring line integral two clearly outline present provide input compact quadrature aim compute combination certain problem thorough mention definite compute lead affine moreover quasi sequence error function opposed sequence quasi continuously frank wolfe linear segment e optimal upper rate gradient size subgradient ascent outline hull set finitely finite still converge follow gradient optimization change certain obtain weighted line lead set unconstrained happen uniform kernel interpret update minimize problematic choice mrfs extreme common convex originally algorithmic perform subgradient ascent temperature frank wolfe might interpretation establish connection indeed max operation cm differentiable natural ascent equal equivalent therefore surprising subgradient ascent reflect change bregman divergence would lead function problem size cm diameter polytope lead integral two conditional see section however affine hull center relative choice step size random search search relative interior satisfied satisfied definition hull orthogonal orthogonality distance cm continuous kernel true cm finite get positively homogeneous construction one show function imply imply compact subspace infinite exist apply projection orthonormal eigenvalue know orthonormal f kx function kx hence cm rate know justification cm context integral infinite expectation kernel choose general example graphical practice possibility minimization exhaustive sample could cm aim assess relationship cm section space th px x expectation basis consist interval exhaustive kernel density middle right randomly em gradient original weight search lead quasi density inverse cumulative min denote compute optimal follow conclusion cm perform good search perform bad infinite set extra projection significantly outperform regular least empirically achieve present explain theoretically eigenvalue cm
validity agree valid agree negativity relaxation negativity absolute avoid certainly importantly objective care much close tend encourage objective deal gradient previous optimize convex valid discuss matrix max row internal norm elsewhere relaxation eq alternatively whether recover cluster cluster problem assume case cluster binary matrix constraint clustering noise match tight relaxation introduce main c v maximum definition inspire notice cluster inside outside follow certainly ensure recovery affinity combinatorial region vs cluster lemma cluster combinatorial small affinity matrix lagrange multipli norm c I unique consider balanced unbalanced motivate clear theorem plot wise tight unbalanced two affinity binary relaxation relaxation section enhance norm constrain optimization norm linkage perform absolute problem linkage power hierarchy clustering node similarity cluster pair cluster similarity fig exhaustive search find separate hierarchy separate check recover solution tight expect max provide trace max balanced max unbalanced scale trace require proportional size guarantee proportional huge advantage provide max two diagonal entry large one entry compare algorithm ideal ideal ideal gradually graph although binary affinity matrix absolute unbalanced fractional reveal absolute objective advantage affinity recover max proxy formulate solver small sdp condition slow scalable change problem convex large reasonably rank update operate unchanged otherwise formulation optimal solution zero gradient likely gradient gradient however optimization problem solve loss iteratively operate update cause numerical precision gradually emphasis close introduce lagrange multipli saddle iteratively fix optimize optimal fix eq q soft ij ij check initialize differently avoid stop positive semi double equivalence two size every factorization norm update trade dual provide sparsity add linkage although tight relaxation trace go introduce relaxation relaxation norm indicate suggest intuition matrix node cluster strict spectral cluster relaxation exist matrix optimize constraint cluster secondly integer run round fix clustering pick affinity replace cluster clustering enhance trace pick experiment explain summarize enhanced outperform competitive like investigate level clustering objective trace counterpart spectral balanced unbalanced clustering describe indicate norm far underlie norm spectral cluster mnist enhanced spectral point construct explain report time complexitie matlab top rr rr rr r closed see equivalence rr exceed contradiction equivalence since sub relation trivial counter strict gray cloud clique link connect cloud construct recover cluster alternative cluster alternative step characterize optimality condition dual construct equivalent max wise residual adjacency guarantee gradient distinguish zeros zero space projection define define yu sharing column space contraction definition theorem eq tu define contain eigenvector repetition variational norm definite matrix sub eq suffice condition condition follow characterize sufficient optimality notice construction entry corner cluster take standard duality norm alternate infinite sum geometrically wise division easy show last inequality projection consider last attain optimize lemma I q proof straight forward conclude proof effectiveness cluster approach partition similarity different cut receive lot normalize symmetric cluster minimize total term disagreement capture member cluster decide unfortunately disagreement np hard formulate matrix relaxation constraint recently nuclear minimize penalty provide propose max tight
element use estimate coefficient vanish toy identify network network find turn element detail max min clique infer independently certain condition basis realistic true grow correspond true consistently unnecessary candidate element subject constrain em lead k ij poisson deconvolution truncate maximize involve linear normalize develop theory result extend unclear writing proof begin give adjacency induce subgraph otherwise expansion satisfied expansion consequence uniquely decompose encode binary symmetric matrix basis constructive practice robust algorithm use basis non basis nonnegative entry encode clique bernoulli maximum encode clique clique include network element basis matrix network asymptotic bind element redundancy never true redundancy scenario redundancy imply regime limit behavior node assume capacity independent basis tend include analysis facebook empirically recover matrix quantify accuracy consider approximation characterize specify basis element exclude reconstruction basis element low theoretical order statistic give basis recursion network red use simulate blue weighted estimation binary without enable comparative pattern facebook us historical datum quantification concrete illustration idea whether simulate parameter basis distribution sample network run th parameter estimate permutation normalize distance reconstruct count clique size quadratic reconstruct incorrectly clique reconstruct table summarize reconstruction last table support pose reconstruction accuracy edge specify basis choice storage comparative decomposition prediction interesting pattern facebook facebook period student analyze r r r college edge american mit berkeley forest summary result salient college include report basis runtime second error incur element range compression substantial decomposition element reduce recently analyze association record history sc record general union network set find non association power associate directly result suggest tight interested developing holds explain spatial organization operate build analysis involve member meta activity association reveal organization show association identify summary exploratory tool capture multi organization historical record take quantify explore quantify clique scale possibly os poisson random contain edge baseline adjacency integer encoding svd summarize interpretable quantify provide overview row network row row box svd cumulative utilize slowly svd poisson consistently achievable svd rapidly reconstruction achievable achievable top support able notion social information social exploratory simple amenable weighted parsimonious orthogonality constraint attain interpretability basis social social summary always achieve element suggest interest parsimonious complex suggest leverage social facebook high compression ratio utilize desirable sensitive miss properly maintain outline power network weight long window element message window period author wish science grant award office grant ex edu weight social scalable combine fashion parameter sample include redundancy expect synthetic facebook association keyword deconvolution massive parallel year media facebook collaborative spirit wikipedia services service include cnn com popular com service momentum social science study pattern interaction tool collect interaction network pair arguably adjacency integer diagonal zero modeling decompose exchangeable value number binary decomposition popular orthogonality constraint attain interpretability multi choose enable linearly inferential utilize infer multivariate decomposition key community multiple scale come avoid adjacency matrix infer basis idea poisson edge weight rate matrix interpret latent induce social interpretation clique identify
precede important interpretation response previously analyze response response integrate spike compute potential impulse process model spike generator impulse threshold however goal nonlinear usually repeat response introduce several nice form avoid yet additional three addition implementation importantly coefficient aspect circuit fit associated response interpret neuron type neuron drug affect processing assess drug prediction spike paper prediction accuracy build sequence aim happen future benchmark high curve high rest organize describe motivate future protocol train light nm code independent ms ms indicate presence spike spike spike ms model behavior hz choose spike potential simplify analysis output spike train consecutive spike next next number roughly neuron spike spike divide part first relate interpretation spline sequence sequence start status mean interval f tf denote otherwise predictor immediately interval form model assume useful fire predictive spike simplify datum datum stimulus validate assumption first function depend spike spike assumption coincide neuron input hypothesis fit contain limited integrate data stimulus special form logit intercept logit model assumption additive simplify well moreover logarithm function formation additive include stimulus transform understand sharp fire leave hand side sharp sharp similar stimulus critical method atom contain component describe characterize spike expensive importantly compare function construct explicitly parametric amount procedure check parametric order three logarithm logarithm transform log relative empirically also response interpretation straightforward last spike pf cumulative cf fit spread sf response spike plot top panel sequence red spike bar solid circle grey vertical grey vertical spike times note spike status design predict framework follow sf sf achieve either sf achieve st f implication intuitive function equal conversely achieve answer fundamental care sf measure include interaction order term predictor high order examine possibility cf sf backward aic history development generalize parametric semi straight recommend plot pf fit value panel validate first time build adopt receiver operate roc unified approximate fire mechanism neuron believe spike comparison design sf r rf matlab svm instead potential stimulus history neural firing ms substitute show model specificity curve obviously advantageous prediction orient auc interpretation limited complexity stimulus filter spike among difficult spike whether spike encourage spike hand provide answer auc behavior provide interpretability simulation rf interpretation model comparison example integrate employ follow potential assume occur fix process spike spike time constant spike thresholding potential conventional box stimulus represent simulate ms simulating purpose sequence spike length randomly ms bins box shape stimulus yield time replication level coefficient confirm capture classical pf half may indicate highly power extract coefficient pf sf frequency coefficient entry scientific insight fit replicate vary cause neuron consider scenario plot fit pf sf respectively fraction plot scenario especially pf drop pf pf sf vary illustrate relationship clear pf increase spike probability due increase sf closely plot pf sf plot sf level spike spike spike remaining use different use measure auc different outperform small se rf svm good performance highlight bold time need prediction cpu ghz gb advantageous fraction identical fourth fast fast c c rf fitting prediction total se well highlight bold modeling draw probabilistic introduce response sparse advantageous interpretation computation discuss capability capture history spike equation reverse exist spike decode glm encoding model involve neuron extend predictor neuron status fit coefficient term neuron network estimation joint likelihood criterion interested explore additional component pf coefficient spike occur address mean model predict spike provide important direction analyze datum interesting provide fix intra follow state hold seek inter point rgb circuit circuit circuit challenge frequency spike high light propose new relationship employ logit provide nonlinear link input output response interpretable insight neural validate spike within ms advantage efficient computation simulation integrate spline approach output specific pattern demonstrate datum summary parameter circuit disease generalize suggest great brain specific brain leave neuron goal genetic light light application circuit input neuron circuit early stage circuit precise circuit I train neuron encode circuit brain cognitive circuit make information thought affect brain slice recurrent circuit light neuron randomly spike neuron record neuron directly activate neuron circuit producing continue stimulus show light output spike neuron run experiment scientific question
active arm km km km last element bandit identification event reasoning make phase make arm active arm eq suppose arm assume active arm happen e induction objective illustrate arm identification accord arithmetic allocation budget report experiment bad mean arm geometric arm experiment arithmetic experiment bad interesting sr arm identification even performance uniform involve single arm fundamentally always outperform tune provable research university edu wang department mathematics edu department science rely successive bad successive one algorithmic contribution tackle particular good face unknown choose reveal agent identify multi maximal note budget possible formulation model want minimize attain accuracy correctness latter history go budget set budget logarithmic notion characterize hardness identify well intuitively good arm sr evaluation paper suggest generalize analysis top contribution solve distribution successive h v nu sr find experiment sr perform involve problem fundamentally one solve author face arm single identification evaluation bandit construct identify arm nu nm km arm identification numerous reader previously cite terminology bandit face arm probability sequential evaluation observe arm select denote correspond mean assume mean make measure mean evaluate fine argue quantity arm easy see logarithmic represent hardness identification useful surrogate ccc arm n introduce multi bandit face identification problem sake problem deal good arm arm quantity problem mh mi bandit arm forecaster sequential form evaluation select agent arm high mean sort weak gap conjecture replace good identification paper shall derive section build strategy tune much gap time round sm sm quantities identification sr successive good feature confident among arm informally end high low decide rely gap precisely accept end phase empirical among empirical empirical good arm compute well sr order gap ki successive arm identification good eq hoeffding probability complementary event bound eq suffice event empirical within respective active arm
hessian product hessian guarantee shrink coefficient guarantee decrease kl divergence heuristic parallel update heuristic modification nature opposite sign sensible prevent case mutually drive mode parallel update result include effect e produce away effect individually optimal assign evaluate usefulness supervise color ten example per cifar patch test arrange overlap train svm pool post process pool image feature compare size vector cifar achieve grid accuracy coding encoding sparse achieve enhance modify seem require patch achieve close cifar semi svm learn cifar train advantageous medium label aspect set yet cause transfer competition competition color unlabele test label class public competition label cifar competition unsupervise demonstrate effective discovery supervise clean dark work figure page span always immediately leave line figure separate extra white page figure draw table table table title one table title table terminal cell acknowledgement discussion computation done conduct computer consider code learn spike rbm sparse code intractable ability dramatically demonstrate approach capability cifar learn challenge transfer latent variable visible generate q sigmoid bias spike respectively precision respective conditional element product constrain restrict variable hide rather understand gate subsequent motivate discovery describe avoid apply discovery coding show achieve cifar object recognition dataset normally factorial prior cauchy choose posterior factorial spike drawback sparse variable merely remain even kind undesirable laplace active little reason believe realistic setting model avoid control unit via determine active control coding control integrate extraction generative boltzmann inference approximation scheme map inference make suited activation feature relevant rbm rbm work think overcomplete small advantage discriminative capability factorial poor generative purpose discovery spike feature supervise e appear contribute kind classification powerful discovery turn exact mechanism learning turn intractable maximize step rather admits close find online work goal variational maximize variable minimize kullback leibler family ensure tractable step step sparse code difference posterior
vertex problem hence clearly arm conclude acknowledge grant dms mm take oppose specifically sequence bound worst additionally achieve way paradigm partial side online include compete regret assumption feedback promise research stock variety field game level prediction irrespective often naturally develop regular successful appear prediction expert theoretic regularity set restriction possible move adversary constructive author provide method focus set linear optimization information carry online reader nature round learner observe regular fix may view nature suffer process word plus adversarial ideally pay theoretically key make constructive game instead close side tight make precise allow deviation game constrain sequence depend short serve motivation statement value find purely well even ahead total advance either one obtain eq discuss interest regret previous move proxy move tend go alternatively one past occurrence get predictive bound provide predict regret property arbitrary write far full linear reveal reconstruct dependence learner require generally nothing prevent appropriate learner bandit yet actual learner study information might contain complete delay set consider optimization take optimistic incorporating missing bandit appear scenario differ feedback delay fall trend advance optimally employ completeness set dependent slightly bad represent sequence also divergence begin modification follow regularize regularizer simplicity next mirror paper minimization variation style perturb present generality provide learner outside compact set loss local shorthand consider initialize observe update notice regularize see incorporate turn correct endowed barrier optimistic expense additive small boundary self modification md respect bregman require consider strongly initialize update recently convex dual convex optimistic mirror descent yield q mention usual optimistic md gradient sequence mirror respectively completeness exhibit norm simplex case feedback serve explore letting mirror correspond term norm hessian optimistic mirror simplex enjoy turn receive simply incur suppose include move external source present simply availability opposite decide follow randomized full f present update observe adversarial appropriate step self let eigenvector eigenvalue uniformly analysis simplify generalize ball regret statistic x sm ti f put aside nothing investigation far see low raise good go formalize process indexing strategy us predict optimally however scenario comparable set example occur goal set stock expert option learner access step stock day regret know loss stock optimally achieve initialize initialize q rely loss unit banach nature optimistic let different set expert forecast give one stock expert expert would guarantee among stock measure see minimize idea run secondary regret replace online full setting get compute scenario number section access compute initialize regret term loss radius round get useful stock provide loss stock day trading day service forecast day provide correspond bind initialize initialize f g tf bandit loss arm limited information improve analogue set banach ball optimistic md process variant set suffer partial information setting early self arm draw update contain radius regret perturb unit norm object notion attain trend present deviation advance learner add shall constrained constraint admissible recover unconstrained admissible relaxation guarantee irrespective easy search sequential computationally attractive eq q adversary small deviation key since supremum expectation might able difficult I verify satisfy may take rademacher norm expect vector view perturbation final process rewrite form randomize ball ball take simple indicator perturb output regret converse implication satisfying draw respectively rademacher variable output expect recognize perturb cumulative couple example set give consist feedback move sm immediate argument immediate bandit delay feedback draw mean full norm write term exactly quadratic expansion obtain couple transition argument work e follow immediately bind positivity use multi armed bandit loss attractive tight whenever thank process barrier set step state case bind interesting gain term good negative gain barrier couple function round arm observe f h ti q q regret completeness trick extend let stand receive introduction let contiguous interval make monotonicity f take fix constant priori regret q box broken phase define eq generality follow assumption regret calculate deal period suffer regret may forget learn function trick full bound instance enjoy factor lemma computable trick optimize unobserved quantity clearly us eq quantity computable lemma imply plug inequality get final optimize argument tight lemma bandit regularize observe hold add side conclude inductive newton f follow self function local e hence prove yield eq convexity tf together technique let project normalize eq diagonal use probability view random
produce table simply indicate cluster c cluster j ht set cluster apparent norm provide cluster well experimentally norm probably favor weighting measure diameter height weight weight pair determined omit age variable dataset aim age contain size unbalance various number value range ht contain characteristic width overall store find ccc ccc cluster cluster document file automatic collect speech normalization entropy individual breast obtain dr observation uniformity predict variable perform well though superior c principal make look gap interpretation argue although partitioning explore variant cluster research tolerance effectively control cause split ignore inclusion identification especially document extract web document extract look along look secondary like thank nsf possible grateful repository website breast detail http uci ml breast cancer matlab document herein variety social business retrieval refer grouping human classify human main work know develop involve eigenvector decomposition matrix center name step word replace word natural gap understand detailed algebra geometry support column trend way component principal spread another define trend orthogonal deviation minimal concept deviation sake development present trace tn vector frobenius norm center lx see among orthogonal onto j locate finding characterize minimum deviation line datum yield trace regardless minimize observe know minimum way line orthogonal line directional direction line line minimal total deviation unless state hereafter understand principal two disjoint line depict front depict side ht distinction front back determine jj center q principal singular compute immediately sign need determine respective evident determined right matrix sign negative sign column assign option partition option examine scatter maximal principal trend separate three disjoint continue produce disjoint project split principal take gap motivate column partition appeal easily superior look gap cluster three distinct gap unbalanced contain address figure create gap human likely depend whether point goal cluster ideal line create tolerance balance ignore project experiment percent particular separate tolerance small data tolerance percent identical aside data project principal split gap maximal entire collection center gap right begin sort step create take document traditionally store mn row correspond individual extract filter word remove matrix text mining term effect normalize document frequency inverse frequency weight scale text variant variant document term q normalization normalize unit euclidean length note discuss scientific document normalization commonly scientific unnecessary stay range evaluation validation important cluster relate important accuracy evaluation measure break
regression hide follow mcmc covariate sampler rwm states x perform rwm update simulate disease x rwm x require adaptive rwm perform disease disease covariate estimate run remain uncorrelated sensible disease proposal accept within take approximately hour start trace chain thousand adaptive algorithm learn shape trace six three estimate plot iteration final run fit examine see choose six disease disease capture mark provide arise bernoulli trial six disease b interest disease whether presence absence affect logistic coefficient whether old affect infection affect correspond via reversible jump infeasible probability effect individual might raise probability indicate probable interaction positivity posterior parameter firstly clearly probability specie evidence reverse interaction presence chance statistic plot specie less infect previous exposure old infection infection decrease perhaps evidence interaction evidence change infection recovery exposure prevent arbitrary disease estimate prior markov dataset describe create check draw interpolation growth year provide disease independent rough correspondence parameter likelihood nine parameter summarie posterior probability positivity interval parameter markov disease always baseline c c main none posterior change affect one effect important analyse longitudinal study record six model offer exist approach approximately methodology cycle disease disease sample regression probability walk part gibbs motivation recursion directly would provide parameter conjugacy conditional mh tuning allow gibbs hide mh efficiency mixing current mh sample states disease covariate disease therefore avoid entirely update logistic parameter disease hide conditional state however lead poorly mixing discusse examine couple hmms consider article figure allow similar article simplify collection practice consider apply regression assume separability computational complexity state gibbs qualitatively allow major briefly biological insight offer old complete indicate last month previous infection specie specie effective acquire date population suggest vary specie interestingly infection appear infection infect recovery next month evidence current infection month find covariate covariate predict finding mention state disease disease treat apparent dependency sufficient without infection group certainly could enough disease appear consider disease state scenario subject might model methodology give state minor west project university support natural gr trust z section definition uk environmental sciences uk institute university specie absence repeatedly regular different incomplete profile discrete disease dependent logistic regression disease time point influence specie probability specie via cycle disease metropolis covariate hidden disease disease parameter evidence pair acquire keyword forward population infect successively e intensity infection impact involve complex allow prediction outcome interaction order longitudinal sequence infection four spatially population specie aside community human disease capture mark study set lead incomplete profile subject contain profile give disease miss response incomplete datum examine occur month first influence covariate logistic realistic methodology investigate disease disease dataset infer covariate disease covariate five disease analyse english cut site forest site near site live comprise day description outcome tag identify site factor lm capture integer weight gram identify tag capture diagnostic infection variable contain least informative transition capture initial chain contain substantial almost half capture month even observe twice month miss derive status despite table table summary capture response c investigate potential six study wish presence disease perhaps infection first month affect applicable disease probability model time disease disease state structure length infection geometrically infect month phase phase disease semi first expense influence dynamic knowledge presence absence disease must lie markov hmms disease e relationship chain capture forward backward consider disease interact couple hmms couple disease chain backward e size size forward backward hmm states take operation naive time equivalently deal certainly chain sparse approximately justify f l hide markov arise reflect covariate parameter simplify presentation hmm disease hide simplification disease chain likelihood likelihood summation hide simplification couple chain disease disease likelihood multiply may useful tool time point detail covariate clearly record unobserve dynamically could perhaps adopt simple whereby weight sensible investigate remainder brief disease dependent covariate dependent drop reference dependence detailed description resource specie human covariate recover third fourth month month majority one month first month likely particular specie might recovery past disease specie model infection infection old infection infection either status hide time govern justify similarly effect relate prevent example logistic month elsewhere disease infect infection say infection recover disease depend infection month use infection infection infection vector connect observation transition relate include disease disease human dataset infected chain transition separate regression relate hide infection infection remain present
volatility standard forecasting demonstrate aid behave mis specification keyword valid non tool characterize fundamentally function plausible alternative nearly iid extend economic financial forecasting rigorously accuracy mis finite allow immediate asymptotic generate popular selection tool aic imagine iid bad respect distribution prediction use natural criterion actually calculate risk generate single neither infeasible call assumption uniformly distribution series expect poor sample model bind assume mild condition existence provide general series satisfy mild average model amount use way beyond traditionally empirical quantitative inspection application really prediction especially mis validation partial exception economic long recognize difficulty pseudo prediction performance remainder procedure approximate bias toward overfitte giving tend true three reason test compete despite already instance recent financial enhance ability c reflect phenomena period often rare robust lead typically employ generalization bound rigorous reliable generate finite broad confidence normality understand report policy interested forecast specification wrong economic case still predict future derive apply setting finance economic engineering science loss make form rarely time dependence therefore ar past assume instance state view show stationarity ar lead worse phenomena plain provide way assess good really scientific explanation evaluation economic policy ask another approximate reality correctness success fit wrong directly past matching remainder background give focus concentration idea complexity leverage powerful idea generalize state prove result conclude toward generalize elaborate goal control datum reader sketch classical setting make poor throughout solely denote independently generalization q word expectation perfectly function rarely plausible index call one pick choose particular indirect ad hoc minimizing tuning fit risk risk big match end reproduce apparent depend fluctuation learn theory control tight quantity unknown allow add accuracy risk hand size observation form variance trade quantify result fit also claim piece bad choose forecast expect risk show standard result proof begin fix hoeffding minimum substitution rf nf powerful say observe training fact function want hold function wish nf achieve extension number function take result model quantify prediction choose functional analytic covering rademacher vc starts collection infinite every every latter vc vc subset one growth subset form growth effectively distinct go definition dimension finite show vc capacity straightforward kn immediate risk inside pre include fit apply explicitly need conceptually right hand fit nothing I infinitely vc illustrate count concentration show average concentrate expectation generalize class must handle dependent information know observe iid empirical expectation exponentially effective move iid series forecasting modification observe observe predict value process wish predict iid sort dependent first reader strict strong stationarity integer stationarity imply unconditional constant limit separate observation restriction convergence occur arbitrarily slowly bound vc give describe sort restriction restriction regularity coefficient process mix many mix definition approach product individual probability overview know mix iid correction background piece mix find prediction bind modify risk forecast preferable tighter risk increase event risk exceed solid black event probability force upper right curve desirable negative slope within budget put form move depend divide odd block block let block sequence mean empirical risk base sequence nf nf statement slight get obstacle vc calculate class ar equally bayesian conservative class forecasting model model linearize place past notation prove grow datum grow memory satisfie triangle inequality fix condition mild satisfied sub multiplication value prediction norm likewise absolute approximation become apply trade effective whereas depend desirable consequence memory q apply state model calculate demonstrate behave well unfortunately linearize stationarity eigenvalue lie unit circle stationarity ensure forecast one conditional like replace maximum forecast form space model simple compute output theorem quantify volatility calculate method standard stochastic volatility observation show investigate long kalman match let log remove return day kalman log minimize actually calculate point upper vc stochastic volatility vc dimension large much error translate forecast produce intercept reduce predict effective spurious volatility fit risk methodology forecast business cycle model forecasting use bring four consumption hour work economic estimation form list unobserved variable map nonlinear set return uninformative parameter penalize rather filter produce grow nontrivial bound finite lag vc dimension deal result course suggest effective calculate order magnitude suggest error risk surprising try get idea address show bind standard forecasting course raise address selection training penalize minimizer risk truth loss describe minimization work iid iid inefficient slightly suppose exactly constant exist erm nf nf k ignore constant iid I retrieve setting pay upper slowly iid case tight since general rule fast characterize ignore choose frequently risk many different minimizer forecaster collection minimizer small course complexity risk minimize bic rely asymptotic penalty risk natural predictor bound allow selection principle knowledge penalty account complexity error prediction volatility forecasting exercise model function vc control vc choose risk procedure demonstrate control generalization forecasting otherwise linearize dynamic equilibrium linear derive hold generate unlike standard selection penalty aic inherent forecast term economic applicability forecasting whose forecast would decay derive attempt nonparametric fully version possible circular review bootstrappe fit sensible coverage fit quickly let alone require obstacle resample method step theoretic forecast exploration gain picture forecasting algorithm want low risk forecaster target forecast well forecast actually realize remarkably try make ignore stationarity
minimizer theorem origin admit minimized inequality conclude remain inferior fact family admit differentiable level broad functional regularization loss q minimize functional admit theorem satisfied sufficient functional form instance uniquely minimized family functional define hilbert regularizer radial characterization dimensional family functional say admit every admit characterization general admit non report continuity regularization methodology ill estimation focus hilbert space endow product enough regularization square machine principal variety kernel space rkhs value empirical desirable solution include framework machine concern situation available space objective functional minimize case dependence definition functional cast regularization regularize regression choice control give label interpret corresponding recover principal monotonically constrained unitary formulation broad class deconvolution evaluate convolution noisy form many classical appropriate finite measure algebra give input regularization functional regularization take problem functional let functional say admit choice reproduce easy therefore obtain admit appear rkh functional early condition also necessary existence existence exploit author relax admit merely satisfy existence minimizer radial functional present radial functional hilbert space hilbert space convex star shape respect point shape radial space hilbert let hold conversely fix generic eq last also pass follow shape origin semi close ball center align unitary subspace orthogonality addition angle span positively sufficiently form
conditional variable symmetric matrix self loop typically notably focus expect degree within commonly encounter practice remove ij satisfy identifiability discrete degree regular community derivation good optimize np bayesian chain method study generally scale million node propagation comparable slow profile likelihood trivially plug speed profile search thousand propose model block case involve occurrence approximation profile give label block correct version consistency estimate truth converge grow fast misclassifie converge one need study belief propagation analyze cavity physics transition establish impossible unless one sparse correlate paper purpose consistency degree grow practice find graph quite contribution likelihood dependency make tractable adjacency symmetry block compression divide look likelihood sum accurate allow million truth overlap purely cluster likelihood connect perform present numerical demonstrate political contain maximize optimize possible assignment convenience true idea group group give quantity along give row th column nr base mutually approximately poisson long ignore among latent write pseudo maximize standard em probability node initial label repeat algorithmic summarize lk lk lk lk l lk lk current label label value return converge return lk il fit valid identifiability stationary label skip empirically fast update result hand important substantial community divide degree design cope situation correct extra degree node likelihood conditional whether matter fit reflect observation multinomial node multinomial pseudo parameter obtain iteration update converge unconditional replace probability turn initialize note full function modularity modal arbitrary option interest way dimensional cluster degree work identifiable distribution pair laplacian diagonal collect absolute eigenvalue omit mean block poorly due many simple surprisingly belong add add matrix find empirically constant value rest forming mean perform need pt know act tx ax constant increase edge grow balanced extension unbalanced balanced naturally call node prove model section undirected block introduce direct adjacency adjacency matrix confusion introduce case adjacency draw loop edge analysis effect direct natural extension model practical direction link social model context direct edge restriction parametrization particular make comparable undirecte allow carry consistency undirected expect mild long condition satisfy soon sublinear growth start label label estimate obtain confusion either key analysis overlap amount overlap important naturally data small fraction community preliminary formally know match value community uniform consistency particular initial algorithm operating depend turn implement plug initial whereas label direct permutation account label community counterpart undirecte denote notion discuss function let adjacency probability consistent think hard balanced simple start update majority intuitively enough initial figure illustrate key grow primary may strong plug direct obtain maximum estimate estimate parameter local maximum truth even normality example undirected probability addition proof theorem condition meet choose recover label positively truth give move positively label vanishing extend unbalanced direct block supplementary initial include overlap initial balanced supplementary material directed investigate unconditional perturbation simulate scenario correct presence node conditional label probability draw enforce regular control determine relative degree degree degree relevant information diagonal otherwise overall expect mutual one think column sum r ij ij always reference matching dc perturbation unconditional likelihood dc initial outer specify figure smaller easy uniform moderate amount serve value pseudo perform poorly limitation scenario apart serve value show pseudo achieve gain poor start give surprisingly uninformative exception unconditional accommodate degree already room well initialize appear good value limitation effectively spectral sc take outperform dc might dc memory consume excellent complexity variety brief belief propagation different time scale rate slow constant bad likelihood political node focus manually conservative treat community direction analyze connected pseudo produce close expect degree result correct block unconditional put low unconditional fitting model correct model restrictive go go match estimated vice direct compression imply define normalization ie focus concerned slight abuse since case start classical bernstein sum ij b complete symmetry note variable rh see supplementary pick follow
I cholesky supervise variable greedy pursuit technique applicable candidate must start advance context propose include process unsupervised time error incur exceed sufficiently current contribution later ignore visualize add basis column k k ti remaining go overall long costly operation reduction forward augmentation exploit addition contribution try subset unnecessary ensure otherwise resource make necessity present square evaluation approximate automatic recursive represent underlie take reward tuple policy problem eq reward indicate problem state w tm h tm tm tm w fix r w derivative square state obtain eqs compactly k mm solve eq mm solve b recursively update currently update reconstruction contribute solution problem decrease add give threshold increase line least square propagate identity recursively compatible add second outline implementation unnecessary repetition equation minor symbol estimate far current cross weight execute observe loop execute simulate check basis additionally either step true currently consider least clarity matrix tm tm mm solution assume tm tm previous computation new observe update tm h tm tm tm step complexity k inverse reveal undesirable go away merely need access past example online tm h eq h already precede r show computed computation obtain via expression update need computation store square residual projecting span current inclusion contribute reduce well represent aside criterion usefulness usefulness take I together carry article use publicly procedure policy reflect transition function represent make actor maintain regularization actor use learn previous prediction evaluate propose ever grow irrespective reflect real amount computation function agent example complete evaluate actor iteration example policy state sometimes greedy recommend action pick greedy random among action observe differ associated value interval guide increase initial experiment show tuple employ suggest action take delta action state choose rbf length factor rl usefulness try effect supervise start unsupervised tolerance least reduction runtime simulation hour roughly hour pc try combination rl set report I scale rbf select basis function real time pc behavior interact agent roughly configuration run curve plot example additionally horizontal indicate code plot hand code consider latter considerable manual par second second code need discover supervise one basis function regard conjecture strongly depend I vs actor specific opposite observation least rl key point basis subset rl effectiveness benchmark overall rl possible superior grid fairly want function require considerable manual carefully specific manually suitable algorithm use online selection basis oppose reduce relevant policy evaluation bellman residual apply rl problem deterministic unified deal use respective demonstrate simulate pose dimensional control arm state rl vs deal state transition rl anonymous comment suggestion let action reward represent add element execute tm tm grow supervise normal tm tm grow tm tm tm z h tm r tm tm tm tm tm tm check reinforcement vs key like mean online require framework evaluation underlie family approximation supervise selection relevant indicate approach obtain square iteration widely accept common platform intelligence consider problem maintain ball region field team try gain reinforcement rl individually ball team team play dimensionality observe rl second noise agent imply environment need model server necessary learn successfully underlie fashion throughout state exponentially exploit knowledge various interact one also state simplify represent solution choose adapt try visit growth solve optimal control square policy ideally suit efficient paper formulation regressor subset select subset employ greedy candidate improvement supervise error incur kernel complexity ensure resource way learn necessity structure brief introduction describe derive recursive third problem long defer end regularization network dynamic time finite maker control set admissible distribution action sum reward maximize denote decision evaluate q discount reward choose proceeding accord discount policy policy greedy choosing achieve good obtain well trivial otherwise function obeys relation bellman solve linear transition reward situation infinite approximation unknown one figure depict parameterized mean practice approximate sound bound infer problem reinforcement carry approximation transition difference td initially prominent linearly parametrize td evaluation converge fast least base descent
control bellman lyapunov case highlight towards develop drive dynamical strategy give concept reproduce space nonlinear rkhs theory tool nonlinear system material define tc minimal show q lyapunov energy pair define make evident follow problem linear norm solve precede question give system define ellipsoid output compute stochastically force stochastically control system input give corresponding dimensional motion sde density evolution density differential operator refer context find covariance uniquely transition mean satisfie dt xx xx xx bb steady bb exactly iff fact steady suggest linear approximation obtain approximation equation explicit lyapunov find solution subsequently reduction system hamiltonian hilbert connection system define introduce rkh give brief reproduce hilbert heavily develop theory rkh hilbert inner product say reproduce kx kernel positive kernel reproduce unique x definite positive compact hilbert property rkhs oppose map symmetric symmetric operator practice typically rkh theorem guarantee reproduce rkh immediately nonlinear algorithm product instead look look eq inner reproduce variant implement rkh balance dynamical briefly however compute lyapunov equation directly prohibitive multiplication implement primal adjoint alternatively linear take simulation extend nonlinear proceed coordinate system orthonormal let response xt see matrix respective response finite interval assume measure response respectively matrix thought matrix observation nonlinear assume linearization origin observable stable rkh quantity map sample sample separately implicit rkh scale respectively operator w refer rkh version integrate refer empirical whose scale map similarly build express important index ill condition svd section introduce empirical energy control system estimate assumption suitable space reproduce rich capture strong space assumption set energy satisfy origin stable origin smooth equilibrium would solve explicitly solution simulate develop unknown nonlinear drive gaussian noise compact former balance approximately datum projection approximation event need impose precede overfitte limit q towards derive computable define operator adjoint associate operator mc check ms c cx dimensional column product let output kx vector product sample collect result kernel energy energy estimator consistent approximation making open make rkhs induce reproduce compact define rkhs furthermore space function separable introduce briefly recovery regular rkh introduce notation rkhs form control involve denote preliminary schmidt covariance reproduce hilbert schmidt establish consistency method adopt fix eq norm calculus make part adjoint expand orthonormal converge lastly iii part requirement fact enough e note precede estimate observable control definition energy function replace obtain sample general validation ellipsoid observable characterize definition ergodic nonlinear key provide great control nonlinear sde however system steady existence steady back formula solution often conservative balanced truncation context play key sde energy theory hilbert estimator measure aware technique rkh control systems parametric invariant density give behave map rkh may control energy measure system nonlinear system reasonably infinite sde dimensional rkh ergodic stochastically nonlinear value linear self adjoint wiener invariant along invariant absolutely respect proceed derive invariant interpret heuristic nature mild sde exist pg trace give condition zero back establish
orient detector different color encode orientation invariance encode orientation interpretation complementary pool feature return pool along pool across interpret strategy pooling bilinear visible structure course possible interaction architecture imagine overlap specify variable interaction believe overlap potentially consider topology exploit high possess building deep interaction intractable variable interact analytically resort approximation conditional standard choose minimize equivalently variational bind maximized take bind variational approximate factor fix depend three stage strategy parallel parallel maximize parameter evident parameter contain negative derive adopt strategy positive gibbs sampler variable outline training latent particularly learn motivate block wise organization structure allow remain local interact interact neighborhood procedure complexity interaction adapt block tractable super hidden interaction structure consequence relatively block extensive consider stack multiple factor local gradually toward local strongly attempt variation effort direction also variable multilinear critical attempt factor variation method interpret attempt extend subspace variation bilinear essentially model state factor factor element tensor thought generalization unsupervise consider separate develop algorithm demonstrate style font content later develop bilinear code additional observation factor develop multilinear simply compose together multilinear ica model face factor illumination orientation image identity people extract pose recent high interaction interaction boltzmann dramatically rely rank approximation employ highly connection interact structured norm interact interaction learn use ability variation synthetic fig compose object vary color color rbm negative bilinear factor color configuration fig bottom learn encoding color would enforce hidden generate perfect deep pool pool row tie evaluate rbm measuring usefulness evaluate usefulness use large spike spike pooling spike iii order arrange standard training performing unsupervise unlabele learn cross label various spike fold valid ss rbm number factor representation product representation representation form consider multiply factor outperform confirm hypothesis successfully learn low factored classifier deep achieve present extension spike restrict boltzmann factor three interact interaction act binary interpret previously future gradually stack architecture datum base high interaction see latent train regard latent datum generative involve complex interaction interaction challenge individual pixel showing expression well identity expression dominate image space seem appearance individual faces importantly interact combine easily affine often raw cope reality variation provide feature appropriate face capable effort cope movement computer vision toward engineering set common source motivation inclusion convolutional pooling induce pool powerful notion filter pool together purely unsupervised feature situation variation expense lose filter pool spike include high perspective interaction variable factor rise conversely see attempt interact factor combine generative unsupervised factor research direction information responsible principle organization learning train subset relevant consideration lead factor discard little information motivation behind spike htb g precision additional gate group extra subscript term visible model progress variation boltzmann spike boltzmann previously invariant pooling spike add variable spike variable constraint model
dynamic programming improve bound tight frequency call methodology represent simplify uncertainty attempt model uncertainty robust uncertainty mdps robust organized mdps function term optimization involve concentration use bound leverage exist mathematical programming benchmark offline paper generate advance function identical section concept require general symbol respectively symbol appropriate element part algebra expectation trivially action sa sake action avoid idea action denote state assign probability randomized policy map choose randomized pair ap ar value policy always approximate formulation mdps mdp solution discount remain state take action denote define value equivalent frequency action measure closely section characterize basic e part hold policy basic bellman frequencie greedy action maximize return respect tie mdp specific mdp solution derive tight bound transition also property bound concentration transition formalize describe analyze sampling simply derive identically programming subset return mdps simplify restrict policy minimize loss return follow rely value simultaneously restrict policy approximate frequency element represent combination column approximation definition follow without reference feature follow show importance suppose represent assume ready paper gain motivate primal slack formulation mdp value upper approximate penalty return invertible ignore formulation particularly frequency equivalently define use representation assume duality show combine separately remainder assume policy generalize state solve program suppose imply equivalent bound readily show subtracting factor involve undesirable rely purely derive follow admissible suppose tight policy mdps hold derivation assumption inequality bound rely require feasible conservative solution well solve np easy practice np completeness feature action mdp favorable property solve mixed formulation solver develop formulation bs computed policy restrict assume take state policy represent linear program np suppose solution deterministic greedy respect equivalence optimal dual separately contradiction solution treat low optimality positivity th element q trivially feasible optimal choose bilinear separable bilinear formulate generic impractical relaxation instead existence bilinear upper optimal assume simplify individual whenever show discuss practical must bilinear similarly identically sec sample estimate reward nonzero zero action section experimentally synthetic chain three offline feature unbounded restrict balance cart either action represent force cart uniform govern differential benchmark arrange grid center term require balancing step average run figure show bar indicate compute poor feature space decrease become conservative optimization become quality factor good single benchmark generalize chance moving feature orthogonal polynomial show randomly outperform propose base formulation significantly strong theoretical guarantee improve solution additional state small encouraging mdps discussion inspire anonymous detailed comment greedy com programming popular approximate curse turn derive analyze theoretical translate performance problem reinforcement programming many study empirical many must carefully tune offer insufficient
supremum depend geometry upper specific algorithm prove adaptive query function class large infimum oracle supremum take depend geometry independent turn applie typical additive gaussian convergence bad dependence restrictive setting function polynomial focus feedback build idea optimization obtain question fundamentally somewhat describe convex point probe plus change time noise one theorem markovian consider priori belong simple easy original minimum query require desire minimized associated query let p kullback leibler ip take simply refer explicitly semi point specify select select define minimum fx proof ham denote element elementary radius ix ix jx differently bound differently accord oracle comparison markovian assumption conditionally conditional note underlie kl bernoulli f f ready apply want choose side definition satisfie apply since use slightly different unit define c iy ix exactly bound section recall consideration corresponding evaluation underlie algorithm depend kl eq compute claim attain oracle pick uniformly exploit guarantee expectation oracle error approximate line accomplish however confident comparison sketch leave supplementary material define choose random zero except analyze direction coordinate sphere line guarantee fx q last gradient lipschitz take respect strongly define lead within comparison pairwise comparison q wish concern minimize wish make operate maintain iterate iterate regardless away close fast gradient let make pairwise algorithm subset also rely heavily unclear assumption bound suffer contribution relate oracle bind x fy fy random fy fx fy fx density equal true unimodal laplace side distribute fy fy fy loose pairwise oracle summarize pick uniformly exploit gradient argument decrease make line accomplish binary line probably correct robust repeating query confident pairwise comparison active line position query estimate pairwise comparison taylor hand convex minimize q recall chosen eq respect law iterate unique minimizer convexity calculation complete proof wish fx pairwise concerned minimize single variable indexing search produce indexing index present assume comparison noise iteration eliminate iteration modification great fraction removal search provide fx fx initial estimate straightforward fx k side fx fx bring request straightforward terminate pairwise loop argument bring request comparison oracle iterate bring comparison error repeatedly comparison active ability adapt unknown mention way want implement efficient subroutine fx fx fy identify request pairwise convenient could result impossible low lemma algorithm ensure see algorithm begin loop repeatedly compare soon repeat sampling terminate strategy proceed noiseless location loop compare repeatedly noiseless noiseless iteration iteration noiseless algorithm unlike request distance evaluation small distance probe consider bad comparison straightforward k subroutine subroutine request call subroutine must union comparison search bring call query subroutine n rate remark definition university usa usa lower rate free noisy gradient situation algorithm comparison evaluation wider pair comparison example show boolean value optimize large tuning evaluate unclear influence thus free rate noiseless method equivalence gradient otherwise function function contrast scale strongly rate evaluation new boolean comparison decide interesting subject pair comparison convergence ambient new achieve formalize strongly convex gradient gradient define minimum query way optimization
n quantitative measure noise ratio widely image learn similarly slightly well interpolation measure purpose perform amazon http www com average win e g bp vs amazon web interface follow people experiment intelligence etc reliability explain measure reliability sampler train online vb super base superior natural ask visually assess reconstruction likely bp vs decision quality ground algorithmic fail select number vb iteration see vb complete mini reach optima vb batch vb find may local optima mini hour gibb burn collect sample distribution take approximately matlab implementation collecting reduce hold db finding super nonparametric training scale code traditional evaluation variational case slightly human speed important online regard analysis ratio capture run human show necessarily build time series generate process super via bilinear n lemma proposition super resolution resolution super beta element determine benchmark natural human assess visual gibbs sampling approximate large datum bayes dictionary need factor variational stochastic optimization resolution denoise compressive sense datum image represent combination pattern may beneficial build accurate superior denoising model derive efficient image sr lr surveillance imaging variety super general possible solution regularization researcher machine train ground image relationship reconstruct image lr patch use neighbor search resolve propose kernel regression another sr algorithms texture match know image super one use scale image via coding regularize dictionary element across position specify dictionary variance difficult assess provide nonparametric method latent provide infer otherwise posterior dictionary many image architecture segmentation multiscale representation paper super feasible super algorithm latter approximate enable evaluation quantitative success analysis necessarily devise human evaluation nonparametric model sr variance couple devise scale experiment assess sr give scalable describe super resolution parametric section present detail use natural analysis denoise model learn e patch resolution build factor analysis lr lr perform super correspond detailed forming depict preprocesse extraction first weighting function interpolation sized patch location lr couple train model element loading local use dictionary resolution lr dictionary dimensionality resolution element use resolution dictionary produce key property frame super inference assignment mean distribution binary encode element activate elaborate place gamma weight level share represent element represent patch illustrate sr phase dictionary lr expectation reconstruct lr distribution weight patch inference discuss assignment use bp binary ibp encodes activate one correspond distinguish characteristic specify priori condition binary ibp restaurant infinite try dictionary element ibp customer enter restaurant sequentially choose first beginning take th customer begin proportion popularity previous customer quantify took consider customer choose record indicate infinite indicate joint final customer restaurant exchangeability observation bernoulli trial draw beta ki observation represent usage inference scalar tend bernoulli approach ibp enough exhibit component pair super lr training variable computation posterior dictionary image pair rewrite couple writing fully reveal dictionary single scale amount posterior patch dictionary lead share dictionary lr reconstruct patches weight score lr image patch dictionary fit determine strength control score precisely set hold patch posterior dictionary step estimate lr stage determine patch correspond wise overlap reconstruction match lr image solve anti follow sampling prior conjugate exponential gibbs observation equation provide material sampler patch sample develop alternative sr scale streaming inference mcmc replace family minimize track see typical variational coordinate iteratively hold sampling replace subsample datum adjust subsample rather scale exploit develop ascent algorithm derive variational easily perspective bayes base field learn become factorize govern optimize observation divergence maximize bind variational dimensionality twice big scale function hold parameter go algorithmic online update parameter identity matrix q variational parametrize ascent update ascent equation variational coordinate free precision gamma variational variational gamma coordinate ascent update variational use nk compute k update develop divide variational image weight optimize per optimize find basic repeatedly satisfy optimum optimum sn new vb noisy estimate write uniformly sample thus noisy channel right element pair locate consist subsample variational sample local setting parameter variational copy image local equivalent fit cost optimize global versus ascent parameter early speed global full online vb mini mini initialize vb useful help begin recent near draw whereas algorithm anneal jump optimum train web berkeley image image various community evaluate sr
empty bin respectively lot careful basically process jointly really want available fact denominator normalize become equivalent set highly unbalanced expect would significantly reduce occurrence bin mention one conduct permutation long bottleneck use permutation chance empty bin decrease increase already review strategy integrate hashing modification bin sec bin e empty zero strategy procedure matrix depend empty bin th believe deal empty bin convenient produce artificial expand always long preserve random slightly accuracy permutation scheme robustness permutation news interestingly code perform extremely accuracy comparison random strategy actually original hashing provide scheme testing convenience logistic small permutation scheme one outperform achieve accuracy permutation reach compare bin code course news h merely test one strategy application dataset difference permutation scheme permutation difference length understand would connect count min sketch length divide evenly bin scheme grouping entry bin permutation connect min cm sketch count cm estimate remove subtracting sketch idea bias multiply whose probability recent show sketch equivalent projection hashing conduct showed believe fix scheme difference major bin empty actually bind bin difference ff possible potentially perhaps variable massive hashing algorithm require accuracy preprocesse hash context estimate similarity binary text method hashing linear learn logistic sublinear near drawback hashing require time expensive e process true preprocessing come efficient conceptually massive evenly bin reveal permutation hash scheme accurate replacement scheme test robustness also merely outperform news summary merely preprocesse efficiently bit hash low bit near search text major drawback hash hash expensive permutation text binary extremely dimension page contiguous mean need substantially increase english word super word practice total dimensionality convenience occur document note image vision either feature dimensional example distinct dataset potentially hashing design equivalently two measure inner product compute prohibitive consumption propose efficient computing permutation permutation probability permutation binary dimension q training regression store low bit dimensionality expand idea million achieve original building hash table sublinear sec example inner product permutation preprocesse hash loading dataset sample format take format second preprocesse document process preprocesse cost hashing nonzero use view vector divide evenly bin vector divide evenly bin small bin empty bin rarely concern set bin bin obtain bin representation example indexing original take modern show empty occur reduce permutation permutation much efficient perspective energy highly especially software implementation process permutation may meet interactive easy number generation cost gb realistic resort universal hashing permutation universal theoretically always hash much easier generate permutation one hash one permutation nonzero datum verify work scheme name basically sample estimation however align unified fashion scheme evenly nonzero align interestingly hash permutation author realize hence sublinear hash table product bit hashing projection work show use extremely sparse small merely projection scheme convenient count min sketch hashing briefly review original permutation large search numerous etc research modern practice often fall general framework lsh performance lsh solely underlie idea directly bit hash hash near sublinear scan permutation bit signature create point signature match phrase permutation bit signature many neighbor use permutation hash query retrieve table example permutation table hash permutation store bit signature low bit signature place apply signature become signature search near neighbor much confirm strategy perform logistic popular package sgd give regularize logistic solve regularization solve permutation binary vector store lowest merely run expand exactly illustrate digit expand fed permutation apply zero never permutation hash encode empty bin strategy later refer confirm search learn permutation figure practical although occur rarely unless vector interesting probability analysis provide rigorous foundation consider empty bin bin define bin see appendix binomial analog scenario often sparse lemma bind approximation expect probably empty significantly impact practical assumption datum distribution q en n q appendix confirm lemma eq show good see exact already derive seem note always original completely empty zero var see appendix probably scheme may slightly merely preprocesse replacement essentially vector appear web l contact web university business book job thorough figure result base repetition verify theoretical curve empty practically hash substantially would usually see strategy bin
applicability complex secondly modular graph simple graph consist node link node link conditional variable concentration determine independence mle concentration modern application observe observation thousand product genetic many behave term point involve trade cost benefit e error error incorrect subset forward backward procedure select predictor regression forward extensively adapt constrain precision penalize solve estimate absolute scad penalize approach alternative base theoretical property penalize derive well establish tend impose symmetry precision matrix reduce introduce symmetry important likelihood restrictive particularly need relatively paper symmetry modelling impose precision model useful random come use idea precision motivating time microarray cell experiment introduce graph factorial correlation detail problem address graphical aic bic adapt factorial encourage simulation edu system understand interaction protein gene develop collect gene concentration rna produce snapshot expression temporal evolve dynamically internal genomic continuously expression response thereby flow genetic dna snapshot profile reveal determine differentially microarray determine functional order infer interaction gene expression aim variable relationship variable motivate describe describe find describe item collect gene level analysis cell produce cell cell line laboratory reach collect hour contain investigation complementary second microarray experiment array investigation compose collect cell cell another set cell assume technical replicate independent replicate replicate two dataset replicate sampling conclusion cell step conduct obtain across firstly normalization systematic due describe dimensional subsection variance illustrative across compute select base partition matrix square matrix summation interpret variance predict behaviour lag self self represent conditional dependent give gene temporal lag cell gene gene lag element last partition graph lag concentration replicate gene across gene measure gene lower read gene education longitudinal teacher across study conduct student secondary school student illustrative purpose control influence proximity score behaviour score measure teacher compatible correlation class student show triangular covariance upper score measure experiment four measure illustration subject prox prox prox empirical matrix precision mean divide root range measure equivalent correlation pure influence indicate independence conditional dependency conditional relation undirecte correspond set gaussian vertex dataset whereas course dataset establish gene measure collect hour hour speak vertex order natural order convenience notation dynamic graph assume conditionally interaction term equation different type third represent line refer dynamic order measure cell gene across change correlation diagonal moreover think time self self across two graph triplet link induce partition type line line colour continuous line represent colour dot colour let partition indicate stand subset vertex partition partition consider lag graph time lag induce firstly indicate colour secondly edge colour across indicate differently tv tv j l f substitute specific natural induce partition I v jt jt ic g partition calculate colour edge resemble eq firstly follow natural partition create couple couple simplify secondly induce subset colour create bring graphical model graph denote ij tv ij couple satisfy property relation undirected connect factorial dynamic factorial triplet natural partition n factorial correlation factorial figure variable f sub partition natural impose design connect element concentration partition associate uniquely mn self partition induce parametrization induce specific equal restriction follow factorial description n diagonal empty empty estimate replicate induce subsection colour time gene generally comparable scale interpretable conditional matrix rescaling show rescale scale precision rescale correlation rescale e factorial structured restrict diagonal partial identical colour class structure graphical model conditional obtain conditional element conditional correlation constrain follow factorial estimate conditional element conditional triangular correlation equal triangular element partition partition total note I c c subject prox prox prox reach considerable interpret vertex connect graph tend density follow edge density could test selection parameter would bring instability penalty concentration induce convex hence variable hence feasible design produce operator necessary impose linear every induce describe order p frobenius want sparsity assumption absolute constraint equality introduce slack variable equal diagonal penalize constraint impose structured priori minimize e standard determinant allow impose factorial optimization solution employ proximal conjugate develop matlab linear graphical package analysis open source code package virtual connection within apply algorithm penalize restriction example represent constraint constraint guess code consider smoothing parameter framework several factorial choose well find false reach stability selection stability bootstrap idea smooth aic bic compare factorial factorial degree number zero e element partition chance fitting smoothing
asymptotically assumption description set obtain discrimination limitation besides discrimination generalization discriminant analysis analysis random help system decade wide speech music recognition asymptotic optimality classical actually go scatter covariance scatter projection thank expect acceptable asymptotic bayes optimality limitation infinity provide quantitative especially address superior increase dimensionality second examine respect w mild discrimination insight severe ability population power eliminate influence influence proportional worth result besides discrimination acceptable estimate theory main understanding draw originally motivated nuclear physics mathematic engineering communication law spectral wishart almost almost extreme singular random formulate proposition whose independently e surely deterministic letting sample notation bold letter space matrix compose first column sphere locate th sort descent denote denote column space distribution class follow scatter class nonzero eigenvector obtain fisher discrimination reduce scatter denote generalization discrimination training infinity power population counterpart however increase infinity discrimination dnn regard linear reduction loss generality eq cumulative cdf replace population counterpart error give discrimination dnn dd unit length sphere independent eigenvector v td estimation ready main total recognition handwritten digits handwritten digit dataset dataset get population discrimination power different discrimination horizontal population minus discrimination properly bound hold high dataset major considerably small problem distribution ratio generalization properly randomly select perform evaluation set classifier ratio result result generalization upper dominate lemma simultaneous theorem semidefinite clearly ii eigenvector must v submatrix u imply therefore I fact arbitrary nonnegative rescale rewrite unit independent unit must unit uniformly sphere give proof distribution wherein fact invariant orthogonal similarity transformation complete statement converge spectral distribution f complete uniformly distribute entry sample uniformly number divide side q first argument know r third nonzero examine hold eq let partial denote integral substitute thus proof lemma scatter wherein e c n x e ei tc q proposition
vector term example list chain loop count singleton singleton chain singleton singleton third singleton combine tail work normal scatter tail find spectral wishart distribution chi square individually collective eigenvalue top eigenvalue generating supremum b eq eq adjoint dimension bernstein semi definite q
decide lower decide similarly full recovery may follow problem page least coincide apply coordinate j detect ideal choice since elementary omit diagram popular include g expensive local way graphical structure guide utilize setting set tuning convenient second easy preliminary situation believe conduct long respectively convenience pe depend tune minimum strength tune insensitive point bivariate high screening use order screening memory need need change choose degree performance much slow case identify first p j case change pe take threshold take depend scad shape mc penalty integer take tune lasso scad ideally depend slightly inferior scad mc comparison fair unclear tuning mc range report average hamming consistently especially three lasso scad perform error conclusion save lasso scad model cd scad scad mc detail p bayesian report ham independent repetition use well cd consider post filtering model simplicity mean penalization post filter end show distance loss difference soft naive parameter theoretically optimal signal equal case address gram graph insight post filter live case stage clean candidate stage overcome employ technique develop focus regime rare achieve rate wide situation limited mc selector well even successfully problem time case covariance screen effective marginal closely relate attempt optimality recent work different motivated compressive sense genetic largely dna variation long series different ill pose filter paper study memory time change problem reason primarily broad coordinate without continue adapt validation post correspondingly modify tune slightly tf pe pe core could long series result largely many rare weak two main long live small isolated connect focus change post decay extend setting post matrix modify accordingly length still extension necessary specific pe paper closely recent dna save leave notion mean understanding operation compressive genome wide sense already motivate change analysis operation try study along apply design gram future acknowledgment david lemma condition example support nsf grant dms sciences national grants gm gm support part award dms major article complete engineering model sparse main separate nonzero fan primarily gram finite focus regime screening induce graph conduct interact decompose separate filter newly introduce give stage fan song candidate remove positive selection measure minimax hamming situation gram successfully apply memory change nonzero nonzero one deviation know generality gram eq often difference signal rare gram ill pose challenge signal well concept signal important largely focus regime signal scientific signal hard find easy partially explain publish relatively element filter removal gram play role application area drive dna copy receive exist major interest piece wise jump relatively parameter jump gram adjacent row display adjacent observed process exchange rate reformulate gram say toeplitz matrix norm gram matrix adjacent row example jump asset example sum example adjacent rank removal rare variable also investigate reveal give consider mild equation fact full rank solution see spirit ground look motivate penalization look noiseless penalization fundamentally intractable decade limit lasso mc penalization method four rare strong recovery penalization fundamentally unfortunately rare first fundamental phenomenon indistinguishable signal weak principle hamming distance appropriate fundamentally signal rare example rare regime tuning penalization detail penalization method imply penalization scad propose show achieve condition clean heart marginal face call relatively see effect motivate application example linear matrix small penalization motivate new multiplying one largely exploit free cause serious issue post filter regular regression model face helpful procedure correlation incorporate essentially sparsity deal cause innovation create rich graphical structure lasso heart screen reduction big date extend model major variate screen remove noise exclude result strong screening overcome challenge covariance screening screen retain almost signal construct connect exclude sub generic screening efficient fundamentally regime sufficiently setting discuss component theoretic three fold develop theoretic framework rare asymptotic minimax partition call criterion assess procedure phase diagram space different diagram rare challenge ill regime practical perspective paper rare regime term distance find tight procedure exact entail derive diagram section need usually associate especially challenging introduce optimality simulation denote contrast hamming shall symmetric arrange method introduce rare weak signal model introduce start filter explain filtering help interesting interaction cause call discuss section discuss complexity feasible asymptotic framework hamming function hamming hamming section apply explicit convergent formula derive diagram section find primary interest weak signal location randomly primarily relatively signal rare weak rare rw remark theory develop tie rare signal extend signal g ising mention primarily gram view reason necessarily filter say memory time change point induce strong filter eq lose reduce matrix h compare easy track naturally motivate three call definition parameter choice sparse exceed unclear sparsity help variable sparse path unclear remove try surprisingly answer lie sparsity support subgraph justify support realization p case many know component one view separate subproblem insight attribute sparsity latter attribute linear filtering tie broad ise largely filter induce discuss linear cause frequently denote form restrict row denote sub form vector explain matrix correlation continue must filter moreover di definition compare imagine fully never validate phenomenon content neighborhood add di capture fisher matrix eq orthonormal negligible phenomenon fisher substantially patch fisher information model although ratio cause case screen information technique summary filter q observe induce sparsity construct signal cause information overcome selection pe candidate signal carefully achieve candidate remove false positive step integer ps ps pe pe step suppose connected denote arrange size tie break toy example subgraph triplet sequentially decide test screen bivariate screening variable examine screen new far additional similarly never variable clearly multivariate screening exclude formally retain stage convention sub step th set fix tuning node node step random thought follow update node td terminate write accept sequentially accept stay course could spirit regression use collection threshold threshold test central circumstance arise appropriate misspecification alternative continue help provide ss property say subgraph subgraph expand introduce ss enable illustrate retain index example retain sure false positive compare reduce original parallel one node view expand recall retain end step graph subgraph component fix subgraph subgraph fix nonzero pe pe pe pe dense address moderate select fast graphical challenge desirable adapt variate desire coordinate test exhaustive force fashion marginal exclude obtain innovation variate subgraph connect subgraph great display variate tuple total tuple variate tuple subgraph subgraph step consist part obtaining degree exceed solve context exceed provide properly study therefore conservative computational complexity section set show term hamming distance section memory time add rare driving increasingly successful impossible relatively critical recalling fix rare weak strength signal reason remove future view prefer properly large neither adapt gram small small sub subset v constraints elementary omit discuss broad minimax risk selection hamming expectation take vector denote correspondingly hamming hamming minimax hamming long hold many assess hamming assess optimality study demand local hamming hamming bound error fix hamming risk geodesic local hamming characterize th exponent notation frequently multi provide index aggregation local low hamming toward recall favorable theorem similar r p p gd example include primary eq section broad properly case prescribe lb achieve focus right side lemma lb hold root constant adjacent satisfied mild condition number series certain singular value relax gram belong nevertheless slight continue use tune pe unless signal sufficiently constant principle depend unknown suggest choice relatively flexible section theorem appropriately subset
dataset report report naturally cast estimation group causality see reference therein dimensional infer adjacency unknown implie give operational causality node past series link causal causality insight knowledge selective make technique focus involve nonlinear appropriate problem l set component index history belong vector set dictionary I sum eqn identify subset node dynamic structure induce note eqn recover temporal output associate source dependency causal life simultaneously stage extract group gene group represent gene insight within gene conduct four dictionary kernel output show four functional input nonlinear predictive imply great imply causal graph analyze approach centrality key activity recognize contrast relate bind bind factor bind activity consistent biological knowledge capture connect protein bind successfully fail recognize relevance likely link suggested estimation provide insight show protein bind estimation separable predefine output optimize coordinate descent method computationally costly work scalar eigen gauss solve system quadratic solver inexact provide rademacher value analogous scalar multiple learn literature also causality high class functional theorem york usa h technology c research new usa lemma proposition framework high dimensional allow broad regularizer induce reproduce inexact solver simultaneously framework high causal cast lead graphical development analysis endow suitable reproduce formalism value far yet value extension popularity responsible complicated value turn polynomial default require associate greater value solver cubic multiply scalable learning become necessity broad family regularizer induce serve optimally combine value kernel algorithm simplicity separable kernel definite formally define full kernel estimate effort one address scalability carefully inexact solver subproblem value hypothesis extend result case enable causality believe state main body multivariate amongst naturally motivate algorithm treat independently reproduce rkh attention value challenging selection problem certainly scalar case contribution kernel technical element exist smoothness select elastic net penalty conjunction square scalable block gradient fista learning solver overall turn multiple classic var traditionally time vector regularize forecasting collection time small induce subset endow causality interpretation benchmark least follow eq reader value rkhs principle overview supplementary value evaluate input nn sense pair solution dimensional sized gram comprise ji identity compatible development familiar general linear value improve though costly scalar comparable input matrix semi definite system eqn eqn solver solver eqn method e solve completeness though point cubic solver diag diag output recent elegant extension eqn separable kernel correspondingly language denote trace denote frobenius denote cone stationary globally propose block fix show large dataset take day standard corresponding vector reproduce reproduce f f j eqn eqn joint scale denote gram individual scalar version reformulate strategy minimization keep minimization termination return eqn subproblem conjugate solution eqn dense prohibitive repeatedly outer cg solver massive cg advantage special kronecker eqn warm previous termination never cg q cg fast matrix fourier l never explicitly cg rapid iteration strong jointly trace eqn possible formalize convergence cg solver current iterate warm start dictionary gaussian scalar strength give material weight value oppose essentially reason mkl immediately close j elastic net type penalty also discussion regularizer structured sdp solver general differentiable successfully completion iteration optimize linearization result eq curvature eigenvector small eigenvalue appeal iteration note unit trace meaningful optimize gradient diag matrix search search direction k adapt eqn iterate g additional small block descent convergent discuss chapter proposition offer rate associate inexact subproblem quickly stock multivariate output first autoregressive var week eqn
handle work particular stochastic obey admm value quite online refer concerned design stochastic admm modification stochastic class function might directly update nonsmooth deterministic call solver consume contribution algorithm structural good feasibility contrast differentiable notation deterministic simplicity clarity denote stack vector tuple semidefinite vector product euclidean ambiguity often assume quantity frequently present algorithm list statement nontrivial constraint augment lagrangian minimize augment least easy alg gauss minimizing alternatively follow lagrangian initialize k variant name linearize admm alg semidefinite alg linearize proximal point note linearization help line alg close solution alg share linearize one line replace approximation mirror stochastic nonsmooth smooth linearize prox scale stepsize stepsize admm alg exhibit convergence first variation utilize probability adapt place able obtain deterministic admm exist literature finding aspect admm prove feasibility admm direct reach convex convex take alg assumption b magnitude scenario function rate besides assumption admm applicable broad subproblem minimize augment structural convex nesterov help achieve rate lemma divergence prox convex scalar bregman proof convexity definition alg eq handle last condition alg convexity alg inequality result operator derive feasible second k k apply therefore order claim adopt implement follow strong last bound
lie subspace might apply vanish limit become utilize input vector small small approximation superior example kernel hilbert semidefinite allow empirical imbalance replace weighted trace weight task theorem hold reduce equal rademacher average sum understand number benefit transfer give excess use risk lot observe approximated assumption see reference therein somewhat consider column marginal without know unclear convert reveal learn sparsity derive consider regularizer closely fall short introduce general elegant regularizer apply dominant read observe empirical task picture expectation convert dominant eq observation different disadvantage simultaneous letter real space map adjoint call unit bound compact finite linear map compact call adjoint nonnegative self adjoint cone operator linear nonnegative operator adjoint number ie operator eigenvector sequel self adjoint eigenvector adjoint operator eigenvalue whenever monotonicity use operator coincide invariant complement without map compact theorem apply represent restrict subspace tr tr x ax ax ax ax e basis case independent random satisfy q k r get kk pass eigenvalue combine inequality prove conclusion follow give slightly almost iii q previous subspace contain range realization contrast span argument lemma show rademacher span contain distribute k implication together jensen divide elementary algebra q line lemma iii excess risk thus total map inequality definition third hoeffding technique bound q q rademacher eliminate loss b tr ss operator old jensen tn v v td q central range expectation verify integer occur rademacher imply cauchy schwarz j b x nj b conclusion since b q together use jensen bounding term help subspace range rank invoke jensen lb together theorem algorithm axiom conclusion condition conjecture summary em height depth ne ne pt pt skip h ne abstract chapter head head head head head head ex compatibility conjecture compatibility compatibility compatibility compatibility false false mu size false mu mu mu th mu mu end align align end end end end end align environment environment environment style array allow false true tag tag true department university college popular excess explicit independent space infinite sum semidefinite fundamental limitation incur sample potential multi sample size small collection sample individual sample machine try theoretical preferable seminal bound trace time consider input probability pair observe risk minimization model z assumption remove replace attribute intuitive
hyperplane separate r coordinate xu coordinate coordinate yu xu xu coordinate away coordinate xu yu yu xu xu yu cycle baseline coordinate xu coordinate yu intersection coordinate away intersection xu xu coordinate away away cycle partition right model fair coin elementary gamble indicator gamble gamble gamble second gamble gamble ff assess acceptable assessment interpretation gamble ff acceptable decompose decompose wish minimal happen happen whenever intersect sided gamble gamble reformulate intersect acceptable cf proposition interpretation point interpretation restriction agent might make second background face assessment answer assessment coherent lead compatible coherent put correspondence remainder devote answer question relation interpretation turns uniquely associate coherent dominate coherent maximal uniquely mean background model compatible whole assessment gamble uniformly dominate gamble use equally subset interesting arise discuss well treat coherent real price acceptable gamble see infimum price conjugate linear conjugate coherent subset gamble ff weak thing previous concern compatible carry satisfy assessment associate criterion avoid confusion assessment purely statement model see notice abstraction purely statement statement still know direction acceptable price translate background correspond preserve resolve add pt scale coordinate yu intersection xu xu yu coordinate yu xu yu coordinate intersection away xu away cycle coordinate xu yu xu xu yu yu xu yu intersection r away coordinate away statement never less resolve coherent reject gamble relate gamble set back tool lower dominate natural extension coherent incoherent avoid sure gamble encode gamble possibility property assume composition positive gamble associate experiment agent transformation rise gamble assessment avoid confusion e distinction background cf gamble gamble ft background gamble status gamble special obtain overview idea paper exchangeability f linear invariant gamble gamble sequence state gamble exchangeability mention finite sequence atom turn expectation extend infinite started allow character directly accept statement preference counterpart even input transform reasonable requirement whether contain impossible basic numerous practical tool inference core need space large part deriving formulate expect bit essentially carry literature gamble build framework know closed set closure deduce polytope theoretic duality know know topological object able illustrative acceptable gamble gamble empty restriction exposition possible formulate counterpart proof may dominate existence finish pair coherent generalise scale take probability also framework aspect correspond accept thank version paper discussion three confusion mu equivalent assume equivalently former gamble change claim linear difference positively contradiction apply claim one q latter lemma scalar hull union latter equivalent effectively former prove lemma identity c give hand side extension follow corollary claim confusion proposition us inequality fact cone positively scale intersection g g cone use claim show confusion preserve non empty intersection empty still assessment set acceptable gamble family note mu maximal closure proposition prove dominate intersection closure operator nature dominate claim dominate dominate mu assessment dominate confusion mu mu onto wise cone mu gamble already furthermore mu proposition direction separately definition f know thank closure acceptable assessment confusion trivially satisfied equivalent therefore whereby confusion whereby confusion becomes I complete equivalent f equivalent f f f scaling reasoning coincide mean g g f section theorem proof preserve empty intersection empty prove equality claim preserve mu mu lemma guarantee axiom dimensional necessity proposition preserve lemma operator cone axiom maximal proposition imply closure proposition follow condition counterpart lemma side mu mu infer existence equality contradiction thank proposition verify confusion thank proposition accept accept gamble avoid lemma repeat add write condition state lemma explicitly go calculate claim lead theorem final identical converse equivalent theorem change substitution equality pp mean confusion contradiction coherent gamble intersect gamble linearity pf pf p assessment confusion positively scale remove cone fact pg ff create confusion proof avoid confusion avoid confusion avoid confusion equivalent prove expression follow distinction separately prove coherence gain proposition homogeneity inclusion separately reject left hand element f f coherent homogeneity f ff prove claim sequentially explicit derivation closure topology claim ff equivalent coherence tell confusion empty empty section reject uncertainty reject preference inner outer fill thick dash ex em axiom modelling accept reject gamble preference relation bridge simplify variant provide modelling statement reject preference tool deal lack phenomenon quantification reasoning uncertainty assessment event conclusion optimal mathematical framework around gamble value uncertain acceptable reject one speak gamble expectation expectation reject framework start consist gamble expert restrict unconditional theory utility express uncertainty gamble useful application arise symmetry character strength type express course expressive already mention representation far work event gamble closely speak gamble expectation experience working gamble attention theory simplify application basic bring centre focus develop uncertainty statement representation gamble incorporate much work theory specify probability expectation introduction concept give clear interpretation available unique precise possesse deal probability correspond preference order event gamble probability gamble may whose partial basic towards sometimes strict preference relation associate preference neither basic separately way preference axiom order preference gamble coherent uncertainty set gamble although use representation exposition prominent set acceptable keep type gamble present strict gamble discuss contribution propose axiom theory add allow strict preference axiom define axiom associate axiom call axiom avoid sure extension probability natural extension school want mathematical applicable analogously g opinion transform discuss construct goal basic conceptual set mathematical course useful conclude investigation section small section effectively symmetry improve give description nature gamble subsequent main criterion confusion consequence scale gamble whose reject axiom extend assessment full incorporate requirement axiom conceptual central assessment obtain rise answer question perform resolve encode accept contain answer important gamble section framework important assume accept shift away framework work relationship coherent probabilistic reader preference framework translation concept relation application theory simplified preserve express separate preference gamble encounter outcome construct exclusive outcome guarantee possibility apart axiom infinite space topology intuition finite formally name pay uncertain outcome pay express precise gamble gamble multiplication constitute space origin agent gamble coordinate xu fy yu fy possibility gamble format example fundamentally negative connect exist concept prove convenient concern operation gamble h secondly gamble value gamble associate subset section object axiom rise room priori agent ask gamble remain agent pay statement statement reject agent force lack experiment pay rejection gamble acceptable set similarly denote xu coordinate yu coordinate xu yu xu coordinate coordinate yu finite format accept gamble reject one extend discussion accept reject gamble thing indicate choice reject gamble mean loss base category accept reject reject accept reject gamble accept reject similarly non acceptable gamble mean rejection confusion avoid axiom confusion confusion although confusion ideally investigate remove confusion number confusion remove gamble accept gamble gamble statement gamble either accept reject assessment wise interpretation give statement agent gamble terminology moreover confusion course gamble neither acceptable operational gamble illustrate later statement make gamble gamble gamble gamble gamble though call accept determined pay express linear precise utility acceptance statement generate acceptable positive utility reject impact axiom dc express ps g extension never remove statement remove confusion assessment assessment mu feasibility assessment finite statement feasible reduce plain example possible remove closed flexibility modify suggest ensure assessment still confusion remove gamble gamble accept take formally gamble part cone either cone couple close gamble acceptable status gamble set cone cone solve optimisation decomposition sum like use complement space sum r yu intersection xu xu yu intersection yu yu cycle coordinate yu intersection xu xu xu yu background xu xu yu yu yu away away away cycle apply three right grey gamble include apparent work gamble amount thing close close necessarily general e even non gamble affect reject remove confusion close one remove rejection statement closure impact apparent I make assessment interested increase confusion close assessment exhibit increase confusion close assessment confusion gamble gamble reject yet thing reject confusion avoid tell closure gamble gamble consider acceptable confusion statement assessment satisfy confusion hold simplify closed confusion f start assessment gamble lead axiom definition increase create closed mean assessment close avoid confusion extension without assessment confusion without confusion result assessment assessment confusion wants remove confusion case confusion confusion set reject gamble gamble something deal something confusion coordinate xu yu background xu yu coordinate yu xu coordinate away xu xu yu coordinate yu xu yu away away xu coordinate yu background intersection xu xu yu away yu xu yu r away away cycle intersection xu coordinate xu yu yu xu yu intersection xu coordinate away extension depict grey gamble dash line example reject gamble agent gamble certain small arguably last gamble abstraction gamble visible accept gamble gamble acceptable indicate gamble example ray ray ray gamble reject gamble support material pair wise inclusion say former component resolve terminology gamble gamble exploit provide terminology component assessment acceptable gamble effectively transaction assessment gamble include restrictive restrict potential scale xu coordinate coordinate yu xu yu background xu xu yu intersection xu away intersection yu coordinate away r intersection coordinate yu xu yu intersection xu xu yu away yu xu yu away intersection xu xu yu away intersection yu yu away r away away cycle coordinate xu coordinate yu xu yu xu xu yu yu away away yu cycle r away xu coordinate cycle away illustrate neither right consequence one union operator play role infimum north xshift south west right ex east yshift west rectangle yshift east leave assessment confusion gamble nine class whether gamble acceptable illustrate label gamble visible gamble gamble symmetry proposition maximal gamble accept reject maximal class empty pt sep ex outer sep ex sep ap xshift ex east xshift ex south west yshift west rectangle yshift ex east xshift ex south west xshift ex north east yshift ex west north east node shift west rectangle shift south regular side outer ex ex sep ap rp east xshift ex south west node right yshift south west rectangle yshift north xshift south west xshift yshift west rectangle yshift north east shift ex ex north south east background stay empty useful assessment category encounter gamble invariant symmetry second show gamble maximal summary concept notation quick reading remainder gamble reject axiom confusion bold plain derive axiom gamble must acceptable assessment assessment close axiom gamble confusion acceptable reject assessment confusion element rejected gamble assessment assessment coincide axiom expression accept reject status accept framework former read despite imply nature relation follow axiom accept confusion closure status translation axiom theorem equivalent status gamble reject accept accept independence reject f vector link two useful two gamble exchange vice say exchange corollary preference f h h preference reject ideally suited decision make gamble accept exchange accept moreover stress gamble gamble arguably gamble latter cause statement instead relation conceptual moving equation pt scale coordinate xu coordinate yu intersection yu xu yu yu xu yu coordinate away cycle yu away xu away fm fm leave fm fm ex coordinate xu yu yu xu yu xu away xu away away intersection away relation illustrate acceptable reject accept acceptable reject reject background assessment go associate impact pair status coherent gamble monotonicity reject need look simplified statement allow framework restrict term allow framework model gamble side inner sep column sep sep ex ap rp xshift north rectangle xshift south node right yshift ex south west ex east xshift ex south yshift ex south west rectangle yshift ex give illustration reject acceptable gamble view statement therefore specify assessment provide set use requirement example gamble loss create g subset satisfy closed restrict attention useful cf accept assessment furthermore intersection structure closure counterpart accept care proposition preserve non intersection intersection proposition set cf coincide close accept confusion mu proposition intersection structure closure satisfy condition lead like dominate purpose accept proposition without status therefore accept status closure accept compact status accept attention gamble assessment accept status background status acceptable gamble cone deal restrict statement reject gamble acceptable end status confusion status would lead reject exclude compatible regular ex outer ap xshift north xshift south west yshift east node xshift south west rectangle east yshift west yshift north east illustration simplify accept restrict either statement statement impose gamble statement something assessment set set situation also remain restrict specific version cf mu yu yu coordinate away intersection xu away away xu yu away away away coordinate xu yu background yu yu coordinate intersection xu away yu away away baseline coordinate xu coordinate coordinate yu intersection yu intersection xu yu yu coordinate intersection xu away yu cycle away cycle shown graphical gamble gamble either conclusion still assessment status framework attention accept gamble theorem assessment model background status gamble form cone gamble linear deal form restriction look theorems extension give gamble background status side size ex none rp xshift north east xshift west yshift north east node south west xshift east south west yshift north east illustration partition work instead general highlight model namely focus lie gamble status f gamble gamble form
depict historical record identify contain set ip ip fs ip ip ip fs ip fs ip fs ip fs fs ip base attribute second variant variant ht c ec ec ec ip attribute ip time period ip table historical table ip unit email count aggregate definition feature machine ip address roughly distinguish high threshold large historical list operation resemble heuristic heuristic machine applying threshold historical another predict train set ip period ip slightly mainly second predict mean ip nearest future whether close heuristic historical window change ip predict ip threshold ip add list auc score consistently examine configuration compare ml set difficulty set tool aforementioned ground past window spam oppose currently black design filtering module list list mechanism incoming instance step cause email accept instance accept pass step order reject processing list contain ip final list heuristic receive ignore never reach space contrast continuous non ip pass operate activate predefine controller address address ht setup l machine history n auc bl I n heuristic min min evaluate use email receive online comprise label un receive online list dataset automatic device claim spam rate partition send set validation mutually exclusive vice mechanism roc curve metric enough load mail potential customer classifier tp tn false negative roc rate versus tn auc equal unity positive classification bad binary example auc auc consider performance discrimination experiment ip time amount resource spend email black translate without inspection white less spend spam filtering within learn mode low reject high result heuristic whereas among update auc auc metric well superiority white drop sensitive stay subsection suggest aggregate behavior create try address least list update investigate window construct construct window induce net instance income email black list per history list window sec bl performance historical time window figure historical improve historical auc score change setting w w respective performance experiment windows window rate window surprisingly decrease time window explain decrease dimension problem k dimensionality notice decrease historical window increase window probably reduce ip contrast black increase window trend interestingly window average since auc window window great nine overall window list increase aggregate email test heuristic email log art effective nine window similar approach another interesting base roughly classification relate g base particular believe multiple boost mechanism base set enough motivate effectiveness clear spam filter frequent execution less power inefficient period tendency spam short explain compare email service activate influence length general window certain result increase efficiency list decrease entire filter system grow auc occur probably dimensionality play mail history historical window small historical increase behavioral select currently due record machine construction end requirement suggest I bayes spam usually list list frequently minimal list content base slow demanding filter list list increase process process ratio error increase update black list white spam email list content filter black entire energy email filtering partially thank useful remark er email service spam continue gain vast send automatic distribute world spam manner propose new aggregated datum encode mail transfer agent historical receive learn build predict mail transfer near update white list internet service spam less effectiveness rely addition black white eliminate content inspection incoming result reduction filter load survey show email spam mail account spam directly exploit computational service mail box spam filter spam computationally sometimes speed spam complement rough incoming maintain gain spam attack flexibility important email receive reach refine strategy learn mail focus extract feature communication elaborate mechanism help spam attack manner analyze try pattern indicate address exploit especially require quick detection mail transfer white list address maintain email target suggest add black list email service email service historical machine create list empirically email setup pass describe encode filtering spam pass positive list incoming discuss section spam l level application spatio temporal three prominent approach spam work focus spam roughly divide three content filtering real refer file produce feature upon make email high open model content lot however internet service perspective classify incoming email heavy content filtering costly later disadvantage continuously plain spam spam ip list ip regard accept reject require email computational resource systematically spam keep track report email entirely ip inaccurate heuristic lower allow spam filter mainly low resource spam compute email base extract network social et al system change behavioral reject gain detection change address repeatedly spam usually accurate address email long exploit due high large service create address detect change address quick reaction require email service massive spam attack attack spam finally internet spam know ip range heuristic dynamic currently configuration dynamic address arrange range easy necessity address address ip heuristic furthermore large spam aggregated spatio application extract incoming email share similar domain work base email et classify email ip abuse tend ip another level domain reject email cluster report data spam organization filter al spam new ba imbalance spam spam spam mail attribute compute computationally mechanism positive could unbalanced use ba classify spam spam receiver show precision rate high communication filter trust system network example receiver likely list j al email book degree social email header construct single classifier email white black report white list test set black list list spam mail spam belong network white spam mail example spam supervise collaborative approach network email user assign high close friend assign friend network create white recommendation low positive several email domain email email record establish received investigate accuracy mechanism spam fully receive spam less another focus solely geodesic ip email receiver comparable ip liu assign return low volume accurately spam maximize west introduce name spatio potential spam email author identify spam pass positive work rate scenario week email email service hour email email maintain spam white email pass list set way address implicit ip inter notation ip ip ip nr relational represent line mail nr address error spend process email mail classification spam present property example email cm spam ip ip ip ip ip machine ml base see play significant fact able address pass email nr pt rank instance spam respective list list roughly art log attribute target attribute spam spam instance frame hour apply directly email heuristic base solely send past continuous exclusive ip denote ip start spam send ip window ip ip email window behavior history behavior ip time window describe objective black white list manner behavioral
control communication sensor rare optimality transmission observation especially large analyze suggest sensor positive message communication accumulate time suggest positive integer eq rewrite message fusion whether transmission fusion fusion fusion information likelihood ratio suggest detection continuous treat separately however structure design asymptotic suppose continuous close exactly sensor center value center kt corresponding eq fusion piecewise time centralized recursion fusion frequency transmission sensor moreover possible communication sensor threshold attain target value specification simply previous clear preferable correspond centralize practical excellent additional centralized test proof threshold sensor asymptotically preserve want transmission interesting claim walk increment n triplet order setup set need threshold j suggest specification resp interval resp clearly threshold computation follow unbounded establish centralized moment bad centralized change equality exact could time fully recover sensor small number bit main remainder design sensor communication j j admit expression simulation rare overcome k find j efficiently ratio account unobserved fusion fusion receive center approximate clearly quantify delay actual k lemma way fusion likelihood prove connect alarm period play role establish result consequently asymptotic ready communication order u divergence sensor come value latter quantity write k desire additional go infinity rate bit transmission order asymptotic achieve bit transmission low communication imply asymptotically optimal optimize magnitude emphasize alphabet bound away first optimality communication accumulation quantization error source alphabet large alphabet elaborate depend sensor minor become statistic measure contrary quantization period bit period I become obvious ask bit fusion center preserve quantization degradation quantization scheme increase reduce quantization almost illustrate conclusion simulation study take independent normally observation variance every k r bit per message period compare rule use bit communication communication centralize require transmission sensor curve centralize optimality operate asymptotically scheme infinite communication correspond period operate characteristic parallel optimal use two per observe similar gain sensor fig novel decentralize detection order performance bound rate show order optimality preserve decentralized detection independent need centralized decentralize assumption sensor indeed guarantee form adapt path sensor imply sensor correlate dynamic long indeed case every wiener square invertible b b log likelihood decentralized model sensor observation appendix subsection stop use choose ct eq sum consequently increase pair divergence equality proof define time due threshold following see write numerator kullback leibler sensor variance local ratio line bind observe furthermore classical walk whereas inequality u alarm numerator change jensen inequality maximal average take k proof goal connect threshold alarm period order elegant proof center indeed message fusion center fusion fusion correspond identity message receive message sensor simultaneous message label fusion fusion receive stop stop time relate physical term message fusion denote instant fusion center connect follow indeed number message every instant sensor message center suffice write component clearly epoch message justify observe v iterate expectation random variable ratio martingale obtain clear change outcome repeatedly proof change whereas j one conditional jensen vanish I u complete eq k z third expression imply moment order asynchronous eq consequently suffice upper clearly borel triplet identically distribute integrable stop prove take eq summing obtain inequality relationship lemma science decentralized sensor fusion bandwidth energy center responsible detect change possible novel communication bit parallel order scheme loss use observation asymptotically false go fix communication sensor optimal finally superiority decentralize rely class suppose area number fusion sensor occurrence sensor fusion exhaustive review refer however rule application mobile wireless communication surveillance system low device characterize order preserve necessary limit load transmission activity sensor constraint center sensor fusion sampling decide constraint rule contrast observation paper decentralize literature e sensor communication shot sensor fusion bit decentralize detection enjoy optimality asymptotically suboptimal asymptotically decentralize rule sensor fusion center communication summarize sufficient message decentralize testing call time establish order false go optimal induce low fusion storage capacity require additionally k transmission communication sensor fusion center sensor possibility impose decentralized detection feedback rule detection rule intractable make simplify compare decentralize rule attain
office grant de acquisition mapping present mapping object consist presentation together target word context exponentially presentation human discrimination limitation acquisition child learn fix two event human capability viewpoint theory child mind although demonstrate exhibit remarkable difficult account book review mechanism child issue human statistical principal mechanism necessary endow capacity object statistical term observational idea learner determine place cross mechanism build coherent mean confirm evidence whereas essentially counting occurrence object albeit much sophisticated counting contribution derivation explicit mathematical characterize learner performance study strategy work minimal word must object context refer occurrence object store count time occur word pick great co frequently paper expand et offer analytical expression minimal scheme finding effort monte compare performance human learn performance algorithm discrimination capability introduction law human somewhat act law eventually match organize sect introduce scheme counting independently learn complete word sect subject understand storage capability describe finally sect finding conclude remark word object mapping represent object distinct without name word object represent integer mind entity event say distinct object form present learner guess refer multiple ambiguity word list wrong association decrease exponentially learn subject never experimental mechanism deterministic selection stage may frequently selection guarantee word trial error mean utility dictionary basic point represent integer entry yield zero object appear confidence increase trial update consider correspond word simply select recall correct mapping trial fraction sect reciprocal easily detect intersection realization minimal immediately adapt incorporate magnitude ideal scenario salient minimal learning learn update aside rescale sect together word associate word sect whole event object state may incorrectly always equal trial possibility unchanged appear reasoning trial know trial q chain yield absence object word pick comprise hyper since transition main correspond transition leave unchanged eigenvalue recall write term produce analytical et derive analytical episode choose maintain notation mention curve result circle carlo simulation lower choose first time n total deterministic easily introduce integer deterministic variable rate rate single accordance intuition task complete fact learn monotonic reach increase sampling scheme lead non whereas realistic situation context linearly large find much random sampling produce result simplicity minimal analyze previous contain powerful regardless second perfect always large confidence regardless closeness say relax perfect ratio magnitude accordingly select object probability differ among identical confidence decay implement subtract word may difficulty rise responsible context relaxation perfect sampling analytical frequency compare aim algorithms map training episode episode comprise presentation word ref condition word divide word appear time condition divide time time time allow episode carry run modify result show fig straight line deviation discuss fig subset fig minimal quantity excellent agreement get excellent agreement model intuitively appear frequently outcome actually depend fig find frequency experiment subject three may considerable improvement direct consequence discrimination discrimination select trial imply discrimination large course large show fluctuation consequence ad hoc procedure summarize fig storage discrimination capability human introduction confidence bring sake minimal limit sect nearly impossible subject attention focus sophisticated thing attention learner describe briefly confidence adjust accord index run entropy word weight entropy ref explanation comparison experimental performance mapping simplify sect I circle summarize finding selection al reproduce symbol independent slope large insensitive minimal realistic conclusion vast extensively original contribution conclude although study determination scenario quantify learn learn author various inexact recover non pass introduction effect capability minimal previously object episode fix learn except word dependent
pi ji easy calculate mass expectation consider expression follow suggest require expectation outer calculate availability collection miss context target observation em need infeasible cardinality resort know em section p monte carlo particle iteration update involve step weight know stochastic em satisfy choice calculate calculate average particle smc em respectively da proposal available author smc obtain approximation z sequentially weight alternatively resample accord weight degeneracy contain index I particle resample particle proposal connect th path carlo step smc good sequential proposal show implement monte complete situation update receive alternative smc linear point statistic compute require expectation see memory time handle showing evaluate expectation functional also allow density technique intermediate eq additionally possibly dp b smoothing recursion follow else calculate smoothing treat separately perform show intermediate quadratic scalar ie obtain ij ij sufficient appendix calculate dy notation simplification show calculate extend target newly target time detection association ti tb step observe follow convention irrelevant multiplied handle statistic recursion target target target recursion k z mx recursion write subscript ij r ti variable initial conditional sufficient calculate variable filter expectation sufficient conditional sufficient respectively calculate store target terminate write function hence online rule implement short represent q forward recursion ti ti x birth prediction th th detect update recursion ti ti z tm recursion save expectation include completeness availability manner online average update estimate online update z modification reflect illustrate recursion k expectation calculate finally calculate add stability notice smc em implement z nj stochastic obtain b nm z list expectation conditional monte recursion incremental statistic th expectation target th target target tm compare velocity velocity assume estimate velocity target life expect around target per method batch smc implementation smc assignment sample association set smc appendix mcmc run statistic smc implementation use estimate use use follow x mcmc em respectively smc mcmc iteration estimate generate comparison execute result need give density I see smc em around mcmc mcmc move fix dominate average target cost smc dominate good assignment one smc far slow figure true approximately iteration mcmc slowly induce step preferable careful choose influence smc smc often without may smc online vs pass start smc cost pass roughly though introduce concatenation e target target free surveillance survival parameter depend crucially however little em run ii smc large target em use good assignment complexity em execute comparison create window surveillance approximately target time mcmc em bayesian em well worth stationary em converge iteration axis demonstrate first create scenario number target surveillance estimate slightly target stationary assume velocity trajectory create length target except use burn execute note target effect target thus online mle parameter initial negligible parameter distribution estimate batch online target estimate converge value particle estimate online estimate compare leave suggest target check comparison initial robust k bold plot experiment velocity target generate length take figure show around estimate quickly set true know birth death death indicate horizontal value bias namely illustration monte dash illustrative every precisely poorly run em birth drop indicate bias birth death unknown target hmm therefore birth death generally speak smc birth death tracking per finite integer variable assignment bias birth track assignment detail expect smc good obviously accuracy smc raise particle online em unknown address issue smc simulate smc target bad else stop compare birth death track present infer comparison offline implementation prefer smc disadvantage slow convergence smc batch cause boundary smc target clutter time budget smc em easier restrict smc parameter birth death accurately particle verify recommend consider use track estimate linear gaussian extended mle merge target target per need allow bernoulli measurement gaussian gaussian distribution target centre surveillance region type non monte sampling state provide require useful intermediate q smooth update online modify update multiply rest mainly section present exclude particle calculate perform calculate prediction assignment produce assignment score assignment infer eq q simplicity true smc filtering particle constant track average simplicity approximately batch smc trajectory sufficient statistic bit smooth expect batch apply data size us length particle track close expect time term batch iteration online use calculation whose stationarity cost smc stationarity solid example offline static target present expectation algorithm monte carlo assess simulated scenario estimation provide concern move observe extract motion traditionally surveillance problem biological framework comprise independent move surveillance region fashion arrival target birth initial target target observation record generate say spurious observation generate target collection generate record without target target extract motion trajectory record highly challenging problem large algorithm track move target handle association origin record tracking variant association monte methodology carlo smc mcmc smc implementation filter huge developing largely rarely case know include derive poisson target central filter additionally likelihood static target gaussian henceforth batch linear model exact novel development tracking stress virtue false measurement target generate false follow distribution mean value uniform superposition
shoot similarity use guide knowledge visually object however category contextual likely contextual exploit category object detector boost update round pixel work separate first step property non detector feature different category crf segmentation select previous thesis support exception early classifier task task hyperplane select feature perform logistic task likelihood adapt heterogeneous leave optimize cluster area visual huge underlie basic idea transfer important small allow quick thesis topic visually recognize rough suggest know approximately day learn appearance new visual performance machine method especially recognition thousand transfer learn technique try still gap human vision illustrate analyze category possess therefore number annotation common expect category illustrate plot phenomenon scientific current art detector detect want extract rich representation image try car etc likely rely category recognition weak logarithmic law similar database problem real expensive impossible arise identity person allow security obtain hundred person especially illumination consume hold verification handwritten recognition another scenario preference rating generalize rating high suggest probable application solve example simply improve rating competition lead amount relate case explain section prominent application vision quality production piece check manual control need localization arise obtain kind use example underlie challenge example common traditional machine assumption insufficient notion task database visual task classification cope clutter illumination diversity category intra class variance number representative learn high task number intuitively polynomial severe require require broader deeply theoretical bound possibly infinite hypothesis achieve refer sample task valid follow example sufficient achieve regarded complexity measure hypothese vc close regression vc variate polynomial exactly nearly indirect manner view say inherently try solve pose possible role reduce possible advance important want visual recognition often regularization smooth solution mostly possibility utilize unlabeled datum estimate manifold number unlabeled datum thesis machine vision amount preprocesse reduction classifier manually software visual classification task automatic reader detection increase expert prior knowledge decrease need however manual goal transfer knowledge new task automate advantage traditional build new manual effort obtain concentrate label example transfer tb concept recognition concept domain transfer linguistic interference quickly learn new thesis conceptual difference transfer contrast transfer single task rather emphasize rather class distinguish even though exploit transfer example recognize quality image digital relate body try handle lot superior example manual knowledge domain training concentrate principle transfer section survey perspective journal paper comprehensive machine also cover broad early development task couple incorporate answer application transformation manual supervision face benefit e illumination rotation try gray histogram target approach text estimation important directly apply without align face image current important relate metric piece early technique metric find metric minimize single maximize distance category apply use near learn mahalanobis identification task find distinguish car work classifier instance obtain training database adaptation apply gaussian decision appearance term shape transfer share base modeling combine target classifier learner boost transfer learner achieve preferred specific concept propose classifier extension try learner lead reduction slide approach evaluation grow logarithmic category benefit difficult assumption task valid estimate text news subsequent eigenvalue incorporate latent framework allow use approach transfer visual object inspire lot common target prior appearance share instance part lot manner connect prior distribution bayesian last regularization share task relate assumption share feature transformation regularization instead work idea propose optimization jointly utilize machine allow modeling want hyperparameter marginal idea categorization task powerful perform parameterized product compare
unary marginal extended even calculation partition bipartite interaction homogeneous sake simplicity bipartite discuss previous assume order first contribute factor reduce unary computed recursively graph partition invertible unary yield value bipartite vertex unary partition complexity show partition sum efficiently field complete homogeneous similarly partition sum homogeneous pairwise expect computer vision expect evaluate compute good class field unary factor outer width unary marginal thorough improve original manuscript author wants express special valuable strict motivating technical receive degree physics research interest probabilistic article accept publication issue journal note attention ise homogeneous pairwise potential unary potential probability bipartite homogeneous potential class large field provide markov field unary inference pattern vision markov gibbs restriction underlie structure marginal probability ise estimation propose new improved calculation class field ise homogeneous pairwise potential unary potential similarly polynomial complete bipartite provide homogeneous potential equal sum contribution unary factor homogeneity pairwise unary recursively sub label either vertex label remaining label dynamic programming size mapping unary probability calculate unary generalise field give denote notice assumed vector respective value kronecker contribution
comparable bioinformatics system science perhaps exponential possible match suffer place mass empty complete graph propose modification degeneracy non density applicable relatively small modification degeneracy describe family constrain around principle discuss distribution network moderate density model via outline possible future exponential distribution describe contribution variable base exponential I canonical open assume denote ff dirac family satisfy exist unique provide poor family find mle modeling assign little mass family approach function place around distribution heavy tailed approach use degeneracy case choose black red line non exponential tail use combine family add symmetric positive indicator version indicator hypercube center side kde g kde match accord empirical kde bandwidth dimension preserve determine multidimensional assign smoothed tuning canonical augment parameter ne constraint point know induce allow moment satisfy set aware capability turn learn undirected vertex loop complicated commonly model degeneracy place graph empty away modification parametric family degeneracy statistic feature motivated graph hull degeneracy mle find place mostly complete graph place c tb degenerate red bar right blue mass low address degeneracy information degeneracy attempt address degeneracy specific edge exponential suggest approximation truncation degeneracy interior due modify geometry towards degeneracy modify geometry space category modify uniform neighborhood graph family probability preserve function eq concave several challenge fit computed form mle sg however graph computationally expensive equal result modal close mode predefine illustrate constraint normal df simulate vary sample sample parametric assume schedule kde determine choice obtain exponential choice kernel kde kde estimating kde gmm density quickly consider model show sample increase similar moment schedule estimate mix sparse bad enjoy parameter whereas schedule schedule statistic non zero estimate normal number distribution zero normal zero estimate mixed compare discussion fit make commonly fit measure degree proportion share proportion edge common geodesic proportion connect pair minimum tb kp business datum set ad datum kp fa ad unique max dash line red dash line line close black line red triangle first consider tuple compute mass function put large vary since gibb use package mle sample generate whereas graph goodness test sample run markov chain graph tune scale find need variance able sample sample count triangle need state graph suggest considerable family capable fall empirically increase arbitrary continuous density control sparsity fall estimator global constraint require mle optimization raise challenge constraint adapt graph degeneracy bandwidth acceleration acknowledgement help proof nsf theorem augment augment eq augment statistic respectively prove convergence half kde expect kde uniformly positive kde kde f density identical draw ne assuming family parameter nf n n ne family chapter w e exist sure additional constraint match constraint small reality rarely condition though constrain regular canonical closed subset space pointwise parametric inequalities kde ne kde ne kde ne kde kde kde density kde continuity density n ne satisfy continuous combine lemma corollary
cast video subspace include subspace particular ssc datum lie union challenge unlike unified framework corruption membership subspace spectral cluster contaminate noise point perfectly sparse nonzero error free perfectly lie solution entry sparse sparse precisely vector entry linear magnitude square root dimension subspace program collect objective function program hence programming tool optimization hand corrupt eliminate deal noise consider datum program similarity infer cluster incorporate corrupt separate correct datum incomplete fraction incomplete cast fill datum follow apply cluster graph coefficient drawback fact know case cast miss complete see define index denote ambient column ssc keep row complete base nonempty address problem problem lie motion segmentation involve subspace naive way deal case cluster subspace fact include origin drawback increase origin indistinguishable deal fact subspace solution lie subspace comparison subspace subspace well also affine ssc representation nonzero datum point representation datum subspace direct operator figure leave space span every intersect origin show subspace intersect independence converse characterize two define principal union subspace underlie cluster minimization specifically n recover subspace without price subspace disjoint subspace appropriate recover consider investigate intersection restrict also solution restrict subspace except supplementary ssc succeed subspace intersection strictly norm precisely result minimization recover subspace sparse successful explicitly show program depend angle establish datum minimization successful sparse hold subspace angle polytope nearly degenerate reach I n I I I hold nonzero recover subspace speak sufficient principal certain subspace recovery norm subspace segmentation datum apply ssc norm subspace subspace condition always recover closely property difference optimal instead feasible violate feasible solution still optimal successful result subspace number relationship column hull polytope reach interpretation nonzero small polytope recovery middle figure sparse reach degenerate close orthogonal hold note sufficient translate subspace angle section study program recover sparse result point lie get synthetic always hence component verify subspace subspace great odd graph subspace right increase row common similarity correspond graph program connectivity subspace principal angle subspace angle merge add indicator hence number row choose common sparse np consider connectivity point trade add term representation demonstrate term regularizer objective subspace angle well solution recover uniquely show figure connect feasible representation hence minimize sparsity regularizer show success principal angle consider embed dimensional ambient subspace reconstruct linear generate basis two subspace addition subspace angle equal verify change sparse data increase probability normalize point subspace program error denote sparse I inside summation norm subspace correspond choose subspace choose sparse coefficient spectral ambient subspace trial predict decrease also success minimization recover representation note small increase undesirable spectral also subspace error imply exactly component ssc real face ssc art lrr ssc direction multiplier whose supplementary material variation optimization constraint choose experiment face experiment use program general solver coefficient spectral art dimension face dimension subspace segmentation lrr face note lrr accord ssc processing similarity effectiveness sparse rank objective investigate lrr lrr processing method motion segmentation alm variant subspace priori laplacian noisy set comparison subspace segmentation sequence video cluster b face subject subject lie describe present experimental present first pair motion segmentation face subject small principal certain range subspace dataset angle principal angle always principal angle degree subspace compute percentage show subspace dataset show subspace b near near plot principal principal angle challenge subspace lrr h median motion segmentation refer multiple region scene track nf vector stack q segmentation refer trajectory affine subspace lie trajectory motion segmentation lrr lrr ssc median median ssc problem consist video video frames trajectory frame singular dimensionality underlie video perfectly lie linear subspace dimension apply trajectory subspace table conclusion case ssc separation subspace angle feature success program number inside cluster ssc without step normalization help cluster result ssc perform post lrr error post try find representation function error dimensional pca projection close trajectory onto dimensional structure close error table effect ssc dataset ssc result ssc coincide random addition datum subject acquire pose vary illumination consider fig subject illumination subspace image well extend face face image acquire reduce memory treat plot value value dimensionality face singular showing corrupt image corrupt lie cluster algorithm devise divide subject group first three subject consider choice apply svd plot corrupt correspond cast image model sparse deal validate corruption face due error importance remove face subject subject experiment performance datum subject remove sparse conclusion ssc zero suggest ssc deal face ssc angle small ssc lrr show lrr relatively show post process low always improve lrr cluster number subject neighborhood addition neighbor subject number c lrr lrr ssc median median mean median apply collection point cluster make ssc low ssc obtain subject respectively come bring common low angle subspace increase respect table lrr lrr ssc median median median algorithm process conclusion subject respectively ssc incorporate corruption model lrr deal corrupt large increase clear corruption lrr general hand post step help large ssc try find rank lrr deal corrupted neighbor subject right lrr median time subject equivalently drastically high complexity lrr fast time supplementary material close union technique ssc build obtain show succeed recover datum ability deal affine experiment face video show effectiveness superiority research investigate theoretical sparse miss datum extensive real point form similarity graph point probabilistic optimization applicable future author like support proposition optimization theoretical optimization solve optimization program q side inequality achieve solution program equal equivalently theorem representation I prove write substitute right independent intersect contradict optimality obtain desire proof subspace also minimization restrict point every strictly succeed recover n I recover contradiction lie vector solution optimal imply second strict inequality sufficient contradict optimality optimization program prove contradiction exist nonzero intersection program select subspace contradict subspace condition norm full column rank thus establish bind norm multiply leave recall definition subspace angle column subspace except rewrite optimization optimization solve solver solver typically implementation alternate first eliminate optimization equivalently overall lagrangian respect primal maximize abuse vector entry admm set terminate update auxiliary whose help penalty vanish make variable introduce lagrange multiplier lagrangian admm consist follow kk kk system large conjugate gradient method thresholde act element give return return fix gradient ascent lagrange multiplier repeat iteration iteration achieve denote dual summary implementation program h ssc subject computational different b function code author mention lrr ssc fast lrr make lin r liu linearize rank fast propose zhang compressive sensing scientific corollary many world collection video text web microarray dimensional several category union dimensional among point point subspace subspace solve sparse optimization program succeed recover desire propose point near intersection subspace entry miss incorporate program demonstrate motion dimensionality programming spectral angle motion segmentation face area image bioinformatic video million web document affect effect insufficient respect ambient commonly refer curse often lie uniformly ambient recover low reduce computational memory improve inference recognition category trajectory video subject vary illumination write digit rotation ambient therein problem separate numerous image processing compression computer vision segmentation illustrate arbitrarily centroid spatial take structure algebraic spectral subspace alternate assign fitting subspace drawback generally require dimension initialization factorization algebraic segmentation thresholding provably independent assumption violate principal fit contain mixture probabilistic cluster subspace expectation maximization drawback sensitive dimension choose enough subspace noise know however must exponentially theoretic
although similar analysis conduct interact colour line mixed computational seem work good explore relationship cs ac collective classification attempt instance tend pattern interaction make instance class pattern interaction blockmodel link alone simultaneously provide interpretable well understand datum explore network long focus predict attribute leverage traditionally modern shift exploit improve performance attempt assume e stochastic blockmodel instance stochastic blockmodel efficient update relative investigate understand introduction new collapse inference avoid long diagnosis sampling update expensive social analysis refer zero occur adjacency interaction role belong respect role role interaction assignment usually accord attribute create role link paradigm blockmodel role infer attribute stochastic unsupervised role role assume homogeneous linkage pattern behave dirichlet name extension model lda dirichlet corpus topic blockmodel derive identify distinction role possible heterogeneous linkage pattern role membership supervise examine standard blockmodel sbm assume role interact interaction indicate absence application blockmodel role thing blockmodel sbm network draw role role interaction interact role draw draw absence link mixed membership blockmodel previously extend blockmodel role draw node interaction role interaction receiver draw role interaction provide py use inference sampling convergence inference method variational introduce approximate role field evidence conjugacy bernoulli model integrate collapse posterior tight bind implementation collapse variational expensive practice taylor collapse inference first order approximation implement standard stochastic blockmodel updating role count receiver collapse remove count role reflect inference class occurrence oppose sbm occur receiver assign rather assign role interaction receiver role involve receiver represent softmax conjugate conjugate gradient predict unlabeled rest label involve generate four examine citation web citation popular collective classification comprise represent paper represent frequently occur novel link network corpus corpus american english category occur noun result link classification type namely primary investigate maximum role varied sbm macro measure f tp fp tp fp negative harmonic mean recall macro use f remove
current target evolve system new position position stability depend sample evolve locate matter center case inside trajectory apart stability extreme small hypercube length center lyapunov hypercube approximately linearize pass well eq lyapunov eq trajectory although converge point stay around example suppose dynamical define high value dynamical however two situation stay rare case trajectory stay eventually could jump yet stay htbp converge stable hypercube strong situation small small q close thus evolve stability fix experimentally converge text especially demonstrate question connection neuron indicate text neural instead flow certain pass boltzmann machine message pass field thus structure necessarily algorithm directly encode neuron propagation satisfy point encode inference encode neural inference semantic learn exhibit parameter joint generalize backpropagation boltzmann machine preserve symmetric correct semantic context local uniquely determined term conditional methodology neuron variable neural network must way conditional boltzmann mapping situation text imply neural without machine neuron binary allow hidden neuron represent support information happen rigorously even though continuous neural preferable boltzmann machine question continuous encode efficiently approximated distribution encode number digit representation digits evaluation approximation inefficient look eigenfunction relationship fourier eigenfunction unify investigate binary preliminary question neuron bayesian model external introduce explicitly neuron correspond correspond observation choose part parametrization I I term hand contain two term recurrent neural roughly speak integral deterministic counterpart get well close match deterministic
limit something usually data collection tweet km radius uk twitter search able retrieve recently publish tweet km location english twitter feed format collect content feed library store request retrieve tweet database collection minute location retrieve project aim collect daily basis obviously short period tweet collect twitter user reach able collect tweet per sampling uk st publish user twitter rough number tweet day daily increase together publish reader plot volume collect tweet cumulative equivalent collect acceleration collect million day side increase tweet maintain representation twitter corpus tend tweet see method library basic field ir machine project deal content library engine index tweet word frequency document create index base fact database library source code create type indice another useful project software handle large scale implement machine algebra operation singular key paradigm operation computer operate retrieve implement comprise tf tf stop rarely cosine similarity need least software library text perform stop space representation well type like college practitioner weather uk weather read uk whereas use present chapter start characteristic twitter service serve main project twitter attract human massive platform opinion mining twitter retrieve tweet provide report collect tweet cumulative refer tool give version grind throughout project mm tracking epidemic reduce impact help plan various employ monitoring report monitor capable disease population twitter hundred thousand tweet search automatically identify turn test uk score obtain preliminary completely independent commonly hence provide epidemic extend version tracking monitor social medium detect infer monitoring epidemic population population insight health intensity epidemic existence demand affected give phone call network drawback delay engine health query engine query extend monitor content web tool twitter micro website option status phone device character mobile text twitter uk quantify various region country twitter reveal stream tweet hour delay furthermore entirely method early stage one twitter life provide evidence social marker marker word form phrase gram topic pick social could health datum validate daily marker appear tweet otherwise contain marker daily twitter corpus tweet divide tweet eq day tweet divide daily retrieve score truth report provide statistic base scheme metric express number diagnosis uk region retrieve equal twitter expand former period assign day week expand expand smoothed peak reflect epidemic uk marker temperature infection score time point move express tendency twitter score day linear coefficient twitter series correlation respective present whereas investigate correlation move smooth locally smoothed ol across implementation present smoothing window produce correlation move average well ccccc south e plot twitter high two notice actually build twitter weight marker tweet q number weight twitter daily compute tweet day marker twitter content twitter marker retrieve twitter unweighted vector unweighted day smoothed move ol time smoothed expand learn weight infer remain region method time indicator retrieve correlation train linear rate region region similarly report various smoothing interestingly smoothing correlation window day smoothing decrease perform produce fact correlation differ point produce interpretable tendency perform slightly well leave investigation avg point assess aggregated score epidemic form use correlation p test fold randomly decide correlation case score unseen train period train together experiment marker relate region uk obviously choice marker deal concept select enable effort make marker extract marker select keyword correlation form create candidate informative create marker relate use reference discuss preprocesse stop removal extract marker topic formation choice marker discuss justified section form candidate feature daily twitter day twitter represent tr day array total move smoothed rate order advantage produce sparse solution discard candidate prove redundant estimating least subject optimisation task shrinkage time region validation percentage region five choice lar apply table possible choice high correlation setting region comparison infer choice non extract marker majority spread heart mention page home ill counter check water confirm phone cancer train avg remain three whereas aggregated day period region respectively day axis day read follow page check home member exist cancer spread ill far aggregate data form percentage remain data value retrieve infer point marker automatically infer candidate signal correlation target unseen automatically marker uk experiment choose smooth marker move tendency unseen use overfitte chance lasso especially inconsistent try issue propose formation gram corpus index reference choice partly justify day period gram stop word appear twitter corpus retrieve gram smoothed move truth expand smoothed ccc epidemic movie movie epidemic movie epidemic movie get quick datum correlation ground correlate marker correlation statistically word derive series word something ccccc media market player school live record death public level bank wave award know region top twitter movie music popular international possibly popular name something signal behaviour target also select day cross weight excellent every term cross something another safe conclude set effect matrix estimate shrinkage bound expect euclidean nonzero proportional sparsity show bound denote loose sample shrinkage norm intuitively result solution holding assume conclusion increase well e far rate turn satisfactory dimension solution reason chapter present epidemic uk twitter give early various mostly plan base micro calibrate extend message send mobile device besides privacy concern character form keyword phrase normalise manually score discover actual ol retrieve one next achieve regressor initial use twitter usually approach ir form feature target wikipedia page health orient nevertheless majority choice justify firstly secondly understand error affect characteristic learn marker scoring function propose truth specific generic present methodology methodology limitation infer phenomenon explore rich amount service twitter study consist benchmark location infer tweet rate effort detect epidemic build lasso amount study show significant different learner core investigate chapter extended web learn mm already chapter detect twitter content core inconsistent variable close gram bad feature english language character improvement use gram chapter hybrid combine short span smoothing something improve application important lastly event epidemic allow finding commonly economic inference regard magnitude refer recent past reference consider magnitude event variable entire content observable proportion life web aim observable detect infer magnitude event tp event entity include web see learn observable logic web web twitter make content concentrate web content type paper paragraph manually relate related keyword application sentiment predefine phrase map sentiment report rate pick similar query furthermore sentiment apply effort extract vote office daily average exploit location tweet detect approach feature method apart reduce human minimum tend great later track component average subset query keyword high value preliminary chapter twitter content automatic rate incorporate detector tool infer besides retrieval distinction lie principle regressor handle candidate feature search methodology abstract summary use observation datum general chapter vocabulary gram marker marker extract web candidate target connection consider gram gram combine propose unseen flow web static behaviour document tweet facebook combination aforementioned comprise know gram indicate extract stream gram depend gram least subset inference gram take candidate marker suppose user stream raw count marker limitation tweet character twitter special setting value marker score marker represent occurrence stream interpret tweet marker tweet score candidate marker keep eq matter depend duration day hold interval restriction time retrieve variable candidate bootstrap attempt shrinkage solution shrinkage lar inversion either continuous range explain look algorithm feature space vocabulary array perform ol scalar bias term strict application feature go soft feature fraction refer consensus ct strict ct decide computational phase include ct retrieve ols regression ct optimal ct definition sample define iw naturally definition essentially try plus square outlier investigate literature research square comprehensive metric present result ct suppose threshold validation denote index testing phase take consideration rate infer zero perform filter testing ct track well go truth ct explore stop reach assume form perform lar entire set lar set uniformly lar end nonzero select ct next decide ct change lie fold cross preferable feature aim evidence characteristic therefore shall compute ct correspond ct feature train p train mi b mi jj train p class candidate investigate gram hybrid gram gram gram interpretation topic base context gram gram consist piece tweet character daily close zero tp v train I exploit advantage form combine try approach form hybrid consensus gram gram hybrid gram class denote tp p q I j q hybrid gram explore gram gram b hybrid combination proceed note gram gram follow class perform something class gram gram entire run unified name bad hybrid dimensionality without training perform feature mainly differ via describe training feature pearson coefficient candidate feature training rank retrieve compute fit incremental top correlate performance select evaluate disjoint set train train train twitter infer daily availability daily weather since piece available majority twitter various tweet easy due non marker study weather relate web wikipedia page language course weather vocabulary weather terminology marker probably good semantic marker twitter study gram keep candidate likewise gram year twitter datum form location million collect size bootstrap complete soon one stop criterion meet essential execution amount cross start month validation fold half marker five location ct principle marker ct batch retrieve inference selection select round validation present ct select turn able rate large number select month month tweet discuss day create weather capture evaluation table interpretation numerical rate equal range outperform total achieving comparing indicate feature interestingly feature denote validation testing fold cm present result validation remain cross validation month select feature gram outperform cloud comprehensive majority gram table semantic connection without direct connection act weather use weather orient positively select gram semantic weight particular multiply cc pour multiply cc cc air light stop wind weather cc cc air weather pour light look cross round inference term marker frequency location unlikely gram cloud font weight word content infer uk base uk region truth information result diagnosis accord baseline daily report interpolation rate consecutive compute factor day within duration week several reference web health service follow general case extract gram gram count tweet involve time collect twitter data period rate epidemic period feature assess datum randomly apply region evaluation tweet period tweet million previous study validation day round fold train fold day ct rest testing contiguous randomly day contiguous wise round fold study feature either significant period gram topic tp ccccc cm cm hold cross comprehensive interpretation equal deviation range class term method improve approach multiply cc cc cc spike huge public remain rough team behavior member perform wave health wikipedia tp multiply cc need site health fit head visit health care loss worker home channel take care multiply cc gram gram gram stage attempt web care check code check site rough health health care worker health home visit wave item wikipedia font proportional weight negative present round class gram body water solution large surprisingly gram relate large feature mind significant generic one tp figure fold cross correlation provide additional significance present carry contiguous day datum test remain formation epidemic period inference outcome smoothed inference move induce trend class experimental method inference public web medium distribute phenomena property uk rather unstable daily whereas evolve smoothly figure picture hard discuss weather precede forecast especially weather example bad day location exact average day gram day one propose overcome select figure high truth randomly turn major base experimental target applicable event draw similar extraction ir technique imply entire instead focused reference event justify lasso section span limit train risk avoid focused event candidate constrain dimensionality training issue nevertheless small word daily tweet location feature gram proportion properly contribute experimental process make manual gram rare stable ct operate adaptation application strict ct would apply ct tp methodology ct validation event twitter linear end predictor weight value choose either function motivation behind fold firstly want metric secondly investigate whether task cart tree nonlinear apply cart cart mostly latter intermediate comprehensive series tweet candidate learner cross validation identical measure performance month fold training month fold either pruning cart ensemble half tp three consider gram gram concatenation gram gram experiment threshold percentage level prune equal cart retain full insight infer cart produce datum gram gram word topic weather derivation replacement tree go ensemble decide validation ensemble average prediction feature present pruning cart cart elaborate bootstrap rmse gram gram combine improve error fold optimal tree use ensemble keep tp level tp tp c min perform variable every subtree importance give importance ensemble table important investigate select bold feature weather identical irrelevant gram bold act bold cc cc week wind like start ic weather meet al feature bold cc cc series uk rate fold day training day datum decide rest experimental apply pruning tp min max tp cm method rmse table cart case ensemble ensemble gram gram drop rmse bold cc cc low school warm live bold cc confirm check light switch care contract bold cc cc live feature feature class marker topic epidemic important ensemble learner conclude function feature far target concern reliable even hybrid ensemble irrelevant might outperform nevertheless select gram seem correlation initial amount must define manually preferable feature ensemble outperform incorporate affect ground section examine probability general limitation infer spread region operate vice versa inferences bootstrap see standard deviation cart divide ci compute multiplying se quantile normal tend quite three similarly model base mean advantage framework inference regression ability input covariance spatial characteristic event chapter methodology capable uk identify theoretical framework already initial insight smooth truth score different location respectively mm median mm likewise deviation previous rate unstable rate time zero rate never rate per inactive tend notable short period try derive likely occur pdf rate histogram well fit pdfs gamma pdf mean logarithm see event normal event frequency tend small therefore inactive inactive active precisely short peak expect topic strong chapter general methodology infer exploring web methodology specific procedure well claim gram gram hybrid combination gram gram give good whereas gram semantic event alone naturally comparison even variant learn interest message propose count frequency disease name well easier exist optimal furthermore obvious actual bias prevent study actual twitter population twitter oriented lastly insight characteristic behaviour generic characteristic look extract norm content twitter focus well daily characteristic daily correlate significant life partially detect uk twitter v uk exact major medium micro website twitter various way range already propose track epidemic infer seem nonetheless explore variation real large via twitter permit variation accept notion certain phenomenon even capture twitter content volume affect day contrary clinical concept load early confirm study show pa rather increase daily effort explore variation user assess period uk tweet within centre bias daily centre text divide bin hour day hour assess level activity namely message text list retrieve priori build select filter way keep preprocesse affect apply extract hour type base interval number twitter corpus normalise interval count derive frequency averaging frequency final scoring time w I iw divide hence interval weighted final figure hour pattern aggregated figure include multiply se distribution interval consider linear correlation high point deviation follow test stability daily sample pattern base day pattern permutation number correlation test eq consider value aggregated aggregated statistically instability consider weight extract scoring investigate stability describe depict first drop start reach apart reach hour much peak start increase reach peak tp produce pattern present drop start reach level something peak steady reach high hour whereas peak slightly reach peak happen level hour tend peak decrease start reach core high p hour tp linear correlation pattern aggregate pattern positively statistically show correlation expect derive apply scoring something table scoring type correlation tp cm tp cm tp section investigate previous section show occur peak unclear extract peak day form histogram peak day depict difference peak frequency tp figure behaviour period occur unstable amongst investigate rarely reach quite step autocorrelation figure scheme lag consecutive hour logic something high affect previous autocorrelation become evident type daily pattern strong significance statistic vector length day compute autocorrelation compute test lag autocorrelation periodic display unstable behaviour term affect apart interesting result justify score overcome bias create life tweet merge na pa pa patterns plot na strong count hour also begin dominate pa slightly affected negativity tp content try pattern level medium uk two main amount time reduce statistical stable result elaborate scheme positively correlate signal clear peak also peak flip peak early combine show peak pa also peak hour na peak take na evolve pa hour signal across fluctuation early peak something na na strong pa pa express express moment monotonicity reach finding state pa states behaviour day recent pattern pa na across day week finding uk one derive uk pa retrieve peak scoring period day scoring extract limitation study people therefore drop collect exclude content bias usually datum characteristic extract limitation arise due general twitter create present claim uk investigate daily similarly focus norm section uk partly affect tweet process million apply interval daily time interval extract daily basis section score count twitter corpus tweet likewise day extract treat equally daily score smoothed version allow retrieve peak smooth time series reveal period affect base norm life turn significant event sense identify time series period explain death people uk reach record security peak international phone period rd possibly day apply yet uncorrelated period month take place uk rest peak happen nd th scheme equivalent output moment signal happen uk day death rd co occur attack side period figure figure produce identify least throughout year significant list year day st day peak much event apart peak uk receive positively event international death possibly weather condition moment occur mark bin death rd death occur attack speed short peak series derive table peak rd period date match characteristic significant usually average minus pa day twitter user together event base peak level negative exchange movement death positive day death across observe follow exist high contrary happen smoothed reveal similarity see correlation might norm set twitter cm cm day lag correlation positive negative bound significant autocorrelation experimental periodic type interpret consecutive day lag consider week indeed large extract one exception autocorrelation happen retrieve include strong day belong principal component scheme see clear separate evident close might also day task cluster day day figure expect moment relate bin death attack uk cluster separate rest new day group also speed observe day probably day uk entirely separate rest place opposite space close form date uk together attack cluster majority twitter stream uk international occur recall define scheme significant event concern uk affect twitter scheme death severe scoring give across scoring day lag lag increase decrease pointing might day day pass perhaps due event interestingly lag equal periodic affected week autocorrelation nd solely cluster daily discover scheme consecutive tend common seem day behaviour identifiable area cluster face importantly acknowledge twitter large bias actual ground fundamental come partly justify moment signal daily affect twitter e one extract whereas second remove firstly twitter content publish uk partly agreement similar finding acquire bias pa vice versa hour show hour length day support affect user figure track event real uk combined death twitter track event day day scoring result scoring scheme nevertheless show two scoring scheme combine identify finally show investigate lag day day likely dependency day week cluster score consecutive day exception negative identifiable preliminary nlp way content uk form similarity show stable form briefly examine twitter slightly something use feature preliminary aim voting inference input medium uk still result serve investigate twitter influence respective tweet content window significant formation share location tweet twitter uk carry already describe centre location tweet area correspond conduct setting document tweet day since datum set comprise day span also reduce day month day document day location proximity twitter answer question pairwise location respective similarity content approximate cosine document tweet retrieve tf entire document stop denote cosine similarity publish total pair km end cosine cosine experimental smoothed obvious content experiment increase relatively experiment seem figure km p window pattern however cosine primary question location necessarily twitter look global pattern twitter uk twitter quickly place regardless proximity country unitary apply multidimensional scaling location similarity well pattern trial datum retrieve computed represent monotonic point satisfactory level minute location nearby location distant might observation uk space whereas location centre try understand content reveal influence similarity within country effort content relation twitter share among propose preliminary pre set average similarity already previous period degree content average cosine pair average cosine ranked decrease top one location infer decide participant else numerical expect comprised month depict proportional degree location something happen depict visible directly observe share read place far likewise embed month minute show average picture network consider month central j degree comparable experiment one month infer quite examine short content similarity location balanced manner nearby include network see significantly increase shared content additionally reveal connection connection appear day score range equal numerical observable dissimilarity form network location network stable therefore well description share introduce form metric consecutive similarity edge dissimilarity use q similarity base distribution original replace consecutive network test switch original ss nan divide justify first month examine whether month minute investigate network plot respectively daily instance always switch might score first low last datum investigate network minute figure draw period second week similarity smoothing move extract hour minute pattern rise indicate periodic twitter user importantly news medium twitter minute line smoothed twitter million tweet volume tweet hour aggregate figure time p work hour p hour twitter hour become therefore activity might activitie stability day set compute correlation also linear say operation day significance value pattern present paragraph divide one twitter day twitter patterns behaviour trend indeed twitter volume peak deviation hour possibly day week hour volume repeat previous test stable respectively least another whether cluster day feature twitter volume interval figure figure day week pattern observe consecutive place position interestingly interesting offer social medium tool majority perhaps easy preliminary twitter finding united uk recent publish discuss discuss three major uk party party extract positive sentiment sentiment voting day vote percentage per political party dense publish day tweet million section tweet tweet political party retrieve political party see per party party name party search gram insensitive latter look match search tweet contain gram search keyword base mainly entity could also create automatic manner repository human approach build elaborate approach sentiment part speech noun positivity negativity negative sentiment weight might appear stem positive negative tweet retrieve sentiment weight note tweet list tweet sentiment ignore pos use motivation behind pos twitter pos might inaccurate however case probably tweet principle pos map particular sentiment pos tweet carry stanford pos log pos sum sentiment tweet term retrieve sentiment score incorporate automatically list list content tweet tweet word list tweet sentiment tweet sum pos identify extend tweet tweet reduce add enhance semantic orientation english achieve pseudo sentiment great tweet method one previous negative sentiment tweet extract keyword political party represent vote party describe remove tweet tweet unclear help later finding experimental setting remove entire also tweet sentiment score tweet subtract score vector ground weight voting percentage bias ol receive value reduce freedom study three major normalise order sum inference thresholded take unless take existence political people vote create comparable normalise vote three major much voting inference unseen overlap learn calibration sentiment score party sentiment multiply example conservative party triplet represent normalise inference voting party denote thresholded sentiment tweet many sentiment vice sentiment tweet sentiment number tweet negative sentiment refer sentiment measure mean absolute infer voting metric easy interpretation read mae unit base voting measure triplet voting error infer incorrect ranking difference correct position since one incorrect triplet make triplet error triplet correct triplet equal range computed method extract tweet sentiment come tweet sentiment remove top unclear tweet retrieve figure table thresholding tend performance mae fail voting properly performance thresholded figure depict good inference ground mae leave sentiment tweet cc ccc one size focus testing tweet retrieve search increase day part close retrieve randomly come test process time count give value significance sentiment tweet ccc present pos tweet good fairly poor performance mae significant thresholding reach mae ccc c c quite publish provide twitter political propose predict word count semantic tool negative introduce averaging like keyword party name orient propose traffic party present tracking voting tweet deal bivariate e two nevertheless indicate specific proven chance predict tool google interesting conduct triplet consistent content medium author firstly prediction aware characteristic justification base try figure form three scheme extract sentiment tweet try twitter particularly tweet enhance similarly aforementione work sentiment try show poor statistically significant sentiment remove contribution thresholding improves begin thresholde far modelling preliminary aspect future research come proof capability choice keyword argue influence automatic mechanism bias sentiment tool also tool expression extract medium content political opinion extend tweet probably vote quite exceed mae derive therefore formation consistently truth issue resolve rich twitter investigate show country capable form uk prove time content uk secondly extract twitter twitter tweet evolve peak occur hour temporal characteristic day etc explain form cluster cluster exception infer voting twitter something topic tweet sentiment apply consider speech incorporate stanford pos third sentiment voting percentage indicate effort public infer several uk display four type set region publication tracking mm monitor important school delay demonstrate web provide epidemic work social web medium predictive infer region uk twitter chapter validation complete automate tool interface use uk twitter data train validate behind million tweet km uk rate ground detail give summary feature gram extraction comprise gram gram gram decide ct select gram gram n gram ol finally weight daily twitter content region daily since realistic large equal inference behaviour also inference maintain focus tweet km get tweet centre retrieve atom format database describe perform offline inference basis website apart three region south central display entire uk detector website chart website infer infer day display separate box patient rate exceed page medium predict stock market implement finding compute display already namely see scoring importance display raw score score divide remainder standard raw score plot figure indicate deviation large interpretation stand negative multiply score negative constant maintain dynamic type day automatically peak carry skeleton website amount maintain region twitter datum different implement basis show chart include type page website entire uk page day display display historical datum interesting discuss peak day mm chapter outcome conclusion research interesting behind scientific primarily amongst chapter operate give primary firstly attribute drive prove store twitter ground truth track epidemic diffusion exploit content twitter manually prove correlated rate propose fully automate regressor lasso extract gram semantic topic accurately rate uk epidemic limitation mainly learner gram possible experimental infer occurrence phenomenon rich amount method bootstrap consensus ensemble core methodology gram gram investigate combination rate usefulness rate hard test datum source twitter content problem extraction pattern assign importance weight analogous text stream remove bias marker investigate type infer pattern partly confirm one pa show vice daily gain support life twitter example observable twitter uk daily period length week additional pattern study twitter could country twitter depend location form uk infer secondly focus extract twitter distinguishing seem confirm match norm report address inference twitter uk general lastly online tool serve dynamic theoretical derivation uk display uk region tool twitter content thesis web text stream reflect part life social medium increase reflect thesis way type web service focused extract life trend inference event inactive discussion could quantify show stream general periodic norm similarity reveal share among nlp twitter vote therefore verification truth compare unsupervised acquire fact abstract infer voting web conclusion become life public effort develop possible side applicable content investigate study acquire target able several signal position platform course avoid twitter early success improve gram increase select combination gram gram improve performance selection bootstrapping lastly content sentiment paper idea short paragraph social stream conference journal recent book newly topic extraction voting sentiment tweet detection environmental twitter topic could extended direction grow something project scientific medium research technique application sophisticated linear learner new information key function hold could concern gram show semantic gram regular aim look aspect incorporation sentiment rational life medium already need quantification detect concentrate english language method language apply framework interest go challenge successfully signal stream challenge combination vast social answer question result path issue impose development carefully consideration propose computer closely quality speech arithmetic sized sample define deviation uncorrelated tend increase make sample identically distribute sample standard deviation transform suppose variable denote way mse broken variance bias average around something unless bias complexity extend reference vector size cosine simply cosine angle sim range point odd number equation move plot move average recover original nan hypothesis hypothesis nature nan trial unlikely contradict hypothesis might extraction principal analysis observation orthogonal space generally direction derive eigenvector component define eigenvalue twitter uk centre set belong list belong read hull york extract manually form marker gram describe particular ccc back infection track retrieve affect gram gram term ir alarm belong care concern favor good great heart like warm weak h bad blue dark tweet political uk character empty space twitter conservative party term cccc conservative conservative party david grant david david vote vote cccc ed vote vote david david david cccc mp mark david david vote pc thesis requirement code research award thesis author signature date ph old matter go grateful person properly primitive another ph thesis course significant scale ph good mind internal ph progress useful remark progress also sc study grateful interesting build sound people negative grateful mit technology special database person project I cover financial support ground computer department university year ph amongst thing ph book thank interact indirect aspect platform company spend write life east east east east art art core google web form unique web approximately time
input cope meaningful propose sparse principal exploit sparsity probabilistic base setting geometric distance neighboring dimension embed manifold volume dimensional performance affect author suggest base smoothed author property technique neighboring point exploit graph adopt either geodesic minimal span arc distance efficient base euclidean usually work exponentially curse reason dimensionality practically available insufficient acceptable manifold raise dimensionality increase edge behavior neighborhood propose empirical correction procedure produce generate fitting correction use local give extract estimate bi compose fit horizontal whose author describe base comparison distance neighborhood distance manifold high locally map manifold smooth mathematical drive spherical neighborhood origin radius restrict describe ball uniformly ball uniformly distribute neighborhood estimator information unit property draw sketch angle consistent locally nonlinear map sample independent uniform nearest j I normalize compute employ translation k q parameter section draw digit version letter alphabet circuit face database three consist gray test represent actually know digit range synthetic point particle motion nine series delay collect value letter twice training group set study compose realization circuit delay contain employ toolbox generator create point unbiased achieved execute contain number resample trial iteration shape distribution hyper table configuration summarize relax selection perform run result report summarize c estimation obtain manifold able embed manifold consideration confirm affect notice manifold embed dimensional space table percentage report percentage error manifold consider indicator good c obtain quality poor geometric approach test affected noise correctly dimensionality well perform strongly correction precision mean promise valuable tool finally w r employ reproduce average curve compose range combination obtain robustness novel call angle method compute synthetic aim employ employ overall really strongly really capability ii linearly embed iii effectiveness propose width dimensionality concentration dimensionality last decade dataset gain considerable deal work devote dataset point lie embed high propose intrinsic exploit nearest neighbor angle neighboring provide close form leibl respective distribution synthetic highlight effectiveness compare art decade great deal lie low manifold embed estimate term element lie entirely information follow use curse representation projection dimension minimal retain maximum useful furthermore suggest reasonable neuron capability depend empirical classification relate recently series crucial structure geometry research focus development present investigate dataset embed high space highlight report fail kind precisely note
comparison substantial effect evaluation recognize bioinformatics inference face objective letter highlight aware future develop protein natural science foundation china interest cn letter comment claim specify argument indeed conduct evaluation inference letter search parameter estimate point learn bioinformatics meanwhile unbiased research approach real application closely relate separate assessment final selection protein problem fig selection protein bipartite input produce protein protein assessment procedure evaluate assessment result optimistic analogous bias search calculate area auc performance ground description source code score allow positive ideally calibrate use ground true label assessment select final index assessment highly check search optimistic conduct procedure
code yet infinite learn learner even odd code notational learn version great result learn book anonymous feedback lemma corollary conjecture ask anomalous distinguished finite hypothesis infinitely often permit thereby answer pose strong learnable learnable failure produce family natural collection subset denote computable give code computable fix enumeration finite letter string text theory string natural depend initial either segment switch order list set use switch wish specify cardinality write mean symmetric computable enumeration machine code learn enumeration n mf sm I j identify enumeration set sm e inspection weak sense learnable learnable might j ci n theorem conclude remark make relationship notion anomalous search string learner code string exist construction string include enumeration never produce infinite difference index learner produce learnable begin need prevent family course string finite collection meet essence string hypothesis extension equal verify require natural thus sequence string extend construct sequence string next string sequence symbol use string give algorithm stage string yet empty possibility string least set process terminate end observe furthermore converge define number define conjunction statement substitute position return suppose
class randomly sample nonparametric specificity specificity sensitivity transformation correctly classify threshold score object particularly aggregate equivalent integrating unfortunately compare example score object curve incoherent relationship total freedom measure reduction way instance weight give proportion population proportion misclassifie statistic proportional misclassifie misclassifie equally etc requirement specify integrate misclassification classifier property misclassification classifier give problem reformulate calibrate score reference cost misclassification brief measure refer object represent total incur total yield rather normalised advance requirement exactly require different classifier classifier although researcher problem entirely distribution tackle distinct one universal response experience researcher wide universal class unbalanced another would rare symmetric would little would everything transaction detection transaction class size unbalance transaction phone call plus call bank likely less cost transaction easily run recommendation researcher another alternative class misclassifie incur object none misclassifie misclassification object class cost size unbalance small detection transaction transaction order magnitude attribute transaction transaction essence principle pick relative unbalanced situation serious mode several way leave open reasonable unimodal extreme result fully nevertheless suffer disadvantage treat understand undesirable form prior switch beta specify value wide mode sensible default universal default respective shape employ reflect parameter higher clearly guess normalise cost place mode cost context expert opinion type misclassification severe class guess invert single place guess misclassification burden specify encourage whenever expressive standard fact early classification solely threshold fundamentally incoherent different rule differently let propose measure choose objective belief consequence kind misclassification give work report belief suggest experience correspondence researcher world asymmetric cost distribution unbalanced c mathematics south college area roc measure fundamentally incoherent treat differently different overcome classifier extend propose match
error similar use correction notice ii minimum quality decrease e fine tune advantageous impose correction summarize base role adjust tuning correction small increase value improve know efficiency algorithm select surface factor surface value parameter neighbor correction structure precision apply mini cf neighbor correction mini r highlight highlight mini surface similarity rmse usage surface quality mostly image annotation sparsity yu absolute penalty group pp j overlap graph fuse svm pp joint subspace selection pp trust region mix simultaneous pp pp pp classification schmidt linear potential pp discrete r plus mit induce norm hierarchical f pp k representation learn invariant map chen compress pp pp plus scientific collaborative filter pp g neighbor netflix pp e email web pa email web international conference separation structure code dictionary learn novel filtering system numerical experiment present outperform several advantage structured dictionary collaborative online service alternative daily decision recommender rs recommend item among recommendation movie book news recommender system underlie rating item cf try rating make user recommender collaborative advances cf dictionary assume representation behind user product usually factorization include analysis one considerably problem observation enough minimal exist prominent approach enforce covariate structure tree sparse application ease sense number life benefit structure annotation group micro array iii transfer vi excellent sparsity code already representation find novel appeal transformation extraction ii background corpus face extend collaborative cf appear online typical reason appear time motivate advantage offline amount case small available rating cope use novel call online dictionary overlap non induce iv briefly numerical draw vector diagonal coordinate coordinate stand coordinate pp gx respectively treat idea structured observation assume give know observable column bound group call hypergraph vector available otherwise belong triple sparse code eq structure responsible quality coordinate similarly average dictionary factor td sphere child hierarchical cost eq manner actual sample estimate representation dictionary variational property introduce z g j z step minimize quadratic convex mean stability smoothing optimize keep fix minimum solving project diagonal b x otherwise update tc j c numerical approximation estimation worth improve update batch cf correction improve estimation cf task optimize group user task miss coordinate rating accomplished remove code representation neighbor may estimation neighbor approach assumption item movie rate cf detail idea individual rating user k prediction observable item error simple modification illustration benchmark detail prefer item evaluate element evaluate two rate remain rate user per similarity row quantity item mae quality popular mae measure square rate efficiency use mae section rmse sake completeness report rmse rmse item neighbor regression neighbor capability focus follow structure purpose sparse similarity correction affect numerical choose weighting indicator
directional spectral address value rigorously valid directional derivative rectangular repeat assertion prove full rectangular singular divergence matrix rank take convex relaxation np minimization norm proximal soft derivative take theorem strictly map everywhere derivative corollary simple provide sure completion encounter system netflix therefore low entry choose approximately rapidly depict function range single gain core proximal nuclear norm use recursively recovery regularization also automatically regularization derive mapping mapping set singular lemma locally write taylor expansion use product sort impose sign constraint directional j hermitian n v denote particular along apply system invertible inverse yx implicit svd neighborhood desire distinct composition function derive use anti fr paris france france paper recursively risk toward spirit sure derivative divergence solution close splitting iterate second challenge potential applicability automatically consider bound operator entail problem ill pose various research ill side minimizer non proper impose low remain largely typically want minimize quadratic risk value stein weak jacobian sure solely way contribution derivative split algorithm estimate finding proximal smooth yy study value svd singular value rectangular matrix main unitary left vector definition symmetric extend subdifferential spectral x u value argument number observe proximity operator spectral matrix spectral follow provide closed value value quantity
propagation iterative message pass compute marginal gain popularity paper new belief usual graph markov want underlie link form node variable together factor graph undirecte bipartite say mrf exhibit deterministic behavior exact procedure generally exponential resort computer procedure graph cycle apply several bethe minima bethe question bp series work mrf unique however multiple fix parameter bethe consist viewpoint looking condition single local quantify know bp normalization section exhibit case message provide bp paper propagation message belief include belief idea factor product contribution come message probability sum belief ensure constant converge follow compatibility often normalize define read worth convenient shorthand ax ib beliefs convergent practice refer express obviously aim policy pointed message belief strictly positive concern lead belief write moreover preserve belief resp resp imply indeed vector conclude look correspond invariance natural plain mapping span vector belief simply convergent iff converge space normalization normalization play normalize belief attractive jacobian modulus unstable modulus orient orient second ingredient node row unnormalized representation jacobian read jacobian plain bp pair element analogous jacobian encounter interesting depend graph simplify irreducible long always spectral cycle graph less positively homogeneous share look differentiable jacobian fix belief read summarize matrix presence message jacobian possible obtain belief indeed admit point generality restrict pair associate combine reference eigenvalue matrix fix associate stable I combine effect local give uniform degree correlation correlation variable stable fix order part consider local q w convenience somewhat I b reversible implie conclude proof deal express average stochastic ai jx ai ai contribution individual eq triangle inequality project ai ai ai ai ai end homogeneous respective level stochastic encode statistical connectivity average
quantile different relate also fail situation exclude covariate quantile consideration penalization special answering available censor therefore novel quantile major challenge censor regression quantile set sequential nature interact fundamentally censor iterative course generalization convergence additionally individual considerably scope applicability paper section realization way censor quantile result finding proof technical censor predictor censor covariate censor instead observe identically aim regard covariate study predictor quantile function combine idea reference coefficient iterative spaced define piecewise censor quantile estimator quantile censor computation directional satisfie present consist wide dependency structure provide point base see relate generalize solution equation definition distribution order negligible view estimate possible case censor version share start behave control survival version reason seem censor accuracy identification correspond vanish devoted penalization vector allowed depend lasso detailed penalization lead rate alternative way penalization avoid section state penalization vary zero technical ft px z ft corresponding generating process intercept maximum norm finite f uniformly denote impose consider possibly relaxed introduce additional leave restriction condition characterization quantile identifiable censor discussion refer contrast approach define converge center follow case setting violate note z denote interpret index vc chapter argument law function overview recent cite therein particular imply soon ergodic impose precisely imply soon converge towards process check establish literature class function dependent case dc mix time soon ready main hold eq equip supremum ball sigma algebra process uniform derive test censor discuss interesting theory q infinity arbitrarily order quantile g condition proof uniform censor regression converge censoring note tu correspond quantile classical aspect penalization process notation maximum non vanishing correspond mapping coordinate remain coordinate penalization power additionally number discussion give particular provide happen suffer impact quantile tf tf special structure eigenvalue statement view index vc subgraph continue c omit sake brevity ready main enjoy property tend coefficient remain coefficient p hold ns integral supremum see denote v asymptotic limit complicated special identity remainder oracle similar estimator technical omit brevity penalization statement theorem satisfy preliminary penalization adaptive lasso investigate condition result give partial answer question show optimal precisely turn exactly dependence sake brevity n basically reflect th enough affected penalization asymptotically coefficient tend imply intermediate assumption define aa hold preliminary consistent penalty encounter traditional avoid consequence quantile analysis quantile covariate impact exclude take use highly flexible adapt conduct presence censor weight weight value note weight weight censor function divide denote index kx minimizer basic idea procedure estimator weight consistent might quantile quantile curve total sum change sense invariance move censor lasso well average quantile finding estimate obtain penalization estimator behave penalization systematically component systematically intercept ht average row correspond censor roughly regression view wise penalization slow probability coefficient observe slight systematic advantage penalization quantile slow shrinking reveal picture suboptimal superiority penalization become ht local proof brief main prove result general coefficient major role aforementione coincide derive ingredient representation give detailed fact dc denote lebesgue sketch j imply constant minimize j kn j simplify minimizer find part observe yield p kn kn kn second part w make begin part simplify ni b kn last moreover part c assumption begin eq combine derive p n remain assertion exist would also minimizer component choose contradiction line kp kn dominate proof uniformly consistent quantity b b c r c n p result point additionally complete proof array random q assumption hold minimize let assumption hold obtain proof point difference major step note prove eq estimator supremum consider grid hold follow classical argument yield omit brevity condition finally front iii iii hold c n yield I n n algebra yield show complete iii combination establish lemma n n nc mc nc mc b nc h nc mc j db obtain j yield v r cb w nu integral denote value ji tw tw assertion induction suitable next summation u tw ji tw w end cb dc yield ji u u v u v nu u k cb finally u proof convergence yield ns v eq p taylor combined algebra statement matrix u v uniformly prove satisfie th moreover separately finitely interval without solution base prove estimator uniformly argument uniform additional consider exist consecutive contain large tend n careful inspection show continue eq quantity c n n note large ns lb nc n nc assertion establish let hold follow iterate ns yield computation nc pn assumption argument whole definition universit censor case quantile traditional penalization adaptive yield regression cross penalization introduce deal censor yield assumption kind keyword phrase weak censor
setting behave misclassification pl di sim former require therefore di slow setting describe detail bernoulli simulation outperform di sim algorithm almost conclusion profile h ccc pl b l ccc km b b describe microarray activation gene distinguish dataset microarray datum contain measure different dataset belong age study relationship expression residual log gene error gene independent zero gene block conclusion hold residual negligible light consider exploratory perform use profile preliminary appear representation result fourth fairly gene visually appear expression gene group high gene group find cluster suggest interaction gene behavior well new individual since vary product identify product application apply result split period individual rate movie sufficient base likelihood apply dense comparison variety situation procedure theoretical operate assumption row simplify selection future reveal finding find high level result review behavior individual movie period sim negative normalize example simulate block row column membership chen class ij ccc c km simulate cluster number column membership matrix row ccc km b n b standardized student parameter conditional column ij h ccc km ds b profile log criterion perform three example verification finding data review movie user rate movie rate five rating release movie gender code retain customer service movie individual already movie rate likelihood bernoulli determine vary movie seven qualitatively movie group seem data assignment cluster report movie within suggest rating explain alone release order movie median release date gender effect mirror face air brain fan extreme group day attack reservoir water group contact return future english patient toy air force roughly speak movie group activity popularity consistently inactive movie rate movie group movie discover suggest recommend review movie individual recommend engine corollary variable popular effective tool apply successfully source range review website currently practice sufficient condition consistency infinity apply broad bernoulli microarray collaborative filtering suppose row example gene patient seek patient profile find group similar activation alternatively indicate review goal simultaneously movie term cluster block co broad help collaborative diverse difficult purely base biological science one gene apply method drug identify association medical several well comprehensive survey goal reference hoc lack notion generalize collect lead different report formalize profile consistent tend infinity assumption notably handle general treatment recent binary cluster consistency network nod belong cluster individual specific effect cluster spectral develop extremely generalized simplify knowledge main text organize setup next finding study microarray remark additional technical result formalize follow determined solely column vector membership unknown refer undirecte goal assign true analogously profile assume bernoulli propose likelihood general simple misspecification element draw family conditional entry respect likelihood deviation derivation scale terminology deviation literature refer though require rate misspecification technical maximizer column constant exist row column class membership vector surely multinomial ambiguity subscript mean depend membership tend avoid confusion proportion entry similarly nontrivial neighborhood bound satisfied long handle element infinity word heterogeneous place restriction consistency variant away moment certain bernoulli data zero setup maximizer true row label f limit proportion row probability label multiple permutation maximize profile log furthermore result specify model misspecification datum could poisson make motivate distributional assumption method example outline constant hold neighborhood confusion sequel version neighborhood around true class proof literature symmetric symmetric binary network work whereas type paper work criterion potentially unbounded lastly criterion continue section notation establish uniform assumption matrix version version sum next locally lipschitz appendix maximize confusion close permutation independent fix inequality choice imply permutation misclassifie set g proportion n permutation study profile bernoulli standardize misspecification immediately obvious poisson function poisson sum equal maximize identifiable try simulation maximize though
jt jt I jx I I case hmm I forward x x x k potential running intersection hence u j although prove theorem x x jt eq partition eq achieve hmm exactly j I I follow k x k x next dd dd dd dd dd dd dd x x dd x dd dd roughly dd dd dd x dd dd dd dd dd hand hmm forward iy I forward since case jt direction subtle call message opposite message compute remain see involve recursion recursion another back jt perform recursion possible message process backward recursion x recursion possibility refer suppose matrix leave establish introduce message center without complete extension hide evolutionary loop sum propagation replace sum max pi lambda lambda lambda x col blue x col forward lambda e tf pi f col col col col sampling ps sample ss ss ss pi ss ss I I ss pf c pf x pf x pf pf k x x x x x x x k x x digits digit distribution x p p k x p p k x p x p x x x cp cp cp cp cp size cp cp sample cp cp cp cp exact propagation achieve provide contrast variant define forward derive introduce message recursive message consider along formalism code provide probabilistic graphical powerful like hidden hmms inference quantity use multiply variant basically compute point message hmms hmms message derive exact message implicitly algorithm objective forward hmms introduce paper follow hmm illustrate approach illustrative result derive code mm simplification pressure denote pressure day transition poisson eq typical trend pressure phenomenon pressure problem convention prove solid I b backward compute eq prove forward leave right term recursion simplify produce quite reference fig posterior l l l h l l l l h l backward conditionally heterogeneous direction backward forward direction numerator denominator use corollary discrete acyclic note thanks acyclic call ex var var var node var node var var edge thick thick thick edge thick thick thick dag parent set generic fig represent relationship loop cycle allele non allele indistinguishable parent allele general population text depth var var node var var var var x edge edge edge x thick thick introduce subset empty evidence hmm evidence disease disease individual affect affected affected disease status individual evidence dd dd dd dd dd dd consider jt connect cover u jt
significance show six code lee filter result rest lee edge consider situation c figure lee look universal quality present ground metric j display filter e sensor band nominal look al look homogeneous lee filter image smoother highlight almost corner visible apply lee stand filter hellinger windows level table assessment filter hellinger order index lr lr lee lee lr h h lr filter noise protocol carlo showing application real present use behave filter yet assessment proposal com paper filter distance pixel compare use filter vary look change heterogeneity window compare lee criteria quantify employ line edge assess optical noise information ta si model development plausible image statistical propose literature datum use multiplicative intensity employ describe et lee filter protocol carlo filter edge index pearson filter quality filter draw pixel outcome characterize specified deal homogeneous intensity image gamma look gamma obtain employ figure filter nine denote central estimate eight account homogeneous maximum look heterogeneous area estimator namely distance test modify level significance whole derive distance include due flexibility look vary kl r n decision hellinger since filter checking come block set reject local filter central around filter assessment performance par technique preserve follow assess two datum version quality filter et sim u corrupt image quality compute assess analyze
demonstrate value go distance two function minima apart shall fix oracle choose learner query domain function apart identify wrong horizon proof lower bind probability prominent active proof convexity state ff f unique return whose sc hence ensure depend ensure part convex maintain interest constant enhance return ix whole distribution correspond divergence p condition identity half substitute give jensen bound large conclude recover exist low strongly setting tight appendix batch proof immediately give tight bound derivative free optimization gradient exponent conclude strong help show satisfy fx fx derivative yy tp tp x f initialize tt tx x et tight query f f may variance keep use radius condition convex derive diameter g fx x fx g fx f appropriate probability use induction first induction probability correspondingly epoch event c e induction like condition probability c trivially g strongly round query alternatively use assumption prove convex class behaviour strength smooth hierarchy natural imply convexity correspond low classic demonstrate proof novel free generalize rate support see behaviour noise jump sign corrupt identify optimum extremely tight epoch gd extremely optimize lot value achieve correct number step noisy substitute strongly recover order proof bound condition point every small large use strong unknown value function old grow elsewhere behaviour bind active give hope intersection active optimization polynomially try hard problem boundary level interested get subroutine generic reduction set give problem conceptual field research support grant thank input satisfy convexity uniform say positive taylor grow go grow least slow need uniformly fx rearrange since eq eq result random variable characterize random around want integral rectangle small height tt px te tight inequality suffice return work divide grid interval x modify initially align ie mx du elaborate avoid grid align complicated variant grid exposition I firstly lipschitz norm bound x growth convex dimension constant ix fx h first q convex special argue rhs argue ty ty fy query complexity determine growth quantify like specifically least strongly function attain tight bound complexity interestingly rate characterize active connection rely drive oracle generality everywhere always deal furthermore convex diameter give function oracle unit let later analogously function repeatedly query estimate optimum question way formally convex small condition mean quadratic everywhere everywhere strongly geometric question arise minimize work determine intuitively connect central concept tight possibly equally point claim determine minimax rate growth optimum strength rate around optimum everywhere later form hierarchy sc state decision relationship seed function precisely rate hierarchy tight algorithm reproduce get free bound stochastic generalization say equivalent weak grow margin literature reproduce version ie decision satisfy exponent
acknowledgment thank manner multiplicative univariate look inverse law reference attractive describe space modify inverse moment moment belong multiplicative describe product variable outcome complex wishart rejection normal deviation correlation verify whose generate transform boundary detection active contour region spline specify contour boundary model area synthetic prove among mention map update ice monitoring among due sensor day illumination sense device weather extent devote application frequency imaging band sir able receive vertical capability provides value require specialized follow description see feature observation alarm treat design work propose al perform em detection maximum image segmentation law approach base description every edge ideal area mechanism property one index harmonic knowledge combine contour operate whole considerable virtue spline curve allow noisy image base spline contour criterion et techniques spline boundary tailor present attractive proposal begin manual specification region spline curve radial around point belong contour give assess increase capability statistical specifie assess edge detection real conclusion algorithm random design receive vertical wave receive ideal subscript indicate subscript indicate signal processing enhance complex q denote look summation hermitian real paradigm product random unitary variability heterogeneity random mean corresponding look exhibit multivariate complex complex wishart present density determinant matrix function integral compute q consider random unitary mean harmonic channel law provide law model distribution close issue impose parameter moment diagonal eq nz ig ig univariate th return total error I necessary relate correspond hermitian display font lb lb lb lb lb lb diagonal account I I feature intensity immediate area heterogeneous forest grow homogeneous area type regard texture intensity correlation three check describe look image exhibit look generate every look intensity target estimate forest target frequency allow htb value derive number look preserve fit excellent figure show synthetic image show background figure assess contour et method real adapt spline convenient representation spline function curve parameter control reduce effort polynomial arbitrarily relate smoothness control control control individually curve spline representation contour work sophisticated could description length present begin rough segmentation manual automatic scene boundary want curve fit spline develop initial specify automatic initialization perform ns homogeneity language text threshold correspond consider region also natural manner block array complexity big window evenly target mark else consider discard form convex compute automatic determination connect whose spline alternatively user search determine centroid proceed belong change candidate angle consecutive centroid interior object contour outside segment discretization array array sp font pt lb segment parameter segment slide mask consider method build curve candidate boundary determine segment rectangular window around segment detect set return spline methodology employ assess edge detection image establish true propose image segmentation algorithm surrogate truth manual mathematical boundary manually create possible option measure local use amplitude point image show right show vertical white dot situation replication situation estimate distance boundary th relative error pixel denote replication big algorithm simulate image find eq situation area model besides indicate horizontal consider three present three channel greatly capability accordance result exception situation fail fluctuation content individual notice three channel easily
several important fact implication theorem unlabele algorithm request example round unlabele unlabeled round learn sufficient passive consider nearly log nearly remove isotropic distribution admissible mm unit p c proof without admissible arbitrary active label passive example fx concave log distribution whose matrix whose prove isotropic concave tail disagreement claim start isotropic log within exponentially oppose argue tail log concave concave ec c u hx whose hx e x fact property desire sufficiently non constant isotropic light tail project isotropic isotropic well isotropic concave center constant small apply whiten see show generalization except since isotropic associate small log passive example bound complexity passive active theoretic procedure appendix small whose covariance center passive matrix effectively essentially consider closely capacity notion use begin definition p x w dr cd disagreement coefficient appendix small constant theorem immediately concrete result obtain dr dd request condition minimize generalize margin constant label example excess use condition passive improve dependence bound improve far appear sample achieve use amount computation unbounded seem hypothesis repeatedly unlabele example evenly split candidate eliminate roughly candidate would label constrain conceptually erm concave localization literature useful part nsf grant microsoft fellowship technology microsoft concern passive pac passive building computationally unit tight infinite hypothesis significant progress separable agnostic nearly concave erm low noise condition agnostic challenge long seminal accuracy pac time well lower improve polynomial open distribution resolve belong include convex area research classification exponentially passive learn answer active exponential improvement remarkably passive connection active case linearly specifically provide new widely disagreement coefficient case form passive label instance label unknown classifier new study play crucial providing algorithm tight especially imply learn example lower explicitly pose providing polynomial polynomial erm intensive local really bad hard rich bind far however suboptimal complexity ignore eq yield complexity polynomial specifically algorithm concave provide interesting non hypothesis function constant complexity gap ask unlabele hope classifier past year mainly availability large amount raw modern application development principle call active paradigm analyse conservative strategy amount consideration label far manner versus analyze active prove request answer passive learning improvement distribution uniform ball even multiplicative dependence inefficient splitting novel characterization disagreement constant region disagreement theorem fact use margin active round learn passive least hypothesis interestingly quite classic analysis erm conceptually run algorithm hypothesis tool active likely show would use preliminary failure later stage presence extend receive passive bound optimal bound label request passive example generality selective learning request previously change upper selective section make p xu throughout nearly log concave play decade area include integration theory extension nearly concave concave isotropic mean origin covariance broad uniform concave summarize fact isotropic upper density mm f ad isotropic isotropic fact universal constant two homogeneous time angle vector project disagreement implicit completeness include isotropic analyze provide disagreement log choose enough hold let imply imply turn ball carefully upper bind include integral eq exploit rescale probability distribution let volume return c complete weak prove unit ball addition tight passive analyze prove fewer passive somewhat ellipsoid except instead update multiple instead one c maintain instead constraint new target old small ball previously analyze ball oracle sequence cut draw put mm hypothesis datum reject label put isotropic lemma
inf inf inf inf inf inf inf inf os inf inf inf inf windows inf inf inf inf inf inf inf inf mac os inf inf inf inf window mac os window window os herein assess table beta except binomial compute binomial poisson student quantile lr em inf inf inf lr lr lr em lr em e lr lr lr inf inf inf inf inf c lr lr lr e e e lr em em e r lr lr em e inf inf inf offer analysis divide level two two seven explicit em r lr em lr em five numerical computing standard test dataset lag quantity none test bad platform respect variation note conclusion turn subtract instability perturbation original input change deal student also cumulative poisson matlab better version al platform produce binomial law normal matlab produce acceptable deal matlab fail return message serious use statistical test six consider matlab windows operational system platform safe result make determinant ill condition operate acceptable logical comparison careful equality interest involve assessment worse sensitive eigenvalue double test latter balance bipartite spectral double comparison equivalent situation km com com com use computational matlab architecture operating system os set function platform conduct verify include data national institute technology protocol include computation basic univariate lag correlation assessment compare digits secondary spectral statistical mathematical modelling rigorous computer model diverse area bioinformatic rely pde ordinary ode number effort final often finite mesh use operation rarely check approximate consider reasonable bound tight computational library function huge structure partitioning parallel need correctness either method assess algebraic graph mesh mesh dual element common associated graph partition eigenvector determinant decision offer library calculation problem little attention assess diversity operational accuracy institute dataset besides statistical employ assess viewpoint numerical scientific operating window sp mac os whenever hardware precision organize discuss assess subsection subsection conclude simulation arise model error computable solution discretization error instability availability limited operational system accuracy author instance measure provide obtain national institute consider measure assess mean deviation autocorrelation quantile base logarithm absolute value error approximately match assessment convention place correct digit inf perfect match platform compute platform numerical check stability propose platform deviation lag compute dataset obtain original platform produce error observe belong second deal first computation propose assess q accuracy logical platform assessment another assessment laplacian graph finite without loop vertex note eigenvalue connect call consider algebraic bipartite two say vertex subset connected cardinality laplacian eigenvalue assessment balance almost example size assessment denote percentage eigenvalue equal correct answer window os mac univariate assess lag nine classified dataset difficulty come world difficulty value precision arithmetic digit compute std st deviation inform return divide deviation n
regression example individual association gene genetic association snps genome association attempt relate variation many genetic variant snp association modeling phenotype explain available phenotype despite opposed genetic variant affect phenotype regression selection assume small affect phenotype performance application expect vary true genetic phenotype genetic architecture unclear prefer sparse mixed hybrid model case capable genetic idea use genomic prediction trait trait jointly compare detail model particularly emphasize benefit parameter provide exploit algebra avoid ad hoc approximation sometimes reliable result thousand thousand marker involve naturally potential use association specifically phenotype marker e genome wide potential underlying phenotype could ultimately application datum successfully combine variety setting outperform model phenotype method motivation begin consider phenotype individual individual genetic marker genetic marker effect vector code allele marker column see detailed marker accurately kind modeling assumption genetic become effect equivalence compute brevity paper restrictive compare usual definition commonly refer statistic combine commonly unbiased predictor assumption dimensional come mass proportion convention variable regression commonly phenotype phenotype distribution put mass variant assume effect assume large include effect mixture two normal assume proportion effect normal regression hyper include assumption flexible special obviously still normal mixture normal include alphabet model name genomic selection summarize specifically effect use component answer effect way model practice term treatment term greatly affect example make flexible flexible among certainly consider advantage although hyper issue tool less turn description specification focus linear induce prior effect terminology terminology refer effect instead emphasize mixture prior discussion specification refer imply equivalent symbol whereas effect application estimate although estimate genetic effect direct interest relate effect component specifically small marker environmental phenotype correlate affect addition environmental measure issue relevant give observation normal flexible natural modifying come mixture point modification change specification specification summarize phenotype control zero control expect magnitude non zero estimate framework chain carlo approximate imagine bayesian incorporation external e individually effect take inference variance q gamma arise limit improper posterior property measurement phenotype seem desirable intend specification extend marker upper limit marker reflect uncertainty put number marker correspond place event place probability prior follow easy term interpretable quantity variance effect phenotype genetic interval reflect predict phenotype snps together well predict alone function depend brief choose one reasonable relationship hyper make ratio expectation ratio variance marker mean matrix text derivation marker effect marker marker genetic term relative error properly scale provide bridge would near datum mass summary term prior distribution rather specify prior use uniform incorporate contrast seem hard directly specify reasonable default prior course note approximation prior estimate directly use definition computation approach introduce coefficient point write sub corresponding use mcmc detail use marginal marginal association sample sample value evaluating burden involve determinant calculation individual incur hyper parameter consequently impractical ad hoc commonly burden phenotype exploiting develop eigen unitary eigen vector phenotype make transform follow extend evaluate efficiently burden study analyze software implement software result assume snps small proportion snps genetic include hypothesis perform real snps phenotypes scenario simulate effect come genetic architecture simulate group causal group number plus snps represent realistic scenario phenotype create replicate variant marker either exclude mean estimate method estimate perform best rmse scenario figure poorly strong bias conversely method although approximately unbiased wrong robust wide true model advance make setting appealing recommend despite meet simulation phenotype phenotype brief mean square phenotype negative accuracy compare qualitatively interestingly phenotype differ scenario scenario close number causal scenario involve small capture equivalently due capture actual distribution effect pattern simulation comparison exclude snps potential explanation much phenotype effect reliably individual assumption poor tend effect reduce phenotype trait trait datum datum contain snps contain include density tc snps narrow trait isolate population tc three trait estimate almost trait explain narrow trait suggest trait substantially consistent substantial attempt precisely additional snps specifically come distribution also environmental estimate generally trait range tc suggest contribution trait disease turn trust case assess include disease share snps seven disease disease diabetes diabetes seven disease split individual perform split estimate treat binary quantitative taylor binary phenotype set auc seven perform disease performance result study strategy expect well number association qualitatively although well disease unlikely human clinical control generally disease relevant quantitative phenotype heterogeneous stock performance compare phenotype small human vary slightly wide five propose phenotype bayesian modification perform poorly simple greatly improve general fix online distribution component obtain computational burden fit similar lee posterior text focus treat control status interpret phenotype prediction one might modify probit probit find slightly bad treat binary outcome quantitative phenotype partly due considerably nonetheless burden remain heavy store substantial computational resource somewhat large currently collect fit moderately hope also help highlight principle characterize certainly rather important often interact conceptually characterize easy predict tend importance size assume normal flexible point phenotype method flexible well less flexible distribution importance estimate hyper parameter phenotype illustrate benefit estimate hyper integrated rather two procedure hyper focus proportion phenotype explain genetic component notation could helpful method use answer distribution underlie sufficiently flexible extent distributional mixture normal mixture degree freedom produce gain seem likely use therefore preferable expense effective flexible component effect directly unclear necessarily computational expense integrate ensure fast difficult predict size ultimately provide nonetheless appeal flexibility performance thank height thank software thank helpful comment manuscript research pi fellowship science program use full project trust award contain height measurement individual datum set phenotype age effect normal quality snps exclude snps allele imputation consist consist individual study phenotype present high tc phenotype quantile normal correct mass index status quantile snp array snps minor allele third trust phenotype data case seven disease case diabetes diabetes well control obtain control result snps minor depend split come heterogeneous stock consist individual family eight phenotype performance phenotype measurement set phenotype percentage cd cd phenotype previously method narrow cd median phenotype correct year quantile phenotype available individual allele frequency phenotype human height simulated phenotype effect scenario phenotype datum causal come fixed causal size distribution size come causal snps snps small medium effect size leave causal snps size addition medium effect explain proportion snps effect error mainly rescale call predictive predicting error covariance snp practice snps linkage snps snps I rescale apply formulae ensure result account random correlation also assess value observation depend q formulae estimate center measured marker marker code allele simplify center throughout unit center affect column assumption marker relevant throughout copy reference marker center matrix matrix center effect generally multiply side result semidefinite expression take parameter extension slight expression simplification expectation approximate numerator denominator rough expectation expression conditional matrix derivation assume center plug approximation prior solve give independent induce marginally decay another nice marker marker effect increase estimation previous use generalization restricted respect evaluate despite derivative approximate q correctness also bayes give identical phenotype estimate e effect multivariate theory conditional predict phenotype observation correlation correlation size present give give observe rao text add method fit fix example software double coefficient denote parameter gamma rate study scale stand degree freedom posterior similar key difference treat pre whereas specify prior greatly sample modify available online software fit fixing avoid update approximation study burden intuitively fix effect size prediction snps base except run due server million assume markov carlo explore marginal n perform decomposition eigen transform phenotype matrix multiply eigen previously calculation determinant iteration mcmc hasting posterior hyper marginal walk proposal particular snp standard rank snps small put snps rank high snp test distribution truncate denote add covariate switch pick covariate step switch hyper add new boundary reflect back proposal use improve mcmc move obtain h dimensional instead sample never normal diagonal simulation formulation iteration sample rao mean q end scheme fit comparable complexity practice maximal number causal text binary refer link trait row cumulative cdf auxiliary vector specification hyper mcmc text posterior posterior sample slightly q hyper identical calculation transformation need iteration probit set transform quantitative trait assign individual higher quantitative half consider approach probit view probit counterpart score exact probit calculate figure difference trait trait treat trait use probit partly reflect burden factor provide alternative derive notation proportion cdf pdf
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image follow another rbm family introduce binary binary representation include layer form binary binary layer refer encoding conditional factorial give advantage layer close match value binary deep ph training train second either cd good train cd generate get inference exactly discuss pass possesse sharing patch texture structure feature field layer hide map across map layer associate layer weight field mm texture texture texture dataset stochastic gaussian rbm high fidelity remain challenging model deep ccccc synthesis bi tm multi tm texture unit synthesis relate model fair protocol rescale original either texture divide bottom half testing report layer task synthesis model size minibatch deep always cd high mix negative relatively begin persistent chain especially possibility advantageous mixing model follow texture synthesis texture layer cd cd inductive impact unchanged filter field imply difficult task handle add convolutional share layer limit show result texture measure texture measure present texture synthesis inspection impact depth inductive versus cd texture help explain improvement mix gibbs chain improves already argue performance train negative sample improvement mix autocorrelation spectrum markov plot sample chain find layer texture specifically worse obtain consistent regard advantage vs consistent sub train divergence visible stochastic something cd cd help make sure information rbms job avoid spurious rbm mm deep texture heterogeneous train layer rbm sec filter convolutional rbm layer generate model apparent sample particularly structure rbm label easier share texture single texture amenable cd suit even effective interestingly find cd low deep integration development finally high apply boltzmann machine modeling achieve art texture synthesis parametric rbm layer layer result deep belief powerful generative capable model single resolution texture human image certain extent progress natural progress area graphic although capture property probabilistic model model model human vision boltzmann previously demonstrate sample preserve would texture model share weight filter convolution particularly choice texture model texture capacity texture concentrate evaluation texture exhibit strong largely consist regular repeating right wide wide decompose use pyramid interact spatial convolutional generative architecture depth part belief texture layer concentrate find training cd maximum persistent offer important similar model amenable cd contribution texture model model competitive art texture synthesis novel rbm stack belief train strategy layer third able encode dependency cd train translate texture model exhibit stationarity help introduce capable label constructing mode texture belief capable purely unsupervised community decade probably popular texture synthesis currently base method typically seed transform patch texture unlikely readily applicable natural statistical track scope rbm quadratic visible activation unit configuration change several student visible unit hide mean hide work multi texture boltzmann tm single oppose multiple texture deep texture impact layer model help texture feature recent stack rbm layer top convolutional ability globally coherent finding motivated attempt coherence three set visible random spike value unit work concern spike contribution energy joint interesting despite maintain bipartite boltzmann machine th consist share conditional restricted use block conditional mm covariance represent logistic h nh root efficiently block gibb also persistent cd involve negative phase likelihood phase maintain approximate negative gibbs approximate regard block gibbs mark important distinction monte hmc draw likely vector amenable gibbs ability efficiently sample amenable unlike
collection index composite term analogue give modify shorthand oracle correspond minimum output procedure furthermore inspection logarithmic corollary application discuss process satisfie like theorem follow establish analogue proposition coarse extend prove grid shorthand sample definition shorthand arbitrary event technical lemma choice replace constant specifically select event somewhat event hold hold strategy use lemma fail union lemma choice analogue grid inequality eq transfer entire suffice substitute theorem nest collection collection consider model forest family collection dictionary wavelet polynomial obviously challenge limit understanding outperform relate model impossible class least must allocate finite approach trade class class selection one quantum time linear box constant f assumption difference penalty successively choose iteration balance compete optimistic intuition behind definition class encourage optimistic formal begin preliminary optimistic budget example n index goal selection excess attain limit choose f gain I strategy non sequel requirement follow allocate quickly proof alg round round definition result fraction budget allocate procedure budget allocate rate vc dimension satisfied constant bind multi armed nearly general see connection multi bandit reward suboptimal plus always intuitive class try distinguish amongst visit suboptimal good oracle quantify risk remainder assume dependent result aggregate function appendix choose round alg define function vc clear factor ignore selection arm set reward force excess must scale split computational budget class roughly small key distinction logarithmic small improvement I split budget across class computational outperform I online nest case apply grid replace theorem similar particular setup risk outperform allocation multi armed bandit start give choose receive small recall notational q q lemma class error occur much risk high risk assumption three sub index choose need thing number alg interpret implication invoke obtain concern indicator markov examine modify multiplier new framework idea main attain computational selection nest take space competitive extension complexity problem exploration obtain optimal certainly raise address dependent penalty adapt instance somewhat would another question ask intermediate selection correspond understanding broadly believe number available performance broad hope risk even computational hope extend affect quite acknowledgement cl version read argument perform support microsoft fellowship support engineering fellowship acknowledge award dms arc award start recall shorthand monotonicity increase inequality monotonicity establish simple establish shorthand usual q finally proof suffice probability eq event occur q choose minimize right eq follow event apply union final empirical second part non combine event complete event far apply early bind inequality occur since define apply guarantee find rearrange lemma let hold recall grid induction inductive q assumption event hold claim claim condition inspection two possibility apply inductive use monotonicity assumption inductive noting complete occur break amenable analysis control event happen three must happen shorthand relationship none eq string event bad consequence q assumption imply probability particular remain recall satisfied solve immediately occur complete proof proof risk e fraction budget allocate model class argument drawback select even model theorem averaging argument frequently online define divide union penalty class differ get note proposition theorem technical somewhat minimization objective multiplier lagrangian infimum derivative see I I px proceed shorthand clutter definition minimizer see previous make drop probability jensen replace sum bound I immediately complete average risk department sciences california berkeley usa penalize empirical minimization practical model effort require compute minimizer budget satisfy setting ff minimal tradeoff class q excess error error common address tradeoff express class choose tradeoff standard approach early compete objective empirical class approximation error rise example therein complexity penalty perform penalty decrease follow output selection author penalty give roughly possible incomplete include paper give complexity trade error optimally drawback limitation feasible low prohibitive modern form key inferential consider rather quantity effect specifically difficult assume expensive computational model rich strategy criterion class allow receive thus overfitte class address issue make consideration provide many different result address order competitive know logarithmic extend rate condition guarantee concentration risk oracle refine regularize risk consideration class share structure novel exploit base prove select computationally paper organize estimator oracle model refine rate selection model technical argument need main corollary work setting class inclusion increase give natural proceeding computationally first perhaps example aspect computation inclusion natural minimizer say computational many estimator estimator formalize give penalty budget formalize notion problem model computational class part mainly result rule degenerate case upper sequel shorthand clear certainly many study penalty e extensively satisfied example loss unless model selection grid particular penalty continue hence large inequality least class hierarchy j class place model main explain furthermore unless trivial must computation brief remark ask oracle access could optimal entire budget satisfie compare hand know optimal budget logarithmic computational budget require priori g accordingly round computational update guess suppose round budget round get budget last round length oracle replace bad ease set theorem arbitrary may turn setting inequality specify process output sample process computation output algorithm satisfy proof slight generalization satisfy notice suffice additional notation generalization choose
metric agreement weight kullback area metric base metric area metric validation mean predict absolute error method reliability relatively less section uncertain two variable random probability discussion derive experimental relatively sample assume validation test eqs test model reliability letting decision decision acceptable within directional modify interval bayesian interval original interval half fail less case quantity predict deterministic zero case model metric propose al attractive capability fully pool validate experimental cdf observe q statistic probability standard match cdf cdf uniform variable small correspondingly observation deterministic applicable reflect directional due cdf gaussian random gaussian small variable distribute area cdf reliability interpretation decide reject accept disadvantage aforementioned prediction quantity predict coefficient uncertainty characterize compare device rf require reliability device rigorous quantification observe reliable device framework quantification strongly affect switch micro available regime micro illustration study major prediction failure mode variability quantification simulation surrogate krige r combine failure failure system integration denominator integration numerator although different test factor testing failure test reason pressure pa pressure pa failure percentage equality hypothesis table observe performance pressure locate pressure pa interval calculate reliability failure compare percentage failure pressure existence directional pressure pa vs failure failure r vs number four computed eqs result pa perform bad directional mean reflect pressure pa surrogate rf switch performance test middle two pressure pa pa whereas low pressure pa failure reliability reliability base metric failure base test bayes reflect bayesian testing base directional bad classical testing equality testing hypothesis reliability validate polynomial predict micro coefficient rf standard directional testing validate entire model formulation test fully characterize experiment hypothesis subject minimize average use mathematically relate reliability prediction reliability reliability incorporate failure reliability device explicitly account direct testing method validation paper upon partly support national nuclear security award na provide quantitative investigate bayesian hypothesis testing validate interval hypothesis consider measure report measure testing properly choose avoid error reliability existence may consistently specific reliability metric mathematically metric conduct frequency surrogate run fully characterize testing process real perspective comparison prediction widely supplement systematically effort application validation development prediction experimental discussion base method unclear validation fully characterize characterize directional threshold validation result characterize experiment input interval expert range partially characterize partially uncertainty input characterize limited practitioner input assume validation term input experimental thing physical variable experiment diagnostic quality input component measurement represent previous study validation experimental explore validation existence prediction physical predict literature deterministic value variable case study validation directional bias directional direction remain vary explore validation account existence directional fourth usually validation datum show condition mathematically rejection various quantitative include test hypothesis deviation entire two characterize characterize bayesian modify account directional classical reliability method present quantitative characterized quantity predict uncertain directional mathematical validation example switch generalize polynomial replace expensive micro simulation device predict quantity comparison experimental response prediction base relate type available partially experiment characterize measure physical quantity uncertainty source uncertainty modulus variation micro structure output input stochastic input report uncertainty lack knowledge probabilistic paper uncertainty value uncertainty uncertainty note quality validity discussion due uncertainty measure experimental limited section h predict deterministic partially interval equality reliability area treat select computing decide accept prediction metric fig validation involve plausibility nan alternative hypothesis could whereas coefficient prediction model coefficient pass exercise hypothesis reject incorrect validation matter necessary hypothesis especially formulation acceptance model prediction physical classical explain detail statistic facilitate hypothesis statistic formulate distribution derive theoretically statistic outside nan good test statistic statistic outside information decision whether make acceptable maker define exceed acceptable correspondingly accept alternative value reject note acceptable significance sample value reject level interpretation testing test equal equal hypothesis come kolmogorov test relatively thus predict normal random normal mean observation datum random freedom model equal mean end obtain cumulative distribution degree freedom reject fail estimate instead standard deviation standard nan tail hypothesis commonly similarly note test directly compare input form interval unconditional unconditional dependent unconditional model uncertainty reveal relation occurrence probability occurrence event occurrence q true calculate probability represent prior validity collect conditional original bayes support factor distribution formulate differently propose base nan hypothese formulate deviation within existence bias formulation distribution parameter either deterministic applicable standard deterministic formulation case two formulation characterize tail deviation may validity moment standard purpose observation random hypothesis derive practical issue underlie therefore calculate bayesian testing divide straightforward deterministic directional set hypothesis fail test directional increase chance combine illustrate combine test use overall datum first interval union support predict formulate correspondingly predict quantity e case observe pdf alternative assume informative pdf uniform affect estimate factor proportional eq parameter bayesian testing pdf partially pdf base report calculate unconditional prediction characterize partially separately equality
occur eq determine far achieve predefine common false run chart occur change positive desire determine classifier predict whether prediction stream stream view detect bernoulli although slight far may situation since affected drift far throughout concept drift increase error rate work subscript standard deviation estimator several modification streaming detection know whereas classification must stream estimator weight less sensitive therefore occur quickly new whenever distance exceed standard deviation standard q implication choice desirable detection positive false likely occur point give deviation keep false positive vary limit recommendation choose post advance drift chart stage limit change decide procedure inverse base carlo method conduct carlo choice carry monitoring begin overhead unknown positive control limit time vary add subscript would whenever update detection chart adapt detector approximation single monitoring begin overhead concept required various stream begin extremely fast approximation follow various monte adequate several derive suppose positive give use misclassification threshold table detect drift call drift stream classification choose classifier value critical sequentially time predict label correct incorrect table current concept modify take depend rest discard begin classifier classifier correctly classify incorrect presentation classifier whenever stream change change store train store practice store order threshold occur stream retain drift classifier later drop conclude false slight equal reasonable occur concept take detect suppose drift detect soon relatively value since pre change distribution recent pre magnitude concept drift generally detect magnitude class distribution switch change pick synthetic drift pair evaluate stream discriminant lda chosen write streaming choose utilize complex boundary knn recursive predict observation knn stream detector detect memory retain change rest discard detector user evaluate detector give performance prefer find give generate accuracy concept drift detector investigate value value emphasis place omit space reason sensitive choose rest gauss gauss compare pc realization change detector concept detector performance without positive occur fast occur less positive change detector highlight importance detector change algorithm method threshold incorporate improve possible performance difference detector detector affect affect difference significant follow use algorithm due simulation less gauss gauss lda lda pc knn knn knn pc knn gauss gauss lda knn knn pc knn previous change however cause change change investigate encountered majority drift change drift move concept drift static perform suitable modify gauss stream initially static drift period previous modification equivalent change point allow drift occur increase moderate c pc knn knn competitive drift outperform suggest choice concept may take drift limitation real world image advance concept drift stream incremental real drift detector serious artificial set desired however change control pc detector key limitation evaluate detector wide setting give good practice varied detector range pc detector free price collect south price market vector demand state label price increase take hour decrease price movement available give sufficient classified drift incorporate increase knn great accuracy obtain knn suggest identical assume setting allowed vary accuracy lda knn classifier accuracy gradually drop critical reason frequently varied drop amount classification perform move slide average accuracy whether contain point detect whether detect accumulation pc knn knn knn knn accurate video contribute early early diagnosis cancer obtain image preprocesse grey level pixel label tumor pixel optimal varied parameter detector set correspond accuracy lda gradually drop result performance suggest frequently appear broken segment change although require verify present streaming problem exponentially move chart stream stream feedback receive memory allow drift experimental performance possible direction research extend monitor entry confusion separately stream method cope known drift detect move chart misclassification stream modular detection overhead fully memory exist rate control time classification stream consist point receive stream meet follow criterion pass process discard acceptable store small repeat store constant increase computationally process stream must observe streaming version unknown predict learn assign literature streaming reveal immediately make classifier receive immediate whether accurate key difference stream conventional offline streaming change dynamic distinction make case gradually change remainder classifier stream significant deal drift fall category classifier automatically adapt stay date classification concept drift concept detector design occur useful adapt concept give occur take thought change concerned detect drift limitation streaming observation receive frequently control concept drift serious case desirable concept drift really drift detection stream contain change detect drift occur detector know positive occur
markov forward backward outcome hide variable adapt backward formulae backward heterogeneous consequence standard marginalization quantity distinct set consequence contribution formulae reduce q conditionally j backward forward recursively recursively forward quantity moreover complexity x forward practical influence real consist model hmm gaussian hmm obtain hidden transition year appear posterior influential look protocol j temperature depict empty dot validate effect five influential segmentation observation remove remove take account period characterize segment negative influential basically segmentation remove influential one bottom solid I point segmentation interpret either particularly outlier underlie statistical application detection linear argue influence effective model outli influence differ posterior condition illustration support supplementary material detection change assume random time outlier sample probability hence average outlier statistic maximum absolute normalize three outli score axis detail supplementary material assess result depict statistic axis individual meaningful play consideration investigate influential protein structure structural point influential measure might reveal property receive ph paris dr interest include propagation computational receive ph university france include network hmms hmm different outlier hmm hmms framework author suggest outli viterbi search follow outlier hmms influence outli hmms heterogeneous model hmms instance hmms sensitive presence sense outli point explain main paper significantly temperature fig estimate function outli peak interpret answer manually add outlier depict peak two outlier top two outlier triangle semi simulation temperature original simulation outlier original outlier obtain sampling average test contain estimated modeling cluster standard deviation cluster base score calculate inverse neighbor densitie dataset outlier sort depend distance parameter integer standardize rescale axis score compute assess auc auc surface receiver operate roc qualitatively interpret fail fair good auc proposition kullback leibler namely sequence hide influence illustration application meaningful datum forward outli local outlier process biology typical markov letter measure start fix sake consider hmms parameter fully set density omit write explicitly observable evidence except measure influence observation influential leibl entropy application distance suggest state distribution speech recognition bioinformatic letter kl one good
media communication generative community detection split stochastically block membership highly flexible represent structure combination asymptotic phase transition tool theory despite block impose restriction network notably block make model many network community fit stochastic low degree community conservative political degree avoid allow history generative edge attribute membership block allow block extend pattern ordinary model selection largely base ratio ordinary correct variable ratio likelihood must likelihood propagation bethe give deal likelihood hand turn usual ratio nature degree correct derive asymptotic recover classic large dense substantial correction correction confirm validity theory network represent want split community priori elsewhere traditionally stochastic entry adjacency version poisson ignore loop former indicating belong assignment multinomial parameterized node thus assign block generates make poisson draw block involve connect twice product q assignment jointly physics ground minimize energy find graph wish total summing distribution logarithm minus energy latent approximate estimate average monte gibbs however review marginal distribution belief efficiently approximate marginal typical rapidly certainly stochastic moreover degree sum poisson degree block lead within block skew heterogeneity block node assign block get number connect recover block provide force total correct drop node belief propagation refer belief likelihood purpose belief expectation belief propagation extend idea marginal update light message node take markov field weight complete opposed network track sparse scalability node simplification distinct reach entire estimate maximization set propagation propagation holding observe bethe energy log chain carlo typically propagation seem propagation numerical selection homogeneous block correct extra heterogeneous without prior thus distribution need pick natural approach nest correct really superior degree converge least stochastic correct model define usual favor elaborate alternative exceed turn nan nan bootstrappe large however helpful bootstrappe nan constraint must must already impose model well central datum functional dependence fail specific dense effective grow correct asymptotic grow recover limit dense correction even shall consequence ratio characterize distribution concentrate assignment major stochastic underlie energy term ground substitute kullback leibler apply k xy axis marginal ordinary degree model correlation appendix show asymptotic limit give analysis small somewhat asymptotic figure simulation mean well fit formula moment simple quantile fig indeed fit free imply concentrate around interestingly fig concentration theory remarkably illustration replicate formula observe ground fundamentally follow really relevant observation likelihood analysis apply graph expect say differently case least extra really ordinary asymptotic see add predict fluctuation test go asymptotic ratio eventually big nominal pass happen would incorrectly must derive distribution calculation consist member make heavily former division classic degree nonetheless bootstrappe asymptotic analyst think twice force block imply mild responsible correct low extent bootstrappe fit reasonably theoretical marked line value theoretical political theoretical gaussian well reject favor less clear political focus political either correction greatly political observe completely agree bootstrap theoretical deviation essentially extreme model network extra capture justify several theoretical
presentation help generalization useful temporal period trial choose trial learner rely memory learn learner fuzzy system represent past fashion naturally generalization delay profile match available trial learner seem natural resemble fuzzy system natural world behavioral human event curve observe invariant human ask reproduce discriminate scale invariance characteristic human wide finding specie suggest world many scale fuzzy memory generic connectivity perturbation instantaneous stable spectral magnitude one connectivity view reservoir node efficiently tune neuron reservoir past fact dynamical reservoir require define reservoir precision input reconstruct net area past reservoir depend connectivity step beyond generic connectivity smoothly exponentially sometimes law recurrent network turn memory tractable least special reservoir net memory increase unless tailor orthogonal feed grow slowly trajectory connect network memory low dense fuzzy memory view reservoir tailor connectivity linear weight input white decay power law net grow relevant interest predictive fluctuation basic essential world learn ignore sake time vary serious multidimensional separately shift storage resource shift sufficient storage general encode technique bottleneck slow strategy fashion complementary feature impose principle low component time fuzzy fashion shift potentially slowly varying activity issue ignore serious application involve ability svm elegant strategy svms prediction slide window shift slide recurrent correspond improve gradually memory step shift svms svms along fuzzy input shift long temporal contain need fuzzy directly svms aim build happen fashion processing datum agent memory resource learn mechanism rely tailor act memory could potentially find signal rely series construct free way information redundancy information fuzzy number correlation representation quantify expression mutual information share neighboring signal uncorrelated long correlation redundancy spread activity simply average integral generated define redundancy calculate correlation correlate node small construct integrating function artificial correlation exist activity turn q decay exponentially even short emphasize shift moment pass without integrate activity reflect temporal activity instant activity response input shift make across mutual share calculated activity individual variance quantity high mutual share vanish redundancy require redundancy need mutual neighboring neighbor happen arrange input long lead correlation k note integral come denominator temporal large shift instantaneous correlation turn summation yield proportional approximately node behave correlation shift mutual neighboring node scale possible eq acknowledgement acknowledge national science air force office fa predict vary maintain recent signal memory shift extend storage resource learner priori many occur scale resource availability fuzzy accuracy free information illustrative fuzzy system time available resource limited purpose learner continuously recent basic arise much maintain past evolutionary pressure minimize resource consume grow generally example occur commonly prediction relevant essentially unbounded learner long range natural signal resource argue reflect scale free uncertainty example second second resource yield grow accuracy free learner way fluctuation mind represent sufficiently evolve rely instantaneous external storage resource inaccurate memory long optimally maximally property optimally temporal scale maximally preserve memory sufficient construct self free past mathematically laplace approximate node lead sufficient fuzzy fuzzy forecasting example ability shift past ability prediction algorithm learn modular system memory use forecasting need value correlation past time present stationary ignore existence long predict axis unknown future shift denote buffer wherein store unique forward store node enter self sufficiently represent shift optimal way relevant prediction maximally preserve buffer fuzzy average linearly center bin bin past represent bin attain represent show prediction information fuzzy buffer evolve self lose compression bin moment correctly buffer buffer actually save resource series though buffer sufficient extremely buffer maximally preserve relevant storage resource linearly skip ahead quantify contain review range correlate since actually behind white integrated generate series long range hence auto word past instantaneous white specifie coefficient term two power series correlate behavior euler formula gamma large sake analytic relevance predict predictive content taylor expansion show restrict linear clear sum quantity entire accurately represent infinite shift contain memory buffer turn shift portion predictive correlate buffer rather single shift theoretic consideration information bottleneck multi variable systematically compress transformation maximize series moment correlation maximize eq immediately single buffer moment buffer statistic simply uniform bin bin examine averaging individually would error extent give functional bin memory summation perform approximate integral bin equal bin turn straightforward pick successive bin completely past buffer fuzzy buffer range fuzzy buffer node buffer bin power fuzzy buffer lot predictive buffer wherein overlap ensure optimally free fashion ideal fuzzy buffer evolve possess fuzzy buffer overlap non average bin extent fuzzy fuzzy buffer mathematical recent free fashion internal critical consideration implement past fuzzy buffer temporal represent scale unlike buffer require resource construct distribute free fashion node estimate decay constant denote gets activate instant gradually decay decay drive functional functional value fuzzy instant ss make resource mathematical infinite mathematical come history laplace transform reconstruction history moment past distant temporal plotted top activity distribute illustrate equation delta input whose width area scale invariance invariant quantify estimate width peak index delta column delta delta input invariance linearly term combination bin spread activity desire bin buffer value nonetheless evolve input node evolve self noise across long constant white noise snr large drop zero signal appropriate delta signal average spread signal standard deviation signal snr positive generate spike activity node multiply delta snr stationary snr long node truly buffer redundancy scale buffer share neighboring node buffer free mutual buffer node accord redundancy spread spread redundancy analyze delta buffer fig activity buffer point value buffer imply redundancy panel buffer activity instead activity whole translate invariance activity pattern pass explain neighboring node invariance activity analytically value eq say pattern inside law buffer except happen whenever integer infinitely condition approximately hold invariance pattern buffer choice accordance redundancy equally distribute buffer redundancy reduce redundancy buffer large moment buffer node redundancy loss moment two buffer formalize delta input past current input successive buffer value min delta minimize redundancy th th almost activity middle vertical line represent dot represent activity correspond activity substantially different imply redundancy imply lose redundancy activity th activity th activity delta past n activity delta input activity value range rough estimate eq require high b attain focus two node close node close intermediate represented activity minimum minimum exist activity great simultaneously information redundancy minimum fuzzy minimize redundancy compare shift forecast goal difference shift self fuzzy learn forecast consider series property generate integrate noise library http com temperature year measure series plot row plot correlation reveal difference like correlation short time short autocorrelation balanced series correlation average year year row row show row self memory red use denote sequentially feed sufficient fuzzy value shifted discuss instant hold exactly self value record coefficient extract standard lm open absolute intrinsic intrinsic variance quantity correlate construct correlation decay decay serie generate reasonable see error forecast reduce error see fig blue shift square white memory shift temperature fig seem small reflect reliable knowing year help temperature subsequent year seem correlation long scale could exploit additional
simulation underlie belong belong thus group tn causal relationship among direct acyclic equation give dimension contain direct mutually group define rewrite block acyclic also include non noisy restriction grouping know merely estimate causal among group observation arrange correspond row difference algorithm explicitly build allowing effect iterate generate group formally group assume acyclic group group generalize find directly state regression follow j ji appendix apply vector combine fisher one hilbert schmidt dependency nonlinear correlation p modify measure infer two scalar underlie point likelihood rx lx ml j meet assumption error within group correctness term measure advantage term section pair meet j consistent total group n lk ix k take obtain group accord adjust third base term among group causality independently base assumption error need measure matrix matrix estimate connection ol estimate direction et direction close suggest trace group infer minimize equation causal set formally state correspond combination group ji residual lemma section formalize causal arrange datum regression approach use connection parameter particular approach apply data variable calculate pick group minimize eq whole set value individual bold legend bold pdf fig black toolbox simulation variant hoc available follow randomly create connection average nonzero additionally complete mix random sample gaussian finally randomly causal order variant ad hoc modify search permutation matrix block yield low triangular finding causal secondly approach distinguish markov graph simulation complete datum generate show size exception fair design dimension perform follow hoc perform dash ica perform seem probably dependent ica replace apply algorithm large mistake make find strategy
simplification take theoretic leibl predictive obtain support take expectation result outcome understand gain divergence decrease amount expect utility typically expect close predictive must bayes outside logarithm introduce nest estimator nj mf approximately term maximize optimal objective need describe stochastic sample require gradient monte simplicity essential idea complexity solve design variable unbiased objective word sequence satisfy one appropriate scaling diffusion section choose technical solution view sensitive acknowledge choose logical converge iterate relatively criterion change fall tolerance successive maximum experiment stochastic fixing throughout entire practice dependent transform random moving example distribution realization transformation practice fundamentally really always shall generality design value application compute large attain low finally run optimal remain within converge respective stochastic true optimality unbiased value across replicate w difference optimality formula optimality decide run approximate still optimality tolerance replicate perform algorithm large compute w optimality post bfgs optimization robustness ease implementation update approximation evaluation bfgs backtracking search find stepsize satisfy first wolfe many commonly fall tolerance variable evaluation reach bfgs taylor l design dimension become start termination initialize termination meet direction search update define u tu kk challenge apply optimal lack gradient introduce simple expansion replace intensive first analytically come word forward expect gradient estimator analytical derivation introduce repeat forward evaluate stochastic enter experimental prediction describe function map discrepancy take intensive drawing evaluating evaluate calculation tractable one would replace surrogate accurate space option spectral expansion approach include recently loop bayesian experimental excellent unlike stochastic simultaneous series orthogonality convergent sense sum truncate common truncation retain certainly polynomial impractical construct pc expansion jointly affine uniquely realization design endow convergent pc expansion computation via approach need difficulty latter prohibitive compute project quantity g onto method treat box solve functional trajectory functional smoothly state employ orthogonality simply coefficient th component model evaluation expensive integration dimension employ quadrature care must avoid significant quadrature polynomial indeed full tensor approximation quadrature rule construct substitute approximation enable gain discuss applie unbiased use finite expect obtain simply unbiased idea availability unbiased estimator optimize instead support new tradeoff course realization monte requirement comprise condition unbiased estimator unbiased require parameterize continuous perspective product noise support case use forward distribution observational meet requirement preserve must term original simple variable stochastic perspective assume diagonal dependence standard normal example translate kronecker delta replace function illustrative extract variable second design incorporate substitute draw realize draw multiplying output therefore monte carlo approximation realization either likelihood expansion derivative derivation gradient estimator orthogonal expansion experimental design tool two place govern square space concentration parameterize center impose along spatial prescribed source location ultimately sensor spaced sample I mean mutually contribution concentration profile realization nd spatial backward integration model parameter use truncation evaluation surrogate g evaluation relative current formulation seek possible accord likelihood five concentration great information gain result numerical explore gain show interpret monte approximations carlo approximations become grow become value flat flat encourage higher solve sa effectively signal ratio gain suggest albeit point lie interior locate corner corner center measurement orientation distribution constant radius sensor sensor minimize area problem happen corner show density source true directly spatial note posterior moreover reason perturb relatively significant construct discretization pde pde simulate treatment understanding impact direction place corner tight sensor center keep realization depend source imagine sensor prior source location result assess behavior method compare ascent maximize figure path ascent direction gradient eventually naturally good gradient increase show instead intuition mechanic set realization find bfgs small subset large realization objective extremely near different objective none encounter maxima corner nonetheless subproblem low table bfgs conduct start location run uniform schedule termination criterion lead similar time histogram major represent result major bfgs different value immediately corner corner around high improve quality bad balance need choose sample size outer overall bfgs intermediate slight bfgs place flat near optimum need close optimum robustness design keep run information variance estimator observe histogram reflect design corner design mode upper sample enough large size shape expect high table show histogram gap bfgs deal sign upper bind low size run bind optimality gap decrease since able corner monotonically consequently bound tight toward increase gap spread indeed objective flat consider occur corner translate sensitivity change expect increase large intuition obtain consider change upper low value increase corner mention early well indicate contour figure choose value certainly indicator surface gap value gap appear another stochastic run table low bfgs take terminate maximum difference reflect bfgs histogram transfer number fold lead account sublinear resource substantial bfgs require take high freedom user stop attractive evaluation expensive integrate quality final design true optimum study take mse reflect plus difference via relate effort computational run symbol four confirm previously encounter solution quality balanced large inferior curve highlight optimal light monte estimator reflect bfgs achieve small mse give effort generalize experimental design paper stochastic employ formulate approach sample couple bfgs case extract gradient gain rather nest evaluate challenge forward polynomial expansion perturbation sensor cast design design perform matrix loop sample size impact gradient estimate efficiency optimization result suggest improve sample optimization size outer loop instance yield little quality consistent necessarily algorithm superior broad difference sa algorithm essentially optimization flexibility surface behave exploit source inversion gradient gradient may appropriate optimality replicate solution use adaptively adjust assess employ stream estimate interval method stochastic outer loop reduce relatively frequent fine terminate without termination present surrogate product method derivative free prefer surrogate increasingly small optimum similar idea additional potential gain point focus open experimental design collect rigorous design stochastic objective continue mathematics air office national science foundation diffusion equation five design vertical range vary htb surface figure corresponding value htb surface htb surface position htb cccc htb c cccc represent frequency cm c htb versus time optimization inner outer loop highlight red bfgs light derivation estimator form method present gradient form respect form constant vector mutually compose component become normal datum derivative condition summation expression model case quantity available
k concern miss feature conditionally word object observe assumption correlation variable generality suppose probability soft class object rating uniformity define use rating object rx co co miss feature motivate parameterization r rx vector similarity make learn metric feature provide version hybrid discuss generate vector class estimate parameter describe rx hybrid semidefinite aforementioned ordinal convex optimization analysis experiment generate data experiment consist ct along rating part project seek develop retrieval detail worth consider hybrid free use instance medical hybrid new explain generative vector detail generative c scan include semantic annotation vocabulary haar extract image simplify remove one feature study rating figure demonstrate image introduce note pair appear lc since cross validation c j co result distance retrieve ndcg retrieve list figure ndcg margin htp co give error hybrid learn measure class element datum fit rating try synthetic purpose medical retrieval substantial gain think generative growing explore combination hybrid broad detail randomly partition validation set ascent compute range determine trial choose optima rarely data experiment model mixture gaussian procedure produce matrix iid nx nz pm theorem proposition van xu distance hybrid learn assign assign base provide encode well real medical image leverage retrieval significantly system search object require assess learn similarity rating representative user kind object distance available serve search retrieval classify provide useful relate rating metric attract attention year propose label refer method category ordinal discrete convex metric classify similar distant method share distant minimize class aforementione literature kind stage similarity metric classifier distance optimization similarity rating medical retrieval problem object object rating comprise consist object feature similarity rating dissimilarity respectively class triplet class reason object give object sample metric result accurately reflect variety discussion choice metric retrieve similar discuss three exist ordinal offer learn ordinal assume object rating obey level together coefficient metric nonnegative rating approach compute rx
partition ng result note likelihood dependent conjugacy gps simply nest fact procedure motivating replicate trajectory allow trial trial parent structure hierarchical motivate application trial relax pt trial replicate trial share parent alternative structure consider independently location replicate exposition compactly denote covariance share show qualitative raw correlation similarity ccccc sim sim sim est sim trial recursive normalize set infer conjugacy q likewise predictive pt marginal decompose easily far assume base importance hasting sample hierarchical solely partition recall tree join neighbor contiguous region think partition balanced tree graph build randomly select uniform contrast setup straightforward point level uniformly undesirable lead highly unbalanced partition function adaptively use search tend correlation partition much space think location correlation partition correspond cut edge weight depict seek cut give set measure connectivity avoid small fig select cost absolute location straightforwardly propose partition normalize cut simply cut generate partition distribute extension distribution weight draw amenable efficient size match alternatively straightforward algorithm iteratively propose accepted proposal normalize cut dramatically improve proposal acceptance decrease discover especially many tree large benefit efficiency proposal first cut adapt encourage search shift towards great balance search particle proposal gp dynamic cut gp computation extension similarity gps collection gps node leave cart partition assume hierarchical inherent perspective instead motivation partition gp aim partition parent partition datum covariance value trait distance common tree define branching application internal gps gps approach fundamentally necessarily spatially define multiscale history cf variable define level process high level share vary level differ gps contiguous gaussian vector vary fashion combine formulate focus gps address inherently another fundamentally latent take mapping finally hierarchical associated tree closely task standard tree partitioning randomize unbalanced tailor pt ccccc sim sim sim sim map trace log likelihood versus chain minimize error assess ability infer propose replicate level set specific simulation demonstrate prior chain run pure search hierarchical estimate especially average region result consistently capture feature sensor replicate visual much well track jump stimulus break display expect see key occur finding processing length visual accurately standard response also compare predictive functional become approach trajectory share information trial limitation restriction grid enable direct word independently trial interpretability exceed pt generative characterize structure examine capture certain accurately previous model provide locally ii global iii change enable theoretical gps focus univariate formulation amenable multivariate naturally accommodate share parent possibly model interesting relate specification represent particular parameterization alternative hyperparameter mix poorly future exploration broadly new framework stationary analysis especially possible extension thank acquisition preprocesse useful suggestion support grant science es marginal compactly discuss therefore c k py pf derive assume additionally drive hyperparameter perform comparison scenario take take fix parameter optimize grid training sensor hierarchical gp assume single bandwidth parameter particular shared function jointly gp covariance constrain study bandwidth parameter gp assign trial setup determine jointly optimize example ccccc account significant variation account trial trial drop fairly rapidly level indicate note trial variability instead sensitive hyperparameter fairly robust reasonably setting marginal sec word sensor word cut amount record smoothed strength baseline prior information expert run prior identical aggregated fig demonstrate dominate related display example test datum sensor full ccc fig mu pp trial fig pp trial fig pp trial fig mu pp trial fig pp clarity slightly large interval observation mean subject give write protocol review record use device acquire pass filter hz filter hz horizontal participant position indicator place separation remove activity begin location filter hz remove contribution line contamination ms stimulus end ms stimulus second subtract signal amplitude stimulus multiply precision replicate iii proposal py detail search root select location hyperparameter previous partition point initialize structure proposal normalize proposal pt z py pt hyperparameter previous partition initialize proposal previous remove normalized proposal w proposal assumption example section definition david statistical science nc propose capture dependency gps nest capture top level partition inherent conjugacy analytically partition allow employ theoretic analysis key challenge insufficient one address challenge employ band covariance gps underlie mat enable smooth function assume adapt vary capture span memory extension smooth shift invariance fundamentally different gps direct interpretability local stationarity grid information across motivate activity stimulus snr key stimulus although similarity replicate key difference phenomenon well correlation ability share single trial long potential gps nested inherent conjugacy gps gps conditional encodes enable correlation generate distance encode stationarity importance science vision sampler global move inference integrate partition allowing
stop dependency increase demonstrate dependency capability capture moreover source reliability expect source reliability extent list assess true risk dependent approach jointly group true source capture dependency collective experimental exist edu collective intelligence low become barrier effective collective intelligence address issue assess reliability observe often source influence share high intelligence redundant possibly reveal latent source aggregate prevent collective dominate source reveal reliability result real respect intelligence collect task participant fairly extensive set crowd article amazon involve wikipedia claim fact object collective infer fact collective intelligence people source may treat aggregate dependency among involved group reliability incorporation reduce observation especially dependent source dominate collective intelligence especially reliable moreover equally reliable may reveal reliability group negative reliability reliability two closely reliability group aggregate individual reliability distinct generally reliable likely reliable individual object vice versa overfitte object reliability consider remainder formally define source sense evaluate section tp define multi sense description intelligence source object take source report infer true source discrete value many crowdsource binary assertion correctness test value however extend topic paper five object object category object alternatively book book value broad g present answer claim notational convenience claim bold subset meanwhile among membership source inherently dotted influence tend object unseen infer knowledge number determine bayesian break construction impact group reliability reliability meanwhile specify object reliability dependency infer generative sensing eq denote discrete generate bernoulli stick break latent membership pg l stick break need group determine capture degree construction determine dependency accord likely assign dependent share observation degree probability belong group extreme yield complete dependency reliability soft specify priori sample specify reliability high general reliability sample generally reliable reliable object sense general risk reliability adopt uniform distribution determine posteriori process observation source conjugate depend reliability detail specification adopt categorical generate different setting group object value hand object randomly object soft count false indistinguishable member source guess ii tend value contribute observation observation process wish family factorize factor determine u coordinate descent update sequentially update form still dirichlet em element vector paper short still q find sum reliability object intuition reliability reliability object object reliability logarithmic beta update line reflect reliability affect estimation reliability reliable group vice versa overfitte estimating simultaneously update source expectation l define index thus sum distribution exist demonstrate effectiveness infer reliability perform book book online web site book author data book science book com book com book extract book total book source book object author object book categorical domain claim name preserve name ignore middle name author book manually book cover list test one
energy use range energy non submodular underlie regular optimize compare multiscale swap energy state tailor energy multiscale framework state family energy apply multiscale optimize challenge metric energy diverse mrf energy energy well optimum opposed move global optimum able multiscale table fig multiscale energy valid integrate multiscale put track running time energy mm swap single show relative close energy cc c swap expand multiscale drastically visible swap numerical swap close energy cc unable cope step unable propagate far fill hand framework step gap numerical single multiscale scale sec sec second multiscale runtime construct pyramid take optimize coarse time serial construct effective multiscale scheme experiment compare energy measure sec three unary min unary unary agnostic within multiscale four representative energy percent different energy unary wise percent lower close bar energy aware consistently label prove example energy pyramid take principled swap energy energy label denoise sec sec use variable unchanged percent achieve relative well time unified multiscale multiscale energy multiscale landscape involve level involve method energy interpolation multiscale energy gap multiscale algebraic multiscale pde solver g allow method develop solver multiscale optimization minimization problem l l n look j unary put together I diagonal represent unary term energy long zero coarse interaction unary easy add unary whenever zero diagonal chapter energy pyramid energy pair energy restrict simplicity energy wise write q entirely wise differ energy energy pair general define unchanged energy agreement neighbor fine denote entry interpolation indicate fine affect ij energy q interpolation issue derive future however yield fairly straight energy variable label energy beyond interaction energy refer energy energy variable q high way order parameterize energy interpolation energy estimate agreement neighbor use interpolation entry scale variable coarse hyper interpolation trivial representation interaction label leave derive interpolation matrix coarse parameterize label mm derive multiscale energy pyramid unify different different energy multiscale framework minimization segmentation denoise apart submodular binary energy develop challenge energy etc discrete energy submodular regular energy submodular energy submodular energy encourage term energy reflect piece popular application vision focus energy encourage hardness energy energy consider energy optimize contrast energy contrast encourage energy parameter learn g functional optimize energy non energy explore optimization method energy minimization method primal act formulate provide approximate solution solution optimum labeling lower bind discrete hard one representative instance energy easy optimize energy show tend approximation tight discrete minimization exploration exponentially make task way toy effective method looking phenomenon scale fine local high multiscale landscape multiscale phenomenon encourage coarse apparent gradually search fine decade vision focus pyramid experience unify directly pyramid moreover energie multiscale empirical evaluation energy come optimize tend well approximation method multiscale space multiscale framework optimization process exist multiscale landscape fast multiscale problem denoise correlation cluster multiscale aggregation take underlie landscape label well pyramid multiscale framework energy include example submodular energy non energy superiority primal method multiscale scale increase optimization challenge multiscale directly prominent coarse pyramid work regular grid energy work propose scale aim unified analyze suggest landscape energy improve optimization method property restricted energy training address principled build energy pyramid decrease number label assumption consider discrete minimization many computer vision cast see restrict submodular energy build pyramid freedom key pyramid interpolation map solution level pyramid principled aware interpolation energy pyramid multiscale landscape energy apparent however counter intuitive semantic correspond label variable label allow discrete solution representation basis multiscale entry find energy label parameterize pyramid description scale wish generate representation approximate variable coarse assignment assignment yield approximate fine correlation construct desirable variable neighbor general may interpolation grouping formula unary dominate correlation naturally account influence unary pair inspire energy energy regular grid generate assignment neighboring variable iterate mode obtain energy assignment assignment choose yield condition random initialization locally assignment give together preserve energy smoothness preserve term allow explicit encode get algebraic select representative every variable correlate variable either consider strongly correlate c add yet select index fine coarse index fine interpolation row leave entry refine compute interpolation coarse eq energy describe multiscale put together multiscale energy fine matrix pyramid explore coarse fine fine scale serve two coarse solution binary round set intensive module framework empirical estimation scale refine solution fairly module restrict pyramid energy aggregation pyramid coarse refine main goal first submodular energy advantage primal method minimization unified improved exist primal diversity energy energy denoise measure correlation sec energy improve swap representative move make overcome non energy follow lower compare energy bind close multiscale scheme consistent multiscale dual discrete energy behind move make algorithm challenge energy multiscale give significant boost swap expand scale percent relative
erm tight online factor length notation mistake guarantee separable establish lower bind imply question bind study integer define number online batch achieve one loss rate hinge loss establish agnostic desire weight vector integer positive universal integer denote predictor associate half hinge margin denote q subscript omit clear threshold monotone formula generally denote case fractional negative refer unknown pair sample expect simply subscript minimizer ds subscript denote rademacher complexity respect draw average rademacher follow variation bernstein draw eq cases rhs eq least range b therefore consider fraction apply find mp unnormalized gradient unnormalize unnormalized theorem vector constant obtain assumption obtain rearrange definition desire sequence unnormalize list batch algorithm expectation x case x r I r three observe set detailed return average set canonical mirror set batch rate achievable next accord empirical w w theorem slow tight hold rate guarantee provide combine propose tight convergence analysis tailor positive label bound rademacher conclude dm dimension remove therefore j obtain inequality choose maximizer consider case otherwise q rademacher denote obtained side lemma define easy mu mu therefore get finally use uniform negative loss constant draw exactly lipschitz translation w combine use euclidean norm cover easy u w rademacher complexity entropy state member draw label follow combine rate give adapt fraction negative theorem case bind loss trivially prove loss eq bound induce get second provide constant trivially hold eq note bernstein prop become hand second since statement erm theorem prop eq light loss predictor factor via standard convergence provide convergence result show even domain restrict discrete pose complexity learn output example construction respect predictor norm construction loss adapt construction hold theorem hadamard matrix row exist construct hadamard exist hadamard eq q transform let every ki one coordinate label z term desire z z zero vector equal rewrite concatenation concatenation n finally h yx bound satisfie show existence algorithm respect statement construction h output verify contradiction show uniform learn theorem rate asymptotically indicate need erm exist prove lemma first class similar distribution denote thus leave define change cause change therefore inequality gs mt constant define appear hand difference cause fs fix walk chebyshev therefore check probability two ready low universal domain suffice domain vector lemma exist z w suffice define since microsoft school science engineering ram institute il science york ny department statistics nj non norm monotone batch minimization rate tight grow classifier empirical minimization low instance change extension binary classifier binary threshold formula formula specify study formula mistake formula analysis general case separable show mistake applying consider fractional note study separable order move vector therefore negative target necessarily separate fractional weight explicitly consider misclassified margin guarantee rely half upper misclassification obtaining translate hinge gradient online md bind statistical setting use rely regardless threshold yield mistake complexity relevant range agree md result mistake distinction lose md erm guarantee
impossible certain mixing consistent obtain select homogeneity upper length choose average linkage sample svm describe parameter definition distance artificial dependent marginal finite countable construct right remove integer time single distribution length min applicability choose brain interface competition subject letter originally one split dimension sequence minute method dynamic base estimation unsupervise svm interest svm unsupervise error time subject education project european program er de concern series cluster homogeneity address algorithms consistency prove experiment real problem algorithm perhaps binary conceptually simple try building algorithm different reducing many start problem formulate underlie theoretical datum hold exhibit range sample may well homogeneity testing binary result algorithm assumption stationary statistic homogeneity problem distribution distribution infinite evaluate consistent building finite dimensional distance apply homogeneity testing change far work time build stationary ergodic measure marginal sum infinitely marginal indeed fact algorithm classification setting choose concern interface present algorithm stationary ergodic distance trick involve reduction explain distance devote respectively condition address homogeneity testing cluster guarantee present experimental measurable space induce sequence time separable limit ergodic I express one study kolmogorov note mild particular separability sufficient metric rest easy apparent consist case slight abuse write ph true generate verify directly trivially set euclidean check high dimension space polynomial series two take decrease two distance e metric marginal q separable vc generate stationary ergodic finite vc stationary ergodic n I consistently find last arbitrary statement problem suggest calculate calculate eq indicator calculate follow dimensional attain minimize conceptually sample time distribution whereas generate ergodic mix memory assumption dependent stationary distributional statement stationary vc dimension generate straightforward instead distribution give sample thus algorithm distribution accord whether cluster asymptotically length point separable vc stationary strict separation separation cluster asymptotically algorithm theorem linkage work distance close assign follow linkage clustering consistent marginal ergodic spirit number situation strong assumption ergodic consider necessity distribution involve general result advance strong turn establish ergodic sample guarantee strong another reason statistical homogeneity provably ergodic shown analyse homogeneity sequence sequence mix process algebra mix ergodicity much generate indicator vc fix choice guarantee use similar bound finite term exposition sake rest letting whose whose decrease eq imply statement respectively homogeneity sample asymptotically stationary ergodic series consistent
galaxy correspond fig mostly galaxy autocorrelation galaxy red appear initial value random change successive factor pf show follow pf log band galaxy hierarchical galaxy formation model explain elliptical exist galaxy step mass formation spectrum average galaxy author thank anonymous impact foundation national foundation department office iii web site http www iii iii laboratory group de de berkeley national laboratory max institute university york university university university university university physics md usa edu universit fr ed france variety galaxies digital affine aim curve curve cloud principal galaxy galaxy measure find letter shape turn define branch galaxy population control low old variation important galaxy variate fit arc power log fit arc length galaxy increase autocorrelation power arc correlation arc length show galaxy decrease allow galaxy cloud red galaxy dominate disk ratio understand trend encode available curve dimensionality reduction support relevant physical scenario merge formation massive galaxy arise formation galaxy blue red constrain drive galaxy common property galaxy galaxy survey galaxy available constrain accurate level estimate wide survey survey million galaxy turning become feature keep increase amount physical want important galaxy dimensionality principal uncorrelated orthonormal base spectra wide galaxy property equivalent emission mix galaxy population show population galaxy enable contain correlation embed galaxy spectra take surface preserve local geometry review see practice span high curve value begin complexity multi rank associated arc classify rich galaxy principal curve belong low galaxy limited galaxy galaxy behavior statistic much volume keep galaxy assign p galaxy physical galaxy galaxy remarkably local universe detail galaxy pca method pca detail sec paper flat galaxy dr ms server galaxy constitute band apparent cut distribution cover resolution selection cut primary appear calibrate imaging field image interpolation galaxy complicate define area cover fractional area galaxy reduce galaxy statistic close great galaxy center correct apparent lower arise contain avoid slight variation limit magnitude galaxy limit subset magnitude limited subsample use study absolute leave galaxy galaxy variety relevant create within color shape spectra age color since color highly correlate computing color correct correction frame correction color color spectrum negative spectra good spectrum computed spectra provide cloud decide include partitioning cloud introduce desire artificial cut magnitude neither strongly feature galaxy elliptical galaxy concentrate profile galaxy dependent band also property degeneracy dim use k apparent magnitude star formation star formation mass mass dr galaxy derive band cover part galaxy correct spectral equivalent full galaxy include build cloud apart deal pd reduction description section component transform dimensionality transformation rotation new projection uncorrelate select highest project onto order low describe span rank pc variance svd allow variance eigenvector respective thus expansion transformation ji ji weight weighted schema point assign one calculation covariance general property might made point divide surface surface go ahead pl l arc compose segment connect projection datum project onto self series expectation equal curve j il practice compute weight cubic regression calculate knot close segment iteration cumulative change p valid first contain galaxy equally survey volume use weighted galaxy galaxy fail galaxy galaxy assign volume galaxy galaxy limit apparent magnitude limit give volume iteratively value directly inside database library curve volume density histogram measure visually outlier wrong property measure simplify calculation avoid galaxy initial ht galaxy orthonormal eigenvector combination property high property pc since ht point order arc line circle text curve present turn mark colored direction galaxy project onto pc separate main cloud galaxy sec galaxy notice contain component combination th th pc pc magnitude expansion galaxy share evenly important correlation high old population opposite sign galaxy surface star formation opposite next mostly expect star form galaxy color probably concentrate star galaxy variance pc furthermore pc obvious galaxy nn increase line population denote galaxy mark group mark boundary color turn value principal unity dimension example extreme change spline unique equally spaced quantile arc length turn across straight show maxima place along red galaxy population principal curve resemble letter regime branch turn galaxy place fix arc galaxy group value onto place galaxy amount branch choose arc branch branch branch branch table galaxy group radial like fine branch follow remain visible see many spectral hand evident nd branch branch although line visible arc turn right hand straight circle within bar span quantile bar quantile red galaxy log na h red red group sec evolution galaxy arc feature present great present shape behavior individual initial arc branch turn curve fig great leverage pc length scatter turning behave differently emission degree arc also galaxy spectra function property strongly shape emission belong proxy interestingly average dispersion make old population arc length present turn fact na inter consequence present high star formation galaxy index determine light profile galaxy disk elliptical emission ratio diagram galaxy galaxy present emission line well measure fractional velocity dispersion km cut reduce go length none emission line location group line cover star form galaxy branch region interestingly branch happen turn feature powerful describe beyond construction dispersion bar include colored dot random group dash pure galaxy composite increase arc blue triangle aggregated sample circle red galaxy ccccc unit group evolution arc computed method explain estimate lf galaxy parameter choose double law fit index define reach power cutoff q whole change curve st tail behave branch branch slope fit shape log st branch mostly constant mostly surface st see function fig give slope expect star form dramatically start continuously start rd branch slope turn st branch long end sharp cutoff end well fit double law since cut rd turning present law section flat fit drop fit since shape belong track eps diagonal solid black auto correlation represent dark left show correlation expect galaxy sample dark second galaxy arc length quantify explore galaxy cross parallel generalize two consider measure cross brief generate use galaxy draw correct sensitive bar circle survey volume sample sec follow galaxy remain group auto correlation solid plot high amplitude large scale color instance amplitude increase object universe small sample power scale suggest compose galaxy scope paper highlight correlation autocorrelation cross galaxy dark information pair galaxy dark matter trend context close arc length expect correlation galaxy distant particular mean less galaxy perfect mix galaxy arc signal galaxy distant dark matter show galaxy mix suggest two galaxy dark matter aa function analysis galaxy group stand aside galaxy sample pay galaxy apart cloud another galaxy whose appear eps curve panel small disk galaxy red fig find galaxy cluster cloud separate build galaxy appear arc precisely dominate opposite cut galaxy image average spectrum accord compose mostly minimal formation component exponential disk concentration galaxy lie interestingly size uncertainty value estimate arise spectra component galaxy around galaxy group member ram pressure mechanism disk red galaxy double power group study galaxy explain choose curve radial partition galaxy datum cloud high core sequence galaxy average show useful able support physical scenario principal curve power curve galaxy arc big
call select intel ram method write misclassification partition operate able fast fail complete memory time versus proportion take unable train hence fast scalability scalability kernel significantly exist mkl implementation result mention goal add technique kernel line kernel scalability support nsf grant thank geometric kernel mkl problem search interpretation combine structural formulation allow mkl optimization routine provable guarantee set empirical significantly fast unweighted principled kernel wide learn jointly optimize exploit heart elegant approach require exist albeit accurate large present perspective mkl view mkl formulation multiplicative primal show primal reveal extremely light sdp call library optimization base provably extremely light optimize mkl technique engineering mkl view work focus add already detailed evaluation nystr om approximations apply heuristic merely baseline mkl nystr om additional scalable baseline since determine early simply train well combination semidefinite efficiency researcher start alternate alternate update mkl cutting plane semi program mkl improve combine cut plane level variant lasso point speed heuristic work mkl study gaussian family regularization base dynamic briefly present learn good classifier recent stage kernel stage stage meta example meta svm solve highly inefficient scaling example scale scale past propose limited scale bad letter bold letter htbp cl semidefinite likewise duality margin short exist convex combination row close point hull express collect define expand familiar prove interpretation stand discrepancy simply search short distance formulation later solver approach store kernel exploit find short fix merely adjust iteration exploit sdp primal mkl forward step rather solve merely backward computation decomposition speedup update allow due convert mkl margin place general constraint soft appear simply ir mr rewrite canonical sdp apply brief overview thesis guess correct attempt either primal guess guess base constraint start end else lack solve program primal guide primal assignment form adjust paragraph backward incur traditional deriving sdp gap role important guarantee sdp explain assign solve denote ti convenient explain backward matrix symmetric unit eigenvector orthonormal equal rest call eigenvector clearly correspond part eigenvalue equality form orthonormal quickly cause rapidly double precision fortunately converge completely multiply certain result also factor suffice merely significance note avoid store ab store n program explain r shall algorithm iteration lemma subroutine p qp mn step wish template prove guess infeasible particular I trace become simply collection geometrically examine point choose correspond positive pseudo process max I highlight consequence produce sparse never need compute I expensive root explicitly find mkl eliminate close multiply side complementary maximize constraint need search eliminate start observe I complementary recall combine I summarize parameter objective loose via sdp much absolute ti eigenvalue absolute always ir require require run bound mn fail result small dataset baseline weight nystr hereafter mkl datasets uci repository medium classification scalability medium scalability constraint subset amongst kernel weight trace par beyond figure entire matrix memory poor employ nystr observation learning perform thought simple algorithm feature nystr om
chart observe lr chart reference choose chart lr scheme optimal value delay moreover change lr sr chart choice average delay attain equal decrease chart exception bad delay observe illustrate chart well small bad chart turn arise control gaussian assume change reference suitably control simulation compare control assume ar criterion chart small depend reference control value equal change lr chart chart dramatically analyze use except chart always chart change lr chart change expect begin small delay author management european discussion comment remark european box derive use sequential generalize give process extensive study scheme variance delay ratio process observe process appropriate tool monitor control cf en many public health economic environmental sciences model time directly series propose transform possibility chart extension exponentially weight move chart dependent overview topic deal mean behavior observe surveillance variance series instance economic detection asset control time introduce stationary process focus independent disadvantage procedure lr sequential deriving suitably drawback generalize lr sr great advantage reference process white explain control briefly new autoregressive process chart derive residual control chart section control section lr generalization approach generalization modify sr obtain scheme autoregressive average sr chart reference reference true sr chart small value medium lr chart chart dramatically generalized delay equal length limit average delay chart must prefer later spc structural examine consider target sequential detect occurrence course target detection increase variance problem arise economic increase asset variance reflect production process since bad stationary process observe process position observe exactly scale control chart introduce page control chart memory point previous former chart value mainly normally expectation likelihood time calculate recursively dramatically simplify practical calculation control design practice fix practice replace fix determine control target practitioner recursion stand equal control choose scheme determination target formula available via assume cf determine gaussian obtain minimize quantity denote chart likelihood assume likelihood occur control q indicator thus j j chart n cm autoregressive moving process follow x x j describe part lead chart recursively describe approach apply chart recursion apply residual chart chart normalize independent recursive presentation statistic calculation delay markov depend thus limit consequently simplify application sr paper change sr procedure prior sr asymptotically sr recently several variable independent normally distribute density univariate normal distribution notation sr statistic know quantity lead derive apply independent length chart j j jx replace chart causal get n n iv statistic change reference lead disadvantage available procedure must section likelihood size change receive spc cf section denote control let hold q obtain q test gaussian mean condition else n chart give causal ar v consequently rule nf cm nf x ccc chart run assume n normally distribute expression value exponential reason eq logarithm increase arithmetic independent variable iid nf x determine concave function attain n iid iid iid r n iid u iid chart get function position run scheme control detect optimality know aim give chart apply situation control choose limit performance explicit formula criterion extensive study run delay determine repetition presentation average delay consume
standardized fan scad penalty zhang mcp mcp big impact see discussion choice variable group coefficient top show estimate norm group panel show see characteristic path quite group mcp path bias toward path solid plot group dot understanding characteristic nonconvex group case group j lin lasso via soft unchanged mcp scad verify group mcp expression group group mcp scad norm mcp scale soft threshold scad similar mcp unbiased threshold complicate mcp pt mcp operator operator scad hard scad family close expression building describe descent fitting kind propose lin way path use fu wu originally scad discuss behind extension group iteratively reach optimize function particularly scad mcp expression group coordinate partially optimize coefficient define l j exclude group ordinary equal residual penalty consist f choose attain maximum group mcp norm bridge composite penalty possibility penalty encourage bridge mcp bi one mcp bridge penaltie mcp limit extent variable dominate norm remain bridge variable whereby draw prevent group bridge achieve individual norm mcp group bridge composite mcp group generate group line represent equal composite mcp three underlie model equal line equal reveal group penalty nonzero coefficient magnitude solid enter phenomenon composite mcp grouping composite mcp zero eliminate mcp perform penalization piecewise unlike although penalty idea still main solve first li direction equivalent threshold operator solution penalty whereby guarantee decrease every com group composite mcp achieve penalty group wu possibility convergence model fit coordinate long procedure preserve transform back propose portion replace likewise penalty replace mcp knowledge explore work performance genetic important age relate control analyze marker suggest relate group mcp marker gene belong penalize logistic fit additive marker dominant analyze single univariate logistic marker cutoff select marker time group composite mcp table penalization penalization gene age chance demonstrate different three clearly see although misclassification error gene marker genetic bi method achieve bridge identify promise snps look mcp global minimizer take mcp penalty condition estimator justification selection reasonable group j index group least denote empty first mcp chi given suppose proof oracle group criterion strictly tucker mcp behave oracle sufficient yu due mcp extension present therefore say identically selection consistency mcp situation example strong mcp strong convexity extra effort descent concave penalty largely equal close zhang zhang helpful give selection method model regression learn genetic genomic dimensional covariate intercept unobserve lin operator procedure reproduce convergence additive high zhang lin van propose variable closely nonparametric distance specify threshold compatibility penalize infinity form condition group lasso lasso spline consequently lasso underlie adaptive lasso select correctly achieve linear mutually exclusive index linear priori property method al rarely covariate nonlinear effect zhang liu determine component regularization closely relate show method consistent special four parameter difficult pursuit approximate spline expansion use mcp show consistent structure correctly time interval error investigate effect covariate measure repeatedly know longitudinal important smooth chen li estimation apply selection selection author li group regularity function great level select important probability vector identically regression efficiency simultaneously multi task author criterion general th variable regression task select drop study selection et matrix group method throughput clinical phenotype status phenotype result pathway gene pathway treat incorporate pathway select pathway gene study marker identify suffer perhaps relatively size effective pool regression arise gene song important identify gene complex hundred sample often control utilize hundred snp human genome powerful select simultaneously separately group genetic discuss selective group progress area important issue well variable determine even include bic schwarz freedom lin freedom least estimator feasible systematically estimate analysis group selection b choose applicable discuss furthermore estimate degree group selection prediction important insight group range whether penalty validation clearly lasso method consider mcp relevant solution descent group group overlap pathway belong multiple pathway extend lasso variable without group liu yu overlap et bridge assumption group appendix chi freedom chi distribution kkt intersection let x use recall j ji jj n define orthogonal projection mn j complete define j along zhang omit event reduce reduce condition prove acknowledgment thank anonymous helpful comment extremely grateful point lead improvement research grant nsf grant conjecture grouping include lasso article give selective concern theoretical pay bi selection additive regression genomic datum analysis wide association highlight consider naturally model write matrix th predictor around desirable unit grouping estimate select variable many statistical algorithm lin group coefficient extension fan studied discuss wavelet coefficient van b logistic yu penalty selection special zhang simultaneous individual bi bridge propose general coordinate grouping structure reason rise goal example group indicator effect basis also introduce advantage meaningful expression gene belong biological pathway group genetic genetic marker gene desirable account grouping
document calculate power well vb give train thorough lda prior hardness prevent infer solution main reason almost corpora fact converge quickly despite loose application reach convergence average quality even much efficiently empirical analysis second prior thm lemma thm integral part non challenge want yield document article framework probabilistic topic flexible easily prior sparsity goodness develop infer latent topic model significant interest recent nature integral np vb collapse variational collapse gibbs converge slow vb fast often development remain common limitation quality second latent representation huge storage lie document lie gibbs limitation goodness inference extend sparsity probability process induce gain suffer severe drawback meanwhile change non smooth inference find expensive common latent directly inherently ignore document meanwhile require consume nonetheless retrieval computer inference sparse representation significance contribution resolve problem way framework model enjoy follow rate incorporate sparsity representation would exist representation add play may solution always provably decide hence sparsity approach theoretical existence fast linear mf interestingly knowledge first mf tr work intractable discuss definition introduce framework equivalent map briefly lda section practical go vocabulary vocabulary appear document term consist document kk k assume corpus mixture lda variant model proportion indicate proportion ml topic problem maximize topic inference proportion maximize necessary infer topic contribute emission document nevertheless unnecessary many leave popular objective situation differ discriminative belong serve propose framework continuously differentiable frank wolfe topic proportion frank wolfe proven moreover find solution face simplex let continuously f frank wolfe find face select objective continuously concave frank wolfe vertex note rate provably crucial mostly explicit face lie approximate provably trade basically flexible replace forward basis selection flexibility objective difficult suitable serve give principle turn technique widely fact vb next subsection property corollary define iteration error frank wolfe replace forward basis want appropriate nonetheless extension open future research exactly frank wolfe next discuss proportion inference endow naturally map besides imply inference model topic reformulate maximization contribute pz j j task maximize need topic assume far density express reformulate p constant would complete proof reveal map exactly function choice topic document h form complete influential topic model us connection concave reformulate problem combine document exist solution allow proportion objective simplex frank wolfe modify consideration lda proportion lda problem function objective come dirichlet topic induce function cause hard require document latent sense lda modification objective log inference lda proportion task logistic correlation note conjugacy deriving inference algorithm inference slow contrary model employ proportion various correspond vector identifiability use recover key reason mostly correlation importantly inference correlation document far use topic proportion reformulate concave task concavity second derivative diag diag diag element diagonal diag semidefinite combine feasible concave interior consequence maximization maximization specify fortunately interior convergence basically exist belief believe derive note inference correlation slight modification objective imply inference exist converge rate objective well define unit simplex slight frank wolfe slight modification original maximize believe easily enable investigate framework sparsity infer proportion analysis library write reduce design library general enough topic flexibility successfully demonstrate work attractive deal application employ design effective supervise analysis
formulae root label different symbol arbitrary possibly generality tree follow symbol label adjacent sequel variable usually leave path root internal subtree root root leaf contain subtree devote relevance hypercube propose basis use denote often treat depend boolean relevance dimension induce assignment relevance exist indicate relevance obstacle correctness relevance hypercube set relevance hypercube subset sized relevance hypercube iff contain least hypercube kind read follow fill sized first subset relevance hypercube hypercube exist otherwise valid relevance sense know agree relevance know relevance hypercube one recursively reconstruct skeleton tree represent equal relevance hypercube check row sophisticated reconstruct use paper assumption subsequent include material arbitrary read relevance hypercube check read vector take read firstly root label symbol aid focus irrelevant need main proving correctness remain aid induction represent section assign colour definition relate function present need sequel projection colour leave subtree colour label symbol colour remove replace projection adjacent node symbol remain root support subset internal reason necessarily boolean variable relevance hypercube every hypercube size internal label symbol hypercube restriction symbol read function denote iff stable existence relevance say hypercube partitioning conservative branching inductive partial subset subset extension take composition need conclude induce read lemma relate structure property reflect variable colour leave subtree colour arbitrary variable conservative since read agree hypercube restriction hypercube therefore one internal colour colour leave colour repeat reasoning place reveal colour colour unique index e take colour root leave colour contradiction partition refinement node great colour leave different leave leave colour symbol colour define subtree short root let induction nothing colour claim set conservative cardinality restriction relevance input label leave colour take except one subtree construct relevance hypercube need definition tree read different depth subtree inductive colour subtree belong put state conservative agree hypercube root leaf label internal colour colour contradict complete read hypercube constitute relevance alternative read agree make tree leave follow mi rule put contain subset subtree colour conservative stable lemma leave lemma agree relevance restriction subtree depend restriction variable relevance set relevance hypercube regard projection recursively end formula read give form classic read reformulate correspondence internal independently transform table circuit tree represent boolean node reveal verify agree sized input concern issue polynomial acknowledgement author useful discussion grant md read variable distinguish read show check contain relevant improved reconstruct variable strong classic representation boolean function read appear check iff read function check iff distinguish unlike irrelevant denote read checking basis small vector check test regard basis hold relevance contribution method correct basis arbitrary checking cardinality previous proof point relate exact firstly constant secondly check regular projection may turn thing consequence learning know implement unknown basis query turn identity basically specify function
htbp absence record number tweet user tweet help recommendation accept item hence divide user small balance precision computational successful recommendation reflect user interest analysis recommender consist popularity rank discovery recommend item acceptance rejection select category reach precise specific group item organize dr indicate popularity possibility acceptance little recommend popular item item function return rank similarly whole normalize user accept item interest active keyword interest class analysis mapping suppose item compute item category satisfy candidate include distance normalization directly storage computational enough collaborative interest search interest let amount search interest return indirect merge adopt comment could happen link obtain sigmoid correspondingly target category include modify increase interest coincide choice affect recommender training supervised present initialize stochastically termination initialize recommendation epoch weight search round eq momentum factor typical update training epoch training control apply instead process discuss performance enhance system experiment record stochastically divide omit update comment computing train process user interaction inactive inactive prefer similar active popularity item train system recommend recall item low due difficulty interest adjust help inactive ccccc item active inactive enhance recommendation user interact preference group accept item possibility base similarity frequently update platform interest interest behavior stochastically gradually track keyword user find target category indicator order respect user initialization good accelerate reduce search competition competition thank wang platform university zhang precise recommendation help twitter design hybrid solution track task user might consist user extraction recommendation performance lead service considerable growth observe dominant since chinese helps result attractive recommendation crucial recommender system influential unfortunately consider little fidelity difficulty stable overcome hybrid recommender generate item mining platform discuss problem hybrid discovery interest order experimental popularity twitter services china lead platform attract lot million million daily dominant china instant service carefully platform nice growth group service embed user platform service write comment website party group china time public widely frequently comment message matter website active user twitter platform recommender real interest accept recommendation recommend item list present prior introduce recommender overcome main help interest association rule huge involve search parallel adopt mining association property support necessity frequent frequency keyword class user keyword database
discuss road represent direct road intersection segment intersection road segment along road segment account segment date trajectory discover sub trajectory trajectory cluster cluster possible proceed graph trajectory bag comparison segment segment checked individually take come individual segment spread naturally adjacent road segment therefore sufficient bag segment implicitly trajectory road segment fashion spatial analogously tf length cardinality second inverse frequency trajectory calculate similarity weight similarity calculation traffic monitoring portion pass undirected trajectory node two assign weight edge choice natural also put emphasis nothing never put cluster since analyze number edge trajectory one tend vertex degree reason modularity community modularity measure cluster modularity community randomly modularity graph perform modularity node discover validate retain proceed isolate sub nest level induce modularity synthetic dataset generator use edge direction start different similarity compare use td weighting achieve weighting vs classic hierarchy top low cluster produce third choose expand sub arrival trajectory along visit road visualization separate produce cluster understand visualization traffic one cluster expand agglomerative calculated adjacency agglomerative linkage comparison conduct hierarchy cut number cluster quality end overlap trajectory assess road segment trajectory share result number trajectory lack table trajectory cluster share road classic overlap achieve linkage come fact unbalanced trajectory linkage linkage modularity behave c overlap mod mod attract proposal include mainly base suppose move hand movement start path like configuration furthermore whole path trajectory might path contrary modularity trajectory road trajectory base cosine define conduct modularity optimization trajectory propose relevant hierarchical future direction exploitation relevance decision make traffic
gradually dimension underlying estimation size estimation neighborhood estimation depict estimation obtain estimation technique technique capable respectively rp precise estimation could cope neighbor value result rp see fig group magnitude obtain estimator among competitive computation also computation computation time slow time see fig reduce rp advantageous work directly rp dimensionality reflect view european implementation cm mathematic ph ph os year award award ph student image transaction method collaborative sc solid molecular sc physics technology os european artificial intelligence review journal review conference paper lead illustration test dataset neighbor method size column size size size column neighbor knn knn rp digital processing original publication tv os department mutual various exhibit computationally intensive consistent distance embedding rp technique method image efficiency demonstrate theoretical distribute problem rp classification near manifold geodesic stream reservoir review relate compressed sensing show recently rp potential patch vote patch naive promise correlation norm theoretical exhibit modal paper apply neighbor color intensity filter texture descriptor may dimensional formulate mutual accomplish randomly straight histogram bin image considerably efficiency theoretical shannon multidimensional differential efficiently purpose rp evaluation approach exploit formulate condition knowledge rp rp conclusion draw task section follow embedding projection transformation transformation describe parameter possible produce close measure measure among thing similarity pixel simple choose neighborhood pixel value color vector q similarity information concept joint shannon differential entropy quantity enyi similarity projection purpose distortion set preserve approximately construction notably property probability random strict cost introduce draw rp length construction rademacher image characteristic compare directly estimation cite reference efficient note pairwise sample rp decrease divide sample group call estimation multidimensional entropy pairwise approximate embed address rp base divide fp ab ng drop estimate entropy rp ed group enable quantitative version green channel version filter version fig choose evaluate objective angle interval ideal degree choose around feature rp draw construction size neighborhood dimension feature randomly project rp outside interval outlier estimate k partition neighbor minimum spanning type
rr outperform tree significant tree previously parameter poor mainly variant right combination straightforward performance project process random forest good appendix show benchmark training configuration project process closely forest variant rr train number perform poorly regardless amount instance unseen combination instance configuration configuration run set split random permutation combination configuration surprisingly benchmark permutation randomly instance l sp pp rt rf rr nn pp rr bound rmse data provide additional pearson coefficient sp nn rt sp nn rt rf inf e e unseen quantitative figure generalize section respectively heterogeneous set poorly outlier ridge case order quite experiment different region space heterogeneous sometimes runtime robust prediction instance observe ccccc ridge neural network prediction vary select correlation remarkably well high hour execute second user execute build ridge variant heterogeneous otherwise forest perform across heterogeneous benchmark remarkably low yield datum interestingly important give instance decision configuration well unseen instance already pure unseen instance rr pp rt rf rr sp nn pp rt rf test test ccccc runtime ridge rr instance long configuration good instance plot benchmark predict visually indistinguishable random forest closely forest ridge ccccc rf configuration figure sort hardness configuration value runtime combination represent advantage forest well heterogeneous appendix instance configuration visually assess work true indistinguishable random forest challenge instance configuration interaction parameter configuration hand regression neural network capture instance hardness distinguish even training instance configuration prediction yield capture instance hardness configuration simple forest instance configuration nd evident finally depend focus forest forest salient improve qualitative capture difference overall increase set yield return entire runtime practice comparison far second second issue forest run unknown censor usual censor runtime censor indicator typical strategy deal censor datum fast really bias bias mostly call building unbiased censor need people people use handle censor survival analysis handle censor regression make portfolio algorithm selection good method apply exist candidate consideration process one likelihood right censor forest describe context handle censor base configuration pdf cdf runtime gaussian maximization rf I denote predictive rf good rf confident censor runtime censor rf variant implementation detail potentially prediction know maximal runtime keep censor subtracting sample absence censor major mechanism need observe experimentally procedure baseline ignore censor treat censor runtime unseen censor different censoring threshold instance specific runtime instance represents generate experimental study section instance configuration prediction strategy measure rest estimate drop censor vary benchmark qualitative treating censor dropping censor yield consistent treat censor yield beyond case vary enable well l drop slack half drop drop censor treat one censor quantify show drop yield improved variant yield top threshold drop yield uncertainty value censor close yield half vary variant yield uncertainty prediction censor poorly figure varied brief respect yield low good rmse censor fix threshold bottom threshold censor benchmark advance combinatorial new building focus parameterize np problem aware algorithm compare forest process whether unseen parameterized previously unseen instance demonstrate setting coefficient small hundred forest far handle run wide variety researcher seek schedule portfolio automate datum source reproduce experimental online thank early thank benchmark benchmark clause preprocesse preprocesse benchmark comprise instance clause respective standard maxima clause comprise instance average variable clause deviation respective maxima clause benchmark comprise category instance contain clause respective deviation respective maxima clause union bound check comprise verification library clause deviation maxima clause encode software verification static verification filter window os clause respective maxima clause win previous mix publicly mixed benchmark instance instance deviation maxima cm comprise constraint deviation comprise spread red conditional decision random generator parameter instance comprise encode winner determination combinatorial select ratio deviation maxima comprise euclidean generator select deviation comprise clustered dimensional instance generator deviation set prominent repository code complete em em predict previously input machine application analysis portfolio parameterize decade much thorough treatment empirical kind span range instance overall new substantially runtime approach algorithm simultaneously machine prediction empirical model parameterize mixed np ai input arise realistic size conversely take amount theoretical objective runtime greedy competition reflect scope include variety practical classic selecting predict sequential setting algorithm algorithm setting basis benchmark generator assess impact forward identify five explain algorithm review past work describe contribution sophisticated forest approximate gaussian arise setting section new novel yield comprehensive advance perform comprehensive runtime date evaluate method data instance parameter section literature analysis handle improvement perform forest extension censor conference first evaluate work gain insight hardness ai plan three planning depend size run long community predict runtime various planning decide runtime specifically success runtime programming winner determination show runtime solver regression include lasso multivariate spline regression end relatively ridge regression subset expansion response polynomial yield runtime ai planning make connection literature survival censor work dynamic configuration publish characterize hardness combinatorial classification article neural also classification yield worse make online fast run good contrast predict meta compute stop act scheduling distribute performance analysis discuss show subsequent refined approach incorporate note runtime correlate mostly prominent fitness correlation autocorrelation exception expensive predict model describe rather number careful necessary issue literature design input generalize core automate literature consideration running instance continuous aware relax categorical regression instance analysis detect use linear quadratic note characteristic control call model much local problem runtime tackle employ suffice yield five feature yield identification mix previously quantify measure interactive use see provide thus detail list computable piece usually domain problem low parameterize configuration real second principle run runtime quality consumption communication configuration combination entire allow nevertheless review configuration set record vector performance transformation easy focus effectively predict log experience find important variation hard discard input across datum normalize deviation relatively rarely handle miss normalization miss respective ridge regression ignore multiply validate mean rmse simple frequently grid fit input simplicity conceptual interpretability combine competitive performance feature predict f optimizer scale gradient minimize error default hide neuron gaussian gps dominant ridge albeit great expense similarity choice input idea correlation general expect take variable subset tradeoff specify observation optimize improve book contrast hyperparameter hyperparameter grow integral gp maximize optimizer optimizer make big limited bfgs gp expensive inverting hyperparameter optimization yield prediction time know handle input tree disjoint predict follow point region zero point square mr regression procedure start binary scalar data point otherwise split sum square region equation continue find training tend branch contribute little fold see predict response variable split point continue child categorical select error split categorical good loss efficient sort predictor datum thus implement fitting tree dimensionality time variable continuous split split categorical partition consecutive split building depend balanced lead good case recurrence experience perfectly balance close bad point tree regression merely need propagate query tree node point categorical store mask member take balanced primary improve prediction parameterize advanced handle categorical prediction partially space new model family base potentially forest regression tree limit input prediction value input section value categorical input extend arbitrary handle encodes categorical domain binary input indicator column column point one rest column treat encode categorical gps dimension
political event prominent expect political sentiment remain sample crucially estimate discrepancy experiment demonstrate algorithm regression thank discussion paper derivation rademacher bound rademacher adapt proof jensen verify associate unchanged sign equivalent analysis discrepancy prove rademacher discrepancy scenario tighter substantially improve upon previous one generalization line batch combination exploit qp demonstrating keyword environment domain pac draw spam financial condition environment devote line scenario allow study label distribution consecutive step generalization scenario long show consecutive generalization achievable improvement assumption constant setting allow intermediate base relationship recently analyze consecutive nearby assign limited paper deal drift pac bound unlike bound generalize context adaptation previous admit favorable scenario use accurately complexity contrast discrepancy tight rademacher complexity vc simple standard batch term discrepancy measure line lead meta combination discrepancy section qp preliminary demonstrating finally discuss natural section definition input output space loss function l adopt complexity sequential rademacher element random draw coincide use even favorable efficiently loss adopt provide measure dissimilarity distribution take consideration loss function set discrepancy introduce adaptation fix consecutive hypothesis distribution define discrepancy already mention accurately finite rademacher h h discrepancy definition symmetric triangle inequality base discrepancy discrepancy straightforwardly advantage fact discrepancy unlabele two learn pac loss scenario environment oppose generalize discrepancy define sample return mistake mistake scenario carry upper risk also scenario pac agnostic emphasize expect depend x first commonly empirical bound loss th side yield rademacher complexity consider additive next good expect return empirical erm combine bind indicate trade value small grow increasingly challenge minimization limit last instead algorithm pac example exactly select rademacher vc fix mc minimize hand exactly many vc discrepancy divergence lead tight informative mh last integral rewrite mm corollary vc track track however satisfied tight base rademacher complexity importantly fine target accurately finite analysis empirical risk bound arbitrarily close discrepancy bound favorable absence distance would uninformative vertex uniform measure measure function disagreement also hypothesis present dramatically favorable illustrative greatly case sequentially assume adversarial scenario distributional return line hypothesis strong guarantee regret minimization combination need main draw form loss bound process probability inequality hold discrepancy independent hoeffding q follow sum inequality statement theorem bound argument hypothesis return process hypothesis least q scenario I sum vanish straightforwardly rhs th statement claim
class ad balance scale cancer rating block digits diabetes tumor segment vote breast variation classifier test difference variation gm multivariate attribute dimensionality uci dataset variation lead classification c c gm ad ad gm ad paper anomaly ad detect various anomaly content document challenge extract transaction automatic extraction content address unique transform algorithm since feature preserve prominent detection association perform poorly dimension comprise univariate model superiority algorithm detect anomaly transaction ad transaction filter anomaly prevent attack machine school science ac il transaction interact transaction attack potentially interact regarded anomaly transaction machine technique anomaly automatic method extract feature transform dimensionality ad utilize central novel four transaction capture system detection anomaly security machine anomaly detection interact transaction may fall attack mistake medium regardless origin especially schema exploit interact information anomalous transaction detect anomalous mean security end security protocol transaction signature mostly take protocol document process language definition language describe document tag format transfer share internet file schema schema anomaly action attack mistake technical error describe prominent generator attack interact web service attack attack exploit mechanism server prominent attack buffer field attack site schema attack attack attack produce anomaly attack stre transaction collection inherently modern arise cause attack attack simple human transaction lead simple way regular simple scheme file anomaly attack document application anomaly detect anomaly document attack anomaly opinion important behind approach anomaly nature would work detect anomaly require elaborate domain indicator attack document anomaly anomaly aim discover majority anomaly spectrum security financial surveillance name anomaly range cluster neighbor theory supervise impractical supervised classifier example pattern moreover easier obtain anomalous domain semi supervise anomaly detection semi anomaly detection transaction assume ht classify label general denote score close document e sx x dx pairwise distance mutual feature able document document anomaly anomaly detection plausible inherently curse dimensionality consequently dimension grow dataset concept another similarity use unable new incur allow localization anomaly pattern detect detection function domain detect far anomalous paper outline detection framework transaction framework section present several evaluation experiment summarize document anomaly follow detection method take supervise al propose occur relationship association quasi functional dependency anomaly previously association information represent outlier rule database rule incremental handle database anomaly describe relationship item occurrence chi attribute anomaly knowledge incorporate et classification anomaly extract accord attribute feature represent introduce liu detect frequent anomaly value arithmetic detect anomaly extract namely adjust software computer original extract structural transaction practical anomaly propose detecting stage extraction dataset generation machine learning transaction define transaction transaction feature extract definition put extraction extract feature element generation transaction aggregate arrange tuple contain transaction tuple add train final stage anomaly stage extraction start acquisition parse definition file numerical define visible contain file carry type etc enumeration range handle process single transaction comprise element store complex occurrence occurrence measurement correspond relate htb produce object string traditionally machine require regard input scalar feature transaction contain contain different item met overcome process consequently rectangular gain determine due aggregate tuple separately complex denote tuple aggregated since aggregate tuple derive rectangular aggregate aggregate aggregate feature however minimize loss aggregate function simple summing possible enumeration dependent enumeration string produce simple word minimal transaction compute document frequency lastly tf prominent word choose extract tf transaction text contextual mention dimensionality scalar effective detect anomaly process essential property translate element standard thousand feature anomaly feature anomaly detection instance represent class inherently compare unsupervised anomaly detection detection capable none cope dataset dimensionality high propose later unsupervised anomaly transform see fact regard source attack propagation localization complete operational meaning anomaly goal anomaly detect anomaly univariate classifier full regard normality dimension anomalous transaction identify low likelihood additionally order section method attack transaction use svm prominent branch svm algorithms algorithm specify seven experiment classification adaptation identify anomaly anomaly near neighbor although ranking problem classify compute neighbor classify point anomaly percentile density near compute compare ball ensure neighbor least one anomaly achieve l c ad mean gm ad gm main classification receiver characteristic roc plot specificity vs sensitivity equivalently fraction true positive positive select model discard one distribution strongly diagnostic use decade auc cross procedure subset utilize second subset role process repeat fold implement experiment pair level difference auc statistically follow hypothesis follow evaluate transaction collection collection transaction real transaction relate anomaly contain transaction capture university transaction actual attack transaction follow standard transaction item ensure collection comprise transaction consist transaction transaction process extraction transaction regardless transaction stand attack computer university attack capture convert duration attack minute tool network traffic computer file collection put process collection transaction differ aggregation evaluate ad experiment transaction document however document contain anomaly system entire anomaly ai module transaction site text attack perform transaction attack represent string encode attack encode attack transaction lastly attack element sentence child book attack ai module transaction element target transaction construct anomalous transaction module one mention attack class transaction avoid ai module attack schema mention attack introduce anomaly rare content anomaly affect system transaction stress train anomaly detection label document anomalous transaction focus determine approach document machine suit regard produce additionally examine capability algorithm examine accord detect anomalous transaction classifier transaction level attack evaluate ad ability detect attack multivariate determine anomaly belong whereas gm classifiers roc depict possible obtain rate gm cn result test distinguish anomalous transaction conclusion process preserve detect anomalous examine gm average achieve ad classifier perform auc close result aggregation dataset dimensionality relatively arithmetic aggregation multivariate svm poorly transaction distance detect anomaly transaction previous substantially well classifier univariate classifier inherently classifier deal detection well produce detection examine experiment classifier system new gm use geometric aggregation denote system indicate classification mu stand ct anomaly detector transaction classify real avoid bottleneck transaction time ad gm auc ct auc ct cn transform multi univariate increase effectiveness improvement mu mu respectively version enough experiment high auc classifier time
follow seminal intuitive advanced via idea behind measure system equilibrium u convenient conceptual parameter possibly convex element whole entropy consider possibly variable respectively parameter support eq strictly increase employ hellinger distance hellinger complexity intensity shannon gray proportional remarkable closely relate e discriminate various hellinger distance universal linear corner summarize provide value discrimination target stem analysis process os platform excellent reveal information model homogeneous heterogeneous appear promise include analytical hellinger sample generalization de col de generalize complexity et functional capture order distance former latter divergence scene corrupt amplitude non require model describe among law validate expressive tractable target assume appear little shannon hellinger complexity expressive identifying type synthetic prominent carry conventional sensor optical spectrum active sense carry illumination source operate operate spectrum price pay automatic visual use image analysis call assess dimensional signal able regime statistical nature law entropy divergence complexity stem feature framework describe stem image formation intensity format every outcome variable truth intrinsic unitary law return completely multiplicative single distribution amplitude format accommodate later description multiplicative reduce distribution law gamma adequate homogeneous eq denote model heterogeneous area law develop texture illumination absolute distinct order distribution system via pdf describe observable series primarily characterize give shannon one degree shannon logarithmic physical restriction represent system micro predict probability
price design reason study table code fitting compute choose cpu implementation use dataset slight genetic optimizing reduce variation result average design choose c table profile log discrepancy implementation precision double matlab far suggest precision arithmetic yield optima stress neighbourhood good optimum double precision run cpu refine large runtime cpu gpu somewhat short double might involve include software package matlab similarly library gpu accelerate possible four gpu cutting currently cpu gpu network like fourth fast world chinese develop execute numerous processor frequently heterogeneous technique popular intensive computing computing architecture graphic unit powerful cost capable modern central great deal intensive statistical dataset demonstrate implement gp gpu gpu acceleration suggest drive graphic offer computing processor complex parallelism modern execute task parallel execution time magnitude extremely purpose graphic unit intensive cpu gpu traditional fitting gp computer consume time model determinant numerous spatial cost determinant prohibitive moderate large moderate dataset thousand run fast speed leverage intend demonstrate heterogeneous computing platform lead remainder organize review basic formulation model intensive implementation gpu cpu discussion study suggest intensive computation gp model terminology simulator simulator model although several choice exponential parameter good power exponential correlation stable alternatively suitably avoid another like mat ern numerically cpu gpu likelihood parameter q log likelihood q hyper denote determinant expensive increase numerical stability qr cholesky system solve implementation viewpoint log likelihood back solve log likelihood evaluation expensive surface optima maximize challenging contour input generate hypercube simulator dimensional example simulator chosen contain minima near though region near evaluation evolutionary like commonly gp point minimum sufficiently involve matter search fitting computing ga require cholesky decomposition seven cholesky ga cholesky solve gp solve fitting use reduce burden heterogeneous gpu moderately intel core cpu gpu intel cpu capable core per core capable per cycle core per cycle intel processor gb matlab hardware slightly old class ram twice core computer multi core processor end fx core dual core cpu write matlab linear systems link performance inverse linear matlab cpu specialized parallelization core platform adopt study intel core core cpu development process matlab programming enable gpu free library step gp gpu library algebra operation routine source gpu block similar available gpu evaluation correlation kernel library include back transfer gpu mutually note parallelization value perform likelihood numerous simultaneously parallelism execute time computationally intensive
biological biological relationship gene biological represent relationship activity graphical distribute element effort focus example problem remain et likelihood develop nesterov method solve al ascent widely glasso call box alternate problem lin solve interior et newton estimate derive condition solution single subject associate block decompose small massive gain formulation observation draw single may work multiple matrix jointly penalty introduce learn et estimate fused grouping penalty admm require eigen decomposition propose node graphical even recently graph decomposable necessary fused modeling fuse penalty estimate graph paper fuse work formulate sequential estimate graphical penalize regularization encourage adjacent similar graph consider order common motivating disease emission graphical nc mild cognitive patient ad expect common identical expect evolve time order nc ad fail key contribution sufficient fuse diagonal duality precision screen computational employ order solve project shrink conduct synthetic effectiveness paper matrix denote bold format tr clear eigenvalue denote form diagonal share rest paper fuse screening present result sample identically variate covariance many conditionally precision notational simplicity covariance take eq furthermore usually dense induce sparse employ fuse simultaneously lead sparse fused encourage mathematically solve model fuse graphical assume throughout diagonal strictly say ensure assumption solution establish statement problem optimal solution sub f region nonempty one observe associate p diag tr kl tr hold u solution feasible compact fact solution order optimality subdifferential convex convenience objective unique strict optimal hence set nonempty diag diag j relation addition addition notice relation together every bound thus conclude hard compact ready prove theorem observe equivalent also problem determinant involve many precision reduce first block precision whole necessary fuse graphical remain derive necessary sufficient solution fuse multiple block graph first demonstrate decomposable derive say block feature exist permutation simplicity presentation throughout suppose k kl l demonstrate diagonal decompose sized latter problem problem address block decomposition scheme section satisfy describe adjacency satisfy otherwise adjacency matlab component partition scale decompose latter let system eq satisfy inequality solution lagrangian dual ready sake inverse order ij ij k ij k k diagonal solution block block ij ij ij ij lemma ij matrix kl kk optimality fact optimal conclusion screen capable partitioning feature size large sized block need compute solving replace problem simplicity simplicity tt iteration optimization iterate quadratic obtain newton suitably monotone show zhang locally computational global rate proximal decompose set sized addition search length ensure reduction positive prove exist let associated solution reduction one convexity adopt backtrack length hold cholesky positive associate term cholesky give advantageous minimize space issue identify single lasso allow update shrink single size quickly shrink improve technique also successfully shrink fuse graphical gradient iteration ki ij follow optimization reformulate ij j fix optimality solution shrink scheme shrink coordinate free fix addition quadratic result summarize determine free backtrack screen rule image experiment perform intel ghz gb memory screen following include comparison admm screen admm admm screen matlab involve involve double loop implement routine fair block ground block structure number matrix draw fuse penalty fix total solution terminate error admm stop admm great speedup time parameter c iteration admm admm experiment way draw reverse edge edge edge distribution precision vary replication fix accuracy glasso high accuracy demonstrate graphical ht attention affect school cost united states project fmri typically child research detail website dataset develop graph region obtain computationally intensive datum screening curve fast admm second utilize demonstrate superiority include h block rough cost block compare high rule identify identify block image ad mild cognitive control nc disease database region brain volume volume automate derive corpus examine whether information common percent group stable edge glasso say nc glasso
road cluster cluster meaningful level hierarchy cluster hand cluster top level top segment sake road classic agglomerative linkage visualization result omit due limitation overcome community participant positively modularity need loose produce two study trajectory similarity measure trajectory dynamic subsequence real ignore movement apply context road network unsupervised proposition trajectory exist proposition framework tf aforementione similarity suffer drawback sensitive parameter ii trajectory rather homogeneous rarely discuss author describe discover path frequently road resemble segment aspect produce flat parametrization trajectory road network measure shortest calculation agglomerative efficient trajectory unlike computationally road network require together trajectory underlie road focus trajectory deal road segment segment often visit compute road segment community present exploration various situation flat clustering produce however sensitive trajectory rarely thorough segment produce index like real trajectory dataset impact community cluster phase universit e paris paris even trajectory attract move euclidean presence underlie road evaluate trajectory two constrain trajectory approach clusters road orient approach experimental synthetic proposition device store thank device become dedicated move light challenge motivated management analysis move database mining road collect along deduce understand dynamic dedicate sensor expensive precisely study trajectory interested discover road segment aim retrieve road segment frequently useful context management planning road trajectory give road basis predict situation guide planning object movement impose address concern proximity road propose interaction modularity cluster discover nest exploration analogy road trajectory segment cluster organize follow present trajectory cluster road experimental finally conclude road represent intersection direct road segment edge road link node trajectory sequence segment e segment connect collect apply produce segment give trajectory road trajectory trajectory aim road trajectory appear segment appear segment belong different unlikely appear framework introduce measure trajectory road graph trajectory partition community algorithm discover adopt segment trajectory segment individually account segment simplification justify underlie direct graph consequently direction implicitly visit occur isolated spread adjacent consider segment totally reflect absence fundamentally exist similarity trajectory especially unconstraine along separate close trajectory road since similarity cluster conduct massive tend especially modularity adopt express modularity edge modularity generate community community discover clique trajectory group heavily intend segment modularity account node occur trajectory great well code graph high significance evaluate modularity randomly valid present clustering apply recursively cluster edge separately stop partition step complete hierarchy trajectory explore detail maximal detection phase rarely practice complexity interest discover proceed analogy previous comparison similarity discover road represent direct frequently empty subset large segment concept segment segment compare road segment therefore equivalent compare visit claim advance road segment considerable road segment versa contribution characterize road road segment eq contribution segment ratio whole
dropout finally globally layer feed test patch image four corner patch horizontal though pass imagenet dropout globally connect serious without dropout art performance validation find describe fortunately main dropout significant help net joint many year non layer lead thing bad cm feedforward typically poorly hold greatly detector prevent complex co feature detector helpful several detector neuron learn detect context operate give big set record artificial adapt connection detector enable correct input correct typically perfectly weight worse test datum data detector prevent co train hide randomly omit rely unit dropout good train computationally train huge network almost certainly network presentation case share present descent normally use grow large length bind update unit division prevent grow large allow thorough start contain unit weight layer compute equivalent network guarantee assign log assign network square network always network effectiveness mnist write test greatly extract without without publish result feedforward error reduce use dropout incoming weight unit reduce drop random figure dropout also generative feature detector discover available train deep describe fine tuned unit code use pre train boltzmann backpropagation error hide enhance apply widely system hmms need acoustic frame fit state deep pre train neural network hmm outperform gaussian test benchmark belong input net advance ms connect hidden layer softmax subsequently merge dropout significantly architecture rate probability neural frame decoder know transition hmm sequence hmm without recognition dropout improve record identity core benchmark hide unit improve cifar object image object use transform data error use neural three pooling layer convolutional layer connect layer imagenet consist thousand image thousand subset roughly per competition six state achieve comparable performance convolutional hide layer final softmax incoming dropout see b recognition decision hold validation assess dropout class contain exclusive document vector count stop feedforward network hide gets reduce try various dropout performance network fully dropout dropout worse throughout input dropout help often adapt individual dropout probability unit performance unit require regime probably improve create statistically expert get fraction implement model give complicated feedforward typically carlo assume eventually equal importance make easy dropout net pass efficient averaging bag different bag often tree quick quick dropout allow apply feedforward neural dropout bagging sharing regularizer standard parameter towards familiar extreme naive bayes input feature predictive multiply together little train feature context finally similarity dropout recent role interpretation theory gene achieve co nearly robust achieve perhaps optimally way gene end fitness co number gene environment decrease fitness learn thank h discussion google microsoft advanced consist digit classify digit architecture sensitivity dropout net architecture dropout visible unit stochastic entropy exponentially decay start apply minibatch multiply epoch income weight update length scale draw momentum momentum linearly stay multiply update minibatch epoch tp f f neural dropout consistently since input pre train rbm bias initialize number sample visible divergence momentum momentum start linearly multiply epoch layer rbms use activation rbm rbms neural dropout backpropagation momentum epoch minibatch hyperparameter mnist needs run backpropagation keep classification objective validation architecture dropout achieve overfitte net train remove momentum corpus volume corpus cover economic social market market account market production service class split category category remove cover divide test frequent dropout backpropagation backpropagation architecture hyperparameter dropout epoch net train dropout improvement smoothly stop million color image web search image search english come noun cifar image among ten class cifar filtering image incorrect cifar varied canonical viewpoint scale image contain instance belong indicate imagenet image category amazon crowd tool roughly class recognition competition object imagenet image spirit cifar resolution difference imagenet imagenet number imagenet feed convolutional network cnns feed layer neuron output bias add filter activation neuron pass filter refer differ ordinary organize bank organization neuron apply filter local extent center neuron decrease pixel object input examine neighborhood neuron bank different location roughly statistic image kind treat equally bank apply cnn neuron become channel pixel determine convolution operation large imply neuron bank unless share reduction neuron apply reduce advantageous convolutional cnns pooling summarize patch neuron convolutional convolutional pooling consist convolutional output bank pooling layer call layer space least apart pool convolutional pooling pooling unit aid pool convolutional unit layer translation pool activity cell exhibit invariance include response layer layer encourage competition activation bank organization neuron order course arbitrary begin real utilize nonlinearity neuron neuron bias nonlinearity advantage traditional time reach nonlinearity processing activity scale take pixel network raw across label initialize positive layer max nonlinearity take derivative input initialize sufficiently simply try initialization attempt neuron help ground reason use stochastic batch size variable objective respect gpu cifar take minute imagenet four equal layer whose power typically
determine sm statistically significant conduct test degree trial suggest sm statistically sm feature virtue encoding collection albeit million report sm feature highly movie baseline purely thick positively dash highlight font color highlight green movie colored topic popularity user interest mean account moderate edge show attribute pair popularity introduction pose question facebook pt interact content interest user thing interest movie distinct interest exhibit sm interest friend social user interest hope sm interest visualization say visualization sm learn subset parameter infer various fact dataset give towards various reveal friend learn topic across interest cosine middle commonly user interest next search topic high count normalize popularity popular show corner friend positively friend specificity valuable exception strong yet positively movie anomaly occur friend presence something fail statistically much popularity significant proportion separate life implication fit unlikely movie actual positively play connect bar south final concern topic notice user positively topic country ac movie within yet possess highly notice topic word highly activity contain modal connectivity speech serve anchor common meaningful inter interest term topic observation policy grow network increase modality distinguish exist model network either outcome text topic network link except asymmetric link crucially infeasible zero link model complexity also work facebook used implicitly sum feature neighboring cast topic text link neither outcome equal result link influence salient interest facebook million system model sm learn aggregate user interest million sm closely combine address challenge scalability phase phase parameter likely likely scale user facebook facebook title video social decade aggregate hundred million broad open problem user interest health social classic gues appropriate content potentially interest provide content user friend increase content consumption game player activity oppose play social interest facebook facebook interact graph content occur interest interest say movie group pattern facebook tool visualize summarize salient aggregated population critical enable individual detail social network turn policy aim network enable growth interest however mostly incomplete text act flow user attain complete view medium ignore user modality view link categorical principled enable capability yet produce address facebook hundred million diverse modality profile status name linearly datum fail presence structure facebook information simple multimodal require treat potentially sharp change require challenge mind present scalable visualize interest million facebook scale hundred key datum early successful certain modality mix blockmodel citation mix sm multimodal integrate user subset millions feature initial integrated feature pass information principled manner collection topic usually coherent assign intuitive g report label vote would topic positively correlate label value label facebook know positively interest movie contrast two motivation friend naturally interest interest degree mutual recommendation perhaps highly mutual driving potential drive prefer friend drive could latent figure label single proceed application four interest movie answer interest justify analysis interest bag facebook context movie status update link denote interest movie movie intuitively want capture concept word interest actor correlate interest movie friend actor relationship network insight nature facebook text pose user instead use learn label sm sm stage pt sm subset network learn text user label tendency towards topic user find topic classifier consist topic represent anchor interest topic topic user frequent give name american american self american friend label user topic might american contain word term user vocabulary desire generative background vocabulary draw dim word dim draw topic user text foreground word ik user topic user draw label shall circle red learn salient value text every dim probability label salient topic high word link friend american coefficient correlation represent status title compare form facebook document hence document exactly draw notable topic tailor paper facebook keyword irrelevant topic generic topic distribution per foreground background draw assign separate class distribution friend let matrix arise topic outcome generate word specifically draw upper triangular bernoulli draw essentially describe topic facebook sparse variable put pp music include learn positively interest user indicator topic sm proceed parameter follow explanation form right detail simplify reduce latent integration link beta binomial conjugacy random remain also amount ensure scalability direct maximization hybrid motivated gibbs sampler like easier implement easily optimize parameter dimensionality resort direct condition derive use background word user function prior term compound distribution integrate word notice importantly time sampling background topic ignore integrating otherwise come integrate background word simple count ji ij clause take care term integrate compound bernoulli integrate algorithm require metropolis hasting draw draw stochastic gibbs expectation whose th gibbs sampler ensure gibbs initialize eqs prediction input parameter randomly initialize user friend predict feature assignment gibbs observe learn sm learn topic likely friend discover facebook facebook user movie relation facebook page interest facebook select broad scope page sufficiently base collect complete million million explicitly mention collect follow text document status status one facebook title document typical nlp removal identification user symmetric list collection across document sm latent th iteration cumulative increase
relax prior family message loop reach loop exact penalize kl w relaxation easy choice part scale old message relax cause side product simplify overhead negligible practice choose expensive kl relaxation relax kl new propagation q w w energy detail update however update ep difficult matching demand exactly match ep ignore reduce final accuracy min max include special max ep bethe relax moment far along approximation rather identically sample input noisy latent gp projection encode smoothness nonlinearity factor factor relaxation share bf initialize converge loop posterior multiple minimize relaxed divergence cumulative obtain release implementation publication ep ep fractional variational case outlier ep increase algorithmic stability change divergence minimization necessary step message size reduce speed furthermore guarantee ep ep speed ep excellent inference gp ep thesis use update possibly improve quality figure belong reflect labeling x classifier dimensional input approximation importance sampling posterior boundary figure outli boundary bayesian decision boundary close decision perfectly power examine exact covariance align covariance ep outli force ep relatively small wide range consistently ep nonlinear gaussian red blue label vary apply tune decision boundary ep dataset set power relax ep clearly ep decision give well ep shape finally decision illustrate reflect curve time sample test iteration convergence run reach ep frequently always converge accuracy flip introduce labeling error low error test five uci benchmark diabetes spam detect heart time diabetes medical history uci diabetes two experiment patient five year breast contain patient randomly accuracy split outperform detect spam partition training test experiment demonstrate various additional error ep method increase ep relax classification demonstrate avoid give ep primary energy q main text update guarantee relaxed experiment ep give outlier relax ep function kl duality condition constraint dual classification divergence optimize line gp ep ep cumulative ep cs cs edu school west expense powerful propagation expectation ep efficiently exact nevertheless ep sensitive suffer inference propagation requirement propagation relaxation exact moment presence outlier relax moment outlier greatly quality uci benchmark demonstrate significant ep ep algorithmic stability inference relaxed expectation process bayesian principled make prediction learn exact expensive address challenge variety speed computation representative approximate generalize approximation discrete continuous posterior maintain gp predictive despite bethe examine model ep good experimental benchmark consistently power performance distribution probabilistic normalization could obtain
differ slightly optimize e constitute sized interior method idea convergence l bad big coordinate increase lipschitz logarithmic precision hence role feasible concentrate projection converge contrast devote obtain different infeasible sequence generate primal dual saddle formulation primal polytope relaxation pair cast saddle lagrangian iteratively dual primal feasible gap l one decomposition subgradient size apply relaxed imply convergence guarantee q none please operation take directly gap reconstruct primal method gradient vector converge optimum reweighte approach smoothed optimize mapping expression optimize unconstrained reweighted free energy optimization tool speed projection preferable approximate smooth optimize benefit lemma analogous converge primal optimization approach solution auxiliary see augment lagrangian base apply problem lagrangian combine descent versa scheme infeasible project optimize projection describe aware reconstruct primal relaxed non descent scheme relax convergence practice concentrate reconstruct feasible qualitative objective point conference benchmark implementation primal algorithm dual decomposition base subgradient ascent sg average sg nesterov accelerate gradient ascent smooth dual library optimize polytope elegant source author site primal potential lp map problem deal infeasible subgradient sg fast optimize exceed gradient require ghz primal significantly contrast integer relaxed problem converge primal bind sg plot line primal dual problem provide generate lp know generation unary potential assign dataset ran project eq suitably function purely potential significant contribution dual infeasible estimate lipschitz proportional plot l energy differ infinite energy correspond energy infinity projection primal one contrary efficient certain separability limitation however particularly overcome construct maintain feasible address support foundation application grant mm subsection corollary subsection definition subsection example end propose problem separability infeasible produce dual often feasible complexity obtain feasibility optimum property apply polytope inference problem markov demonstrate superiority relaxation appear vision often million thousand optimization strong primal primal sub infeasible slow size situation feasible polytope relaxation mrf problem relaxation integer application complementary heuristic search converge guarantee vanish duality determine sound ii basis iii adaptive optimization duality gap selection subgradient smoothing procedure problem cut plane scalable method infeasible separable construct feasible point optimum problem infeasible counterpart theoretically empirically polytope mrf formulate problem separable separable programming special illustrate method avoid numerous detail refer proof special generalization cone integer index subset matrix dimension program standard feasible ni main please due contrary euclidean compute solve assume scalable simplex due size additionally show speed assume polytope unary binary max unary potential feasible primal infeasible energy problem posteriori marginalization require relaxation hull set polytope local polytope approximation non theoretically algorithmic polytope relaxation dramatically simple optimization objective relaxed function maintain infeasible primal feasibility cut discuss message pass propagation optimum obtain infeasible vast conference aware formulate accuracy attain end primal auxiliary attain case grow size obtain slow get closely introduction section case relate mrfs allow reader familiar marginalization mrf specifie optimize construct dual algorithmic scheme dual primal estimate feasibility estimate optimize euclidean space paper notion optimize simplification optimize infeasible get big soon possess projection make computation additionally tree reweighte posteriori primal dual formulation separability construct optimize undirected finite labeling collection uv uv uv unary potential potential label minimize express denote uv uv r uv x relax programming form polytope later denote slightly briefly correspond satisfy iff else vector become collection similar suitably select write split subproblem use optimize local polytope collection primal optimize define simplex sized study example optimize quite grow assign euclidean quite bad demonstrate depict optimize actual value neither projection euclidean pairwise factor control infinite thick red line assume infinitely primal assign corresponding pairwise factor equal primal high relaxation model high local straightforward optimization split representation relax since unary one degeneracy u read follow explicit consist follow coordinate uv possess separability fix latter split uv unary potential improve denote reweighte unary potential constitute possible space optimize equation theorem optimize projection energy influence construct correspond polytope reweighte optimize formulate lagrangian base marginalization section reconstruct primal estimate dual technique base jointly solve
medium although preferable nearby compare one rating also stable characteristic mean less often ordinal number rating bit metric process address scalability inference predict measurement formulate partially complete contain node structural spatial long various ordinal completion great acquisition similarity recommender well applicability recommender simple factorization produce focus network example network contrast completion ordinal recommender describe network result application give end heart metric serve various example measure service indicate rate concerned streaming acquisition metric effort lead tool acquisition suffer accuracy bandwidth path paper metric bandwidth determine choose metric go addition rating reflect experience end performance define rating ordinal paper qualitatively good acquire determine metric belong illustrate evenly requirement need certain metric acquisition versus particularly significant internet objective bandwidth site medium service achieve goal use knowledge find globally allow rating matrix rank entry practice matrix often component negligible approximate completion entry preserve entry turn word try find approximate rank difficult constrain neither continuous rank adopt compact look class factorization mf contrast factorization appealing formulate node network come internet overlap heavily induce enable problem evaluate characteristic value original decrease hinge pt incorporate additional negativity besides divergence value close build prediction machine perform unseen reduce variance prediction combine final mf model mf different scheme adopt stochastic mf pick reduce sgd demand process locally interested reader decentralize ex path system randomly select neighbor sequel path choose trading increase always accuracy measure overhead lead pt ex mf construct complete proper give search optimal empirically one enough keep hand require choose limited evaluation among dynamic hour static adopt common use root square q small rmse dataset partition evenly dataset ms ms ms unbalanced rating learning rate equal regularization equal tune impossible decentralize empirically mf number regardless mf mf parameter predictor impractical mf mf centralize use mf strategy particularly generally perform nmf mf ensemble well mf marginal worth extra I nevertheless adopt sequel c mf c ensemble netflix netflix show adopt achieve three rating row represent rating thus mis rating rating class bad value connect choose optimality performing prediction class rating random baseline optimality perform good rating rating path qualitative cost metric address end rating recommender netflix factorization adopt benefit rating internet application ac li scale acquisition end distinct contribution ordinal rating completion former measurement
fold provide list alternative shortest discover close deal length node direct adopt short probabilistic uncertain relationship directly view probabilistic feature recommender experimental obtain decomposition represent method problem case conjecture exercise proposition world domain mean edge quantify likelihood existence represent likely language node connection two weight link instance link label link l regression predict unobserved real collaborative filtering well adopt last uncertainty lead usually edge quantify likelihood existence strength link reliability pair node connect graph probabilistic probabilistic along play network road concept whose task formalize link likely unobserve link graph important ingredient classical link prediction probabilistic need want whether two interested node paper likely probabilistic adopt tool connection may probability link particular label output correspond adopt learn model link choose recommender predict rating user world achieve induce svd kind probabilistic propose section work edge probabilistic system undirecte assign assign assign represent absence imagine instance discrete sample accord probability treat mutually indicate belong discrete graph probabilistic length edge path follow iff language language simple free path path path formally path randomly sample language otherwise give possibility free discrete path belong existence intensive intractable check every existence path sample discrete accord basic estimator eq check path iterative depth path existence visit sample adjacent stack iterative stop node stack non language path label element link belong train way classification unobserved class eq label link prediction language particular language eq query language consider language particular language real observe x x represent problem huge weight l regression belong solve unconstraine w I tc binary rest nonlinear quasi efficient minimize w score hessian element identity newton validate recommender probabilistic observable domain similarity graph similarity relationship measure heterogeneous entity item relationship assign goal user rate past predict unknown exploit user similarity user orient rating rating similar item orient rating rating let rating preference user item strong preference rate approach predict unobserved follow represent stand similarity user pearson item base formula neighbourhood connect item structure particular entity recently prove technique leverage mining item indicate similarity node kind similar user adopt pearson rated connect v pearson correlation rating rate user v connect item item compute pearson corresponding j rate item score add element belong solve constrain path path user belong language particular predict rate pl belong construct complex query connection belong user classification set computing item rate know map training link classification tool deal probabilistic research university international heterogeneity version range report rate movie user gender code site seven month rating consider fold rating test rating ccccc r testing create rating specific testing adopt procedure resp procedure sampling path language language neighbourhood probabilistic
recent book deal propose present overview context stein estimation elliptical estimation test along estimator deal mse estimator risk conclude square ls test hypothesis special situation elliptical q likelihood statistic hypothesis statistic let sn sn moreover maximum testing mixing get produce calculation probability significance call statistic cdf distribution follow stand insight lemma give possible elliptical see result measurable ease propose elliptical proof expectation see g far obtain addition proceed estimator estimator convex combination indicator percentile disadvantage extreme outcome test stein se se disadvantage become negative bias quadratic estimator consider matrix form determine bias risk evaluate quadratic risk estimator first expression estimator obtain make centrality risk define partitioning qp obtain rewrite obtain r pt f n f n conditionally te n te f provide equivalently eigenvalue easily zero thus estimator dominate square whenever pt use risk vice h pt vice versa superiority difference dominate shrinkage robust model error belong dominate quadratic get thus dominate whole away outside origin dominate expectation outside around origin dominate hypothesis always determine information order stein h dirac shrinkage inverse student remarkably behavior elliptical symmetry theory phenomenon estimator substantial robust elliptical could elliptical important c discussion regard tend regularity ii sp asymptotic distributional bias q pr ratio gd g p n n q r se difference e md dt see acknowledgment like anonymous suggestion put pointing grant ms ed pt maximum criterion stein type estimator inf stein stochastic constraint multivariate regression k transform new york ng distributions york k univariate york implication stein estimator north spherical location aspect new york md preliminary york useful statistics york stein elliptical preliminary estimation error ph thesis pc corollary md mathematics technology box school mathematics department mathematical
follow percentage apart percentage except error depict assess figure heterogeneous present leave draw independent outcome g I region enhance visualization easier hard summarize main look significantly exhibit competitive behave consistently bad mainly rate large peak shift top bottom hold employ error detect area look experimental compare perform edge separate area relatively look become detector display result respect execution many method speed unable mostly account technique replace achieve similar precision lower latter approximately distance derive acknowledge grant thank comment suggestion universit pe issue propose nonparametric detection numerically display datum planning environmental management detection coherent synthetic platform extend surface employ effect obtain direction platform motion remain moment movement throughout platform trajectory energy direction measure intensity send return surface illumination intensity presence acquisition region cloud resolution complementary optical selection band angle receive complex nature store amplitude image possible cover area surface little texture water ice example extremely area surface texture area texture cell employ sensor texture correspond cell heterogeneous resource operate l band sense band object group principle segmentation segmentation identification edge think characteristic occur image degradation illumination signal image degradation characteristic employ make local image viewpoint stochastic law pointwise analyze use information approach attractive perform area homogeneity intensity index look interest many parameter relate power reflect present target characteristic heterogeneous forest area method raw maximum smoothed propose outperform date maximum discuss image present use direction physics formation describe observation field coherent illumination look format model small high expense look exhibit degree homogeneity reciprocal tractable law denote density random l look independent correlate field moment distribution attractive mathematical linear tail display large lead dot nine eight look column image mean right readily find hard task local variation wish approach common define equation moment counterpart correspond arrive system follow numerically show severe numerical analyze related likelihood correction ml estimator effectiveness resample correction show ml show presence corner source contamination image estimator asymmetric motivated outperform summarize emphasis explicitly employ edge statistic additive show usefulness sample image visually detection nonparametric pixel neighborhood test local grey ratio describe detector robust detect noisy whether sub detection variety technique present compete acceptable contour spline parameter control smoothness use amplitude square image homogeneity value follow present feature final belongs exhibit value centroid angle successive around segment illustrate partition candidate small red dot come namely dark area estimation estimator upon consider segment lie lie transition maximize assume along straight population statistical nan identical intuitive two rank regardless possible rank assign population tend small population hypothesis consist population assign rank assign tie population within sample scale ordinal wish difference nan follow tie rank population tie divide different population hypothesis rank consist sample possibly different arrange cccc sample let assign tie follow sample sample respective mutual amongst ordinal either identical else yield large remain alternative sometimes population simplify small moderate equation preferred rank assess sample random sample rank usual equal tie assign tie require state respective population independence rank assign population divide get square range
surrogate truly appealing would need hide variable behave nearly well behave nearly theoretical one suffer thus present reach scale modulus continuity typically scale like accurate minimum nonzero singular component entry variable concentration magnitude scale dimensional well type condition interestingly instance approach robust analysis corrupt corruption totally nuclear suppose completion want norm need precise sample incoherent matrix work establishe recover even fraction corrupt reliable regardless corruption occur essentially error instead minimal require clean corrupted ls model move l survey circumstance powerful begin rich make range latent subtle video surveillance computer application know competitive loss robustness various application application flexibility robustness computer ls apply although low may relaxation practically model perform surveillance important foreground stack frame pixel hard imagine change foreground offer model texture texture name recover low superposition l work reconstruction recover align image component acquisition spirit ls acquisition large compressive sensing rely sparsity explain start two concrete efficient acquisition sequence interest pixel hyper frame thought index sequence important concern image clearly video time scene nearby correlate tensor trace nearly low roughly innovation frame moving object foreground could imagine might acquisition ever suggest image variation image imaging movement quality ct precise imaging variable context static correspond imaging recent potential since support substantial computer concern np size vertex factor easy graph select clique add clique find clique possible formulation difficult modern fundamental area machine pattern wide applicability mention new connection show optimal player game clique think plant clique decomposition represent clique interestingly apply problem clique sparse clique thus recover able break barrier investigate tight relaxation research find superposition introduce across follow reflect low decomposition extension
computationally feasible overview idea encode lattice compact representation need provably distortion codebook follow heuristic explanation attain distortion characterization encoding regard tradeoff distortion successive refinement distortion letter random letter denote denote inner vector unless otherwise mention rate measure n entry gaussian integer whose specified fig compose property property compression rate give find encode mapping encoder decoder produce proportional recover shannon storage construction choose dictionary sparse code satisfying offer trade error source generate ergodic encoder determine pick non zero give choose section section succeed distortion sequence distortion formal contain encoding consist stage imply source sequential codebook length matrix memory source encode module compute source column module compute residual module consist multipli fashion delay module simultaneously encoder computational memory automatically encoder bit indicate position decoder receive bit reconstruct rigorous analysis number law number precisely projection mutually component collection sphere product random deterministic q normalize norm list begin true independent list terminate pick distortion distortion make rigorous bounding distortion stage value sequence variance let define encoding satisfie I rate encoder distortion decay exponentially length enough interval constant choose I chernoff event bl ml np equality hence distortion probability excess decay exponentially large source ergodicity need theorem encoder distortion source line encode source attain small growing grow polynomially storage operation gap distortion illustrative choosing yield gap govern next complexity low convergence shannon propose algorithm essentially exponential gap dominate distortion whose grow polynomially length encoder distortion complexity grow polynomially propose encoding interpret code think codebook act sequence minimize distortion distortion distortion achievable codebook rate distortion attain codebook I codebook follow typical close distortion distortion fall source whose second moment ps redundancy code distortion function bind redundancy successive refinement linearly stage excess distortion tight determine redundancy encoder successive inherent codebook important encode bb focus high rate dash high rate distortion entropy code scalar encoder source sequence brief step selection almost amenable theoretical encoder variance dictionary obtain plot increase keep eq length graph come expense distortion column give well rule bit modify due deviation norm take gain shape source nearly distortion code ec distortion gap theorem excess distortion bit distortion summary include ec gain empirical distortion error exponent interesting work essence value residual step introduce normalized euclidean norm distortion value express statistic I armed goal distortion give hold whose hold satisfy appendix follow first follow due trivially side therefore since q prove towards induction sufficiently obtain apply proof ensemble compression design polynomial block result codebook encoder computational grow polynomially attain distortion rate variance source satisfy deviation excess distortion distortion exponentially block length may successively source emphasize successive unique inherent codebook coefficient choose encoding power choose distance analyze value design matrix design efficient length optimal among code fast expense high complexity simple successive refinement section distortion encoder greedy across sequentially pick successful recovery may approach distortion section explore small storage find distortion establish theoretical performance open communication performance binary codebook encoding analyze al compression communication rate shannon theoretic demonstrate channel superposition key scheme terminal channel framework joint condition collection joint q recall bind p moment similarly substitute distribution brevity column integral equality maximum variable eq generate use eq integral standard evaluate q verify maximum attain exponent dominate claim bound bind attain large go use eq author like thank discussion anonymous comment define linear combination column design successively source
mean termination guarantee check stop condition purpose efficiency sequence accomplish mean infinitely stage number positive n x appendix eventually stop termination n establish rule eventually stop termination sampling method purpose principle construction accomplish apply size sequel construction since variable interval use z x n appendix notation construct n take sequel compute limit give check sufficient sufficient decrease notation x b b I truth virtue follow fact true w w true check critical subroutine low st st limit adaptively interval truth confidence give truth width w I sufficient follow truth statement check virtue fact statement n w w w check efficient follow st let return adaptively interval check method sample confidence variable ready construct sampling purpose sampling stage term inequality inclusion termination take result confidence region mean variance subset c w point solve w method x n x point solve equation w would like point technique interval sequential procedure bound tail bernstein inclusion method geometric poisson sequential bound make information estimation guarantee prescribe rule notation lemma eq lem z hence r z z lem simplicity z z z prove lem nn imply definition n sm n n x nn l n n lem b n nn imply see n b u sm nn n n nn u u u n n x q complete show suffice r computation thus lemma z result imply eq combine lemma u x consequence lemma stopping ensure desire lemma know continue u sn stop n claim n x x terminate theorem preliminary corollary lemma lem lem r notation check inequality z prove lem g notation check z z z lem need since imply l sm n n n complete lem n g n show existence define u g sm u nn x u n nn complete x sm sm eq combining complete r show lemma simplicity r r consequence prove make lemma follow n n x lemma stopping ensure coverage stop equivalent sn continue show fact sampling terminate de p prove lemma pos r r therefore suffice notation r p r complete lem pos lem lem pos n need well existence define x x n n nn n l lem need together see x n p u pos n x lemma pos result define n nu n complete show p z suppose follow x x last chernoff position stopping ensure coverage rule show fact n sampling terminate de f preliminary r notation note p w z z r r r r definition definition lemma l n nu l u x p inequality z claim prove rule know stop rule continue continue need follow n u x eventually terminate z z z z z z z proof define u l x u u n region v follow region u similar prove inequality unimodal respect complete sequential estimation estimation construct sequential contrast rigorously guarantee specify issue estimate overcome difficulty advance evaluate stop accordance pre area estimation wide scheme prescribe level unfortunately nature pre specify level zero average infinity method overcome limitation shall develop virtue inclusion variety sequential estimation cast prescribe coverage use confidence sequence interval must termination situation impose sequential except probability specific principle scheme process include sequential inclusion justify interval variable assume theorem sequential inclusion principle bound
stock stock real construct take car difference graphical model markov network network fire conventional single firing process network manifold fire responsible system geometry joint joint call manifold follow flat flat manifold I replace replace marginal part skip save space bregman strictly denote property manifold role determine kullback algorithms difficulty design close decomposable intractable variable difficulty firstly propose fire information fire introduce algorithm geometry experimental appropriately fire network herein network fire node mean sample assign fire fire direct number fire fire one method cyclic call chain fire subsequence homogeneous matrix random manner markov firing process homogeneous fire empirical distribution firing ergodicity sequential firing ergodicity equally homogeneous converge distribution fire markov network fire process follow fire equivalent gibbs sample give fire aspect fire sequential fire stationary distribution firing note rigorous rigorous lose determine show work certain condition count sample fire fire sequential gibbs fire fig notion geometry fire onto fig gibbs converge incomplete thus converge however thus provide theoretical treat fire part paper define conditional part manifold kl bregman manifold close firing limit illustrate information subtract conditional already determine form good model role machine requirement construct complexity low avoid overfitte show fire trade requirement minimum treat empirical situation table denote define similarly depend distribution section cause rise continue break forward describe dominate source dash goal intersect intersect reasonable independently node situation construct computation form addition node evaluation subtract rarely node eq whole computation numerically transition therefore large learning markov carlo compute distribution numerically draw perform firing node separate part variable
likelihood coefficient newton fisher square replace information matrix continue criterion modern extension framework etc adaptively allow tuning penalize randomize widely freedom calculate nonzero multivariate logistic algorithm optimize handle know million fast smoothing popularity linear importantly generalize enable researcher complex increasingly computer describe member formulate smoothing spline consider multivariate restrict reproduce hilbert reproduce flexible formulate parameterization adopt really ratio project onto spline reproduce utilize smooth iterative square computation multivariate exponential way structure ise capable estimate importantly via significance multivariate property equivalent marginal bernoulli furthermore include statistical inference hypothesis interval extension multivariate logistic technique candidate smoothing spline linear effect clique proof proposition py py py condition distribution regard py follow bernoulli examine expand formulation bivariate bernoulli eq function separable formula conversely term assertion log combine correspond product expand exponent zero appear numerator scenario apply negative formula verify trivial numerator theorem take generating density rewrite separate moment separable assume expand hold decompose c independence multivariate moment dependent moment function node bernoulli form author nsf grant dms theorem proposition corollary consider distribution family regarding demonstrate importantly pairwise interaction among model interaction ise bernoulli distribution conditional bernoulli utilize canonical covariate node structure bernoulli linear predictor undirecte devoted resource study undirecte allow describe thus conditionally edge capture categorical pairwise correlation consider sufficient structure year area focus matrix example lasso determine covariance valid pairwise true bernoulli discuss represent third clique call physics gain popularity several discrete include ise structure study multivariate community extend clique order ise model assume interpretation interaction want variable potentially consider spline model node high graph multivariate covariate section start simple mathematical multivariate clique discuss optimization result bernoulli logistic section provide deferred case extend widely bernoulli bernoulli explore outcome consider take density valid probability simplify inverse exponential marginal bivariate bernoulli bernoulli bernoulli bivariate bernoulli hand bivariate bernoulli vector define log bivariate bernoulli separable give deal variable independent bernoulli distribution among component bernoulli component separability outcome follow defer wise importance component ensure assertion researcher multivariate bernoulli generality block proof defer graph researcher conditional multivariate bernoulli conditional distribution bernoulli state random generating bernoulli hence solely parameter hessian fisher optimization possible indice cardinality say definition reformulate form partial generating derivative respect bivariate bernoulli second order natural gaussian main undirected multivariate ise random symmetric positive define bernoulli ise pairwise interaction clique convert introduction retain markovian continuous ise consider interaction log normalize multivariate kind similarity independence structure ise
q row valid observation identity combination least argument obvious topic every row entry every constraint solve quadratic give define nonnegative projection onto k cx fx z theorem remark subsection depth sciences department sciences california technology paper describe programming computing idea drive factorization salient datum identify constraint select demonstrate similar recent contrast early propose lead experiment minute keyword nonnegative programming computing matrix factorization mining package include heuristic nmf theoretical heuristic correct develop rigorous nmf stem challenge hardness consequence additional hope practice provably nmf meaningful answer potential preprocessing impact practice machine scalable robust nmf problem appropriate contribution theoretical show succeed modeling algorithm substantially may make pose believe independent adapt class major experimental research system solver extremely sgd order implementation minute nmf drive factorization express row appear qr adapt row nonnegative indexing indexing nmf seek feature matrix sum may np whether motivate algorithmic nmf condition uniqueness nonnegative result I separable nonnegative jk pt circle hull circle nmf distance hull definition uniqueness result row hull row nmf call row permutation identify set distinguished row allow justified variety text reconstruct count word localize predict audio bin remain admit factorization well factorization nonnegative constant nonnegative nonnegative factorization robust particular compute although guarantee run undesirable priori knowledge distance third row row lie finally link norm drawback separable matrix key admit matrix matrix element accomplished extract construct matrix extract column approach construct factorization theorem solve single program extract nonnegative nmf matrix find unique row nonnegative sum row factorization suppose state paragraph find correctly identify topic assuming consist topic reasonably far away construct guarantee know show early version contain establish theoretical remain develop solve lp may suffice scale prohibitive describe algorithm large column subset popular qr localization use outlier et preprocesse recovery noiseless improve aspect enable bound rely incremental depth aim minimize proceed lagrangian multiplier set ascent solve lagrangian subgradient minimizer update make little difference minimize project descent index nonzero incremental choose subgradient project iterate sort plus drop result negative item solely break find incremental repeat epoch variable identify diagonal row minimization subgradient separable incremental nonnegative primal nk line parallelism cache critical technique cache intel incremental respect store contiguous encourage hardware choice row cache row similar classical loop join relational curve identical x core ram scale synthetic programming run couple solver stepsize fit incremental gradient combination uniformly add run run convenient performance compare metric profile particular performance curve outperform algorithm give convenient algorithm visually ccc profile correspond experiment linear slow must fed run fast noise achieve low matrix qualitatively code principle occurrence place dataset normalize recognize c feature gb e figure speed serial parallel exhibit hardware cache datum able correctly rmse middle quickly demonstrate dimensionality language svm extract misclassification versus topic right topic misclassification compare speedup versus number horizontal achieve analyze problem pose work factorization theoretical author like support nsf award fellowship
restrict rational coordinate actually sequence fact use addition treat define new matrix write v trace product create analogue formula moment coordinate map marginalization diagram ct cm x cm ct swap ct cm rt node node rt rt swap node stochastic end motivated observation v I e empty truncation index inclusion summarize auto ct cn distance distance right distance co cn co cn swap distance node swap diagram exhibit map direct inclusion map trace denote entry trace map inclusion relate hmms new e diagram auto co node right co swap node swap spc spc swap begin map trace q q write diagram co spc cn co node swap cn spc co generator check may map follow diagram cm co distance cm co cm spc cm swap swap spc node swap swap spc spc swap imply diagram auto cn swap cn node cm cm node swap map factor first take map map marginalization set open w applying complete diagram dominant cm cn cn node distance cm cn cn swap co cn swap co cn generator question hmms turn cut map point analysis reveal negative point light nonnegative test distribution membership reduce ask determine contradiction reduction lie respective induce observe formulae vanish square simultaneous mind stochastic end implicitly describe characterization rational closure assume irreducible rational parameter rational field application extent observational alone different notion rational theoretically identifiable theoretic rational identifiable theoretic identifiable identifiable answer question introduce rational fraction field subset dense open thus stable division show algebraic statistical question parameter coordinate simply q rational since field extension algebraic closure hence extension extension identifiable parametrize model hmm variety transition emission sum hence large exclude complete formula equilibrium turn yield reveal geometry consider readily e visible identification process emission triple define apply yield vanish sign hide alphabet emission identify obtain polynomial one hope state theorem thm theorem lem question conv nb explicit recover membership comprise gr basis coordinate parameter identifiable sense closure projective variety dimension gr considerably hmms selection problem hope hmms primarily identifiability algebraic geometry begin hmms binary hope eventually precise attain provide insight relate branch model series paper e begin hmm use speech well since gene dna biological alignment build measurement kind time algebraic algebraic geometry fully explore hence early investigate algebraic especially attribute interest manuscript gr walk subsequent include algebraic ps dms geometry statistical selection analogue model polynomial equal e encode equation analogous use hidden markov model bm exhibit fact actually modify non verify ideal hmm degenerate hmm verify make new coordinate hmm short reader sense equation generate fine membership hmm consecutive node algebraic analogue generic identify parameterization excellent context causal effect ode parametrization hmm composition dominant explicit hide component identifiable combination recover particular triangular generator gr basis ideal simplicity complicated look like parametrization variety invariant theory triple inside trace element find finally explore result find geometry equilibrium hmms visible thank lin suggestion note hide visible reduce throughout markov distinct hide binary hide denote respectively call hide index node row specify specifie transition probability state emission emission formula read precise set h v compatible distribution binary jointly binary process accord see give familiar model alone chain mind node unobserve vary sometimes allow vary set process closure geometry purely stochastic consist triple cube call triple arbitrary sum complex complex replace convenience sometimes algebraic function write make identification length write complex subscript intend likewise denote generator model simply word frequently reasonable usage image stochastic observe continuous cube markov closure equivalently closure ideal homogeneous vanish homogeneous type closure whose reflect algebraic ideal turn computation simplified become feasible compact indicator function thus confusion write specify subset avoid usually treat ideal moment coordinate projective cut observe elementary even hide alternate writing occur generate moment function respectively change formulae taylor relevant term vanish ideal conversely formulae describe computation run intel core ghz cpu gb ram light understand geometrically homogeneous ideal patch reduce continue long expression arithmetic trying probability coordinate instead extraction generator subsequent run generator generator order generator degree eliminate redundant generator reverse second minimal homogeneous set generator generator low degree save hour generator turn omit apply generator minute minimal generate set polynomial degree generator homogeneous ideal minimal inclusion homogeneous set moment run memory gb compute author place memory try compute kernel dominant uniquely follow c ct ct node swap p rational visible moment exhibit use explore observe swap permutation q generalize permutation essential use parametrization subscript letter string hope context avoid confusion make term fact coordinate act q word sign act matrix factor map introduce interpret consequence eq write diagram dominant distance cm auto cn cn node cm cn cn ct swap ct cn
draw take probability euclidean hellinger leibler bins draw even nan hypothesis repetition probability regard random experimental occur realization realization accuracy remark p number monte simulation conduct p follow trial trial simulation whose corresponding course fraction exact calculate discrete parameterized specifie meaning exist distribution desirable actual experimental assume calculate realization repeat experiment calculate value realization realization repetition procedure parametric bootstrap family discrete distribution highly repeat calculate realization maximum likelihood distribution number somewhat remark favorable indeed point generalization goodness fit generally much classical position remark goodness toy example use draw bin unlikely arise specify exactly goodness detect plot test whether via evaluate simulation draw extremely find bin increase draw bin demonstrate root statistic nearly report statistic bin sure support matter arbitrarily value bin classical goodness fit reliably independently power law behave agree draw bin bin bin unlikely well various fit plot arise simulation empirical square consistently classical little agree draw bin bin draw draw refer text page source english assess goodness consist english repetition come news randomly corpus bins figure word sort frequency statistic significance permutation sort frequency choose bin great number choose bin great among great bin via sort know similarly know bin plot value distinct must list word whose display must substantially large assumption respect goodness fit priori fortunately root digits value follow define contrast p fact one without know dictionary classical analyze analogous list english word repetition list word corpus sort plot compute monte carlo involve distinct word zero please note display though substantially figure remark list list whereas root square introduce great draw truncate law corpus randomly nonnegative method determine fit bin bin contribute aside bin total draw fall three frequently occur maximum turn model calculate via root statistic truncate law fit great frequency follow classic decay poisson report whether datum bin accord distribution truncate bin total depend draw demonstrate experimental bin strongly influence classical statistic simulation bin statistic fit reasonably agreement classical statistic tail whereas insensitive particle bin interval second dot report display plot whose poisson calculate monte carlo simulate four report poisson reasonably good ccc bin square dot poisson line suitably proportion pair expect random proportion nine naturally proportion genome individual give suitably random entail goodness high confidence confident suitably table via simulation negative root blind discrepancy please note classical nan fortunately irrelevant mean estimate inaccurate potentially inaccurate sensitive bin log section refer statistic simply logarithm generalization log facilitate rather good relative table associate log root mean square c match american number match report health generate rating rating value root issue small symmetry investigation modification provide testing root powerful detect root square statistic much powerful detecting detecting mean bin whose associate recommend ratio excellent fair poor excellent good fair poor ccccc fair poor excellent poor ccccc fair good fair report identify exclude experimental figure collect specie sort build bin since sort subsection sort permutation integer method obtain sort frequency please truncate geometric omit complicated discrepancy solely fluctuation log ratio unable root determine significant number goodness detect model quantify statistic success remark actual generate draw p simulation compute present p example subsection step remark cost subsection consideration draw statistic value estimate accurate present equivalent statistic calculate parameter root mean sensitive bin bin desirable recommend square variation simulation draw arise generate define root classical often simulation plot remark square classical furthermore draw root square next specify distribution q require increasingly root square exhibit opposite require distribution specify section specify require model distinguish bin specify draw draw distinguish actual specify distinguish mean powerful begin truncate draw actual distribution remark specify specify eq consider draw draw distinguish model specifie distinguish see involve specify distribution via plot define specify method distinguish define maximum distinguish poisson maximum draw require specifie distinguish q estimate eq distinguish specify draw distinguish actual model specifie distinguish q likelihood method bin great among choose great draw contain remain bin finally draw sort specifie distinguish specify draw distinguish maximum specifie acknowledgement would like thank fan w stein many pointing show hellinger version root nsf grant nsf fellowship fellowship nsf fellowship mark goodness distance test standard magnitude goodness article discuss numerous fit practical euclidean goodness interpretable black program rapidly calculate precise significance square identically model may countable accordance terminology discrete category draw use root statistic distribution rao root root square statistic confident draw quantify confident denote root square fact parameterize correspond draw arise model square avoid number pearson root square average produce classic statistic average involve division zero divide power round error arise thesis classic long appropriate superior computer illustrate conjunction goodness fit statistic preferable classic statistic division somewhat kolmogorov root certain circumstance kolmogorov root way complementary largely easy understand computing mean large continuous maximum underlie unknown value parameter clear devise none standard
mix essence graph number define approximate sampling accomplish state simple call neighbor otherwise rapidly maximum markov sample independent irreducible chain relate chain whenever respective locate appendix maximum moreover independent permutation exhibit chain figure depict grid since grid rapid mix identical follow fact whenever group rapidly couple probability locate refined moreover capture power coupling merely consider indeed strength chain presence hamming locate chain rapidly vertex runtime number leave detection markov people whether grid motivate lattice numerous core processor gb list scale quadratic set generator permutation w markov chain state art discrete implement file dedicated topology degree sampler bottom dimensional depict partition size edge topology center generating partition vertex depict instance generate b I need generator topology run start set require ram create accumulate topology sampler accumulate plot variation gibbs fast gibbs chain computational overhead observable topology summary chain result sampler reader recognize state bi count limitation chain combine generate highly advantage avoid intractable markov exhibit ise markov chain sampler numerous expect improve problem exhibit relational language candidate chain algorithms marginal symmetry detection use break group model conditional design stage markov easily incorporate permutation van discussion remark system markov uniformly state element chain remain hence statement accomplish omit state since state last follow chain p hamming element path argument distance degree five random hx g hx hx hx thm corollary thm corollary thm proposition thm remark remark thm thm example convention convention thm conjecture thm thm thm open thm thm construction com present novel detecting utilize two contribution computing chain family markov leverage establish connection rapid mixing chain probabilistic result effectiveness efficiency algorithm reduce first exhaustive exploration search answer integer utilize work relational formalism logic avoid instead notable propagation exception somewhat principle since design applicability contribute deep symmetry construction color graphical model consideration main first inference compact set permutation group link path coupling make couple chain respective locate coupling argument chain possibility investigate lead mix analytical insight empirically markov art chain applicable begin recall concept brief overview work utilize design logical probabilistic discrete structure preserve group binary element element form group act permutation cycle understand map relation equivalence denote f define give chain transition step chain start chain irreducible converge let markov chain leverage challenge exist breaking predicate symmetry symmetry symmetry put answer programming programming notion probabilistic relational work algorithm belief bi relational contrast applicable large formula large graphical colored undirected formula network weight model counting framework represent partially formula colored graph clause color construction belief bp contain receive message colored permutation acting feature partition variable message receive message notion exchangeability de finite partial state act whenever exchangeability contain every exchangeable draw exchangeability feasible compactly represent generator utilize go marginal via message pass applicability turn inspire previous introduce markov exist chain presence chain range transition original chain require generate act see generator set markov stationary integer state random markov chain chain chain state element element nearly uniform polynomial generating replacement initialize pseudo random perform multiplication verify overhead negligible irreducible base gibbs commonly perform chain follow step time uniformly conditional chain identity permutation equivalent gibbs sampler major property space irreducible irreducible stationary distribution original compatible capture
inequality central banach space banach let absolute explicitly specify state front example electrical engineering edu mathematics air institute edu group responsible tumor focus selection high exceed work variable response restrictive comprehensive low avoid limitation term predictor paper relate fundamental problem relationship sample tumor health problem many great time change observe variable per sample impossible collect imagine hundred clinical world stay paper inference small case response mathematically model sample call predictor compactly term comprise tumor classification correspond express tumor would tumor despite continue inferential focus high linear predictor responsible variation small gene design variable per x exist implication predictor imply model gene biological pathway situation problem group linear denote predictor regression underlie explain one contribution time estimate design nonzero computable establish predictor indice group contribution response basic researcher notable direction despite high report total body restrictive kronecker incomplete restrictive scaling predictor nonzero finally group study paper analyze variable selection addition limitation influential predictor discuss early incorrectly nonzero group coefficient predictor response assume understanding carry manner letter scalar constant denote shorthand transpose spectral finally norm qp qr mathematically formulate bad group result main numerical conclude attention relate comprise column since norm relax error combination group group third performance selection procedure term estimate sort marginal norm sort focus seek characterize performance group impose metric goodness metric time group check suited group total response thresholde dimensional arbitrary coefficient paper coherence property design coherence finally offer price correlation theorem marginal give return guarantee predictor early proportional group expand predictor estimate come affect confirm section develop notation facilitate group correlation yx predictor coefficient respectively marginal correlation require behavior two toward leverage proceed take subset group dimensional vector x similarly dimensional kx x z banach begin note condition concentration proposition upper construct martingale k martingale element eq u v order every consequently nature induce primarily need bind uniform coherence construction martingale banach appendix algebraic fact together km ec appendix begin element follow involve resort argument primarily utilize straightforward construction martingale proposition use together claim union distribute correlation correlation complement recall statement trivially I therefore establish lemma claim condition establish claim probability regard trivially note theorem create generate matrix entry draw schmidt stack orthonormal confirm empty specifically plot four figure solid see figure
upper previous claim incorporate claim claim marginal precisely marginal bivariate covariate marginal denote exposure covariate technique q q maximum likelihood eq general maximize gauss copula copula optimize margin step estimate marginal claim via dispersion exposure time wise vector poisson numerically time compare initial performance confirm finding likelihood solution asymptotically maximization maximum denote hessian partial explicitly bfgs matrix via derivative approximation assume fix copula family nest copula obtain restriction model copula pointwise family parameter difference variance prefer copula standard eq prefer otherwise adjust via joint step claim loss policy positive variable loss mean asymptotic total replace estimate joint numerically group gender car contain number constant offset consist marginal model intercept correspond covariate age draw uniformly last car car c car intercept car claim intercept car car claim coefficient variation copula family parameter fit copula repeat follow regression coefficient estimate r r freedom independence model prefer aic claim center policy bottom aic center display narrow display result run display error bar upper display square policy square leave significantly relative square low copula base value display value joint e ignore panel display total dash respective systematically loss confirm conclusion three family confirm make display company car year covariate exposure categorical analyze ccc description gender drive age year car analyze average claim poisson significance interested asymptotically normally interval addition adjust claim covariate age year marginal respective covariate fit joint copula family table result display copula indicate select select c gauss frank gauss frank gauss frank frank frank indicate model model conclude copula preferred family remainder continue family aic model comparison base moderate claim number claim equal copula considerable investigate impact copula total loss eq already dependency indicate conservative take appropriate rating company size claim tend skew theoretically incorporation estimation incorporate claim allow flexible class family extend study copula extend family overview appropriate marginal dimensional mixture continuous random copula construction see e average size claim portfolio severe us car policy covariate marginal moderate avoids portfolio claim bivariate family relationship cumulative cumulative univariate far kx dt gauss u gauss v u v display gauss frank number center leave aic center display narrow claim policy loss leave score center display width display narrow average size center right bottom estimate aic score display standard narrow joint copula claim accommodate restrictive skewed multi independence show efficiently test optimal copula usefulness car keyword claim size portfolio pricing base compound model independently quantitie restrictive effect portfolio propose dependency average claim combine paper derivation policy often skewed policy mean usefulness copula popular year book bivariate triangle management claim covariate overview claim separately claim flexibility extend copula generalize family gaus copula propose extensive study loss reliable total confirm bivariate copula bivariate distribute importance theorem state bivariate copula copula conversely marginal separately define copula monotone measure association measure denote joint x xx yy copula parameter relationship truncate gauss frank parameter illustrate copula family result claim claim copula claim size equal display mass gauss probability mass value due conditioning average panel copula tail dependent shift compare size
fmri order clinical score level propose linearity simulation formulation capture recovery brain pattern fmri formulation bold capital dot target paper vector prediction value brain map visualization pattern matrix form design volume target word voxel stimulus presentation lead regularization parameter penalty fmri linearity construct pairwise operate pair predict sign index restrict image associate pairwise hinge use successfully information retrieval loss pairwise extension might know close misclassification distant discuss implement regression solver input library via learn library tend appropriately major logistic simulated volume consist voxel real cubic size choose mean coefficient leave model regression cross validate square equivalent ridge non linearity correlation times linear mse dimensionality second correlation coefficient unlike increase paper result pairwise linearity pairwise assume unlike zero pair case equivalent adjacent coefficient pairwise logistic ridge unweighted ridge appropriately major set mse dataset describe sentence complexity simple sentence high sentence consist informative validation cross choose class adjacent computed score pair image keep superior temporal tp inferior order estimate project regularize parametric weighted linearity vary shape region exhibit investigation however trend decrease bold tp effect well p function effect region projection region apart investigate pairwise improve pattern learn fmri benefit fmri presence brain choice loss extension brain via grant ga team le de france france france bt france team paris paris infer functional specificity image fmri general glm remain
therefore cauchy lemma argument follow number compact ball center origin remark detailed every dt dl constant depend consequence get every deduce yield simple proof turn interest let step subgradient subgradient cauchy schwarz use twice thus would combine get proof fx q unit e divide side let directional directional eq dx assume notice derivative equal convexity right satisfie vc satisfie imply f dx dx work derivative right proof indeed replace cover l deduce existence whenever remark corollary section cover number convex function lower cover term constant uniformly summarize cover provide alternate implication study optimal numerous convexity constrain estimation risk minimization hausdorff kolmogorov metric pack metric entropy space play central area information theory nonparametric cover uniformly dimension relevant know covering function uniformly convexity characterize optimal convergence rate empirical implication regard convergence numerous convexity nonparametric study study crucially concavity estimate receive machine motivation prove section cover relate independent convex cube specifically value function uniformly bound absolute theorem sufficiently logarithm b convexity estimation usually uniform function difficulty cover number convex function deal class cover number metric space totally metric motivated number recall metric function work cover cover simple loss dx fy b cover b ingredient direction specifically real value x b euclidean constant inequality result bound ingredient section metric constant denote p f f also p rectangular clearly restriction function belong existence depend positive obtain set coverage cardinality suppose integer p r r coverage restriction belong v cover cardinality small observe cover cardinality prove exactly induction induction involve interval however elaborate control simple exist dimension every scale rest prove sufficiently pack cardinality subset whenever positive integer satisfying v uk vi vi k di ds vi h let fix vi j vi j vi let let property last distance dx sx integral ii ii vi vi j alone deduce
think least easier think regularize situation clearly reference example computation pruning remove section take improve exploit generation add know practitioner use speed effect truncate model well precision shrink iterative stop neural gradient addition ill pose problem pose distinction sharp divide line effectively assume away modify add smoothness involve modify objective box modify step prune preprocesse inside noise preprocesse empirically randomization include approximation small vector iterative heuristic lead sort computed truncation three via three way problem underlie theory exist difference interested compute smooth regular answer generally database graph many web eigenvector interest mix direction partitioning task importance node eigenvector theoretical characterization let weight graph strong riemannian manifold ad weight appropriate graph variability walk diffusion graph one whose reason lead nontrivial eigenvector lx yy interest compute quality scale medium box solver n I na improve entire ram sophisticated eigenvalue instance look vector preferable perhaps ranking page extensively computation surprising well advantage multiplication laplacian multiplication take advantage distribute indeed relate necessary consideration design constraint much perform rank root assign positive negative probability node evolve dynamic canonical evolve heat seed evolve move neighbor evolve input stay neighbor evolve power time seed walk control quickly choose seed carefully seed could randomly draw vector seed prevent state run value assumption early depend seed set justification typically expensive answer result well formalize regularization approximation procedure actually exactly nontrivial something source code follow program stand trace operation relaxation optimization distribution program versa solution sdp question shown arise regularize entropy determinant certain conversely run act ridge respectively three dynamic reader issue assumption implicitly formally result characterization algorithm compute lead exactly optimize partition objective cut cut balance set weight balanced formulation article interest cut variant say formulation study wide load balance vision theoretical primitive algorithm social network interested find meaningful community empirical social large scale noise property structure analogous cut spectral region locally class atom generally show short length spectral base light size node social scatter plot cluster flow spectral flow except nontrivial gap low axis correspond spectral cluster interest explicitly perform regularization explicit procedure figure present axis cluster interested axis clearly cluster reasonably aside tradeoff basically except two explicitly term algorithm essentially space approximation locally biased observe sense partition heterogeneity empirical function formalize intractable thus approximate flow implicit locally bias near nearly every small certain find cluster nearly linear sort modify usual nice property original formalize locally bias eigenvector use biased graph partitioning modify locality vector seed set parameter version usual nice exact quickly running perform bias like guarantee result seed find achieve good node volume great contain objective medium approach biased wants find cluster operational sort procedure either procedure use prove seed locality diffusion base nearby seed walk approximate modify heat small simply maintain thereby number need probability concentrate change take fast time output base web characterize cluster million though disadvantage complicate interpretability actually procedure problem thing seed root consideration relationship informally provide similarity thresholding commonly terminate equal manner application simply operational interactive practice characterize cluster body use diffusion scale dynamically evolve conclude general discussion modern theory completeness theory guide qualitative computation guide etc thus success problem status neither think algorithm near analogously bound often practice bound qualitatively analogously qualitative insight useful practice realistic reason combinatorial input emphasize apart application nearby reliable geometry like meaningful construction encode question describe involve go worst address question heart perspective heart notion entirely central algorithm computation implicitly either science practitioner implicitly suggest treat consideration rather much secondary practice analysis inferential algorithm implicitly violate exploit bad interest database database practice domain computer term statistical adopt scientific computer learner term datum aspect lie heart output property nearly every apply empirically fact implicitly lead exploit way scale database also inferential predictive several overview computer science adopt roughly area scientific specific make effect reliability thesis application domain share common force extremely way view algorithmic consideration box quite medium perspective scale understand exploit term statistical property bad genetic note couple computational datum seem particularly appropriate generally emphasis detail gap modern massive set analysis store variant traditional database relatively note thus way become increasingly resource complementary ability analyst understand analyze insight work noisy poorly hard imagine quite heavy extent big massive application preliminary main place analyst play test hypothesis unfortunately thing traditional database issue lie algorithmic perspective notion traditional intuitive idea basically objective optimize constraint measure analyst reason apply objective regularize thing interest nearly regularization form apply approximation perspective case scalable statistical inferential non computation either approximation computer sense heuristic decision early stop practitioner implement real often sort implicitly different interested give intractable approximation mean depend approximate amongst practitioner experience surprising practitioner algorithmic perspective perspective datum I three case illustrate phenomenon way involve lead nontrivial eigenvector laplacian second involve compute partitioning third computing approximation locally bias version problem implicitly smoother regular characterize sort analysis purely perspective scalable algorithm ignore likely challenging research front practitioner datum obvious reader perspective rich put look set help financial transaction hyperspectral sense microarray nucleotide web engine video etc implicitly root structure hardware input generation etc consideration computation interest appropriate algorithmic typically keep close enough process flexibility flexible enough intractable inference problematic several way view two member column member array another variant model science machine well scientific view continuous problem consist sort entity pairwise entity geodesic distance permit theory alternatively theory eigenvector notion pair vertex value encode correlation area scientific interest science rather array transformation space dot product orthogonal semantic table relational structure way dna model string matrix graph relevant discussion researcher probably familiar relational various extension rule logical suited datum see therein database application automate record keep system correctness reliability sophisticated rich ever either original noisy poorly dense obvious graph arise arise numerical algebra look good problem fairly easily almost algorithmic tool ram reasonably ram run reasonable sized datum detail access etc graph pose substantial science construct make cpu digital natural science extent area economic science provide source although typically involve something interest crucial secondary motivating roughly pose exist solution depend topology especially numerical sometimes amount input well condition pose ill pose answer draw occur split field compute numerical business tool versus logic lead two algorithmic mathematic use turn work condition problem form solve exactly poorly presence introduce truncation error error represent
multiple similarity unify representation modal technique establish improve knn classifier challenge question investigate knn grateful constructive comment suggestion support correspond author axiom attract relatively little method particular analysis rademacher sum specific derive similarity estimate complexity similarity norm statistic machine g neighborhood mean neighbor depend mean algorithm distance measurement retrieval rely devote automatically learn focus square similarity successfully various problem verification although devote metric learning address strongly frobenius offset similarity deal general metric reduce rademacher sum sample refer complexity similarity complexity similarity method rademacher similarity technique review metric learn term conclude paper notation let trace time denote frobenius dual input output space denote pseudo large bilinear similarity similarly report pair point require main term appropriate overfitte performance expect offset equal optimisation way overcome upper hinge upper order overfitte enforce term restrict emphasize general linear putting error lead metric formulation instance favor norm encourage wise trace analogy consider similarity frobenius encourage rank detailed similarity novel analysis form q analysis true metric z similarity sequel since similarity exactly follow brief comment modify offset firstly example label z problem desire label j x b z j jx j consequently imply rademacher average sum sample related reason refer complexity worth rademacher complexity divided let z b apply inequality hand equation technique b mx z rademacher q contraction property average see lemma inequality side inequality put estimation consequently put similarity replace exactly follow formulation metric without loss focus popular norm frobenius frobenius norm rademacher solution frobenius complexity estimation back dual singular value frobenius norm mention hold norm mixed follow obtain average x b norm estimate right hand side put back let put complete mixed norm mix side inequality apply lemma yield put combine main give key mainly less see x estimation dl mix deal high secondly similarity different frobenius estimation norm simplicity omit paper regularize learn rademacher sum analysis norm input technique rademacher analysis mention question firstly
relaxation lack symmetry outline optimality feasible feasible exist feasible solution tucker condition see vertex compose solution let sample plant multiplier explicit multiplier use satisfy theorem multiplier block solution multiplier provide explicit multipli complementary satisfied condition satisfy desire multiplier multiplier block explicit multipli block vector perturbation know feasibility optimal semidefinite extremely establish show dominate diagonal block high block imply semidefinite sum compute satisfied rearrange use see desire apply choose next delta otherwise c linear perturbation linear know perturb obtain use establish nonnegative symmetry c c column sum satisfy equation equation span nonzero satisfie unique sa si yield r scalar q define provide solution density disjoint clique clique subgraph vertex matrix sample plant accord choose nonnegative exist clique subgraph disjoint subgraph nonnegative show positive fix decompose rest index similarly index optimal sn kx unique contrary lie fact imply w q solution mm plant nonnegative tend sufficiently begin derive entry repeatedly hoeffde variable hoeffde independent distribute satisfy upper holding tend scalar r p apply obtain show substitute yield c r combine proof consequence tend large exponentially approach q eq sufficiently tend exponentially exist scalar combine bound show scalar union mm tend feasible tend lemma state give also sr combine show positive uniqueness tend next ensure feasibility derive upper impose I k c provide scalar satisfied fix equation define system form eq next show choice bind hold satisfy rearrange satisfied small since bound finally also identical remain bind follow solution c consequently combine n bound scalar depend nonnegative nz unique u u establish uniqueness sufficiently q tend exponentially random independent identically I accord distribution choose q correction block invoke rectangular random distribute sample c probability tending exponentially verify heuristic generate symmetric plant distribution entry plant partition similarly compare solution planted sample plant solve multiplier admm comprehensive manuscript recent survey represent apply x minimize augment lagrangian yx successively dual multiplier via seem work subproblem current subproblem let eigenvalue decomposition unitary subproblem admit see admit apply solution moreover solve efficiently stop relative duality gap particular admm augment lagrangian convex program k solve augmented admit simplex projection update apply project duality gap desire variety follow repeat rw model bernoulli variable success probability stop tolerance subproblem solve tolerance admm block successfully recover solution figure average sample plant cluster accord sharp perfect conservative observe trial satisfy repeat bipartite draw plant weight plant return construct successful behaviour heuristic predict theoretical guarantee institute mathematics science research grateful david suggestion especially grateful implement admm trial suggest relevant reference early thank anonymous presentation organization true theorem claim identify significant wide cluster clique disjoint clique densitie subgraph clique ensure clique solution semidefinite consist semidefinite seek simultaneously partitioning graph density result bipartite subgraph disjoint clique empirical goal group similar cluster fundamental play wide retrieval biology optimal depend fitness clustering pose intractable heuristic cluster practical unfortunately evidence usefulness heuristic ensure separate establish ensure certain approach graph similarity two object weight correspond equal similarity representation clique connect clique different obtain identify sense subgraph sum edge subgraph induce clique unfortunately np sum subgraph maximizing relax semidefinite program employ paper although establish relaxation certain input show relaxation include concentrated collection subgraph establish object co aim simultaneously feature call exhibit obtain seek respect expression across gene grouping topic customer preference recommender overview partition bipartite graph dense subgraph feature bipartite disjoint bipartite subgraph establish semidefinite instance correct recover special disjoint bipartite subgraph input give set feature build regard paper heuristic cluster optimization recent paper yu paper appear release sufficiently cluster relaxation matrix construct concentrate heavily disjoint subgraph datum consider early result spirit relaxation hard establish certain convex sum nuclear relaxation result generalize nuclear norm transform sufficiently subgraph matrix matrix clique adjacent every pair node clique subgraph disjoint subgraph vertex subgraph clique disjoint concern subgraph clique subgraph compose clique densitie subgraph equal n subgraph maximize find partition minimize minimize potential partition np hard note disjoint clique subgraph subgraph outlier column matrix characteristic disjoint normalize slight column complement use notation formulate quadratic combinatorial constraint constrain semidefinite replace variable component equal thus vertex subgraph column program q nonconvex program far ignore nonconvex constraint clique define feasible objective equal feasible clique plant disjoint clique ignore sake efficiency due entry unique suppose sum equal therefore point semidefinite approximate partition although think nuclear relaxation indeed semidefinite matrix x semidefinite obtain ignore nuclear feasible set equation minimum nuclear norm prove analogous relaxation clique subgraph ideally identify cluster correspond graph clique subgraph weight connect node within low clique focus clique disjoint subgraph vertex compose disjoint clique entry independently one entry satisfy ji independently satisfy c variable distribution plant clique mean say matrix plant plant model plant disjoint subgraph stochastic add within plant dense subgraph independently add edge clique plant disjoint clique subgraph simply bernoulli follow yield draw plant plant disjoint clique clique subgraph find probability disjoint graph random symmetric plant distribution delta exist scalar maximum clique
importantly construction see orthonormal great column row bound small outline row column infer discussion column permutation admissible compression matrix matrix block permutation respectively row column matrix diagonal still uniformly thus toward ensemble bound condition recovers dimension feasibility focus simplicity exposition condition rise utilize next corollary accord accord generate orthogonal probability suffice exact guarantee course outline throughout presence compression sparsity mainly dominant cf limit increase column recover deal solve non convex optimization accelerate originally success fista estimator stable offer attractive notably addition dimensional arise build eq square zero lipschitz f optimize p quadratic algorithm iterate aspect form convergence rate accelerate result stem possibility subproblem employ initialize decrease geometrically reach terminate drop tolerance detail conclude section respectively p generate f tt augment lagrangian especially suit prove tackle task directly subproblem challenge common auxiliary formulate equivalent tackle associate next positive splitting entail iterative comprise implement block augment lagrangian variable minimization singular thresholding likewise update via unconstraine program whose closed termination inverting fortunately perform line reduce inversion svd compression matrix I ad optimum iterate converge p converge k k b f c x trade convergence ad attain need appropriate predict extensive test considerably choice g ad tuning suit resort component bilinear draw entry column entry demonstrate solve wide range cf c c l suitable depict recover namely value apparent succeed sufficiently observe interestingly hope recover omit avoid unnecessary performance list less accordingly significantly challenging last though therein still possible l superposition feed poorly mainly due inaccurate flow experience change result service term external network failure attack service anomaly engineering network traffic challenge since load superposition next consider represent graph cardinality number layer flow large single intend accordingly connect carry time compactly traffic flow across flow traffic flow periodic period relative measurement suppose instant anomaly column link matrix identical whereas measurement central protocol overhead wireless network power constraint centralize fashion raise robustness since specific isolated motivate anomaly whereby carry rely directly subject algorithmic put general regularize cc line marker estimate anomaly agent place less see comprise flow candidate reference I collect internet internet usa record week internet internet comprise flow traffic multiplication internet acquire trace superposition traffic truth therefore apply ground dominant low anomaly compare whereas anomaly anomaly free unable anomalous flow scope anomalous slot framework enable identify anomalous flow instant assess anomaly across roc curve depict apparent false case illustrate across depicted internet pca effectiveness flow deal matrix via task traffic monitor video surveillance principal optimization local sufficient exact intuitively require incoherent compression behave isometry operate sparse condition matrix ensemble algorithm nonsmooth globally complexity real data traffic anomaly flow several extension provide new challenging research impose room obtain study datum naturally broad e pick satisfy nonzero must rearrange obtain value r l express eq w w upon arrive linearly suffice nonzero triangle b fact assumption value obtain nonzero follow establish property hand side fact inverse b orthonormal substituting bind suppose upper complement term summation e r ignore arrive bound r j p r j plug move rewrite sequel element inside th similar deduce index summation put together r suppose begin desire presence compression stage proof emphasis distinct bind norm r former establish lemma parameter u hold higher less build eq addition recall plug readily note apply sampling distribute bind j eq utilize inequality constant b argument completion omit put infer complete cm cm ann edu pt superposition deterministic low fundamental arise traffic anomaly recovery encounter leverage ability nuclear sparse program confirm say isometry ensemble exact high first nonsmooth provable traffic flow time outperform identifiability traffic let sparse observation deal anomalous flow network vary foreground identify region fmri fundamental plus decomposition absence one signal principal component refer pca sufficient special superposition matrix challenge identifiability presence stable early deal recovery lead guarantee therein cs variant corrupt noise last noisy nothing else pca arguably main contribution recover solve nonsmooth parameter nuclear aforementione norm surrogate respectively albeit natural np optimize compressive put assess capable real practice line first identifiable notion incoherence component restrict isometry constant ensure succeed small sufficiently subset recovery cs condition result duality valid challenge dual procedure distinct show sparse compression draw solve accelerate direction ad claim effectiveness traffic anomaly interesting defer bold letter set operator pseudo inverse spectral singular cardinality scalar th zeros n issue address identifiability matrix space still problematic sparse span space denote singular svd subspace l w notational henceforth element help solution admissible assumption put unique identifiability uniquely nonzero perturbation l belong contradict zero uniqueness h h locally intersect denote simply orthogonal onto row complement subspace addition identity throughout build derive ensure identifiability incoherence holds achieved readily arrive give u exactly high arrive matrix exact sufficient condition state next element exceed context row together last condition guarantee recover solution determine reference therein essence express deal lagrangian multiplier constraint subdifferential nuclear optimality f iii may contain dual r p deal construction dual tight existence special construction compression matrix lt express b arrive b find incoherence guarantee bound follow instrumental assume contain nonzero b v attractive norm substitute yield notational brevity substitute q hold tight back note upper eq cauchy schwarz come column exceed r become upon hold straightforward condition necessary requirement equivalently
fast aspect toolbox inform choice scheme informative take pursuit gp input induce regressor give fully independent approximate attractive conditional recommend development prediction substitute original equation substitution k index subset solve show set training input use toolbox mix match adapt prediction reduce basic without point outside close smooth surface example physical simulation important method recursive cluster random onto split sized recursively median median test split concern hyperparameter cluster cluster joint cluster likely implement toolbox modification sum method cg multiply dense argue method alone cg iteration preliminary ad hoc termination rule see section necessarily examine test termination recommend exploiting iterative provide fast certainly scale firstly linear construct memory hard dataset disk reproduce computation expensive storage computation dense implementation concern many possible regime identify piecewise provide meaningful method truncate series apply machine implementation matlab software open firstly dense program automatically secondly usually eq review time complexity fast compare mean predictor effect compare point lie fixed training comparison hybrid hyperparameter phase final phase argument expect superior explicitly explore experimentally sec hybrid mean compare full evaluate gold could computational alternatively small predictive option square relevant definition time count mind visualize help understand difference unless synthetic alternative compare approximate approximate freedom hyperparameter e approximate likelihood although hyperparameter second base implementation matlab suit efficient matlab hybrid implementation toolbox comparison problem approximation use usually variance dimension require accurately input space dimensionality irrelevant detect equip automatic relevance ard another noise level datum underlie noise average variation much large distance check level hyperparameter model isotropic irrelevant dimension ard parameterization appear fast cg cg dd mm horizontal residual diagnostic iteration reproduce however long measure later directly iteration test plot add insufficient iterative true cg direct cholesky either cg heavily ghz core machine multiple increase matlab intel cholesky multiply focus provide isotropic square share dimension randomly show known indeed standardize regression width direct implementation accelerate mm mm kernel ard power range power small memory intensive local range five hyperparameter leave four four make selection choice dataset individually learn although run overhead matlab always slow four dataset account plot local look plot leave column monotonically test represent predictor turn well improvement take optimize hyperparameter result inferior pattern monotonically approximation trend often operate comment specific function fairly obtain good turn fashion make error difficult slow function select randomly superior converge convergence local detail slightly select induce randomly joint separate training give report result joint consistency give random mm ccc cm mm dataset hyperparameter setting learn fix although learn bad small setting fix outperform produce generative well prediction computer implementation suffer behaviour keep track thousand careful code might reduce problem combine tend dependence could investigate dataset attention control optimize induce point improve expense potential area balance spend select move develop context challenge use require phase future approximation code gp compare try difficult run run give test behave monotonically varied comparison iterative cg dd comparable make provide dataset assume hyperparameter dominant simple hope act gp approximation approximate require subset partition work dependence scale partition scheme small recommend anonymous thank many discussion european community publication reflect evaluate gaussian regression rgb computation neural california institute technology usa ed ac ed ac school st ab uk even straightforward process approximation assess prediction compute baseline subset empirically problem comparison parametric component task unsupervise dependent gaussian algorithm understand recommendation quality obtain varied different approximation baseline subset set dominate publish code encourage baseline structure follow specific practice section issue select approximation direction task dimensional input function test
question tangent space norm point necessity fisher base condition graphical experimental describing raise consistent space identifiability condition geometric piece lie intersection underlie fundamentally identifiable quantify identifiable piece ask interpretability nature relate condition bound incoherence component row latter low completion choice regularization parameter method emphasize respect qualitatively experimental estimator stable observe ideally regard experimental end expressive synthetic experiment graphical reason tie useful upon marginalization g connect observe graphical intuition synthetic hand setup seem generate maximum average degree randomly latent may rank decomposition beyond observe direct spirit condition variable broadly similar domain beyond optical decomposition subsequent wish highlight practitioner sophisticated package develop package invoke convolution scale essence matrix also atomic viewpoint could decompose two suggest fundamentally composite highlight role broadly regularizer trace model gaussian interest algebraic major challenge specifically marginalization categorical understand intractable decomposition describe quantify effect careful reading would like thank contribute statistic attention several across processing comment also challenge variable latent gaussian specifically represent disjoint represent possible note observe variable sparse marginal latent gaussian graphical effectively uniquely individual study algebraic sufficient identifiability give statistical interpretation decompose sample graphical observe variable induce induce penalty induce rank important fisher formulation discussion constrain seem promise certainly investigation formulation develop provable computational drawback existence formulation potentially optima behavior rigorously nonconvex one property go solver well solver scale involve scale several analyze rank apply step follow trace sparse follow component roughly two cycle low piece also reader must vanish ensure pure rank away thresholding sign entry contrast component low rank rank reason vanish broadly analyze possible minimum entry component norm residual norm norm selector selector contain selector relation unclear might suitable selector study bring consistency rank spectral population purely low nuisance application structure quantity imagine quantify deviation low via weak may estimate rule careful investigation suggest component minimum magnitude nonzero entry consistent typically build bind one regularizer require issue polynomially note probability polynomially whereby desirable search sparse graphical consequently hypothesis affect may far refine fast factorization regularizer apply joint induce viewpoint sum
comment provide separately condition result low bound constructive prove sm sm si f function give f tm design sparse estimation relate signal leibler alternative density I make proceed distance distinguish choose interval subsequence pass subsequence arbitrarily high estimate eq kullback leibler lie contradict adaptive sense present difficulty instead bound accuracy kullback equal argument proceed mass allow ok n function q finally adaptive sense single sequence passing contradict denote accord k k contradiction prove apply context j k n make similar argument set j j loss k k n decrease slowly follow c sm similarly attain involve lemma choose discrepancy target let constant include dyadic index r discrepancy since would would since remain true design operation noise subject p region deterministic j k j thresholded k eq conclude little effect coefficient constant k j show term wavelet series lie support say n f parent fashion also make two estimate accurately deterministic depend result j e nj cn eq separately part may effective nj hold part argue f coefficient parent place design jt wavelet proceed induction j c eq n inductive obtain nj prove induction requirement choice conclude attain spatially point choose f may noise nj thresholded fall two eq bound suffice time contribution definition pick instead obtain instead r uniform condition tend old point time hold consider coefficient estimate satisfy nj k n effective quantity k nj depend cauchy schwarz thus constant f may lemma constructive proof b old class deterministic tend n holds tend sm proof wavelet thresholding ac uk g secondary sense design adaptation spatially far restrict paper sense applicable problem spatially improve spatially function likewise design typically advance design sequentially describe adaptive combination adaptive sense literature regression classification f author convergence describe adaptive standard intuition place heuristic justification know regression argue lie rather consider spatially hard seminal kind problem scale estimator function obtain improvement region rough overall describe show obtain rate contain class adaptively class adaptive find standard insufficient achieve might often assumption estimate gaussian sense adaptively significant thus sense nonparametric unknown spatially sense implementation finally describe algorithm varying design move design wavelet give need compactly wavelet wavelet vanish width l term wavelet f estimate wavelet distinct observation yx suppose scale coefficient change apply wavelet ik wavelet design consider literature transformation simultaneously accuracy choice note estimate depend yx k therefore coefficient index nj ease notation j able converge adaptive thresholding hard f hard otherwise fix adaptively select initial stage point space design density choose estimate ensure later point unknown shape thresholde decrease factor least jk wavelet know coefficient accurately fix constant split p exist n simplify point x design require design effective nominal least n grid describe density design regular effective approach require pick calculate tm sm thus wavelet lie must scale however establish indeed similarly locally old follow topology sm discuss begin result sense old sense ny together ny spatially attain sm upon sense f sm sensing sm two feature first rough place commonly measure old benefit achieve r factor sense uniformly f benefit enough rough behind class tm interested f function smoothness least restrict quasi locally old likewise restrict require fundamentally however obtain achieve eq log n bind design rough otherwise improvement easy locally derivative sparsity achieve spatial adaptation sense confidence even spatially asymptotic sm choose simplicity wavelet penalty wavelet set finitely many convert back nx evaluate ik I post thresholding transform uniform predict point describe standard always make nj tend error build make look resolve slightly different definition observation
generators generator extreme subset anchor influence simply I need residual point cone expand depend behave norm support hyperplane cone eqn variant rand variation may expand residual henceforth implement simultaneous matching pursuit greedy response variable explanatory solve r norm count nmf explanatory variable relax solve penalize variant sparse note greedy guarantee well concerned optimize albeit refinement selection separable refined alternate frobenius typically build one hold alternatively introduce anchor extensive scale benchmark parallel propose alternate code respective author tend cause perform less compare remove similar baseline perform guess nonnegative combination show image possible closely separability assumption minor image contain region factorization non learn version noisy select image add pixel normalize recover correspond unnormalized recover control noise separable entry generate dirichlet choose zero std correctly average noise range good robustness anchor perfectly resolve perform competitive turn competitive expand crucial h evaluate human label commonly tf representation document normalize column henceforth identify anchor unnormalized correspond cluster cluster significantly report sake figure perform almost classification obtain document term restrict select dataset inner dimension ten local search mean accuracy feature rest testing scenario induce representation base unlabele classifier select separable greedy outperform dataset initialization rapidly advantage minima audio checking cart absence economic absolutely gold activity approach backward benefit birth propose convert tuned clustering assign cluster refine solution iteration figure converge optimum initialization error bar indicate propose require run traditional nonnegative square consist similar nmf tf popular empirical task previously separable nmf conduct effect anchor column svm setup early show normalization tf require separable nmf predictive quality none white house site house inspection united team anti constitute contact matter united checking grow hour country constitute tune space weather gb gb c e anchor tend table word topic extract eps share distribute parallel parallelism run multi machine machine parallelism use library em parallelism parallelism message performance core intel gb ram run experiment co twitter relate present report greedy depict detect second sequential complete factorization core core believe speedup amongst dense architecture omit brevity state run head characteristic example per epoch execute iteration nonetheless dataset run shorter comparison separable nmf offer scalable minima streaming medium content acknowledgment provide thank condition edu com university college usa ny separability tractable find extreme hull new empirically several favorable relation scalability share say admit matrix nmf provide appeal signal topic geometry comprise non column cone geometry know np hard np hard g treat convex heuristic nmf elegant approach develop certain separability nmf geometrically separability cone small informally modeling problem association separability existence anchor identify across usage separability investigate show uniqueness nmf scale place right nmf assume normalize norm find hull attempt feasibility lp involve scalable robust knowledge estimate formulate exactly nmf robustness general specialized algorithm incremental descent parallel primal dual size asymptotically relate qr noise cast multivariate derive sensitive perform certain preprocessing scalable produce correct separable exactly require closely hull procedure simultaneous matching pursuit sparse variant quite point correctly cone multiple residual demonstrate superior experimental emphasis scalability robustness implementation correctness work short background relevant concept notation convex linear ray generate ray ray generator combination finitely element point see state point finitely possess extreme hull extreme generator vector express nmf contain pointed generator uniquely index non admit separable nmf dimension extreme cone support generate matrix I cone solve problem e residual th vector consist
issue individual estimate multiplied call mean relative variance individual factor simplify result page next two piece grow schedule approximation sample case schedule schedule interested primarily huber integral area distribution run approximation worse issue issue deal involve analyze far possible add assumption change gibbs difficult drawing describe show schedule piece algorithm piece always nonnegative early various gibb together generate return form process operate page different way output combine fractional keep know simplify page initially get enough chance even value value exponential enough concentrate balanced interval independently pair follow let half length interval eq q similarly calculation calculation show involve fall final concentrated tight concentration relative long balanced schedule repeatedly draw copy concentrate pair product oracle generate output time sort successive value hx iv iw complex value algorithm run time schedule well third part produce failure poisson run concern least poisson come bound run repetition always make failure get exactly exponential number k union z make chance least fail schedule next value relative I control interval come fashion increase th slope schedule regardless hx hx last case proof analyze w e w w chebyshev say chance successfully create schedule give union mean z complete average variable sample average concavity jensen inequality
distribute sample approximate bayesian recent filter image fusion segmentation mcmc paper generate posterior result adapt mcmc assume gibbs generate according successively draw conditional associated interest resort metropolis hasting generate proposal reject generate sampler metropolis gibbs sampler mcmc base see accord generate whose voxel quantity compute mrf gibbs finite use loop accord dependent discuss generally coordinate gibbs mh move sampler depend acceptance previous term present require possible intractable likelihood introduce carefully tractable sufficient target auxiliary discrete state tractable statistic generate introduce variable mh proposal replace simulation discrete auxiliary despite explicitly exact likelihood free mh acceptance candidate vector addition sufficient limitation address abc mh henceforth abc mh abc mh use statistic restrictive tolerance distribution discrete generate accord mh auxiliary constitute limitation perfect sampling costly million alternative gibbs state infinite perfect regardless small explanation choice candidate approximation around improved expense result abc correctly even small explained previously define statistic function generally fortunately mrf belong statistic neighbor voxel sufficient exact work euclidean distance crucial drift center eq period strategy walk medium pixel become sharp walk summarize know lastly incorrectly observe fix incorrectly either htb htb correct c c true mmse mmse mmse htb conditional associate th order assign fig map burn period correctly label jointly column fix ease interpretation second scenario highlight blue observe incorrectly considerably show mmse propose good agreement show map simulation actual class label mrf propose fig incorrectly classification obtain htb c c classification htb mmse mmse mmse apply acquire depict acquire resolution sum mrfs extensively model mixture detail experiment class result empty noise preserve contour markov burn period remove interpretation observe optical clear boundary classification inspection topic depth problem use couple mrf class report mode image acquire probe contain outline rectangle vector mmse result fig show contain layer produce boundary within half observe obtain fix corrupt hand spatial yield fig viewpoint surface tumor cut towards deep htb gray light gray fig fig c gibb jointly unknown bayesian image set cross validation standard require normalize within method abc likelihood hasting intractable rejection apply experimental obtain synthetic unknown incorrect value successfully relax reversible jump include texture field project region acknowledge optical grateful parameter jointly mcmc perform normalize constant estimation hasting obtain synthetic unknown value know incorrect successfully apply estimation sampler image processing field mrf recognize tool capture mrf classification segmentation generalize vector segmentation precisely define whose hereafter generally process fully unsupervised monte carlo mcmc powerful tool square mmse analytically mcmc mcmc method directly indeed compute design recently expectation avoid possibility eliminate define posterior although analytically convenient lead poor notice normalizing implicitly depend paradigm base analytical development strategy combination state bethe approximations sampling langevin mcmc recently expression compute analytically recursive mrfs base integration expense estimation arise apply monte integration approximate generally recent precisely technique recent statistic hasting introduce random auxiliary distribute hasting require computing admit unfortunately suffer acceptance ratio application auxiliary several sampler however substantial costly image application alternative auxiliary monte carlo sampler arises provide possible normalize free substitute intractable likelihood metropolis hasting auxiliary auxiliary step require distribution although abc study density increasingly regard processing contribution propose abc included particularly adapt show easily exist previously remainder hybrid gibbs posterior bayesian synthetic vi report th voxel n make associated stationary class fully follow characterize map observation work enhance relate amount spatial adjacent introduce coefficient exist segmentation priori consider jointly framework conditionally associate characteristic mrfs mrfs conditionally pixel pixel conditionally properly pixel voxel neighbor neighborhood structure neighborhood eight near represent neighborhood near voxel neighborhood consider clique horizontal neighborhood mrf variable indicate voxel whole random obtain pixel nn result
loss determine regression loss nonsmooth differently work guarantee nonsmooth satisfy necessarily magnitude objective convert objective bregman indirect approach regularize relation vector draw minimize strictly full feature scenario linearly separable case achieve small tend direction drawback usually candidate thereby aforementione nevertheless explicit sparsity addition regularizer logistic point denote sparse remain matrix self suppose assumption every none matrix henceforth imply adjoint index obtain furthermore thus possible choice union r regularize constant probability implication grow achieve provide include depend knowledge extreme average independent psd matrix well achieve glm error go eq hoeffding variance previous subsection also probability interestingly obey synthetic algorithm sparsity constrain induce one though optimize difficult characteristic model accuracy dimensional datum independent autoregressive parameter datum adjust data pursuit logit effect effect additional logistic elastic mixed available produce fair net measure label furthermore net regularization label logit omp include average evaluate parameter furthermore evaluate see correspond indicate net elastic logit omp convex method nevertheless set step application involve type identification gene interested cause greedy convergence guarantee base notion restricted smooth cost stable restrict linearization cost generalize square known isometry recovery square concrete requirement logistic function lipschitz incorporate broad medium scale optimization whose hand especially running dominate grow deterministic ordinary improvement gain coordinate introduce beyond scope analyze establish series operate lemma lemma turn main eigenvalue large symmetric definite respectively concave convex jensen yield q imply symmetric index thereby q tt operator jensen introduce q coordinate apply inequality proposition q use ft invertible multiplying c ss easy b therefore yield recursively part basically deduce proposition inequality divergence prove lemma prove proposition l inequality definition bregman divergence obtain bound eq yield inequality subtract q expand obtain side always inequality subtract sum discriminant desire denote corollary write eq current q proof vector involve th algorithm corollary remark constrain applicability learn processing feature compressive vast constrain application estimation linear fidelity relatively effort sparsity involve quadratic pursuit approximate minima linearization paper produce approach generalize known arise compressive sparse compressed demand decade bioinformatics finance datum often use infer circumstance inference ill significant structure exploit regression acquisition regime structure usually cast formulate requirement computationally optimization extensively receive area norm regularizer induce nonlinear elastic frame analyze provide work restrict function domain emphasize sufficient condition generally elaborate coordinate descent minimization formulation increase stand alone operator nonlinear consider residual error show hard formulation scope apply forward impose target multipli never multipli stop condition multipli magnitude pursuit sparse arise nonlinear prove characterize sparse canonical hessian analogous analysis guarantee smooth smooth derive function symmetric hessian prove unless complement letter v nonzero depend indicate equal except coordinate e capital transpose restriction identity indicate one outline overview work formulation present provide guarantee prove example algorithm apply logistic present gain attention title pn measurement consistent absence entry every solver np solvers proxy approximate solver require equivalence hold measurement high ideally measurement necessary interior method scale several another category cs convex orthogonal omp pursuit iterative subspace pursuit name greedy constrain least square residual iteration successively position zero exhibit convex though requirement accuracy meet restrict rip understand satisfy rip small number cs show rip accurate reader guarantee algorithms cs provide incur acquire variety desirable broad everywhere rely induce sparsity show et statistical decomposable glm shown establish rsc basically curvature curvature descent constrain guarantee different rsc provide majority regularization bound albeit implicitly logistic regression recognize sufficiently short desirable assume glm restriction restrict estimation perspective mention generally perhaps corresponding mention impossible desire convexity property know rsc quadratic location may investigation rely proxy fidelity appropriate acquisition application right significant advance prediction generic cost replace desirable approximate pursuit generalize broad course even simple combinatorial np cs obey certain accurate guarantee analogous rip framework smooth introduce notion restrict hessian smooth cost introduce linearization bregman divergence restrict canonical subspace somewhat restrictive requirement compute c algorithm maintain estimate first evaluate nonsmooth large choose index merge current sf c terminate change specifically gradient reduce proxy step reduce step form study minimize invertible step relax restrict approach large descent thresholde square variant characterize restrict function linearization rip standard basically require function stable restrict continuously differentiable q sparse say stable kf twice continuously condition number cost square condition sparse vector special condition equivalent rip function canonical subspace everywhere determine eigenvalue across therefore cx k tx diagonal norm ct everywhere convex notion nonsmooth function subgradient restrict subgradient
convergence rate method step accelerate sg default inner successive gradient stay constant convergence remain sublinear sg gradually transform achieve convergence specialized weight strongly numerically treat pass increase sg batch iteration rate oppose sag identical sag cyclic rather sampling consequence convergence convex derive treat pass proof involve lyapunov experiment show sag robustness suitably fast also emphasize sag perform show assume gradient lipschitz fairly average function strong regularization term transform strong convexity exist initialize result depend gradient norm optimum denote finally convergence consider internal constant sag proof sag achieve sg method albeit advantageous sg pass example large sag use sag assume iteration sg subsequent sag iterate contrast sag sg experiment sg rather minor sag appear analyze performance convergence know example basic fast sag much number indicate sag substantially proposition primal wise case rate sag reduce excess cost problem depend rate true may sag step size far indicate use lead sag iterate lipschitz describe sag requirement prohibitive form since store scalar rather full reduce cost sag update parallelism sg example mini batch sg iteration mini mini batch sag dense storage requirement since batch mini batch reduction storage early iteration sag uninformative far modification analyze sag outperform sg sag objective like rather approximate recursion regularizer proposition satisfy conjecture size start lipschitz instability allow high smoothness multiply extensive sg dataset dependent tune em full cubic step satisfy strong wolfe condition hand nesterov method base publicly limited newton method linear art sg projection contain regularize dual method sg sg average strongly sg nearly constant describe either sag estimate logistic section sag ls sag ls initialize theoretical suggest strategy sg sg method pass several sag pass l bfgs indeed website causality website website although differentiable regularize half testing add regularize standardize one testing effective pass datum measure divide practical pass cm cm colour experiment sg carefully sag compare basic sg accelerate multiply size stochastic describe use size iterate gradient cyclic average among power method bfgs sag show select step size cm bottom view colour observe trend performance carefully size substantially pass sensitive steady bfgs eventually pass sg sag seem make steady progress sg method sag reach cost often translate reach line search fix perform similarly method show surprising sag deterministic large size sag learn datum equal derivative optimal regularization asymptotically negligible regular batch enough high covariate aware satisfy parametric contribution though setting speed decrease step size pass unlikely testing cost sag sag iterate reach sg sg slow terminate step paper sag iteration sag advantageous termination sag termination specify sequence sag sufficiently le schmidt european fellowship science engineering proof proposition compare coordinate sag regularize pass proof convex function gradient denote unique stochastic k probability nf ix k definition n concatenation block diagonal proposition prove step shall lyapunov rate shall dominate case proposition show stochastic desire proof denote I generate choose affect pass datum stochastic gradient run sag section provide descent sag one use sag variance denote k take average convexity quadratic term cc denote compose diagonal diagonal diagonal throughout moreover side third side equal equal last expectation summing term n b rewrite conclude proposition sag size lyapunov lyapunov quadratic n n bound term positivity guarantee convexity third gradient line stem negative vector use condition convexity convexity give since function dominate x complement verify dominate eq eq lyapunov rate sag shall lyapunov goal express term positivity guarantee convexity property q get complete square convexity follow finally get finish positivity previous gradient prove minimize ng x stochastic use descent constant gradient initialize truly reflect apply sg sag use eq formula lipschitz maximum square column similarly dual convexity argument large fast speed determine basic size q fast determine independent primal prefer yield dual primal dual size iteration could apply cost coordinate apply descent happen variable sufficiently cost coordinate dual depend sag sag rate coordinate slow determined sag highly sag l pt pt plus minus pt schmidt sup paris sum convex incorporate quickly problem machine redundancy example class structure sg method allow learn sg sample average focus ix b ia example binary smooth extensive include smooth use minimizer cost fraction fast rate method
advantage importantly invariant vary component allow across task learn contrast change different state sample additionally free distinguished rkh make operator scalability note linearly mdps problem applicable discuss formulation path infinite horizon control section integral wiener system control act subspace good respect terminal expectation start require scalar give path dynamic consequence express make challenge point admit finite specifically use take dynamic cost hierarchy evaluation express term necessarily treatment basic concept function positive definite kernel embed constitute direct commonly consideration interest lie give allow express rkh although conditional embed generalization specifically variable q rkhs product scope paper satisfie argument conditioning directly proceed introduce delta treat formulation analytical immediately practical discuss supplementary function use e rkh embedding apparent convenient allow pre computed evaluation require evaluation kernel term expectation practical regularized kernel turn joint distribution representation empirical obtain denote hadamard take hence representation importantly note weight recursive use fine furthermore function material transform lead poor characteristic also overcome laplace mode small basic drawback relatively inversion subsequently per additionally policy overcome alternative weighted supplementary employ gram schmidt reduce novel choose discussion address opinion efficiency instance repeatedly optimize single reaching movement interaction change generally monte method require sample trivial state sampling case allow efficient repeat necessity evaluate infeasible estimator base ideally arbitrary set particular change importance estimation g reinforcement incur transition expensive explicit expect turn distribution sample agnostic often appear concentrate guide context incorporate amongst task execute task situation invariant relate cost look stay balanced path integral expectation path invariant component stay induced well objective abstract appear framework explicitly implicit cost double demonstrate monte close enough highlight task concern move velocity coordinate affect position aim position final avoid dimensional weight consider I true reinforcement learn access sharing time sample start low supplementary exponential kernel median comparison firstly monte reinforcement see b approach carlo compute knowledge true policy starting position monte capture optimal lead without approximation similarly new starting would affect dependence sample refer highlight recursive sample sharing equivalent embed yield empirical estimate specifically apply finite representation rkh
neighborhood filter filter pixel neighborhood nine parameter eight estimate likelihood looks give base within filtering whose si tie ti hermitian definite increase choice divergence divergence simple scale good prop er sample respectively eight test al derive distance yielding express transformation possible visualize color color decomposition green channel laboratory evaluating generate look counterpart use figure figure original figure window whole albeit instance present hellinger fine detail preserve employ directional instance structure within forest maintain image selective produce strong fine linear texture proposal decomposition assess iterate filter close equivalent homogeneous com sense complete produce consist hermitian matrix suffer noise literature preserve analogously cone color propose describe type type smoothing technique visualization detail visualization sense achieve prominent imaging coherent provide image interpretation segmentation analyse lee principle preserve signature filter average neighboring ii filtering execute element homogeneous neighborhood image good efficiently simulation statistical covariance organize smoothing distance complex wishart visualization kind imaging result intensity relative phase datum possibly vertical wave signal characterized medium complex multiplicative circular module determinant transpose besides hermitian definite characterize
ideal assignment sequence redundancy probabilistic generating quantity extra bit would code assume ideal exist statistic diagram perform average partition average distribution form employ average temporal partition know complexity propose partition still guarantee redundancy member method reduce temporal partition partition utilize advantage kind partition begin partition although restrictive temporal partition temporal partition possess later exploit parameter define define map tree show partition notice weight datum number associate depth weighting application special contribution specific computation equation clearly equation lemma run data main overhead bold tree update kind traversal need summarize stack significance many base memory describe space compute routine change context bit number bit bit representation differ effectively run keep dx depth model ni tw ti weighting limitation minor state omit base almost partition arbitrary model almost partition complete temporal particular precise terminology end segment begin partition say start segment abuse binary construction add ba bit q verify base source redundancy piecewise class whose redundancy negative monotonically non redundancy source stationary furthermore combine see partition contain segment redundancy definition concave jensen gb ga aa b complete part partition half tree leave segment decide include result require choose restriction n I overhead due advance appendix time depth tree extra straightforwardly modify idea size one copy relevant boundary alternatively justify weighting overview consider successive trial denote observe one kt use weight code n use next stationary source accord segment generate redundancy estimator base upper show technique perform segment relative complexity synthetic average iii heuristic exponentially decay non illustrate number effect report average uniformly point segment get slightly linear algorithm runtime already time book paper replacement compression binary source performance context bit consistently improvement file already considerably slow method also alternate set memory model empty switching model switch compose index map sequentially add string define convention symbol boundary bound memory bit possible switching define state redundancy switch x x number logarithmic tracking generating difference method prior towards small point prior arguably job ensure together however line remark tree unfortunately find would require follow derive hold partition tree weight efficient automatically piecewise extension closely redundancy match literature thank helpful technology note trivially equation depth dx j x j x x third step use inductive use prof allow derive low note ax dx x dx take negative ex corollary example introduce weighting algorithm piecewise source bayesian average large segment prior compression code distribution competitive redundancy exhibit superior trade
limited available logarithmic regret corollary di understand reward moment order surprisingly moment refine empirical achievable armed introduce agent action select one agent observe independently past survey variation vast majority assume gaussian moment generate variable factor reward hoeffde similarly assume ucb variant basic strategy satisfie guarantee ucb eq reward sub order know bound distribution weak sub suffice disadvantage gap become sub tail regret heavy tail weak may even infinite moment logarithmic though gets refine robust construct upper confidence strategy factor dependency availability guarantee various rough behind ucb sum estimate interval sub common factor arm q order accuracy know significantly wide strategy appendix property heavy tail handle tail replace estimator need like show mean precisely need let finite suppose also factor interestingly may recall estimator subsection satisfy shall requirement mean define reward arm ucb provide distribution assumption exhibit mean suppose distribution estimator regret least interesting v n assumption suboptimal precisely step prove notation round equivalent either three inequality v assumption union bind n false conclude hand old next regret heavy tail truncated estimator mean moment q bernstein computation conclude follow distribution satisfy regret assumption distribution bind even worse robust ucb find remarkable impose strong moment still regret replace armed satisfy suboptimal problem furthermore fix exist satisfy take bandit distribution bandit two bernoulli formally follow kullback divergence directly along computation arm run bernoulli straightforward satisfactory invariant strategy shift reason center quite around raw indeed possible truncate possibility mean estimator section next section divide data one standard median empirical show certain block size property robust ucb let let computed empirical distribution binomial next consequence rather moment distribution satisfy ucb median satisfy elegant well optimal however good term unique prove additional requirement framework robust strategy upper large distribution proof arm reward maximize policy satisfy modify satisfie well median validity heavy tailed bandit reward arm mean
phase synchronization coupling identification include phase shift couple identify synchronization synchronization research phase synchronization weakly relationship long history back synchronization decade interact study context nonlinear dynamical type synchronization self weakly couple uncorrelated coupling phase whole synchronization dynamical synchronization experimentally electrical circuit biological phase synchronization synchronization regard couple single activity couple activity cell area abstract year single eeg study human demonstrate phase synchronization memory synchronization phase brain support memory memory act communication mechanism gamma synchronization grouping synchronization investigate discover phase activity within pattern synchronization pattern review synchronization synchronization frequency prescribe also ignore work sound mathematical theoretical treatment tend form theoretically couple model dynamic phase test synchronization coupling identification include argue theory good investigate drive key describe physics transform spectral transform try directly process synchronization theoretical synchronization insight research ground process concept discovery aim trend consumption short run exchange level market introduction develop researcher become develop theory diffusion give heuristic elementary phase ti hilbert phase error white drawback adequate external keep phase increment simple case iid drift substantially phase evolve brownian motion stay relatively model exactly concept remain heuristic formalize appendix phase synchronization focus synchronization interact stochastic view brief describe statistical phase definition synchronization show aspect want address may proceed model example mainly start synchronization start synchronization traditionally synchronization define hilbert rotation center phase alternatively one system periodic represent amplitude latter signal determine transform hilbert obtain hilbert cauchy principal amplitude method estimation transform model particle filtering next couple phase shift usually know practical behind definition rescale stay move synchronization synchronization synchronization coherence close indicate synchronization often synchronization several article advance distribution present directional statistic phase measure find directional surrogate mutual phase couple drive mention many directional type synchronization process provide introduction large population interact phase white discretized diag mention fulfil specific us insight synchronization network namely obviously synchronization sometimes load phase synchronization synchronization testing representation matrix full synchronization intensity equilibrium lead test addition vector drift equilibrium phase transform meaning contain example section come linearize synchronization relation eigenvector synchronization stochastic structure eigenvalue synchronization almost synchronization process situation phase synchronization non test look fit addition parameter synchronization relation coupling etc synchronization difference replacement difference employ phase employ phase zero phase happen section see usual paper autoregressive identically distribute random restrict var e call process integrate stationary univariate thorough specify univariate synchronization long straightforward calculation decompose full stationary representation multiply stationary show move equilibrium force illustrative another move l rd r representation process component component process side walk multiply stochastic trend drive representation back occur illustrated detail synchronization belong correction eq stationary obtain x obtain true relation next back drift process deterministic contain use phase intuitive dimensional walk discuss simple intuitive restrict take identifiable synchronization know synchronization mean reduce suffice describe joint example confirm since increment usually correlate note one phase relationship test distinguish relationship synchronization synchronization relation inspection system usually factorization synchronization relation shift represent direction couple meaning section lead likelihood theory ratio theory test determination decomposition multivariate situation much challenging classical ratio integrate lead example strongly recommend reading presentation read reader keep mind inference lead drift role drift term least consider maximum system essentially first step remove eq lead view canonical denote equilibrium correlation increment thus belong synchronization zero belong synchronization split belong discussion drive force way determine call recursion call hypothesis significance level stop test hypothesis section test test standard know disadvantage test common restriction power alternative significance device construct test asymptotic superior inferior strongly p uncorrelated prefer test test also single conditional assumption solely error unknown follow important residual lagrange multipli test trend finally test next use synchronization idea context phase synchronization describe stationarity propose stochastic difference always back stress discuss model signal phase increment phase integrate sometimes stochastic trend occur trend occur see integrated synchronization phase process linearly non identifiability phase synchronization also rank relation phase integrate term usually drift increment var process still phase pt case synchronization synchronization give may relation may linearly example recall process stochastically error correction phase synchronization relation uniquely reflect r synchronization span testing view phase space difficult equilibrium process provide drive remain nothing know priori phase must produce phase cause external mathematically trend I intercept trend stationary regard phase cosine cause daily specific higher sufficient conclude additional fit var intercept trend test lr generate trend phase shift know testing phase synchronization collect approach present phase instead phase regard adequate regard process phase example often test lead stationarity instead consequence finding author maximal synchronization perfectly proceed phase ignore phase phase mean model unobserved phase model challenging consuming give original phase use phase prior specific conduct lr prefer bottom situation synchronization one hypothesis synchronization particular result slightly however lr test perform bad estimate significance significant calculate difficult testing coefficient software synchronization direction coupling investigate adjustment drive force synchronization force mean ratio whether hypothesis adjustment chapter software appendix detail additional system coupling strength corrupt couple inclusion couple integrate step burn point subsequent strength make realistic stationary estimate trajectory plane rotation symmetry plane rotation center exist figure phase apply window estimation plane coupling strength b coupling e corrupt case noise system I phase plot minus plot stochastic h indicate fact slightly ascent ar part coordinate line corrupt trend phase bottom ar simplicity discussion reject lr rejected accept thus reveal phase eigenvector system test c synchronization index coupling strength dash statistic critical test hypothesis reject plot determined value nearly indicate plot explain turn see appendix agree apart jump difference lr test penalize highlight plot whether black red blue procedure specify synchronization decide synchronization line dot critical coincide remarkable test tailor penalization mention include lr little synchronization relation dimension tendency prefer lr given analyze directional coupling std significant except agree strongly reliably identify coupling residual purely correction coupling parameter informative discussion modify system illustration theoretical set nice obtain calculation value obtain representation representation meaning get active soon equilibrium case process coupling correctly implicitly ahead mean phase mainly direction precise phase identical significantly deterministic part trend one keep mind hilbert smoother investigate vary frequency point connection theory process phase detail coherence process equilibrium relation shift couple couple synchronization network phase synchronization system regard essential coupling new come enhance coherence triplet investigate confirm potential synchronization perhaps synchronization point understand fit towards equilibrium synchronization little seem interesting important phase cover phase consequence couple calculate whose move regard contrary recognize inclusion structural shift occur close guess situation occur set topic topic economic framework detail able chapter iii possibly mutually simple amplitude vector meaningful phase synchronization prefer write form see shift trend full phase transform term smoother estimate technique course effect hilbert transform significance need investigate scope everything smooth one become conditional rescaled asymptotically argument regular stochastic unit lr heuristic inspection proof test stochastic term trend remain estimate phase hilbert transform original unobserved consider heuristic synchronization relation remain group different phase synchronization check stay transform obvious integrate stay integrate phase stationary process transform hilbert constant cycle reflect trend hilbert emphasize argument mathematical stay highly derive proper synchronization likelihood ratio hypothesis clear derive strong particle filter base furthermore misspecification generalization space signal periodic function typically unknown need may amplitude var nearly clear increment positive increment happen mainly overcome case autoregressive duration extended bivariate generalization lag smooth aspect module http www com www net package project web package index http project package em phase hilbert pt type none pt chapter correspond use appendix pt vs beta beta type drift specification correspond statistic quantile give quantile correspond lag ct beta ct lag lag em critical segment plot remain seem aspect simple successfully phase synchronization reason modeling choose aic synchronization bic test bic reasonably synchronization situation investigate develop least present highly situation true accordingly bic lag nee high approximation
rule soft rule accuracy compare predictive soft quantitative trait molecular provide prediction phenotype numerous genomic community ensemble genetic interaction marker highly contain short day line available variation website second evaluate display r r size l soft l soft rule rule output variable might use two combine lasso combine input unsupervise weight least obtain kernel rule model develop ensemble interaction importance dependence tool soft example rule importance rule involve interaction naturally ensemble big adjust scheme thousand loading time acknowledgement award section property article soft importance soft logistic rule soft boundary various soft ensemble hard approach challenge view provide simultaneously focus stable ensemble whole influential bag bootstrap aggregating involve sampling model produce although model use classification response tree soft ensemble attack problem rest post explain convert section soft conclude comment discussion ask predict variable restrict model consider framework additive base approach arrive involve obtain use combine code eq bag adaboost boost special generation procedure bag value boost fashion takes recommend mn w decrease include final learner precede section article indicator region variable tree terminal input say r lk lk subset categorical space terminal many terminal maximum tree rule logic rule call boolean logic statement involve construct logic cx logic construct input logic logic rule show logic express ensemble weight l ensemble random forest bag boost hard value rule include interaction variable explicitly model good subset term minimize perfect response explanatory likelihood finite deal separation approach recommend produce bias correct jeffreys fisher logistic ni diag g g iterative utilize linear product build model structure smooth model use calculate hierarchical child terminal function fuzzy decision perhaps model utilize tree build cart rule soft approximation rule n l estimate lead final soft ensemble prediction ensemble hard soft make slow however fast fast programming language part accomplished summarize soft rx convert rule soft sx soft additional soft hard set line recommendation several algorithm recent solution minimize rule final
detail start simulator hypercube design take al optimal vary simulator sequentially add ei evaluation ei vary complexity e estimate minimum simulator output scale lee benefit ei uncertainty appear guide sequential design utilize search trial point global point place uncertainty e global shape repeat time add point ei figure present ei ei ei performance realization h expect outperform shoot slightly outperform ei clearly gp ei run estimate global yx global minimum find replicate optimum yet function spike make densely spike sequential add ei median realization run global outperform naive shot performance ei ei result mean ei rate ei ei dominate ei sometimes wrong spike mean paragraph prediction encourage ei detect discovery region add follow ei summarize ei method median curves global attain replicate well suit negligible ei candidate engine situation localize surrogate still competitive optimum run simulator procedure configuration impact et suggest depend complexity simulator run design long notice piecewise utilize tree example employ average effective continuous experience effective carry run course possibly nearby simulator possibility response effective engine evaluate ei candidate since computationally identify relate sake current allow input comparison explore alternate way ei find provide support student award discovery centre hypercube package h additive c e spline bayesian application computer model lee case model computer optimization experiment guide stable gaussian mit multimodal multiple global optima th international conference optimize sc thesis mathematics ns computer mathematical represent physical computer experimental virtual phenomenon think consider goal simulator feature simulator simulator minimum utilize version simulator ill application tree improvement universe sufficiently complex scientific impossible impractical constraint phenomenon enable study place electrical observe mathematical computer take input consume input situation relationship simulator model model predict surrogate consist world phenomena simulator simulator run value result tool minimize sort collect sequentially accurate location popular overview response closely model krige interpolation quite rather strong stationary response entire limitation lead extension lee recursively partition rectangular model upon refer tree give capture relationship also deterministic noisy monte turn enable computation guide design identification organize surrogate discuss experiment briefly outline surrogate power improvement et al review adapt playing conclude future brief detail also discuss lee assume simulator value error output vector shorthand terminal return different tree terminal rectangular denote terminal flexible use variable model capable incorporate adaptively individual tree place split point area allow nearby say interpretation place gps place interesting receive mass gps way biased low continuity making response spatial correlation gp imply stationarity enable curvature locally model inferential tool importantly uncertainty ingredient design wide real considerable noise simulator error choice emphasis modification surrogate modelling simulator follow general distribution specification priori ii node generic computer believe quite observed tree give point accomplish scaling recommend distinct within shrinkage g output less prior mass accomplished chi recommend percentile default percentile specification response mcmc consider motivation lee inclusion residual approximate gps estimate specification parameter ensemble allow fine grid axis discard first quick observe cover involve put mass terminal large tree later make base briefly specification replacement offset instead terminal lee terminal node accommodate model computer modification setting default r package c indicate predictive stop specifie model comparison package r describe power modelling description give wang basic predict produce km wide high generate approximately power placing collection suggest energy consider simplified simulator place control simulator location along scale shape considerable flow location area thus optimization problem display panel simplify study exercise pre evident spike panel panel display prediction mean power display rapid fluctuation deal consistently observe fit case model unable consistently introduction consider kind optimum computer simulator switch maximization let improvement unobserve less maximum form expression normal first local seek improve currently identify minimum place uncertainty nearby variety similar involve predictive unobserved response account parameter employ idea mind calculate framework unobserve incorporate ei calculate ei directly straightforward get posterior ei incorporate uncertainty strategy calculate ei new ei maximize possible simultaneously hypercube point
description comparison simultaneous development microarray techniques genomic bioinformatics microarray gene expression mass simultaneously study thousand biological feature protein nucleotide snps challenge determination whether biological differentially reference term biological address use throughput include david review far modular concrete precisely whether biological category reference exist approximate test test test follow gene differentially express gene microarray go category statistical representation total apply value prevent positive incorrectly reject hypothesis gene control fdr value biological category reliably category application well false biased estimator application microarray gene expression microarray thousand concern category appropriate reliable adapt nevertheless broadly protein biological pathway section arrange concept gene cancer conclusion recommendation preliminary describe state de go category binomial proportion go category nuisance unconditional consideration represent go total go nuisance probability mass nuisance conditional justification concern nuisance interest statistic nuisance parameter alone contain variation independent minimal nuisance therefore statistic little parameter false discovery go q let ii multiply odd precisely category go category alternative bayes significance category represent differential go modification go tend maximization likelihood usual microarray probability parametric let mass observed category satisfie odd go category equation go eq mle normalize equation go next develop stand base odd hypothesis alternative eq mle member denote minimize regret mass define estimate combine equation guess category ratio bayes mle breast applying cell human breast cancer contain u file microarray gene gene microarray criterion de de gene variance gene estimate theoretical hypothesis bias normality expression presence absence hour exposure define go go molecular least de thereby go sided mle display bayes factor indicate tie h nan hypothesis alternative component component grey comparison grey commonly threshold evidence simulation study de conduct separate study behind assess symmetry odd ratio equation odd go category asymmetric randomly fisher exact odd ratio sequence time performance simulation estimate low moreover bias affect whether odd furthermore attain equivalently category differentially bias conclusion go category thousand gene study medium number go go category adapt mle medium go cancer sensitive component medium go breast cancer mle formula show sensitivity left plot component plot component nevertheless figure bias even mle number go category estimate category equivalently component freedom recommend number go interest mle simulation category finally recommend
benefit life challenge arise challenge utilize code hundred target follow assess introduce validate estimator estimator demonstrate policy robot constitute artificial intelligence stream generate interact period big diverse stream approach acquisition verification reinforcement life mobile expressive represent interaction al represent reward pseudo termination condition policy semantic intervention scalability computational represent massive mobile robot several operating form consequence policy great learn parallel learn sequential long learning challenge policy scale life sampling td size computation per memory available life long robot arguably make life recent highlight big et policy robot robot hundred prediction policy policy require place computable progress policy target time robot scale formulate agent model discrete system observe characterize note access environment current environmental step transition produce conventional predict discount special receive formally particular policy control discount factor predict action select reinforcement estimate common use action policy policy value behaviour behaviour policy td become unstable fewer reliably td policy function incremental td except secondary complexity td projection onto weighted bellman operator discount whose policy thing policy capture architecture pose generalize predict th scale predictive question happen behaviour different behaviour effect velocity action substantially dramatically learn million distinct easily sensor architecture consist triangle update prediction set sparse previous parallel mm provide parallel architecture et desirable characteristic run linear scalable distribute ability learn modular specification behaviour depict enable compositional td question support scale prediction physical build mobile robot see sensor detect entity ambient field internal status acceleration velocity robot mobile operate raw coding produce binary constant comprise pair sensor code produce always white fair evaluation present policy evaluate period direct evaluate robot update behaviour five reverse counter stop new execution behaviour execute action normal occur select one action policy five complete spend second centre behaviour robot spend learn generate reverse stop sensor question discount robot pseudo termination scale value specification step result learner parallel correspond match identical total computation ms entire ram dedicate wireless present prediction hz axis stroke profile return yield percentage non shape test evaluate online performance precisely execution truncate return test normalize configuration normalize square use weighted represent percentage question illustrate policy prediction physical robot scale share identical divergence substantial return learn important demonstrate policy experiment test considerable mass infinite estimate evaluate place scale policy frequency execute trade change step frequency ensure closely approximation propose measure relative ignore question never go quality representation common technical converge provide performance evaluation rewrite expectation use stationary expected approximation expect backward view trace additionally target agree incremental sampling current exponential trace update second storing first compare chain end episode begin middle transition incur determine scalar expensive mdp incremental sample estimate well experiment behaviour move measure provide simple compare change learn episode secondary trace online change simple chain provide match magnitude episode primary weight vector random online estimate online aggregate curve average task robot chain experiment run exactly prediction six hour time step effectively effect every question reaction error measure quickly vector time exhibit trend bellman storage predictive illustrate important validate policy behaviour estimate slowly occur also incorrect affect long quickly scalar estimate significant learning efficiently estimate consume twice estimate limitation prediction question number policy magnitude policy maintain action action parametrize policy copy otherwise offset random random assigning independently test architecture scale policy robot experience question policy correspond second evaluate learn cycle ms ram requirement result many learn strongly availability prediction environmental square bellman heavy denote estimate curve learn progress result provide policy online performance measure literature idea prediction along option temporal abstraction policy approximation develop et exhibit slow learn progress policy active relate al real sensor
common identify combination subject throughout describe study motivate use identify quantify upon cd cd cell present recognize recognize activation future constitute total cd cd production activation match label cd cd cell define must collect ensure rare subsequently cell maker threshold phenotype difference subject generate subject difference real might detect analyze hoc rule method test contingency test subject share observation develop name cell explicitly model binomial across place unknown proportion binomial multinomial dirichlet mixture sharing help count specificity multivariate model help detect cell organize alternative method model multivariate use expression observe subject two un cell negative measure cell classify positive combination count un cell vector define gene expression use set generate part trial dna modify boost boost goal assess multiple analyze consist il measure envelope ease cd sample peak subject cell isolate flow subject tm measure array pair control present vs handle multiple correction present analyze combination count combination univariate cell alone measure everything loss case one case summarize contingency un depict positive proportion call concerned identify expression follow sample proportion un pair respectively subject compete un proportion cell alternative proportion cell alternative parametrization describe sided proportion indicator subject share whereas marginally jointly two treat miss expectation maximization also hyperparameter via markov section propose em mcmc greatly calculation utilize nan likelihood conjugacy likelihood available form give eq beta miss define complete likelihood side specification involve calculation normalize calculation web log unfortunately form implement start em step convergence initialize exact realization mcmc except give short pilot run present iteration burn leave margin result compare performance compare expect false discovery false incorrectly analysis specificity side compare sided fisher exact base observation negative positive potentially day sensitivity specificity exact ratio detect comparison fold change alone give comparable compete panel il il consistent web cm x north west font b bar bar east bar west font font west font cell use side detect model figure specific difference proportion gene inter pattern evident probability preserve difference proportion fisher fdr adjust reveal identify distance south east west font right bar bar east bar west font south east north font examine performance unconstrained mixture primary event ten independent realization evaluated sensitivity specificity identify roc fisher ratio log fold side examine nominal fdr properly fdr unconstrained compete include specificity b additionally estimate reflect fdr compete web figure auto anchor south west south north right west bar bar south bar north font west east north west font anchor west south north west font proportion maximum web beta multinomial beta replace si unknown proportion respectively share subject exchangeable proportion dirichlet algorithm estimation mcmc binomial simplify use close nan function similarly marginal likelihood estimation procedure base dirichlet except initialize positivity call test em greatly although slightly alternative mcmc appendix b cell express expression increase upon approach sum cell il would appropriate change count different population decrease cell difference apparent condition combination advantage perform require adjustment power auto south east north font bar south east bar north look eight assess fisher table multivariate significantly power test auto east north west b bar bar south font eight compare fisher roc curve discovery color article mass quantitative routine sequence single become development become fisher result inference quite small importantly bayes microarray addition nature generation use multinomial count experimental condition share across non beta increase fisher underlie assumption violate figure univariate base beta binomial allow constrain proportion cell sample proportion match sided motivate example type cell set ignore express cell pre filter ability demonstrate broad importantly demonstrate cell gene c cell identify change cell population protein homogeneous cell population correlate disease context study increase cell detect express differential count functional iterative perform positive univariate satisfactory test might preferable detect true select combination side population differential expression sensitivity specificity compete condition multivariate figure c provide experimental design limit
om motivate introduce formally error appropriately nystr om suggest adopt study nystr om good rank use frobenius implication closely error frobenius find application pca low manifold improve norm understand application om domain stand spectral frobenius compare bound bind term scenario frobenius remove diabetes rbf blue vary overall indicate large explain dependence examine rank legend varie observe combine nystr om motivate error nystr om analysis nystr om measure exploit learn incorporate nystr development nystr om world well nystr om submatrix select row usually computationally expensive submatrix need let nk I reproduce space rkh endow eigenvector approximation nystr compute nystr om b already large inequality integral eigenvalue integral eigenfunction eq easy eq integral schmidt concentration integral operator hilbert sure probability l result last base two additional adjoint h n bind key large normalize quantity theory perturbation result foundation theorem eigenvector consistent unique eigenvector allow assume found indicate sufficiently imply eigenfunction compute approximation eigenfunction q appendix approximation nystr norm last hold obvious l besides nystr approximation approximation try bridge gap effectiveness nystr om nystr om base concentration plan nystr om proof j simplify nd eigenvector therefore decide relationship matrix matrix define canonical
joint usual positivity getting map scalar ensure usual recursive message relative column denote linear belief update output preserve non simplex immediate consequence basis wise negative fact instead track negative belief output prediction via relative column form basis express form right term estimate form inverse invertible substitute statistic estimate observation representation convenient maintain appropriate mapping note box method hmm spectral hide mit document markov though slight notational original maintain usual map necessary update limit analysis order give denote relative basis indicate
decision reject classifier binary towards biased two define risk surrogate solution solution framework margin stage without feature acquisition subject closely cost sensitive infer take acquisition cost study decision cost propose increase utility classifier regressor discrete deal consist million develop learn formulate multi reduce cost stage reject rejection decision rejection classification stage reject classifier investigate reject stage framework introduce classification class posterior rule context machine reject lie small separate hyperplane criterion implementation stage reject stage medical margin always meaningful illustrate degradation happen measurement margin meaningful reject cost cascade multi stage reject cascade much cascade cascade introduce reduce classification cascade detection alarm partial decision pass stage whether belong detection cascade architecture cascade primarily concern binary classification problem decision classification decision stage conceptually distinction fundamentally cascade unbalanced negative goal stage admit false negligible associated problem well rejection decision problem medical diagnosis detection argue sec item performance detection cascade stage average misclassification example stage sensor modality reason develop cascade nevertheless goal cascade reference therein perform cascade sensitive detection system decision see learn possible ensemble base classifier size ensemble current cascade distinction set partial stage computation classifier full label extract acquire th truncate th fix reject classify delay modality standard system classifying indicate active misclassifie incur analyze fundamental simplify minimize risk arbitrary decision minimize appeal remarkably risk risk stage optimization simplify derivation c account conditional optimally dynamic program possible conditional expect minimization rewrite optimal eq reject strategy class implication ignore stage minimize single multi optimization modify remarkably modify replace stage meaningful risk meaningful reject htb classifier reject option classification clear expression express two reject modify term agreement disagreement namely claim thm binary classifiers nf reject classifier eq minimizer inspection illustration compactly use derive derive decision assume task n wise parametrization stage go try learn wise empirical go training boundary cost recursion eq optimize function constrain decision minimum equal stage simplify due stage stage three system refer appendix program ensure terminate multiplicative step stop n correspond direction hypothesis minimize maximize classifier margin employ two system consist q thm reject subsample f sf proof approach complete detail please refer appendix compactly st stage nd margins stage interesting reach nd reject st second goal large early stage life stage confident reject base reject binary margin infinite case reject use measure attribute acquire attribute acquire work normalized margin measurement utility denote estimate measurement parametrize limitation compare error unclear set cost stage possible reject vary reject system nd reject datum stage classifier classify sign split averaged iteration subproblem surrogate outer c stage dim nd dim rating diabetes test bin image ir level synthetic dimension nd dimension uci machine attribute extract stage rate discrete employ opinion demonstrate utility approach possibly estimate htb three dataset diabetes dataset uci consist body mass index two expensive hyper attribute intensity spaced fine acquisition stage coarse course various device image modality ir wave extract many image training carry either clean fast modality slow c name centralized mix diabetes global margin marginally point reject nature classifier utilize nd multi imply half expensive opinion without medical resolution ir whether require person slow medical diagnosis penalty easily adapt risk reject roc high reject high incur lastly three algorithm generalize way cost stage ir vary rd color fig map correspond make decision horizontal example modality point system ir achieve reject rate one demonstrate modality every vertical ir avoid together achieve centralize ir htb ir strategy achieve error rate general sequential parametric principle erm optimize remarkably subproblem turn practical minimize surrogate several formulation multi fraction security ed term average drawing choose express equation size expression chernoff bind second nx px define rewrite note iid use collect two system minimize keep constant term get keep minimize sensing modality cost unnecessary classify majority reduction datum seek average acquisition cost risk minimization erm multi reject wherein stage classifier measurement next acquire cost erm show reject parameterization global surrogate framework generalization medical demonstrate substantial achievable classification sequential learn security medical diagnosis compose stage sense less informative sensor fast sensor expensive measurement practice allow modality make scenario attempt classify consume th modality make example detector could use slow expensive wave consume inspection stage ct scan procedure many example salient sensor measurement order sensor sensor modality stage consume situation sense modality case sense action become methodology primarily fix stage modality justify account come across consist sensor sense modality
vertex add boundary add integer order write size find order nearby insight thin minima must maxima order order give increase find component leave order right zero htb order thin width find globally thin see minima order property note flat consist width flat greater contain respectively flat write minus vertex eq let adjacency unweighted calculate sum index go subtract sum particular non locally maximal locally minima note locally integer shift weakly maximum ordering loss shift remove add vertex remove slope add shift could increase imply substitute find maxima ordering irreducible shift rearrange maxima shift helpful seek minimize finitely strictly guarantee reduce one maxima strongly irreducible come strongly irreducible cluster cluster prove terminology proof slightly add small add large boundary sequence remove create strictly boundary cluster vice versa increase complement remove portion irreducible order contradiction cluster e fail concave fail replace minimum flat define find look index locally minimal slope adjacency vertex slope negative slope vertex calculate respect remove vertex maximal slope find terminate every vertex negative slope define terminate follow sequence add increase decrease concave intersection sequence increase boundary add decrease boundary complete proof lemma vertex slope respect remove increase final convex sequence vertex increase final vertex vertex slope slope add increase boundary w reduce boundary contradict conclude convex repeat convex therefore terminate irreducible contain trivial discover correct increase useful irreducible stop irreducible choose irreducible ordering order irreducible repeatedly find strongly irreducible initial shift consecutive vertex minimum vertex order shift leave repeat argument shift irreducible middle irreducible order fact exists give repeatedly remove vertex zero none cluster non empty core say core proof lemma terminate cluster every core core contain assume contradiction remove sequentially first contain slope core slope strictly negative slope strictly remove remain thm thm translate subgraph thin knot dimensional manifold mining search large set dimensional point measure point geometrically fall category determine make cluster book algorithm underlie gaussian center way assume restrictive less effective euclidean less hierarchical building place tree flexibility final construct partitioning translate point reflect point define subgraph separate whole close cluster encoding generalization knot manifold thin surface thin position translate quite sense begin vertex define width look improvement criterion first start order cluster initial ordering improve repeatedly examine future
naturally period behavior take place short important behavior carry relate time period description integrate access pattern page indeed dynamic nature change influence week factor event economic volume profile start take author date involve investigate semantic recently outline discover episode partial periodic temporal majority period consequently datum interesting may short period domain period case web behavior evolve consideration give rise study analysis concern summary carry evolution objective change stream usage article organize propose include benchmark external criterion well report suggestion future modelling cluster evolve evolutionary dynamic splitting entire period sub period use year sub discover reveal cluster sub specification store datum prototype reflect individual subsection describe strategy correspond traditional order cluster contain sub period month year put partition period period us partition period process contain cluster prototype belong time cluster repeat period common completely allocation run request repeat request percentage repetition duration average duration request number request page site percentage request site semantic duration request adapt contain row real column except obtained present analyse apply external implie evaluate cluster match gold cluster f partition cluster second partition cluster combine cluster contingency measure formula element element correct rand cr two partition cr agreement cr sensitive number cluster cr indicate object rand index range indicate perfect agreement near confirm correct rand index near indicate partition obtain month figure value reveal local cluster cr mean partition word versus dependent local detect also confirm notice result conclusion local dependent use refine appear cluster dependent detect cluster local value surprising clustering similar whether carry suppose reveal summarize occur domain issue highlight adapt extract knowledge evolution devote handle evolve article propose divide regard scale offer change nature use change split set regard hardware overcome cluster acknowledgement author grateful cluster usage domain suffer classification detect occurrence change distribution ignore concept concept make inconsistent regular necessary usage necessity evolve solution article summary evolutionary sub period validate propose occurrence change mining appear end use order extract web
summary reference satisfactory result describe summarize reference automatically reference color achieve perform channel illustrate process detail subsection representation comprise hidden patch patch size much small goal gaussian size reference fix densely overlap associate patch image coordinate independently mapping pixel patch map possible interpret patch behave traditional map mapping take suppose mapping eq indicator equal true maximize function patch graphical maximize follow e respect mapping get parameter eq coordinate alternate reference applicable train sift convert channel represent dense sift patch divide orientation calculate grid dense vector train channel sift channel dense sift channel sift represent sift sift learn graphical densely cover patch mapping patch essentially take channel channel color channel channel patch specify patch denote overlap averaged patch channel square half reference densely horizontal figure result image convert image parameter color sift compare color reference level matching result produce continuity whole image contrary learnt contain sufficient color patch respectively top much spatially transfer target natural result user intervention segmentation exploit color information appearance result effectiveness htb htb department electrical computer engineering institute department electrical national university color image lack segmentation eliminate develop automatic use appearance effective train reference image image contain information scientific image illustrative color part manual consume suitable manually image automatically automatic reference target interaction selection approach biological human human
hierarchical interest represent scenario maximally away maker internet link parameter link depend moreover grow slowly increase child height grow go infinity aggregate majority non bayesian contribution derive bound decay break break consistent rate bad achieve achieve fast rule breaking rate mean therefore gap almost achieve rate certain pass message explicit convergence majority alternative majority explicit span binary organized leaf agent circle agent independent true measurement agent decision measurement forward agent next agent exception agent receive parent final hypothesis receive leaf represent tree infinity agent hypothesis alarm ii error answer question type ii error explicit function yes way aggregate binary agent binary break aggregate information daily life fusion explicit probability towards root derive probability turn characterize know error probability associate new binary use locally sense bind probability majority rule use rule derive explicit asymptotic divide odd type ii agent wise kind type type root degree branching agent tu binary fusion parent majority odd suppose message type ii ii binary message probability pair decision pair fusion induction suffice bound h mx km mh claim monotone monotone binomial mx jx moreover monotone monotone decrease proof analyze wise analysis error root odd majority rule know moreover consequently increase proposition odd integer rule fusion moreover brevity note addition explicit error substituting suppose majority integer upper bernoulli show assumption relax g type use suffice apply majority achieve derive break suppose broken bernoulli distribution bound proposition low level tree suppose root useful balanced binary remain provide analysis binary ratio agent agent characterize unit ratio globally reduction notice odd unified simply derive low root majority rule provide total underlie hypothesis easy type root rule bound majority recursion maximum type ii probability leaf agent test level majority identical majority characterize explicit error evident majority achieve gap describe negligible branching tree total change convergence rate test rule break agent level eq integer recursion omit c suppose brevity type alternative total n p decay surprising ii rule decay sufficiently difference small likelihood rule break break repeat consecutive decay combination contrast strategy involve strategy asymptotic alternative fig decay purpose alternative strategy large majority random breaking gap htbp section allow agent rich pass message alphabet alphabet tree investigate decay integer agent th level message u tm k leaf message alphabet agent subtree root agent leaf message leaf parent sum receive immediate child agent integer message number subtree alphabet know subtree leave star leaf message directly star decay height I hold fig leaf measurement intermediate sum message agent e see message alphabet enough agent abuse height strictly scheme know subtree leave connect subtree subtree ignore fusion rule aggregate subtree leave message agent henceforth simply rule throughout decay easy leaf connect agent recursive agent connect agent directly equal regime decay decay simplification agent tree tree rate upon simplification desire result notice mean theorem alphabet decay quickly change decay depend furthermore sensitive compare achieve almost exponent test convergence n strategy majority apply argument make proof corollary proof omit brevity message pass agent message leaf agent binary interesting scheme strategy suffice message height message agent agent size bit q large convergence rate average bit still small fig show message size become tight study error characterize convergence majority majority break suboptimal pass convergence probability alphabet many question remain usually topology branch different agent complex question involve conditionally another question communication agent reliable communication ideal ratio know inequality q proposition p network consist agent root aggregate binary make measurement hypothesis leaf message member summary agent type provide scheme involve binary characterize exponent error bayesian learn decentralized hypothesis testing attempt jointly consist agent measurement underlie past neighboring agent hypothesis decision fast respect network answer depend structure structure primarily feedforward agent private decision restaurant movie go popular appear behave market private decision agent structure common political structure online social e sensor network decentralize concern sensor message measurement compression maker goal typically compression associated intractable complicated decentralize focus process economic biology physics review relevant aforementione make equal make measurement agent recursively decision agent decision belief agent decision rest copy ignore phenomenon decision asymptotically agent unbounded private ratio decision problem change sequentially observe immediate agent identically bind belief derive observe decision randomly agent relevant situation network complicate wherein decision study agent structure largely network exhibit hierarchical structure concept hierarchy extensively structure network human
get p bit mathematically instead use qx qx qx qx qx likelihood row ensure one google gram gram vocabulary word vocabulary token take minimum small definition draw blue shape red thick thick red thick co occurrence frequency triple fast gibbs sampling reduce generate vocabulary elementary accuracy estimate either bound put condition useful drawn vocabulary widely often either slow optima zhang show hmms theory close onto value decomposition svd adjacent project observation state perhaps surprisingly co pair triple observation sufficient accurately course observe hide make precise generate require gibbs sampling observation count return result know advantage et mapping size alternate replace vocabulary introduce consider emission give observation joint vector length create everywhere else observation operator theory predictive state representation effectively distribution state directly certain exist fully observable operator fully orthonormal mapping moment derive several way show consist singular value property discuss detail estimate observe conditional probability observation recursive key take current produce map et al b observation equation dimensional reduce easily relate also invertible correspond singular detail material observation edge observation edge et edge h edge x terminal way reduce estimate achieve sample transition check instead focus relative know compute state thin lose estimating main reduce reduce term generate estimate define empirical third namely iy iy iy yy variance word tensor likewise version ij small relative estimating quantify sufficiently zero ability accurately invertible one infer require become identify hidden problem estimate fact close absolute relative large fortunately discuss shift rescaling term accurately basic scale bound usual hold everything enough state triple proceed two corollary correspond theorem proof theorem bind proof write x product write scalar b scalar state prove accuracy product thought relative allow give error generic show complexity rely know observable convert complexity hmm triple hold state lemma basically say element accurately use accurately give hold happen know hence since exactly bind hand involve know know interpretation carefully discuss different hard contrast address couple way improve rescale increase estimate word word basically would accurately challenge compute unstable basically significantly improve ratio probability could fact help change relative improved fact htbp vs vocabulary slope slope show internet corpus gram frequent gram gram occur figure find supplementary material increase singular small decrease straight complexity idea direction hmm learn discretization provide analogous transform rr learn observable rr extension preserve tensor transformation recent replace contain contrast reduce proof simple simple discrete discrete hide approach way present improve hmms tensor observe tensor
utilize q rhs u beyond simplex euclidean denote operation define generalize respect set function consider denote cube give bind low calibration note inequality define also hardness fix follow remain strong polynomial relevant complexity strong calibration desire consideration thm thm microsoft research new existence efficient calibration calibration existence nash equilibria unlikely weather forecaster calibrate every predict day approach probability forecast numerous application first randomize subsequently existence calibration think round low thought achieve critical calibration calibration forecasting henceforth calibration outcome forecasting explicitly pose net calibration widely utilize namely implication main suppose exist run ram polynomial cumulative notion randomize nash game widely round inherently concern compare sense statistical distance rather norm outcome outcome outcome forecaster distribution interior simplex constitutes forecast strong calibration least random use motivated primarily convenience restrict minor distinction literature standard forecaster strongly consider grid normalization satisfy cover one weak weak cover manner partition set intersect common face lie simplex simplex specify associate uniquely write neighboring test weight uniquely q say diameter hull test cover manner continuous opposed indicator weak note normalization tx define ram main base use calibration equilibrium game let nash bi matrix player payoff respectively mixed strategy refer simply equilibrium ne abuse notation payoff definition equilibrium mix approximation entrie particular r player game along outcome denote distribution smooth response uniformly sample law
prove guarantee computation fact often multiclass coding code develop show possible method interestingly naturally extension generalize large loss rest follow section discuss simplex analyze respectively supplementary copy describe semantic belong bx bx classification rule erm bx ir base section multiclass binary modern algorithm consider relaxation erm indicator surrogate consider often suffice special suggest misclassification label side step sometimes margin exponential machine square surrogate effective risk functional section replace misclassification effectively denote rule pay relaxation generally misclassification risk relaxation quantify requirement surely suitable depend random possibly possibly misclassification risk margin decrease case particular loss suitable x q polynomially multiclass multiclass mention interpret kind relaxation interestingly scheme multiclass seem experimentally performance sophisticated general large multiclass notably extension inconsistent classification rule apply map function consider problematic make come paper study fisher give impose constraint section natural function explicit cm label label right b thick thick dash red red dash red code strategy turn label decode return note implicitly treat naturally strategy simplex c correspond separate simplex figure natural geometric map vector hence close code propose binary follow simplex non bx yx bf f satisfying definition svm function several function classification naturally extend multiple code focus extension generalize misclassification induce one consider expect x svm ls l loss fx df sc l long theorem sc respectively square notion generalize noise say satisfy arbitrarily exponent simplex l tradeoff exponent constant constant svm ls enhance separable inequality derive misclassification risk min max technique svm loss could know work computational hilbert brief introduction kernel span choice discuss particular kernel induce kx set version expression consider loss notation simplex functional n classical value computing close loo loo loo hadamard leave df eigen standard see qp kronecker delta derive omit observe convex especially formulation fact programming iy formulation train bad consistency optimization simplex first sub finite offer neither matrix fit memory descent point ix iw computed x decomposition interested computing note convenient perform system explicit feature simplex versus simplex solver complexity complexity omit constraint path algorithm simplex essentially particular class conduct performance online uci list simplex boost uci raw hierarchical map selection leave loo l small eigenvalue width pairwise distinct point collect classification accuracy letter sc l ls simplex consistent omit sc achieve see good among method include simplex boosting compare rbf ls rbf I mit mit edu ai mit multiclass define strategy simplex code relaxation relaxation develop constraint derive provably class tool
compositional paradigm risk intuitive play maximum deviation zero indicate expect understand affect case classical candidate reduce one e outli risk task one thus task argument approach maximum minimax focus replace example consider formulation hinge new h task trace norm nuclear formulation paradigm time convenient constrain fair comparison ep involve task ep education analysis digit pairwise minimax ep regularize solve line multi attain various problem mm deviation relative mode draw uniformly sphere radius parameter task isotropic take task draw normal iid center normal mean task ball radius mean wise risk train tradeoff minimax maximum risk risk homogeneity plot respect risk solid minimax minimax visually school appear predict score student root mean rmse normalize point learner moderate share capacity minimax objective outperform specific capacity minimax stopping risk improve stopping build normalize mean relaxation capacity good capacity relaxation overfitte fold fold red minimax green compose rate computer training price etc maximum rmse objective rmse norm overfitte obtain low rmse objective minimax outperform rmse regularize norm differ mnist task classifier use component class training intuitively path node minimax figure confirm minimax capacity minimax competitive establish formulation newly formulate lie relaxed formulation introduce compositional paradigm operate risk induce empirical outperform build overfitte relative make formulation proper effort study even area minimax college technology ga usa ny usa task wise empirical risk loss compositional include spectrum minimax formulation minimax loss compositional vector relaxation case tends newly test encourage essence exploit observe learn inductive learn task connection include collaborative multi admit admit provably task random task know present challenge store future sensible ensure learn scenario music company business star rating would assign training company learn predict rating company limit quickly user rule company preference song ask user rating company leverage learn predictor produce rating task minimize risk task classical assume test notably em user fix user user rating adversarial preference company minimize negative feedback business rather mean whereas much locally outcome notion cast spectrum multi end minimax hardness full classical compositional paradigm equally goal admit hypothesis future handle exception theoretic contribution relaxation compositional include third empirically task learn criterion finally core empirical across training task level learner theoretically show future likely maximum risk short restrict attention subsection iid let empirical h hx focus observe risk newly bound emphasis bound expect expect markov show risk newly decay risk control certain space control tail cardinality theorem argument iid cf observe task high decay goal minimize task motivate empirical risk oppose singleton risk I test h union least decay probability minimizer take uniformly particular high probability risk minimizer empirical risk b maximum next section introduce compositional paradigm task benefit tx additionally define task loss compositional paradigm minimization typically convex yield
difficulty meet vary depend approach approximation error closely appear random identity fx fx identity role compact depend g g fx assumption old desire yield generalization h f minimizer diversity candidate function e var hx h follow algorithm q theorem var f e tell var h h right term cause small large quantity f decompose h f v eq estimate technical possibility h f h immediate follow error ratio probability almost bind get sharp involve almost constant almost eq see z u g fu u apply tw h old inequality fx f q fx power e q fx position f know whenever satisfie bind prof apply cover know small solution write understand part prove constant complete proof g fx gx n f jx bernstein cover h bounding side e enyi entropy scale parameter infinity explain value tune however entropy approximately estimator converge consistency far know consistency open question analysis instance consistent shannon error setting identical sampling process question require research topic understand relate minimize empirical differ extra vanish analysis mean page random r enyi explanatory variable random predictor entropy shannon entropy enyi erm variable power decay number projection interval generalization least generalization fx x ft remark minimum criterion empirical enyi present explicit novel entropy overcome technical difficulty arise square error analyze related keyword enyi theoretical inspire introduce machine paradigm framework several attention processing engineering datum systematic area find therein minimum entropy principle supervise maximally blind deconvolution topic idea extract much entropy entropy average shannon enyi explanatory become predictor use principle searching contain entropy variable principle classical minimize error put might still principle since moment order error account entropy enyi neither explanatory explicitly supervise reflect explanatory relation entropy approximate gaussian approximation shannon entropy enyi empirical computable principle decade effective foundation mathematical yet scaling parameter large enough algorithm tune convergence converge impose algorithms barrier analysis possibility detail consistency require effectiveness application minimize empirical enyi focus set input learn marginal explanatory identically aim r enyi appropriately erm learn let continuous define compact space computational purpose scale erm adjustment approximate requirement adjustment translate boundedness decay study weak constant large error approximation space regression pair hypothesis measure number exist radius center shall assume satisfy reproduce associate smooth cover number together concentration scope simplicity adopt cover prove section assume cover almost confidence explicitly bound small simply get cover small enough error bound cause method know adjustment theoretically well approximated uniformly bound heavy computable quantitative proved tell imply replace estimator put power index appear
derive respective fourier relationship identical derive close cross stationary kernel author smoothing function sound latter powerful develop gp model gps key insight identify smoothing convolution smoothing expression smoothing lead point express separable e g basis function identify form separation say universal smoothing function consideration derive kernel demonstrate auto covariance equation give equation expression auto equation gps scale dimension eq dl respective base gps dimension input term exclude refer length exclude task incorporate covariance provide flexibility task gp learn hyperparameter system learn length scale learn hyperparameter adapt filter experiment conduct datum make real sensor x chemical composition apart hundred deep east depth element element correlate capture metric fusion perform evaluation experiment gp conventional quantify compare kernel prove aim ten fold motivated fold validation estimation world set trial performance paper gp find appropriate subset identical point auto auto covariance gps optimize context resource modeling comparison vs independent vs cross block version idea rather block robustness point uniform block collection represent fold fold testing fold together small fold e follow estimate parameter result fold point e empirically implication cross fold number fold min max cross away block c comment fold min x cross x error prediction x prediction least high various mean value fold represent mse popular metric paper variance var outcome se var suggest confident inaccurate estimate outcome correspondingly var inaccurate uncertain prediction extent include gp gray mm ms se var nlp se var nlp se var se nlp mean mean mean std std std std std std std std std gp x gp c size method kernel mm ms kernel se var nlp nlp var nlp nlp mean std std std std std std std std std std std gp nn kernel mm ms se nlp nlp se nlp nlp mean mean std std std std std attempt hour ms attempt hour nn attempt attempt iteration hour individual reasonable converge well nn test figures nn produce se estimate test competitive marginally small block consider size lower confident se nn nlp nn base nn nlp remain mm rise nn cope incomplete ms competitive mm nlp kernel three gps optimize gps fusion figure figure derive independent se nlp demonstrate benefit heterogeneous source table gp small intermediate size x x gp gp improvement model correlate data ms figure prove independently se nlp perspective ms gp gp prove trust option meet nlp gp result attribute inferior element perform well nlp metric prediction se inferior figure e figure equivalent se nlp se ms se gp inferior uncertainty nlp trend element ms kernel gp nlp se small produce se prediction basically confident outcome nlp block prediction nlp trend large variance nlp overall poor limiting case mm use poor poor minima investigation stationary metric poor confident attribute correlation profile decrease increase support point interest trend able cope large support across block se take analysis se metric provides describe important understand nlp metric confident poor prediction increase bad outcome meet study answer fundamental model kernel I mean universal cut substitute statistically develop past domain model numerous sophisticated gp beyond scope develop experiment good question instance could hyperparameter metric ensure check question confident uncertainty derive gp availability ideally se significant I prediction net nlp metric test check metric behave would optimize gp datum hand purely base domain way treat modeling optimize good information multi gp multi representative g model kernel network kernel gp depend model suited competitive local minima sample reasonably gap modality paper resource individual gaussian gps demonstrate individual modeling problem effectively integrate heterogeneous benefit integration use resource modeling quantify validation gps neural competitive robust size acknowledgement centre paper evaluate fusion resource modeling demonstrate information superior estimate model provide powerful simultaneous modeling multiple consideration experiment perform fusion resource exploration resource flexible high fidelity challenge deal problem capability application limit applicability collect datum expensive collect consequently kind characterize numerous resource handle spatially correlation homogeneous model also gp ideally suit spatially use multiple correlation totally different quantity together quantify quantity evaluation understand simultaneous fusion independently kernel effective gaussian gps powerful handle gaussian list gps produce multi continuous domain desire gps manner spatially interest basically perform work large scale stationary kernel standard interpolation method applicable alternative stationary find superior stationary least work build address heterogeneous modeling usage correlation improve prediction process context presence attribute uncertain set entity model preliminary address former demonstrate model output available learn attempt generalize random source fuse scope integrate heterogeneous protein fold representation separate gp use idea composite soft human source domain kind information kind information sum encoding represent common heterogeneous demonstrate source heterogeneous model heterogeneous qualitative heterogeneous separate combine sum fusion gps derivation covariance heterogeneous example multiple former use gps incorporate surface spectra spectra map environment use implicit surface incorporate modality prior inside demonstrate context gps gps address sensor quantity extend heterogeneous idea refer gps one improve cross covariance idea dependent gp terminology analysis fusion focus resource picture relate practical heterogeneous together work study work provide resource use perform gp gps independently gps quantify benefit correlation quantity paper fusion modeling modeling present recent survey discuss multi regularization perspective focus review specifically address process convolution present efficacy broad perspective tie together process gps wherein jointly characterized specify resource modeling coordinate model necessary appropriately relationship random variable stationary square diag north l depth quickly change east direction east nn augment vector augment diag east east north constitute nn layer input use show universal tend kernel kernel smoothing equation east north depth model east north east depth kernel test free hereafter yield estimate eq nn kx n covariance evaluate likewise learn equation hyperparameter learn various approach posteriori maximize latter likelihood
offer useful representation well operator order number order th element denote clear integer absolutely median normally x obtain function max operation take function examine fourier method regard apply conclude combinatorial optimization programming consist discrete programming e question discussion minimize always difficult express graph surface produce illustration objective minima clear practical determining conclude problem subset condition minimum hyperplane intersection correspond always suitable produce sketch express e firstly function arbitrary local clear consist index solution coincide include express illustration independently minima active successively difficult represent minima outline problem produce clearly objective low solution define formally exhaustive consider actually reduce depth search provide examine point exact time dimension within reasonable branch design problem search subset would draw valuable discussion kind I paper arise method replace equivalent residual give local local dimension observation algorithm recently robust estimate regression follow optimization ip ij regression parameter optimization minima efficient design problem
define classifier convex wise maximum simple convex conjugate interested finally keep put appendix nonparametric model could discrimination margin latent dirichlet lda task project probability express model later constraint model example whose graphical document unobserve latent build mix index average mle posterior large principle correspond minimize constraint minimize insensitive loss n learn intractable variational auxiliary distribution q nm nm nm nm optimum z optimum although problem develop additional primal iteratively solve alternatively approximate solve approximate present mt discussion unknown fully framework ready present model ideas ibp margin machine svm simultaneous feature parametric automatically latent choose define likelihood task basic setup latent binary feature multiply specify dimension resort nonparametric bayesian let ibp review ibp successfully break construction efficient column sample independently stick generate decrease ibp property row ibp mass number column harmonic almost finite ibp task discriminant stack zero bayesian profile matrix need discriminant widely discriminant explore inference post want infer include place define arise expectation move well also zero ibp prior essential dot reader generic banach unnormalized practice problem finite level see truncation increase expectation note ensure fy finite impose latent gaussian super principle row deal vector denote definition fy fy fy ny penalty minimize hinge averaging prediction surrogate squared clarity cost predict label set besides perform may linear value loading replicate together form p arrive generic divergence kl divergence unnormalize although infinite finite ibp property prior decrease increase exponentially make feature truncation level detail show truncation marginal distribution exponentially hard deal duality develop algorithm solve second operator th ny g proof defer conjugate optimum distribution distribution conjugate make together impose true label datum likelihood prediction generalize share ibp cast impose problem hard resolve label inductive set latent feature train standard typically single task statistical strength jointly develop task particular representation prove relationship infinite mt brevity task let mn multi easily na task mt introduce discriminant one dimension projection view let couple share projection alternate learn pre latent unbounded projection use multiple task mt type pass transformation act input project latent representation projection view fully factorize discriminant possibly rule naturally impose define likelihood mn illustrate conjugate case similar minimize hinge rule equivalently derive mt deferred conjugate mt mt z similar detail duality strategy inference test margin test datum parameter priori appendix discuss perform mt formulation obvious basically method perform bayesian dual infer distribution however primal dual intractable mt hardness mutual desire method field dependency form impose constraint apply mcmc sampling infer iteratively estimate dual approximate e expectation sample mt idea apply easy stick ibp furthermore truncate truncation duality constrain duality problem lagrangian margin lp lagrangian step appendix corpus output given solve minimize theory adopt alternate solve margin update equation constraint large margin infer latent optimum scene closely stick break augment infinite e nonparametric mt ibp use ibp share among multiple mt mt ibp inference detail comparison macro micro dataset perform nonparametric mt ibp separate hamming loss actually equal mt art evaluate cccc cccc ibp mt ibp mt school compare bayesian multi learning report mt mt ibp concatenation mt mt ibp svm mt mt well study mt mt ibp separate classifier feature input boost mt ibp mt change parameter constant mt insensitive school insensitive affect mt training see mt achieve run mt change initially estimate maximize objective explain school explain present regularize rich formulate theorem induce concentrate develop learn task explore margin principle constraint latent automatically appear bayesian bayesian work plan constraint represent regularization nonparametric bayesian contexts social collaborative preliminary investigate direction mrfs hard plan investigation room far nontrivial normally inference truncation resource important adaptively determine feature preliminary progress along extend deal acknowledgement thank anonymous manuscript nc national projects cb cb national natural china china science foundation nc support fa fellowship appendix beyond bayesian network inference implicitly direct undirected general undirected model boltzmann could mle frequently practitioner figure need presence instance may mle lda choose parameter unknown inference converge specifically optimum bayes mle objective holds direct many formulation distribution illustrate scenario subset subset connect via graph connect observe direct property chain graph know factorization mrf hybrid boltzmann insight undirecte problem formula h put together generally problem formulation latter unconstraine normalization constraint discrimination dirichlet dirichlet result discuss loss unobserved topic topic build generate document proportion nm n index average nm expectation principle quality minimize insensitive use introduce nm posterior replace nm constraint nm optimum conjugate approximate primal formulation margin iteratively solve inference mt unknown dirichlet interpretation move proof appendix adjoint ex cc dual respectively regularity inequality take infimum infimum conjugate duality attain supremum infimum get q put appendix proof definition first easily eq therefore moreover together claim appendix easy infimum equality eqn eqn conjugate prof appendix similar operator function conjugate eq q conjugate appendix variational conjugate optimum posterior distribution normalization expectation conjugate constraint dependent algorithm mt mt consist truncated mn mn whose individual term q write finally effective discriminant easily adopt variational belong simplex normalization denote replace q tb input data mn mn mn iteration binary svm learner eq outline mt lagrangian constraint constraint specifically update duality theory solution although assume factorization form normal obtain solve solver light priori objective fix mt develop inference stick ibp outline constraint feasible truncation nk unconstraine duality dual let effective discriminant n computational bind get kl procedure solve computed mean equation cm testing datum absence solve duality normal dual efficiently update primal form update hyper g mt hyperparameter easily chen bayesian rely prior knowledge discover improve prior affect posterior bayes impose arguably direct natural regularize posterior post flexible procedure network undirecte markov network formulation chain induce concrete infinite margin present efficient report study benchmark dataset contribute interface largely keyword decade bayesian remarkable popularity field partly desirable wide heuristic practice determine unknown gaussian process beta restaurant crp bp describe ibp nonparametric grow traditional parametric recent relax homogeneity exchangeability dependent process exchangeability correlation structure structure dependency common share rely define case encode posterior accord bayes know principle offer arguably richer flexible rich structural etc hard capture prior denote parameter define mle latent post necessarily bayesian induce rule regularizer certain latent variable posterior drive type obtain augment mle parameter interest going regularize entropy discrimination discrimination dirichlet allocation give rise knowledge make impose nonparametric data nonparametric formalism bayesian apparent formalism formalism learn posterior ibp style constrain formulation objective enable regularize formulation desire kl minimization solve feasible appropriately perform parametric interesting impose slack formal description normally define formalism wide graphical chain operator solve allow regularization distribution source datum remark facilitate integration complementary largely treat disjoint core machine maximum discrimination extension markov maximum discrimination lead successful outcome flexibility bayesian handle develop machine infinite feature readily master large employ ibp unbounded feature infer pre specify procedure high optimization technique discuss relate present duality section preliminary finally discuss research arise scientific domain popular book mainly recently bayesian attract many develop much rich datum subject constraint estimation label development generalize discriminative entropy appropriate label empirical expectation normalize unnormalized expectation criterion alone usually maximum conditional likelihood al expectation supervise auxiliary propose measurement label model prediction satisfy auxiliary maximum entropy principle expectation moment match minimization special case later general banach draw paper develop regularize bayesian duality generalize present extension preliminary max nonparametric example infinite assign dimensional space multi feature share task however regularization mt popular theoretic atom endow borel algebra measurable absolutely continuous exist assume dominate exist reason introduce variational formulation density background assumption clarity optimum q q function distribution soon additional post sequel distinguish posterior post omit post posterior imply project bayes special mainly interested inference define model model variable input datum feature ibp feature input datum e capture knowledge effort optimization rule straightforward generalize rich replace normality constraint unconstraine regularize generalized constrain problem regularization impose subspace satisfy constraint besides normality rewrite master u kl unnormalize soft penalty type expectation expectation possibly dependent post make solve nonnegative interpret slack soft cover hard indicator figure feasible convex illustrate lead higher shall unconstraine u assume induce function infimum versa well many mention domain expert discrete natural expectation expectation sx yx general use unnormalized expectation supervise learn choice indicator please summary
freedom classification change planted confirm validity eq show assessment condition fig indicate mode mode imply plant mode solution plant free vanish become offer great hold change appear number linear perturbation limit dominant begin mean plant solution learnable negligible allow plot phase represents plant learnable dl region straight even letter nonetheless assess correctly plant dl use replica early learning plant square per sufficiently measurement show characterize rs dl solution replica symmetry break correct critical energy assess rs pure keep vanish promising framework refinement ratio work aid energy formula assess mathematically rigorous difficult replica evaluate limit take analytic precisely evaluate valid perform come evaluation identity constant mention nd eqs stand whose q saddle lead identity auxiliary saddle x ax exactly lead restrict candidate dominant saddle analytic xx follow replacement convenient handling saddle eqs rescale offer temperature intelligence science computer technology observation engineering material science major relevance number shannon sampling measurement recover arbitrary band signal match find attention signal fourier component amplitude wavelet basis property exploit sense cs enable cs rely considerably basis look therefore signal basis primary signal learn denote represent dl sparse dl characteristic trend compactly represent superposition strength specify dl efficient processing dl storage database standard fail patch respectively dl dl uniquely dictionary dictionary answer algebra plant learn sign permutation element zero column size dl supposed enable substantially respect consequently relevant assess central carry systematically replica keep replica compose pure free give x ph ph ph density transpose mean performance dl variable mean linearly represent effective body concern element statistically randomness effectively th active inactive zero active fig b thresholding base accurately sufficiently feature divergence fig increase degree ph suppose combinatorial
autocorrelation dash seed figure optimally curve marginal figure modify version online meaning perform topology illustrative figure restrict region component marginal restrict triangle marginal distribution distribution autocorrelation pdf pdf ordering pdf pdf consider build walk empirical variance precisely balance incorporate acceptance restriction area represent end run agreement solid certainly happen pdf pdf ordering suggest skewed roughly center respectively also distortion non marginal efficient chain ht pdf significance make resolve non adapt diagonal explore severe permutation require lead consider tractable mcmc target mix efficient key proposal proportional display obtain run example topology examine indicate criterion look seed autocorrelation figure autocorrelation tune walk seed gaussian perfect adaptation covariance restrict achieve result examine become nonlinear figure yield much nonlinearity seed selection change see select form remark pdf multimodal multimodal ready suitable region marginal mean moment restrict target prove stable density lebesgue compact invariant group organize behavior large number ergodicity theorem proof find cover ask lebesgue endow product tb tb euclidean real toward convergence stable give family adaptive approximation adaptive follow value history tend condition transition invariant stable satisfied soon almost resort approximation step design establish procedure rely existence lyapunov sequence make technical mcmc respectively markov compactly support prove eq also show algorithm one hold conditionally h q expectation covariance ingredient lyapunov lemma differentiable satisfie w lyapunov stable bound one mechanism make da x term sequence stable almost surely projection surely convergence neither experiment pointwise convergence accumulation large compact assumption almost except group simultaneously hold lemma lyapunov give field w exist symmetric consequence suffice continuous lebesgue dominate x show first term continuously w upon w th negative iff conclude symmetric exist matrix ib conclude note assumption denote away lemma process let prove measurable measurable define construction projection mechanism guarantee lemma rhs qx x dy similarly detailed balance definition separately hold conclude poisson equation p exist solve poisson assumption lebesgue cc h l l l pl tx p x tv characterize require satisfy three u hold u u set case tv lebesgue simplify coordinate remain generalize coordinate borel thus consequence circular line outer assume without enough proof regularity transition kernel rhs dy dy dy enough establish rhs repeating yield dy v cx x dy establish let result exist rhs lemma respectively prove counter respectively enough exist function martingale increment eqn term cc consider hx conclude q wu wu wu u compact see uniformly choose small conclude differentiable u set compact finite pick small finally taylor wu wu dt yield item remark lemma almost surely proof contradiction projection assume define prove induction hold addition induction eqs imply conclude induction contradiction item theorem along line thus check measurable exist tend almost lemma exist surely conclude limit consequence lemma note use almost surely permutation measurable tf continuity finally eq come projection recurrence f eq adapt stable replace research proposition lemma invariant permutation monte face marginal mixture adaptive address switch associate strategy devote consistent variant cope switch step base adapt metropolis online compare compactly distribution ergodicity consistency secondary generic explore many inference common face serious target know permutation difficulty algorithm visit inferential particularly usually situation model invariant component favor specific ordering component additive model represent superposition recover know reversible visit hard due vary address switch posterior conjugacy specialized assume permutation physics computation commonly defer code permutation also restrict finite know kernel space value distribution past internal along iteration reach proposal research last year paper adaptive metropolis hereafter variant aim identify hasting suggest tend minimize matrix unimodal proposal target empirically section difficulty obvious switching shape also solution switch identifiability post satisfactory post simulated trajectory adaptive improper issue algorithm manual prior target manual poor purpose provably variant cope idea idea step adapt proposal require loop small permutation modification heuristic scope evidence couple establish mode beneficial distortion identifiable restriction posterior marginal distinct recovered permutation problem early previously online quantification presence differentiable variant showing marginally identifiable alternative illustrative example convergence detail artificial define px explain mechanism stable sample also denote mcmc point center implement permutation permutation minimize draw uniformly minimum achieve possible usual hasting sa term toward penalty
rate spike train dependent method drawback dependent effort train dissimilarity code vary spike train spike complementary real variant essential difference fire spike distance track spike code spike distance relative spike respectively obtain distance piecewise dissimilarity profile value arbitrarily maximum whole spike train code spike material supplementary figure www user spike spike train profile instantaneous accord ratio identical train distance absolute deviation equally since rely thus knowledge spike causal real dissimilarity temporal analogous coupling train spike train flow unit patient experimental independently medical clinical criterion verify computer eight typically micro diameter mn micro ground channel digital filter hz continuously unity phase sort pass hz useful thank european project regard joint laboratory foundation award human grant education science fm foundation spike train measure rely instantaneous spike dissimilarity lead instantaneous like fire improvement allow localized train selective averaging comparison define instantaneous dissimilarity past train extract valuable monitoring train applicability model eeg synchronization train spike spike train tool area potential application evaluate role fire pairwise within quantify role synchronization bind spike train determine spike train spike adaptive complementary quantifie dissimilarity fire profile spike track difference spike kind circuit spike move certain compressed compare successive hand application resolution neural track understand fire code research pattern involve phase generation termination van distance spike spike capability spike train temporal instant van calculate yet spike temporal correctly trend move spurious instantaneous consider spike spike separate difference precede spike spike spike situation record spike train promise rapid online signal patient medical treatment aim train distance overall spike lack even causal instantaneous train spike spike future modify instantaneous train causal new three improve spike variant generate train measure able synchronization within spike train reliability visualize instantaneous follow pattern train selective subsequently train activity record distance fire involved termination finally train lack capable address application variant time preprocesse eeg series synchronization hoc mixing distance profile spike train give instantaneous measure dissimilarity train define temporal average profile e bivariate spike train show derive dissimilarity spike spike bivariate neuron case calculate instantaneous instantaneous result identical train profile bivariate subscript improve henceforth replace three spike take locally average labeling time spike train spike instant fig precede spike ambiguity interval distance precede spike spike spike spike precede spike instantaneous base precede spike divide instantaneous achieve normalization scale invariance train bivariate quantity dissimilarity profile improve dissimilarity eqs order average spike close dominate previous spike neuron weight locally make profile train average distance reliable neuron simulate allow reflect change already reliable slightly frequency spike train gradually half train spike spike close spike expect monotonic instantaneous follow monotonic decrease trend recognize short high spurious value background half evenly fire event without noise spurious spike still spike precede small spike spike spike among spike dissimilarity profile train spurious high firing reflect spike bivariate train space consist background start spike correct compare spike spike rewrite sake brevity omit dependencie j j give expand definition precede spike first spike spike precede second spike train wrong happen restrict spike resolve restriction flexibility corner spike train define analogously see dissimilarity four replace furthermore previous spike interval separately weight spike read analogously spike train contribution difference corner spike spike take care train difference fire spike train locally lead improve definition precede train precede spike spike lead spurious spike spike precede vice versa instantaneous dissimilarity profile modification spurious desire dissimilarity monotonic instantaneous monotonic decrease actual change match train bivariate presence less less drop half evenly spaced fire precision reflect rather decrease instantaneous reach ms vertical dash fire introduce eqs desirable consequence spike spike train fire spike spike information information spike position length interval like variant difference corner spike spike restrict spike precede spike spike local weighting divide corner spike interval precede indicator spike spike train gradually spike spike relevant go back precede go back spike half difference period decrease instantaneous desire sign common slope precede difference regular average calculation causal dissimilarity exhibit move average show reflect spike interval match regular trace depict move green dissimilarity profile attain maximum spike decay well move average reflect trend profile period high original causal spurious multivariate precede spike spike train perfect ms drop reflect delta spike successively event short interval spike start increase spike quick spike train narrow reflect decrease peak denote become prominent fluctuation eliminate move decrease increase reliability peak reliable spurious event first ms neuron correctly reflect average dissimilarity available train turn reliable spike train reflect rapid decrease accordingly decrease contrast causal peak spurious yet reliable identical spike train except omit second part necessarily neuron recently second evident neuron rapid firing reliability decrease spike piecewise linear one spike pool spike piecewise constant localize visualization per pool train even decrease interval actually small imagine example angle pool spike spike three step spike spike train instant spike train distance spike normalize closely correctly reflect dissimilarity selective temporal averaging spike train right selective average marked line average average certain idea significantly something obtain train external influence occurrence internal statistic exception neuron five correspondingly spike train spike spike train represent spike train spike spike train spike train fire five spike train mark horizontal red follow denote triangle time spike train regular selective whole tree external generate spike train neuron noisy half external stimulus vary stimulus amplitude horizontal green green triangle dissimilarity depict bottom f train stimulus level task neuron via visual inspection unable identify spike dissimilarity right c construct spike train identify link shaped line height mutual close repeat iteratively require minimum train obtain average separate dendrogram modify spike train train easily external stimulus relate spike tool code neuron stimulus stimulus representative neuron reliability minima order half trial stimulus varie spike half stimulus dendrogram decrease stimulus train consist spike train reliability supplementary train instantaneous selective individual interval example see spike group train manually assign correspond dendrogram respective dissimilarity average group spike train movie see supplementary train b dissimilarity average two separate regular spike train patient previous train spike spike knowledge university patient monitor prior description train patient later confirm formation brain variability exhibit fluctuation high amplitude fig remarkably time subtle continuous increase among neuron role spike variant multi patient formation spike train dissimilarity profile spike period offset marked vertical brain line la accordingly right selective temporal averaging depict selective interval horizontal bottom line obtain brain spike distance reflect record spike distance indicate brain fire neuron finding thus mechanism termination level provide understanding spatio functional spike region may insight variant aim series reliably instantaneous dissimilarity instant time discrete arise data dissimilarity signal resolution beneficial synchronization scale example observable eeg transform event characteristic potential spike shape carry minimal response spike maintain spike eeg time maxima make apply profile profile causal calculation average spike eeg duration transition perfectly minima mark red color dissimilarity profile apply trace trace move top ten signal generate channel mix mix identical eeg spike variant respectively black average box validate measure synchronization continuous internet eeg series www science university hz introduce independent maintain channel identical although decrease adjust maintain appearance regular eeg signal illustration application bivariate eeg use short channel depict profile alternate variation mix towards capable monitor zero identical transition observe easy identical signal deviation independent three contribution dissimilarity profile spurious obtain like fire add extend continuous eeg instantaneous reliability elimination capability track synchronization spike train fig possible visualize cluster supplementary evolve pattern spike train instantaneous pairwise spike dissimilarity instant continuous fig external non absence channel resolution neuron internal certain detect converge fire code localize stimulus reliability another possibility average application population response whether train successively reliable spike train possible spike train measure aim measure similarity control situation protocol stimulus repeat trial spike similarity approach trial select appropriate spike similarity response evolve extent train regular spike time spike past future spike spurious reflect uncertainty demand monitoring upon latter half regular spike capability track train possibility several role tool rapid decode brain device monitoring ensemble could lead furthermore
determine optimal quantization due oracle inequality co exchangeable array accordance co blockmodel blockmodel estimator dy b eq misspecification blockmodel serve proxy approximation stochastic blockmodel profile minimize divergence limit value increase specify blockmodel profile polynomial logarithmic growth generative long necessarily blockmodel leave open consistently sparse simulation suggest behavior blockmodel qualitatively similar across model theorem essence array yield underlie admit author furthermore time algorithms generative blockmodel suggesting exist clustering recently universal replace objective assume quickly absolutely respect implication accord blockmodel fit yield main technical enabling interpret clustering may clustering generative formally optimize yield appear relate co clustering bipartite adjacency partition empirical co piecewise define index proportion edge span subset mapping us encode use index co clustering partition binary array analogously resp partition subset k introduce co clustering admissible def nonempty zero objective mn mn mn line establishes least profile ab closeness least square risk profile likelihood average kullback leibler divergence equip serve establish exchangeable array accordance exist fix hyperplane support point closure convex combination point functional equivalently triple functional formally follow geometric hausdorff distance frobenius I schmidt metric base metric measure short subset nonempty hausdorff natural recall norm see h k since imply hold h also h result relate dense broadly speak analyze term hausdorff metric detail estimator require fast convergence estimator involve optimization turn hull optimize function interpretable exist yield connectivity criterion profile l yield x yx xy b ab ab dy ab ab ba kl ab ab expanding ab ab accordance choose apply blockmodel even misspecification identification cluster array co cluster equal size connectivity theorem attain polynomial regularity strongly dense small regular rademacher adapt ranking allow improve describe cover support fixing exist claim result upper lipschitz inspection measurable analogously count section st conjunction bind generate come rademacher devoted proof f nu clearly introduce rademacher complexity generalization sec analogy q serve define co blockmodel partition prove lipschitz condition bound random whose element without replacement lemma q construction comprise expectation invertible range exist analogously choose ready define complement complement compare well approximate set g expectation eq due hoeffding sided interval set satisfie rademach arbitrary generalization quantity similarly frequency lemma n auxiliary mapping close metric expand upper statement q quantity mn lipschitz iv claim e mn h f fp le h k least claim brief misspecification misspecification form parameterize monotone curve normalization maintain area regime introduce outer product separable stochastic blockmodel top percent relative excess risk decay zero row kullback leibler decay toward optimum grey horizontal figure separable array graph column regime suggest blockmodel despite misspecification array blockmodel profile algorithmic initialize coincide blockmodel class blockmodel contain excess percentage toward normalized divergence decay toward represent quantity obtain expansion blockmodel behavior square omit brevity address group class relations blockmodel blockmodel achieve limit rademacher nonparametric pairwise ranking set important seek thought covariate describe represent covariate take never effectively co misspecification rather assume real value value co utilize discrete otherwise technical consistently case blockmodel know number derive elegant proof implies give instance blockmodel setting good blockmodel able convergence prove first respectively lem prove support directly apply theorem satisfy ab ab l h ij ab ij satisfy dx dy ab dx f ab ab dx dy k dx dy follow similar mn b ab similarly dy ab dx dy h provide support use establish section state generalize definition arbitrary induce generalization rademacher lemma u k proposition support lemma lipschitz chernoff q result conjunction union recall mn h h let statement proof lemma st st tw ij lipschitz st st differ term sum st w therefore union statement recall deterministic variable respectively imply uv u recall definition uv r expectation side argument q nonnegative identity define expectation show observe tt tt side substitute yield statement first pt ax bx ax bx informally either group attractive latter analogously symmetric follow define follow induce likewise otherwise increase may interpret hoeffding I uv convexity mn pt equally convexity linearity expectation mn independent argument mn r I mn expectation induce fix apply conjunction vary result partition vary analogously partition hence dt mn hold rademacher hoeffde apply condition argument theorem theorem prop tr f lipschitz imply h proposition prop hyperplane prop choose k c excess quantity l lemma establish plus triple blockmodel show involve maximization learn ab establish analogous argument let maximizer exist assign quantile class quantity pt line third definition corollary office award nf award w nf uk sciences establish ep ep fp ga european establishes address co partition array exchangeability provide rate correspond profile minimization blockmodel nonparametric asymptotically connectivity underlie generative class tool wide rapidly grow use
pointwise poor alternative average choose minimize leave right table improve adopt determine regularization ridge although perform bad variability result highlight make expect perform draw intractable loose criterion mean neural bic well outperform entropy winner critical influential analysis comparative review one order qualitative dimension comparison relative illustrate compute relative example bic square outperform affect choice summary requirement suitably hyperparameter sensitivity uninformative understand abc perform efficient abc carlo carlo sampler implement practical decision dimension potentially bad sampling derive example chain monte carlo base arguably necessity evaluate evaluation diagnostic leave procedure serve comparative acknowledgment discovery dp national width width comparison observe summary implementation summary statistic rather full central question summary observe minimal article provide split exclusive subset regularization subset selection ridge regression reduction three typically update refers family computationally intractable feasible become statistical research see distribution first ps ps summary p informative ps intractable approximation construct standard density argument explicit intractable rejection sampling term proceed draw summary straightforward ps ps intractable posterior avoid weight suffer dimensionality help algorithmic approximation weight consider sampler chain monte carlo practice typically smooth small may balanced much effect allow small turn burden markov sequential adjustment aim explicitly simulate summary achieve trade summary highly dimension statistic abc choice comparison reduction characterize exclusive ii projection iii reduction within bayesian regularization involve evolutionary fitness mutation drug production article section classify review reduction outline comparative section conclude abc informative candidate loss dimension information analysis information vary exception establish statistic implement data summary dimension broadly classify exclusive class evaluate criterion contribute criterion criterion argument subset demonstrate importance choose final consider combination statistic layer abc framework whereby response transform predictor include feed expect consideration abc use however summary shrink regression toward uninformative contribution discuss idea adjustment strategy abc suffer curse dramatically increase adjustment aim avoid explicitly describe regression adjustment notational simplicity assume adjustment use eq draw prior zero assume estimate ed ed adjustment control variance trade reduce effective accept flexible conditional variance residual ms estimate flexibility produce adjust bayes alternative adjustment summary dependent variable subset simple summary statistic exhaustive enumeration infeasible especially monte result greedy principled reduction statistic contain note relative score content justify practice implement conceptually whereby posterior differ threshold specify particular procedure various follow definition summary statistic acceptable determine statistic choose sensitivity aside obtain statistic observe statistic unlikely generate considerable especially continuous discrete acceptable restricted reduction entropy shannon w minimize subset entropy posterior sample write denote empirical remainder posterior w work weighted minimum entropy summary stage assess author root eq compare component scale component wise naturally parameter unknown summary treat leave minimize likely vary observe minimize simulated set summary statistic minimize subset summary directly posterior know truth computational expense procedure stage example occur distributional tail informative exclude suit stage argument consider entropy easily alternative technique reduction abc criteria suffer require technique combine nonlinear transformation order informative advantage well number statistic handle uninformative addition accounting disadvantage project interpretability method projection seek explanatory high correlation dimension explanatory suitably cox response optimally correlate summary choose square leave validation response inspection minimum argument disadvantage square aim global relationship draw distribution orthogonal component truncate ps pi region pilot feed nonlinear generalization regression propose network learning dimension regression unit q first transform statistic combine neural neural neural network possibility hide unit rather dynamically hide specify parameter infer minimize least neural term decay neural network idea regularization criterion use theoretic previous reduction summary ensure p require good parameter assume model prior noise vector potentially use function consist power fit base truncate prior ps pi significant posterior pilot sophisticated alternative regression summary summary inference considerably regularization approach overfitte abc adjust uninformative summary regularization could greatly inclusion ridge adjustment alternative dimension abc method include comparative adopt summary abc specification computationally tractable idea likelihood spirit use estimate pilot regression adjustment call seek good bic evidence attractive contain bayesian choose develop genetic weight contribute equally simulation uninformative ideally negligible finally recent find way accurately full avoid find conditioning potentially joint density likelihood weight variance component depend induce marginal practically useful writing construct py dy computation ap conditional simulation analysis last hide iterative analysis processing importance whose claim curse abc method derive aic bic construct second modification fit available information goodness statistical maximized likelihood measure py determination equation aic number define effective simulation arbitrary insensitive favor uninformative aic subset summary kernel residual size aic replace involve could g adjustment element vector aic seek summary minimize variance adjust criterion section select however construction include predictor requirement informative range outside uninformative criterion section linear weight describe univariate notational clarity exposition approach weight least adjustment adjust uninformative summary avoid implicit reduction regularize square regression adjustment toward impose magnitude note consider alternative regularization procedure additional pt ridge weight minimization pt deal generalize average comparative reduction study analyse literature include evaluation mutation drug production clean simulation draw performance set evaluate value divide parameter standard first comparative ease summary adjustment score within abc euclidean statistic simulation nonzero choose summary size keep slightly adjust follow exploratory analysis fit use take pointwise obtain summary median function start choose chi model previously propose strategy focus scale mutation scale dna sequence individual site mutation mean distinct examine abc dimension reduction six genetic table site show regression regression method pt c perform regression adjustment substantially bad summary summary exception pure next reduction relative reduction regression adjustment contain obtain adjustment supplementary parameter combination reduction aic criterion partial pls networks loss ridge ridge obtain adjustment summary good row c c pt third example integration achieve statistic substantially uniformly improve partial net adjustment six population improvement standard adjustment reduction aic partial loose dimension compute adjustment relative follow adjustment bic net perform stage entropy gain evolutionary plausible mutation fitness transmission evolutionary incidence marker transmission relative parameter transmission transmission drug mutation stochastic statistic number diversity proportion case replicate reduction technique univariate abc well analysis particular proportion marker mutation distinct regression adjustment inferential performance argument favor marginal whereby replace estimate precisely margin reduction aic bic substantially adjustment depend aic result increase reduce clear improvement almost aic regularization mostly comparable improvement partial square produce result adjustment partial ridge neural posterior loss
event diagonal get put complete subsection take define detection assume sequence test discriminate converge pt price pay multiply minimum gap remain rate therefore tight sdp simple large inspection proof hold arise dual indeed control show proposition statistic pt employ hard quantile dataset behave sdp one strictly diagonal support problem seem qualitatively rank unnecessary indicate plant large sdp parameter section original functional sparsity notion order enough zero exclude h make notion sparse detection hold test old let vector exist unit vector eq generality otherwise v p p appropriate base let approximation proposition since statistic achieve context relate examine maximize quadratic argue solve quadratic form choice hold follow step discriminate define value sub coefficient I nx condition p replacing hold probability similarly two observe rely fact weak set attain optimal assumption constant initial situation necessarily probability n low take one discriminate test difference phenomenon method proof limitation hardness rip attract lot complexity theoretic impossible statistic arbitrarily clearly suffice achieve reason first polynomial approximate input hereafter develop another plant clique problem inspection level prove argue unlikely let q hinge optimal rate contradiction bound polynomial plant clique hereafter argue algorithm unlikely vertex place connect pair edge plant clique problem call vertex goal h consists planted traditionally plant surely appear even algorithmic relaxation fail provably hard see bring toward hardness note capable finding plant currently tensor yield polynomial lead researcher hardness plant clique include dependence equilibrium vector adjacency rademacher random take construction vector long qualitatively manner subsection illustrate method argument complexity begin show plant even plant cardinality surely otherwise begin side variable n display side lem inequality would detect presence empirical valid soon replace one clique probability k k pc hypothesis subsection propose approximation sdp tolerance recall plant clique control appropriately soon constant clique consequence way one clique intrinsic sdp reach level intrinsic polynomial computable control element sdp achieve dual however polynomial prohibitive statistic latter problem grid purpose illustrate empirical compare x x vs unit dimension yield matrix density hard distinguish particular statistic discriminate set almost discriminate discriminate long bring evidence consider tight factor need discriminate soon minimax behavior illustrate phase test different ii close predict reciprocal setting exhibit suboptimal scaling exist hold actually transition parameter sharp choice q vs vs distribute left level compare demonstrate critical indicate scale existence pay technical appendix various concentration inequality ce variable generalize sum sub gaussian chernoff tail follow bound expansion chernoff define chernoff e st st display support wu fellowship support grant dms level sparse principal high covariance optimal know np alternative prove detect principal theoretical science bring evidence inherent sample set sparsity belief explore high pca classical pca produce inconsistent namely relie explain assume identity sparse assumption contribution word may ask principal explain statistical answering consist construct detect associate variance ii prove test level lot vector propose detect shift sub encode two focus aspect problem detection close shift detect diagonal plant extend direction sparse precisely minimax usually notable exception unlike extended linear refined exhibit qualitative light important size testing capture although construct optimal raise prove topic consist overcome reference therein solution statistical address introduce call programming sdp drawback direction semidefinite matrix eigenvector notable allow simple near importantly bring support evidence evidence build conjecture plant clique well conjecture vector robust model discuss notion sparsity distance covariance sup follow discuss link probabilistic spectrum wishart detection particular test spectral principal probability prove sometimes become computational relaxation true whenever th row denote define norm form define radius trace rank functional denote usual identity submatrix real number copy random objective h covariance sparse euclidean large two imply perturbation weak critical minimum coherent hypothesis testing existence direction exploit submatrix affect perturbation matrix define eigenvalue equivalently behavior hypothesis govern nan begin statistic enough alternative probability unit sparsity define use matrix general satisfy condition indeed support eigenvalue wishart adapt desire bound q net sphere easily q since yield subset subset cardinality fix observe cardinality get complete sufficient lambda whenever follow previous subsection find h regime take provide discriminate converge goal exist joint hold infimum find distribution let proof get determinant moreover formula substitution display q eigenvalue together equality substitution desire turn proof yield inequality expectation
however per rest denote hoeffding follow principle rational multi armed high establish use action high must pure exploration arm bandit stationary work well rational policy heuristic policy instance armed bandit reward arm multiplicative ucb aware sampling computer particularly engine extend aware ensure aware average per engine engine still player advanced phenomenon win vs node outperform work suggest monte policy outperform pure largely aware estimate use policy differ decision principle david il monte carlo decision ucb policy armed mab minimize cumulative search differ mab arm move reward value empirical evaluation comparison ucb mab instance go appear although method successful appear successful past job trading exploration exploitation statement ucb simple principle scheme outperform adversarial search nature good computer go show improve monte suggest approximately optimal decision mdp explore trajectory termination cutoff multi armed bandit problem mab mab ucb near state art search get reward policy mab
problem cast shown adaptively rate sparse set general tend precision belong liu lin good conditional cv theoretical unclear contrast procedure minimax begin later attain precision norm loss rate observe nonzero need consistently bound rate convergence expectation constant p n theorem z p z z I depend analogous lemma hold depend thus prove condition similar way theorem mainly rate wise wise graphical rate norm theorem hold norm p pp cm p precision define theorem attain collection indeed establish minimax estimate discuss technical outline section particularly suit treat directional matrix generalization le tensor r lower alone parameter like bound maximum estimating belong need state density dominate measure variation affinity r rd vary establish non zero matrix observation projection r aa aa vary remain equation apply sharp major norm detail difficulty essentially finally calculate factor leave sec shall major yield precision satisfie make clear component sum less equal later associate equation immediately equation p lemma separately lemma comparison define q sparse collection put put theorems collection ball ball define w graphical enyi random random uniform generate end additional select choose sample obtain parameter select minimize cross liu various setting table loss compare tune three ccc band ar ar ar norm tend risk sparse covariance cf important distinction depend difficulty precision mention introduction equivalent undirected independence contains order indicate additional depend imply threshold recover context liu adaptive dependent thresholding recover sharp var much difficult gaussian liu transformation apply graphical neighborhood selector adjust correlation detailed leave prove key technical begin collect technical technical lemma px n suffice great ik jj jj jj j jj jj ij I ij classical general matrix proof contain set yield w p belong feasible p pn lemma belong n prove theorem np cm pn cn pn prove estimator minimax estimator projection inequality follow establish lemma p ph w n yield without assume affinity affinity without case prove constant affinity chi calculation triangular matrix possibly leave single normal r normal precision form nonzero element submatrix observe number row value upper triangular p place prove chi proof straightforward omit precision simple total immediately determinant subset j simplify notation r ready prove recall overlap equation imply equation last establish enough equation equation equation show v special simple first eq eigenvalues jk n k write nonzero equation drop upper triangular thus n p nk equation follow definition wide multivariate high convergence adaptive minimization propose parameter space implement step establish derivation sharp directional technique minimax low optimal convergence sparse adaptively minimax collection space simultaneously minimization graphical primary secondary fundamental many high knowledge analysis furthermore broad application portfolio lin precision tool relationship scientific recover equivalent liu liu distribution recover draw recent attention b matrix introduced estimate lin study also et fan et selector precision liu rates spectral many various unclear estimator estimating term serve fundamental benchmark goal present establish matrix norm parameter sample variate matrix ij order consistently zero graph symmetric spectral equal matrix model ball away collection range particular low liu thresholding variability entry paper introduce detail study estimate precision establish match minimax rate compare frobenius discuss connection difference proof minimization minimax establish together adaptively begin notation vector
differentiable neighborhood statement twice em satisfy let continuous q lipschitz subdifferential exist bring great deal design nonsmooth continuous obtain nonsmooth locally lipschitz computable certain arbitrarily hold every lipschitz subdifferential everywhere imply next divide optimal value relation conclude hold hence rule lipschitz continuous everywhere eq recall relation immediately iii every statement subdifferential em nonsmooth lipschitz arbitrarily eq statement hold lipschitz f lipschitz fact imply statement iii iii em nice find computable let order stationary moreover nonzero let contradiction definition contradict substitute multiplying use immediately second minimizer statement differentiable hold low immediately ii let minimizer notice order satisfy proof local solve also variant subproblem moreover unify variant subsection reweighte propose sequence vector q close provide unified analysis suitably solve apply minimization choose arbitrary weighted accumulation point first point accumulation hard q k kf x k f side definition stationary em reweighte whose computable variant subproblem simple form g minimization choose minimization k go go set outer inner criterion iteration th modulus x relation inequality imply outer k next accumulation order problem suppose accumulation stationary subsequence continuity f conclude optimality side x stationary generate accumulation multiplying conclusion present variant subproblem solve method minimization method scalar apply end scalar accumulation accumulation exist subsequence side equality problem iteration variant close moreover unified present em ik next establish accumulation suppose accumulation ks argument k ks increase combine inequality accumulation subsequence definition g p fx eq complement claim indeed see imply k kx yield observe equality stationary em reweighte computable subproblem simple form choose I step go iteration termination criterion inner iteration strong modulus inequality g kx kf kx kx whenever em show accumulation generate stationary problem accumulation fx fx converge outer argument let subsequence optimality take limit equality I proof study parameter dynamically adjust approach whether iterative reweighte share adjust address variant subsection locally computable certain stationary view iteration remarkably establish accumulation choose minimization ik point accumulation moreover nonzero satisfy bind n gx q accumulation subsequence monotonicity gx limit side recall imply substitute obtain stationary monotonicity result order rest sequence computable arbitrary arbitrarily weight go outer inner inner termination let th outer iteration modulus last due yield f sf similar accumulation point satisfie accumulation order hold nonzero accumulation point subsequence monotonicity denote value outer iteration proof condition limit rest conduct numerical datum generate convenience matlab mac spectral gradient choose supremum terminate accord integer one denote generate vector zero deviation choose list value column second give mention cpu objective former cpu c rr c c generate instance cpu report point objective instance method similar cpu much rr cpu c minimization minimization derive bound entry order stationary local variant unify addition lipschitz develop threshold value remarkable reweighte dynamically computational new generally stable cpu li propose minimization solution regularizers method discuss reweighte regularizer pt corollary support grant study minimization nonzero order iterative reweighte minimization solve subproblem unify addition continuous develop show accumulation remarkable iterative reweighted dynamically update demonstrate exist cpu key iterative reweighte minimization reweighted finding system regularize nonlinear programming observe minimization iterative hard penalty widely function one
scheme towards towards towards give short summary two stage stage stage lead trivial corollary prove np prove np hardness trust relaxation lagrangian quadratic give scheme well consequently well scheme decrease towards bound towards dispersion relaxation scheme stochastic framework successful uncertainty environment context interact efficient strategy design policy maximize reward bad adversarial receive observable generalization implicitly return consider compatible deterministic mostly environment aim policy strategy suppose know optimal occur work uncertainty robust studied formalize state discrete whose element denote element finite action horizon instantaneous r sequence action return optimal action far mode dynamic action transition set transition learn transition sequence u originally exist finite action lipschitz coincide system return intuitively action initial u subject bold min generalization aim bad return lead focus design resolution set action generalization problem assume e horizon step two case relate want safe clinical clinical subproblem section f u p x u matter drop argument refer bind two subproblem r subject obtain solve correspond stage except action trivially prove provide u drop first relax constraint u solution u intersection size also lie exist k depend triangle inequality replace satisfy index replace satisfy focus subproblem x p focus resolution element u intersection lemma cardinality l maximize u constraint low order side equal solve constraint prove introduce problem determine np feasibility feasibility n z px np reduction exactly similarly belong compute equality prove linear exist q point hyperplane choosing obtain suppose satisfie imply prove hyperplane dy prove ball tx furthermore inequality prove hardness choice space follow hard hard np hard equal time optimally solve general stage aim approximately obtain want propose tractable drop problem show provide relaxation constraint interior prove scheme relaxation scheme prove computed relaxation converge dispersion section two subproblem solve straightforwardly cf theorem problem drop suggest constraint drop index relaxation give u tr u radius minimization determine hand side literature value lie lie distance equal family relaxation consider combination relax relaxation make constraint trust tr tr f trust region provide exact original optimization tr way lagrangian section second multiply constraint variable constraint dual value decompose square observe quadratic fix soon interested never unless follow inequality never convex closed formula rest proof useful come write convex trivial write ml observe notation linear know formulate ml compute propose address return p l bind compare case relaxation first two k region bind always great u definition possible k tr construction equal prove due k bound two corollary let x tr f u tr always preliminary p u u u k tr tr f tr relaxation transition k u u x l x definition lagrangian relaxation bind u u section tr b consequence theorems lagrangian derive problem introduce action bound dispersion decrease zero dispersion equation dispersion intuitively see visit lagrangian dispersion u u tr f u u tr end f denote trust maximal maximal tr tr b also sequence action u u tr tr action lead return dispersion trivial u u tr us u sequence j bind satisfie relationship j b c u f end notice scheme dispersion suffer curse action actual lead state trust lagrangian bound benchmark x reward u denote th lipschitz state c u tr transition follow f u maximal maximal trust relaxation transition f tr lagrangian relaxation great trust equal ia ia computed transition optimal action cardinality order quality protocol cardinality space f kb maximal tr resp transition tr c u u tr sample transition observe lagrangian much trust bind close sequence policy show problem relaxation extensively perform min continuity reinforcement imagine environment continuous would interesting scheme problem space acknowledgement scientific research present system office author nesterov point relaxation author trivial trust tr r x u tr f f quantity
jx functional right hand homogeneous compare fx jx jx continuity exist open vx x jx jx jx jx let mx x mx vx x p vx un ni ni closure mx un I vx cover define around jx jx kx j sx ds gp may projection orthogonal g vx vx p fx vx x compatibility fx fx vx sufficiently kx jx x jx p kf mx mx hence kx take jx I c jx exist jx together property dimensional contradict cone sign place continuous kx b depend sum kx kx b kx twice interior since qx dt much jx c jx ds satisfy denote exist bx constant distinct satisfied constant distinct number positive constant x lemma fx fx n n thank p na fx ii replace lemma sufficient formulation worth state toward distinct number j j j j lemma p p p n x n dt n fx x dt x l fx dt note fx fy fx fx c n complete fx fx dt fy fy dt c n sufficiently n q complete admits support sequence compact set continuous observe proof lemma sufficiently representation replace n change line k bn ii give proof consequently conclude condition iii replace implicitly proposition proposition replace supporting imply c c x x x sufficiently way large exist constant x c constant p x exist c x n n event fx lemma original integration keep order exist help dimensional type limit inequality answer field connect newly manifold proof theorem obviously satisfied shall constant diameter may sn kn sn sn k sn sn sn sn subspace process tb wiener increment wiener theorem use brownian sn integral aid continuity boundedness old inequality yield bn sn sn hand term x x u ss qx sx sx sx sx eq consequently process likelihood examine asymptotic maximum value bayes estimator estimator independent deviation good bayes maximum estimator c mean c example previous standard deviation maximum type estimator compute likelihood estimator behavior furthermore small theoretical point view estimator performance mean mean criterion derivative shall remark qx c jj lipschitz x p satisfy theorem open ball open may call sufficiently sufficiently neighborhood moreover continuity contradict sufficiently intersect fx cn large remark theorem standardize write du nu later du rigorous time n dt eq complete dt cn dt u ds sx lemma terminal discrete martingale I n h every u h h every eq j easy proof check regularity satisfying yield complete every exist nu nu q nu obtain desire use consequently q note z nu nu u nu calculation yield dt dt b formula dt one decomposition second enough f k since k ds j n k nj ds theorem show bound random depend u show finitely show family r see nu nu complete proof satisfied condition setting complete checking existence obviously support support neighborhood necessary logic depend location increment choose continuously topological support besides side necessary effectively together theorem general derivative include start move quickly possible derivative fx bx sequence pure york diffusion coefficient estimation multidimensional diffusion ann coefficient normality diffusion r posteriori york conditional de observations yu rao exposition identification dynamical yu inference spatial york yu statistical inference ergodic le l asymptotic york le asymptotics york observe process journal statistic quasi jump appear statistical process rao asymptotic square process estimation jump discrete observation infer asymptotic differential dynamical estimation martingale ann related process n inequality quasi ann deviation institute mathematics deviation u definition lem example university sciences university mathematic quasi type deviation become index nature make na I require quasi estimate key phrase normality convergence observation deviation classification secondary study paradigm analysis p normalize factor tend tend continuous simplify condition p estimator since sup u sup u pp asymptotic mixed likelihood convergence separation moment derive process include serious process prove convergence exponential type place serious likelihood statistical field connect stochastic example mix drift convergence dimensional wiener respectively value dependent course finance asymptotic theory estimation unknown volatility frequency study refer rao ergodicity diffusion perturb limit score become distribution mixed quasi likelihood scheme highlight field appearing find prove wide applicability see limit correction divergence distribution obviously necessary standardized provide kind high statistical large deviation enable valid high estimate field quasi develop analysis quasi quasi likelihood bayesian give type inequality locally carry process le le notion setting ergodic diffusion difficulty besides exact expression calculus complicated jump involve rate associate drive evy positive index moment likelihood diffusion process jump deviation sampling polynomial field meet question difficult na I far quantitative author part differential equation dimensional wiener measurable process bound locally boundary lipschitz true asymptotic mix normality convergence contrast bayes unknown volatility type main context lx n set admit extension qx sx sx process admit ds take wiener time admits support function exist vx j vx jx satisfy qx fc jx relax strong condition degeneracy unless random thank exact available inference quasi satisfy bayes estimator dt standard article fulfil growth fulfil polynomial growth following suggest degeneracy easily satisfy differential dx qx fx case thus ii obvious w stop necessity introduce x satisfy stochastic wiener value differential dx sx sx degenerate neighborhood sx sx fx function x difficult fix time I however na I devote proof likelihood random
robust describe fista solve cs extension efficiency finally vi conclude paper matlab reproduce website sense cs compress compressive setting recovery pose establish reliably strength various solve cs recovery cs cs induce cs recovery hold normal cs inefficient cs underlie additive achieve quadratic g ir huber penalty derivative huber penalty outlier detail linear actual solve trivial cs option literature mr reweighte outer involve iteratively double loop limitation one loop powerful framework suitable thresholde fista optimization effectively smooth sense share fista variant minimization update involve historical couple easy solve smooth decompose univariate analytical trick fista variable approximation historical update subsequently loss ensure proper second convergence original fista readily loss see remain compute constant domain imply mixed quadratic part negative cost alternate multiplier admm simple powerful optimization large arise develop advanced compute different recently unified framework robust technical challenge couple make extra easy tackle variable element wise group use know admm framework term smooth tie affine cs objective augment lagrangian augmentation improve numerical augment update dual variable concerned treat thus optimality lagrangian iteratively particular operation due iterative burden robust update difference admm algorithm solution admm cs essence replace previously major oppose iteratively modify quadratic recognize fista choice converge see inversion hence inversion update subsequent case state admm update approximate converge discuss stop update affect residual residual dual vice versa penalty trade admm generally emphasis primal residual admm error primal dual due work case examine detail terminate primal small please fista present tackle slightly angle fista simple admm regularization fista admm split quadratic requirement numerical experience tolerance fast fista realize need cs make fista extension simply case would model priori cs lagrangian equality solved previously replace primal sufficiently residual e requirement stop wish regularization may realistic quadratic loss g elastic robust formulation treat special recognize convert solve many efficient trivial modify admm indeed update step fista quadratic fista need induce operation likewise case generalize solve slight modification mix propose admm huber select suitable framework restrict huber indeed type note fista easily differentiable overcome difficulty non apply framework introduce rewrite thus scale rewrite easily principle solve yield exact recognize stop vector include k k cs reveal basic sparsity exploit far knowledge exploitation sparsity effectively requirement comparable recovery compare slight cs haar wavelet image could example video circumstance similarity consist background move illustration sequence bar later exist cs likely task perform denote recover cs task cs formulate g l loss along reflect prior correspond wise respect single quadratic formulation extend fista soft result prove solution lie point satisfie e intersection minimize shrinkage generalize I I write solution meanwhile rewrite thus decompose row immediately step resort loss admm update k task fundamental cs formulation framework example affine set investigation practice robust recovery constraint construct residual exist become residual meet happen robust cs robust combine coarse fine search find robust bar admm variant inversion image inversion directly much algebra exploitation reduce dimension cholesky accord direct fact lemma note cholesky factorization whole regularization cs recovery bar admm compare nest nature double loop nest robust cs cs solver loop select solver speed know expensive numerical experience show inner solve high solver robust cs case see iteration inner outer compare approximately computation shrinkage examine take former insight iteration respect solution iteration take threshold solver inner cs termination tolerance within loop residual nest cs algorithms matlab roughly bar db noise gaussian time versus plot achieve initialize indicate cs considerably respect loop converge normally clearly profile iteration accuracy admm fista need number actual take tolerance advantage algorithm nest interested tolerance admm fista algorithms nest fista robust bar image top haar wavelet show method minor different robust next examine much improvement cs affine robust cs formulation formulation know examine robust bar example select error take admm robust take reach minor affine formulation random bar recover behavior affine cs formulation huber noise random noise huber new remain behavior admm cs show behavior right observe slow accuracy compare huber loss admm theory next examine cauchy fig cs use nest respectively completely fail meaningful recover nested maintain recovery quality regularize admm cs computational admm cs behaves model use previous show formulation provide db huber thus despite less favorable cs well loss admm robust cs demonstrate usefulness robust separately multi cs formulation share bar frame bar move block across frame obviously wavelet bar distinguish clearly illustrate image haar wavelet bar image common admm multi task show fig cs cs recovery cs recovery every clearly robust cs formulation
validate moderate modification amount noise motivation component modification could nonlinear difficulty component robustness measure different different noise curvature validation base nonlinear term nonlinear input explicitly model find set weight attempt auto nonlinear preserve validation network problem non nonlinear ability pca model model miss approach adapt neural pca nonlinear pca complexity lead error validate control new datum validation exist method validate optimal validate curvature even nonlinear pca perform supervised architecture unsupervise validate nonlinear pca able component fit decrease distance effect use function plus standard error datum large fit clearly specific unsupervised label target correct onto curve error architecture compressed generation locate curve pca use perceptron mlp auto encoder auto force square extraction generation hide layer nonlinear mapping layer middle hierarchical validation pca done generation mapping lose input nonlinear control decay add coefficient change component validation fail evaluate nonlinear missing specific nonlinear validate estimation sample reject nonlinear decay get nonlinear pca repeatedly initialization network validation miss demonstrate ensure embed nonlinear component circle projection blue dot complexity high consist lie manifold loop embed factor apply network optimize weight parameter loop miss incomplete per standard test validation repeatedly newly median missing run compare approach able minimum validation small complex fit contrary supervise fit validation optimal setting coefficient weight decay force nonlinear linear force solution well impose pca whether unimodal multimodal e ni nonlinear incorrectly detect inherently unimodal point illustrate obtain important show validation miss validation plot set nonlinear pca test contrast optimum absence test fit produce incorrectly distribution unimodal miss validation dimensional unimodal multimodal nonlinear set nonlinear give increasingly complex validation lack class complex pca high describe complicated cover space complete moderate onto restriction pure validation contrast curve remove target predict test datum validation remove unsupervised validation validation auto pca behind miss unsupervised remove validate correctly availability software pca auto neural nonlinear pca neural
constant nx kernel define fx nx constant k asymptotic confidence hence investigate band procedure profile elliptical galaxy base simulation simulate bivariate unimodal include dataset convolution correspond choice slightly bandwidth pt nominal nominal computational feasibility scenario preliminary motivated bandwidth band normalize figure illustrate decrease normalize average unimodal correspond small coverage band wide band decrease significantly figure bivariate band set increase precision band often spread use methodology profile elliptical galaxy pixel galaxy image model convolution sharp optical detector digital couple device pixel space dimensional elliptical galaxy band nonparametric estimator narrow centre galaxy centre normalization uniquely determine control curvature profile model coincide de detail already classify galaxy consideration galaxy analyse give probably point source middle advance bandwidth choose band unknown profile gradient see due partly cause bandwidth whole order region galaxy restrict check band satisfied parameter extensively dd jj dimensional direction j vector contain pt l one axis give example pt b b alternate b regard increment rise assertion decompose sum part intersection intersection axis auxiliary throughout convention nk hx proof weak brownian detail precise n da limit independent variable rt l assertion accomplish purpose introduce g notation brownian next consist replacement process quantity regard increment u step pt q pt negligible pt assertion representation straightforward nx da df df z integral r n n z nx ff hx r fx x x r n f h h da da n f embed observe convolution integrable f proposition application taylor h nh kernel hold h inspection show omit brevity replace precise calculation n rewrite increment sum express increment give n da increment accord account least dx rewrite sum index sum n application version recall obtain imply existence x du nx h k n du nx n nh nh nh nh nh jx bb bt bt bt db bound part wiener rescale brownian u nh x nh pt nh nh pt increment correspond pt nh u nh nh nh g nh obvious manner modulus brownian pt n bs bt dominate argument nh modulus continuity obvious sufficiently nh nh nh nh da nh law logarithm nh nh nh nh v uniformly negligible brownian nh nh nh nh nh u brownian nh nh nh nh u lemma contribution quantity obtain nh k u give k definition yield x p b p lemma da limit depend repeat lemma b df note pz df set z df px u minimum maximum observation yield n desire argument proof error partial sum replacement fold away helpful stein parts manuscript technical support center foundation style graphic remark asymptotic band multivariate convolution method increase set particular confidence band feasibility elliptical galaxy provide band application impossible observe inference problem type medical physics biology express well emission involve heat reconstruction imaging device closely operator study deterministic physics example numerical reproducing space investigation importance point view field construction convolution square recovery device biology observe image slightly physical propagation light surface application true mathematical observe image approximately convolution real call problem convolution regard problem become nonparametric band develop band dimensional motivated dimensional typical reconstruction also spatial behavior spectra like variable star depend refer predictor univariate investigation weak sup appropriately construct band periodic reconstruction biological device often extend construct band independent supremum center derive bootstrap confidence band band band local bootstrap van deconvolution band asymptotic confidence band asymptotic cut estimator indirect limited estimation univariate regression author band band standard extend straightforward manner multivariate multivariate case inverse band require present band convolution predictor originally motivate asymptotic band small strategy long sense restriction whenever constant kind convolution distinguished convolution determine degree class moderately pose decay precise subsequent existence b specify understand difference fast enough relevant structure result remain decompose two sufficiently fast completely illustrate h h kx definition f ff kx example condition factor polynomial imply odd expansion obvious even situation h taylor h give x nx straightforward dimensional exponentially tail assumption transform square assumption fourier transform weakly differentiable integrable derivative satisfy moreover
walk reversible add weak connect vertex direct particularly become diffusion graph place diagonal orthonormal eigenvalue place order diffusion state euclidean distance embedding time approximate depend spectrum top computed case align eigenvector respective distinct alignment extend union embedding define allow thorough policy choose piece low cut spectrum direct determine cut sequence cut establish partitioning sophisticated also free version truncate walk evolve restrict find eigenvector eigenvector small trivial cut range small eigenvector give side minimum identify state state side cut bottleneck give stop meet algorithm subgraph cut recursive application bottleneck partition associate spatial scale determine scale scale discover mdps pre compressed mdp transition matrix strongly type diffusion policy region light structure mdp accord goal strong multiscale multiscale suitable mdp mdp coarse mdp original event mdps relative indeed think inherent rather mdp view coarse fine coarse describe carry coarse coarse mdps learn detail bottleneck explain mdp priori simply choose suggest determine mdp coarse compatible fine coarse scale mdp tuple brief primary mdp subsection scale section bottleneck bottleneck discount collect trajectory reward along trajectory chain optimal fine coarse compressed respect scale vice type allow instance related part may always accomplish computationally simulation compute computation parallel prescribe cluster compute interesting relevant computation eventually multiple independent comparable section analytical proof multiscale read skip multiscale mdps obtain construction fine policy small diffusion everywhere regularization partially follow initially towards coarse iterative process fine transition guarantee except situation cluster boundary bottleneck relevant subgraph induce restriction p incorrect task error easily lead compressed boundary equal policy cluster correspond system discount sx ss compressed mdp refer restrict run execute policy mdp large circle bold fine vertex intersection undirected coarse dark gray cluster fine policy state execute execute feasible reach back top hand execute reach compress dependent discount even problem cluster loop path start reach compress another fine neighbor well blue draw left loop edge fill draw node draw fill rgb grid fill height loop edge loop loop edge leave edge two well thin two loop show per coarse coarse reward action see detail refer action trajectory restrict determined state cluster action reach either carlo compute analytically trivially advantage parallelism box simulator need slow develop light quickly separately solution form consider bottleneck respectively compressed may obtain non solution computing action discussion appendix derive reward discount reach define define expect discount reward trajectory bottleneck state cluster associate cluster repeat process example scale mdp depend action compress mdp coarse computation involve give accumulate variable start bottleneck concern accumulate reward discount reward solve scale action correspond denote characterize p aa h q system course discussion consequence length apply calculation negativity compress mdp still insight involved path necessary characterize hierarchical discount coarse scale experience consistency precede coarse fine depend length coarse mdp coarse discount correct discount path length transition partially correct simulate fine impose coarse uniform discount discount incorporate coarse compatible policy accelerate procedure policy cumulative discount tt proposition discount negativity coarse correspond denote markov chain boundary discount scale let compute non linear discount fine pair start coarse jensen imply ts loose connection length discount potentially context simulation suggest calculate proposition average previous subsection solution course value law terminal mm bottom font skip thick black every join style corner style style terminal solve coarse sp xshift sp row xshift xshift row point join p sp join sp join sp join branch sp join join join join join join sp join join sp join join sp join join loop join branch loop join join inner mm inner multiscale mdp mdp obtain several suggest flow fine update fine coarse update coarse coarse bottleneck collection independent iteration fine fine scale solution may update represent loop latter outer loop pass update instance bellman appear asynchronous iteration compression fine policy solving successive fine coarse compression iterate allow move fine therefore repeat transfer pt mdp policy coarse save fine bottleneck rest fine global policy set criterion meet state repeat average bottleneck criterion meet way recursive application particularly top bottom reach bottom compatible mdp path goal enforce coarse fine coarse update information update local readily handle successive describe localize fine bottleneck give comprise fine compressed absence specific reliable policy coarse action agent involve coarse cluster bottleneck vary depend provide additional action coarse mdp algorithm mdp convergence coarse bottleneck fine local cluster determination step think boundary cluster bottleneck interior explain solve require interior interior new policy updating bottleneck state use globally combination interior bottleneck averaging boundary execute convergence guarantee value modify monotonic condition note find policy iteration cluster bottleneck rapid hierarchy distinct piece efficient algorithm fewer primarily reason multiscale start coarse fine good otherwise iteration multiscale treatment offer since give policy constrain mix time cluster slow within rs context solve hierarchy coarse fine fall compatible range conjunction coarse realize precisely optimal fine local fine compression coarse mdp likely close coarse summarize cluster behind coarse get fine bottleneck cluster large reward outside may advantageous visible locally thus policy cover alternative cluster r denote geodesic bottleneck law original modify terminal state rr mdp truncate modification discount original mdp rr range find give rise r action scale bottleneck adjacent trivially cluster across within involve mdp fine mdp fix coarse cluster solver inside outside illustrative policy initial least problem boundary poisson problem x bound boundary seek solve system system capture label interior label fix interior state given solve unchanged policy bottleneck state update block towards determination cluster need another iteration completely converge tolerance cluster reach along value require bottleneck quickly function process value bottleneck may update asynchronous cluster update step comprise outline bottleneck iteration modify iteration variant analogously determination characterize need partitioning capture bottleneck interior state fix known assign system ideal virtue likely product furthermore connection likely solve likely asynchronous policy scale policy scale pass per bottleneck satisfy interior policy unique first value interior state fine may determination infinite challenge assumption modify asynchronous converge unique corresponding initial coarse algorithm coarse mdp zero everywhere fix shift asynchronous policy find discount operator direct reference transition define section recursion repeat substitution fix large ensure optimality convergence action bound reasoning policy length l satisfy determination partitioning compression analysis assume step general use approximate stationary iteration however eigenvector sparse solution cost far example compute randomize approximate compute find sub graph accelerate substantially compression make compression local space requirement assess complexity complicate negative transition iterative square guarantee generally reach newton et al competitive previously classic routine practice unconstraine linear system appear proposition negative complexity per cluster coarse transition probability coarse coarse solve constraint necessarily negative involve time complexity naive case iterative linear example reduce solve multiple side side system determine complexity reflect quantity element complexity solving solve mdps dynamic iteration iteration number solve entirely assume significant coarse bad solving compare bottleneck markov chain fast successive scale everywhere step cluster construction update policy interior bottleneck state assume since dominate collection solve multiscale determine complexity strongly dependent gain reasonable good multiscale structure ease scale equal scale scale say scale iteration require scale level multiscale assumption require relative relatively attention challenge substantial impact broadly problem second well policy refer depend type transfer several form systematically knowledge argue novel systematic mdps multiscale discuss hierarchy hope meta transition part appropriate former imagine problem sequence distinct identify part already solve global remain part solve conceptual distinction transfer policy value reflect global structure problem easily translate much applicable easy consider assess section compute occur coarse scale single element quickly compute candidate dynamic sense potential respective advantage reward globally transfer systematic identify transfer general proceed problem whether transfer potential operator step detail section give multiscale mdp compute hierarchy select cluster hierarchy transfer match pair algorithm globally locally match pair cluster proceed matched transfer already remove possibility correspondence section determine operator solve problem retain policy solve algorithm variant start respectively section respective transition discount cluster index quantitie truncation restriction context interior boundary question otherwise refer appropriate throughout information case special case within scale consist explain scale see single within transfer diverse restrict belong match address match situation correspondence focus attention transfer cluster cluster boundary receive regard policy bottleneck transition cluster bottleneck form involve decide keep entire coarse scale simultaneously correspondence close metric natural average wise embedding underlie state respective map embedding sign alignment problem match cluster denote weight subgraph e occur scale auto draw width minimum height width cm ap ap bp edge ap bp draw ap draw bp r ap edge bend node bp w match subset sub exactly database transfer possibility policy knowledge compute mapping either take denote subset resp j aspect transfer action action action policy action policy entry uniform previous result used summary transfer everywhere compute local coarse hierarchy instance derive policy function apply initial coarse transfer correspond either diffusion policy apply previous scale stop return suppose coarse scale operator compressed discount exposition entire scale moment correspondence development idea equation sub scale value j define system collect reward expectation apply back compute reward operator action reduce situation unlikely determination operator advantage subtle initially result compression guess compatible coarse determination knowledge eliminate transfer potential determine obey correct scale markov provide warm optimal policy pair match cluster detect transfer policy whether diffusion current policy use warm correspondence check support transfer give current uv function compare describe equation improvement relative computation transfer underlie take heuristic warm cluster boundary assess assume dynamic collect transition dynamic transfer reward transfer proper boundary policy coarse remain reward probability possibility problem operator current usual dynamic discount reward determine comparison quality use establish correspondence transfer graph state edge graph instance goal graph establish role play state example terminal state desirable able match terminal state terminal sense representation coordinate could seek abstract specific match similar role across several section computationally otherwise actually require matching solution match restrict attention correspondence detect poor choice transfer coarse scale collection compute involve build median apply match research several min cost give diffusion either locally cluster match illustrate dimensional problem base domain cart move cart balancing position finally carry sequence various setup compression reason dramatically per propose multiscale iteration small global several different boundary update iteration algorithm update summarize multiscale algorithm name interior boundary convergence boundary interior possibility either convergence iterate fall per make comparison fair iterate cluster local iteration cluster every local mdp boundary regardless interior construction transfer algorithm well suit whether initial question initial fine initial initially scale type iterate boundary information allow propagate throughout interior value modify interior solve coarse mdp compressed coarse iterate interior immediately solve coarse diffusion policy interior iteration otherwise propagate slow solution considerably initial coarse initial impose fine scale fine exist pool augment coarse scale ignore current fine action initial coarse increase action result coarse reward proceed coarse scale involve compression diffusion policy discard coarse policy amount diffusion cm range fine coarse zero mdp transition link state direct terminal mark correspondence transfer agent grey grey circle terminal action movement reversible otherwise agent obstacle fail reward state reward bottleneck detection compressed guess policy figure detect marked character adjacency compressed transition original successively mdp graph plot cluster successively compress determine cluster policy result coarse depict direct path state right right policy reward goal multiscale cluster though fine scenario fine section advantageous discuss transfer problem correspondence cluster source partition correspondence color gray connect cluster pair match exception orient detection safe thing matching pair general priori identify problem share orientation pre solve store database solution scale involve detect valid particularly match transfer candidate cluster coincide early visual intuition source map mapping procedure receive transfer give action action mapping action identical original transfer without modification figure place cluster corner grid transfer transfer enumeration state scan right create cluster impose optimal policy constant likely reason transfer alignment transfer relatively mix cluster various transfer comparison multiscale fine initial initial coarse mdp pool policy appear impose convention plot value give indicator progress plot description algorithm label transfer choose action various multiscale fine policy transfer scale vertical warm coarse solve mdp initial coarse compare curve clear coarse initial iterate cluster boundary trace ms trace update ms trace corresponding policy ms ms involve transfer multiscale multiscale figure correspond figure compare iteration difference start multiscale transfer multiscale able multiscale transfer curve iteration transfer reflect improvement due multiscale multiscale algorithms optimality ms non monotonicity multiscale give compressed multiscale fast policy iteration rather time reason iteration policy entirely fair multiscale section intel matlab release parallelization optimization call evident multiscale iteration ratio general identify cluster cut cart cart track balanced force cart cart location hold balanced must balanced cart force cart subsequently step additive deviation three state angle vertical along span end reach note reward task default move receive cart hold cart move otherwise able move balanced start balanced reach goal transfer task position simulation end region cart opposite carry balance transfer ability balance apply task multiscale achieve continuous control furthermore domain partition graph matching circle terminal see detail mdps simulation diffusion ignore episode hz input cluster net approximately pool state discrete mdp reach simulation separately cluster terminal boundary clearly discretize apply separately terminal default transfer plot show coordinate dark circle sample cluster define reward statistic claim state fine mdps boundary sample terminal near step boundary translation coarse geometry helpful approach different natural partitioning balancing without place default transfer state cluster mark gray circle interface bottleneck partition cluster compress uniform discount next fine fine across already partition match place fine matching assume across form correspondence coordinate map interval define coordinate across correspondence distance neighbor fall match coordinate fine policy default match match multiscale scale policy multiscale list text transfer fine policy policy experiment transfer choose force cart multiscale coarse coarse respect diffusion appear plot euclidean axis iteration optimal figure multiscale pi fine scale plot compare warm furthermore attribute slow policy improvement convergence still small multiscale figure evident error give improvement transfer visible compare policy converge slowly relative confirm near origin bottleneck fact converge suggest either fine initial datum contain initial policy pool simulation show give performance non bottleneck near gradient object room ball music switch action succeed marker object look music flip light switch order latter action must unless note modification period switch light stay state object currently domain illustrate compression pair task remain goal gain environment towards solve another build mdp simulate episode trial reach action small sample transition spectral stop potentially cut multiple subgraph assume fine task transfer coarse default sense compress policy optimal pair section light turn period switch marker alternatively flip default playing must music music place marker ball ball music turn switch begin marker random object flip light switch music goal look flip light switch action switch agent look music place look flip switch default transfer text visualization two appear nearly difficult mark ordinary terminal bottleneck colored cluster although embedding diffusion task cluster seven scale transfer default operator correspondence determine step canonical scale execute action map execute fine cluster match transfer fine multiscale list table match matlab confirm match match list default across algorithm list appear update non transfer coarse mdp policy natural correspondence error fine action compare multiscale canonical iteration vertical label error plot detail description algorithm without compare transfer warm start multiscale transfer transfer reasonably assume reason coarse iterate ms trace algorithm boundary update ms trace involve ms exhibit ms family since coarse value fine scale transfer setting conclude would expect initial coarse entirely potential good good absence ms involve although ms trace policy reach multiscale policy mention multiscale involve discussion figure ms without scale fine scale plot demonstrate coarse transfer transfer confirm fast fewer ms maximally comparison multiscale see policy problem look light marker turning differ task goal music play music look marker look ball light switch flip switch position music playing must music turn music place marker look switch flip light switch episode begin look random object random object default leave although pair lead compare previous scale match map two mark ordinary goal see default state vice versa direct match coarse scale assume hierarchy transfer possibility hope transfer portion fine scale policy deal music sequence music immediate goal confirm possibility color spectral case music provide illustrative purpose plot next section music music cluster music music intersection plot point magnitude next algorithm separately correspondence role problem assess blue trace label inside axis plot clearly expect everywhere follow music state suggest apply policy problematic goal either problem conclude transfer cluster absence transfer helpful state early correspondence relevant underlie map mapping generally priori change post transfer place marker policy state thus policy state default task policy match compression text compare plot correspond euclidean intermediate iteration true optimal trace pi transfer trace label pi correspond fine policy pi transfer pi plot except trace appear obtain fine transfer policy purely greedy policy condition compression pool diffusion specify collection pool respect value pool multiscale iterate coarse opt boundary several conclusion draw impact conjunction coarse fig policy condition robust policy multiscale fine even transfer multiscale compression fast emphasize true problem policy iteration suffer take compression figure impact initial coarse bottleneck figure optimality ignore improvement multiscale come improvement lead contact comprehensive comparison similarity several much literature multiscale principle approach strong may treat depth many ultimately require option generalize beyond coarse fine separate mdp notion multiscale impose improve plan reduce planning structure combine macro multiscale planning multiscale fundamental result transfer knowledge potential policy generality detection compression locally policy solve mdps analytically model estimate specific expense conceptual follow flat review address challenge hierarchy process essential literature divide similar complexity throughout biology abstraction coarse macro broadly extend primitive action framework reinforcement learn pre collection macro closely model formalism option problem hierarchy discovery employ option discuss option hierarchical flat abstraction consider policy specification option end termination element option important difference distinguish option option approach decision aspect substantial option framework macro action specifically objective mind option primarily plan macro exploration scheme macro action receive considerable attention yet coarse macro fundamentally take view couple macro planning action couple conditioning macro time fast mix coarse localize multiscale decomposition globally distant multiscale scale solve contrast option introduction additional necessarily add plan indeed macro sequence scale particular language option bottleneck terminate whenever markov termination coarse lead option always remain markov option direct terminate one option option problem branching avoid loop desire leave direction execute bottleneck direction action context able guide store transfer option query closely way goal expense ignore want differ decomposition able broad transfer enter picture option macro possibly option robot room constrain termination policy coarse scale carry scale bottleneck drastically partition time accelerate problem mdp problem solution policy policy scale arguably core option option state execute option scale planning may describe top problematic reason expert specify determine multiscale option accomplish hierarchy option option across scale lose define flat user avoid either burden burden repeatedly difficulty example embed markov termination could clear problem lack necessarily branching problem reason level contrast strongly multiscale impose action multiscale organization invoke coarse solution scale transition combine transition macro trajectory suffice option interest single abstraction transfer learn construction coarse transition discount advantageous length represent keep operator probability sub preserve mdp mdp trajectory rather constant concern distinction reward termination aggregate reward pre average possible serious start state path span coarse contain little fine definition multiscale sense keep track approximate analytically termination macro multiscale consistency policy layer specify recursive policy optimal overall hierarchy impose consider global optimum markov policy policy scale recent explore learn paper speed interesting learn partially observable learning macro emphasis macro open sequence coarse action loop state important literature mean interpretation must provided assumption provide paper automatically goal exploit locality create transfer primarily algorithm researcher discovery characterization top framework option approach overlap work literature automatic macro organize three aggregate state simulation bottleneck define state visit heuristic frequently visit state algorithm search policy trajectory occur approach intensive automatically create analyze relevant compact representation specialized maintain principle consistent intuitive notion bottleneck spirit definition appear although emphasize around graph identify online employ former advantage cut neighboring reach option separately reward similar consider successively explore without global option correspond graph cut incorrect learning option prescribe bottleneck state may base node define must high alternative technique give substantially graph mdp compressed third represent discovery reference coarse decompose piece solve guide accelerate abstraction domain reward salient outcome serve option learn policy layer abstraction manual specification event salient differ sub task learn tailor development nothing objective store method seek particularly query paradigm closely problem solve goal around previous consider constraint coarse bottleneck terminal terminal scale chain localize geometry localize function likely capture lead geometric tree level abstraction goal however isolate efficiently locally consistent several reward bottleneck goal principled way learn partition boundary may clear dependent scale finally mdps approach hierarchy share like type coarse deterministic decompose discuss solve maintain product coarse algorithm similar bad multiscale self mdps trading
p max sub band b pair level allow interference channel gain sub band spectrum represent realization constraint low want achieve goal controller know network require decentralized information formulation normal joint discuss function monotonically decrease consumption player achieve minimum utility ne state empty refer player profile player game reasonable mean link formulation nash equilibrium follow ne theoretical formulation modelling scenario player individual game follow equilibrium se due use k define efficient equilibrium player may effort minimize effort trial te characterize system scenario te implement state triplet reaction action content action new decide evaluate otherwise benchmark eq opt player increment observe e contrary stage step turn utility noisy opt notation pure nash pure nash equilibrium approach play player employ te nash maximize sum equilibrium play te converge action profile player te number player minimize employ k te corollary channel snr te te state study approximate allow expect converge ne ne se spend state player simplify define sec approximations restrictive player action satisfy optimal w r study te interested frequency ne ne player individually transition probability list description omit ne bounded euler proportional freedom nonetheless ne pd pd pd pd c shorter ne se fig se level achieve state theorem se bound ne thus increase converge speed validate second validate general channel power employ two two experiment summarize figure channel model prove precise formulation second converging analytical increasing bring possibility note convergence ne compose te fraction player curve employ player reach player satisfy power employ scenario inefficient configuration study decentralize able work receiver consumption knowledge bit feedback realization analytically expect simulation general channel work carry theorem proposition conjecture count minimization device operate signal interference introduce decentralized trial statistically meet exist local stable use framework form converge point correspond equilibrium nash similarly provide trial nash sharing receiver bandwidth divide several orthogonal subject mutual interference device service plus device design achieve stable operating operating achieve device power transmission communication centralize neither information current device manual inequality centralize game nash equilibrium interference water noting work
point box dash line kernel ascent find reasonably corner error rbf kernel maximum interest tb p local show hinge approximation compute clean report completeness black circle effectiveness attack mnist digit focus distinct vs vs nature handwritten digit provide attack digit mnist properly normalize pixel consider regularization validation respectively retain class digit tb plot class point attack plot display attack appearance attack reveal prototype appearance final segment segment become round increase show due nonetheless cause classification rise initial error attack iteration cause svm attack point multiple extended point randomly set size respectively steady attack effectiveness attack point variance toward security attack arguably surprisingly svm present also reveal assess carry function previous algorithm realization strategy implication future address method restriction interesting large solution simultaneous point attack sequential attack individually optimize attack simultaneous every optimize effect use attack selection heuristic allow improve regardless optimal attack significantly practical limitation label human user message ground although arbitrary message guarantee attack attack satisfy side work understand side attack incorporate problem desire handwritten digit world image many spam mapping smooth solve attack crp r scientific innovation acknowledge von financial carry solely necessarily attack machine attack motivation attack datum come natural behave adversary extent ability datum attack enables experimentally demonstrate ascent identify good maxima convex technique tool infer hide complicated adapt behavior help security problem behavior fact solution security task spam unfortunately generally adversarial adversary achieve goal adapt spam detector generally adversarial learn explicitly attack attack attack exist database often collect example attack family attack security machine assume know algorithm underlie learner real setting instead surrogate capability construct accuracy base solution svm technique depend smoothly program attack attack formulate optimization performance subject retain although surface reliably maxima surface gradient dot point attack latter strong disadvantage since practical ground impact indice contribution product gradient maintain introduce kkt q attack cause smooth restriction composition remain us solution eqs component rewrite use substitute objective attack gradient particular expression gradient common rbf dependence avoid substitute provided enable extension section present attack initialize class principle point margin adjust attack point may progress tb label attack attack final c incremental svm p crucially preserve classical search
act panel fig problem mind fig signal reconstruct reliably case domain normal illustrative package make domain signal segment periodic condition early one consequence overcome extend correlation leave unchanged reconstruction influence influence reconstruction rectangular pixel scale pixel deal scale bin appear formula fourier whose bin reconstruct bin rectangular spectrum choose reconstruction example exhibit reconstruction good agreement region far fig reconstruction relevance interpret manifold sphere parameter power spectrum angular spectrum often last case replace part effectively around resemble field version panel show gap structure infer fig wise fractional approximated region seen around observation expect neighboring infer infer normal wide use field power extend simultaneous reconstruction field develop apply field past field logarithmic power suit add formula implement associate prior stress derivation formula depend spectral expect satisfactory signal field reconstruct demonstrate reconstruction spherical pointed observational represent additive incomplete demonstrate information field matrix represent incomplete convolution make applicable mind reconstruction galaxy thank anonymous helpful perform medium correlation structure examine smoothness broken power shape logarithmic behave like tend toward double derivative change logarithmic value prevent prior sigma event break special case pure power arise motion function arise example transform space spectrum double scenario study remainder field statistic regard spatially calculate spectrum drop employ spectrum reconstruct reality therefore choose spectrum become support example triangular stationary spatially field transform power spectrum power double case correlate correlation keep finite add introduce signal case addition smoothness signal variation finite pixel sec spectral field potentially problematic discuss smoothness logarithmic derivative spurious variation true function latter spurious correlation feature present spectral smoothness prominent spectral line usage smoothness grid two exponent spectral exclude approximate derivative represent entry opt lin develop infer random affected normal fluctuation amplitude order use formalism free signal reconstruct gibbs bayes maximum posteriori power smoothness scenario reconstruction demonstrate validate degree reconstruct continuous field branch physic develop reconstruct logarithm arbitrary log normal suit physical positive order magnitude spatial correlation consist discrete often arise field cox field finance study often log field density universe simple matter galaxy per volume approximate well underlie motivation intensity come magnitude spatially log description natural bring observational uncertainty regard number apart deterministic relationship log normal accommodate observational setting field lead fouri transform field correlation description aid knowledge field region observation field homogeneous correlation describe spectrum wants simultaneously infer several technique formalism employed tackle infer log normal field gibbs spectrum equation noise field power uncertainty arise reconstruct spectra turn redundancy spectra sharp drop spectrum occur actually neighbor interact one prominent ad hoc smoothing follow idea show spectral smoothness introduce appropriate spectra feasibility gaussian field discuss applicability smoothness estimation spectrum formalism smoothness prior derive formula way easily additional demonstrate smoothness formalism formula log smoothness reconstruction case sec assume analyze signal field discrete continuous response instrumental instrumental response fourier contexts sec case linear arise incorporation signal restrict notational straightforwardly field statistic necessarily symbol quantity limit argument regard least field straightforwardly space realization call formula serve desire continuous reconstruction physical covariance priori problem reconstruct signal principle unknown overcome formalism energy formula signal formalism briefly sec homogeneity unknown harmonic fourier sphere mode harmonic dimensional space sphere angular stand signal spectrum symmetry formula q wiener see sec detail subsection regard unknown formalism reconstruction eqs iterate hoc part usefulness smoothing case signal sphere periodic spectrum form power realization field pixel homogeneous uncorrelated noise formalism realization fig reconstruction iterate eqs smooth fluctuation reconstruct scale reconstruct power chance reconstruct conjunction option load package graphic explanation terminal graphic macro ltb lt lt lt lt lt lt lt lt bp spectra dimensional black solid dash draw green power dot package color conjunction terminal option explanation use load package package graphic explanation terminal graphic macro ltb lt lt lt lt lt ltb lt lt lt lt r r solid show datum smoothing dot wiener show illustrate especially interested wiener filter power spectrum see residual reconstruction spectrum constrain point way property signal spectrum power exhibit fluctuation turn position rapidly power thus reciprocal length enforce formalism present incorporate spectral eqs posterior spectrum posteriori solution eq solution consider posterior define spectrum accordance spectral location value tune priori turn inverse jeffreys flat logarithmic always limit filter straightforwardly derive calculate signal datum spectrum negative logarithm hamiltonian definition power homogeneous isotropic hamiltonian make derive mean maximum posteriori power effectively delta procedure bayes note formalism spectrum rough hessian curvature regard complex formalism develop gamma dot horizontal line level power scheme top spectral green hoc power dot line spectrum dash terminal option either load color graphic terminal graphic ltb lt lt lt ltb lt lt lt lt lt bp power spectra signal top black solid line power spectra dash line realization draw reconstruct spectra smoothness power spectrum hessian dot conjunction option explanation load graphic graphic macro ltb lt lt lt lt lt lt lt lt lt bp r r correspond top panel signal correlation bottom show triangular signal realization solid reconstruction green sigma uncertainty area bottom plot different add realization calculate reconstructed fig plot dominate power scale dominate reconstruct spectrum increase dominate scale reconstruction wiener reconstruction slight excess residual reconstruction ad power spectrum window width ad hoc reconstruction outperform smoothness discuss signal triangular eqs power spectra serious practice figs reconstruction spectra respectively divide pixel power spectra reconstruct reconstruct scale power rapidly reconstruct stay flat triangular say spectrum drop zero see dash line stay reconstruct panel small represent reconstructed spectrum well thus accurate power magnitude signal dominate irrespective reconstruction perform even power spectra power priori case mild noise smooth neither priori significant feature problem reconstruct logarithm log simplicity interpret pointwise linearity posterior highly even signal add treat formalism free energy gaussian mean covariance calculate gaussian sense leibler obtain gibbs last definition covariance term approximate energy internal energy posterior hamiltonian temperature give around remove posteriori reader depth discussion temperature assume calculate expand logarithm appear expectation expansion ensure vanish simplification calculate gibbs confusion integral appear derivative respect yield right hand appear eq together last posterior self estimate correspond many uncorrelated response purely equation simplify somewhat eq sec reconstruction spectrum power maximize interesting employ formalism reconstruction eqs regard assumption viewpoint inclusion spectrum power accord derive package explanation load package graphic explanation graphic ltb lt lt lt lt lt lt lt lt lt lt lt lt lt bp r r r dimensional low realization draw panel field blue solid
generalized bregman divergence detailed maximization converge finite two variant divergence use divergence divergence different parametrization fix bregman partition select df intensity raw poor preliminary performing give bregman notice mean significant iterate lb lb lb lb denoise collection collection collection noisy image lt noisy lt two variant poisson initialize draw standard normal normalize euclidean norm euclidean rule constrain axis initialize many stop iterate algorithm long u add dictionary since coefficient whole remain information among instance average also investigate variant aggregate noisy poisson bilinear interpolation level significantly reduce computation patch computation deal count comparison bm method real image summarize result follow visual metric house bridge simulation fig find comparison variety way bit improve upon poisson multiscale low light interest tend transform classical pca section patch test generalization image cube comparison ease mae cluster patch suit count quality provide channel square intensity different homogeneity across spectral patch respect third dimension practice level legend legend I legend legend I bm legend I legend I bm I I legend I legend I legend I legend I bm I legend legend legend I bm I legend I I I legend I I legend legend I I legend I bm legend I legend I legend I legend legend I bm I legend I legend I legend I bm legend I legend I I I legend I bm I legend bm legend I legend lc c house bm bm bm bm bm bm c peak bm bm bm bm bm patch channel reconstruct adapt generalization denoise image find patch space logarithmic natural logarithmic might ask several see work alternative pca power else improvement theoretical question minima interesting reduce acknowledge award fa nsf award thank provide spectral anonymous propose straightforward hessian case origin l gradient computation use approach method numerical one possibly ill condition step inverse hessian inverting transform row introduce write hessian matrix u multiply lead follow usa electrical computer university nc universit france limited imaging sensor array limitation image vision inherent removal yield introduce denoise limited combine dictionary method employ adaptation poisson develop sparsity method help reveal conceptual base highly light regime broad detector certain imaging available limitation arise environment spectral produce cube location spatio count wide variety tool particularly challenge limited available intensity poisson algorithm challenge count designing bin create resolution spatial resolution contrast exhibit pixel generally whether detector conventional fail transform variance transform count suffer demonstrate advance dimensional modeling sparse poisson resolution combine poisson principal poisson pca sparse intensity non nature state art particularly principal introduction denoise approach extensive combine pca approach white extension image directly use data fidelity singular direct suffer recall relevant property exponential iteratively computational conclude acquisition device poisson mean intensity accurately represent concatenation patch combination one interpretation self describe setting denote overlap patch similarly intensity patch many method represent collection space spirit pca framework pca aim represent row element element wise uv different similar assume allow issue relate goal assume restrict act representation family though use poisson gaussian idea analogous comparison purpose space equip dominate measurable exponential r parametrize call independent know parameter gaussian possibly poisson distribute identically wise eq counting parametrization usually parametrize proximity analysis exponential kullback leibl divergence write function define gaussian unit zero divergence bregman bregman divergence non square size observe let patch patch weight dictionary element patch approximate th patch factorization exponential criterion bregman divergence amount matrix minimizer intensity pca remainder solve intensity estimate dictionary solution atom ensure keep follow jointly fix newton algebra entry represent diagonal appendix update propose introduce transform precise definition update appendix updating rule row tv
policy bandit agent ask produce reward great immediate exploitation action immediate great paper bandit reward thompson allocation bandit thompson attention optimality achieve give choose observe receive possibly choose past observation minimize expectation arm prove e must suboptimal leibler divergence reward policy satisfy equality asymptotically efficient ucb et al efficient good use confidence past reward optimistic arm choose action high suboptimal imply contrary kl ucb confidence asymptotically confidence successful thompson policy idea model yet fundamentally frequentist regret thompson choose action thompson reward recent investigate optimistic version propose provide theoretical guarantee extensive scope generalize without consequently major scale like meanwhile strategy policy optimistic sample analysis ucb thompson corresponding quantile bind thompson draw precisely thompson asymptotically experiment thompson optimal policy ucb present assume thompson bernoulli uniform arm arm explicitly cdf resp binomial link beta q binomial trick several stage resp ucb index quantile distribution computation inspire analysis index policy policy arm round index reward aim draw suboptimal arm arm suboptimal optimistic k ucb argument ucb also optimistic ucb directly apply analyse thompson since optimistic confidence close convention regret analysis sample proof explore sampling exist tell distribution concentrate inverting convexity last inequality lemma draw let occurrence optimal arm play suboptimal arm extract range play play concentrated around fact say action e lf interval eq fact hence condition also sample interval proof fall mean adapt j suboptimal arm arm last comes draw small hold suboptimal arm depend conclude inductive hypothesis complete induction show event begin arm interval deal range arm arm bind size get f induction complete summing hypothesis I link mention give exact lead bs bs bs j bs therefore median binomial inspire rearrange write affine function negative whenever clearly constant conclude thompson bernoulli thompson ucb reward different gap thompson ucb well ucb small figure display cumulative trial ucb ucb close thompson bernoulli incorporate reward index sometimes thompson implement kl ucb costly produce posterior regret various dark show central asymptotic bandit policy simulation show thompson outperform policy idea control rather valuable control
scheme despite simplicity mark application rational tree open empirically develop well non root meta reasoning principle helpful current complex address future successful go pure extend specific select current move really fair dominate nevertheless adapt aware acknowledgment science foundation computer sciences center production management theorem david computer university ac state games process ucb multi mab cumulative differ actual move rather oppose regret arm bandit ucb stage optimize although theory apply trivial theory propose aware achieve outperform formula numerous although past job exploration exploitation correct bandit ucb principle issue game nature action exploitation action close pure exploration selection problem cumulative small cumulative regret begin ucb propose scheme sampling principle suggest another paper generate scheme improve monte search initially suggest policy decision process mdp mdp reward explore termination condition cutoff reward arm associate action reward set literature bandit reward ucb scheme maximize time far search describe algorithm mdp adversarial get reward policy problem reward arm great minimize simple upper exponentially respective ucb polynomially theorem yield regret sampling control principle bound description rational maintain current root gain cost factor ideally select intractable simplify action basic non concave utility algorithm frequently multiple many case insufficient change current get rational define goal achieve scheme amenable analysis concept examine polynomially upper bound bound uniform confirm greedy scheme q substitute ucb aim finding move outcome mdps adversarial choice move optimize root deep tree move choice internal optimization far mark suggest combine rest select ucb stage select step line select statistic update root realization sr cr employ node depth node node next next node reward expect stage less tuning exploration ucb need step rest achieve choose maximize infeasible current incur change equation gain reward minus current exponential action sample one denominator aware maximum stopping leaving appear amenable analysis section demonstrate low result empirically verify armed bandit tree experiment sr cr scheme comparison bandit return search compare reward sampling vs average ucb dominate ucb large b vs perform sampling scheme child anti symmetric cause uniform incorrectly give tree exploration reward exhibit dominate everywhere advantage greedy ucb tree arm
explain automatically attribute complicated mechanism provide explanation empirically cutoff explain alone quantification within population increase replication series widely mechanism increase respectively regularize incomplete gamma last let acknowledgement office award fellowship ep fellowship award ep uk award fp integration grant european union theorem derive product enables realize network large approximation form enable quantify across population achievable expect need attribute complicated mechanism considerable study neighbor nod connection statistical construction exhibit expectation quantify variation remain important realize effect observe count select node sequentially valid edge loop lose requirement replace edge bernoulli trial success probability nonnegative normalization degree th equal p connect assign edge assign statistical insight recognize degree sample condition near later eq neither loop ij poisson fully graph study enyi reduce determine describe variation introduction parameterization understanding parameterization range weight increase regime sample property sequence network proposition degree edge trial ij proposition parameter strong realize degree expectation behaves scale thus nd th decay directly distinct maximum variance dispersion whenever poisson variate dispersion aggregate go dispersion cauchy schwarz quantify difference eq dispersion must theorem establishe obtain element accordance edge trial comprise degree exchangeable conditionally degree discuss natural treat deterministic decaying change relate inverse large choice characteristic mean consider node binomial counterpart indeed iid yield provide moment degree degree express dd dispersion compare network dispersion depend dispersion match unity dispersion conditional manner illustrate notion variability variability ignore network binomial heterogeneous either source ignore correlation degree arise specify special case equivalent classical homogeneous study lead binomial simplify early direction entire map analysis quantify laplace theorem act integral regularize beta concentration toward step transition admit lebesgue weakly whenever later series entire significant quantify incomplete variate imply law standard normal verify apply tail hoeffding quickly whenever choose relate standard describe survival dash transition region analogy survival observe fig censoring follow near examine simplify covariance counterpart natural begin decay decay power law relative magnitude grow small ni treat regime realize network polynomial decay variability distribution significant literature characterization proportion degree sequence law constant value converge individual drive growth growth argument euler law variate order chebyshev sufficient central behave variate grow degree nd become describe degree decay main specify relax form overall polynomial deviation control hold introduce redundant variation overall decay constraint via variability essence retain permit deduce total convergence law suitably whenever nd power law whenever I apply establishe claim ii numerator balance ii converge prove expansion gives constrain grow consequently element value er multivariate outside power law set I possibility section vary accordance estimator exploratory refine suitable provide toward understanding goodness fit network important open challenge highlight realization wish pareto restrict amount correspondence nn degree form pareto pareto uniform multiply k dt k expand f reflect n incomplete restrict degree within well unity accordance visible behave region dominate decay censor law panel average f b panel show taylor exponential cutoff effect investigate censor hand fig empirical rapid censoring power decay hand panel scale match censor cutoff motivate explicitly effect exponential cutoff law frequency infer n k due purely derive valid must generally enough admit second binomial dense expect degree linearly eventually edge essentially determine distribution quantifie later extend twice differentiable random strictly attain taylor expansion carefully choose multiply side recognize truncate establish hypothesis may write form taylor remainder substituting mean n hence q part identity implication follow concentration lemma expansion essence kf nn tail choice reveal via fine survival come taylor formalize split integral mixed binomial reveal essence multiplicative degree recall discussion mechanism impose beyond decay accordance lemma network substantial heterogeneity heterogeneity explore theorem variability marginal variability degree generate regime variability well understand smooth reverse occur limit recover apparent behavior achieve behavior total network law study regime studying realize concentrated heterogeneous degree variability require direct control turn sparsity smoothly network notion scale affine degree corollary parameterized constant mean behave property regime shift scale map admit limit mean converge grow recover set importantly achieve dispersion product characterize dispersion network instead estimate realization mechanism size expression natural rescale enable set rescale degree grow overall magnitude grow collective heterogeneity depend refine simplify complementary rescaling extend yield simplify expression every eq fixing thus recover leave effective normalize exceed toward begin shift regime simplification follow set corollary restrict distribution variable assumption likewise n first lagrange yield multiplicative understanding n unity latter formulation censor prominent corollary smooth beta gamma especially unity corollary highlight statement incomplete censor degree characterize binomial survival extreme serve heterogeneity limit must consider previous per q expand multiplicative exist eq
wind linearize equation become cart cost first far state reflect seem simple specialized capture application could robot receive human plan action try plan issue future wind control use issue commonly minimize square historical result forecast predictive control room improvement observe particular beneficial sometimes empirical minimization compute minimize historical historical certain technical infinite available future work main contribution fit regression direct large improvement controller aforementioned relation direct series regression forecasting control build present control static decision work become unstable whereas static efficient employ new devise slide cross validation evolve minimize period definite gx u simplify problem control horizon time subscript action observe horizon formulate time new solve produce process order expect future simplify objective become dynamic govern involve linear quadratic generate control forecast idea extend generate forecast becomes arbitrarily grow nonlinear drive scalar useful predict form noise conditional straightforward forecast event produce forecast policy generally discuss coefficient l note sum begin compute linear square coefficient observe time forecast natural forecasting model capture generate assume principle whereby empirical problem compute minimize historical action start easy ergodic historical optimize appear generally however study experience initialize detailed implementation appendix challenging parameter mainly evaluate gradient require linearization typically include reason sum practical significantly past decay influence future nonlinear one dominant motivate refer coefficient typical iteratively forecast fix allow set validation ht algorithms kind hz employ around derive physical therein keep minimize parameterized hence period ta b experiment policy accordingly ensemble series follow sample ar five coefficient tend five variation sample generative model helpful forecasting objective differ generative another choose control naive dominate select forecast five feature approximate extract period forecast perfectly generative context recognize sort misspecification practical sequence zero mc give mc clearly generate aforementioned five cost form generate focus presentation excess subtract influence cost choice control plot excess qualitatively rather small suffer prevent l outperform interesting gain substantial compare quantity interpretation cart system reach cart angle certain threshold twice root simulation period initialize state reach generate policy consideration mc ls failure summarize result much close mc l ht direct coefficient predictive version namely forecast methodology transform problem linearize slide cross validation technique applicability paper forecasting hope observation work involve broad consider paper multidimensional system involve relate idea perspective algorithm controller model control transition aside autoregressive conceptual fact ls analogous study data method hand discriminative provide combine generalize offer approach broadly combine method adopt weight square put emphasis critical component series develop forecast use decision line misspecification method well gauss newton carry backtrack search determine take iterate whenever hessian add spirit
standard ccccc low p low residual technique spectrum spectrum respectively root function sd innovation standard subsection calculation produce unit provide red empirical implement interval whether root outside incorporated root ar polynomials lie disk stationarity whether class standardized alpha residual decay function behavior typical asymptotic label plot formula converge make asymptotic normal measure sample exercise mobile forecasting forecast three function alternative two sided contrast estimator begin summarize difference vs mle mle three consider value significance mle value increment horizon parameter model tend zero tend increase prediction test execute package addition provide base evaluate evaluate property capability diagnostic forecasting would improved root long series grant em em depth autoregressive move cl mm de cat de mat cl mm exhibit persistence lead development account decay autoregressive integrated move package implement contain ability impulse life key memory variance impulse package long introduce key event package example package memory produce estimation density noise mle describe time series model forecast package offer forecast estimate package aforementione unfortunately computational implementation severe calculate variance forecast impulse response discuss develop function fitting via asymptotic aim paper package exist development package finding devote parameter impulse response estimation provide assess apart describe package also illustrate life memory beyond walk root autoregressive account long general process define autoregressive operator root expansion mean innovation asymptotic usual gamma function process unit solution stationary singular decomposition unique purely express measure purely absolutely lebesgue spectral write polynomial mle calculation likelihood normally give associated parameter likelihood calculate two indicate version minimization minimization newton type condition distribute da estimator assess ar plot include figure behavior consequently tend large function especially near inverse tend near discuss covariance useful tool make inference approximate likelihood calculation alternative computation hessian base explicit obtain spectral partial derivative sequence kf true consider density identically covariance spectral observe yield I j p analogously impulse commonly find outside disk invertible write expansion coefficient relationship expansion dy give assume illustrate general consider parameterization function term process autocorrelation dd root one cd correct absence ar formula reduce eq behavior mean asymptotic exact autocorrelation approach prediction square rmse forecasting presents rmse statistically difference paradigm prediction carry diagnostic base observation nan difference prediction test horizon windows estimator obtain simple autocorrelation consistent function life http area display persistence analyze illustrated tree series specifically h usage package fit covariance hessian mle study function r na option implement allow
hand side random q numerator eq follow current status censor mle smoothed status censor theory expect reasonable plug locally estimator purely mle actually hold mle indicate large mle mle bivariate censor bivariate status bivariate censor independent maximize formulation first second observation look rectangle tucker df generators mle maximal mle compute mass study status censor mle discussion mle datum put treat minimax low estimation consistency put determine mass computing somewhat energy spend intersection mle corner corner really effort mass bottleneck censor determination intersection mle put phenomenon simulation place prohibitive want behavior computer ultimately determine insight mle section attain determine smoothed asymptotically observation study accordance extend censor end conclude rate rate basically current status monotone diagram observation interval derivative simple model sometimes represent variable leave deterministic derivative derivative bivariate distribution diagram seem process incorporate duality condition optimization analogously status duality square hide point mass deal denote process coincide necessarily status must function fact must dimensional status complicate deal equality mle bivariate current status status surprise preserve get estimator gets argue attain construct purely converge locally remainder appendix define twice suppose normal point rectangular way necessarily negative turn nevertheless build leave get correspond seem seem research picture plug nonnegative support treat boundary representation conjecture order unless happen bias compare behavior distribution r indicator ht value mle mle plug mle sample letting coordinate coordinate randomly uniform permutation integer mle place mass put picture different plug put inspection intersection put similar rather mle reduce ht asymptotic theorem value mle r mle plug bias time standard deviation sample suggest vanish detect present theory confirm get active still usually partial vanish uniform achieve uniform distribution bandwidth plug correction bias bandwidth actually attain mle extent suggest censor censoring model eq define similarly maximize q bivariate probably theory measure introduce belong unbounded finite vector consist zero otherwise nonnegative optimization treatment analogous aspect experiment quantum frequency belong observation frequency change bound give upper put mle put apply package c mass facilitate literature preliminary follow mle convention mass mle right corner close correspondence apart package bivariate censor picture mle picture figure mean occurs mac see picture df mle ridge coordinate level I
equivalent imply rr preference specify agent alternative concave concavity log concavity hard normal log make location normal specifically p concavity p technique maxima concave follow necessary maxima bound alternative subset least sufficient maxima family suppose hold unbounded alternative increase simultaneously alternative suppose alternative edge path conversely follow edge alternative follow path log concavity bound estimate correspond unique maxima strict contradiction maxima contradict never reveal mle belong ef motivate domain determine iteratively proceed iteration previous compute conditional latent maximize log simplified w bx e aware analytical approximation sampling sampler approximate gibb choose accord tail gibbs rao jx ms tx tx l k x rao reduce tx j x k j e step x tx j bx I exact mc uniform mc e however application long remain large safe gibbs sampling sub ratio factor suitable ratio similar gibbs number rao make arbitrarily mc datum namely set preference order simulate use utility equal method agent iteration infer order middle well increase get case axis middle panel mn access obtain full interval method public collect partial candidate non rank rank alternative experiment alternative entire truth since find adopt compare data sub variance subset case fair great public kind mc parameter synthetic variance guarantee variance part parametrization variance rather variance fit model criterion log likelihood predictive aic bic value whereas negative model fit predictive likelihood likelihood sense data compute randomly choose vote standard deviation summarize htp c aic aic significant compare fitness fit well log outperform mc linearly agent intel ghz gibbs acknowledgment grant compute anonymous suggestion property sketch alternative draw alternative rank alternative case receive attention general enable mc concave global maxima world simulate scalability capability metric aggregation social part aggregated g crowd rank ordinal consist rank single select formulate true rank alternative report view noisy regard preference alternative agree rank otherwise maximize mle pairwise comparison agent preference pairwise preference winner random economic utility alternative denote naturally c illustrate systematic economic seminal economic alternative adopt cyclic preference correspond strength preference parameter alternative l term shape likelihood function analytical extensively apply recently em recently propagation p variant social p model use closely model quite characterize strict preference inferential particular preference set order agent voting preference profile win ranking tie win include singleton mle social unobserved truth ranking agent independently mle maximize preference report vote estimate ground truth determine win preference ground truth win rank mle return parameterization belong ef ef format eq e x x
assignment link determine entry q respect flexibility represent type structure link affinity core effect ccc connect membership topic belong close membership modeling flexibility feature section task nod network membership discover member link member member member likely affinity member member opposite behavior link likely non core affinity c membership model rich belong predict derive evaluate data friend citation document play increasingly modern machine provide friend scientific focus link decompose kronecker product structure though powerful network ignore node text document represent connection along complementary use simultaneously exploit network link node powerful link traditional provide predictive node predict node profile new link predict link keyword predict connection reach belong multiple group membership art assign node latent belong membership share group membership series truly belong multiple contrast membership necessarily membership link link link combination membership group entry capture link affinity link affinity understanding formalize illustrate value membership beta parameterize multiple membership node affect limit
evolve describe characteristic function functional function function instance make robust easy interpret extract raw functional extract give approach maximal inversion regression drawback extract easy link original feature feature original approximate piecewise functional solution us interval set term functional feature extraction consist feature context consist original unconstrained functional pca feature interpret support analyse elsewhere basis wavelet unsupervised basis I discard different version select spline orthonormal compactly dependency localize easy drawback wavelet compactly consequence function interval spline suffer length lead representation large function without among il norm support get approximated index partition satisfie order belong also minimize square quality find agglomerative dynamic lead propose bellman single iteratively quality good class initialization iterate minimize index keep backtrack class piecewise prototype loop use compute search precisely interested store array accord recursively j compute recursively cost storage scheme use quality measure rather problem restriction piecewise prevent continuity condition search continuous spline expect piecewise one measure evaluate need exceed reach particular leave quite noisy rely overfitte basis leave give constant segment leave maximal variant basis readily accord l leverage basis reasonable could solution likely select behavior htbp illustrate dataset mid range cm spectra along spectral consist support example solution unable pick detail peak spectra approach
challenge receive attention field include recently solve equation previous recover compare major approximation consider noiseless case recover exactly noiseless sharp omp real hence complex complex description model reformulate reconstruct organize classical omp solve omp geometric parametric set conclusion section obtain omp iteratively atom greedy fashion current add omp assume atom normalize e nonzero pick atom atom otherwise slight notation use atom submatrix omp detail e residual atom counter span element stop achieved go otherwise go reach iterative ls return algorithm key stop rule noise noiseless natural stop achieve construct maintain expand atom atom choose maximally approximate construct fall algorithm terminate require two effective omp non l omp select atom step among omp note gain insight omp main behind proof provide technical light properly noiseless meanwhile omp noise case well gaussian extend meanwhile propose restrict isometry rip omp signal beyond pose success omp equation nm guarantee generality solution entry begin decrease computed error eq utilize step first thus must h h substitute hand right side exploit mutual definition hand derivation sparsity imply decomposition atom I omp residual onto atom atom current operator span element residual clear omp variable necessary imply lemma u select denote omp recover correct atom coefficient satisfy lemma due complex b noise gaussian distribution real part part directly vector entry bound complex theorem suppose meanwhile suppose k I rule obvious easy however imply suppose support give cardinality besides success mainly vector dictionary fundamental unchanged application complex widely literature description image ideal center characterize center light correspond stochastic measurement measurement obviously problem carlo via application subsection range ghz ghz band start ghz five scatter locate contaminate noiseless respectively mutual incoherence situation atom small determine support omp far calculate present noiseless finally result recovery r sparse noiseless remark relative understand fig noiseless scatter recover sparse dictionary incoherence among support omp meanwhile also matter inter atom incoherence interval support mutual incoherence obviously incoherence condition fig paper omp approximation dictionary present incoherence atom interference provide sufficient omp theory omp importantly new complete omp make omp confirm omp consider yet see fig fact adjacent case second scatter
area high anomaly simulation look anomalous phenomena anomaly iteratively refine highlight regression value functional sample nonparametric edge fail geometry edge sample nearby embed nearby place first randomly learn skewness set display predict enyi rmse covariance drawing element obtain matrix rotation rotation outline repeat lebesgue weakly divide indeed accordance consistent grow whenever case estimator special rearrange factor ensure lemma conditionally limit th power vanish thank empirical law couple serious gap lebesgue imply handle strong concept need uniform lebesgue put density slight cm corollary algorithm treat individual object machine operate suggest treat employ define pair set projection cone semi enable machine anomaly dimensional embed numerical simulate extend classical area economics bioinformatic consist especially wish use quantity quite algorithm consider function density observe directly finite treat approach space represent point application representation traditional machine technique problematic wish survey certain group difficult similarly collection collection develop classification map set svm define machine anomaly detection focus individual joint goal detect finally develop handle problem distribution map generalize generalize machine estimator nonparametric consistent publicly generalization kernel exist kind drawback product family fit inner product base fisher family rare belong method inner product contrast provably certain nuisance divergence intractable optimization space reproduce sample sample convex programming distance appropriate functional computation demand fix small able kernel define affinity evaluation embed kernel hence generalize operate either set deviation show kernel svms closely euclidean appropriate speed empirical vision domain image problem collection local method descriptor represent image composite kernel pyramid feature histogram use intersection histogram curse bin problem near distance bayes integrate bag result improvement comparison class principle define possibly object functional improve limited available approach expansion b inference recent use scalar functional functional value functional vector although develop functional classification directly input density sample need product sufficient general machine generalize consistently divergence include enyi hellinger estimator problem however inner classifier divergence estimator investigate inner divergence consistently extend classification image show anomaly formally investigate unsupervised let supervised seek ideally review discuss perform multiclass vs classifier vote final experiment approach inner product map form eq gram matrix tool quadratic available idea class svm anomalous consider sample around elsewhere supervise set scalar functional mutual point reliable intensive alternatively size use problem look transform hilbert perform nonlinear preserve dimensional generalize distribution map local geometry distance kp p j kp j local characterize reconstruction vector reconstruct locally dimensional nonlinear algorithm include method require distribution estimate positive semi definite gaussian try r divergence divergence limit enyi divergence satisfy lead definite gram show ccc distance hellinger enyi set tool estimation enyi divergence euclidean neighbor th nearest neighbor provably consistent condition dd ball estimator thus plug formulae consistent estimator therefore increase however definite r enyi divergence instead euclidean gaussian asymmetric therefore gram project frobenius discard similar eigenvalue amount flip negative instead view noisy approximation jointly support find small discard separate euclidean associate analog tucker provide yield well well projection example kernel theorem might valid gram crucial solver approach predict kernel although computationally experiment see inductive approach label p calculate discard calculate predict applicability experiment mention publicly consist class point handwritten digit error algorithm achieve raw classification even distance easily dataset however euclidean distance become become necessary normalize intensity sample image take grid penalty degree try times kernel pixel perform use divergence near neighbor accuracy evaluation rate enyi indistinguishable hellinger polynomial slightly bad accuracy raw mean ht inductive raw raw enyi enyi enyi hellinger visually structure image multidimensional scaling digit preserve letter raw euclidean uninformative help explain turn kernel particular whole object extend set word might visual correspond dependency visual ignore unique give vocabulary patch feature category area extract concept category vocabulary patch represent category within image histogram chi square histogram discount root transformation extract patch stop bag extraction extract dense technique descriptor grid toward scale range extraction feature feature location produce use normalize unit sift evaluated image tuning applicable perform project enyi hellinger estimator extremely poor test enyi approximate twice kernel mixture fit gaussians monte previously chi quantization consider pyramid vocabulary pyramid kernel variant recommend computer vision group I two let wise match kx tune way pair get computation good match dominate see basic performance individual image angle angle image view color bin fix pixel invariance reduce preserve therefore test cross result show enyi well properly tune std std pair perform worse bad may hard divergence precisely show enyi near divergence performance r category dataset horizontal show scene method nonparametric kernel method dataset total show goal classify color location patch mean local allow location image
although direct reality assume case optimal access give access cope polynomial time fashion calculate dynamic simulation show good behavior link setup uniform sampling among case difference product find bound behavior set context thm thm remark examine various active sampling analyze normal distribution expectation setup address estimation although solution functional efficient structured functional induce yet obtain probe delay along efficiently delay accurately follow eq network assume move link possible trace trace reverse trace trace draw frequently trace since probe equivalently surprisingly trace result survey work experiment section formulate discuss section structure walks walk setup recent bandit chapter suggestion research similar pose study overview grow solution sdp insufficient complexity depend size recognize functional particular field concern resemble multiple focused estimating parameter al include weighted square similarity several difference entirely formulation focus efficient al exploration restrict although great interest assess address sampling issue effort solve specific present concern network deal overview al field find suggest parameter field effort find fine rather probe minimize another examine possible multiplication add subtract possibly infinite estimation logical define complex entirely clear new throughout situation occur since multivariate minimum diag diag take inverse information matrix mse like option believe appropriate option discrete time setup ease I addition shall restrict denote simplex formulated find optimal combination achieve formulate different sample setup per functional abuse notation row fit convex sdp programming minor variation restrict budget formulation affect solve nevertheless tend contain inner motivation describe sampling part start case subset variable natural functional functional one solution functional xx nn mb km diagonal entry tm nc diag generalize yield well kk solution show small suggest k combination least optimal functional show theorem entry unit functional dag direct acyclic node order normally estimate scheme delay delay actual path characteristic consider dag sample w exhibit instead functional middle far many path go counter suggest solution semi program impractical grid path large concern example grid distance subtract key problem although suggestion deal mse solver cope relaxed efficiently programming simulation show quite relaxed introduce product leave specifically denote leave easy recall distribution source path give product path optimal produce effort drastically order efficient solution calculate without calculate theorem use programming first time path sequentially finish node follow equation likewise start employ trajectory pass distinct leave probability minimize sdp specific entire even compute globally
separable constraint wolfe context structural svms subgradient wolfe convergence practitioner size need proper criterion size pass small pass review svms goal structured tag map information pair dependent slack surrogate loss th slack rescale unfortunately combinatorial nature one define structure replace structured hinge input efficiently many paper subproblem call non unconstrained formulation maximization subgradient respect maximizer decode subproblem write dual associated column tucker optimality ta cf finally b objective frank wolfe algorithm optimize thus wide iteration corner find picture next size start corner previously representation suitable case exponential second linearization yet compute linearization feasible special explain readily current convergence importantly choose theoretically sound stop know duality convergence hold appendix affine measure yield see formal apply dual structural due section frank wolfe main subproblem frank wolfe augment decode dual since potentially iterate frank sparse previously visit need maintain previously solution augment subproblem keep size use kernel maintain maintain structural primal obtain frank corner augment subproblem start linearize duality lagrangian problem gap compute maintain duality gap stop simply restrict give algorithms difference feature wolfe appendix obtain svm duality oracle since prove imply svm primal accuracy surprisingly frank wolfe equivalent primal frank choice search equivalence notice step subgradient wolfe batch step apply seem wolfe subgradient quadratic identity cut frank wolfe augment decode dual towards corner frank wolfe cutting optimize task program exactly visit result frank inclusion plane simplify ki disadvantage frank pass call oracle generalization frank wolfe maintains appeal wolfe algorithm block idea method affect method write pick uniformly block frank wolfe interpret nesterov svms need subproblem k k theorem obtain duality appendix partial assumption appendix compute structural svm classical wolfe curvature algorithm either predefine f h expectation block duality k apply frank structural dual maintaining see observe significantly vector analogous wolfe formalize svm bad new block specific search theorem overall rate approximate structural duality gap single oracle call require constant number duality gap cut plane subgradient frank allow compute iteration pass duality gap terminate unlike cut call use interestingly hold minimizer subproblem minimizer give accuracy duality gap theorem generalization improve applicability scale decode decode possible kernel maintain combination cut support xlabel effective ylabel line legend legend pos north east index header col comma product ls style thick mark triangle mark header col sep lambda product thick mark repeat option index header col comma lambda opt txt header sep comma lambda product txt densely thick mark option solid col comma lambda product ls header col comma datum dataset lambda confidence color solid repeat index header col sep comma include txt thick repeat index comma include lambda txt index header true comma lambda txt densely dot thick mark repeat header sep lambda table col comma color gray solid thick header sep comma txt densely style mark repeat mark option index header col sep comma lambda densely style mark repeat mark option solid header false comma txt xlabel pass ylabel area line legend legend pos north table index header comma lambda l txt color solid thick mark repeat header col sep comma txt solid style thick triangle repeat header col comma opt txt index col sep comma datum product l txt densely dot style thick square repeat mark option x index header true col sep comma dataset lambda l txt header col sep comma dataset lambda txt color style thick mark comma lambda style mark index col sep comma dataset lambda header col sep comma include data dataset lambda densely dot style mark repeat col comma lambda header col comma include lambda txt mark repeat header lambda color densely dash option table sep comma lambda txt densely style mark square mark mark index header col comma include txt xlabel pass ylabel false normal legend north sep include l confidence txt solid style mark triangle header col comma lambda product style mark mark repeat option solid header col sep data lambda opt txt index index col sep comma lambda product l txt densely mark repeat mark option header true sep include dataset lambda product l txt header col sep comma lambda txt style mark mark header sep comma include dataset lambda color header comma data lambda txt index col sep comma lambda txt densely dot style thick mark index index header sep comma header col comma lambda txt mark col comma lambda densely dash style mark repeat option table header col comma densely style thick mark repeat mark header comma include dataset txt xlabel pass ylabel false style log legend legend pos south true col sep comma include lambda color solid index index header col comma include l txt mark mark header col comma dataset lambda product opt header col sep lambda product txt densely dot thick square mark table header col sep comma product l txt table index header txt color solid style thick repeat index header col comma lambda color header sep comma txt header true col comma lambda txt densely style repeat index header col lambda index index header true col comma lambda color thick mark repeat index header col sep comma lambda txt densely style mark mark repeat option false sep comma include lambda txt color densely thick option header sep comma lambda txt xlabel pass ylabel legend legend pos col comma lambda l thick header comma lambda l txt solid style triangle mark option header sep comma dataset lambda opt txt table header col comma include lambda l densely mark mark header col sep comma lambda header col lambda txt color green thick mark repeat header col comma lambda txt color densely dot thick repeat header true comma lambda txt index index col include lambda densely style index header sep comma header col comma lambda txt color gray solid thick mark repeat header comma lambda densely mark mark x sep comma include lambda densely repeat option header xlabel pass ylabel primal e area legend west index header col sep comma lambda l blue thick mark triangle mark repeat header sep comma lambda l txt thick triangle mark option header col comma product ls opt txt header comma txt densely square option solid header col comma data ls index index header true lambda confidence color green thick repeat index true sep comma lambda txt mark mark repeat table header col comma dataset match lambda txt header sep lambda txt color densely style mark mark repeat header col comma lambda txt header col comma lambda color mark repeat header col comma match lambda txt style mark option index header col sep matching lambda l densely thick mark option solid table header false comma match lambda txt achieved randomized solver parameter solver batch objective setting pass result less e though surprisingly compare task frank exist labeling augment decode viterbi third sentence different structure bipartite marginal label decode flow frank wolfe option stochastic subgradient also kk iterate call recently converge instead analogously iterate method efficiently also provable convergence different several criterion dominate superiority especially practice term average sometimes slightly despite amongst investigation optimization objective reach specific accuracy measure gap regularization minor ignore cm cccc saddle gap bregman yes primal primal slack duality subgradient yes primal frank primal dual duality coordinate svms sequential structural svms output space estimate method oracle factor parameterization optimize obtain expectation degenerate wolfe actually dual exact accomplish search appear descent applicable loss include svm despite motivated perspective option exact direction provide rate method implement method summarize reach svms regret effect cut plane approximate unclear cutting plane frank wolfe oracle appropriately bound propose randomize frank block potentially iteration duality wolfe structural svm online issue address need optimal closed cost pass frank wolfe duality gap experiment structural svm realistic setting pass possible exact oracle guarantees computable gap available maximization oracle although expect frank wolfe helpful national ms partly support ms fellowship constant appendix frank structural svm provide appendix self frank wolfe main linearization duality interpret additional experimental approximation correspond affine form upper lipschitz diameter affine ambient frank wolfe lipschitz invariant wolfe generalize coordinate curvature expansion partial time diameter global block frank curvature structural maximal length proof twice plug taylor hessian two identical product upper time column formally curvature norm svm domain define maximal feature e block curvature block restrict block notation augment block analog whereas zero taylor word vector tight worst indeed exactly frank wolfe problem exact frank loop product simplex reduce corner solve decode vertex variable vector show simple linearization equivalent duality gap lipschitz continuous wolfe algorithms primal original meaning error difference n argument eq define structural convergence fw obtain iteration cost frank wolfe g paragraph iterate either predefine step use satisfie f furthermore algorithm iterate duality gap different block follow structural obtain approximate duality gap cost duality gap guarantee predefine step write convergence error f analogously plug dual detailed lemma bound predefine show stay valid plug additional part argument make duality add requirement restrictive structural need around comment practical aspect structural additional e wolfe svm sometimes get fix usually frank section duality criterion fast partial gap single pass necessary obtain efficiency reason soon convergence wolfe structural frank wolfe e block recall contribution denominator maintain mention subproblem duality gap duality gap paragraph instead primal combination number upper dot search I size implicitly dot product suitable term begin self contain wolfe prove two part fast line appearance know rate method good scalability large result optimization product domain I e converge let pick scale lipschitz domain structural nice example big difference variable subproblem loss simple variant solve approximately multiplicative coordinate pick single random leave unchanged wolfe gradient converge minimizer quality candidate additive error parameter default determine candidate still attain th error internal oracle together predefine step step candidate line maintain convention analysis average iterate duality iterate call appear getting prove fix say storing fraction iterate know stop alternative mention maintain uniform round trick round towards round motivate weight average prove irrespective frank wolfe paragraph example use coordinate descent block difficult prove strongly convex cyclic already use frank block could prove frank crucial duality namely sum domain linearization notation product curvature sum block show obtains iterate exact algorithm q approximate variant eq predefine search block iteration crucially lemma let move within direction towards obtain expectation choice block duality approximation take block condition write simplify curvature convex duality eq definition lead good therefore claim improvement random gap exactly follow idea variant condition hold line use directly duality gap deterministic inequality expectation side choice previous block q appear wolfe convergence difference expect similar argument slightly base immediately definition bind particular f induction simply rearrange analogous primitive quality exactly argument occurrence moreover combine easily additive accordingly convergence hold frank variant additive remove independent precisely give case match frank wolfe define curvature constant base get well dependence search solution duality frank wolfe look one true gap without iterate duality duality gap scheme iterate example quadratic dual concave affine function concave imply primal either predefine line iterate duality gap approximation define use average simplify expect duality gap start improvement variant begin take expectation lemma duality gap idea handle combination bind upper existence duality iterate combination coefficient use see master convex example consider predefined size master master proof iterate last proof primal induction line weak thank start convergence iterate line subproblem multiplicative quality primal decrease primal k subproblem additive quality replace variant decrease soon fall leave enter subsequent soon equivalently always plug recurrence recurrence appear solve recurrence monotonically decrease differential equation side q complete proof multiplicative rate finish hypothesis definition induction crucial simplicity structure need search eq even size yield recurrence could induction advantage schedule know frank wolfe step schedule case know batch ensure proceed frank wolfe primal line search get theorem duality gap duality average make use theorem master gap appear valid master iterate otherwise choice master fast amount conclude k follow argument similarly part upper integrable ft n nk work complete sign strictly decrease b complete original convergence predefine shift note fully weight average regime primal refined speed forget case begin gap use optimization crucial differentiable linearization give arbitrary subgradient position ff immediately crucial duality gap differentiable function subgradient duality additionally interpret duality subset give f indicator conjugate directly assume inequality become equality choose subgradient last subdifferential f f detailed explanation refer reader summarize simple linearization duality gap restrict particular position slack lagrange multiplier scale multiplier variable multiply correspond primal attain finite saddle simplify also wolfe w set q expression lagrangian lagrange problem negative claim provide experimental frank wolfe method compare simple col sep comma lambda product l txt header col sep include lambda confidence txt color thick mark repeat solid header sep comma lambda l txt header confidence txt green thick mark index header col sep comma include lambda txt densely thick mark mark index header col comma lambda txt index true comma include txt densely dot thick mark repeat header col comma datum dataset lambda index header col sep comma lambda confidence color black x header col comma include lambda avg header col comma lambda txt gray thick col sep comma lambda xlabel pass ylabel legend pos north east col sep comma lambda confidence solid mark repeat header true col sep comma include lambda txt table index header comma l txt style triangle repeat table header sep lambda txt color densely thick mark triangle mark mark header col include lambda opt txt header col sep product l txt color densely style thick mark index header col sep comma lambda product l col sep dataset lambda confidence txt solid style repeat header col comma include lambda color densely thick mark col sep comma dataset lambda header sep comma lambda confidence txt color densely dot style mark repeat header col comma include lambda index header true col sep comma lambda avg color black thick mark mark repeat header include dataset avg index col include lambda confidence color thick header col comma lambda txt xlabel pass ylabel header col lambda txt color mark repeat x true dataset lambda txt index header col comma lambda product txt solid mark mark repeat header comma lambda txt color densely thick mark option solid header col sep comma l opt txt x index header true col sep comma dataset lambda ls color densely style header col sep comma lambda l txt table header col comma include confidence txt solid thick mark repeat comma include lambda txt color densely dash mark mark col comma dataset lambda txt header col sep comma lambda txt densely mark table index header comma lambda x col sep comma lambda avg confidence mark repeat table header col comma data lambda avg txt header col sep comma confidence color gray solid style thick repeat index col sep lambda txt xlabel effective pass ylabel primal legend pos east index header col sep comma confidence txt thick col sep comma lambda product true col sep comma lambda color blue thick mark repeat x header sep comma lambda product txt dash style triangle mark solid header col dataset lambda product opt index index header sep comma lambda color densely dot style thick repeat option index header col comma include lambda l header comma lambda green style index header comma include lambda color densely mark header col comma txt index header col lambda densely dot thick mark table index header true col sep comma lambda txt header include lambda avg black solid thick mark comma include lambda txt table col comma lambda txt gray mark mark index header col comma data dataset xlabel ylabel false area line legend header col comma red style square mark repeat index header col comma lambda txt index index header sep comma confidence txt color solid thick mark none mark mark col sep comma include lambda txt densely thick mark repeat mark option solid index header comma lambda ls opt txt header col comma product densely dot style mark mark repeat option index header col comma include lambda product l sep comma lambda solid mark header comma txt densely dash style true comma lambda txt index header col comma lambda densely dotted thick mark mark true col comma lambda x index header col comma lambda txt color solid thick mark col comma dataset lambda avg txt col comma lambda txt gray mark repeat true comma reach increase thousand early xlabel pass ylabel log legend pos north east index col sep comma lambda txt color thick square mark repeat header col comma data lambda header col comma txt color mark triangle mark repeat header true comma l densely dash mark mark solid header sep comma lambda product l opt txt table index header col sep comma include lambda txt color densely thick mark option index col sep comma txt header comma include lambda txt color mark repeat header col comma dataset densely style thick col sep comma lambda header comma include lambda txt dot repeat header col comma dataset txt index header col sep comma lambda avg txt thick mark x col sep comma lambda txt comma lambda gray solid mark col comma include lambda xlabel ylabel area legend index header col comma lambda txt solid mark repeat col sep comma dataset lambda txt col sep comma lambda l txt blue style mark none triangle repeat index header comma lambda txt color densely dash mark table header col sep comma include lambda l opt table index header col comma l txt color densely mark square mark header col lambda ls txt header col comma include lambda confidence txt solid style thick mark mark repeat dataset lambda txt color densely thick mark table header col sep comma txt table x index header comma lambda txt color densely dot style repeat index true comma datum lambda txt x header col sep comma avg confidence black style thick index header true col sep comma lambda index index header col dataset txt color solid thick mark repeat index header true comma txt xlabel pass ylabel area style legend legend north east index header comma lambda red thick index header col comma lambda header true col comma dataset lambda confidence txt color solid mark repeat header comma lambda product txt densely style thick triangle mark option solid col comma opt txt header true col sep include product txt densely dot thick mark option solid index true col comma lambda l txt header col dataset lambda confidence color style mark header col sep comma lambda txt densely thick mark repeat x index comma lambda txt header col comma lambda txt densely style mark repeat header true col comma header comma lambda avg txt mark mark repeat comma lambda txt col sep lambda txt color solid style mark repeat x col comma lambda xlabel pass ylabel legend header comma red thick repeat col lambda product txt header col sep comma include lambda confidence none triangle table index header true sep include txt dash style mark mark option comma lambda col comma include lambda l txt densely dot repeat mark solid col sep lambda txt header comma lambda confidence txt green solid mark mark repeat col densely mark header comma data lambda col include lambda txt densely mark header col sep comma include lambda txt header true comma avg style thick mark header col sep comma include lambda col lambda solid thick mark mark repeat index header col comma lambda xlabel pass ylabel legend legend north east index header col sep comma include lambda red style thick mark x index header col comma product txt table index header sep comma txt color solid style mark mark header col comma lambda product densely style mark repeat option index header comma lambda opt col include dataset ls txt densely mark solid col sep comma lambda txt index header sep comma dataset lambda txt green solid repeat index header col comma lambda txt densely dash sep comma lambda index sep txt dot repeat header comma lambda txt header sep data avg color style thick mark header col sep comma include lambda avg index true comma include lambda txt repeat header sep comma lambda txt scale xlabel pass ylabel false area legend header col sep include thick mark repeat index header col sep comma include lambda table index header true col comma include none triangle mark col comma include l densely style thick triangle repeat option header lambda l txt sep comma color densely dot thick option comma include lambda txt index index col sep include lambda txt color thick mark repeat lambda densely style thick mark repeat col sep comma include dataset header col lambda densely dot style mark mark repeat index header col txt comma lambda thick index header col sep comma lambda txt comma confidence gray thick mark mark header lambda xlabel pass ylabel primal log legend north east index header comma dataset lambda red mark header col sep comma lambda product txt header comma include l txt style thick triangle table true sep comma include l txt color densely dash thick option table header true col sep comma lambda txt table header col comma lambda txt densely dot style thick mark option solid table header col lambda header sep comma match lambda txt solid thick mark repeat table header comma lambda color densely dash style mark mark repeat x header true comma lambda txt index header col comma lambda densely mark header sep lambda sep comma lambda color thick mark index header col comma match lambda avg header true sep data lambda txt thick mark repeat index header col comma match lambda txt xlabel effective pass ylabel log area legend index header comma match lambda solid thick mark repeat header comma matching txt index header col comma lambda l txt color thick mark none mark index col sep comma match lambda l densely dash style thick repeat mark option table header col comma include dataset match lambda product l opt txt index true col comma lambda txt color densely dot style square mark sep comma include match lambda l header col sep comma include color solid style thick mark repeat header col comma dataset densely dash style thick mark col sep comma index index header comma match lambda txt color densely dot mark index col sep comma include lambda txt comma include avg txt mark mark repeat header comma col sep matching txt color gray repeat col comma include lambda xlabel pass ylabel style log legend legend east index col sep comma lambda red mark square col comma include lambda col sep include datum lambda product txt blue repeat header col comma lambda l txt densely thick mark mark option header col comma lambda l header col comma dataset match lambda txt color densely dot mark mark header col sep comma dataset match lambda l true col sep include match lambda confidence txt color solid style thick repeat header col comma match lambda densely style mark mark repeat header col comma lambda txt table index header col comma include match color densely dot mark header col comma lambda txt header col sep comma lambda color mark repeat header lambda avg sep comma match lambda txt gray mark table index header col comma matching lambda xlabel pass ylabel area comma match lambda txt red style thick square repeat index header col include dataset match lambda product txt header col comma matching l
domain domain unlabele formulate knowledge statistical relation bad include remove audio comparison propose multi modal relationship application access domain task word recognition greatly benefit availability audio verification much accurately modality major multiple label domain example wish audio gender label label audio set limited subject although straight train image alone clear good modality pair audio collect nonetheless predictor even practical availability aid machine suppose age audio audio training unlabele audio visual help solely audio domain unlabele domain unlabeled set use design predictor alone account label case set view tend agree unlabele construct setting observe label I I make view framework assume availability mapping assume exist implication lack multi infinite however suffice determine give mean error mmse x situation label predictor domain fall unlabeled example domain available suited view extract handle via framework nevertheless setting pair unlabeled transfer scenario data modality audio video perform one construct label want classifier operate audio audio audio share construct alone instance want train visual unlabeled audio instance cross recently link instrumental also highlight show cross representation learn example one label statistical assume mse estimator use domain domain unlabele every label multi course unknown bad set problem available insight multiple domain particular help underlie illustrate audio visual word application demonstrate convert neutral pair neutral face recognize available recognize audio access preferable paper regression capital bold letter xx mathematical expectation px inequality wise xy several give access label training independent assume approximated nonparametric label zero relation label associated mmse say accurately term test treat ask predict base solely include label datum several fig double circle correspond continuous line multi share also label adopt generalize statistical l independently practice parametric try rich accurate come bias sequel estimator one moment yx xx sample correspond form predefine arbitrary set k ki j quantity linear reference trivially borel measurable case zero set label degree even number set determine joint suffice compute mmse uncertainty able specifically determine predictor accordance note predictor one valid example predictor many unlabele label estimate depicted assume nf illustrative suppose linear pdf restriction valid infinite mixture eq density requirement consistent construct predictor domain tackle domain estimator give alone access domain set possibility include two predictor alone worse minimize mse close joint triplet know belong subspace function mse attain expectation since depend concept estimation interested next find mse distribution solution need mmse estimator among possess simple form demonstrate utility mse approach performance sometimes area bad difference mmse depend bad minimization mse insight characterization namely show multi minimax solution fig consequently solution show come unseen bad perspective predictor distribution perform stand contrast basic help agree context view amount suffice identify predictor collection consequently imply moment unlabele training similarly determine apparent however orthogonality respectively carry knowledge formulation case suffice situation k orthogonality suppose determine domain good predictor mse particular twice obtain stage technique unlabele u l www yx yx yx case base minimize rely concept innovation innovation mmse estimator denote make regard minimax appendix innovation optimal consequently denote x conclude approximated training domain second linearly interesting improved operate proximity joint seem help improve accuracy however narrow distribution design belong optimize single regret coincide minimize bad determine minimax estimator hard bad remain goal determine next describe single domain minimax explanation sense measurement alone interesting example illustrate fig intuition boost solely label framework set situation domain minimax imply cross unless cross learning classify isolate unlabele visual fail boost audio bad nothing illustrate fig second naive feed predictor base rather mmse approximate unlabeled minimax predictor single predictor case generalize coincide good predictor case yx yx yx strategy suppose numerous x x coincide u innovation highlight come train unobserved help recall term vanish q rich need meaning already determine mmse example jointly linear consequently moreover jointly imply coincide indicate another switching role estimator label training second vanishe happen example concrete example function x therefore x x x mmse therefore satisfied domain example form rather concrete side approximated covariance approach derive result illustrative preprocessing aim remove observed database may include variation illumination demonstrate produce neutral face straight forward appear twice neutral unfortunately collect access appear relation neutral face impossible obstacle view example pair manually several predefine location image database triplet annotation neutral may unlabele annotated face neutral employ design predictor construct third image neutral face apply regression discuss depict manually annotate neutral annotation template appear automatically unity kernel tune constant root correspond image show result neutral fourth result set face neutral training comprise neutral perform side information equation rmse attain low set lead additional training indirect seem worse apparent sort average joint example neutral ever able know relate point annotation relate neutral half people neutral set direct nonlinear share claim output digit video corpus structure sentence sentence isolate distinct set label visual
inspire problem require allow variational applicable close approximate mixture mean approximation principle resemble perform allow introduce variational approximation solve new look variational posterior represent idea make generally applicable extension separate discusse improve section despite generality competitive conclude effect observe upon bayes distribution proper kullback leibler kl divergence almost divergence posterior analytic monte choose factorize often solve call factorize make optimization restrictive conjugate specification assume posterior independence block parameter factorize parameter shape bayes usually member vector statistic normalization base often call approach optimization parametric analytically entropy analytically determine derivative respect posterior often factorize able evaluate restrictive generally allow factorize next draw parallel problem variational bayes linear regression limitation notational write assume vector family distribution rescale unnormalized px qx respect expression minimum implicitly identifiable exponential family insight maximum dependent matrix maximum associate unnormalized consider analogy analogy notational base measure analogy continue residual base model right depend constitute optimization section require computable allow extend impose simplicity mixture approximation variational regression carlo update px equation iteration unnormalize approximate initialize guess match prior approximation initialize draw approximation px tc n inspire research start seminal relatively update step size taylor give comparison divergence first pre since stochastic usual pre close algorithm family analytically suggest approximated analytically first seem approximate condition turn approximate draw dramatically reason optimal use randomness term particularly example approximation exactly gradient vanish finite sample exact exact functional still deep contrary literature algorithm statistic asymptotically gradient infinity conclusion e decay fast choose either take long reach rate put order vanish final algorithm grow number remarkably average match optimization excellent average half averaging rather remove use previously use mc give mean early expect guess define negative pre converge unlikely good help pick good guess help quickly difficult guess rough choose curvature guess default prior increase sufficiently stable variational divergence unimodal optimization optimum multimodal unimodal variational mode desirable guarantee minimum kl approximate variational functional normalize variational precision strategy analytically bad run option even option option toy approximate functional dash use give serve kullback leibl divergence give likelihood depend measure leibl essential performing model comparison present final kullback follow intercept recognize vb integrate marginal eq convergence jensen inequality tell log everywhere lower often kl however often comparison oppose marginal evaluate monte carlo show approximate marginal something use regression perform relatively log plus capture approximate kl approximately equal relationship come square exactly highly dependent curvature comparable scale curvature posterior therefore across set interpretable application mostly moment therefore construct relatively moment tail kl divergence good propose bind calculate approximation quality different approximation efficient extension discuss example conclude approximate expectation draw work remarkably well explain form certainly approximation often well addition problem version probit classic literature exist gaussian cdf gaussian performance method major much wide model simulate make use expectation calculate analytically compare message excellent performance classification implement inference approximation negative otherwise variational use thought truncate univariate posterior prior parameter often extra correlated reduce variance approximation implementation stochastic dependent g language look since synthetic performance quantify root rmse mean score posterior finding estimate present well see rmse assumption make introduce rmse indicate approximation give significantly average variational indicate rmse score identical digit give factor hessian transformation efficiency linear section slow convergence term run draw ep converge small ep rather infer matlab section introduce implementation probit regression additive factor optimality condition regression equivalently regression conjugate example know analytically approximate importantly separate often regression low factor projection contain property correlation control subset remain sufficient zero course omit low regression approximation store probit depend write link normal approximate distribution py f control approximate particular transformation use variational new parameter transformation jacobian make combination hessian usual parameterization stochastic approximation approximation working parameterization express gradient average algorithm appendix storage new unnormalize twice posterior total iteration initialize guess matching prior initialize step calculate gradient storing invert induce dimension parameterization hessian store invert even probit structure also twice still log log additive unbiased learning step since calculate large tradeoff subsample many successful gradient probit implement divide process update minibatch reach converge require pass much approximate relax approximation typically analytically approximation often still moment mcmc indirect identify strategy two block posterior choose analytically easily freedom capture practice conditionally tractable block type bayesian hierarchical suggest exponential joint normalize conditional precede conditional optimality condition still variational slight modification derivative setting sufficient leave far intercept normalize approximated straightforwardly carlo still perform separate conditional section alternatively approximation mixture allow dependency precede approximation exponential signal vary variance extremely finance vs model govern autoregressive specification hierarchical prior q exponential q choose form examine notation prior normal therefore multivariate conditional approximate approximate form statistic block optimize approximation make fact respect kalman direct indirect effect case indirect little application give expectation calculate analytically kalman sampling element belong posterior inverting precision expensive still matlab ghz processor magnitude mcmc figure similarly posterior latent show indistinguishable ht stochastic volatility characteristic importance auxiliary regression infer mcmc method able leverage hessian posterior approximation optimize volatility approach ability posterior need significantly approach construct flexible family letting analytically tractable approximation mixture enough mixture arbitrarily close note multimodal suffer kl q requirement integrate auxiliary joint density fortunately approximation mixture normal apparent section equation auxiliary thus posterior practical normal consider problem estimate death cancer pair city cancer occur city binomial model follow beta binomial precision give improper prior skew eq bivariate gaussians auxiliary assign categorical outcome approximate adapting include discrete approximation fit rao expectations advantage statistic equation mixture posterior therefore use expression update introduce extra expand apart different mixture use examine quadrature tail density gaussian mixture apparent consist square clearly advantage much rich true quite especially unimodal optima good global optima approximate arbitrarily py qx lower height solid line interpret newly propose section thus usual inspire regression sufficient approximating since unnormalized density distribution show nature software package along net would considerably wide range automatic beneficial automatic selection appeal despite applicability demonstrate problem hessian rare without factor present extension important use hierarchical practice mcmc chain option mixture tune trade mixture family analysis model variational usually conjugate collapse integrate adapt work collapse complex model methodology collapse mix conjugate conjugate closely variational message algorithm infer net conjugate maintain convenient pass formalism specifie perform variational approximate integral expectation quadrature secondary unlike monte open acknowledge anonymous thank organization scientific support college microsoft stanford center cancer unnormalize condition q clearly normalize recalling give tx tx tx tx px solution divergence point notational variational conceptually mean kl divergence regression q usual mean parameterization parameter differentiable identity q parameter stochastically point author perspective sampling convenient distribution share specifically choose variance result evaluate sampling importance practice process stability condition identical fitting posterior important tail variational posterior unless recommend might directly instead suffer offer way combine monte aforementione recent literature collapse aspect
institute mathematics natural university institute consistent kernel machine generate finite basis sample fourier carlo induce fourier distribution correspond parameter domain identify band prediction correspond kernel scalable learning methodology multiple produce fast apply problem radial different linear combination complicate practical vision shape texture descriptor role category principle easy represent descriptor base combination category learn scalability combine kernel limited consequence capable insufficient massive million image imagenet article principle costly preserve non embed span infinite simplify notable kernel learn massive seminal radial focus kernel histogram empirical kernel largely multiple combination produce conduct recognition difficult object order learn linear operate broadly classify single kernel combination kernel prove learn definite involve attempt track conceptual model kernel semi definite programming reformulate norm regularization cone medium scale kernel combination direct search perform difficulty multiple relatively example scale become formulation within framework carry scalability wide applicability large attractive kernel offer substantial central dot point become infinite feature mapping visual popular trick quadratic example direct usage impractical dataset methodology linearly random methodology fourier transform every transform define j frequency carlo linear explicit feature two monte dimension dataset hundred thousand example train motivate interest class approximation randomize dimension htbp c u methodology achieve goal need criterion parameter fast due rely procedure kernel optimize parameter hold validation training overfitte sequel denote input wise organize target drop map kernel ridge cost hinge logistic etc eq optimize square rr q regard expression expansion regard th kernel kernel identify band useful prediction effectively expectation parameter change although conceptual difficulty become parametrize far away start resample potentially nevertheless convergence belong q fix product throughout entire process generate automatically feature obtain respect smooth norm formula parameter fouri optimizer parameter similarity introduce besides usage embed advantage kernel number invert connection interesting practical far section fourier methodology opposite lift group formulation initially fouri lasso prove approach feature well store organize wise target example concatenation would component apply overlap formulation write feature quadratic insensitive regression perform solver standard present build gram define set element primal associate theorem drop simplicity rewrite trick multiple shown observe group lasso formulation set equality happen well restriction order insensitive il standard insensitive function logistic group assume algorithm maximum solver dominate computing matrix approximate become constant could slower thousand kernel learn compare random fourier methodology counterparts accuracy challenge image image without follow
plant community lin first optimum run heat iteration refine ever possible initialize stability try isolated community focus community plant keep community assignment make concrete block vertex label label correctly label fairly assigning classify degree probably connection near equally initial correct unable degree well assignment structure help somewhat separate degree vertex degree even broad degree exponent work randomly initialize either assignment kl follow generate perform non correct work undirected network block degree distribute orient plant asymmetric choose work simply grow correct edge block versa network precisely generate degree classify vertex since distinguish block network run step vertex separate consist common novel corpus american english many corpus news contain network ever version occur size size news accuracy correctly label everything noun tailed degree asymmetric english often roughly block noun david news rs r likely block e compare model apply ignore direct start random lin heuristic find optimum run heat naive heuristic label noun correct assignment david news independent execute network bold sbm dc david fairly moreover mistake zero since away correctly block well fail bad degree separately edge full case thing even broadly slightly news despite likely assignment connect news edge suggest though kind look try give model assignment kl news take base degree refine low use random help stay accurate optimum condition follow step news sbm nh counterparts distribution heavy moreover vice degree assignment r r block noun table let improvement slight well alone degree generation benefit optimization let good kl step oppose degree give assignment generation converge avoid kl degree block deal network sense help block allow heavy orient partial total degree degree assume generate prior unlike direct able capture account degree put unlikely degree high direct subroutine impose community illustrate small block generation appear handle without benefit kl depend heavily know correct form degree truth assignment difficult variant correct degree infer structure network bad strength kind kind select meet grateful foundation constraint q r log likelihood partial give orient correct undirected number direct edge poisson check impose r rr ignore rs rs recover rather eq bayes line denominator however prior analytic call likelihood degree generate distribute proportional product gamma natural gamma gamma hyperparameter block multiply family uninformative reduce integrate focus estimate continue posterior simply posterior calculate integral algebra numerator denominator hyperparameter optimize empirical approximation integrate long degree result integral impose degree likelihood derivative setting get com tool infer community world degree degree correct block accommodate degree vertex vertex generalization direct tail english text high correct connection social daily contain link political speech vertex way feed relationship functional crucial highly generative block vertex block let capture many community community independently membership distribute solely membership block degree lead model heavy degree distribution community conservative degree undirected parameter vertex control degree accommodate degree community remove tendency degree vertex correct classify vertex explain may recognize separate dc direct generalization block two parameter degree english rarely vice versa strongly membership leverage type strength correct block able utilize degree asymmetric include stochastic treat law whose likelihood assignment interaction community structure community degree separate community experiment well network community enough vertex classify vertex block relative community notion degree strongly asymmetric partially capable take call try version direct point direction generate undirected degree correct oriented writing giving orientation direct inter possible substitute combine log add obtain poisson special case force edge way utilize domain exponent us membership tail degree help vertex unlike maintain degree generate specifically independently depend block
connect paper represent clique node degree increase rank capture paper characteristic observation degree connect connect share explain mean link connect connect biased connect connect plus connect branch implication observation use take difference solve generality consider eq normalise evaluate recursively ambiguity dependent tend unique evaluate rank scheme reflect evaluation entropy minor neutral network average obtain circle neutral line see statistical fluctuation evaluation also degree nan evaluate show two quantity z j link nan notice score closely link degree degree nan nan degree entropy degree rich many tend correlation good model closely reproduce nan rich connectivity lagrangian network rank sort decrease order useful comment suggestion restriction rich two share link describe nan approximate network topological pattern behaviour occurrence mechanism perhaps responsible complex statistical method surrogate reference nan purpose intrinsic preserve connection node basic procedure surrogate sequence pair link create degree procedure create sampling bias network self loop many example bias take consideration generate random technique order nan shannon widely describe attack shannon case entropy approach way maximally maximal entropy attractive nan probabilistic measure use model degree correlation community use explore walk equal law method structural like degree difficulty share degree lack heavy measurement insufficient limitation node degree classify limitation correlation classifying network limitation nan ability could method construct rich coefficient connection rich correlation accurately distinguish node degree node degree rank scheme node divide node notice enough allow multiple rich rich probability connect loop impose link formulate represent solution formulate normalise interaction interact via maximal probability maximal certain common use via link give constraint show network five network label degree link give map relationship inside node label inside read probability link correspond pair constraint sum lagrangian multiplier nn mf q equation define p
proposition noisy exclude therefore verification lasso replace still valid w I mu tm central limit probability tend normality tucker kkt j j shown verify note tm n constant verify slightly modify th replace replace right still tend end iii fan show consistent assumption verification assumption scad suffice event satisfy scad convergence assumption x es ij ik ik jk ik jk lx lk ik iw lx lk lk jk fact jk chebyshev eq together jk meet validate condition pattern q side tend specifically p cn sign th j addition uniformly tend independent satisfying end end verification definition wang wang supplementary material fourth scad assumption fan I suffice follow inequality follow finite chebyshev prove lemma I regression adaptive scad slight modification existence verify subsample magnitude element accord c select active
compare horizontal scale multiplication precise indicator tc tc bc k bc bc tc bc k bc objective zero conduct see iterate convergence slightly due conditioning reach precision overall gradient hand intermediate require loose suffer slow sublinear begin behavior vary keep big variation affect stage suffer sparsity rough last vector interesting implie still mc bb tc tc tc tc bc bc introduction homotopy study l reconstruction separable use strategy reduce iteration initialize satisfied say accord method able exploit restrict acceptance stop homotopy either monotone employ call bb option bb precision numerical figure demonstrate number stage number bb result loose option reflect figure use intermediate overall pg bb tc tc bc bc inner figure nonzero convergence monotone version bb low bb set bb method look high homotopy see homotopy counterpart instead mc bb tc bc pursuit experiment choose generate independently I transpose transpose soft modulus bp ls use bb remarkable converge requirement become small always reach loose homotopy recovery confirm stage default reduce bb stage proportional regularization stop constant homotopy path advantage homotopy homotopy recovery multiplication homotopy method need track break multiplication homotopy important strongly hence show iterate homotopy method structure objective become convex thus geometric homotopy accelerate automatically exploit convexity discuss convexity obtain convexity ls accelerate come explicit nesterov give suggestion direction strategy periodic investigation lemma assumption zhang square iterative slow nevertheless exhibit fast local homotopy l approximate stage warm although assumption homotopy iterate homotopy function effectively find solution interpret rate convergence solve l vector term learn statistic receive interest year build representation recover measurement l recover measurement relate regularization great investigate rip constant recovery denote nonzero element depend rip achieve closely paper numerical recovery focus sparse sufficiently assumption method provable complexity algorithm follow nonnegative appropriate choice however parameter directly nevertheless efficiently lagrangian root find similar nice survey practical briefly summarize complexity relevant ls minimum among use ls bottleneck approach normal cholesky factorization prohibitive large solver method conjugate multiplication involve computational cost multiplication conduct l problem shorthand iteration search procedure rule include appropriate iteration complexity establish strongly unfortunately sparse submatrix rip fast stage g nesterov minimize l accelerate typically statistic complete exploit piece linearity examine condition submatrix break quite general bound recovery homotopy idea gradually reach employ adequate approximate serve homotopy use e result mostly hoc sequence method provable along algorithmic geometric value precision precision nesterov parameter gradient target sufficiently final solution sparse satisfie like exhibit sufficient universal satisfie overall imply convergence homotopy proximal gradient stage always start parameter path stage rip imply along homotopy path effectively geometric nesterov homotopy track scale much small target number homotopy method phenomenon complexity also confirm empirical compare iteration term theoretical homotopy strategy path square condition near provable solution interior insensitive important free compressed l sufficiently special paper parameter homotopy analyze extremely difficult free pursuit technique demonstrate proximal homotopy path significantly simpler necessary homotopy main devoted proof map key nesterov ls condition nesterov close solution call proximal quadratic lipschitz see q satisfy composite mapping directional eq therefore presentation x b x correspondingly add subscript specify fy fy call ls problem gradient lm l k x composite develop variant proximal gradient accelerate primal need choose optimistic line try lipschitz iterate algorithm line ensure lemma objective unless mapping case criterion optimality l additional depend differ need line search although repetition nesterov search let lemma example nesterov complexity find strongly convergence complexity nice function whenever case objective strongly complexity though convergence observe nevertheless homotopy strategy enforce iterate rip homotopy geometric explain condition characterize throughout practical purpose constant message without special attention optimize natural pick automatically satisfy condition concern basically start precision geometric simplify presentation symbol denote restrict satisfie roughly proximal choose q condition step k hold eq terminate total output satisfie measure optimality precision reach let let achieve gap global restrict global plus last globally particularly interesting sense bp algorithm gap informative convergence suppose outer outer global geometric bp proof subsection iterate algorithm prove theorem appropriately homotopy lead iteration complexity direct sparse eq let subgradient achieve approximate ta ax z ax break part fact combine gives q rearrange arrive obtain x inequality prove since subgradient result much suppose assumption proximal optimality monotonic objective establish iterate small expand ax ax ax ax therefore ax splitting x drop side claim inequality q hold argument imply last eq mean much sparse operator eq nonzero truncation part inequality last exist eq take side inequality since achieve eq side prove recall take form monotone iterate satisfie search terminate imply lx smoothness termination search accumulation point accumulation accumulation restrict x fx accumulation accumulation especially stop procedure x second relax restrict segment eq homotopy proximal approximate eq k fx prove desire ready give estimate homotopy method within optimality km kx smoothness restrict property theorem obtain g k criteria eq prove homotopy outer performance follow moreover outer instead method terminate apply desire bind numerical support first property method implement follow solve pg nesterov line search propose described nesterov accelerate method pg replace dimension entry choose dense uniform choose
receive node receive recursively certain construct node imply consist ratio threshold error equilibrium fast rate rate slowly slowly respect backward search network integer node asymptotic law fast scale scale let network converge assumption distribution private proposition derive q much rate probability follow situation p j c scheme length least cn n kk consider situation node observe backward scheme length away converge moreover converge converge sequential testing model channel recall symmetric channel complement k probability flip symmetry section easily general know probability decision suppose exist exist strategy observe immediate decision immediate k xx ratio rewrite linearly without henceforth obvious become type error bound regime appendix error presence probability converge condition probability converge describe way suffice type I strategy map corruption away ensure contain true mean decision increasingly uninformative move recall denote belief belief eq easy derivative monotone follow ratio martingale converge surely expectation martingale almost public belief almost implication public belief completely wrong public bound away public everywhere imply almost everywhere suppose statement independence eq continuous contradict almost surely prove converge convergence proof generalize scenario away variance bound convergence generality generalize denote belief converge surely property assume exist loss assume characterize polynomial tail density public fast rate establish bind converge rate sufficiently type I decay probability decay substitute evolve characterize non proof away non constant belief fast outcome surely sufficiently small almost error probability decay linearly decision asymptotically uninformative maximally completely uninformative belief condition public belief evolve characterize sufficient negative logarithm exist subsequence show show contradiction eq public converge limit represent decision among public limit surely evident nonzero away truth hence provide learn nonzero belief collective arrival explain let event decision arrive collective false phenomenon use occur lead hope nonzero expect depend relate scale law condition surely almost surely almost surely provide error guarantee necessary condition nonzero space sequence infinitely proof wrong decision e sufficiently public belief convergence uninformative fast signal public infinitely slow belief exist lead exponent expansion density theorem necessary density assumption tail derive explicit establishes law belief uninformative tail probability almost asymptotically uninformative probability surely k kb k decay ii assume eq decay type failure bound decision case go relationship observe decision converge necessary characterize relationship convergence derive event nonzero multiple similar go decision want generalize technique network topology moreover failure expect scenario finite snr martingale convergence snr go go infinity signal snr error probability node observe decision threshold combination suffice ratio decision equal decision always small ratio exclude hypothesis testing node decision ii xx p xx xx j k see treat recursion ode ode exist two public belief plug establishe claim iv ode establish satisfie rate fast outcome result probability condition q conditional decay recursion public belief know decay condition event positive q iv solution turn thank anonymous reading manuscript constructive comment presentation node sequentially make give hypothesis observe decision two failure node subject interested ratio decision number bound immediate decision show grow number decision converge probability converge consist test node previous decision probability channel away locally convergence error converge decision locally underlie truth explicitly relationship probability learn decentralized channel social sequentially decision node measurement receive immediate decision true converge sequential single sequentially collect fix pre maker stop hypothesis decentralized sensor network case spatially hypothesis decentralize network resource sensor aggregate sensor sensor failure case fail message moreover channel sensor noisy message subject aggregate spatially truth sensor increase another agent case underlie also state learn action decision agent use objective locally minimize bayesian ratio social feedforward customer decide whether restaurant previous feedforward private customer receive represent reveal might decision decision converge rate e converge converge explicitly convergence error use denote nonzero denote denote node scene decision hypothesis decision corrupted carry make decision immediate find sequence making tend e proceeding definition assumption algebra assume history mutually probability algebra absolutely ratio
product evaluate whenever term follow constant occurrence notation although statement asymptotic defer ne intuitive become replace proof exponential tail sum concentrate around essentially guarantee term hence contribute term chance confusion us range affect verify product convention member order pair countable define let q application produce ready give functional brevity break proof original product part form denote first term apply give low matching rhs choice eq q lemma appear product second lemma get first apply rhs bind let proceeding bind miss equality desire remain delay r accord correspond edge break two e break edge total category obtain product extend sense term bound technique remove index describe put piece asymptotic behavior throughout conditional convention appear condition definition r tn intermediate nothing dependence equivalence collection q irrespective whether n ne en en prove theorem regard asymptotic behavior far bound eq long bound polynomial say n u ec obtain nothing boundedness take result proof elaborate proceed step various time shorthand notation introduce drop en event hold c bind long boundedness state large sum e break follow move couple throughout drop implicitly lemma statement n n I en ne en statement boundedness let eq prove lemma definition proof ne ne result log e u ne ne ne p ne ne p p note sum sum u ne u establish early see note lemma move introduce e sum rule take term apply ne recalling term en since lemma u r u get graphical stop change functional subset thresholde message delay direction change restrictive simplify place impact phenomenon delay detection rather interesting analysis approximate message pass algorithm computational scale constant fast approximate message theoretical publication remark I decay seem likelihood break complicated simple piece irrespective tail behavior whether one derive minimax consider node argument show independence priori depend notation convention product divide obtain hold definition one drop term nonnegative rhs version enumeration version introduce version let eq dropping separate start prove pick throughout pm pm knn kn hold shorthand kn multiply b n b b side interval nb nb bb b probability imply b fix change thus enough small integer satisfy previous assumption c generality denote assumption polynomial moment lemma remark ram discussion support part grant sequential detection present detection rule functional delay graphical formalism rule protocol linearly size demonstrate classical detecting time decentralized data rule constrain take paper interest change may detection traffic concern spatially treat change sequential accounting reduce alarm probability detection delay propose formulation multiple change point network adopt estimate globally locally across statistical site associate point another detection message pass network computation interestingly express leibl divergence define simulation rich sequential tend variable formulation sequential diagnosis associate take across array interesting distribute line impose severe constraint site use usually understand denote integer denote link sensor variable represent time sensor fail interested point take endowed central ingredient formalism specify linkage observe collect cf vertex associate share distinction aggregate observation denote nm nm ccc graphical depend hence often density density specification summarize I change dependent despite independent priori primary change eq algebra detection stop detection rule trade false delay pearson false alarm minimum delay ingredient formalism represent rule aggregate primarily fashion pass message one neighbor graph conceptually communication play two coincide obtain multiple crucially rule geometrically highlight case simplify asymptotic delay delay eq delay see delay decrease counterpart independently result delay regardless whether edge asymptotic basis give basis posterior counter result even moderately access one geometric exponential allow tailed brief state ingredient heavily message pass mp single star node pair point together false alarm tolerance depict node sequence post prior geometric mp posterior node private single change ij limit expect carlo change take single relatively alarm though slope suggest mp single access pair mp mp high guess regard nonzero observe establishe concentration ratio term cf marginalization asymptotic graph complete
decade progress make design efficient undirected graph dimensional observational method penalize regression work feasible excellent theoretical property normality copula copula replace extend normal transform smooth almost study represent relationship encode I conditionally invertible towards zero vector independent property pair construct random define kl kl also appear free explicit easy compute formula become easy procedure change compute code calculate energy construct pair ordinary invertible invertible computationally singular construct correlation p computation package code language graph tuning parameter os enyi graph os random graph construct relationship manner os enyi structures node recover edge section huge implement huge estimating copula stock huge price stock day market stock select instead force graph
grant exploitation low final gradient q filter term account second gradient know easily weight weight class difference convention author tool useful discrete also divide interest range real algorithm algorithm filter superior near knn learn support maintain conditional retrieve proxy demonstrate pair find sense return process retrieve visible tool map ice help water road ice pixel pixel instance pick produce require retrieval inversion classification vector know could count might correspond surface forest road water etc input represent normally likelihood thus near knn skip domain large necessity real important feature include filter describe advanced retrieve water retrieval comprise million fast first classify surface extremely ability confidence absence knowledge true value unnecessary probability estimate super section gradient knn estimate density perform classification work pick determine voting refinement probability th coefficient justification carlo except multiply essentially arbitrary assume roughly constant isotropic distance simple metric q q region tail region sample weight unity filter density fall wish filter substitute correctly select produce width valid integral quantity knn scheme bandwidth estimator filter width varied pointwise give width conditional advantage scheme knn produce desirable discrimination make classification class sample adaptive coordinate test derivative class search discrimination time necessary giving describe class size lie simplify probability behave assume pdfs roughly filter iteratively coefficient filter iterate square th width obey q call obviously take filter width target weight part twice contrast need computation member large marked removal member add tree mark removal operation member filter apply find follow paragraph must calculate least arrange element large one repeatedly large unbalanced tree actual great element entire new tree tend two belong solve locate find dimension considerable consider two evaluate sign least aid convergence estimate estimate repeat true within certain tolerance combine newton hence library use solve cubic solve rank linear synthetic consist class illustrate triangle centre minor axis dimensional knn implementation cost second compose member class rather class sample compare compare advantage quantify return conditional knn codebook search classification algorithm represent fitting ccccc classification correlation knn n show trial uncertainty affect true retrieve class quantify column visual figure figure compare main thing much four important need datum phase much fast unfortunately necessary measure accuracy classification prior retrieve come exact region codebook design codebook coverage high surface map seven channel simulate colour use automate discrimination statistical dense complex one use image manually le surface water le whose landscape resemble training six resolution minute minute minute minute fairly winner codebook imply section give speed since codebook utility water mis choose correct perfect job classify lie water world monotonic inaccurate fortunately rarely case figure equation right side classification accurate direct knn compete single whole efficiency algorithms classification time generally increase increase weak actually include bottleneck algorithm select although overhead calculate efficiency coefficient svm training read output phase issue codebook return equivalent svm number phase bit redundant appear bit well sample specialized extract classify surface type general statistical get local try discrimination resolution sample minima optimal neural classification generate value overcome variable retrieve dividing range govern range threshold class define follow conditional fine retrieve compare probability value water retrieval seen replace tangent robustness impossible result see normally expectation moment side formulation characteristic robust estimator
generative correspond reasonable validate consider boltzmann unit top initial bias visible ab initial bias handwritten threshold medium gray size handwritten digit feed binary persistent keep track particle gibbs collect tend shape iteration alternate datum independent likelihood rate minibatch particle persistent divergence epoch variant average stochastic reduce estimate k randomly representation build sampler take measured residual curve see produce gaussian perform estimate estimate use update update see center generative advantage strong generative simply discard highlight center fast stable center eps eps eps eps eps epoch mm eps eps eps eps eps eps mm top layer epochs eps eps l eps eps eps eps eps figure layer filter learn tend varied learn suggest rich center top form manifold may still contain digit digit lot useful discriminative balance modification boltzmann center involve center optimization criterion modification boltzmann machine center show machine learn fast center addition center deep boltzmann discriminative training layer still require many understanding come despite initial conditioning hessian exclude dynamically solution within behave investigate tu de machine boltzmann principle extract datum unfortunately layer without greedy largely initially activation zero learn hierarchy abstract datum powerful extract unfortunately jointly argue great reason mapping activity sigmoid default function center offset condition estimate boltzmann gibbs invariant train center deep boltzmann top useful learn fast stable center backpropagation centered center boltzmann boltzmann following sigmoid denote parameter operation element boltzmann machine connection unit contain bias associate denominator function center boltzmann associate unit center boltzmann derive unit x sub sub sub sub sub sub show argue center hessian center vector random equal project dependent independent absence dependent conditioning therefore datum due cause visible center think well perturbation behave perturbation show get limit show stability quantify ratio eigenvalue interpretation hessian project condition hessian number boltzmann unit initial offset ccc conditioning sigmoid conditioning machine ii iii technical reason introduce restrict unit typical unit deep architecture specific role unit layer structure easy efficient alternate gibbs sampler boltzmann figure associate take group sample alternate state persistent center cx w bx x x mb present representation evolve layer layer layer think aim approximate implicit measure kernel match obtain number sample close error counterpart represent respectively q build kernel sort good representation certain measure task lead lead kernel principal component empirical obtain residual task curve curve determine necessary relate shoot ask hand label information representation rich encode subtle purpose represent compare architecture analysis
minimax thus kullback leibler observation discrepancy outline rest supremum latter hold whose th column define chebyshev last find suitable f ip ip side lemma need hypothesis infimum test procedure center coordinate point connection dimension large satisfie convention condition ensure lie sphere inside construction let coordinate case element depend appear simplify moreover point equality apply crucially large lie exactly show f enumeration q nonzero coordinate nh nh hold q n mr argument use nh follow familiar previous non coordinate big satisfy relaxed r set cardinality ensure also ensures state collection ex shannon follow finish fix take observe n low symmetric symmetric r l r ng require lemma concern relate devote square remainder derive view quantity kronecker conclude also q entry tv tn expectation since product vanish verify calculation together conclude next argument convention define l since proof theorem select suitable nb one utilize eigen submatrix ensure involve perturbation eigenvector possibly aspect require establish step follow take theorem careful expansion show term expansion accordingly lemma together q entry rest enough since appear rhs fact upper uncorrelated na appear upper bound computation deduce q event use inequality complete three probabilistic deviation follow nc suppose follow role small pp variant implicitly ta sa bound I discrepancy eq log give respectively contain kullback leibler fx x rhs orthonormal exactly I nonzero cardinality shannon entropy set satisfy I fix demand coordinate nonzero coordinate last lemma expansion involve involve though explicit proof uniformly suppose b n sequences asymptotic principal large old result universit paris functional principal penalize dr temperature north estimation smoothed principal component estimator wang functional f active f operator random matrix banach space c stein l unconditional estimation bayes estimation l normal convergence principal functional regression pca break principal chi mathematical statistic book manuscript principal stanford university nonparametric similar component l financial correlation l stanford stanford university principal component stanford I component stanford university w york comprehensive identification cell cycle microarray statistical multi channel wireless communication e component root multiple root j recent sensor determination minimax convergence longitudinal form cite incorporate dimensional covariance observation rate sparsity constraint pca estimator regime usual principal widely technique reduce traditional pca applicable reasonably two observation distributional eigenvalue various among eigen covariance fact sample approximate large however advance acquisition dimensionality individual nearly magnitude increasingly article collection face typically e student contain taylor outline class analyze act think example relate number size study sometimes thousand datum collect factor hundred stock theory optimal discuss several consider aspect indicator monitor commonly refer dr give extensive treatment theory communication measure point consist person subject study time patient cell gene gene researcher interesting eigenvalue population covariance certain type wireless deal estimating grow problem eigenvector characterize consider cell two pca systematic identify pattern structure smooth corrupted element vector often practical scientific interest advantage time address develop estimation theoretic standard reference optimal model give spline eigenvector eigenvector viewpoint observation gaussian assumption though essential make risk mostly nature even effort make explicit assume triangular individual partly motivate similar analytical estimation mean loss eigenvector relevance sort chapter describe behavior scheme show class constrain parameter suitable regularity lead eigenvalue population eigenvector framework subspace scope suppose triangular assume observation covariance matrix identity vector notice among sign notice identifiability rank mean infinity think equivalently describe term invariant notice relate convergence main focus exposition eigen matrix asymptotic refer ba problem strictly appropriate interval suitable consistency still probability goal assess task specify model invariant sign specify loss invariant specify eq norm useful loss denote l two approximately bound remain assume l identifiability condition big eigenvalue l possible relax condition issue rate growth risk address question investigate gaussian among deal measurement uniform note result circumstance rate analysis restrictive moment consider eigenvector restriction eigenvector parametrization notion later theoretic behavior space positive satisfy sparse formalize demand belong think wavelet sufficient membership coefficient basis refer treat motivation impose impose restrict ball appropriate radius need reverse vector space appear large sphere center fit inside space q natural kk low minimax state condition applicable infimum estimator hold take take hold flexibility hyperparameter appearing notable aspect become clear bind bad suggest strategy able extract coordinate rate convergence subject possibly regularity principle c consistent part hold eigenvector th reveal contribution expansion certainly weak whether sufficient model assume propose henceforth practice datum entry serve slightly biased true sparse rescaled multiply observation slight notation eigenvalue eigenvalue motivate follow scheme sample variance coordinate index submatrix correspond eigen eigenvector n choose good choice convergence single scheme scheme view appropriately assumption structure extract lot focus diagonal suboptimal minimax point covariance case view one coordinate way former contain small preliminary among extract stage refer final stage eigen follow constant later n ok n om tt thresholding normalize vector except derive risk
express generalize usual optimality continuity lipschitz two monotone drop obtain lipschitz symmetric curvature proximal subproblem many approximation reduce usage generally strategie hessian function adapt choose proximal positive adapt handle hessian modification multiple subproblem proximal cause become skip update express proximal type search direction newton composite let scale mapping proximal exist unique strongly convex subdifferential scale proximal mapping cauchy schwarz h express direction composite mapping use deduce newton search satisfy newton type like composite subgradient minimizer direction substitute positive definite zero minimizer minimizer directional derive close newton search usually must resort subproblem exploit iterate decrease accept direction strategy arc arc relative lie low arc drawback trial lipschitz sufficient descent condition lipschitz choose substitute choose subproblem search gx td kx proximal subproblem approximately inexact counterpart indeed implementation proximal newton subproblem efficiency reliability implementation use heuristic inexact affect describe adaptive analyze inexact type conduct condition commonly adaptive function require side generalize composite require approximate near desirable perform exactly optimal solution near preserve convergence newton subproblem solve accurately direction implicit subgradient connection inexact newton stop condition k fx inexact proximal stop condition sufficient descent subproblem inexact converge unit length linearly readily composite generic result separability generic implementation state sufficient descent inexact newton converge linearly generic proximal newton show newton converge globally subproblem show proximal show inexact linearly choose composite lipschitz uniform convexity convexity strong smoothness constant proximal optimal solution neither general proof assume closed function uniformly method e exist length cf exactly decrease exist eq sequence must decay length sufficiently attain descent decay search cf converge newton smooth type twice usually relaxed constant purpose assumption relax proximal exact converge strongly constant auxiliary result add strongly decay descent since mapping lipschitz proximal newton twice continuously differentiable convex constant require quasi newton unity satisfy sufficiently proximal satisfy criterion step appendix definite bound search direction q side side eigenvalue eq x satisfy substitute q proximal quasi newton converge assumption quasi converge assumption lemma length many eq denote proximal thus substitute expression substitute expression rarely solve analyze adaptive stop inexact proximal linearly term small continuously ii assume iii iv eventually prove auxiliary kx non eq combine next gx express multiply obtain substitute semidefinite deduce inexact newton linearly depend sufficiently inexact method step length converge linearly decay inexact proximal unit length monotone cauchy inequality apply substitute deduce converge sufficiently converge decay finally accord inexact proximal newton inexact suppose accord inexact show substitute expression proximal converge explore inexact direction behavior proximal avoid subproblem proximal newton statistical show suit expensive smooth evaluation suppose seek covariance q wise avoid goodness fit probe collect patient gene convert match solve problem proximal bfgs method update bfgs would computationally expensive rule decide condition solve subproblem stop fista solve subproblem plot versus figure met empirically figure proximal bfgs characteristic quasi newton dataset fast ignore expense condition per fast adaptive like third stop yield convergence convergence affect condition number slow logit goodness handwritten digits feature news scale match value proximal fista nesterov problem relative figure figure proximal bfgs fista bfgs versus fista expensive exp log evaluation nonzero entry dominate bfgs method expense shift subproblem whose function nonzero entry evaluation make total bfgs optimizer en opt publicly available laboratory share interface software compatible generator website popularity function composite analyze proximal newton rapidly produce solution level set proximal convergence stationary point disadvantage cost subproblem fast subproblem hope result proximal minimize composite acknowledgement thank lin lin ed schmidt anonymous suggestion lee support fellowship stanford fellowship advanced program er national institute medical national health award gm
summary pa usa em introduce novel reduction find multivariate orthogonal financial macro economic successfully discover informative forecasting well available dimension simplicity transform transform dr dr orthogonal uncorrelated signal keep ica recover autocorrelation forecast accurate efficient dr forecast bottom daily return tree package thus forecast sake finance economic even value interpret return propose move dr technique find provably eigenvector dr find low subspace relate review computation simulation autocorrelation daily return stock efficiency growth correlate intuitively month highly temperature warm warm warm zero uncorrelated k k k stationary transform since non negative function white spectrum example temperature right peak represent year qualitatively vice recover inverse fourier distribution spectrum spectral density inherently eqs uncertainty deterministic shannon thus future entropy logarithm support flat indicate flat spectrum white noise contrary processing literature require forecast characteristic perhaps function q eq leave indeed return guide estimate normalize scale fourier frequency consistent package along multiplicative n gx sample fourier fourier density see future classic neither plug intuitive property estimate iff perfect yield expect qualitatively estimator differential future rely capture temporal dependence similarly rarely linear multivariate make property seem trivial intuitively combine general make simple side efficiently equal contrary univariate complex ib reduction k ts ns semi return eight financial series important structure discovery portfolio stock portfolio bi plot pca almost equally movement gold pc fourth energy though pc water almost twice water water vs gold vs china mining financial autocorrelation lag overall wrong fig sf fast large component latter still reveal white low indeed day month simple study nd omit lag lag autocorrelation spectra absolute growth last growth region baseline state economic help times scale know business well affect global display statistic autocorrelation relate correlation analogously lag lag forecasting compare portfolio density intuitive contrary drive cycle much easy year fit adjusted cluster space interpretable unlikely lag business cycle clearly visible rate similarly maximize lag correlation cycle face find important forecasting pc show period whereas look somewhat associated loading quite loading separate series literature analysis fit var recently another dr separate extent moment bss average forecast bss especially ica entropy fit contrary principled measure furthermore quickly use technique point differential entropy equal proportional coincide g
subtree document advance document allocate topic subtree select subtree stick construction document child subtree construction dp select variational optimization requirement maximize fix objective considerable freedom regard greedy variational update outline dp stick dp j kk switch probability greedy variational subtree sequentially currently activate activate contain subtree subtree determine add great increase document specific beta distribution prior variational distribution topic indicator remain question activate subtree great increase objective reason restrict distribution calculation iteration parameter subtree candidate fall formal description atom dp meaning define subtree document starting constitute activate add determine create subtree include point subtree include pa increment subtree note two greedy though stick break document dp atom advantage atom stick break subtree atom candidate atom change incorporate subtree select document optimize variational subtree structure start value update familiar beta though form independence closed update beta variable j level break proportion statistic pass second parameter document path transition stick construction switch stick break slightly statistic expect topic terminate select take global break data batch discussion section scale corpus size old new document reflect increase parameter datum value batch use sub batch see old variational wang nest topic batch result three set verify multi improvement nest crp stochastic perform million rare giving initial level dp hyperparameter slightly encouraging continue greedy subtree stop low fall continue iterate fractional distribution fall hold process top variational test percent split variational portion form document portion comparison evaluate hdp adaptively probable probable predictive unseen number document processed see hdp give document topic equally contain show word topic use level per three allow entire adaptively figure node probable accord four child relationship meaningful hierarchical subtree branch issue incorporate document learn distinguished topic datum single computer set topic wikipedia comparable figures improvement hdp increase figure topic word would york times though ultimately sensitivity topic wikipedia corpus base typically similarly dp impact posterior sensitivity hold fairly robust structure prior quantitative quality model well relative believe reasonable setting nest chinese restaurant observation follow stick construction new specific path hierarchy entire content variational inference scalable compare stochastic hdp engineering university berkeley university ph electrical focus develop involve wang capital management project department receive thesis award research focus david associate receive university focus image electrical california berkeley research focus graphical model statistical processing computational biology member science member engineering member american american association name international chinese restaurant node share document express derive massive collection demonstrate million million document bayesian process thing natural store might along decide well option end focus develop representation topic topic corpus inferential topic close become tree build chinese restaurant bayesian select limitation practical drawback document drawback example consider com compare article journal type word area yet structure relevant document rather analogy document result need multiple place subtree perhaps child topic dominate statistical compact prior perform restrict path drawback develop nonparametric objective access entire document specific make refer dirichlet allocation mechanism whereby define base collection dirichlet let hdp area separate topic thus corpus significant development inference lda model continue development ability efficiently corpora amount section dirichlet chinese restaurant process hierarchical present review scale well massive document compare hdp dirichlet build bayesian nest chinese restaurant process constructive process foundation nonparametric model rely model set statistical trait effective trait mixture process eq q respective parameter potentially infinite induce partition atom dirichlet briefly probability measurable distribution application representation chinese restaurant crp avoid integrate dependent unique display property give insight evident unique limit crp analogy chinese restaurant table select proportional customer select chinese restaurant work stick construct draw construct broken stick index expectation ig write mean inference crp chinese restaurant tree extension crp crp analogy accord crp restaurant uniquely indicate arrival customer act crp restaurant table occur potentially table tree number start child probability crp constructive construction dirichlet rise stick develop construction later nested tree li paths stick break child equal stick construction child index index child sequence atom topic document leave nest crp generate corpus start nested document produce prior stick break dp standard document drawing topic draw drawback therefore corpus specificity create appear many possibility topic document learn situation document three topic insufficient multiple occur bayesian document corpus path key path allow paths goal hierarchical hdp hdp dirichlet base continuous dp place probability mass atom base atom advance hdp dp use great topic nonparametric lda explicit representation hdp representation rely stick break identical difference framework document access still coherent ideally document main toward word path topic path share tree nest hierarchical first path path document share global stick break construction dirichlet transition dirichlet follow index dirichlet document dirichlet atom tree represent dp stick breaking eq eq lead dp randomly mass place one tree node since probability allow document place atom first proportion high probability leave probability document high node delta recover generate document draw base draw beta topic path mean allow atom path stick breaking determine stick break specific act currently determine topic node continue select breaking imply document imply equal path select independent aid development nature stick break evident along subtree tree al share cluster within summarize text corpora large slow currently index million article last year fast essential bayesian idea inference field inference variational variational exploit variable share brief stochastic work splitting group local one group update
network break opinion certainly affect opinion social agent element adapt environment self separate self interact political influence international effect point simplify control change illustrate figure simplify description htb political political opinion political political support exchange member interaction strength characterize interaction property number strength independent evaluate extreme left extreme simplify reality quantify depend distance environment average effect party way respect opinion interaction equilibrium mechanic temperature enhance interaction party party clearly equilibrium system environment interaction differently right political leave political viewpoint keep spectrum effect essential opinion political party viewpoint attractive character represent party opinion party party age two party opinion splitting way proportional interaction party party large merge age composite party merge split two mean party stay keep original agent one concern belong belong party contain graph opinion gaussian uniform political probability agent evolve belong party randomly another different party agent belong party depend party randomly agent belong proportional respective maintain contact agent opinion political party party party change opinion opinion effect take interaction replace dynamic time ratio characterize difference dynamic opinion correspond agent isolate allow party political opinion uniform set evolve stable range political rule interaction whose hence middle political merge decrease merge party mainly middle range accordance reality empirical study mainly european political spectrum party website datum locate political confirm influence decrease quasi stable state show straight line mean evolution relaxation quasi political long reasonable large evolution opinion whose number reason remain go splitting figure exponent initial large fix later certainly party union enhance since small observe sharp record empirical collect website empirical dividing find law exponent close heavy tail influence merge party link add opinion dynamic quantity party party agent link agent agent typical evolution indicate formation political character opinion maximum quick increase around sharp happen time period party sharp period period determined large initial value political increase agent formation political period agent almost evolution party degree belong party divide degree agent maximum average change despite formation political average degree result link evolution belong formation link link respective political time semi delta distribution formation political consider interaction strength evolution agent opinion social role opinion formation environmental force temperature temperature small average opinion large yield large strength show opinion result without opinion initially shift however always end opinion opinion leave number final number remain party accordance result remain political remain locate side political jump happen agreement effect interaction shift right political weak exponential small close temperature relaxation time work propose opinion compose interact mechanism opinion merge split link interaction political contact environmental influence dual opinion take party exponentially initial alternative period initial exponent verify state regime parameter exponent political regime place temperature opinion temperature numerous remain end political matter mainly strength dynamic close middle political spectrum agent accordance agent political topological structure political formation character consequence merge splitting dynamic extremely period political characterize degree effort realistic opinion network agent opinion party influence feature dynamic strong weak consensus dynamic accordance phenomena party exponential phenomena
compare covariance select note mean tend pattern sharp within band interval drive series wide band dependence market main index stock tool allow synthesis various stock underlie trend market specifically index weight stock weight equal market date date application stock index select behavior motivate logarithmic return national stock index distribution return heavy trend financial recent typical financial obtain well characterization market discrete induce latent perform rescaling sufficiently rapid figure miss solely burn trace show rapid dynamic volatility occur world financial european rejection budget usa jt accommodate heavy tail slow rapid ability capture economic finance line quantile usa european without dot quantile correlation usa failure mae mac financial financial save peak european rejection grow financial instability correlation national posterior immediately clear economic world european systematically south showing confirm consideration exhibit economic structure colored place problem update covariance predict update ahead return correspond future unconditional b mean step ahead previous gibbs return national country become I compare improvement analyze ability obtain characterization play crucial formula calculate present multivariate dynamic simple conjugate tractable moderately model heavy tail key enable fast useful study highlight flexibility respect multivariate show dynamic capture economic financial market demonstrate analysis financial reference heavy mean sharp iv update data vi possible explicitly incorporate prior dynamic detail sampler l equation simulation apply factor recall property state joint reproduce dictionary element equation relate recall lk I vector variance equation smooth latent state space finally b I kt k I I I ji I combine model lead eq online update distribution posterior lk I h smooth initialize state te similarly g te ii conjugate usual I important allow certain rapid inference prediction across smooth slow mis calibration interval substantially narrow wide time continuous allow smoothness covariance dictionary evolve linearly usual online algorithm bayesian assess simulation locally long multivariate nest stochastic analyze collect application monitor key characterize dynamic covariance cycle period rapid change model smoothness single restrict inference rapid event multivariate particular interest vary smoothness meaning couple smoothness construction locally factor formulation vector generalization autoregressive var polynomial spline spline implicitly locally local include spline perform well location knot selection via mcmc computationally moderately flexible kernel separate estimating time evolve weight move see use extension accommodate vary maintain flexibility covariance generalization variant demand step time formulation however uncorrelated evolution fall flexible vary lag missing tend application inconsistent varied join machine effort propose improve wishart process scalability continuous generalise wishart regression via trajectory covariance behavior process sharp wishart motivate rich dimensionality base model apply approach dynamic reference cite therein recent multivariate loading evolve dynamically sparsity latent lead improved portfolio utilize vary autoregressive result computationally proceed validation emphasis develop process accommodate covariance regression loading factor combination dictionary miss data scale moderately motivate assume stationary period sharp difficulty nest process reduce involve locally smoothness latent computationally accommodate specification explore updating compare study finally stock market across examine focus take probability variate realization stochastic assign algebra subset propose process broadly inference article focus equally spaced miss simplify accommodate substantially observation modeling rely dimensional successful advantage frequentist characterization loading residual recent article large cite interested vector comprise lk lk restrict attention generic process dictionary smoothness derivative expect center represent induce vary gaussian noise tw otherwise induce smoothness base markovian imply key advantage extend along instantaneous lk lk element represent nest kt kt kt crucial aspect firstly grid relate discrete time represent I scalar time factor kk time loading sparse linear set play crucial reduce model computationally tractable process time loading loading evolve closely refer loading characterize accounting loading vary independently nonparametric induce fundamentally carefully modify unique potentially constrain enforce identifiability undesirable dependence response follow avoid identifiability ensure identifiability induce characterization hierarchical approach maintain allow mean respectively lk inverse gamma k b great infinity shrink towards lead element inverse prior independently simple smooth sampling parametric outline draw instantaneous mean variance recall shrinkage hyperparameter realization state smoothing consideration mean minimizer ty ct respectively lead less suggest hyperparameter allow variance strong discussion structure smoother require state respect associate smoothing technique allow smoothed series estimator procedure choice recursive formulation approximate posterior available computationally tractable online initialize ahead moment latent computational conditional state sample I overcome problem previous initial drive initialization use online study multivariate specifically pc assess whether extent accommodate sharp vary flexible setting locally technique structure smoothness process subsection covariance time simulate shrinkage posterior use truncation higher find concentrated around parameter element proper initial place accord via similarly prior state balance rough mean also hyperparameter set heuristic outline run discard burn sample implement choose minimize mse estimate covariance formulation correct explicitly process directly therefore spline reproduce mainly simulation specifically truncation reproduce smoother without iteration reach discard burn analyze procedure splitting
value median bound vary highlight near box concentration four absence shall axis limit link period curve around day day vary fourth value peak day introduce step incremental continuously offline unlike synthesis functional streaming series european study abstract summarize content abstract appear user read chapter unless particular series book file chapter template file manuscript plain appear stream gain lot sensor use monitor physical variable traffic necessary potentially infinite flow store computational nature require development incremental exploratory processing cluster method deal identify stream stream usually cluster stream second stream homogeneous cluster observation moment aim find usually method accomplish identify centroid synthesis homogeneous since process synthesis point development stream cluster approach strategy name micro stream keep micro macro reveal final micro algorithm basic come sensor record variance group multidimensional extend introduce streaming allow fine streaming keep account five third median variable site grid propose incremental description track dynamic evolution incremental sense maintain datum receive stream overlap window approximated method micro splitting incoming overlap window associate window update appropriate micro macro cluster micro first incoming overlap window frame subsequence subsequence form subsequence jt jt residual zero summary streaming jt jt w jt jt jt min jt max w jt bind functional analog vary band modify band order band functional statistic envelope region analog band central region envelope represent box classical box envelope dataset vertical formally sample subsequence th band depth curve curve integer third micro functional jt jt f l jt jt dt min min jt jt h max dt allocation unitary increment micro cluster centroid keep micro allocate micro structure however depend choice high involve micro contrary bring generate micro deal value threshold micro allow propose compute micro cluster centroid micro much exceed alternatively discard micro merge micro update store parameter micro micro long reveal stream micro line get predefine snapshot micro snapshot collect update instant order user slot identify snapshot close upper snapshot remove micro effect occur snapshot micro allocate functional removing slot snapshot snapshot store centroid allocate item become process provide output micro macro summary interval heterogeneity macro iterate attribute cluster whose minimal centroid representation macro mean result available record daily make corresponding weather locate period weather way daily local record hour reason observation unable observation record accumulation since see high day could day second daily stream overall trend phenomenon help behavior along observation period represent functional assessment
c approximation term however universal require non result performance use order action verify effect design observation functional could show rmse converge smoothly exp become less exp rely every preferred automatic exp another implementation assess automatically exp move sophisticated partitioning become strategy look ahead practical setting evaluation partitioning space strategy behave dimension great idea across core decompose vast streaming thank lee discussion shape gray increase parallelism set particle filter second popular instead analytical inference paper proposal implement ahead decentralized implementation ahead outperform particle filter bandit enable particle filter problem example paper rao particle filter one two group condition compute group analytically decomposition analytical solve relevance however happen analytical derive approximate iii instead group group state automatically answer ii filter name decentralized effectively group particle use particle conditional parallelization parallel filter graphic gate array resample chen filter stream chen approximation performance obtain carlo ahead resample exact rao ahead describe significantly well pf domain context look bit involve monte experiment ahead standard question pose continue appear future paper paper involve online bandit lead future pose ahead strategy la paper state decompose x n z govern noise equivalently nested subproblem subproblem interact depict diagram implement filter nest except filter algorithm must employ subproblem use pf particle nest subproblem state derive look ahead reader mathematical involve derive pz ty jj n j n evaluate j time importance approximation eq importance weight markov process numerator express quantity numerator marginal replace approximation importance importance sample mechanism note require achieve expression rule proposal mechanism yield application use distribution marginalization approximation detail compute step careful keep track index aside filter common proposal intuitive complexity derivation considerably take prior proposal take importance optimal simplification importance predictive expand draw importance suffer major require ability new calculation importance step finite jump carlo approximation next expand follow hope particle require moreover small resample pf gain mention possible optimal swap resampling enable intuitive benefit illustrate look px pz multiply discard j pz decision decompose system play role subproblem decomposition group large execution change reason choose adopt state group consider choose possible action probability proportional reward action repeatedly yield gain different closeness observation predict real action jt initialize get select tt jt tt drawback try assume reward introduce overhead especially exp stand exploration exploitation information try call subroutine exp receive ensure associate exp simulate vector fill try code vanish regret conduct experiment compare performance ahead algorithm meaningful benchmark adopt multi stationarity kalman filter fail study bandit z ty e experiment standard pf simulation interval
could prune alternatively sequence contain pruning pruning let shorthand max score prune q th good position condition eliminate position fact immediately eliminate sequence definition max possible pass max true sequence edge pruning path label position equation suppose prune standard framework recursively label definition max vertex main pruning method apply give pruning shorthand average thus prune edge pruning position transition hmm reduce q per similarly sequence pruning wish perform update define eq sparse inefficient instead marginal reach decompose three piece piece piece prune logic decompose along hmm require position pruning modify definition pruning separability condition mistake simply prune average marginal achieve pruning vertex pruning pruning possibly suggest enforce choose threshold give follow pruning always base least strict pruning function pruning represent dataset pruning train pruning eliminate reduce vertex marginal position pruning pruning reduce volume respectively observe training development observe training lead set volume development example gram gram new test reduce go reason lattice pruning go run order baseline send cascade ms pos state n
consider monotonicity stress recommend implement overhead section theoretically section serial give precise solve list separable new paper devote h set block grow speedup towards situation grow partially speedup various summarize depend structure block choice framework encode c comment gradient serial serial random parallel new distribute continuously serial give deal partial block euclidean column small knowledge strategy fix uniformly name assumption treatment hence much beyond scope processor compute update doubly uniform nice binomial see definition three monotonic appropriate section later text select update ht bb bb bb l thm bb thm bb p thm en prove deterministic separable believe interest decomposition stochastic program derive method property name speedup situation well paper analyze coordinate algorithm set hence algorithm somewhat variant restrict value moreover lastly serial depend hard compute involve computation huge scale express easily computable quantity turn use hence necessary complicated allow update unfortunately coarse theoretical serial special update iteration uniformly involve node core achieve speedup variant control small set linear svms illustrate theory match reality open efficient implementation publish http code com iteration formally realization subset brevity characterize assign subset technical block proper sampling proper sampling block far say proper rise refine doubly generate equal whenever definition different uniformity uniformity let uniform whenever du characterize nu set nu partition note nice du processor core iteration block block assign block processor block independent ic ik processors core processor choose block random processor block two processor block parallel fast approximation nice du probability independently parallel iteration equal processor nice processor assign processor available iteration easy result binomial sampling nu update block du nu iteration block intersection du nu thin precisely du nu serial nu du cardinality nonempty doubly must n serial consequence sampling statement next sampling separable positive semidefinite w five identity let doubly uniform nice yield eq trivial doubly happen precisely eq vector choice recall choice explain need give answer step update fx k k one eq choice use minimization update could separable separable easily form replace w vector conditioning far note indeed separable recall h block update bound natural set monotonic separable proper admit indeed inequality step explicitly analyze monotonic sometimes behave monotonically enforce derive useful monotonicity separable form h fx fx analogous write intersection argument last combination weight vector j h plug convexity besides usefulness derive reason also say respect q j h separable argument formulate doubly proper doubly uniform let fx fx n could alternatively write combination nice nice sampling give serial prove descent composite monotonic result convex cover monotonicity restriction auxiliary hx fy eq convexity hx fy w fx w inequality ii prefer use finish complexity let nonnegative lastly property constant choose extend aid update deal inexact proper choose level accuracy eq fx random iterate otherwise since monotonic message serial see expression parallelization speedup du expression ht parallelization factor doubly binomial serial compare end fully parallel partial separability especially answer question speedup separability speedup sr ss processors illustration processor separable wish stress indeed huge scale ht choose target level accuracy letting since take finally could proof able hence convex force resort situation term w strongly parallelization speedup serial factor approximately average update parallel quantity separability show core problem hour sense tight predict speedup generate scale instance row enable particular exactly separability ht e e e e e e e e e e e e solve nice utilizing take around gb implementation version core iterate iteration reading proceed another current update soon compute second good nice allow processor pick block show table need coordinate total number coordinate coordinate serial coordinate would summary represent choice one conjecture phrase work turn roughly magnitude trivial find old coordinate slow slow choice ht progress coordinate update e move last nonempty show remarkable degree though method nonzero place local come parallel utilize core serial consequence observation core turn excellent figure b follow utilize roughly dimension separable demonstrate parallelization necessary reach equal measure iteration nice sampling value core core solid line marker seen figure theoretical reality remarkable partial degree separability useful replace far deep phenomenon experiment svm hinge coordinate coordinate nice sampling coordinate minimize box dual dataset gb storage separability e sharing feature theory nice add level processor nearly mean primal observe gap training decrease increase test small beyond say processor expect parallelization h ccc experiment dataset contain divide part training set testing cca gb depend small use nice depict regularize marker correspond epoch processor less achieve nearly offer processor similar work number however utilize processor job approximately speedup accuracy stop increase beyond rarely method h epoch ff section sec sec prop prop block associate block constant vector weight n x w x w update iteration update algorithm parameter central fx h ix fx tx ix kk theorem claim exercise author grant ep ep author centre numerical software grant ep publish http code google com dc parallelization apply speedup serial iteration simple processor separability separability case processor mode processor show involve hour core coordinate partial separability huge expect interest suitable optimization topology sense usually grow capacity grow dramatically decade fact currently challenge aim task algorithm big domain broadly strategy single block coordinate drastically reduce memory arithmetic certain iteration amount multiplication necessary usual classical smooth problem tolerance variable gradient balance observe author serial compete point truly huge necessary ever availability build root massive parallelization combine scalability coordinate topic answer observation minimize variable simplicity assume trivially processor task solve parallel processor parallel coordinate serial processor use separable would amenable acceleration parallelization acceleration reduction separability one message explicit simple formulae speedup many naturally mean promise big regularization unconstraine simple individual variable alternatively certain solution account reader simplicity next decompose precise assume throughout nonempty depend develop analyze need give many objective encounter big elsewhere block e ht logistic depend separability th function th partially separable section reader example nonsmooth partially function arise cut paper recently complexity various brief reader cyclic attempt complexity method update analyze independently regularize smooth unified refine paper separable develop linearly constraint include inexact analyze separable independently study writing aware paper appear various aspect topic building include describe block problem establish subsequently detail parallel contribution selection update development select develop expect separable tool promise computational instance gb core hour objective value iterate conduct problem come formalize summarize
form order optima denominator close close bound line quadratic behave distinguish hard bandit convex characterization setting result difference focus dimension extremely function provable bandit derivative free open bound focus strong convexity performance strongly smooth exist upper function table exist upper seem achievable quadratic function derivative free noise bound message algorithmic generic assumption still specific setting concrete classic set ridge sample derivative setting think query give draw equal set consider noise attain good useful round jensen round contradiction identical thm quadratic q letting get convexity fact substitute result term substitute expression rearrange simplify quadratic root eq root know nonnegative recall average regret remain take side specify let later expect deterministic strategy explain existence expect globally minimize smooth w w calculate substitute simplify get q exercise verify expression require item fraction verify use identity finally substitute require upper kl noise see q coordinate pick ball global ball satisfy thm microsoft institute science convex optimization bandit feedback gradient year upper low literature bandit derivative free available precise characterization imply trivial low moreover set require mild gradient rate despite imply contrary convex free convex consider ability query realization optimize important query free among solve unconstrained framework due datum learn kind study armed bandit optimization decision setting repeatedly choose realization error goal roughly value well bandit problem simplex refer discuss attain regret gap study mild convex query inherent free exception multi bandit query common non armed bandit away bound linear theoretic prove rely construct deal function much originally year seminal optimization pg quite strongly function enjoy known bound respectively bad dimension investigate complexity bandit free focus contribution prove strongly convex three question performance sharp bandit scale optimization linear provide convex analyze function attain error sharp show fast free may quadratic explain later establish low impose result fix domain show bound prove average derivative bandit distributional assumption stand armed free minimizer boundary domain however argue restrictive shall clarity exposition begin low tool tackle smooth set summary performance consider natural additional proof appendix quadratic tt combine table dependence strongly intuitively everywhere curvature subgradient intuitively fix curvature minimizer prevent consider lie proceed round pick realization bound wish domain free case uniformly noiseless pick noiseless set goal minimize namely possible derivative simply get return inequality bandit hard round interesting research extend quadratic definite away behaved assume spectral easily rescale see smooth appear provide insight case convex derivative knowledge free nonlinear oppose however achieve mild least one attain boundary query strongly common case actually situation assumption crucially strong convexity assumption must always optimum discuss free actually decay hold natural utilize gradient value function whereas query interest function structure query relatively far away use known point query simplicity input convexity initialize query quantifie return return iterate oppose average family derivative free instance repeatedly estimate feasible region since generally rate fast lemma appear algorithm perform bound average averaging plugging bound obtain tight namely bad derivative order besides strongly scale case function provable round possibly satisfy eq optimum must ball despite domain allow e bind appear sense see exhibit quadratic expectation attain imply uniformly gaussian random strategy deterministic deterministic generality randomization hold randomization informative value measure kullback determine kl divergence hard distinguish large none coordinate support return represent two use upper
yield eq together whenever l f g inner iteration iteration hence conclusion em kkt g g g hold constraint cx xx kkt monotonically bound monotonicity bound hold replace use obtain replace imply continuity lk arbitrarily find step hold accumulation kkt point notice k k k k pass subsequence assume side kkt convenience u I together modular lx q inequality due boundedness addition due conclusion statement dual view subgradient approximation concavity author attention pt section proposition remark assumption support grant establish iteration programming suitable accumulation generate kkt lipschitz associated result mention establish programming consider nonempty set convex necessarily throughout make continuous nesterov convex solve al order necessarily convex et al another special lipschitz ball recently convex smooth dc difference see program widely view q nonempty close scad ht ht l reformulate observe comprehensive order program suitable assumption accumulation generate kkt variant exact function establish outline paper establish finally inexact inexact solve convergence point normal directional along convex subdifferential establish constraint q eq satisfy condition observe contradiction inequality second xx every hence feasible point sufficiently minimizer contradiction assumption positive eq eq indeed suppose convexity hold define eq q convexity eq sequence notice contradict minimizer definition satisfy finitely cone differentiable convex convex exist let propose convex also variant introduce exact arbitrarily end em accumulation kkt proceed several lemma smooth continuous close differentiable lemma provide inequality close nonempty closed exist cx I cx iy x I iy iy iy I cx cx g convex em accumulation sequence generate kkt k I k exact method hold assume cx xx kkt since cx conclude cx I I k relation k I w w w immediately imply large relation yield verify let z relation continuity px x hz k side since imply kkt accumulation I v generalize em let hold
turn post provably separable near allow oppose robustness analyze dataset focus admit sufficiently permutation imply proposition prove proposition theorem close paper follow factorization permutation first proposition broad nmf hold q necessary apply theorem tight multiplicative condition multiplicative finally hold well synthetic post processing deal matrix th transpose matrix zero identity denote context diagonal denote trace cardinality sufficient column nonnegative imply sum constraint linearity simple would bound could improve stick formulation admit let combine equation eq fact distinct noiseless allow let one column dataset solution solution index imply show complete imply algorithm able reconstruct corresponding must process remove constraint discard since feasible relaxed program correctly tight bound multiplicative believe possible match theorem multiplicative able research apply rank nmf trivial admit separable factorization bad column belong section necessary construct matrix extract permutation investigate variant see provably entrie feasible large post noise bind prove section recall aim identify nmf give actually distinct distance extract twice desirable moreover nmf approximately let large optimal solution sufficiently large column extract construction contain construction theorem take extract extract matrix solution discriminate separable return matrix four extract original reason algorithm construction prove optimal separable construction condition et al column algorithm r loop entry distinct exactly equal correspond take checked column entry enter loop multiplicative multiplicative storing point negligible storing necessary hold vector distinct typically loop perform processing keep compute far design sophisticated extract relaxed variant processing use pre point noiseless robustness hold th datum neighbor discard processing et input topic algorithm inside assign show proportion claim open robust theoretical th let separable eq et bounds bind dominate differ arbitrarily column ill condition dominate factor et however know efficient perform synthetic conclude algorithm provably experiment superiority near run ghz lp develop construction theorem three avoid towards objective separable display column correctly extract extract one column even algorithm large level corner identify identify extract process computational post negligible needed propose provably prove tight also design computationally seem polynomial noise input section bind input impractical seem acknowledgment author would point also belong hull program larger small although simple let program parameter depend feasible feasible optimal solution linearly interval note since one large lemma imply show clearly q verify let let result construction let let optimal first x j r give another equal z last solution therefore problem subdifferential subdifferential slope minimal show equal q mm proposition corollary question institute universit de la nmf recently separability span column problem particular r nonnegative program programming refer new general provably process separability programming one express nonnegative combination nonnegative space aim weight np show separable separable column nonnegative trivial aim small cone generate mx mr reference therein separability sense situation correspond document approximate correspond bag discuss practice separability row row require see discussion separability use imaging refer separable easy vertex see robustness author rather paper general provably variant noisy also noisy exist nonnegative matrix submatrix permutation notice column column et propose separability prefer identify distinct read weight equation use column column entry interesting always
empty two sample respectively empty wherein linearly two column one row option row z z w option wherein apply empty state q apart active part reduce mention complexity complexity find one compute I move accordingly active maintain remain thus become newly modify place replace previous doubly sample consistent maximization hard margin elimination criterion combine lower publish related formulation misclassification additionally feature elimination soft sense utilize radius sense whereby therein whose minima small leave combine radius low light devise qp qp publish method novel herein outperform attain low several feature promise especially one employ tune herein herein information support vector svms interested reader brief manuscript label f w hyperplane act sign decision boundary separate optimization svm linear optimization svm solution associate lagrange multiplier efficiently since svm discriminant express solely kernel trick svm concept maximization hard sense call pick margin reduce retain step actual index leave margin anchor alternative second counterpart margin pick elimination objective elimination every classified elimination elimination hybrid slack little hard sense basic parameterization pose standard elimination decision herein refer solve intuitive lie right intersection point two feasible region lie thin dash accordingly region cone bound slope slope incorrect region define cone slope maximum slope label e generalization curve svm fewer particular elimination feature comparison note author current manuscript include fact discussion currently place result somewhat explanatory isolated explicitly specialized active provide feature feature elimination randomly split select fold validation trial trial average hundred highly probable separable feature hundred herein eliminate separability separability lose half qp feature hyperparameter select candidate denote classifier perform elimination jointly incorporate hyperparameter n mention specific highly form many zero discuss page great e optimization refer give assumption work qp mathematically utilize solve fulfil row need column discuss restriction lemma introduction enter picture highlight point intuitive description central point doubly keep doubly doubly move along active complete movement whereby less movement case movement doubly likewise doubly like single direction summarize movement possible call decrease objective sample movement movement switch movement take place switch margin active represent wherein pair row top half respectively elaborate row block two top category type sample category doubly property elaborate restriction raise nontrivial constraint initial initialization recall piece multiplier classifier ii generator explicitly identify particular would doubly generator assign albeit scalar multiplier set scenario brings assign indicate set wherein via multiplier less sample margin unfortunately within reliable determine extra measure determination find generator provide classifier multiplier discriminant sample positive normalization measure whereby utilize scale large sample paragraph possible provide active numerical make proceed describe essentially switch I move along current doubly movement new constraint require least constraint requirement address fulfil theorem ii qp fewer long summarize part qp qp extend elaborate introduce index row half bottom note contain index plus sample index specify sample index whose constraint e mean active denote three structure cc nm row linearly final row appear row placing
convergence epoch basic weak form locally restrict strong convexity rsc condition allow pool convexity rsc note namely low nontrivial pair rsc strong direction relatively sparse statement whenever application condition convenience namely simplify f choice prox generalize allow length apply dual epoch length regularization suppose expect satisfie epoch length cardinality q theorem return corollary apply iterate dual bind state epoch suppose parameter universal cardinality valid relatively average central proof optimal note proposition equation choice addition epoch future reference note make elementary inequality simplify version rsc condition condition average simplify equip broken case perform satisfied control I proposition recursive epoch hold epoch epoch upon substitute set simplify eq ii feasibility throughout run appeal simplify recall bind error cauchy simplify equation bind epoch algebra bind simplify result net epoch summing series give order convert error epoch let complete computing epoch set inverting allow deduce error bound term geometric inverting algebra observe bind stay epoch thereby bound long feasibility epoch epoch since large hold set early epoch epoch feasible algorithm epoch lemma epoch prox length equipped specifically least observe thereby corollary set yield calculation minimize order must demonstrate rsc notational simplicity shorthand yield corollary know consequently rsc suffice quantity matrix study result appendix rsc complete corollary main theorem lipschitz calculation since check satisfied plug early bind epoch iteration epoch early iteration epoch additional development prove argument ability error epoch epoch may continue epoch become enough encountered theorem error increase treatment epoch simple fix epoch challenge behavior epoch feasible first epoch lemma know constant lemma apply setup epoch epoch complete substitute recall exponent straightforward duality norm perform algebra refer behavior epoch well optimality rsc lower combine bind yield far simplify rearrange yield establish allow translate pair since lemma write q triangle substitute eq rearrange tolerance final use inequality equip begin shorthand result prox function form al upper bind simplifying require provide appropriate assumption size eq control start guarantee valid assumption bind gradient lemma control random early q inequality complete convert bind simplify rsc minimizer hold feasible minimize combine yield recall shorthand close statement application conjunction result convert cone result observe consequently piece universal substituting notation invoke rsc apply rsc yield rearrange recall notation complete inequality equip ready inequality lemma rsc inequality hold explicitly rearrange combine involve term upper triangle inequality yield identical early different finally prox center control error assumption recall meet epoch length suitably complete proposition state straightforward algebra remain result exploit martingale tail particular et sequence start lemma suffice recall plug statement complete turn moreover measurable deterministic satisfied h old use update condition satisfied plug state suppose condition proof result combination lemma feasibility epoch schwarz argument second upper obtain previous rsc second proof straightforward epoch fact consequently update repeat obtain assumption note epoch center construction guarantee recall substitute recall establish completeness establish universal ensure establish condition appeal define eq substitute rearrange case result assume ensure epoch calculation finally inductive claim inductive identical argument complete inequality universal epoch thus obtain bind accordingly rsc condition translate rsc assumption recall shorthand second f combine yield use inequality combine condition translate analog yield respectively eq simplify error least useful theorem lemma simplify simplify rsc technical main ccc department department institute technology university california berkeley loss optimum approximately yield objective dimension successively nesterov establish iteration sparsity locally include square loss statistical result rate effectiveness confirm baseline square stochastic desirable feature accordingly intensive several reference therein efficiency provide sharp function rate strongly optima sparsity precisely enjoy extremely sparse result significant convexity boost square type approximate prove useful many application overview paper reference many mild logarithmic precisely solution entry stochastic mirror descent converge loss linear improve rate attractive slow opposed rate strongly encounter exhibit convex optimum approximated enjoy type rate mild optimum use meaning substantially build multi new sparse optima objective regularization quite natural sample later stage effective appropriately close many example summary development structured simulation confirm regard convergence superiority compare algorithm direct method inferior regularization critical keep presentation restrict multi step note similar result mirror accelerate optimization focus extend structure subset optimization stochastic access vector goal design suitable optimum zero sparsity inequality upper involve optimum well choose appropriately contribution involve epoch sequence number specify length specify q epoch update geometrically upon termination r averaging averaging operate I prox describe average prox choice prox used previously see g example space prox feasible appendix schedule initial prox prox initialize worth subgradient inspire nesterov composite minimization prox norm compute prox mapping prox mapping prohibitive dimension discuss stochastic begin lipschitz instance sequel gradient sense goal fast locally strongly step local convexity concern local convexity refer strong statistical theoretic analysis sparse concern produce stochastic subgradient tail sub gaussian gradient constant component hold sub tail satisfy previously help apply scenario loss consist pair covariate predict decision rule minimize appropriately choose common hinge distribution either strategy illustrate assumption satisfy thus bind exponential marginal zero let minimum taylor expansion sampling pool convexity bound relatively simple verification instead boundedness sub tail maxima problem least covariate condition proceed condition local covariance exploit old inequality semidefinite via likely hold tail hold rsc condition establish sub need assumption tail vector obtain sub analogous calculation position property result problem rsc radius globally hinge logistic example apply square somewhat treatment dual averaging algorithm state epoch base choice suitably radius met outline use result algorithm iteration convexity lipschitz involve subset q measure sparsity result length work objective predict overall apart scale sparsity second concrete optimum bind yield compare perform stochastic gradient assumption choose bound find suffice choose similarly converge scale fails exploit sparsity descent regularize strong key exploit sparsity convexity mirror prox break epoch fails exploit strong obtain inferior convergence close spirit approach decrease schedule consequence overall guarantee procedure understand nesterov use fixed set initial tend reduce factor improve slowly regularization note work zhang assumption allow follow section set cardinality corollary specify high recall simplify setup corollary least directly note result generalize factor develop assumption approximately notion approximate sparsity formalize enforce magnitude small ball vector assumption simplify corollary capture update constant range range rather interesting show precise convergence sparsity seem rate obtain rate leverage phenomenon exactly match dimension sparse optimum convexity translate
theorem usual nonparametric framework construct rich packing construction independently risk norm norm tail bound important correspond tail random variety upper potential eigenvalue need nsf science fellowship dms nsf grant anonymous helpful comment relate canonical angle addition identity immediately bound principal lead eigenvector belong ball bound sharp obtain constrain pca exceed statistical increase necessity goal intermediate either size affect principal pca perhaps situation tend consistent situation sparsity eigenvector perform application error bound lead eigenvector wish look uncorrelated dimensional subspace orthogonal projection eq optimally reduce reduce pca spectral sample eigenvector analogously reduce pca eigenvector pca inconsistent beyond without addition enhance interpretability sparsity notion effect statistical inference research work add maximization relaxation form pca iterative fashion sort reason lead eigenvector ball provide appeal notion concrete correspond hard sparsity number entry vector small soft realistic nonzero statistical difficulty feasibility eigenvector fundamental statistical thus gap tractable main consideration usually constraint formally sphere large satisfying ingredient estimation distance unique frobenius non uniqueness eigenvector ambiguity turn euclidean asymptotic effect dimensionality highlight low minimax use base construct constrain maximization estimator solution feasible active constrain ordinary since replace optimum interesting point coincide difficult subsequent efficient pca analyze semidefinite sdp eigenvector magnitude matrix estimate notion imply wide covariance imply consistent covariance eigenvector condition imply minimax ball remarkably sequence model inspire aware provide bound ball attain allow optimal condition guarantee non proof lemma mainly technical proof
easy task relation treat uniformly closure inductive database also embed turn true programming language possess usual construct input way embed require language comparison method text categorization page computer task category student figure diagram relationship web page represent make totally even feature domain flexibility soft match occurrence page count text categorization problem additional signature common course page contain string follow page leave setup table research course f p course course research course task essentially encode domain exploit atom signature enforce mutually exclusive scale conjugate gradient describe wide result report mc collective result slightly low identical extract capture contextual lack page page page anchor link report table worse attain although accuracy comparable requirement term cpu core intel core take iteration attain internet cast working picture focus predict box office task interpretation directly necessarily interpretation notion allow overcome ground interpretation disjoint endowed total natural movie production e associate atom entity give reasonable training use ground ti ti ti similarly ground atom create movie outside discard appearance movie datum summarize model signature movie actor production additionally signature count movie involve movie year test curve summarize comparative three software ground introduce together hard query year design capture actor false case use exactly signature definition discriminative implement obtain fact fast complete phase follow consistently high year language logical inductive logic kernel underlie inductive logic programming sense variation background probabilistic entity al signature play role programming combine r provide diagram graph prove helpful hand framework signature predicate need relation inside inductive logic system relate relational logic logic logic program relational input partial interpretation prediction statistical relational implicitly logic program process network diagram statistical distribution learn case result tie particular occur template match statistical learn former really knowledge feature use feature relational commonly collective combination structure iterative incorporate em another inference try also kernel see distinguish graph obtain relational rich symbolic kind represent regard define represent specify collective learn relational relational approach relational graph explicitly employ derive resemble several mining employ graph relational encode bipartite whose connect atom atom usual employ learning mining typically label relationship node approach single representation three logical language base address language processing perform extraction exploit arbitrarily connect field dependency sentence dependency indirect logical logical relational kernel method constitute principle statistical relational rather use relational learner perform label original formulate wide learning task state art statistical also programming step relational aspect possibility perform collective semantic follow predicate collective amount important case signature complicated system example dependency regular structure programming exactly conditional field collective graph search kernel eventually inefficient develop purpose interesting open collective produce provide therein set another potential make use define concept neighborhood nearby combine generate consequence informative dense induce case kernel bias library integration therefore important future though implementation implementation issue similar employ develop vision acknowledgment analysis partially sf research partially start grant anonymous associate constructive comment code augment keyword signature header one clause clause signature connected header predicate mean domain additionally predicate measurement comma period signature signature signature header header level role level sake closely notation use call variable vertex bipartite vertex every root adjacent indicate radius edge whose subgraph neighborhood radius vertex repetition symbol denote map edge denote preserve preserve graph edge invariant follow give x semidefinite semidefinite eigenvalue kx converse reasonable always kernel call extension valid kernel kernel sum iff part denote yield demonstrate instance decompose q valid convolution way kernel part instance root neighborhood construct apply present overview hope efficient non graph assign hash graph hash sort edge hash hash hash hash sort vertex novel create compact molecular tool space present hashing mainly efficiency gain introduce vertex edge vertex sort sort list root vertex list root edge label edge label assign edge triplet uv encode vertex plus label graph resort construction list integer pair vertex note space hash naturally tradeoff space hash remark di di novel approach expressive logical relational specify build interpretation entity logic programming database access rich technique first graph entity diagram mix numerical symbolic program inductive logic system apply tackle popular collective logical learning learn database relational fairly representation alphabet difference logical distribution interpretation machine addition probabilistic type method learn vector amongst popular notable relational receive lot commonly accept markov probabilistic programming logical language wide task ultimately fill introduce logical contribution relational similar inductive logic base kernel call space eventually whole discuss relationship logical learn specify type logical relational interpretation entity object relationship naturally employ constitute start logical relational unlike difference instead feature immediately define formulae construct graph entity relation background formulae comparable much rich learn level specify logical relational description describe structure system interpretation transform equivalent knowledge system also conceptually represent graphical turn lead logic tie together capture important flexible specification entity completely graph employ subgraph reader keep mind incorporate similarly mainly variant learner situation section discussion main domain formalize assume interpretation background statistical position context system logic mn formalize problem section report finally relationship system detail illustrate real demonstrating capability anonymous science entity include student paper specify e activitie certain task student e binary student come atom logic function allow atom tuple relational interpretation logical ground interpretation correspond different interpretation false person post post person person person person person person person person person person disjoint database domain entity diagram two unary diagram diagram frame number leave position p student example signature see signature bias system annotate list explicitly file clause name list argument entity signature numerical categorical atom regard connect atom constant procedure signature signature introduces entity signature powerful database may argue co work together activity signature signature associate signature frame student paper student student signature predicate signature effectively paper student publish together relational method similarity map ground atom bipartite undirected graph node connect entity former interpretation figure predicate signature e base use procedure give section respectively person clarity still signature predicate e predicate accept signature specify learn statistical interpretation logical set atom interpretation unknown natural statistical direct indirect inference sake throughout case think consist statistic response framework distinction reflect ground interpretation partition atom query goal supervise associate interpretation e fit statistically motivated measure may interpretation cover number relational binary categorical naive share model yx actually optimize hinge maximize conditional output svm hinge bayes cite conjugate rich relational naive hmms logistic extend sequence hmms setting output simple three every pair feature discussion move suggest alphabet mention begin natural extension hmms stochastic logic expressive relational logic generalization context free investigate discriminative linear relation markov logic loss mn cope adopt rich contribute language generate relational feature relations logistic crf discriminative svm svm hmm table arrange share embed represent learn signature predicate signature g program specifie call library predicate specify formalize object atom name tuple learning expression name signature contribute every atom predicate semantic way take logic specify predicate clause introduce knowledge inductive logic hence ability specify translate maintain fashion opinion key reason relate logic order ensure well subsequent introduce assumption perhaps clear keep separate directly build function relational consist atom distinguish key distinguished relation introduce relation relation job relation signature must entity column represent domain see relation ground ground atom partition dependency predicate interpretation partial consist atom require interpretation output accuracy ground several situation learn minimal ability analysis simple implicitly predicate attribute interpretation construct notational atom every every atom ground appear uniquely denote atom map assumption degree degree vertex unbounde grow e web diagram diagram template atom template expand ground graphical network markov logic semantic quite perform interpretation requirement meet practice efficiency size phase flexible bias variety interpretation graph several exist literature reference therein interpretation tuple entity additionally subgraph kernel suitable sparse discrete edge propose deal soft match large graph whose tuple mix discrete introduce extension kind decomposition kernel part pair detail see integer let root let also introduce identifie pair neighborhood whose root pair neighborhood radius root family denote indicator root neighborhood center vertex section reason consider extension impose upper kernel furthermore g distance build countable illustration neighborhood subgraph bottom neighborhood subgraph fix radius neighborhood application dataset potentially induce section maximally discrete tuple allow case atom maximally obtain concatenation signature attribute follow denote feature transform correspondence subroutine problem whether algorithm bad degree hard high entity play relationship prefer efficiency distance spirit idea function likely number pseudo count neighborhood subgraph match express vertex exhibit likelihood neighborhood match quickly yield dominant overfitte well relax type match subgraph match base match cost give subgraph multinomial structural subgraph replace vertex pair close vertex either neighborhood subgraph dot histogram illustration soft generate kernel represent assumption vertex label vertex tuple extend allow match tuple discrete mix extension mainly focus categorical characterize define tuple subgraph write map vertex name set atom ensure signature tuple six discrete choose property treat atom vertex contain structure match jointly match concatenation tuple formally labeling receive uniquely identify summation correspond finally tuple successful match match tuple match product tuple fashion match tuple label value vertex product tuple collection index analogous labeling canonical identify neighborhood label vertice identity real tuple standard dot convenient efficiency reason knowledge select subgraph signature equation like vertex kernel neighborhood vertice kernel interpretation table conjunction plain kernel machine svm move involve entity tuple entity induce alternatively convert subproblem simplicity job relational ground atom intuitively target query specific entity web page section tuple link viewpoint highlight define figure illustration define graph center finally
first integrate laplace determine covariance discrete integral marginal formally equally give interesting illustrate consider case marginal pass diagonal procedure entropy product marginal constrain marginal easy lagrange multiplier constrain integrate independently express turn eq find actually I another integrate write lagrange multiplier correct covariance note form condition thus conversely end somewhat first trivial entropy marginal nontrivial correlate gaussian marginal index drop notational gaussian easy correlation distribution concavity maximize unique satisfying requirement prescribe marginal establish upon integration generalization expression joint marginal perturbation expansion power rather write power write correct average take guarantee positivity highlight corresponding maximum distribution expression immediately conditional thus result second order perturbation hold assume prescribe marginal construct distribution prescribe marginal entropy straight couple result hard exception gaussian among encode case marginal scheme moment perturbation present grateful valuable discussion grant determine specify marginal consider condition unique choice correspond maximum entropy maximum entropy complicate around product marginal moment give far know assume marginal arise range financial economic eeg system mechanic name finance become marginal copula describe two random variable frequently measurable physics ill infinitely distribution none marginal lift joint maximize
onto use line adaptively prevent maximum simple modification constrain specific project onto technique onto tf project orthogonal complement find basis subtract original rewrite decomposition projection q position constraint un tf f tf line convex program easily subproblem column approach individually subproblem program original value nature difficulty b inside conventional similar direction set brief total derive formulation bregman split tailor z augment lagrangian solve duality coincide method concave g b fix soft formula inversion computationally expensive inversion significantly simplify identity iterate parameter also particular x x k synthetic part concerned almost reference ask care exact true operator effectiveness imagine reference explain certain detect obviously learn mean variance need project onto make project randomly component reference normalise chose corpus project subgradient iterate time check operator reference trial plot practically start point far reference check equation investigate role simulation set population simulation show locally identify less distant way demonstrate consist optimisation program alternatively respectively whether satisfied repeat pseudo plot respectively column row many locally base haar imaging community figure select dc pseudo select constant choose enforce operator learn seem operator choose overcomplete haar operator haar learn objective base algorithm optimum problem use provide generating checking indicate learn synthesis use average dct dictionary row demonstrate synthesis well dictionary svd lagrange multiplier use previous average trial synthesis dictionary slight db dictionary framework marginally behind synthesis promise field denoise find reason leave row right middle ht learn synthesis omp paper analysis operator fact want select appropriate introduce canonical constraint suitable practically show relevance demonstrate recover synthetic analysis operator enough give two image class piecewise operator similarity finite harmonic type observation select part toolbox program optima program avoid check minima recover operator close neighbourhood rewrite optimisation admissible provide pair solution actually check optimality achieve tangent constraint subsection local condition optimality local zero objective tangent constraint see definition positivity violate easy contradict optimality complete lemma condition proof upper hand note x local lemma similarity separately term parameter separable may framework analysis unknown system operator proposition proposition dictionary learn synthesis operator identifiable follow note act identifiability use variational identifiability let n l v x constraint r nontrivial constraint training st statement iff follow positivity x c secondly identifiable identifiability cardinality produce pattern give variable theorem sufficient identifiability optimisation two constraint reformulate right multiply al presentation mi collection approach sample synthesis counterpart transform use overcomplete optimisation l optimisation optimisation exclude trivial solution answer conventional constraint normalise frame project demonstrate ground training set image noisy signal realistic optimisation often two practically identifiability learn representation low model dictionary learn appropriate actually involve describe signal example include music speech edge video low dimensional modelling signal overcome decade many achieve natural crucial dimensional imagine signal class dimensional possible map restrict admissible simple option map ask familiar structure synthesis set overcomplete call index cardinality processing perspective easy task live union subspace subspace one look satisfy come computational aspect look match operator zero row compatible interpretation structural synthesis sparsity signal work synthesis play ask signal design reader refer survey dictionary show dictionary perform dictionary wavelet learning method also norm preferred formulate admissible learn coefficient reason exclude trivial solution signal modelling scale dictionary atom frobenius norm ball make optimisation different sensible often alternate objective upon designing transform decade many harmonic wavelet fast transform perfect reconstruction signal relatively adaptation model current signal application learn counterpart formulate optimisation perhaps surprisingly analysis problem even scale report exclude compatible operator wave atom give optimisation subgradient implement practical open cope give local propose optimisation nature operator develop idea primarily remarkably concept pointed synthesis approach already learn optimisation trivial solution become trivial recent k operator promise specificity explicit optimisation expression analysis clean bold letter vector bold capital present list correspond subscript frobenius abuse notation absolute operator canonical norm space denote whose subscript iteration cardinality representation similarly cardinality x subscript original parameter elsewhere bar complement c c pf signal synthesis operator operator optimisation briefly constraint introduce optimisation algorithm simulation scenario synthetic operator appendix local optimality analysis purpose I dimension problem solve new adaptation problem optimisation use introduction future unconstraine exclude solution admissible assume thus formulate prefer lagrangian multipli simplify reformulate problem noiseless candidate suitable exclude solution space operator signal simplicity constraint individually exclude solution combine normalised frame subsequently constraint smooth differentiable optimisation find optimum repeat enough full operator close closure admit sequence function complete resolve ill conditioning geometrically separate let case ai admissible constraint admissible optimisation constraint actually construct call manifold tight constraint trivial empirical orthonormal zero cause element signal without insight operator operator bring compare motivate apply choose uniformly normalise row yield rest intersection normalise frame manifold frame manifold guarantee intersection two manifold dimension fact optimisation feasible global optima
snps mapping ad gene map identify multivariate trait phenotype measure arrange response phenotype arrange minor allele count snps record minor allele count snp arrange additionally phenotype snp square square include square eq give p q find reason singular thus invertible uniquely equivalently predictor snp regression ideally like exploit estimation boost limitation address restrict rewrite relate phenotype capture th predictor row vector back dimension represent set predictor set model set phenotype rewrite q vector relate phenotype obtain reflect response optimisation exploit response image derive structural correlation phenotype calculation computationally instead simplify approximation group map snps pathway begin pathway snp coefficient denote snps map pathway observe minor allele count snps corresponding snp coefficient small snps sense effect causal snps functional group assumption illustrate fig causal mark grey causal pathway pathway generate sparsity snps causal seek identify impose constraint snp vector htbp causal phenotype box grey causal occur pathway particular snps vary phenotype coefficient impose respectively apart benefit enforce also pathway sparse model correspond penalty whose weighting group depend penalty setting pathway snp coefficient enforce pathway expand note solution eq optimisation amenable coordinate hold additional response pathway group lasso tailor variable group accommodate coordinate descent descent grouped obtain successive estimate pathway pathway constant snp descent within group use h w increase pathway box estimate snp coefficient tend box full algorithm box group p pathway wise concatenation p require pathway tendency lasso pathway snps phenotype bias arise example variation snps snps tuned pathway select pathway factor iterative weight pathway unbiased biased begin phenotype weight adjustment maximum reduce pathway frequency continue even gene phenotype expect pathway snps gene pathway pathway pathway selection presence snps mean pathway selection probability reflect causal snps spurious association nan snps capture one tune substantial gain specificity causal rank capture association phenotype factor repeat factor rank need typically large snps pathway pathway method parameter determine variable choose drawback approach focuses select vary across establish importance alternative resample bootstrappe sense select adopt calculate selection frequency repeatedly relatively insensitive subsample snp pathway pathway measure across q indicator pathway selection snp voxel constitute phenotype euclidean subject voxel dimensional verify voxel discriminate ad classifier gaussian covariance figure accuracy sensitivity specificity optimisation primary concern figure cite use associate ad longitudinal phenotype pathway rank order pointwise pathway min l name gene pathway ad gene snps pathway pathway infection cycle disease fc gamma disease drug identify gene pathway snps rank snp describe lasso pathway lasso subsample accord size map range maximum sd pathway subsample max sd r snp snp map rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs pathway rank phenotype pathway select frequency explain subsample selection pathway pathway ranking aid interpretation pathway ranking pathway rank select rank phenotype study extent pathway nevertheless interesting pathway calculate rank ad take average pathway gene ad gene sum ad gene gene score pathway ad tend compare empirically obtain pathway ranking empirically compute empirically chance permutation htbp pathway trait modelling show sparse modelling approach detect pathway approach begin model snp phenotype pathway biological pathway image phenotype identify model disease signature disease investigation especially case phenotype poorly ratio advantage extract image highly disease pathway function pathway link aspect gene drive identify snps gene phenotype ranking select irrespective capture signal salient pathway drive pathway detect highlight lasso activate driving selection pathway rank link expression formation ad brain aside validate ad rank gene occur ad particular amongst pathway suggest pathway drive small effect interestingly pathway single subsample turn snp ranking table gene gene identify major ad high gene phenotype allele study pathway pathway pathway disease select allele snp allele ad phenotype disease pathway rank presence pathway higher ranking snp snp also pathway evidence interaction ad fact snps interact simplify uncorrelated reality phenotype correlation association assumption gain multivariate phenotype finally tend pathway effect conventional approach demonstrate limitation pathway know pathway snps exclude pathway annotation reflect gene interact annotation different pathway rapid pathway suffer benchmark fail complexity pathway take snapshot biological acknowledgement ms collection project health national national institute drug discovery life company company la company development scale company clinical site national www organization california institute research education california laboratory california grants foundation present associate quantitative trait image characteristic longitudinal change disease pathway genome single nucleotide pathway pathway rank resample exploit finite application study whole genome mr snps map pathway voxel image signature characteristic ad change rank ad associate include secondary driving pathway identify validate ad gene disease imaging pathway regression grow genetic variant allele identify great date study augment phenotype signature effect image genetic phenotype highly common identify gene genetic molecular pathway rather highly multiple variant act pathway association reveal genetic otherwise variant individually help biological association example understand important biology patient care first able accommodate phenotype pathway genome single nucleotide snp voxel longitudinal change characteristic ad pathway collection functional pathway represent molecular interaction reaction accommodate source group gene network rely snp test univariate quantitative phenotype snps assign within snp pathway pathway account pathway linkage disadvantage snp consider snp snp consider snp effect accurately identification false identify pathway trait sparse lasso grouped demonstrate snp pathway show high sensitivity specificity detection pathway great gain previous accommodate phenotype incorporate sparsity regression identification pathway reduce incorporate phenotype single account voxel snps follow begin description voxel use map characteristic ad discriminate ad map pathway describe section pathway resample discuss strategy address pathway gene rank section imaging use study edu institute institute private year public primary emission biological marker clinical cognitive early marker early ad intend aid effectiveness well time clinical trial longitudinal public database serial ad individual control subject follow standardized mp protocol acquisition ms te ti ms flip angle acquisition yield mm isotropic voxel et et linear align longitudinal mutually align international brain exclude brain generate extraction create structural brain change time globally scale scan scan inverse optimize cost code jacobian create brain spatially across jacobian template comparison analysis conduct clinical subject inform participant experimental cognitive screening ad year ad cn detect causal pathway phenotype brain characteristic interest extract univariate quantitative signature voxel multivariate signature individual cn longitudinal exclude voxel intercept dependent change screen slope change voxel voxel ad stage wish clear imaging derive signature ad select discriminative voxel final set phenotype correspond voxel voxel age subject study database snps detail snps variant original
vector adaptive projective half wave pd projective angle consider practical adaptive ease update quantum rotation regard issue experimental scheme consider quantum concerned state reduce optimality see appendix measurement high need must deal possible constraint candidate discrete much simple candidate estimation could probable open consider experimental design measurement optimality state rank projective estimation trial perform first trial conditional include trial fisher include state unconditional calculate unconditional even converge essential unconditional design experiment conditional fisher view lie heart adaptive dependence direction radius solid black dotted spaced line dash light spaced dotted radius peak plot expect six six make thing easy average two thing less intercept effect combine create error quantum adaptively measurement outcome measurement determine optimality general nonlinear analytic projective reduce numerically show optimality criterion standard successful experimental implementation quantum protocol state confirm theoretical perform obtain involved statistically assume finite copy quantum operation mathematical always error quantum key classical precise quantum quantum estimator use context identically choice raise trial trial setting outcome design potentially performance adaptive design criteria fisher calculations shannon report propose experimental perform measurement large state state decrease update however infeasible whose sec terminology experimental brief review criterion analytic update analytic updating use numerically precise quantum sec experimentally appear adopt sometimes subtle formalism terminology apply survey interest include quantum measurement hilbert acting space know try lie include pure state measurement trial copy distribution quantum measurement quantum outcome trace operation represent sequential copy sequence use trial outcome measurement th th trial previously obtain datum let denote trial correspond choice maps nd likelihood specify estimation quantum sketch quantum evaluate introduce minus fidelity random variable density call true density latter least eliminate namely find combination average rao optimality space denote density nd nh operation respect rao estimate eq exchangeability limit derivative converge quantity converge interpret lower square known regularity choose explanation stand describe wish consider approximation parameter estimation must criterion necessarily I measurement adaptive require computationally sum measurement th probability trial consider relationship conditional unconditional estimator appendix optimality q nonlinear analytic state identify rank measurement trial denote projective axis whose parametrization projective measurement hilbert schmidt schmidt trace distance system measure calculate measurement behave hand behave maximum behave indeed make outperform briefly measurement terminology subsection update every trial simple step require measurement trial include trial calculate choose prove mathematically weighted mean squared result adaptation criterion er rao optimality criterion use cost disadvantage incomplete experiment explain paper projective first shannon distribution simulation pure projective maximum estimator perform set measurement behavior expect fidelity estimation pure state projective measurement average expect fidelity show criterion bayesian shannon propose distribution projective numerically analyze evaluation criterion eqs involve cost integral cost mixed projective state unbiased basis separable average expect scheme precise numerical optimality even least projective measurement analytic projective rank projective rewrite analytic sequence minimal equivalent basis invertible scheme choice arbitrary perform projective projective matrix rank minimal span vector dimensional space span third measurement formulae schmidt eqs schmidt eqs perform simulation design detail schmidt selection square schmidt distance adaptive optimality subsection measurement trial projective measurement trial projective measurement scheme selection randomly accord haar three fix projective fourth choose design know minimize estimation computation estimator use likelihood point sphere choose subsection schmidt sec expect black schmidt line blue dotted nu space include scheme measurement sequence carlo integration analyse routine statistical expectation approximated figure expect loss function number axis logarithmic integrate integrate via integrated four schmidt optimality large pointwise hand gap adaptive gradient begin behave consistent result asymptotic design explain sec average criterion estimate fig state time least expect nu n horizontal axis scale schmidt distance expect three state respectively measurement calculation analyse error pointwise number trial vertical axis logarithmic scale schmidt expect depict become plot schmidt square hilbert schmidt
produce fraction decrease eigenvalue covariance used similar test never measure move short autocorrelation value sample must take measure efficiency integrate autocorrelation practice performance reason long autocorrelation must time affine invariant reasonable calculate module module implement author project useful understand key goal metropolis hasting home build reason write tuning build read build package say make similarly piece sampler start spread reasonable another ball expect close practice find effective chance get low mode modal landscape expand fill autocorrelation time third go continuously increase anneal autocorrelation mcmc short acceptance range principle fraction low reduce mean acceptance fraction modal mode expensive human disjoint single small burn number require thought accomplish merge big one mcmc produce ensemble sample recommend hundred matter initialize obtain mistake run run couple way plot less autocorrelation try autocorrelation run time sure huge dynamic contribution think step long modal autocorrelation short mode mode time ratio autocorrelation target modal become happen long good proposal direction acceptance quickly fairly general application actually important move move implicitly linear true parameter value equivalent linear avoid case useful might package chen mit price helpful contribution grant er david address center particle new york ny usa pa nj usa institute university st york ny united states test affine invariant carlo mcmc open already project literature advantage excellent call implementation parallelism ensemble permit user core without extra effort mit package appendix visit page past decade gain come method example problem experience directly often expensive procedure find often understand pdf detail approximation space application probe arguably important advantage bayesian nuisance process otherwise little marginalization process integrate wish nuisance marginalization list nuisance value addition marginalization result regime valuable many evaluation statistically pdf present efficiency slight modification introduce chain center position normally something tuning dimension sensitive proof burn short chain tune extra call computationally look highly density regime principle tune hyperparameter h make sampling calculating get bad dimension transform much sampling isotropic motivate invariance equally insensitive covariance extend invariant sampling hyperparameter implement effort already project summarize decision outline complete discussion beyond interested reader classic like key concept draw necessarily quickly compute model important expensive compute available histogram subspace span implie value sample trivial process analytic representative simple commonly iterative position proposal position distribution sample proposal parameterization multivariate center tune worth step accept previous h converge level implementation need efficient short autocorrelation draw xt expensive informally move shorter autocorrelation clarity involve evolve ensemble current space position position clear
eps alpha vs eps pt pt eps clear experiment sgd converge fast average strongly prediction accuracy error still three exception fig accuracy reason inexact serve second stable rate discuss pt pt vs eps pt vs h vs eps alpha introduce composite nonsmooth propose convex propose batch well nesterov technique possibility exploit link promise negativity proceed clarity smoothly composite expectation smoothly approximate left inequality second bregman appear identity let scalar denote bregman divergence ready lemma smooth convexity subtract side inequality inequality due denote rewrite line easy inequality line alg term alg verify q definition substitute take l want minimizing analytic simply satisfied q ignore bind separately follow q college institute technology minimization nonsmooth convex central machine algorithm exploit common nonsmooth include svms achieve minimize loss confirm subgradient include issue computation nonsmooth nonsmooth sparse support vector regressor make nonsmooth absolute norm widely reconstruction nonsmooth theoretically difficult optimize nonsmooth stochastic learn fast nesterov deal nonsmooth smooth function apply nonsmooth function strongly paper nesterov smoothing propose stochastic nonsmooth combine analyze new algorithm name ml method minimize nonsmooth averaging converge nonsmooth still numerical dataset compare algorithm perturbation nonsmooth complexity serial cast function proceed paper along notation assumption first access stochastic subgradient basic propose objective drive gain variant recently composite convex nonsmooth model proximal stepsize composite idea rely nuclear norm machine loss nonsmooth become nonsmooth separability hinge absolute insensitive tackle study stochastic composite setting lipschitz smooth clarity focus generality drop analysis simply stem exploit propose smooth stochastic approximation attain optimal convergence analyze choose numerical present proof paper nonsmooth nesterov structures nesterov minimize formulate equivalent point form map convex non obtain smooth approximation nonsmooth crucial property continuously differentiable continuous drawback method present subsection dominate still affect take g keep mind nonsmooth indicate initially solution change eventually nonsmooth example sec turn alg denote retain rate without one fairly small without rate call dominate online typically theory batch call batch many employ contrary free effort convex bound e rhs term bound achieve achieve one retain perspective b inferior optimality nonsmooth stochastically approximate use hinge convex surrogate express objective g take hinge enough compare pt pt binary suppose ik alternative popular square regression express maximization look huber green curve fig available sim alpha classification regression dataset sim alpha use rest sec approximate hinge classification absolute regression compare
partition size sketch partition intuitive give motivate partition function group moreover g result immediate corollary lemma consistent consistent consistent example sized consistent partition figure group go probability go find instance problem rational encoding previous size group rational variable every j encode state stochastic define encode show correctness termination target teacher goal learner maintain proceed round teacher update reason mention introduction whether positive tree consistent simulate receive every make consistent receive sense unfortunately sketch unknown describe adversarial teacher keep teacher long hypothesis target teacher enough behind teacher execution mapping teacher seen return never converge able implement teacher interesting execution mapping fail teacher teacher every teacher execution long positive already become negative teacher pseudo treat line transition return least sized first partition make active terminate condition teacher line sketch iteration loop induce execution mapping sub structure least sized future new return hence current lift add finitely many terminate active learning teacher sometimes desirable learn hypothesis need impose every state learn algorithm guarantee condition teacher teacher keep return transition positive self loop teacher return always always partition terminate least correctness original style simulation reasoning verification large increase scalability check environment reasoning guarantee use reasoning plan investigate relation result compositional verification check investigate new checking david answer question suggestion union union simulation clearly induction height tree leaf define height maximum height leaf trivially height let induction therefore conclude induction strong without generality partition bound exist disjoint strong strong simulation arbitrary definition therefore strong let suffice definition choose transition also create initialize induction imply define ds g l ls l sr simulation mention begin furthermore imply say define furthermore induction define distribution way follow definition uniquely eq simulation therefore impossible target adversarial teacher begin round loop teacher act adversary far teacher return fail return fail teacher label compute update see teacher hypothesis make show round add definition add hypothesis label let state argue contradiction construction contradiction conclude step intuitively whenever beginning learner keeps never converge unknown arbitrary def learner def return positive example impossible teacher begin active describe teacher keep generate start initially teacher transition clearly tree return teacher return execution mapping teacher strategy keep transition state dirac loop latter finite return teacher figure force consistent teacher round state return action teacher proceed round initial negative every execution return except hypothesis b round force keep tree pp consider deterministic strong traditional partitioning use teacher query teacher fail latter problem conjecture semi intermediate assumption automate guarantee tree study label transition guarantee compositional verification yield stochastic shape appear compositional verification promise successfully check simulation work language automatically compositional reasoning exist acceptance tree language motivate language tree deterministic consistency verification want learn good space partitioning result partition stochastic partitioning result number algorithm framework active teacher teacher answer query membership target create transition probability assume teacher teacher equivalence target return check assumption teacher fail restriction conjecture restriction also turn algorithms teacher pa probability arbitrary verification system difficult readily intermediate guarantee style framework fully automate compositional verification moreover safe preserve compositional propose check probabilistic trace variant deterministic use sound terminate complete guarantee terminate lead complete rule guarantee termination using infer previous generation separate trace check guarantee separate make query equivalence query probabilistic state study algorithm stochastic acceptance trace say approach discrete probability specify explicitly number dirac e non label tuple distinguished action use circle figure transition give restrict dirac classical thus notion deterministic state finite alphabet finitely branch label system belong simulation belong pair belong pair support belong distribution relate state probability transition one match match possible general notion matching match possible follow split achieve check compute appropriate equal definition explain bipartite giving follow iff exist match iff iff simulation strongly iff simulation simulation definition immediate simulation simulation simulation check great simulation initialize iteratively pair terminate show start state remove exist simplicity characterization discuss exist def height state strong make definition conclude problem learner diagnostic two denote iff trivially l l useful check moreover preferable simple serve tree state make example structure simple execution mapping execution mapping state tree mapping execution condition impose teacher regard execution mapping finite stochastic say constitute sample goal say tree transition trivially tree trivially hence exist consistent checking reason guarantee infer trace state partitioning learn insufficient unknown enable obtain traditional partition equivalence state drop always positive map partition every iff consistent show iff consistent reduce always sketch simulation tree generality induce iff upper lemma size c way general merging na I find sized consistent partition rational arithmetic exhaustive I equivalence boolean say partition start encode consistency want introduce quantification say every fortunately boolean action expression transition add encode sl encode use iff nest quantification variable weight encode subset encode hold image detail encode constraint every correct
prove fx integer magnitude weight may integer weight nearly almost constructive approximate stress quantitative result dramatically previous self elementary algorithm heavily spectral correctness careful rather fairly theorem spectral rely turn required argument proof main theorem relate simple approximation boolean linear type contribution detail subsection give fix form domain need generalization gx simple finitely fact straightforwardly x rational f nf induce function e contrast familiar g fx gx familiar hamming basic structural converse function main may view statement sort bound relationship say require proof also improvement remove dependence carefully extend critical recent whether constant conclusion possible inspection argument theorem paper already ingredient construct construct construct approach substantially find bound structural apply find closely convert small main first define bound function exist gx state algorithmic main vector v proof polynomial imply e much simple algorithmic roughly work identify candidate first inefficient correctness sophisticated result make approach complicated novel much dependence limitation note boolean return approximate apply close order close distance accuracy contrast require weight linear optimal idea use idea distribution learn polynomial formulae fouri like similarly less throughout give section main structural algorithmic ingredient put prove main result present conclude paper let ess explicit ess independent denote cumulative cdf cdf say anti denote I elementary inequalities b ab b arithmetic geometric inequality obtain say closed combination affine letter hyperplane whenever explicitly dimension denote span space contain element regard exact version present time completeness write x first translate separate gx fx gx first second identity use fact strictly start point unlikely solved even check correctness intractable interesting naive variable one requires improve albeit problem target goal function constraint encode gx aforementione since linear linear programming truth obtain straightforwardly time solve provide overview main proof proceed explanation subsection clarity restrict attention boolean suitable weighting near theorem statement prove theorem use good anti extremely roughly moderately good radius original crucial small anti subject seem quantitative improvement instead closely approach use perturbation measure closeness ham direct high explanation modify statement hyperplane relate boolean hypercube statement hyperplane geometric statement hamming function euclidean center differ euclidean distance mass classify correctly closeness key geometric depend dependence heart geometric critical statement completely roughly analogue geometric lie euclidean hypercube stronger bind weak guarantee pass point arm careful iterative another prove theorem hyperplane euclidean recall show euclidean hyperplane hyperplane neighborhood element provide subset eq lie appear fact strict quantitative improvement theorem application lemma depend independent necessarily something require later technical reason give detailed lemma underlie proceed reduce follow prove later instead lie reduction critical apply define definition coordinate effect space tail case tail property index contradiction index ess behave like deviation norm entire anti concentration ignore tail early view refine generalize function various detailed proof key ingredient index implicitly explicitly define regularity fix regular helpful regular ess tell distribute particular anti gaussian intuitively regular quantity inequality tail weight critical apply inequality repeatedly technical use ensure continue lie use define idea coordinate tail tail coordinate tail tail norm thus essentially ignore tail n conceptual clarity continue k n w w cardinality least union bind also x q hence h deduce hyperplane h condition apply absolute ensure recall plug verify use side bound thus sufficiently indeed hyperplane h head tail suppose sake contradiction ess theorem get q consequently w contradict existence statement rest proceed hoeffding turn w x h w w deduce alternate v I true notation sufficiently sense upper ii claim prove case conclude essentially refine two modification generalize allow key thus quantity concentrate x inequality dt ready assume n fx gx generality satisfied give record proof characterize exclude degree bind low proceeding projection line orientation set whenever balanced balanced opposite side x x inequality suffice term write second first recall otherwise ni hence separate contain unit hypercube irrelevant orientation line p lower suppose bound project projection separate balanced x satisfied n obtain hyperplane satisfied proof use j beginning imply j hypercube lie side orthogonal parallel eq analogous say obvious j v set define x eq x plugging establish claim construct dimension nu jx jx jx nj get w n w ny j j j j n contradiction proof theorem convenience formal main algorithmic every boolean probability output intuitive overview motivated reasoning since function course parameter quite try close operation almost exactly current necessary modification ensure correctness proceed namely difference g boolean potential measuring arise add vector overcome counting argument result potential generalize follow process accuracy value tn vector coefficient eq add prove g tf claim proof observe ti operation tx part tx tx tx tx tx g tx g tx g tx tx x tx g g tx tx g tx claim stop algorithm time ensure therefore find g ti tx ti tt claim threshold markov v iw integer weight absolute boost close reverse distance ham everything polynomial together start give ff bit operation output representation proof run appropriately definition polynomial get write n x complete simple theorem produce proof note define integer satisfy iw weight input simple structural w w iw f satisfy gx complete proof say whether conclusion conclusion strong v f nf contradiction integer must application david restrict learner access uniform set receive label specifie bit ask sufficient integer weight satisfy accuracy early computationally give first result give learn running consequence much bit properly section show main briefly review agnostic hx agnostic learning give label arbitrary agnostic guarantee boolean give access representation probability operation describe correctness estimating algorithm theorem run f triangle dominate reconstruct approximately degree context paper give provably nearly existence open complexity believe intractable conjecture exact intractable attain upper believe show integer argument imply indeed output weight improvement improve run quite theory boost approach generalize precisely fouri degree specify degree boolean even bound algorithmic straightforwardly immediately hyperplane pass point give fix pass assume integer x dx easy also observe whose two final hyperplane pass prove prove element prove claim let denote span denote show first obtain th subtracting belong let position sign belong subtract previous imply suffice exhibit easy claim eq whose alternate subtract odd position subtract prove claim technical require extension lie hyperplane conclusion extension need deal find define da w w convenience reader proof towards contradiction greedy integer argument prove homogeneous nn w rescaling define comprise solution w w system system integer new
machine generator suppose ideal device disk make bit desire come resort physical process probabilistic physical device unnecessary attention rather string scientific community procedure possible obtain make bit familiar generate string distinguish determine binary string actually statistical numerical physical generators b generators deterministic mathematical resort physical generate binary string binary test randomness binary string pass criterion use string random string familiar ambiguity digit string digit either occur string generate entirely algorithmic however digit absolutely digit suppose example digit digit vast person string simple explain make digit mathematically generator number string mathematical whose number random binary string root technique cubic root number hundred thousand use mathematically root whole root amount digit discard proceeding justify discard digit digit root digit digit digit digit digit comparison specify figure either generate course neither two contribute binary appear h string figure compose order two number number consider increase appear consider number permutation permutation refer leave call subscript indicate permutation permutation letter first refer number second subscript use indicate number belong refer element order belong permutation subscript third subscript indicate belong order op long op p place long binary string desire result first op concatenation operation place exist finally string make last previously obtain accord successively continue carry root root carry precede permutation obtain permutation regard carry root pair specify permutation permutation different express clearly concatenation specific binary summation specify add sequence specify factor multiply concatenation indicate string product sequence may give triple likewise concatenation number consequence permutation respective characterize assume subscript number subscript indicate belong successively reach want describe order next operation binary already operation express last string base generate sequence discard first course random string tendency four different tendency present refer likewise expression string fulfil string tendency eight tendency seven possible two etc consecutive bit stre random choose suppose string tendency three transition comprise length calculate perspective probable different transition actually string naturally chi square determine give numerical theoretical consideration use significance difference show level case transition string length form use generator pass fail pass test fail fail pass fail fail pass fail fail pass fail test fail test excellent distribution generator string quantity curve string dot solid approximation assess let standard respectively normal area curve specify h area interval binomial previous article index randomness string string string feasible probable randomness string string none tendency generate string correspond aforementioned string index binary carry randomness randomness index probable average index randomness binary string potentially tendency tendency mention string compute average find randomness compute precisely likely generator string string follow carry computed string calculate already table randomness index generator quality ccccc max avg max precede generator binary string obtain generator previously article answer question notion kolmogorov randomness string generate length string comprise develop result algorithmic make possible sequence digit accept string digit generation program exist less bit string whether randomness currently suitable computer approach regular regard h approach vary string string computable close accept truly string two large comprise comprise string randomness string compute index randomness obvious string vice versa class index randomness high randomness string procedure digits root digit discard specify different permutation apply belong want simulation string describe ambiguity generator find mathematics www quality string statistical section generation document publish institute technology find test develop randomness long hardware software generator
approach marginalization process combinatorial category coherent calibrate model computational joint tree method separately particular one alignment infer alignment inferential alignment recursion recursion numerous infer iterate tree theoretical understanding getting calibrate problem finally analyze time considerably simplify disadvantage base joint approach substitution surprisingly relatively major equivalent tree exploit notably poisson computational computation obtain number description local treat fundamental description refer new light evolutionary incorporate poisson value space generation literature lexical character string remainder present formulation exact empirical view subsequent stay unchanged interval mutation substitution sequence achieve exponential correspond mutation character character position string exponential small winner substitution exponential event mutation multinomial variable substitution rate determine character multinomial determine character parameter stre tree visit single root stationary conceptually along infinitely long reversible useful arbitrary make description relate poisson additional continuous point equal branching set edge length whether branch write rooted subtree root dropping long model position character character identically length type mutation aspect aspect character string consist step atomic mass hence root atomic mass population branch black poisson root sequence deterministic two visit point x visit direct subtree root location character whose substitution path single character define single character along set substitution symbol homology path homology see construct symbol thereby observe comprise homology leave character character character point root probability generate homology random description subsection fact description string state characterize distribution substitution denote rate reversible substitution local describe section coincide appendix depend character follow establish reversible poisson advantage geometrically distribute stationary protein life adequate consequence process characterization allow discussion likelihood marginal likelihood condition homology homology unknown number analytically follow capture homology second way pick observed homology character symbol leaf drop exchangeable simplification internal appendix formula alignment partition computation depend locate column precisely look common character correspond occur location simplify fact chapter denote compute slight recursion subtree root derivation appendix column get total time system bayesian use benefit relative separate benefit infer tree infer benefit infer datum assess reconstruction reconstruction platform infer produce inference implementation evaluate frequentist precisely estimate frequentist study four type potential resample compare resample resample increase tree fix produce resample quality infer sample branch exponential two exp yes yes yes sp edge measure quality reconstruction alignment measure quality reconstruction partition difference metric metric quality difference term f report relative improvement relative relative improvement baseline joint full tree see test system generate baseline baseline large versus f substantial tree note simple metropolis suffer fortunately work literature potential sequential monte exclusive driving force sequence atomic segment also prominent consequence bias biological bias undesirable length gap relate poorly align sequence homology evolve substitution permit poisson exponential poisson role pure substitution knowledge representation vary vary sequence align similar length freedom indeed even note process genomic constant actually also acknowledge neither accurate biology inference notably account topology nonetheless useful hope motivate goal specie also view biology inferential example significant model capture arise model regard wish poisson extension superposition make non local poisson approach inference indeed superposition follow second superposition variable design irreducible integrating create bridge pair parameterization character handle assumption replace quasi stationary cox process large variation site branch acknowledgment would thank comment suggestion partially grant office grant health grant kp section proposition condition degenerate leaf length homology ix n exponential therefore conclude proof equivalence edge description rate distribution location fall global description hypothesis base construction character global measure poisson assign parent hypothesis well establish hypothesis mutation give yx item follow item standard random n pn dependency simplify find nucleotide nonzero weight leave c v compute use standard programming survival joint carlo object topology branch proposal auxiliary partition two use pairwise hasting irreducible possible link group move proposal brief involve
small time backward couple minimum histogram htp histogram acceptance eq way accept class ii accept pt discrete continuous extend hasting perfect broad applicable purely work motivated give decade appearance seminal perfect community immediately variation extension appear area statistical physics spatial operation powerful iterate transition law perfect simulate motivated mixed allow possibility become impose mixed result mix mixed adapt perfect somewhat relax assumption partitioning state describe single class remain possibility member second decompose space metropolis hasting algorithm extension example mcmc enable value create shorthand quantity involve usually simulate directly behind perfect law converge come meet couple zero path since path forward refer achieve perfect interested metropolis hasting algorithm way construct transition limit satisfying generate potential time evolve transition hasting metropolis simulator candidate transition evolve density verify sense borel stationary hasting unique limit distribution ratio run even generally metropolis rate study extensively hasting perfect need generate density metropolis hasting perfect ratio leave write ordering attain role move accept move state accept couple description start lower path accept move probability accept move upper move time otherwise state continue point hasting time time monotonicity candidate consequently move upper path accept candidate identify illustrate htp grey arcs grey represent start solid path ultimately perfect state inclusion transition candidate eq candidate partition sample density dirac delta concentrate hyperparameter draw density replace draw make hypothesis report interpret detail simulate hasting long regular proceed approximate start perhaps arbitrary q move arrive draw turn perfect backward care random path meet figure non htp path couple dash line potential path series possible acceptance depict circle image indicate achieve start depict coupling propose non meet note realization fail coupling htp open circle transition dash actual realize candidate value sample time accept stay accept achieve box fail starting determine must acceptance number negative path accept density since minimized likelihood mle include early serve reading coupling depict couple r achieve figure go two forward use exist path start accomplished set target backward path start stop reached extend algorithm point class point two overlap case modify candidate value include point support example propose support propose candidate ii density backward candidate point single atom hyperparameter know priori independent simulate candidate class accept class always ii occur sample path least step store compute coupling rx n k ix rx k require accept candidate
behaviour distribution inferential key aspect duality geometric relationship duality relationship consequence relationship existence family call parameterization affine across useful behaviour give vector set q dimensional change exponential image special thus advantage strong tool limit family multinomial possible polytope converge family converge boundary zero family limit geometric family dimensional define dimensional family base point embed span consider direction give one boundary first first change impact determine see redundant consequence fig eps family clear four ambient via problem htbp logistic full lie simplex design dimensional change explanatory inside space explanatory convenience distribution pass direction htbp plot clearly global geometry immediate sequence list also turn instead datum go infinity explanatory reach form identify effect uniform simplex high geometry region continuous geometry geometry derive geometrically suited use asymptotic formulation perhaps formulae rather approach ideal course deal two fundamental issue stability matrix formulae secondly boundary careful understanding multinomial use first like logistic accurate expansion act diagnostic considerably point contour widely example variable sample bin enough key understanding information control moment bin uniformly support continuously uniformly bound geometry exponential information loss inference two family make family uniformly measurable partition choice bin label geometry approximate particular norm satisfy skewness tensor corollary likelihood arbitrarily fine fig base base partition application affine unlike mixture example geometry numerically concern survival day diagnosis illustrative purpose censor give distribution family embed htbp geometry censor censor value interest mean width dot panel raw censor exponential inferential show skewness asymptotic full example fisher variability datum simplex paper mixture may mix polytope exist convex existence low exploit use positivity hull open generic tangent regularity family representation apparent contradiction curve lie subspace space discussion purpose motivated idea local approximate identification issue additionally usually space show considerable hull hull triangle within convex hull measure quality prefer far norm likelihood maximum turning point condition small likelihood euclidean unbounded change hull determine positivity directional turn appropriate finite far convex respectively simplex appendix evaluate couple examine hull hull decide singular segment consecutive theorem mixture model laboratory show relative variance fit binomial mixture good fit circle mixture proportion directional lie hull binomial node simplex joint observe easy include six opposite edge opposite face model dimensional unlike hull affine multimodal structure aid correspond model h bi convex polytope maximum structure maxima internal see construct approximate surface surface boundary opposite surface slice surface hull convex slice paper focus multinomial area geometry geometry allow numerically geometry geometry support finally build act paper look dimensional careful discuss selection thank ep decomposition notational convenience bin omit without loss vanish trivial eigen eigen denote order span eigenvalue expand claim much close approximated positive indeed proof theorem loss partition thm exist partition away uniform measurable far follow direct follow finally equation q image share preserve eq general suffice positivity expansion account directional zero hull directional finite cone directional case directional accordingly change log directional non taylor n small likelihood least hull maximum satisfie lemma draw statistical extend manifold fundamental thereby support start computational proxy space investigate selection model uncertainty vary far briefly indicate geometry different differential geometry largely focus multivariate exponential differential geometry reference include include consideration important demand modelling highlight insight fundamental inference design approach geometric contingency hierarchical family connection wide algebraic well review paper objective tool distribution framework family geometry home numerically implement information traditional start act use selection conceptual goal detailed development find implement confusion name topic learn practice statistical handle naturally varied illustrate aspect development example geometric issue issue full family look geometry body key model sample space distribution dimensional space space three describe variable accordingly possible variable simplex together vertex structure act identify extend work comprise working represent geometry geometry important nontrivial many consider expansion curvature dimension reference briefly describe author one provide type datum look neighbourhood multinomial space place neighbourhood computational algorithm censor look continuous response censor survival section consider theorem curvature reduction illustrate model response direct fig internal hide member model apparent likelihood inherently appropriate developed case mathematically excellent foundation world make precision arguably fundamentally categorical object treat thus carry treat categorical fig add select bin day plot inferential choice explicit geometry relationship information discuss look understand closure embed limit estimate numerical discuss use expansion look geometry issue geometry proof discussion key inferential construct affine linear duality fisher affine call family pointwise construct geometry introduce construction introduce categorical addition subspace affine geometry derive characterize computationally family dual affine hard subset mixed parameterization furthermore inverse bounding shown extend first sparse multinomial embed maximize shape boundary third order rarely across boundary simplex extensively graphical multinomial geometry geometry fact impractical explicit close simplex panel probability vector line terminology geometry immediate boundary line lie direction show panel natural geodesic panel show relative straight strict positivity strictly simplex parallel line everywhere orthogonal panel line find base geometry limit parallel connect simplex curve closure limit parameter triangle explicitly structure geometry multinomial
shannon uniquely finite countable redundancy finite must exist cx qx evaluate positive sided exist redundancy redundancy eq example express adopt convention infinite minimax theorem conjecture parameter symmetry enyi divergence exist divergence eq hold see term behave evaluate obtain understood finite eq infimum reverse trivial capacity achieve require denote achieve redundancy theorem enyi order read may extend enyi regard order seem nevertheless order carry negative people may avoid negative order property avoid skew symmetry identity follow tend remain identity skew order application symmetry enyi semi addition processing infimum property notably remain true enyi divergence order exceed remainder proof skew symmetry enyi continuous p enyi divergence nonnegative follow symmetry pd theorem divergence enyi divergence enyi divergence number let convexity argument imply contradiction logarithm divergence construct converge topology enyi enyi divergence locally behave root triangle square review include continuity infinite order identity particular enyi leibler capacity redundancy channel order acknowledgment gr anonymous useful comment author van grant van originally perform degree position obtain grant scientific research universit paris france interest statistic include minimum learning divergence state sufficient thesis es es receive mathematic art science university work hold various position mathematics visit network business college journal theorem conjecture remark r r enyi kullback leibler shannon satisfie axiom kullback depend enyi equal leibl review property enyi kullback divergence limit order channel redundancy continuous channel leibler divergence kullback fundamental success many concept numerous never code enyi divergence es gr operational characterization enyi code compress ar operational testing divergence crucial recognize computation throughout hellinger enyi use enyi enyi entropy study enyi throughout literature establish treat divergence detail kullback leibler preliminary version independently publish work finite another read adopt space density sum member long set positive case definition enyi enyi shannon kullback leibler relation whenever interval enyi densitie enyi r divergence long compact relate enyi r denote let tending tend information recover enyi wide value enyi instead enyi divergence kullback leibl order define enyi distribution chain square np enyi divergence imply enyi consistent normal kullback extend r enyi divergence continuous space enyi via definition show enyi extend behaviour section convexity property enyi generalize contain several treat chernoff hypothesis application channel minimax redundancy order order divergence violate enyi enyi divergence present power presentation corollary often skew symmetry often p jointly jointly quasi let p transition markov lower upper semi semi continuous pair outcome separation eq chernoff side conjecture capacity redundancy dp q achieve countable distribution side carry often continuous measurable write algebra may interpret event respect measure absolutely may absolutely continuous common none result mixture countable restriction capital letter g refer take px qx definition order formula order motivate definition arbitrary formula define order sample generalize space follow simple define divergence variance definition ensure relation square distance distance hold order may change integration depend dominate case whereas respect may subtle consequence count eq direct argue ability enyi divergence eq inequality hold stem theorem form conditional result process mass induced verify consequently enyi absolutely proposition expectation theorem enyi divergence countable let supremum would enyi extend use inequality converse inequality bin let supremum countable enough supremum finite end suppose countable find inequality continuity enyi divergence define order divergence divergence enyi original always present exist enyi enyi aa equivalent turn order result follow verify express closed form extend verify sufficiently case exceed dominate convexity since remains p p also kullback leibler q doubly standard way continuous opt limit leibler proof q q hold hence alternatively restriction convexity derivative qp monotone convergence remain argument imply q q monotone together random px countable essential supremum ordinary supremum finite extend range finite aa q completes imply divergence continuity extended order enyi extend fix r enyi varied relation supremum partition convexity order finally various positivity suppose jensen equality extend qp inequality study property divergence order turn continuity equipped topology event topology topology strong sense variation topology countable enyi divergence function topology simple p q continuous semi continuous continuous semi simple imply semi property extend divergence topology enyi skew particular symmetric inequality symmetry hellinger distance equivalent total variation relation enyi enyi inequality x yx enyi divergence q qp theorem enyi divergence total space topology convergence expression enyi topology discuss weak topology continuity mean metric measure metric equivalent metric topology weakly need interior consist algebra apply proof book imply topology q consequently set compact convexity quasi enyi measure continuity relatively compact since denote restriction processing tend depend finite partition imply analogous sequence partition theoretic theorem may hold member part evy proof without probability let q np semi continuity enyi evy family imply similarly md theorem extend nx q jensen expectation follow last processing theorem nx p evy q uniform relate integral continuity probability term r enyi divergence absolutely mutually continuity mathematical tool absolute continuity equivalent enyi version absolute hold space event subsequence entire separation relate continuity mutual theorems theorem equivalent p p continue relate theorems infinite algebra np p call divergence sequence nn special enyi countable pair countable give eq extend observe theorem product distribution proof symmetry letting tend countable theorem raise let arbitrary question space n consequently repeat argument also state hellinger responsible hellinger integral r observation contain p parametric know regular taylor interior argue property aware exact technical probabilistic explain equal relate leibler involve q convention density define infimum uniquely interpretation enyi divergence kullback leibler divergence formulate already identity heart ar equivalently hence infimum uniquely secondly suppose consequently infimum equal either reader hence inequality follow converse infimum corollary concave wise infimum extend continuity concave monotonicity enyi convention addition introduction version survey distribution eq continuity hand lemma continuity
consider turn increment classic give remove fix instead cholesky generality inverse cholesky motivation appropriate detail notation estimate column notation kalman incorporate dependent front see role minimize barrier would like generalized composite function objective step objective composite wish way rewrite nn nn formulation argument propose apply matrix specifically cholesky factor give explicit setting diagonal matrix case factor nn eq block v twice continuously vx nn nn strictly entry write composite nn gauss newton form difference function criterion search whenever approximation sure component great derivation gauss subproblem assumption well provide estimate statement subdifferential order stationary point ignore rewrite portion sequence rewrite q exist eq subproblem rapidly motivate gauss input termination criterion step follow counter gauss newton terminate set return algorithm terminate finitely point point sequence subproblem dependent direction burden addition require always closed form q block algorithms recall equation q advantage kalman present simulate denote model main innovation smooth take cholesky measurement full period time discrete equally spaced noise true state vary situation phenomenon sensor report particular easily belief cholesky function estimate take behavior simulation extend kalman smoother thick red dot ground show kalman filter thin kalman pick last important magnitude make kalman fail know paper formulation model know variance control modeling rise structure gauss newton repeatedly extend kkt optimality preserve smooth inversion inversion definite immediate useful lemma corollary algorithm every require invert corollary terminate eq finitely necessarily contrary let triple condition multiply condition third condition find first q since subsequence generality q work ff take contradiction lem lem conjecture proposition definition medical trend filtering require know covariance matrix framework depend estimate gauss inference apply implement efficiency kalman approach synthetic optimization perspective include smooth nonlinear kalman smoothing system know angle interest kalman modeling error dependent bar choice tracking depend turn turn rate state noise portion idea motivate extension measurement variance
infinitely converge true sparse fraction force sampling population outcome period coincide sparse policy equivalence e observe appear theorem prove result period yield least true supremum bm minimization denote period force possibility q outcome period force order rewrite eq q law case exist flexibility construction consistent sampling collection force section refine notion affect furthermore sensitivity perform simulation convergence criterion almost sure convergence assumption unknown purpose outcome absolutely I e imply outcome population period period appropriately ensure overlapping exponent length obtain period slow intermediate difference period follow force frequent desirable population soon optimal population also frequently force outcome long period force rare converge linear solution intermediate preferable well estimation value avoid non population evident seem address comparison base region correspond scenario narrow figure accurate valid expect value long another issue arise slowly entire duration actually preferable average policy outcome complete policy incomplete information infinite horizon period exceed population outcome could use switch constrain policy horizon reward however constraint period behavior allow specifically neither consistent high outcome converge appropriate amount sampling budget long something impose cost constraint unknown asymptotic cost development sparse force period unknown period employ estimate instead compare convergence sense weak since exceed albeit importantly currently another towards instead assume distinct independent I markovian decision case importantly efficient policy extend technology research program author thank discussion remark gr independent population outcome upper distribution construct policy true mean compare policy independent objective maximize period infinite period exceed introduction introduce dimension tradeoff control incomplete population maker population well versus effort view problem incorporate mathematical programming allocation linear programming advance maker policy ensure learning area armed sequentially population order maximize horizon generalize construct incomplete simpler base policy normal horizon armed bandit model constraint bayesian propose heuristic estimating benefit computational analysis bandit bernoulli beta must allocate approach adopt approximation idea population randomization adaptively modify observe outcome period type population outcome markovian construction family per period converge information mean sense organize reduced cost degenerate basic optimal cost express linear combination define solution cost constraint optimal population characterization policy randomization rational specifically set satisfy incomplete actual admissible restrict attention policy depend outcome specifically let outcome period observation history dependent eq number sample period eq reward desirable policy feasibility feasible requirement first long exceed outcome per period
discuss de impose define us positivity ab plug begin bind k get count follow constitute upper principle simple generate bound look put together nan probability dimensional subspace satisfied useful section rule causal st test fail causal look explain direct vice versa action history influence formation mechanism take range tend prefer attribute formation step attribute influence however st reproduce coefficient influence st time consider node observe estimate causal give due calculate try binomial excess coin extreme comprehensive distribution plug test base apply confidence like hoeffding want check empirical sampling central verify expect average empirical less would fail reject htbp fig trait circle state repeatedly pick node appear tendency become explain static square red generating identify sampling fig longitudinal study type neighbor co worker identical section heart pair survey actor way median mention test include unbounded identify causal effect use rule relevant observe sequence material day combine analysis future observational social mark begin address response center modeling sensitivity central latent pose barrier st identify causal effect bound seem primary practitioner paper take tool geometry possibility long convex relaxation algebraic geometry make approach lp define lead optimization address ultimately allow causal presence come human every reason important obtain causal strength effect could situation whether model unbounded find law restriction public act cause correlation outline way statistically causal social study dynamic correlation common validity future effect heart equivalence exchangeability partially initial count call markov chain partially depend state transition average arbitrary transition matrix property preserve chain pe requirement chain prove property nan ask chain see notion exchangeability differ pe pe possibility sequence back see ccc satisfy long type solve exponential theorem long may another nice condition eliminate sec nonzero remain check differ typical conditional independent perspective order markov look solve influence otherwise think instant influence choice time influence previously delay model equality list read pick maximally violate particular delay acknowledgement grateful provide access thank definition test social trait demonstrate bound causal effect observational study unobserve trait assumption associate effectiveness previously methodology preliminary heart tie year constitute observational social review latent act impossible non necessary identify social intervention impractical even formation action measure central study social method bound limit well additional actor attribute time step formation attribute refer latent edge formation could actor attribute result rule employ shorthand capital letter ambiguity arise pa pa ab principle action surprisingly attribute presence mean observe pick determine term unique sequence lp bind become successively tight increase size lp representation compact g sx ix negative integer rhs quantity ensure positive require impractical become algebraic proceed optimization
initial low purpose gram threshold use refined simplicity still refine resolve gs reason follow generate vector nx screen refined repeat record hamming step gs gs use experiment due critical parameter satisfy study unknown open problem model fortunately study experiment specify amount procedure reason experiment assume know iterative screening tuning iteration refine iterative screening estimate comparison hamming contain goal experiment gs lasso investigate fix fix design block wise block pair iid sign equal repetition gs behave behave lasso recovery hamming error yield lasso prominent similar comparison gs interesting due become increasingly less strong gs become prominent screening screening screen gs investigate fix gs differently depend part experiment alternate adjacent index index equally setting block say ai j I average hamming gs outperform additionally gs effect grow gs prominent less include investigate experiment comparison use sub block combination choice coordinate hamming ratio procedure c c strength equal screening experiment generate way experiment performance use sign alternate across average hamming repetition diagonal first sub sub hamming repetition experiment generate function entry pre specify non value multiply draw hamming ratio suggest consistently htb experiment gs tuning experiment reasonably investigate mis specification whole experiment experiment use experiment experiment experiment independent repetition sub gs know rr gs experiment replace counterpart result suggest mis specification effect reasonably experiment mis issue use experiment result summarize experiment robustness mis specification experiment except sub experiment predictor presence correspondingly experiment assume iid sample df hamming ratio repetition suggest reasonably robust binomial v definition right recalling follow let introduce follow prove section corollary short elementary show corollary subset equivalent claim trivially condition corollary odd diagonal note em odd argument case monotone give argument hold equivalently case monotonicity inequality show case use second coordinate combine odd definition consider similarly j eq write remain strength remove generality assume definition quadratic rewrite similar achieve must satisfy v v claim show event p subgraph form support unique maximal subgraph event depend note I sufficient triplet eq basic realization I I p ty ty dominate basic non recall direct calculation em stage constant retain subgraph lemma hold remain omit hard notation connect integer let I exactly partition gs screening design empty singular score ty I basic direct ty iy j differently iv I definition relatively easier induce constant begin accept algebra I I second w magnitude algebra show coordinate finite towards end definition path connect path subgraph connect connect separate effect matrix path connect sparse g wise calculation attention apply subset subproblem gram matrix tune ss lemma suppose apply aforementioned subproblem ham r q ss last proof reason omit proof em symmetry need see hamming short region plane c calculation bad q equality rv v r j em acknowledgment thank ji nsf award dms partially grant dms dms grant research resource theorem theorem xx xx zhang zhang zhang row large regime selection challenge gram relatively naturally induce sparsity decompose subgraphs clean mle main innovation guide selection procedure hamming optimality minimax compare support demand hamming naturally error broad hamming procedure penalization utilize convergence simple tuning http www software gs matlab graph gs weak signal clean sparsity design vector th motivate big datum though take small proportion nonzero nonzero entry identify selection regime regime signal find genome generation sequence despite regime rare individually focus rare regime appropriate rare evaluate rare weak signal weak appropriate focus regime hamming throughout gram approximately factor row simultaneously find reconstruct gaussian often induce network gene normal concentration available security edge strongly key insight signal subgraph size decompose emphasis paper case lemma related discussion motivate screen gs clean screening identify let subgraph form large subgraph split many remove objective fundamentally correct rare paradigm gs diagram especially rare weak popular idea reduce noiseless model equation mild solution must viewpoint frequently variable selection design arguably vector idea penalization fundamentally computationally intractable selection provide signal signal decade tractable approximate penalization scad say upon component truth penalization fundamentally unfortunately framework signal rare weak fundamental uniqueness noiseless long valid signal many vector perturbation indistinguishable long regime principle recovery consider function unclear penalization correct rare even wise tuning ideally since benchmark development imply optimality univariate call sure well column univariate variable correlation inner corrupted happen snr reason univariate effect refinement screening paradigm scenario positively association test bernoulli signal said mostly screen stage challenge expect overcome screen straightforward integer consist series phase increasingly phase variable retain significant candidate screen unnecessary threshold test order control positive candidate order screen challenge computationally screen gs screening major carry gram node generic fix gs clean method consist graphical screening step step screening retain end gs decompose separately dimensional gs sophisticated note counterpart gs discuss gs step gs screening able aforementione sparse definition hand side subgraph arrange subgraph one tie hold certain ise recognize gs efficient screening first design gs overcome little situation long small moderately computationally feasible moderate explain first frequently span set matrix restrict row sub rewrite model key orthogonality test near orthogonality happen negligible effect gs able retain way overcome gs sub whole already retain differently discuss node gs step g j moreover carefully tune gs counterpart size complexity stage moderate organized follow gs achieve hamming rare diagram visualize optimality gs gs attribute optimality separable two main discuss exist extension gs proof notation use denote occurrence occurrence vector hadamard denote spectral submatrix form restrict column sorted write fix graph strong notation occurrence denote graph involve maximal call great gs section gs rare diagram gs penalization section gs gs step initial sub let list contain broken thought subgraph list step index accept currently update keep continue finish retain index keep forward principle depend subgraph order give different difference usually alternatively example step finish screening theoretic gs would produce result reason skip gs sophisticated choice screen screen tune properly sure screen negligible fraction signal say integer regression small solve unique case whole minimize eq vector gs gs gs paragraph list p k u p u sometimes design matrix gs main exclude obtaining contain gs properly choose two separately step computation subgraph fact least size subgraph graph subgraph computational gs contain part break component part broad design tuning choose probability total great complexity gs moderately use univariate screening complexity gs implement screen subgraph complexity latter multi gs model hadamard product see primarily weak constraint mainly technical reason need discussion let fix range range cover successful impossible recovery correlate fix design presentation translate design notation see gs fix introduce row unknown assume call compressive security fixing model increasingly necessary successful selection rare call many loss weak hamming asymptotic rare hamming respect misclassifie component misclassifie aggregate bind global configuration resolve exploit new favorable estimating geodesic distance j well favorable risk th coordinate define denote hadamard reject type ii negligible survival risk v shorthand stand occurrence denote q risk favorable overlap graph pick appear uniquely non empty intersection p claim hold model hamming calculation say one grow large derive conservative improve neighboring discussion holds replace elementary calculus v especially broad favorable neither gs need hold sub constant coordinate approximately equal gs subset choose step set fix consider model model model gs gs gs screening set h gs p interesting range properly h gs say gs achieve optimal adaptively note influence convergence remark narrow start influence choice replace base procedure accept parameter hard gs need pay primarily interested skip turn form strength coordinate model q approximately number selection small successful universal extend tight across suppose additionally j pi pi corollary rather relaxed surprisingly rate satisfying condition red red panel red middle part illustrative complicated part dash panel illustrate corollary together corollary implication diagram partition whole different region difficult almost full signal recover recovery minimax general last illustrate figure blue line get strong end integer follow p p side long hold partition corollary view assess weak phase partition region inference diagram region rare selection full recovery rare achievable high sense incorrectly region infeasible signal nothing sparse find boundary non recovery region effect area deal issue deep insight variable broad penalization select functional aic bic good theoretic computationally computational relaxation limit scad mc take select penalization penalization problem surprisingly optimality lack adapt design optimal subset selection save subset scad mc selector mathematical block eq convenience broadly consider optimal continue relax represent subset parameter ideally bad ideal depend exchangeable hamming scenario form p p follow model gs set subset set ideally h strict exponent expectation gs outperform selection representative prominent increase gs phase plotted phase partition region exact almost region interestingly separate last separate vary calculations elementary boundary lasso hamming much recovery slow subset selection optimal region subset adapt matrix correspondingly one subset separate ideally explain optimality penalization lasso remain rate save leave gs key innovation guide force computation gs exclude overhead obtain utilize sparsity design
write database program may likewise program characterize input endow database phrase characterization privacy differentially achieve whenever privacy important relation require take simplifie rarely mention quite randomized definition discrete subset topology completion field small open value sensitivity bound privacy mild sensitivity mahalanobis distance demonstrate privacy explicit argument dp let ds ds ds second third finite dominate hold bound privacy via output definite symmetric satisfie output dp lebesgue dominating consider eq exceed multiply leave zero say examine pz tm py c define final proved give privacy mild sensitivity global sensitivity nothing sensitivity mahalanobis previously definition provide sense almost private private procedure remain unable analog case differential adversary database note observe private adversary determine database set private comprise exclude concentrate obtain analog let achieve dp note measurable obeys constraint implication sense incorrectly reject release raise section measure function space essence countable deal dominate exist space differential privacy measure finite family cube dimension randomize algorithm nature describe set borel take prescribe b operation collection field page closure intersection namely contain differential hold sense eq argument approximate dp release differential priori release satisfie claim guarantee union extend countable due theorem countable set hold family satisfy form set towards every obtain conclude dc principle computer would privacy guarantee computer point continuous description borel field therefore function differential lead throughout summary every projection differential privacy every countable explore achieve privacy entail suitable demonstrate differential privacy conceptually appeal remain challenge may sense normalization time consume bring maintain consist subset hilbert k kx x correspond I closure present form require finite distinct release field mean reproduce private rkh density gaussian differ gaussian upper rkhs trivial process zero differential demonstrate assumption bandwidth rate normal respectively evenly spaced grid remain peak hold kernel kernel case positive require employ fix privacy establish sensitivity process usual depend rather order necessary differentially private composition typical way employ leave choose choose maximizer private citation compact priori rule determine private technique private algorithm range work rkhs live function space amenable demonstrate broadly use whenever rkh detail determine level necessary privacy rkh take fold tensor see detail reproduce completion set form obtain substitute complete define derivative gaussian isotropic dimension x dx dx dx choose lead technique attain ad grow high estimation privacy developed use desirable determining among error bound require differential privacy vector privacy kind take classification interval place prevent confusion admissible call whenever lipschitz demonstrate minimizer adjacent norm swap rearrange minimizer functional inequality use together combine turn loss function term plus find svm transaction alternative former request evaluation batch online online typically machine batch nothing multivariate finite evaluation condition q track request inversion answer request general time proportional problematic consider always span evident take position algebra whenever fall kernel mean sort increase sort doubly link list link list constant differential privacy preserve functional analysis theoretical address specifically ask release differentially noise preserve privacy address real count value technique case mutually absolutely bound complicated quantity kl matrix assume operator term nothing statement restriction subspace thm axiom thm pa usa pa department usa differential framework summary database discrete develop appropriate yield privacy rkhs rkhs kernel machine reproduce hilbert suppose consist measurement release without privacy rigorously basic thought induce due coin datum define single vast privacy develop boost parameter logistic svm possibly consist differential privacy measure norm sensitivity rkhs rkhs motivation value fold growth
various primarily minimax precisely magnitude feedback supremum infimum allow strategy feedback definition adversary interested mirror idea mirror discover descent idea see recover upper bandit armed bandit much one question exponentially forecaster suboptimal minimax magnitude bound bound introduce paper discuss average suboptimal online general bregman strategy appropriately choose section magnitude regret obtain bandit case discuss combinatorial simple combinatorial independent minimize perhaps forecaster strategy exp exp correspond expand bandit exp bandit strategy regret derive combinatorial derive see author mirror section obtain latter parallel expert question improve exp provably suboptimal exp defer coordinate bandit instantaneous bandit game ai ta bandit notation expect online mirror mirror descent mirror descent update formulation origin idea perform descent descent tailor properly strategy thorough treatment subject convex differentiable fx depend norm bregman say moreover understand space bregman divergence example study presentation suggest respect switch barrier convex hull section detail study case specific semi bandit describe inspire computational complexity polytope indeed step jointly get interesting selection ranking span acyclic graph also however note role moreover gradient loss detail conv np aa feedback feedback project weight improve satisfied unbiased satisfie bregman divergence w fa divergence fx obtains conclude invertible third technical difficulty bandit discuss prove regret word ti ti x define computation use fact one obtain note second already appear basic armed strategy expert inf forecaster logarithmic notion potential potential every define paper restrict potential simply potential regret hold pseudo bound order direct inf coincide basis negative recommend optimal direct u desire non fact let old combinatorial optimization feedback bandit sublinear exploration component algorithm exp play distribution exploration play exploration contact point polytope ellipsoid polytope feedback exp q next less argue conjecture lower correct idea limitation exp exist bandit infimum strategy adversary information conjecture regret bandit low conjecture contrary far self barrier polytope admit barrier self exist open problem sake multiple hypercube q choose disjoint second half coordinate parameter exist exp adversary regret lower define adversary q put coordinate choose interval interval begin odd vertex pick expert uniformly expect even select precisely select loss exactly add fix remain proportional consider different adversary adversary half matter round identity big sum low important conceptual contain heart argument bandit class leibl divergence prove bound kullback leibler simplify multiple identify word play game attention strategy arbitrary definition time adversary follow word adversary small expectation action denote player face distribution adversary play adversary loss coordinate adversary denote kullback leibler moreover symmetry concavity square q third I since deterministic empirical conditionally probability forecaster play adversary respectively chain kullback iteratively losse sum bernoulli distribution plus see sum conclude proof player plug fourth let denote randomization thank realization randomization previous apply probability c k numerator denominator differ show denominator use hence q numerically verify decrease come fact imply bernoulli usual abuse notation kullback leibl distribution easy consequence chain leibler jensen inequality introduce nonnegative ec grant paris est imagine fr financial university edu action maker difference realize pick magnitude regret assumption maker model bandit exponentially average provably suboptimal combine mirror inf forecaster able discuss lower bind optimal framework setup maker player
reconstruct transform autoencoder htb mi v one regard representation static contain transformation form assess multi capture hide temporal taylor three frame sample epoch equal present include generate frame result dimension dataset see repetition outperform identical give temporal representation achievable dimension along reconstruction single architecture mean frame delay frame delay frame natural movie investigate compare implementation pixel frame normalise unit initially data convergence full temporal learn temporal projection past visible activation plot run complicated unit unit visible layer unit past projection delay whereby delay starting point unit unit choose unit activation unit repeat map hide x layer define trace display likely delay represent represent delay temporal filter field model display row temporal see learn different unit delay force allow weight learn filter show forward select active transformation translation motion call motion movie evolution learn temporal interpretable help well understand effort adapt represent propose learn temporal track employ visual interesting generative support would thank universit department intelligence technology bernstein computational characterize deep enforce constraint little investigate might domain investigate temporal suggest temporal feature usually fashion barrier feature think yield advance unsupervise feature extraction center machine train large train yield unsupervise even set autoencoder boltzmann rbms feature learn shape system explain shape primary visual property natural redundancy rbms structure supervision structure back field find brain vision focus find natural understand filter develop restrict show able movie dataset machine rbms autoencoder prominent variety machine learning field paper briefly two consist visible visible representation represent activation activation autoencoder deterministic weight hide visible layer respectively perform layer mean reconstruction evaluate layer evaluate activation additional abstract representation rbm boltzmann conditional boltzmann restrict boltzmann rbm whereby feed include hide visible layer visible visible manner rbm learn dynamic box see denote energy q given denote delay amenable stacking rbms boltzmann contain hide connection visible step hide architecture rbm likely rbm model dynamical motion capture data model rbm temporal connection denoise autoencoder temporal mark able outperform deep natural gain insight movie possible
version auxiliary preserve probability new topic possible state represent allow choose empty count assign go return pool topic parametric author topic root guarantee topic topic j q assign random top level sample eq one work lda author represent yet necessary propose choose consideration wide serve prior article author present non parametric extension available possible overhead interest author attract model community seminal modification feature set little work article complementary represent topic much valuable extension allocation lda available two document via gamma word author seminal author unknown topic condition condition assignment author topic collapse markov x transition iteratively exclude variable di ki
time competitive dark gray gray image internal meet image represent pixel normalize range near neighbor local point random display randomly fidelity point top white identify perfectly average li segment perfectly report run pre density approach tb datum set handwritten multiclass interface set automatically handwritten project approach require graph scaling implement iteration per fidelity obtain four fidelity confusion good correspond digit computing large mistake try distinguish digit unsupervised laplacian ratio version normalize cut sophisticated setup eigenfunction self neighbor report upon require r interface method obtain local scaling construct adequate exploit multiclass ordering fidelity method condition configuration produce method fidelity representative hold rely less choosing label work investigate interface conjecture variational type exact nature functional support air office scientific pt pt institute mathematical sciences usa air physics sciences china rely quantify meaningful framework desire category devote cluster binary problem alternative interface equation phenomena functional generalize graph minimization cut sense graph diffusion function segmentation cast label combination ranking iii recursive partitioning consist successively cluster desire number reach datum considerable contrast propose interface simultaneous multiclass modify remove value smooth binary interface multiclass setup expression locate graph integer represent multiclass classification smoothness kullback expression involve interface binary multiclass base functional quality segmentation characterize state let scalar denote double minima gradient term variation potential hand adopt discrete label cluster minimize towards region double potential segmentation term potential interface goal transition interface norm slow interface approximate variation tv formulation produce interface transition tractable minimize approximate tv norm furthermore calculus tv interface graph undirected neighborhood represent segment vertex weight symmetry neighborhood set close feature vertex follow calculate I I express graph denote component connect among neighbor represent convergence limit allow label potential li integer periodic multiclass laplacian equation effective calculation class state small multiclass framework contiguous class label label compose compose jump analogous vertical jump interface reason interface undesirable manner require symmetry class determine thresholding boundary class integer correspond unstable well near half difference difference vertice state half whereas function correspond speak nevertheless regardless correspond used fidelity include little effect functional give represent ht dt rand nu u u half integer use detect change minimize term fractional gradient descent update preserve vertex represent fractional term summary multiclass interface segmentation consider real three class construct analogous binary half circle center bottom center circle
dms grant regularize cox existing linear model empirical risk function censor survival neither lipschitz negative partial likelihood iid derive asymptotic inequality lasso cox tackle iid model receive attention regression oracle inequalitie nonparametric include example provide g classification hinge loss regularize survival censor sequence iid covariate largely parallel material let covariate cox cox eq hazard likelihood log non ease theoretical intermediate replacement unknown population negative partial view loss cox excess desirable non inequality cox generalized nonzero iid intermediate lipschitz similar boundedness tackle pointwise type error negative additional discuss assumption assumption identical assumption assumption regression replace censor strictly subset k denote typical constant q k denote version cox base rademacher sequence fix supremum find calculation cox f b contraction follow yx f yx e k k n satisfy triangular inequality obtain f k pz jt nx jt directly pn e assumption e te ii ix ie u ix te xu thus theorem r constant let xx argument k te f ix te ix r mp n te r satisfy event te nr te mu nr lemma show lasso pointwise let assumption f impose ii f meet hold exactly meet pi bb w n triangular obtain remain follow lemma van de meet I pi suppose meet since meet exactly slightly oracle
avoid find consider consist nan repeating thereby seek exist find find common orthogonal extraction present n kk control equivalently component extract basically signal simulation briefly optimize optimal truncate svd page q fact tv converge cca sense form basis require actually restrictive datum high low dimensionality state end section subsequent reduce reduction indeed extensively column basis completely principal independent distribution otherwise robust pca estimate canonical give vector maximize high specify center end bound low high cca cca give wave standard normal distribution mix red extract project cca sense cca another component high extract state propose highly satisfy due relationship cca pls principle basically seek principal span similar whereas pca seek quite useful relevant signal alternate truncate frequent datum hence consume method intuitive large use big issue extremely large significantly random solve follow first use case may lie fortunately rarely examine orthogonality orthogonal want onto desire property uniqueness source bss bss bss mix permutation denote bss bss recovered mixture remain permutation ambiguity knowledge mix attractive extraction pattern source consequently mixture bss actually version obtain desire sparsity independence impose penalty bss link bss bss multi block link perform multi high correction require extract group correction word bss contrast link bss ordinary bss discover case require nonnegative run component common low nonnegative low nonnegative example rule wise division see detailed convergence feature helpful top analysis bss method however method process reduction task visualize low discuss discriminative neighbor etc much unify careful stage critical purpose extraction flow diagram common belong category denote extract matching share th evaluate accuracy achieve although improve version pca close see extract almost achieve separation particularly accurately efficiency individual specify actually second converge however consume instance consumption detect bound component intuitive nf different term observation multiply tend justify base extremely ray accurate detection energy ray image diagnostic task early unfortunately subtle separation diagnosis ray extract separate soft four source respectively component interference different uniform highly separate ica random ordinary nmf algorithm mixture run soft realization component extract satisfied extract desire propose nonnegative equivalently pca c information avg c c avg face human face datum face expression database gray distinct ten expression close dark subject position take different pose illumination expression process face image randomly repeat first split form face image different feature common number method influence time widely accuracy information compare improved cut justify completely due original cluster detailed clustering significantly investigate change stable parameter fortunately cause correlation become individual small propose describe two different individual analyze scale contain view space equally category category share feature different make widely adopt evaluate classify see object category compare three classifier knn classifier neighbor svm fold validation search interval interval database run require sufficiently ensure covariance train training feature routine split two image score classify sample accuracy derivation plot robust well among naturally extract orthogonal whether extraction investigate propose exploit common simulation show able real show promising clustering task study analysis remain feature important manually discover group also quite achieve common task object tailor task incorporate feature high extend feature promise direction restrictive flexible beyond block link often common feature time collect propose scheme common individual block discover common accord feature two algorithm extraction perform common incorporate dimensionality blind separation propose encouraging synthetic correlation link source separation increasingly science tool purpose interpretation retrieve multi attract increase attention encounter take subject various subject design collect individual device diverse naturally link share common due collect possess link devoted promising block canonical correlation cca propose maximize variable datum later cca datum blind separation extraction square pls maximize image framework tensor tucker decomposition therein method name variation simultaneously good knowledge fully study individual analysis multi block main include common detailed method cca discuss interpret high pca propose separation bss nonnegative nmf space separately desire block extract individual able rest organize extraction cca method discuss justify validity finally conclude remark suggestion set follow consist necessity assumption pca component ica method treat naturally component
summarize care occur horizontal axis logistic explanation h limit h rbf kernel eliminate cause huge selection prediction svm experiment cross validation fold cross accuracy validation illustrate th cross validation accuracy accuracy figure validation point one validation ten validation accuracy table lr std prediction regression respectively point model validation well result work actual contrary serious big hence prefer interval mixed analysis percentage composition mainly method svm lr expression expression stable real keep compare apply collect predict application estimate normalize select practical select variable lr one interval concentration respectively computational expand end concentration situation validation numerical get applicability svm lr practical security special purpose situation kind prediction true application induce lead consequence meet practical security like thank zhang comment company support cb chemical primary problem essential machine logistic lr lr get range meanwhile effect occurrence chemical process bring people life property occurrence chemical become lin classify year cause effect prevent occurrence include existence source control principle people naturally source prevent occurrence past since investigation come cause people get static remove turn prevent controlling reach certain range lot give fast mixed storage production influence factor pressure density h development intelligence chemical technique like artificial network genetic failure since happen machine lr efficient tackle decide intelligence interference classification accuracy logistic regression consideration lr clear explicit interval structure basic penalty comparison briefly give preliminary company essential control tend prevent lot medium concentration occur reach establish mining base prediction suitable know pressure special pressure room device concentration experiment density different point pressure concentration pressure whether occur transform another experiment several get pressure p average h c c pressure prediction svm lr incomplete variable introduce incomplete remove datum monitoring scale normalize avoid attribute object summary quality subsection chemical correlation variable quadratic variable original kind feature algorithm extraction transform original dimension extraction reduce transform feature subset variable variable slight characteristic effective knowledge chemical nearly consider serious reaction pressure take end procedure algorithm svm technique classification acknowledge quadratic generalization capability optimize margin briefly speak decision class kernel space view decision bias correspondingly hyperplane slack user penalty error change separation
multiclass adaboost mh mr work gibbs believe multiclass confusion matrix error error positive good generalization confusion pac heavily rely sum introduce bound obtain pac framework sec multiclass basic briefly main contribution bayes confusion future work present consider output class unknown distribution family set find make prior pac kullback leibler otherwise class transpose identity concentration self adjoint matrix case adjoint give symbol call maximum preserve hold element pac classification label consider learner aim choose bayes true empirical bayes classifier risk classifier predict draw accord return true bind pac theorem gibbs eq ab b bayes consider matrix say multiclass learn act sample example draw I example class context confusion consider confusion build upon inherent desirable f correspond confusion example zero outside recall objective learn generalization guarantee give conditional correct consider kind discard element confusion misclassifie confusion every sample confusion confusion equal control confusion classifier task precisely aim confusion small small main multiclass prediction confusion difference remark might confusion p inequality norm op risk formally relate confusion space family example belong defer q inequality fix estimation gibbs classifier upper risk gibbs depend minimal close empirical confusion matrix risk gibbs bayes way proposition well multiclass measure risk inequality defer imply confusion bayes relate defer appendix theorem deduce confusion matrix classic generalize carry adjoint central namely confusion rarely scalar almost surely scalar self adjoint matrix preserve eq sequence fix adjoint refer I ia invoke confusion matrix confusion iy confusion every naturally p equivalent clarity give example one verify notation demonstrate thank note value random negative q second give recall leibler let last complete theorem lemma jensen f calculation leading simplification equation right obtain remain substitute focus give multi class boost mh adaboost mr adaboost take theorem confusion may risk bind instance depend weight vote similar would algorithm sound would probably besides might might kind extend possibly also propose new pac bayesian classification confusion couple confusion confusion concentration self adjoint matrix self adjoint empirical corollary risk empirical moreover classifier interesting belong e confusion gibbs true matrix
eq total mass em theorem form ranking I order example top book publish week york ranking application voting discussion rating product player ranking stage item list appear select proportional denominator item item sensible pool infinite list build work pool rating parameter probability say case formalize representation atomic measure note item probability normalize draw first remove pick basically partial biased problem particular gamma process atomic marginal introduction suitable law give accord simple gibb develop effective time aforementioned start describe take infinite nonparametric operate throughout paper ranking simplicity length item pick th stage interpretation follow item arrival arrive em apply derive ml multiple ranking difficult directly alternative parameterization item shown factorize k factorize gamma conjugate w km carry vb algorithm update occurrence item appear ranking distribution instead previously update rating one model appear ad hoc sampler structure infinite object show gamma choice item random atomic measurable atom iid base density poisson concentration parametrization view easily extend nonparametric partial item arrival item arrive arrive give arrival joint gives mutually exponentially depict visualize top list visualization sort order consider partial ranking consist atom list observe item express law gamma model auxiliary inter arrival times list f generative item unique ranking construct marginally suppose atomic random measure law suppose law gamma conditionally conditional law coincide conditional law tc maintain reverse atom interpret strength dependence measure examine mass mass atom never atom base atomic atom give recurrence trivially extend partial ranking size extend sampler item atom atom propagate time posterior item item define write fix atom atom contiguous interval outside variable gibbs proceed update condition latent note extra integrate directly along parameter previous constant desirable evolve measure measure distribution define able except advantage remarkably discrete section parameter applicable apply discrete york category book list correlation assign account publication date book burn popular book category book appear book book list represent enable category characterize quantify pn estimate respectively atomic generative exactly characterized stage generalization process also continuous york useful density insufficient list path book prior publication evolve event foundation totally neighborhood instead notational work
age true order random nonparametric aim simplest suitable diagnostic simply explore exhaustive nonparametric conceptual simplicity practical popular exist beta roughly speak although truncated age set age counterpart bandwidth discrete beta parameterize accord kernel bias commonly offer neither smoothing technique note kind find commonly cite exploratory tool number time smoothing regression organize estimator adaptive bandwidth aspect preliminary logit computation pointwise interval section relevance package datum section kernel age hereafter since age equally spaced mean move hx define parameter particular dirac delta uniform effect smooth spurious fine get large speak beta possess characteristic firstly automatically change graphical display match age assign outside order near property counterpart discrete restrict vary reliability measure convenient way thus reliability closely reflect reliability small smoothly old characterize depend reliability sensitive reliability number adopt reliability local extreme reliability old case ignore reliability give estimator calculate age age beta vary mode exposure risk death age allow likelihood reliability variability relative variability x take exposure bandwidth regard bandwidth drive vast without simple validation simultaneously fit compare omit description validation minimize xx age remove difference commonly write notational compact rate parameter consist last simplify qx qx perform available plot obtain par par par par par par par old interest cross validation video default six observe rate fit plot rate cross validation residual prominent visible know literature especially probably activitie notable simultaneous representation statistic residual residual display r arise fitting rate minimize cross statistic residual inspection add say need create ex alpha produce plot bandwidth pointwise pass code argument default specify confidence argument allow pointwise interval naturally alpha produce plot note manually pointwise interval manual specification pointwise confidence useful take code bar display great exposure usefulness change visible range bandwidth ex alpha par par par par follow cross residual specification estimate interval code ci observe one one logit bandwidth vc alpha par par par par par par par par par par par flexibility result ht residual apply confidence compute preliminary logit specification joint logit beta e e summarize quantity package package conceptually easy nevertheless several option among formulation validation score minimize may residual plot either confidence interval
characteristic cut regular see follow bound connectivity scan scan statistic bounded let spectral scan symmetric solution fact supremum convenient derive single component alternative refined hypothesis scope article indistinguishable rely square supremum heavily reduce expect recall generic intersection intuition behind spectrum phenomena statement concentration result univariate asymptotic spectrum topology property asymptotically distinguish attain previous logarithmic attribute generic performance estimator naive detector reject reject distinguish examine take result stem distinguished contain balanced result log factor analyze detail topology lattice kronecker scan depth constant signal least cut binary tree spectral scan signal snr strong versus give estimator figure subtree level size gap regime structure improve power dramatically lattice vertex volume laplacian root bound low statistic test acknowledgement support grant nsf grant statistic sufficient normalize constant nan eq statistic function set proof optimal pearson lemma alternative parameter snr indistinguishable versus consider error vanish testing testing remark result distinguish fix let theorem write contain eigenvalue vector use let show intersection ellipsoid unit ellipsoid supremum supremum simpler able hold recall supplement reduce q distance orthogonal dimensional subspace orthogonal fix centrality symmetric minimal alternative square tail proposition test corollary study spectra really notably particularly give algebraic balanced depth small give characterization characteristic characteristic satisfy general balanced kn h completely characterize kronecker graph cut cut cut edge cut enough control spectrum kronecker base sum eigenvalue stochastically bound choose geometric change piecewise analyze generalized likelihood ratio statistic relate find laplacian call statistic depend topology theoretically scan outperform naive thresholding concern fundamental noisy observe activate comprise induce highly variety scientific area surveillance inherent difficulty specific condition algorithm anomalous scan statistic intensive entail individually anomalous graph understand determine computationally tractable cut believe realistic objective cut np mind relaxation call spectral laplacian importantly derive performance scan combinatorial estimator main tree give logarithm spectral scan balanced trees model verify scan dominate simple contribution define cut reflect way topological show cut develop tractable statistic scan statistic performance scan explicit scan notable topology superiority detector thresholde formulate problem realistic scenario involve composite literature subject relate fundamental statistic portion area incorporate life problem normal mean problem generalize test graph unclear extent topology theoretical mention feasibility fail match subspace problem cast nuisance structured statistic formalize change observation possibly nod eq constant within formalize write thought nuisance independent noise snr true cluster bi partition easily formally c interested problem gap regardless value cast structured composite q join alternative meaningful detection separation hypothesis analyze asymptotic testing sense far though explicit establish snr spectrum hypothesis distinguish limit asymptotically indistinguishable exist notation terminology theory central combinatorial laplacian adjacency matrix diagonal degree denote take eigenvector algebraic connectivity study asymptotic testing unbounded nuisance comprise composite hypothesis index feature apart
player expectation take internal adversary move end performance player dual play information loss specific bandit feedback start expand exp geometric expand applie assign weight action use armed bandit loss vector mix exp choose uniform action idea geometry finite exploration attain factor moreover hypercube exp linear bandit expert without action regret say provably suboptimal strategy order address class study mirror paper grow rapidly set observe understanding mirror adapt suggest basic universal picture semi feedback bandit feedback see mirror optimal strong fundamental bandit successfully apply mirror descent seminal compact descent barrier barrier hypercube suboptimal ellipsoid note implement mirror polynomial contribution canonical correspond euclidean euclidean efficient ball euclidean ball euclidean ball study paper exp stochastic descent regret exp show corresponding discuss briefly section show computationally hypercube briefly template exp mix np bandit scheme let round play z bandit feedback exp several equivalent way write usually implicit regret recall bregman write action differentiable proof randomness induce easy exercise example indeed differentiable differentiable mapping computation bregman exploration propose first online feedback use geometry ellipsoid minimal scalar exist contact simplex preprocesse action follow first combination space work ellipsoid ellipsoid minimal x translate everything product loss play ellipsoid ball xu drop ellipsoid product slightly account product eq also estimate outer mapping valid tp moreover clearly unbiased exp product easy bind p ta remain let thus conclude ta proof small eigenvalue one conclude use discretization argument exp use compact set ellipsoid basis specify factor ellipsoid replace computed efficient implementation lead programming consider bandit expert exp suffice turn preprocesse build correspond straightforward detail omit time observe loss contextual notable special armed suggest corner algorithm achievable simple bandit expert restrict attention obtain weight distribution section regularizer random perturbation minimax interior sign rademacher check perturbation optimization regularizer exchangeable hessian precisely follow differentiable thank theorem term involve bregman divergence computation obtains prove fact inequality satisfied v consideration identity elementary conclude j ti ti e p tp remain small inequality restrict attention action exp problem one attain regret order precisely motivation regularizer come perturbation interior bernoulli else easy check perturbation modify key figure
q diabetes version appear ds relatively ds lag thresholding property mild prove lag consistent lie lag discover control noise proof theorem parameter play effectiveness lag lag adaptive provide well one let variable effort mainly divide stream procedure aic show nice property impractical shrinkage algorithm lar computationally favorable introduce gap discrete selection attractive lag application lag improve group lag path thm thm mathematical stanford partially support wang fellowship call selector estimate nevertheless program control weight lag enjoy attractive lag thresholde property asymptotically consistent discover lag identify predictor computationally simplex lag superior compare soft mean gaussian tail predictor initial bias enhance explanatory play ideally complexity coefficient eq coefficient omit keep coefficient one zero discrete process subset limit large convex also zero substitute net lasso selector relate soft processing hard soft select model spread attempt scad penaltie eq sparsity concavity show enjoy variable relaxation optimal short least selector discrete influence program q dependent surprisingly geometrically analytically demonstrate penalty choose hard cancer rest organize present motivation lag connect lag highlight selector provide section discuss lag demonstrates understand design orthonormal ordinary ol notice operator jump soft p coefficient hard regression interval change interval pseudo function motivated lag matter selector write equivalently another hence expect choose name lag gradient zero mean substitute heuristic mean explanatory variable less otherwise abuse pseudo hard ol suggest ols detail section lag formally redundant column furthermore condition c condition actually covariance us define eq c lag consistent hard property proof role effectiveness pseudo interpret lag ty iy flat break otherwise direction break case still minimizer even illustrate toy minimize break similar increase exponentially provide discover true lag move break another breaking fortunately lag possible selector ds similarities lar numerical inferior penalize pc nonzero coefficient significantly stand indicate uniformly nonempty condition lag identify probability ol accuracy lag quantify lag program condition b analogously take section tx upon contradict b imply hard thresholding eq therefore w lag normal hence enough prove py denote cc tx c mean two satisfy subgradient q lag convex lot interior selector lag reformulate solve lag equivalent linear constraint linear program relatively recommended routine absolute lag pass routine treat response lag need criterion validation solution warm justification technique simplex try adjacent current one new solution find thresholding property cancer weight age logarithm amount seminal percentage score set profile lag quite predictor exclude almost discrete lag penalty indicate jump jump include segment include unchanged profile role cause however role right lag piecewise constant demonstrate trend though prediction lag contrary variability diabetes predictor fit lag compare lag lasso lag lasso fold table lag variability parsimonious lag nonzero product lag effort notable magnitude nonzero lag lasso relax relevant consequence predictor weight select relatively argument verify
exponent ensure exponent measure affect high measure affect major guarantee high measure alternative obtain sequential estimation I measure firstly constraint problem estimator max I extreme dependence root square cube size come property denote constrain magnitude support strong percentage exponent htbp cb c integrate square deviation benefit unconstrained improve exponent bivariate conclude limit explain estimator exponent measure constraint beneficial max superior especially moderate dependence large improvement high measure promise acknowledgement financial definition proposition extreme link dependence margin inequality order exponent measure measure subsequently exponent impose estimator max stable exponent inequality estimator stable arise appropriately scale maxima identically throughout algebra vector fr margins max simplex satisfy margin due loss fr margin place structure copula function invariant transformation practice random dependence stable value back max stable positively imply additionally max strong dependence pair non review property max stable distribution although aforementioned dependence class distribution implement paper incorporate max constraints essence dependence stable inequality result nonparametric exponent satisfy inequalities simulation study conduct assess performance arises appropriately scale consider identically fr margin representation exponent extreme w jj exponent measure max stable completely eq measure describe completely margin exponent measure additionally homogeneous I homogeneity property dependence quantity define maximum measure term complete dependence trivially interpretation coefficient inequality bound set stable terminology refer respectively set yield satisfy ease notation set positively dependence property max tight bound high coefficient easily inequality analogously terminology introduce subset ms new exponent negative uniquely function set exponent negative line simpler consistent consistent modelling model limit extreme value identically distribute vector unit fr margins normalise maxima cumulative fr dependence estimator
name allele family identify assign classification tree describe name name recover insight drb assign drb assign give broad type type assign drb broad type medical biological assign solely dr dr dr dr dr st st st st assign name allele drb confirm reasonable check cluster make st allele cluster tree contain drb drb group drb drb allele dr broad dr dr exactly drb broad five dr dr dr drb drb dr drb drb drb drb classify drb drb allele table classified st drb drb drb dr consist split dr dr dr drb drb dr dr broad split dr dr st consist st drb drb drb drb consist drb st exactly drb st small drb drb consist dr broad assign confirm st st st st classify nine classification structural experimentally bind binding overlap classification group propose synthetic panel et seven drb drb drb drb drb bind specificity group nine algorithm result classification work classify drb et seven similarity study dr solely datum drb drb drb drb drb dr drb dr drb drb drb dr drb drb drb dr drb u drb drb drb drb drb drb drb dr drb drb drb drb dr drb drb drb drb dr drb drb drb drb dr drb drb dr dr drb drb drb drb drb dr drb drb drb dr drb drb drb drb dr drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb dr drb drb drb drb drb drb drb dr drb drb drb drb drb dr drb drb drb drb dr drb drb drb drb drb drb drb dr drb drb dr drb drb drb drb drb drb drb drb drb drb drb dr drb drb drb dr drb drb dr drb drb drb drb drb dr drb drb drb drb dr drb drb dr drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb dr drb drb dr drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb dr drb drb dr u drb drb drb drb drb drb drb drb drb drb drb drb dr drb drb dr drb drb drb drb drb drb drb drb drb dr drb drb u drb drb drb drb drb drb dr drb drb drb drb dr drb drb drb drb drb dr drb dr drb drb dr drb drb drb drb drb drb drb drb drb drb drb drb drb type column remark stand marked expert indicate allele shorter c dr dr dr drb drb dr drb dr dr drb dr drb drb dr drb dr dr drb dr drb dr drb drb drb dr drb dr drb dr dr drb dr answer final statement question bind otherwise study look comparative scheme n extend framework protein suggest hope think chain author drb contact drb sequence marker suggestion helpful evaluate influence bind adjust topic future improvement explain capital h lc l see recover vector obtain coordinate precisely www toolbox composition mm mm mm proposition edu live com edu edu kernel substitution use machine allele allele dr allele classification relate chain biological offer simple powerful scientific effort certain string bind classification major complex place substitution simplicity find gap help bind bind bind hadamard power square contrast machine former emphasis describe chain inspire local vision begin finite may think positive definite let basic life protein string definition substitution align represent think symmetric addition check let next index x still recursively string q string kernel chain define chain abuse count occurrence occurrence kernel normalize correlation basic sometimes work place alphabet penalty even gap numerical experiment indicate context reason one limit choose protein protein realization individual protein set related ii drb simply drb subset play central bind body chain allele strength bind string substitution cross also span contain define vector refer ii available operating curve roc auc binding refer contribution nn auc drb drb drb drb drb drb drb drb drb drb drb allele square list auc use thresholding bind non auc mark weighted weighting size set contribution sequential generalization bind allele allow detailed drb dr groups important coverage gene variant overlap bind group accordingly bind specificity health organization assignment base work throughout world international exchange et assignment especially certain drb drb sequence exclude marker drb allele allele locate marker location occurrence location last occurrence reason marker constitute drb encode bind site reason position allele occur range allele transform normal different order collect normal form list since may impose call derive sequel family exclude drb family cluster construction clustering separately drb drb drb base order base instead cluster st st st st st st clusters play neural network sequence type check application string motivation relate string function bind work similarity powerful bind affinity value available output function bind allele take string symmetric call definite positive definite kernel subset kernel also x iii product follow normalization function space euclidean product linearly general kernel refer finite reproduce reproduce hilbert notion usual define example alphabet name semantic collect align find region greatly block block occurrence indicate frequently another eventually normalize indicate occurrence construct qx take logarithm round simply obtain analogue spectrum hadamard matrix positive entry hadamard hadamard logarithm value symmetric hadamard hadamard logarithm conditionally hadamard logarithm define fact follow iii imply verify hence index k semi index index count explain sum give cl definite elsewhere since know k positive discriminative string stand review write space call represent linearly coefficient see underlie sense run guarantee fold validation suppose partitioned assume paper fold one train division division compare define special refer leave cross tune affinity strength ic score ic ic determine bind bind bind ic bioinformatics usually score introduce ambiguity sequel ic bind affinity benchmark publish cover drb representation bind allele compare state art nn align allele allele step leave partition five fold merge training leave cross validation every geometric optimal predict bind aa pp since affinity label datum divide bind allele define allele value bind matter hence measure bind auc table relate reflect modulus dp p list cc allele allele drb drb drb drb drb drb drb drb drb modulus continuity big neighbourhood bind motivation allele bind huge phenomenon refer job experimentally allele put bind datum datum allele help sequence use step introduction sequel parameter define allele kernel bind bind affinity choose allele reduce allele allele kernel allele drb bind affinity whole divide follow contain datum suggest index define five validation division merge part testing leave cross validation employ except test auc affinity transform show l allele rmse drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb weighted auc row mark bold use table table allele rmse auc allele rmse auc drb drb drb drb
basis spline find short severe reduction burden five level difficulty task compose essentially experience distance go minor member help effort compose task assign percentage difficulty difficulty illustrate shape discuss base posterior quantile plot covariate equation gender find negative imply bad condition line attribute disease posterior estimating count age use easily report severe cv quantile summarie entire aggregate age age group assign reason variation specific depend course age region difference population two influence evaluate classified fourth difference probability extreme curve age group l age call cv cv obtain motivate estimate population survey latent addition use unit latent proportional odd mix whole together covariate adopt estimate spline allow propose basis inclusion number knot p keep low avoid recommend spline simple people categorical person accord survey design reliable quantifying amount challenge adopt base estimation latent belong change influence penalize basis show mcmc posterior estimation spline base health survey task responsible health organization planning classify monitor people medical investigation medical person request provide classify population people public department important alternative hoc survey expensive survey statistical proportion european source health condition survey national institute every year recent employ account international world health organization evaluate activity survey person exclude degree difficulty person even aid item maker perspective definition loose tv burden health mainly plan status dependency approach measure score equally contribute measure may obtain functional national survey classify different classify extend tackle locate center large proportion region health department responsible health organization planning classifying level health age label probable latent within health exploit variable range population count population recent rao challenge latent hide area develop item summarize datum probable membership dependent value latent covariate error propose tackle classify get area hierarchical status capture form influence therefore mixed covariate spline spline area use cubic spline radial spline basis consequence provide via spline spline random mixed representation spline basis property behave chain time datum spline specification final procedure selection checking provide conclude population complex follow consist large self consist secondary divide population proportional systematic sampling member survey live involve divide health group age domain make technique account international organization conceptual health loss restriction lack ability perform manner disadvantage prevent depend large difficulty person aid item type accord difficulty movement difficulty one home intend people difficulty show problem rest bend something difficulty daily concerned independence basic get tv volume aid recognize ordinal difficulty group four categorization category home go capable effort lot effort capable category activity get category effort take category category volume capable capable speak use tool perform assess item latent conduct line nine provide particular remove raise merge cognitive addition redundant discrimination latent reason employ reduce nine item aforementione first continuous logit increase probability equal interval effect age functional assume smooth compare parametric approach functional form relationship estimate relevant application interest specific despite numerous area model inclusion area incorporate connection spline spline mix incorporate nonparametric mean unit spline function cubic spline radial spline knot desire make little property p spline framework thin spline avoid coefficient penalize shrink accomplish thin mixing eliminate posterior step implement ij td let triangular invertible cholesky decomposition symmetric transpose ease arrange fashion nature identify orthogonality order intersect axis transformation eq q w rewrite w c rr need b item select level consider prior informative computationally convenient flat real assume health effect independence effect representation thin equation normality hierarchical use spline effect task critical sensitive influence smoothness root component choice along exercise area obtain software count different age combine draw count specifically count small index run subset case deviation quantile summary proceeding choice specification multinomial logit vs purpose nonetheless model problematic consider index membership estimating count draw frequentist cdf cdf information use left gain emphasize propose tool people characterize level age health goodness fit clear interpretability pattern count estimator careful consideration selecting reason suggest class adopt lead class interpretable class approximately class interpret sample lead unstable count small area comparison odd specification model number class odd exclude status covariate give contribution slope parameter around knot spline keep model place value sensitivity exercise interest rule smoothness spline sensitivity exercise section proportional odd class therefore age three popular see detail parameter
consider kolmogorov type statistic xt asymptotic take schwarz c hx formula simple sequence research author grant grant definition xt goodness fit test statistic characterization family belong deviation asymptotic nan efficiency parametric goodness one problem distribution function family inverse pareto often g service period system economic pricing reliability goodness test far test base completely way introduce monotonic characterization characterization find family state goodness general alternative also introduce resemble statistic describe limit efficiency certain need rough large deviation asymptotic refer efficiency kolmogorov hence applicable describe value kullback may n statistic change non statistic degenerate first give theorem deviation degenerate get sufficiently pr generally alternative three contamination alternative density need kullback nan consider nan composite establish general dy elementary calculations eq eq efficiency moderate maximum equal slope admit kullback leibler third obtain therefore integral locally kolmogorov statistic depend calculation get projection point kernel degenerate limit develop show complicated distribution random find statistical modelling statistic bound use deviation degenerate sufficiently eq q calculate local f deduce formula representation eq kullback large admit asymptotic leibler local alternative slope statistic admit know information efficiency alternative see goodness fit testing section local sequence maximal local
rating aspect accurate sentence simple explanation aspect highly correlate I star refer unlikely rate rating text relationship aspect depict conditioning encode eq encodes penalty star vote co occur star vote prevent unlikely possibility occur eqs minimize used multiclass dependency rating section aspect corpus require choose review accuracy fraction rating scaling range large experiment hour randomly test corpus label algorithm test try baseline middle rating prediction high square show seven semi supervise upon learn fully supervise possibly due unsupervised english simple merely english supervision review address seed supervision baseline train correspondence sense occur user overall toy review acknowledge aware million require hour train dataset outperform supervise label unlabeled explanation semi optimize likelihood fully optimize certainly technique supervise extend lda topic lda unsupervised supervise like unsupervised achieve close five dataset consider aspect rating aspect rating task review naturally evaluation train using predict rating rating prediction rating rating per sentence review rating inaccurate capable predict rating text fact model outperform aspect predict rating text fail explicitly largely address combine rating text decrease baseline combine rating level supervision impact rating surprising affect significantly case rating prediction aspect label inaccurate unsupervised learn rating ultimately predict must rating mean rating rating task intervention aspect sentiment model learn match aspect high sentiment parameter confirm sentiment expect role different study old warm discusse interact decision sentiment separately aspect explain star normally ingredient mass account review surprising among introduce corpus million source system provide rating multiple product sentiment determine part review rate aspect review highly sentiment readily corpora yu intel fellowship microsoft fellowship online consist plain text together understand aspect contribute well know help product paper build rating system separate aspect introduce corpora review rate six prediction task rate aspect sentence rating rating dataset recover rating art scale moreover content automatically content well aspect task product sentiment make mean evaluate describe feature naturally answer different website opinion influence may body body refer aspect answer rating aspect aspect provide rating opinion figure tree dark light head excellent dark light thick body dark finish presence actually nice ccccc consist text one example six sentence look overall rating form supervision sentence discuss rate aspect medium thick warm may describe choose review well explain rate third rating content goal corpora million review interpretability modeling approach aspect separately describe interpretable sentiment new name short introduce corpora million review rate three handle dataset training predict supervision aspect supervision manually label addition unlabele supervision label human ten corpus find benefit supervision aspect sentiment discover separate review aspect recover find require explicitly similar author aspect term aspect paper discuss though sentence multiple review aspect rate recover rating discuss unlike topic topic aspect aspect sentiment review body describe aspect describe aspect manual intervention domain critical due complex five multi ten manually annotate million review period time adapt throughput highly aspect highly review I exist source much simple approach lie model contrast learn neutral word aspect simultaneously learn sentiment aspect contain consist plain feedback score many application include discovery sentiment among understanding demonstrate well task nothing good user preference nothing corpora approach topic frequently interpretable representative attempt way aspect multi aspect rating corpus identify highly correlated vote assign aspect sentence deal examine explicit aspect rating aspect sentiment assign sentiment corpus review manually specify sentence sentence publicly rating many system rating rating model model relationship work unlike whose aspect word distribution separately word sentiment aspect review considerable manual intervention learn automatically language review finding consistent aspect look amazon audio rating user aspect rating aspect rating review rate price service product review quality category user feedback rating product author dataset cc aspect price service briefly provide rate obtain use review label aspect rating aspect determine human sentiment label use sentiment label negative neutral index brevity wish label manually label review sentence irrelevant annotation crowdsource amazon review sentence coefficient agreement unfortunately annotation correspond random sentence evaluation address crowdsource service individual expert question approximately hour expert score score review expert review corpus review corpus annotate corpora expert publication model aspect word appear sentence simultaneously learn rating might might word appear discuss high word sentiment intuition closely corpus review divide rating overall aspect though aspect sentence sentiment encode discuss aspect index aspect aspect aspect aspect word star rating use expressive power alone separate critical would find sentiment word beneficial sentiment aspect aspect sentence independently entire learn aspect maximize learn scheme level supervision increase supervision must match aspect optimal highlighted segmentation task five node five ensure remain unconstrained include aspect aspect aspect rating unsupervise latent aspect assignment ascent optimize merely consist maximize sentence concave proceed square ascent sensitive aspect initially assign aspect name aspect parameter initialize randomly high among subtract constraint interpretable issue encounter aspect assign single notice word aspect regularizer perfect word reduce reduce address need prediction encode aspect enforce aspect subject aspect discuss application example encode must
setting however slow arrive finite certain assume many fall see quickly past major direct estimation space among aim direction collect give value temporal difference bellman bellman temporal basis function neighbourhood dimensionality technique construct td bellman angular provably approximation extraction online extract binary function approximation keep candidate combination td generate small td simplicity regardless bellman feature space like assumption space bellman assume linearity bellman derivation projection projection norm preserve class type projection scale project appropriate target exponentially immediate bias induce sparse space probability ok linearity help error fit linear value mild approximation bellman angular estimating bellman discuss reduce control proper return thus prediction error much compressed light result propose simplify variance simplify version compress weight eqn error determine subject projection guarantee algorithm memory discuss fast robust respect gradually complexity adding stop point easy set generation validation start provide tight work bind suffice bellman error td compress eqn let collect td bellman linear detailed appendix sketch bellman control constant transition third simplify concentration norm expect projection preserve bind due variance set trade compressed discuss arbitrarily bellman policy estimate careful benefit state contraction measure bellman satisfy add either challenging goal optimal rp far direct regression original future poorly small importance type sparsity randomize extraction e projection projection compress apply state size compress rp method choose consistency initial guess fail exclude address rp rp rp error rp among fix vary number iteratively minimize method small compare fit solution seem two minute work half million sample space prove projection preserve linearity generate projection reduction analysis projection memory behaviour happen common empirical analysis confirm extraction projection avoid car regularization rl complex course iteration validation selection tight theoretical help analytical closed answer question linearity bellman avoid linearity simplify concentration rapidly mix give base time markov measurable hide need characterization fast past transition kernel ia operator dependence concentration bound large value small chain markov uniformly past major result consider generalization inequality random homogeneous follow application proper constant write td bellman error term series martingale point bellman wise td bellman regression lemma term theorem prove lemma probability define x sum vector chapter satisfy union equation line set substitute simplification probability bound inequality base variance exist element element taylor expansion inversion inverse column see thus orthonormal basis union line second apply lemma line finish proof finish lemma bellman contraction contraction due corollary cs automatic bellman function evaluation function rate logarithmic error parametrize estimating generation reinforcement lot aim bellman current estimate bellman exact unlike fit iterative space feature generation bellman method rl bellman error fair many fail use project exponentially difference help determine size discretized code feature many feature iteration logarithmic time provide minimal code type also need
problematic markovian model trajectory backward simulation section detail novel backward backward development truncation non markovian generic applicable study severe accuracy simplify slightly extend version unnormalize assume would particle time independently proposal kernel implicit write adjustment auxiliary sampler weight sampler construct kernel validity assess mcmc sampler extend smc auxiliary variable target construction admit density indexing point notation x sequential procedure sampling appear know form smc index maintain furthermore sampler straightforwardly never sampler incomplete gibbs ergodic chosen however observe especially poor degeneracy cause collection new fundamental gibbs value draw verify correspond collapse leave plug numerator realize difference introduce break strong set ta tm mx x suggest far explore explicit pass discuss problematic markovian distinct forward sequence step pg backward simulation via problem markovian show compute backward computationally prohibitive backward sampler markovian model make progress markovian markovian hence possible replace follow backward backward fx sl c appendix compute backward weight result quite even accurate inferential know truncation weight evaluate start discrete tv truncation level exponentially decay move remove requirement specify introduce parameter easy change whereas stop cost computational trade well illustrate typically evolves figure example right adaptive early still line overlap dash markovian efficiency sampler state apply smc sampler rao approach rao since marginal process nonlinear density fix trajectory kalman backward particle smooth model evaluate backward weight thus kalman filter complexity employ truncation computational backward generate dynamical naturally model state admit dominating consider smc sampler straightforwardly apply problematic address smoothing particle reason backward degenerate normally simulation degenerate markovian see static appropriate definition degenerate markovian exploiting discuss deterministic markovian backward method see markovian smc inference work sampler markovian application tree node smc fashion agglomerative smc generative markovian observation give markov employ section exact run smooth state first rao particle th system e run coarse approximation discard state make five sampler accurate magnitude less would decrease suggesting cause dominate even simulator smooth forward particle backward computational sampler serious particle term iteration computational sampler affect truncation evaluate use function toolbox take order simulate step system noise sampler iteration use particle consider level system well posterior mean discard sample rmse relative mean box tend increase probability truncation representative th setting run track smc great success g see scenario beneficial instead problem target surveillance behaviour apply state v standard g acceleration straightforwardly parameter state system assume affected acceleration noise use matrix arise discretization state target uninformative arise visual tracking initialize process singular care apply state suggest space u sampler hasting random walk deviation density sampler target worth point sampler complicated sampler model natural implementation sampler particle conditionally similarly sufficient backward state backward discard truncation truncation employ sampler particle discard first smoothed sampler provide inferential despite weight problem tune
form task multivariate experiment outline h initialize construct nonparametric accord marginal minimize solve system equation kernel equation close former optimizer advance summarize base prediction jx classification new want separate class likelihood portion shift series accord learn via basic dirichlet suppose need infer assume parametric parameter belief limitation scope type inference approach dp base positive gb almost hierarchical specification g n fx parameterized generate partition interpret grow new limit mixture dirichlet prior dirichlet dirichlet concentration previous addition calculate variable approximate choose find distribution minimize leibl current factorize analytic truncate stick fix treat follow variational following probabilistic minimize graphical distribution z term expand lemma denoting z e recalling observe see distribution tv e hz hz j z j z j equality j couple observation observation yield hz j calculate hz previous identical z j previous log derivation exactly derivation summarize except calculate calculate equation implementation make extend implement function function fit stop difference demonstrate process st jt jt sample vector sample randomly finally task amount model fix mixture group expectation propose individual variation learn jt jt square qualitatively trial see task poorly st well st area estimate close figure recover set local yield similar st show rmse draw conclusion work st increase converge dp truncation dirichlet dp concrete motivate research star category research number star survey dramatically collect hundred classification star survey base behavior periodic especially study periodic star notice characteristic time light share shape series periodic star experiment explore dataset addition orthogonal therefore context experiment time filter sample point interpolation normalize standard series pre mixture near series slide series point take separate order cv result see densely gmm series aside effect vector gmm kernel hold place flat effect minimize j algorithm set learn consider observation modify covariance estimating put together form equation compare mean vanishe covariance accordingly view extension phase similarity share feature separate investigate truncation dp show bic data dp bic equivalent performance reduce bic many choose learn correspond develop apply star part preprocessing reject million approximate insufficient propose provide process performance propose time start simulate survey feasible rely series normalize linearly nn gmm rbf use order moreover classification close centroid series per sample perform additional discussion performance dense explain obtain interpolation sample notice plus inside improve third sample increase gradually become plus achieve densely sample difference use expressive getting gmm non reason attribute share gmm gmm constraint setting model come summarize densely sample couple perform excellent dp potential three label test assign initialization accordance run dp dp obtain accuracy result order focus point pure without label discovery time attract year range potential example diagnosis diagnosis speech common choose time fourier wavelet coefficient feature fall category base generate model closely detail framework across algorithm parametric mixed task task furthermore effect along different process mean allow gaussian second mixture share kernel focus stationarity regression divide infinite network different closely approach substitute curve series parametric incorporate shift shift multidimensional also treat e curve alignment parametric cluster hidden admit parametric regression handle series differ model shift estimate handle full parametric hand grid variation allow treat nonparametric task model sum group prior extend number variational learn experiment discovery particularly sample several work survey system therefore issue develop appropriate extend context drawback avoid perform cholesky wang wang share among shift periodic novel capture group task learn periodic lead infinite selection experiment use particularly sample multi variational world task study drug find good regression instead measurement leverage patient population individually multi intuition simultaneously attract approach medical diagnosis framework formulate multi value reproduce hilbert exist statistic share learn formalize share build extend include aspect modal second task shift original shift invariant group mixed alternatively shift random regression unlike exist extension proportion adaptively rather explicitly inference propose posteriori map parameter insight hide variable identity solve phase addition dp show complexity alternative gaussian mixture series interest primarily regression utility several synthetic series yield superior gaussian bic introduction interpretation main assumption model report relate conclude idea throughout scalar vector function extend fx fx n reproduce hilbert keep simple generalize framework convex relate large term complexity assume square function term time sample center normally shape thereby variation arbitrary known require section shift individual variation one hidden prefer get concrete train l separately given provide explanatory application excellent classification j analytically intractable couple sum distribution problem resort iterative algorithm iterate step local maximum stand last difficulty couple
come get irreducible reversible state ji reward player remain successful without player pick arm successfully frame sample player n jt max need prove reward player property match occur I j jt jt jt jt jt take eq index know frame decentralize markovian reward per md mn arbitrarily sequence arbitrarily skip bound event theorem display false p max max md put together bind desire previous refer bipartite one distribute bipartite arm know matching maximize match note goal optimal preferred bid player determine player price high arm high find reward player arm less divide frame frame allocate bipartite matching algorithm run phase length describe phase decision time determine channel match change change counter need phase phase figure bid channel user know bid allocation determine bid run return go successful transmission slot frame channel bit indicate bid precision frame match b successive reward channel markovian user bernoulli run show cumulative bold curve upper curve well even markovian case reward generate chain reward e performance algorithm ii much well decentralize achieve usually difficult progress markov state algebra fx department electrical engineering university california usa support grant fa nsf distribute player arm bandit mab pick pick arm get reward I model unknown markovian model arm model irreducible reversible unknown stationary arm reward player neither dedicate control player costly add regret index achieve expect motivation come secondary user cognitive wherein must pick wireless channel different user introduce armed essence player environment player learn suppose choose arm reward density find minimize type policy expect grow order arm regret asymptotically reward come much index policy asymptotically unlike reward come propose markovian irreducible represent single arm evolve probability play passive play liu extend problem trivial show policy chain continue played show space hard policy long high reward employ notion regret compare reward always arm high reward policy priori knowledge propose policy propose deterministic exploration author arm bandit partly spectrum access channel yet know nothing channel statistic bad channel expect channel learn explore formulate channel must channel show must decentralize setting I policy regret markovian weak assume underlying chain polynomially surprisingly regret decentralize armed discover propose index heart operate round price bid value impose communication must nevertheless growth expect achieve know organize section single mab markovian decentralize reward decentralize case reward match decentralize numerically wireless arm correspond channel want pick arm dedicated control among player potentially pick arm regard player play reward function generality mean unknown player statistic play player reward get reward share manner get time play player tn namely time maximize reward time match bipartite match reward pick policy policie decentralized player formulation reward markovian player model irreducible represent x j assume generality player use emphasis order unless otherwise first bayesian bandit later player player pick identically play time denote reward play jt x jt ucb pick instant useful analysis subsequent index every epoch switch space together yield variation schedule unknown decentralize mab might question various affected resolution high index pick square pick positive monotone computation algorithm computation give ucb positive ucb slowly asymptotically slight modification due precision player arm become c follow since p also last equation know get choose regret grow trivially bind regret scenario reward arm come reward model reversible markov finite matrix x distribution ergodic max xx max jt playing satisfy small contain follow derive markov min xx jt jt ts jt nx jt jt jt jt take jt jt mt concentration chains bandit markovian cost ucb sequence consider suboptimal start start jt event j event arm number observe event j jt j max inequality skip fx jt jt c fact similarly event summation eq jt jt max exponent mp yield computation formally section choose expect armed bandit reward precise computation choose monotone omit decentralized multi armed reward wherein player pick arm neither get bipartite bipartite matching however distribute bipartite incur computation exchange time bipartite matching frame kind decision index bipartite determine phase exchange player change frame frame length decentralize define successful play pick arm denote frames time player yield bipartite expect reward initialization least counter match use signal player change increment counter decision player matching
parameterization modify complete may interest pmf amount spend failure formula pmf state state cf calculate ft ft ft rv ft discrete computation ft dft expectation variance ccc desire pmf extremely quickly accuracy handle probability computational dft function dft author thank david pmf hold sample transition distribution pi pi fourier transform f st f pmf proof remark em height em depth interest field analysis find use characteristic inverting calculate unstable general lattice quickly calculate discrete fourier extremely fast able useful inversion characteristic rely existence central rely property general seem approximation calculation partly difficulty case science make fourier preference draw area process another way representation branch represent state label characteristic detail see know calculate challenge cf transform dft dft support term negative binomial fouri inversion secondary easy calculation dft dft deal lattice dft code software package q transform fouri transform understand dft fast transform dft fourier control dft dft rv discretize evenly spaced point adjacent dft similarly ft evenly interval transform domain eq dft lattice nonnegative pmf lattice dft dft approximation ft dependent dft pdfs lattice able essentially information variable include dft sampling rv pdf class lattice essence little census locate allow dft accurately ft sufficiently lattice properly lattice integer argue nonnegative slightly derivation lattice need transform let transform control pointwise dft less ft great dft accurately transform difficult transform demonstrate inverse dft forward dft dft fourier rv evaluate pointwise dft less couple notational help integer negative large imply therefore express forward dft assume approximated dft pmf ft nonnegative bind replace thus error introduce fourier primary estimate division occur address additionally also part primary dft one pmf overcome one bind moment primary case turn transition exist find impractical another approach
straightforward verify hold large fast solution q stem involve supremum diameter tuple see combine condition term within agree obvious correspondence appendix derive bound theorem constant far completely term situation invoke scaling term proof hold modify theorem parallel analog column eq column dominate selection orthonormal select error problem process unlike matrix radius dr proof theorem slightly weak sparse match corollary result optimal reason enough eigenvalue recover individual eigenvector dependence low shall desire remain focus assume appendix proof bind packing set particular packing give k covariance fold lemma minimax gaussian principal remainder correspond packing consist metric packing set reflect post two bind careful maximizer ingredient bind maxima traditional purpose elementary tool functional curvature let span eigenvalue alternative perturbation distance use inequality variational lem function g span satisfy condition suit ideally constrain combining recover frobenius parameter solution corollary sharp nontrivial symmetric obtain thus argument probably detail eq hold let index quadratic form sharp recent advance corollary standard gaussian see proposition hence subspace regression correspondence sparse pca far final minimax theory let penalize play consider penalize subspace analogous u entry row penalize simultaneously enter multivariate encourage sparse go beyond rate independently condition achieve constrain straightforward rate bound theoretical appear intractable rather correspond optimization challenge although minimax noise knowledge modify technique exchange along case whether exist without either thresholding seem front restrictive possible outside one approach follow estimate principal penalize detail future consequence state essential separately capture essential subspace vary subspace give theorem row satisfy eq universal constant every otherwise hold see let exist every imply sparse satisfy constant estimator setup subspace prove theorem tool minimax low generalized measure measurable leibler measurable satisfies calculation multivariate let consider subspace kl exact fold divergence divergence find pack small diameter subspace frobenius prove method construct packing let allow packing diameter conjunction follow generic minimax estimating define define derive allow analyze subset satisfy absolute v lemma relation fix unique verify guarantee thus substitute packing set pack packing choose element manifold representative pack subspace pack manifold satisfying exist lemma proposition convert bind frobenius invoke let necessary constraint eq side substitute low bound f c n capture hypercube hypercube orthonormal support combine pack packing approach however pack product may differ pack require packing without final packing achieve apply round code th coordinate code specify place column packing differ pack set column modification substitution brevity outline major step definition I satisfy small letting every invariant estimator remainder high order term proof ignore dominate orthogonal write quadratic term cross term lower singular spectral norm universal bounding norm v v q v v eq tail controlling involve universal eq thing together condition q angle subspace span ii eq inequality precede conclude write determinant thus f proof lem subset satisfy ii right belong hand argument brevity u since e apply conclude orthogonal symmetry w e f e ef ef dedicated cross proof variable satisfy constant matrix net duality exist schwarz inequality u j number g q side inequality minimal choose satisfy v recall u entry constant follow remain norm thus inequality minimal net constant acknowledgment article complete university thank comment support nsf fellowship dms nsf grant sparse principal analysis high number estimate subspace span eigenvector sparsity bound sparse subspace class estimate optimal without restrictive interestingly appropriately employ early arguably know use dimension numerous diverse much early serious difficulty inconsistent lead conclusion correspond estimation assumption matrix development focus methodology concept eigenvector include development limited result lead establish individual eigenvector open pca span facilitate assumption variation mathematically convenient imposing conceptual principal unlike obvious present complementary notion sparsity norm row sparsity orthonormal intuitively mean subset crucial existence frequent rotation practitioner help principal component principal estimation sparse subspace constructive wide nearly illustration row measure section allow intuitive explanation subspace variable corresponding basis advance subspace sparse estimator iterative thresholding assume perturbation showed nearly achieve eigenvector track dimension subspace work covariance model two reason simplify analysis enable exploitation gaussian possibility equal variable covariance case technical ingredient subspace novel variational corollary useful recent advanced bound framework construction pack develop key distribution similarly satisfy estimator subspace generalize angle subspace canonical angle projection additional angle subspace orthogonal orthogonal denote familiar frobenius subspace singular singular nonzero singular twice identity convention relate subspace ordinary euclidean orthonormal minimal orthonormal achieve constant additional subspace convenient manifold matrix compatible show problem maximizer subspace define essentially lasso estimator idea chen sufficient know convex replace convex
prior event c p bind paragraph right combine proof n start cc cf hellinger dc nc bound n verify condition display vanish equation constant mb nj ng extreme union kb g j nj due theorem last j contraction n proof accordingly involve construction volume hausdorff general position vertex hull lie adjacent distance vertex g rest multiplying depend vertex g kp p g q display distribution v mp g cn kp cn side display employ part use kp g kp n proceed polytope pg mn lower remove regularity induce ki second exploit k ki g k k n conclude proof l kl note dx gd g ray angle ray edge bound intersection ray resp intersection contain support lie intersection segment x x x b c repeat extreme g segment cone b dx g gx h proof fix satisfie soon say general either lie form face among hyperplane center map stand pg c show pg continue bind face hyperplane contain contain omit key ingredient establish existence measurable hausdorff two lemma test highlight hellinger begin existence density vary overcome packing consider pack nh nh tp n g consider inequality dd g existence suppose every every exist packing proof maximal set tt take take way p due monotonicity every probability lem satisfied hold n display thank display term vanish third rule bind denominator I p last bound acknowledgement support grant author motivate work style graphic g mm nm pt distribution population polytope also contraction population kn n population polytope mild contraction hausdorff distance support extreme nm quite interesting rate root density datum entropy layer continue fall effect additional datum set sequel additional contraction contraction compare exponent differ algorithm recover population nature extreme polynomial exist establish consistency contrast concern specific contraction paper asymptotic geometric technique exist proof tool asymptotic continue useful estimation section formulate abstract contraction main endow hausdorff metric opposed endowed hellinger quantity indeed hellinger fundamental analysis tie amount hierarchy paper devote feed hausdorff divergence functional kullback leibler induce relate integrate technique geometry come derivation remainder main basic geometric assumption consequence abstract contraction whose verify present provide leibl neighborhood theorem technical radius g diameter throughout affine polytope denote leibler hellinger variation define independent convex call distribute accord use dirichlet kp detail equation write multinomial specify I ij px I give relevant variable dropping indexing individual n count joint mp product draw nz z early amenable significance work g polytope behavior infinity number population polytope serve bind purpose parameterization consider contraction behavior access contraction distribution g give population metric minimum matching hausdorff mild metric family concern behavior boundary p regularity dirichlet characterize contraction relevant section satisfy j j priori symmetric let hausdorff consequence relatively widely symmetric technical require kullback hausdorff fact replace statement relatively dirichlet difficult try use establish technical asymptotic primarily establish point far endow contraction exposition contraction suffer even total amount appearance root kl divergence distance appearance proof viewpoint highlight provide issue discuss manner knowledge order rate kullback leibler situation lemma estimation technique always posterior contraction rate gp incur independent since spirit regard g hellinger rate size generally derivation still yield rate remark iv remove trivial role exponent intuitively large mass near boundary locate boundary equation enable variational distance hausdorff extra exponent convex polytope hausdorff entail extreme moreover share rate hausdorff convergence positive constant convex generality spherical ball convex polytope lie vertice p part broad polytope arbitrary polytope moreover either note obtain asymmetric role hold state dependent abstract contraction hierarchical whose detail specification irrelevant integrate notion hausdorff ball eq useful hellinger hausdorff metric hellinger metric simplify hold kp g choose around leibler fix suppose scalar define scalar g certain condition defer note applicable model choice hausdorff remainder devoted condition hellinger low prior define kullback utilizing describe total variation metric hellinger population hausdorff metric minimum matching hellinger inequality let contain spherical gp g convex polytope polytope either g gr satisfy tighter allow consequence lemma lem respectively main test order distinguish organize nx ii ease measurable probability b outer gd g g c ii pa ga hold pa g regularity ga pg px define ga gb ga p n conclusion proceed way extreme suitable existence transformation contain polytope obtain hyperplane away extreme solid radius share sharp corner pair two hold ga g g g extreme pa iii latter proof proceed support neighborhood alpha regularity dimensional hausdorff definition reduce l kk rank variable chapter induce dimensional hausdorff jacobian map admit hausdorff kl l dirichlet give density imply regularity concern behavior near weak kl divergence endow structure
play crucial role medical trivial incomplete consider complete thresholding noisy measurement exceed projection matrix correspond fourier describe image speaking setting less ambient dimension acquire impose physical resource recovery high essential cs pseudo projection examine extensively essential typically compressive compressive measurement perform comprise extension design measurement analyze complexity set eq q candidate design dyadic risk surrogate risk idea behind formulation excess concern estimate compressive measurement proceed manner spirit proxy rather comprise careful reader detail examine level set compressive measurement quickly entail tv effectiveness compressive remainder optimization briefly tv fast shrinkage simulation section estimation total tv initially denoise tv minimization collect projection acquire representation operator either isotropic function give tradeoff fidelity measurement quantify tv fr g pose term tv take estimation slightly less value procedure optimization gx fista approach rely quantity fast gradient dimension termination specify terminate successive sufficiently www mapping en wikipedia show test bit format take random I several setting deviation outline target approach algorithmic evaluation excess fig excess measurement estimate art approach proxy define noise achieve noisy measurement figure provide estimate level set several different estimate parameter excess even small ambient reconstruction correspond reconstruction
prove convergence part decision tool section error bayes loss lemma q derive optimal defer closed center solution include eq norm respectively sample satisfying covering supremum norm eq defer appendix constant present hold moreover infinity dataset show error classifier figure replace tend assume risk sufficient existence proof first assumption first continuously differentiable second continuously open interval function calibrate existence lemma increase z condition note loss existence confirm truncate quadratic easy order differentiable condition exponential hence use similar correspond hence yield note monotonicity convex depict bayes consistency general numerical algorithm devote linear regularization outperform hence method examine benchmark variant simplicity subset unbiased train function clearly unbiased outperform unbalanced function parameter regularization evaluation prediction suppose identity way negative define distribution show test estimator svm unbiased unbalanced estimation setup balanced datum slightly well comparable ccccc unbiased real uci benchmark diabetes heart see show dim test denote sample test average performance respectively unbiased ellipsoid covariance correspond exist take loss uncertainty input uncertainty set benchmark deviation function unbiased bold letter test small opponent except breast cancer result close indicate compare achieve unbiased unbiased superior covariate covariate uncertainty capture distribution point appropriately achieve unbiased dim test breast heart conduct unbiased bold opponent significance svm heart paper binary show conjugate property parametrize uncertainty present margin parametrize prove statistical consistency uncertainty proof exclude nice regularization investigate hinge loss modeling optimization theory frequently apply datum see work acknowledgment grant aid support aid ts prove existence accord argument restrict function form problem column property vector kx shall subset compact infinity tend negative compact therefore next since slack also satisfie inactive ensure max gap lemma ps negativity satisfy derive function subdifferential subdifferential present hold convex exist satisfy attain real subdifferential hence corollary hold clear marginal ensure first satisfied reproduce negativity optimality subdifferential derivative monotonicity negativity subdifferential bind mean number subdifferential less define lead hence choose satisfy high choose appropriate positive hoeffding inequality lead direct exist eq exist large say boundedness property lead hold second calibrate sufficiently inequality respect fix fx b equal indeed q hold respect depend vc number higher choose independent uniform respect probability condition observation inequalitie q hold high hence tend follow infinity positivity condition uniquely determined fix continuously differentiable derivative convexity decrease negativity convexity non negativity positive satisfy inverse fix choose straightforward confirm satisfied cm corollary classification uncertainty loss vector machine represent classifier loss property use hard margin mini set employ regarded margin study application field learn study two conjugate relation learn problem goal input training sample label sign algorithm logistic support property base understand intensive another statistical label case picture convex subset hull learn hull label hyperplane hull point paper term optimization minimax machine set study concern set learn focus function uncertainty conjugate uncertainty learn apply uncertainty accurate uncertainty specify margin uncertainty set recover consistency exist relation uncertainty devote illustrate way uncertainty nice kernel consistency proof notation equal otherwise vector space describe face euclidean hull element denote drop subscript measurable function set short supremum estimate binary associated input article sign refer binary decision evaluate e expect measurable low denote q subscript drop hard learn use computation tractable hinge adaboost minimizer surrogate avoid empirical surrogate loss classifier adjust loss computational retain base robust uncertainty determine probably solve optimization application uncertainty design sample high label confidence resp consist convex sample resp hull hard margin svm robust bad uncertainty solution decision illustrate problem appear hard svm algorithm briefly minimum minimax probability propose decision play minimum margin term decision estimate line estimate boundary achieve maximum uncertainty accord margin uncertainty direction margin follow set introduce accord investigate relation euclidean trick obtain rich constant point hinge loss formulation negativity confirm negativity last inequality come dual minimum see lagrange index equality change resp introduce uncertainty represent uncertainty estimator tool analyze uncertainty expression theoretical tool understand property uncertainty uncertainty parametrize section uncertainty exist kind parametrize level set convex proper q uncertainty z replace base popular learning decision constant consider primal derive give representation uncertainty derive correspond via gap primal primal training empirical thus tool consistency learning study algorithm develop primal function keep conjugate define condition resp eq uncertainty define apply shift reason description follow theorem suppose equality equality prove lead statement projection statement impose unchanged formula h p uncertainty set q empirical property optimal tool primal show uncertainty function tc definite matrix matrix hold uncertainty solve vector illustrate panel right uncertainty q uncertainty estimation set phase slightly brief uncertainty define depict hold quadratic positive misclassification decision boundary original present variant use necessarily let reproduce space endow decision
well less measurement number bind suggest indicate evident compare model part l competitive knowledge confidence argument need simulate know need exactly confidence compressed measurement formulation objective restrict quantization error interval finally utility formulation previously along variation attempt acknowledgment foundation grant dms foundation fall competition award grateful constructive comment version lead develop high estimate assumption define yes appear mistake something inconsistent check ok yes assumption least define similarly compressive lemma find completeness proof give set complementary without intersection indicate entry feasibility suppose satisfie ij inequality eq yes checked factor factor yes complete noise prove inequality fact consider leave claim inequality complete subgradient existence prove assume inequality hence prove claim hold satisfied hold take error measurement matrix entry draw submatrix row define choose threshold satisfy large note similarly minimal prove probability large note inequality working ensure exponent dominate conclude derive discussion iv quantity dimension since definition claim check problem could counter counter lem counter counter reconstruction finite thus quantization near outside account explicitly quantization lagrangian consistency appear date show extensive formulation variant compressive reconstruction quantization consider sense cs represent define quantization measurement round near record quantization round represent lie beyond represent namely setup compressive sense hardware architecture remove sense matrix unknown without noise record vector part correspond recorded matrix quantization particularly quantization interval choice weak quantization variable course robust replace add quantization region viewpoint play least square appropriately clear inclusion rather apply reconstruction hold art place vanish eliminate place require code cs norm sign two recover though recover magnitude reliably norm denote mention nonzero use submatrix discuss order notation positive write dimension write denote depend zero partition matrix norm eq quantity place admm solve discuss property bad comparison formulation conclusion claim describe simple combine constraint way specify write lagrange multipliers lagrangian primal turn gradient increase proceed ax ax ax break update conjunction many algorithm accelerate fista primal admm section property obtain difference true solution certain assumption sense formalize quantization error distribute quantization asymptotically variance selector assume recall rip estimate would I vi hold converge constraint place happen specifically p formulation formulation compare five variant formulation try variant constraint omit recovery performance lasso remain alternative appear objective x hence comment discussion notation discussion need I page keep report actually constraint consistent bind ok formulation enforce actually enforce add constraint l define signal satisfy certain familiar selector notice result synthetic entry location draw independently repeat choose yes essentially admit tradeoff confident since tight determine tight simulation elsewhere quantization error note make proceed generate numerous empirically similar seek dimension bit confidence percentage figure value average trial recover extreme formulation give performance lasso tie continue plot
spatial dependency inspire advance motivate different towards predictor hyperparameter stick control proximity identification compare break remainder organize stick break datum dependency section nonparametric cluster dependency relation method efficient derive dp dp characterize scalar refer innovation dp draw subsequently independently draw random joint show exhibit sample new exist draw allocation proportional previous draw allocation concentrate let draw sample variable discount innovation distribution expression g dp rich rich sample assign assign draw particular value scale dirichlet number unique grow characterization unconditional random draw break two collection beta distribution regard stick breaking notion introduction dependent prescribe position measurement arrange dimensional lattice sequential naturally associate lattice depict temporal e point case location prior comprise predictor observable location eq select spatial close location become typical kernel kernel aim cluster relatively feature far location space nonparametric goal sample draw distribute value value take location assign th innovation random kernel stick break construction discussion location innovation valid random purpose follow three summation independent share common idea stick bound unique instead consider beta prior stick employ discount bound kernel location unique cluster contrary great stick stick hence predictor appear bayesian typically mean variational monte impose gamma innovation q hide th impose q also impose conjugate formalism consist family variational innovation inference fix c n c parameter prior hyperparameter comprise hyperparameter impose innovation derivation approximate maximization iterative free consider exponential configuration take functional read let eq q obtain regard hyperparameter maximization determination location assign energy latter
recent exploit similarity nevertheless dictionary unlike dictionary adapt image approach penalty well indeed promise process svd error intensity pixel scale present denoise penalty bottom well bold natural develop well undirected dag breast cancer consist gene breast cancer goal versus argue graph regularization objective improve use identify gene disease lead yield well prediction task interpretation necessary drug penalty regularization statistically significant improvement asset function improve help medium solve time internal procedure gene gene obtain interpretable connect component pair link arc approach require transform dag treat direct choose arc arcs dag arc denote pre prevent penalty goal formulation negative intercept gene expression include ridge elastic penalty link arc variant equation use place norm arc structure penalty problematic vary cope issue try strategy every penalty describe unable select prove penalization number belong logarithmic elastic net penalty select internal set repeat report average table start answer penalty sparse significantly improve interpretable trust preprocesse acyclic report processing run observe concern seem produce connect similar sparse well assess difficult number result unable test statistical rigorously without experimental neither seem experiment try graph structure table conclusion task hand none statistically prediction ridge ridge select form sparsity accord asset aspect stability sparsity method introduce strong regularization structure provide stable solution half experimental whereas believe claim penalty without encourage select improve stability solution biological knowledge connectivity test connect component pair dag random balanced deviation internal fold sparsity gene connect answer question proximal relatively fast ghz cpu intel core slow small value solve proximal reasonably fast conduct reasonable amount proximal structured subgraph feasibility link flow penalty variant flow framework problem encourage connect able non penalty valuable context find convexity helpful prior knowledge approach fast interestingly penaltie dag remove like exploit induce extend lasso allow encourage group line encourage group link arc clear experimentally set task dag exist group every vertex dag define vertex similarly contain encourage intersection subgraph dag number contain connect unable make useful flexible suffer belong concrete arbitrary reason penalty structure proximal compute involve graph group inclusion relation different terminology encourage sparsity pattern predefine decompose latent support encourage desire regularization give show equal introduce define equivalently g g g addition index j g strong decrease rewrite r g g satisfy p interpretation interpret theoretic code inequality cover well uniquely inequality show weight lead interesting interpretation source code length path define unique vice versa denote bit indicate start bit indicate cost weight uv path matrix walk prove definition equation associate flow path cost cost flow vertex minimum achievable flow constraint satisfied flow path flow arc integer classical say cost flow arcs integer tell decompose flow path flow unit cost flow keep capacity therefore flow equal use proximal pattern proximal therefore rewrite uv flow constraint exist flow solution loss generality integer replace constraint fix flow rewrite uv flow piecewise minimum equation without loss generality easy see sign accord proximal p j uv yield j u w w program place define rewrite identify path show converge correct ensure zero moreover unique give g l g update solve operation w w convergence easy solve condition lemma p denote stop select want stop trivially stop p check observe h old inequality yu yu fr bin yu berkeley department university california berkeley usa embed gene gene knowledge easy new computationally feasible select connect direct acyclic sparsity dag path penaltie exist interaction penalty tractable path experimentally show connected subgraph network flow much priori inducing well nesterov supervise available assume predictor gene regularization automatically identify subgraph connect component group gene involve two connectivity improve connect easier involve feature arc risk formulate order choose cardinality encourage subgraph good penalty connectivity graph category involve pairwise interaction vertex arc encourage simultaneously usually tractable connect penalty approximation problem penalty long graph interest group induce regularization encourage sparsity pattern penalty similar strongly two go beyond sparsity new challenging example define subgraph solution address greedy vertex encourage long range interaction define connected subgraph subgraph subgraph soon large connect naturally direct acyclic build upon penalty pay allow function penalty go beyond interaction long form subgraph number path path dag even though dag flow thick draw blue minimum fill blue mm minimum thick draw black fill red minimum thick gray rectangle fill thick dotted line width auto yshift mm yshift mm xshift mm yshift xshift mm xshift v yshift xshift yshift xshift xshift yshift xshift xshift mm yshift mm yshift xshift mm xshift mm yshift xshift mm v yshift xshift yshift mm xshift leave mm xshift yshift xshift v yshift xshift mm right yshift xshift v yshift xshift yshift yshift xshift v mm yshift xshift v v v v v sparsity form subgraph represent cover bold development flow technique chen structure regularization relate flow penalty terminology overlap group complementary encourage encourage connectivity link arc however obvious effect discuss question summarize penalty feature penalty involve optimization deal flow tool implicitly exponential allow penalty operator polynomial long interaction tool brief flow penalty optimization technique devoted experiment genomic scalability conclude penalty flow since concept brief topic section proximal gradient popular define arcs satisfy arc except figure flow remark appropriate give admit variant go arc vertex distance group v yshift yshift mm group xshift source xshift node node node bend angle yshift group yshift xshift mm leave xshift yshift mm xshift yshift mm yshift xshift yshift mm v yshift mm yshift node v cm bend angle auto right xshift xshift node bend auto yshift mm group yshift v xshift source xshift yshift xshift yshift xshift v yshift yshift xshift yshift dag send along path flow flow send along along cycle flow attract attention engineering physics exploit structure prove minimum cost interpretation always flow unit send cycle unit flow cycle figure flow build upon interpretation flow e flow locality neighbor vertex graph locality exploit flow capacity penalty respectively iterative thresholding similar mirror descent insight problem classical w interpret algorithm deal nonsmooth efficiently solve formally able compute regularization denote efficiently represent bit select complexity count code count encode zero selection easy interpret selection isolate sparsity group consider nevertheless equation np g otherwise boolean abuse denote consider relaxation problem addition penalty union group define norm introduce link norm need variant penalty detail large much send every path assume configuration equation flow uv uv enable flow turn equivalent flow solve hold cycle flow acyclic mapping proposition formulation capacity handle flow solver vertex equivalently vertex link arc carry main proposition proof compute consider define flow capacity constraint strongly definition penalty challenge general hard variable exponential graph difficulty convex define obtain define unit path select context methodology define cost flow constraint relaxation address optimize regularize operator flow proximal cost flow cost u ff optimum equation establish detail minimum piecewise equivalently define cost g u cost flow cost cost solve flow often instead choose indeed weakly polynomial require transform cost cost flow problem integer round denote worst appeal heuristic possible even proximal algorithm appropriate recently inexact proximal increase deal cost complicate fortunately relaxation modify handle cost keep describe relaxation convergence come refer issue constraint classical obtain uv uv formulation interpret algorithms network perform update dual primal present instead chapter algorithm active adapt efficiently define additional length nonnegative prove g g l acyclic correctness complexity op appendix complexity loose empirical complexity compute norm useful problem l adapt experimentally allow solve medium instance subgraph call observe time close subproblem easy solve check update accordingly ensure use warm l shortest acyclic u subgraph update solve subproblem stop relaxed could duality guarantee enough behave practice interface make available open software package proximal fista convex available duality gap stop duality gap experiment typically choose grid solution initialization warm follow decrease objective strategy far penalty consider sensitive fine validation validation offer connectivity tune prevent overfitte find choice good coarse parameter detail experimental experiment control since network different partition edge mail arc accord order structured compare ability regularization design less entry remove component strategy sequel perfectly recover snr choose different penalty criterion denoise outperform ordinary ol ol change ols shrinkage penalty improve regime snr also parameter solution path penalty exhaustive testing
successful completely free away major year reader experiment correlation perfect anti perfect correlation exactly bring argument mechanic mechanic statistical theory physic explain individual instance actually happen happen cause driven science spin would influence particle place outcome measure direction particle measure spin correlation merely cause origin common source locality quantum mechanic strong outcome physical merely reveal act quantum mechanic describe aggregate exist nothing physics mechanic instead mechanic deriving turn head assume principle quantum mechanic agree though actually measure run exist may locate one imagine discuss observe take assume locality bring freedom allow analyse tool base like assumption free true way one justify another understand randomness well coin pseudo generator use kind clinical design really want observe come physical mechanism coin another jointly deep otherwise unnecessary never value mechanism completely unknown ensure know measure really know three negative think tell discard super mean discard observational everything explain nothing freedom possibility reject coin click past invoke instantaneous process extremely subtle effect quantum seem phenomenon surface prediction mechanic causality difficulty I force please actually admit outcome like limitation experience physics universe notion evidence cognitive birth framework data experience experiment module algebra module notion interestingly module causality agent continuous agent act together I build create every physic causality quantum physics causality physic quantum imply say problem also quantum mechanic isolate normalised length evolve quantum outside laboratory equal square system together much evolve reality outcome lead framework mechanic opinion von device respect quantum mechanic hilbert quantum unitary inside look operator hilbert space space past copy picture change observable family special exist value definite unitary evolution determine joint fix quantum notion restrict back abstract quantum field theory hilbert traditional unitary evolution mathematically continuous measurement initially invariant yet problem reader work two cite solve world protocol source distant randomly implement long time almost call particle know perform pair huge many event experiment detection correspond thought detector easily overcome necessary explain experiment mention pair pair detection event classical particle leave correlation source agree another like measure decide desire pair generate outcome draw advance choose go probability successful setting particle identical aim advance want experimentally without statistical miss context put history result take call outcome vice versa detector turn sharp efficiency suppose particle particle amount set detector give particle correlation sharp ab ab everything else perfect limit allow quantum go publish journal fair close albeit experiment actually simply analyse good question close indeed never perfect repetition actually logical experiment nothing quantum mechanic rely logic move ball nature take necessarily find implicitly business statistically inequality section clear argue picture inequality worth measurement party outcome unbalance role study case hyperplane polytope everything correspond hidden vertex mix mechanic joint quantum party etc take measurement party outcome elementary specific whether generate table define space sum probability moreover theory quantum mechanic sense marginal marginalization local also list restrict positivity positivity constraint euclidean space euclidean sub create polytope model quantum contain set successively large combine let special class constraint satisfied imply party outcome prescribe correspond thus mixture distribution measurement party joint distribution pick model convex polytope mixture also describe intersection hyperplane model describe affine space full nonempty polytope form strictly slowly picture small within outer square quantum circle boundary picture vertex extreme quantum body generalise polytope sub exclude corresponding positivity boundary hyperplane nontrivial polytope nontrivial hyperplane correspond inequality outcome indeed increase new turn turning mean old measurement grouping outcome seem exercise try classify nontrivial generalised actually violate mechanic pose open question probably counter see generalised inequality turn statistical connection design much material cover excellent author second specific challenge researcher make forward correlation find hard get become serious paper publish use local classical computer protocol past computer measurement message coin course request output setting outcome collect computer computer contain huge table number share course generator randomness exponential like computer I enable challenge two us enforce discover program look inside computer thing difficulty computer detect challenge quantum http www alpha challenge science phenomenon see wikipedia cut necessity communication verification long write post internet news spread prefer style detector computer produce result reasonably run make save program past must accept pair successively neither correct seed exploit force know pair actually pick row seed could run seed calculation model everything claim correct section correlation get single seed style local program simply seed internet draw internet devote discussion quantum interested repeatedly create classical physical system systematically inequality thereby stop round replicate experiment physics local program create instead communication neutral learn think kind rating carry test demand price interested ensure implement generate pair keep run repeat time publish internet appendix theorem inequality red ball red ball number equal probability row ab ab n ab ab ab ab ball replacement contain ball rewrite similarly rewrite exceed three average last exchange role everything eq possibly q want make choose home translate trivially restriction acknowledgment grateful especially thank conjecture suppose establish physics experiment aspect work theorem issue causality understand thereby also conjunction principle refer locality causality match direction influence need propagate spatially connected causality reasoning statistical observational study assumption properly match statistical unobserved issue challenge discuss locality quantum mechanic fundamental physics short name locality world supposed demonstrate mechanic consequence force reject principle hinge outcome measurement spatially separate physical outcome precisely principle outcome outcome actually choose measurement super form implement freedom assumption design experimental actually outcome existence actually experiment existence mathematical phenomenon refer model reality somewhat position adequate physical reality outcome experiment demand object kind physical constraint quantum natural physical locality often consider principle call imply mathematical local phenomenon important thing classical even statistical causality note time statistical observe unobserved correspond independently use coin setting outcome measurement also observe represent grey unobserved might arbitrarily validity causal place restriction instance imagine fundamental familiar life keep mechanic locality randomness become irreducible world primitive merely position cat causality principle mechanic claim interpretation quantum opinion entail reality actual path turn reality wave reality cat observe average purely label column label name denote fair row outcome either one four product product equal former case latter possibility ab b original expect average average sample expect inequality binomial distribution completely traditional limit mean value conclude original four identically suggest conjecture come back quantum mechanic locality want mechanic must hold theory least principle purely certain quantum physics detail mechanic reject freedom almost freedom matter change locality argue ask go positive assertion randomness randomness dependence initial variation explanation randomness quantum randomness irreducible reality purpose need understand mechanic behind quantum call model refer predict spin spin half quantum fortunately understand two distant measure real dimensional space unit spin away direction outcome time imagine repeatedly binary setting description quantum prediction separately pair particle state quantum mechanic correlate product angle outcomes location pt
exchangeable assume replace exchangeable I understand idea testing imagine capital place outcome never risk reflect acquire capital unlikely exchangeability martingale idea fit previously measure one rather deal neighbor many case pp way example example different euclidean accord example symbol line calculation produce tb value follow resp exchangeability produce exchangeable uniformly independently value exchangeability calculating focus martingale calculate sure indeed check martingale define construct denote tb martingale grow shape grow might reject exchangeability use martingale martingale avoid function implement density since unit poor point boundary get boundary point p reflect kernel calculate normalise integrate rule conclusion plug martingale update recursively take evaluating take plug martingale life dataset current plug martingale asymptotically growth rate infinite sequence sequence borel say thing exist us eq inequality contradict investigate martingale compare martingale life test exchangeability tb tb service example handwritten life code attribute gray scale exchangeable reasonable reject experiment merge testing exchangeability sure example set mixture final plug martingale show plug power martingale plug martingale power final advantage level difference old suffer great flexibility example pixel picture spectral band indicate classification label exclude describe reject exchangeability arrive martingale plug martingale plug martingale martingale figure close second peak flexible end old assumption commonly threshold mixture martingale plug martingale would construct testing exchangeability stable martingale life dataset approximately generate introduce idea lack exchangeability look therefore small regime start situation exchangeability p observe ideal shape several kind predict kind value exchangeability martingale past plug plug involve prior integrate efficiently yes whether ac uk cs mm machine learn exchangeability assume distribution independently devoted exchangeability arrive would like measure degree exchangeability construct exchangeability finally investigate testing benchmark former know satisfactory new flexible become necessary many deal algorithm standard exchangeability assumption although violate satisfy popular independent meaning testing exchangeability exchangeable permutation exchangeable distribution exchangeability traditional challenge previous employ theory calculate exchangeability testing proceed two implement generate mode previous exchangeability deviation martingale grow rule exchangeability reject propose construct second exchangeability martingale use martingale assumption martingale martingale namely former grow exchangeability
depend despite convergence ordinary form incorporate choose suitable distribution b gaussian easy result integrate eq mode determine model put b p good constraint see prior vanish asymmetric z q posterior one minimize quantile consider novel form analytical uniform prior pp informative derivative order improper pdf proper improper lambda p seed sort add sr mx sr mx mx mx mx save txt save save txt save txt save se add set sr mx sr mx mx mx mx seed bad proper plot lambda fx ms add mx growth sr mx uncorrelated mx mx label proper growth sr mx improper prop improper proper improper lambda seed sort add set sr mx sr mx mx mx sx mean save txt save save save save save txt se sort add sr mx sr mx mx mx colour typical prior place dot thin dash black value medium sized iid proper correlated grow thick solid proper uncorrelated size thick dotted prior thin improper thin red thin dotted spectrum phase kk anti correlate natural prior spectrum use derivative uncorrelated improper fig concentrated frequency kx k g spectrum f typical unbounded growth super decay spectrum dynamical gaussian decay f law decay get f improper predictive likewise predictive taylor approximately assume fulfilled suggest fulfil depend become eq treat jeffreys prior transpose w transpose demonstrate basically fig iid spaced dot derivative use improper whose trajectory much basically reconstruction delay reconstruction deal compare colour true true derivative cubic spline noisy thick true thin line black dot simple gaussian polynomial alternatively low behaviour plausible although seem also sum fouri linear derivative open source software implement acknowledgement education environmental author thank estimate several unknown real function correspond error correlation find posterior move error special case observation analysis value several real argument noisy occur scientific task classify parametric regression krige regression nearest distance weighting although usually rigorous theory guarantee function belong justify available various form heuristic easily interpret reliability several measurement error also identically iii argument derivative estimate amount correlation minimal diagram estimate assumption contain precise plausible error apparent uncertain however highly measurement knows basically shift direction influence dash line uncertain uncorrelated right diagram retain three error dash show amount error colour effect error article estimate reasoning exist special limit move interest taylor data belief update uncertainty become measurement mixture gaussian derive function simple argument see reason prior article covariance value derivative derivative parameter large want distribution approximate interpolation regular relate also move interest bayesian motivated approximation plausible generalization bayesian weight covariance interpret take account estimate estimator method locally weight smoothing estimate infinitely infinitely differentiable least either measurement infinitely smooth depend smoothly distance method nonsmooth nonsmooth change involve also counter intuitive argument g suppose result argument involve reflect instead four measurement slope slope switch measurement distance uncertain seem use rely correctness rank near measurement extreme might increase robustness might exist tailed use smoothly decay distance variant special common include kernel method near interpolation suggest lie outside maxima consequence weight individual overfitte prior derivative fast estimate vary resolve reliably overfitte derivative order continuously convention component subscript index symbol denote x ii estimate magnitude measurement error belief variability make later I involve plus order derivative taylor polynomial convenient integer derivative relationship note many choice vector matrix linear model estimating error regressor interest determine usually evaluate variance translation multiply constant unchanged multiply ki ic input case get online interpolation lagrange local cubic polynomial uncorrelated intermediate rational vanishing become dirac delta free rational remainder derive eqs improper case quadratic formally regression compute yx w improper also trend proper absolute derivative mean absolute increase vanish q derive substitute care numerical nonzero exact I k addition I vx local j g dot method away though weight approximate behaviour choose prior proper variability one negligible bayesian although truly addition possibly measurement error regression ns yy xy xy estimate square study already symmetric exchange variable absolute vanishe least total give independently cubic eq bs xy show total function vice versa linear cubic one independent b symmetry get I polynomial case minimal choice prior measure weight kf give distance weighting value error occurrence eq smoothing note rational square exactly show directly iw dash dependency vanish word interpolation exponent give interpolation power implicitly assume derivative vanish vanish plausible example feature coincide derivative estimate slope sharp order spectrum certain obtain rp j j simple constant function argument vanish since j hand mean
direction contain residual stress depth one stress illustrate stress profile residual setting component output krige desirable krige analyzing caused especially collect intensive principal instead simple create single computational krige datum irrespective naive extension krige response include liu west experiment conduct nine stress difficulty correlation krige determinant correlation total computationally intensive numerically unstable experiment inversion determinant calculation iteration optimization extremely consume overcome issue krige extension liu collect location run grid sometimes practice dynamic force model tool profile collect different tool feed engine report liu al profiles truncate point functional profile also degradation profile fail especially organized krige modeling illustrate experiment summary conclude remark krige computer response collect note run differently location krige point mean separable tt ir tt case space output universal krige r tn nr I krige ij et eq q et al algorithm employ hundred make prohibitive krige response krige stage profile mean initial interaction compute location collect define location define ij response functional model selection forward blind krige function ix tt td initial stage implement collect grid output regular location liu dramatically reduce result krige x complexity reduce al grid equally spaced functional response collect intensive loss functional location evaluation predictor degree smoothness correlation write evaluation predictor grid response necessarily kronecker problem miss datum grid develop extensively create miss overcome moreover maximization introduce carefully framework combine I view collect negative datum kk e efficiently expression construct direct application overcome monte carlo em main conditioning complete th sample datum distribution al tt I ii result q eq q proof appendix mean three profile except th weight determine evaluate conditional efficient involve observation calculate also easily simple result extend z attain convergence simplify speed conditional iterate expectation z expectation worth note I update simplification complexity per argument proposition n ik indicate row miss assumption recall I side decrease suggest accurately miss simplification make equivalent implement update herein collect regular grid em procedure estimate miss add prediction krige miss z k g approximation convergence guarantee wu residual stress experiment originally al apply optimize turn challenge al cut force turn residual functional output branch hypercube nine cutting stand respectively angle level consider cut edge categorical isotropic al rest factor experimental residual sophisticated element software wave range degree cut angle tool mm cut speed min depth run residual stress originally collect procedure observe assumption side percentage truncation profile theoretically residual stress towards nonlinear profile fit average stress profile least transform original functional fitting h htbp fit krige al apply suggest variable krige ready move build second iteration reasonably substantial em take ghz pc naive krige extension four htbp confidence pp interval illustrate select predict near evaluate matrix lee seven hour ghz computer concern evaluate construct augmentation perform I widely naive krige four basis identify eigen profile entire propose krige krige augmentation marginal propose krige method prediction surprising augmentation completely lose severe quite optimize objective optimization minimize objective maximization large predict stress experiment regular predict predict stress illustrate solid regular sensitivity effect interaction et effect feed cut shape cut stress clarity residual stress surface stress increase feed speed stress profile positive explain physics attribute increase cutting conclude remark computer experiment functional scientific output extend pose article krige challenge successfully stage building procedure incorporate product application sampling run run illustrate hard turn response stress factor setting quantification possible fully bayesian incorporate uncertainty application fully infeasible inversion tackle stationarity assumption relax al additional effort due recursive leaf acknowledgment valuable comment grant dms also laboratory office contract w nf proposition objective q normally
consider estimation component proper approach per monte carlo estimate bilinear measure define include evaluation section among conclusion analysis partition extend factorial tensor domain attribute historical use orthonormal product haar together term hoeffding represent subset clarity complementary disjoint recursively subtract sub df jj l quantitie exclusive complement satisfie easily introduce way importance generally subset introduce measure denote index index count subset important investigate sensitivity index mostly unnormalized work publish translate computation hybrid share component jj jj special result f I nf n I show get I asymptotically small bias important q default nf n require unbiased type relation cube method quasi monte use plain effective expensive approach attractive way another importance drop interaction set black box component dominate version behave differently affect superposition fs easy superposition effective offer well instance identical follow index evaluation fixing subset uv product subset specify integral vanishe vanish vanish equation describe dimensional function generalize square negative compactly mostly zero compute bilinear rapidly computable element element bilinear call combine pair write bilinear low combination one column convenient row equal linear combination variance write representation pair di derive expression dominate evaluation require evaluation product f function evaluation row column sum unbiased nonnegative nonnegative index square square express measure version among classical mixed every expect contribution variance measurement error square coefficient next equal vanish lead degenerate result sum index twice bilinear evaluation begin variable helpful illustrate omit follow simple bilinear require function evaluation evaluation presence evaluation treat variable bilinear use per integer evaluation require let choose put alternate sum otherwise zero bilinear argument theorem bilinear estimator nonempty contrast disjoint w equal coefficient vanishe otherwise equal therefore estimator mean solve carefully evaluation reduce u index along index remainder present value next second contrast q eq therefore way evaluation respectively evolve value nearly value initial set strategy apply index great index empty variance component combine yield truncation contrast measure extent distant lag contribute also base pair segment evaluation per require function per biased estimator proposition bias correct suppose nn first put di much great biased require additional need sum bilinear calculation interval average space estimate desire combination original simple asymptotic estimator matrix importance estimate bilinear ideally variance fourth unknown hard component series papers rao comprehensive version quadratic proxy variance component effect two effect interaction however set write eq proxy choose minimize original former cost proxy argument original estimator q third fourth moment evaluate example non minimum function evaluation answer specific set bilinear formula dominate function symmetry compare large sample estimating error bilinear half standard bilinear analogously bilinear ff estimator modification ff correction asymptotically negligible difference require evaluation per base combine available I involves thus expect small looking subset pair range size report superiority contrast neither always hence proxy reliably specific product deviation proportion last contrast time ratio deviation square little square theorem square advantage way interaction bilinear advantage
approach difference neighboring enforce assumption frame frame apart achieve sparse smooth code advantage primary classification availability label label unlabeled semi supervise without respectively motivation datum visual pattern hence high level domain propose useful motivation supervise useful base well low task impose code use simultaneously collaborative modeling hold datum via weight experiment semi dataset face dictionary described dictionary dictionary sparse code supervision dictionary compare code two conclude describe data ssc incorporate space smooth traditional substantially measure modify code stage add enforce incoherent speedup order scale framework computation temporal supervise transfer number generalization way approach extend kernel bandwidth interesting generalization bound analyze regression iterative alternative scale experiment challenge large recognition variation illumination use extreme conduct consist range various experiment sift patch vary medium resolution follow experimental human perform record consist category challenge video illumination resolution etc randomly densely video sample extract smoothing experiment conduct smoothed person dataset similar technique flexible incorporate norm furthermore significantly dictionary propose speed use improve code code popular paradigm standard code independently dictionary propose smooth incorporate user sample principle construct traditional sparse coding express parametric framework type kernel encode image space incorporate temporal relationship unlabele learn setting fall alternate scalability remain novel code regression traditional alternating univariate regression step speedup traditional code order statistical show proposing smoothing incorporate coding provide sparse code efficient successful various lead improve computational speedup local smoothing lasso temporal achieve locality difference dictionary along regression propose use speedup technique computationally notation correspond vector dimension vector use correspond frobenius notation dimensional explanation code solving correspond denote column object patch sift scheme representation encoding every encode sparse vector structure sift feature class close sift class neighbor apart propose mechanism smooth convenient function capture smooth traditional sparse coding encode encode code square smoothed empirical common kernel gaussian cross validation choice kernel input advantage bandwidth may standard distance may express code code spatio univariate bandwidth video kernel frame feature alternatively fuse approach datum smooth sparse computational fused penalty capture frame may immediately situation sparse handle regression least constraint marginal regression obtain regression proceed calculate least threshold operation provide algorithm ii dictionary code learn dictionary analyze smooth coding provable global prove learn code main obtaining begin represent smooth let random q empirical population true relaxed constraint also unit dimensional ball bound cover every element subset function cover prove concern covering cover generalization code derivation length differ term incorporate factor demonstrate approach speed code description experiment synthetic examine regression regression relative different relative reconstruction report regression significantly less perform method spatial sec ssc ssc mr conduct several demonstrating setting face static code obtaining locality perform experiment recognition video space image generate sparse densely sample pixel step sift code max process repeat report per together base cross validation select smooth result report previous result unsupervise code scene ssc size scene demonstrate scalability classification increase dictionary
aggregation denote coordinate pseudo product function finally exponential define hold arise natural paper design gaussian regression paper concern exponential extra averaging suboptimal exponential weight regression optimality heavily case particular weight natural ms aggregation problem take call aggregate onto hull difficult design high coarse fine would oracle expectation investigate term heavily use fact random design j jx f jx n font throughout rest weight write n first nm later moreover jj scale employ use homogeneity may simplicity treat choose event temperature small enough complement j j note yield eq rely construction ensure nm jj projection aggregate hold large note homogeneity observe n I remain probability old get since choose aggregate correct aggregate automatically minimize interpolation weight family estimator ii put weight fit function equal identity know leibl weight call estimator note weight choose deviation restrictive condition noise variance denote eigenvalue variance proxy directly imply hull interested aggregation simplex aggregate unlike aggregate explicitly solve ms aggregation restrict infimum nonetheless pointing focus aggregate estimator satisfy probability employ uniform may original aggregate aggregate estimator see moderate purpose approximately aggregation aggregate appeal simplicity exact algorithms term quantity go generalization valid minimizer approximate hereafter convention pt variable gaussian choose q satisfie moreover type entropy correction paper nevertheless uniform order aggregate vanish simplex design term optimize range overview simplex promise especially become method sparse case sequel focus greedy take selection property aggregate make explicit appropriately design greedy achieve dictionary simplex statistic greedy type aggregate option algorithm th algorithm add iteration output iteration resp reduce resp refer former use frank wolfe literature minimize value purpose ms appear give approximate solution classical frank wolfe two unable achieve produce j optimal aggregation therefore achieve disadvantage result imply bound dictionary proxy eq aggregate simplicity although completeness prop since relatively fully counterpart advantage convergence fully additional essential update optimize subroutine description order uniform prior solve ms aggregation optimally assumption bound importantly deviation order obtain later accurately achieve remove boundedness variable aggregate q theorem imply deviation bound decrease indicate critical careful inspection aggregate require case easily sufficient achieve ms constant beneficial run confirm experiment stage greedy aggregation paper star stage ms theorem beyond construct iteration constant usual allow bound avoid boundedness problem vanish technique problem greedy suggest increase reduced constant configuration identify define standard identity regression define draw oracle om temperature tune remark random due mis good scenario lk experiment repeat replication order avoid detailed regret exp star indicate aggregation star surprising star regard averaging estimator keep consistent good stage although regret replication noise model nearly good beneficial procedure aggregate ms aggregation analysis illustrate ms aggregation compare greedy bad small theoretical variant sparsity hard display yield eq replace expansion n jensen plug get take follow allow high f successively jensen f observe complete prove chernoff bind q combining identity f j eq use convexity yield statement high proof taylor expansion expand side apply plug q induction second inequality complete assumption conclude vanish simplex similarly j f therefore theorem complete lk c replication suggest example section section proposition notation part dms e function aggregation regression model problem aggregation surprisingly sub limitation lead sharp oracle minimax design produce aggregation performance class statistical learning considerable past
light tail random path cost edge moment path replace hoeffding bind represent consist establish classic sequence evenly exploitation select estimate set compact efficiently path subset dimension consist path construct path belong exploration past observation integer define l dt da hold independent exploration cause th exploitation contiguous denote th period exploitation show hoeffding hoeffding arrive exploitation sequence time mean property select exploitation sequence prove hoeffding define minimum constant exploration well increase cardinality sequence amount lead arbitrarily theorem follow regret exploration difficult value consider increase replace enough proof distribution moment exist cost draw construct exploration sequence base sufficient exploitation exploitation arrive eq path mean regret general special dimension action minimize expect xt identify short repeatedly choose gradually action run epoch epoch epoch epoch epoch cost choose policy point arbitrarily stochastically state tail link propose problem I evolution stochastic lemma property california email edu adaptive short wireless unknown stochastically problem aim optimize path randomness uncertainty dynamic quality vary quality path delay quality observable formulate armed bandit dependent arm arm dependency maintain regret policy ignore furthermore include find ad hoc wireless quality stochastically vary model reward evolve source communication end link policy total model armed bandit mab mab liu player play offer cost draw unknown arm cost player since grows treat share dependency path policy mab naive growing linearly preserve optimal logarithmic specifically propose algorithm light tail horizon show arbitrarily allow performance tradeoff network state stochastically mab address logarithmic asymptotically regret cost light tail light tail tail sublinear tail linearly incorporate exploitation arm term preserve generalize assume fully observable link cost choose adaptive achieve link paper study bandit adversary deterministic achieve sublinear formulation version support regret nontrivial algorithm achieve term cost achieve online bad address stochastically consist
distribution invariant sampler element formally proceed next repeat write sum exceed threshold entire chain rare large increment big jump increment average mix fast big jump really invariant invariant goal update step preserve stationarity conditional invariant clear preserve stationarity step th borel observe produce ergodic ergodicity follow borel number particular order expression display bound complete integrate borel distributional step random rare event essential become section assume tail sequence vary tailed big heuristic step q note view correction factor variance vanishing complete cover heavy tailed elementary sharp algorithm long e prove efficiency computing interest queue er company density integer value sum problem additional difficulty update ease description draw step suppose k nk ty updating step proceed thus j markov chain invariant step possibly preserve stationarity write ta write last order summation expression equal b equal equal eq definition equal desire complete uniformly ergodic q require modification consider distributional rare event property therefore ease reasoning impose generate sufficient instance vary density n vanish normalize denote depend cover wide range random study present efficient algorithm complete section literature include algorithms et monte li find appear al estimator label construction simulation mcmc single variable carlo step get fair computer pareto batch record consistently event asymptotic become well event become rare geometrically batch also different estimate observe indicator well runtime batch consist importance sampling carlo mc approximation avg std avg mc avg est e avg c mcmc est std avg mcmc avg est std avg time mc avg mc avg e std e avg est e std avg avg est std avg batch consist simulation est avg per avg est avg e std avg est std avg avg est e avg est std avg generate mcmc draw lemma corollary theorem research g foundation rare underlie rare invariant generality heavy tailed walk exceed reciprocal whose vanish sum illustrate numerically importance algorithm heavy tail primary secondary carlo mcmc computing rare basic use mcmc algorithm probability outline full heavy tailed walk rare sample repeatedly simulation sample assume rare consist satisfactory unbiased estimator useful inefficient consider relative popular instead original importance exist unbiased depend distribution distribution distribution zero unknown serve select easily sample likely distribution unlikely proving error propose sample unbiased construct part ergodic roughly speak event control ergodic ratio normalize however well study context event simulation methodology problem compute heavy receive paper present markov estimator form suggest conditional belong elementary complete sharp restrictive assumption tail outline efficiency computing event describe computing contain efficiency computing present mcmc exist mcmc well importance sample rare markov heuristic fashion general computing section precise efficiency measure ignore upper lead decay emphasis normalize small choice mcmc determine dependence markov dependence high computational irreducible chain fundamental irreducible chain condition mcmc sampler ergodicity sampler study metropolis hasting xt markov chain geometric uniformly ergodic exist q rigorous markov ergodicity sampler condition ergodicity condition theorem mcmc reference therein walk section gibbs ergodic argument lead desire property estimator unbiased estimator approximate conditional choice appropriate gibbs proposal metropolis hasting geometrically ergodic problem compute somewhat assumption lebesgue integrable special density analogue distribution eq
loss context analogous coverage criterion criterion one well minimize length subject never value scalar interval utilize uncertain prior confidence period statistic theory nd theoretic motivation york w journal american statistical mm definition decision theoretic interval mathematics interval assess coverage apply find good interval interval combine bayes confidence loss combine bayes usual interval I behaviour keyword rule interval estimator author la university mail pmf pdf also good point estimator define expectation pmf pdf pdf improper finding estimator much criterion namely volume decision directly loss include pmf pdf expect yield however point lead poor confidence set set behaviour introduction prior pdf choose solve consider modification prior however generalization contexts iid let also usual tn use new section rule usual since u denote expectation distribution expect distribution posterior distribution marginal subject minimize denote substitute minimizing
convention classification semi version base classification gaussian model amongst early multivariate paper multivariate include skew normal lin lin lee distribution recent paper development gaussian clustering nevertheless certainly introduce skewness also location mathematically elegant straightforward methodology maximization outline illustrate approach work variable inverse function property dimensional asymmetric laplace density skewness distribution list prohibitive application address asymmetric density mahalanobis theorem f density shift laplace recall cf mixture distribution th parameter g incomplete computation latent step step maximize parameter iterate attain common em heavily dependent overcome deterministic annealing deterministic conjunction cf give type algorithm fit observation likelihood covariance formulae maximize specifically mix n ig ig e iterate update parameter update consist cf convergence probability cluster classification anneal increase sequence transform likelihood surface improve find determine many iteration anneal anneal ten high acceleration log estimate log value analysis herein iterate must handle specifically happen term create computational overcome search value proceed take overcome restrict acknowledge quite thorough exploration suppose membership use membership estimate model k nk membership therefore value classification analogous fashion maximize observation herein prefer specifically ig ig reflect group assess true group membership adjust rand index rand introduce partition plus apart total multivariate component skewness shift mixture perfect run ari give relatively poor ari component four component ht ari gaussian ari htb depict typical correct instead comprise argue inspection resolve old national skewness use many skewed fit classification sensible group classification associate contour skew also fit contour htb htb repository localization site development result expert illustration span alm content distinguish two protein terminal cf component mixture model choose three performance ari superiority merge gaussian ht htb would perform well criterion poor produce classification ari I compare within membership know table outperform counterpart I model site take known raise around efficacy flexible merge anneal select base clustering illustrate give whereas gaussian consistently furthermore notable model component old give membership contour reveal model capture shape far counterpart protein present difficult illustrate good performance ari outperform counterpart ari force modelling give worse expect ari model give excellent ari surprising gaussian location know question efficacy gaussian datum merge decade case substantial paradigm skew elegant computationally away gaussian mixture type mixture mixture however poor merging section introduction describe mixture g g g explore cluster conduct selection compare cluster skew university statistic
update message step exponential compute log normalize constant update repeatedly ep ep moment context cavity return moment posterior message ep compute log solve form measurement dynamical nonlinear predictive covariance respectively nonlinear dependency intractable approximate relationship functional effectively use accord explicitly linearization otherwise inconsistent backward partition gaussian z cavity obtain influence derivative I implicit linearization guarantee describe message outline require update division partition measurement make intractable solve uncertain input long however gp cavity linearization explicitly implicitly ep update remain backward take account coupling lose respectively integration inner matching instance approximation implicitly analytically approximation similarly forward involve partition take see propose directly message true approximated instead suboptimal ep context filter tp series update moment via log update dynamical write z new pass ep generalization exist g kalman message log function update dynamical also kalman smoothing prediction linearization compute influence derivative moment cavity pass formulation general solely prediction propose synthetic set art specifically linearization gp generalization ep ep evaluate measurement measurement infer latent absolute mae norm covariance subsequent ep ground truth dash quickly reveal ep matching iteratively closely average space track control angular velocity mass order control covariance measure z train trajectory length trajectory start ccc mae ep ep various ep ep across especially ep iterations linearization ep numerical typical cause exclude moment numerically coherent approximation capture remove observation process trial learn state datum ground label focus tb trial ep occur region red embed period acceleration summarize trial iterate backward infer tight ep bit ep generally pose capture test trial ep trial present dimensional inference motion example moment matching linearization posterior ep trade approximate improve generalize forward improved prediction comprise inference dynamical system inference include investigate alternative linearization computing message acknowledgement receive european grant agreement institute advance thank wang review approximation analytically iterate expectation target dimension eq law iterate target additional dimension term I take determined see conclude match kl divergence distribution conservative mass alternative way approximate posterior equivalent linearization linearize apply gp diagonal evaluated linearization approximation optimality kl lose linearization beneficial largely due simplified college uk complex series system financial market video phenomena diverse require gaussian appropriate message pass backward smoothing thus accurate latent structure result improve compare hence pass filter challenging application economic series often dynamical dimensional noisy approach advance art estimation develop novel inference parametric generalization allow time series gp dynamical broad interest develop contribution pass inference message recover dynamical backward inference system evolve measurement central model explicitly gps mean input distribute gp non restrictive assumption compare dynamical transition measurement train target function function determination covariance function account gaussian dynamical task analyze recently bayesian smoothing analytically intractable previous state
boundary origin triangle intersection triangle contain prove help reason triangle instance follow lemma proof completeness arbitrary must consider origin angle contradiction exactly I tangent suppose parallel tangent angle contradiction edge hull row conversely row intersection ray entry unit hull triangle row map nonnegative nonnegative rank question characterize submatrix satisfy condition acknowledgement thank discussion thank comment stage lemma theorem corollary question question meta conjecture definition definition theorem support part nsf dms nsf innovation fellowship solution inequality notably task inequality algorithm motivate algebraic implication need decision nonnegative yield exponentially yield establish form algebraic entry additionally large submatrix nonnegative fundamental arise admit equivalent nonnegative write nonnegative nonnegative hull nonnegative refer inner application machine complexity machine factorization factorization representative entry compute nonnegative factorization express matrix retrieval exhaustive nonnegative combinatorial polytope subject rank give polytope construct slack column slack prove slack exactly extension quantum remarkable polytope polynomial formulation conjecture deterministic communication complexity polynomially equivalent formulation nonnegative boolean boolean relationship computation nonnegative biology economic process range dynamic historical name modeling resolution priori observe equivalently system polynomial treat entry entry variable valid factorization exactly nonnegative inequality well known finding appeal say run exponential time np hard run crucial reader polynomial polynomial maximum run sense number explicit analogy lemma inequality give exponential run run exponential time run exponentially doubly exponential algorithm proof perhaps system inequality remarkably expressive believe polynomial reduce drastically many polynomial inequality probably make nonnegative rank think result nonnegative inner dimension rational bit r notice exponential exponential normal nonnegative matrix dimension crucially inner dimension determine submatrix submatrix row column row row exponential transformation apply reason previous run doubly number variable dominate transformation exponentially transformation algebraic among transformation particular ratio share normal number also nonnegative submatrix submatrix play crucial admit characterization subset netflix like yet nonnegative differently nonnegative submatrix nonnegative thought though equivalently system infeasible constraint contrast inequality infeasible denote submatrix column submatrix row small respectively call affine contain recall call notion nonnegative requirement want admissible admissible subset row nonnegative always stable preserve inner dimension nonnegative factorization inner approach lemma update update preserve throughout process terminate column ever accord order phase update admissible nonnegative satisfie hence end update analogously throughout procedure maintain dimension support monotonically decrease update row column strictly decrease order nonnegative inner dimension throughout demonstrate need furthermore list row invertible linearly set show column vector break part otherwise could strict would stability set row restrict row identity support next prove main two zero outside u u u combination w j vector support yet identical early early admissible contradict update support early linearly role encode variable attempt ensemble factorization conversely valid dimension ensemble apply vector immediate description completeness boolean vector I u c j compute nonnegative tie output ensemble stable immediately establish uniqueness vector vector nonnegative dimension factorization combine variable nonnegative guess guess semi boolean function semi empty run bind sign configuration need configuration call boolean polynomial degree large vertex cross need linear transformation hull doubly exponential semi stability somewhat exponential reduction number algebraic dependence ensemble column invertible matrix row boolean sign degree determine q algebraic polynomial output matrix compute algebraic algebraic non empty moreover lemma factorization inequality describe expense extra give polynomial inequality run extend also return approximate formulae algorithm algorithms oracle boolean maximum bit apply r
base give bind scheme point study work employ column base bound work recently pick bb vector hadamard eq proof give high rr bb observe follow lemma case vector rademacher hadamard mi mi subgaussian lemma possibly subgaussian lemma union statement choice together substitute obtain combine acknowledgement thank pointing error reference note give randomize form subgaussian mb straightforward prohibitive product specify property
plan allocation mass discuss elegant roughly vice quantity instance parameterization l evy say reach multiply time return expectation know usually large time lt calculate long hold however difficulty equation pass calculation theory illustrate movement patient heart attack patient patient movement planning resource quite analogous inferential movement patient bayesian method focus patient assume nine state care post intensive care unit care home represent quantity q quantity euler integration calculate euler give column demonstrate many computer begin calculation consume calculation time calculation discard way code spaced hour state probability line color patient home calculation obtain hazard denominator conditional reach show conditional hazard process ht markov consider interesting likely state visit show care take define choose calculation interested base uncertainty estimate full summary sample frequentist uncertainty account repeat band alternatively bayesian solve obtain wise credible interval apply parametric datum substitution transform show bellman bellman numerical laplace american network space model thesis university inversion report national laboratory nm partition implementation new york survey comparison journal physics introduction laplace inversion laplace transform relate journal inversion application transaction introduction ii york york health monitoring report national nm semi markov finance york movement volume york evy international mathematics semi process reliability j flow graph institute preliminary markov finitely mathematical r theorem process mathematical existence uniqueness york inversion advance apply sciences numerical fourier inversion concern stochastic process mat time problem transaction american transition department statistic air force base edu david national laboratory markov rich implement capability quite develop exception simple solution numerical practitioner demonstrate implement patient rich model rigorous theory semi provide rich process chain process process birth death process name survival dna real require reader list closely laplace transform property make useful pdfs go theoretic detail make state roughly transition state constitute chain time depend origin contrast state exponential continuous time state may make process time I explicitly depend instantaneous state transition transform self state never return one survival reliability analysis death process recurrent state distinguish start find state count transition occur denote notational also define operation product computation move simplify solve opinion necessary transform unfortunately numerically support indeed book bellman inversion ill inverting transform much discuss experience confirm inversion transform numerically tune polynomial need close numerical numerically euler euler method q lt without integration contour contour emphasize avoid detail exist always uniquely fundamental usefulness process convolution multiplication lt solve transform extensively paper price lt however obstacle way numerically term area research improvement paper euler euler inverting probability smooth cdf two smooth possibility euler euler approximate lt inversion integral associate euler coefficient accuracy terminology move semi process reliability might quantity time reach average spend question application probability probability process find probability demonstrate application amount state period reliability figure interest lt integral easily therefore spend item distribution
matrix know classic reconstruction task carefully orient several dl belong track track use classification discriminative inspire meta learn et classification orient incorporate redundancy noise information recognition computation sparse bottleneck focus al method contain class dictionary sub dictionary et al sub share common visual use reconstruct identify incoherence dictionary independent incoherence term I see atom different independent incoherent derive incoherence n jj incoherence dictionary feature repeat almost exactly dictionary different reconstruction absolute detect represent product atom ignore reconstruction track track discrimination track discrimination track need learn dictionary present dl propose dl logistic conventional dictionary logistic enjoy prevent wherein bilinear wherein k zhang li discriminative achieve desire support discrimination add conventional dl position element scalar control two fuse term drop original final fast image label lc method discriminative dictionary q suitable occur input share term discriminative representation encourage linear mechanism fast discrimination dictionary atom correspondence structured specific associate denote training solve derive discriminative fidelity discrimination coefficient c ii jj nearly fidelity sample code coefficient fisher criterion maximize intuitively bf unstable elastic incorporate issue relate convexity discuss utilize track review representative dl classification track track sophisticated conventional dl dictionary dl discrimination lagrange scalar balance weight mean matrix propagate make track omit note example indirect besides concern seem trade classification consuming extend research ex l l l pt minus pt minus minus edu cn artificial intelligence college technology china present representative dictionary dl sophisticated framework dl pyramid rather concentrate dl classification deal meaningful representative roughly divide category dictionary future learn dl sparse signal aim view signal sparsity dictionary classification code face example track lc dl expect extension c face
reader refer optimization application criterion aic criterion degree freedom estimate specifically bic denote freedom estimate generalize least regularize degree freedom justify local glm collect regularize demonstrate path third implement ode glm conditional completely know scale normal logistic quasi glm assume variance include case appropriately reader refer classical text quasi know predictor general form fuse trend many denote matrix matrix glm formula simplify penalty impose coefficient introduce penalize show standardize leave bic predictor enter match detail reveal monotone effect book nonlinear predictor market estimate criterion naturally fuse lasso trend five predictor penalize linear assign define coefficient simply tool monotone impose order point translate uniform I concavity translate inequality computational theoretical complexity generalize propose linearly constrain derivative loss list provide estimate constrain availability predictor market estimate undirected lasso likelihood variance negative mle precision correspond propose determinant concave apply recent attempt make primitive predictor approximate ode symmetry part column exactly nonzero calculus omit brevity derivative graphical initialize penalize objective cone explicitly incorporate path solution minimize cone symmetric path constitute penalize imply path algorithm student score mechanic algebra display path choose part attract much year involve nontrivial demonstrate applicability maximum likelihood univariate concave estimation estimation elsewhere recently active besides provide solver offer whole solution solution log estimate middle prediction error flexibility nonparametric attractive include gamma density recent review log concave logarithm concave give iid unknown support nonparametric mle continuous piecewise linear knot outside obtain consistency mle prove pointwise mle notation objective derivative derive recurrence facilitate recurrence recognize recurrence symmetry unconstraine estimate article generic implement matlab provide whole solver linearly arise application regression nonparametric extension twice objective huber robust estimation quantile generalization regularization require formulation regularization propose penalty penalty convex pose difficulty continuity fortunately enter promise constrain corollary machine prevent overfitte problem term encourage constraint lasso method develop method penalty exact solver ordinary penalty regression path along goodness fit practice couple aic bic tune generalize regression nonparametric keyword concave density equation quasi restrict framework lasso generalize linear ease tuning yet extension nontrivial propose efficient solver smooth dimensionality denote sum fold learn broad application recover encourage fuse smoothness late equality incorporate properly second encourage estimate require nonnegative regression achieve complicated constraint occur shape nonparametric regression example finite consequently path unconstrained end special regularization homotopy angle lar linear illustrate goodness sparsity sufficient linear expand wider devise generalize least generality quadratic concern attempt long piecewise propose predictor ordinary exact penalize derive loss regularize separable restriction penalty important encounter generalize aspect convex loss equality equality fuse example restrict log last path acquisition constitute whether company seven market book return record company intensive study effect company target exploratory regression explore nonlinear effect quantitative covariate vary adopt predictor say use discretized covariate chance monotonically book log utilize neighboring illustrate flow impose monotonicity market covariate enforce concavity market covariate regression covariate regularization specify equality constraint estimate increase cubic end natural cubic trend regression filter contrast bandwidth semi regression parameter tuning location knot fashion coefficient gradually become monotone covariate book covariate large enough line unconstraine constrain estimate dot line availability solution easy choose show criterion constrain path within second revealed finance company unlikely target hard meet heavy burden company high flow possess hard minimum unique third stay turn piecewise article affine e lead optimization formulate constraint residual principle convex beyond scope loss strictly convex relaxed residual along segment configuration imply continuity coefficient path lemma throughout call inactive ode segment interior segment index increase amount difference active keep active segment ease notational burden lead correspond multipli c cc solution f result path segment configuration solution ode side constant quadratic recover study two inactive vice versa detect constraint differential equation detect type event happen track coefficient active boundary relaxed segment admit current segment active solution stationarity multiplying side current active readily end set inactive set initialize ode inactive constraint p summarize proposition segment extremely use ode ode matlab mathematics numerical notably path algorithm specific solve correspond ode explicitly path ode burden develop path regularization repeatedly evaluate multiplication cost available cardinality computation avoid organize inverse suppose th diagonal new matrix kk k ij ij kk operation
uncorrelated mean study introduce mcmc initial function increase prior momentum operator hence eigenvalue operator posterior observational noise parameterization sampling parameterization target observational pick evenly inverse observational precision method alternate explicit sample show precision consistency distribution wave fouri expansion momentum figure observational increasingly numerical arise condition article paper space demonstrate standard langevin bridge ia method figure purely reference measure mcmc accept stochastic dynamical highlight langevin hybrid carlo metropolis method problem mean modification exist mcmc demonstrating efficacy method incorporation idea body theoretical desirable property highlight technology applicability wide suggest possibility numerous development modify desirable prior arise acknowledgement support fp grant st college support grant grateful financial article support skip section assumption department united result probe bayesian condition diffusion process mcmc typically mesh modify target speed mesh value algorithmic truncation prior part model flexible modelling tool range example assimilation mechanic design principle formulate preserve lead modification exist range process field prior statistical bayesian theory function stem recent problem add grow use concrete directly generate field evaluate draw probability efficiently variety purpose focus expansion draw simply exploit eigenfunction series introduce characterize either application covariance advantage readily prescribe distribution specification particular advantage interpretability g advantage context create difficulty since project pose major challenge discretization apply involve credible law express equation naturally reason I computational ii notable convenient adopt specification specification markov tool simulate application prior approach condition diffusion study generalization prior arise prevent overfitte determine informative bayesian naturally evaluate posterior almost kind mathematically infinite model become curse aim devise strategy devise dimensional possess early within modification monte substantial algorithmic speed approximate furthermore interesting construction detail prior likely absolutely continuous respect typically dominate derivative likelihood dominate natural absolutely continuous end idea introduce random differential employ proposal metropolis equation preserve reference reject lead know major algorithmic algorithm later slight walk outline end random pick variable reject pair metropolis hasting reversible u standard differ slightly type modify clear proposal reversible speed discretize natural accurately infinite robust walk field assimilation mesh dimension b show acceptance appear standard modify walk imagine acceptance curve mesh refined mean mesh refine probability curve mesh refined obtain acceptance mesh implication difference acceptance new independent mesh use represent old grows rapidly mix become great refined providing key proposal carefully measure approach new speed apply space possess measure gaussian set bayesian inverse condition diffusion goal range require respect explain principle derivation lead illustrate give deep throughout paper assumption part model detail mcmc generalization walk mala sampler metropolis hmc action section estimation arise shape problem brief denote euclidean define range give rise measure density field thus evaluate numerical method refer increase tie finite certain common property algorithmic arise frequently unnormalized form specify mainly methodology variable positivity forecasting frequently pde dynamical gain namely equation field nonlinear pair relate velocity later weather forecasting determine inverse encounter equation thus flow head water height require positivity solve pde domain measurement interest observation database inverse second dynamical match approach inversion describe outline methodology curve orientation preserve constrain curve family field choose length appropriately hilbert dynamic define solve dynamical euler lagrange dynamical initial momentum scenario j noise thus form precede concern methodology employ readily extend reference field diffusion process find solve brownian motion end arise ii brownian signal eq arise zero drift three describe primarily field denote gaussian covariance make way truncation result sum mesh building motion prior operator mutually exclusive problem possible representation naturally tool develop numerical approximated similarly naturally concern link efficient inversion develop highlight possibility eigenvalue form orthonormal draw sequence trace conceptually think sequence coefficient thus grid efficient computing exact truncation refer gaussian prior draw indicator surely formula gaussian useful conceptually think make coefficient formulation express dimension basis switch write eq consider sequence conceptually practically unknown random infinite vector gaussian coefficient prior mesh refinement demonstrate generalize walk langevin sampler hmc preserve potential reader take state arise discretization generalize langevin incorporate parameter choice algorithm introduce idea part section specify expansion briefly describe interested define space adopt transition measure take precede equivalent sense reversible walk langevin reference operator potential motion identity may square root via refer rapidly function intuitively mathematically continuity either condition generalize case application target paper show assimilation finally mention derivation limit proposal drift lead known walk certainly fine mesh lead singular move reject return method alternative irreducible possibility drift discretization spatial rearrange cn find operator also form algorithm target g prior mathematically fact look number note absence suggest far familiar argument motivate proposal many way specification efficiently proposal via write form clearly dimensional numerical show proposal improve upon naive method similar cn proposal show explain designing formula cn reference cn acceptance probability sense cn define via accept entirely happens walk truncation parametric contain sde behind metropolis adjust mala reference invariant measure function acceptance require ia define u u proposal draw measure special introduce choice proposal give proposal simply prior probability mala give useful proposal context emphasize proposal section random write hence cn proposal generalise accord function thus eq view formula metropolis sampler partition formula robust mesh within sampler base behave mesh alternate updating word alternate give consider base independent u pi precede metropolis hasting method propose walk satisfy detailed balance respect acceptance monotonic simple random purpose equation posterior measure modify gibbs conditional acceptance move slightly remain pt move mode proposal discretization measure hmc hamiltonian introduce extra momentum velocity appropriate add result remain break walk type behaviour method hamiltonian flow reason nonparametric various sampling employ example introduce highlight standard technique compare walk goal advantage algorithms partition second goal model algorithmic truncate gibbs sampler density recall indicator approximately chance two gaussians dimensional density sampler computational graphical case reader comparison quantify well experiment truncate former use covariance independent draw integrable facilitate fair tune proposal acceptance around require probability refer accept proposal mcmc importance acceptance pt correlation function integrate auto correlation significant treat function decay chain integral determine variance path average integrate determine integrated autocorrelation mix mesh mesh comment prior reduce runtime table improvement primarily cause reduction number due adaptive proceed problem nontrivial arise assimilation distribution start size behaviour proposal aim velocity lagrangian observation posterior observation observational case time restrict divergence spatial follow note adopt basis power easily eq lagrangian scenario hence surely figure term observational use velocity solve fourier find lagrangian integrate particle initial particle lagrangian evenly space evenly lagrangian condition comprise lagrangian observe evenly spaced volume
effect edge highly shrinkage shrinkage impose enjoy benefit spike prior formulate hyper pairwise plus log bias enough act uninformative represent actual distribution equation hierarchical insensitive simple setting parameter sparsity parameter learn necessity validation unlike structure edge detection mrfs due mcmc langevin jointly reversible method mcmc parameter posterior mrf fix drawing fix use mrf mcmc among langevin carlo brief noisy posterior step langevin hmc matrix isotropic discard usually accept reject detailed decay step mrf f langevin run step langevin algorithm modification markov motivate persistent change slowly sampler approximately stationary allow step every scale size hessian prior average burn period sample persistent chain help suitable momentum langevin dynamic know inefficient random draw momentum update momentum partially preserve thereby similar fashion hmc multiple correlation significantly improve mix figure mean however value update correct hasting tb pdf fig pdf momentum typical parameter langevin change reversible conditional add edge exclude degenerate proposal edge restrict explain jacobian accepted metropolis represent likelihood distribution ratio however ratio partition approximate approximation origin reduce problem estimate computing derivative expansion ratio partition log centralized persistent chain unbiased estimate variance consequently plug equation unbiased logarithmic domain unfortunately domain transformation taylor unbiased estimate ns nf consider replace variance estimate acceptance cause acceptance small jump grow set acceptance proposal truncate add estimate since demand sample enough parameter change accept move much sampler jump parallel set indicator inference partial momentum proposal initialize momentum draw draw assess real ise bias convert boltzmann exact sample two consider equally group within group strong positively couple truth ground datum mnist digit convert gray pixel thresholding value pick image pixel necessary compete bias result always neighbourhood output max min implement lasso mle use specify one bayesian pick single posterior bayes pm insensitive use use momentum proposal set subsample exact accept reject decision method sake consider vary induce evaluate validity exact marginal show sample four pick title approximate well bayes exact level value component parameter exact tb validity deviation set task recovery precision quality evaluated hold compute general intractable instead validation randomly group grid lattice I train bayesian remove edge typical precision recall tune tb fig lattice tb lattice fig edge percentage edge fully chain edge suggest sample size tendency dense range sparsity almost peak contrast data mle design mrf parameter estimation model inference generate exact bayesian model regularization sparse result dense globally fitting prior instead another automatically selective spike discuss bayes bayes phenomenon density deviation fix include real edge figure decrease however return bayesian release hierarchical improper automatically vertical fit level harmonic high corner lattice exist ground truth learn dense model method bayes pm robustness although sufficiently model e get quality without produce qualitatively compete tb fig horizontal pdf limitation experiment bayesian able parameter need hyper performance computational complexity grow clique mrfs turn regularization model cost bayesian mrfs prior achieve langevin reversible attempt mrf structure prior work present bayesian mrf search selective shrinkage mrf fitting fitting set strength bayesian insensitive hyper automate choose sparsity method upon department university california california automatically regularize model expensive parameter suboptimal regularization induce bayesian spike prior regularization suffer langevin dynamics reversible conduct hyper induce highly graphical know mrfs large variety domain social subset automate relevant become sensor attribute increase help overfitte
background recognize color able thing use good create location construction detail store location maximize usefulness percentage corruption address address indicate location corruption update location signal describe address far identify need percentage corruption create could surely satisfactory performance original reach strength hamming distance reach strength hamming wave percentage test demonstrate graph figure increase counter counter increase decrease hamming flexible storage necessity location explain previous justified wave generate input say corrupted background retrieve white corrupt white background remove algorithm identical summing pattern test radius pattern q intend give test pattern corrupt original static straight percentage corruption new see visible important emphasize number address pattern retrieval self address retrieval little pattern use retrieval go retrieve pattern percentage bit error signal retrieve highly test well decay recognize highly corrupt pattern purpose come get biology trying say way enhance memory recognize rotation signal overcome inspired human brain recognition dynamic hard location process article replicate human moreover capability promise reference neural chapter university sparse memory fully memory suit connection institute di alphabet short long architecture permit binary pattern retrieve partially match efficient non capability recognize approach purpose create signal try memory dimensional consist
geometry gaussian model intuitive determinant suitable meaning grow recall undirected model markov field family case factorization translate precision random say graph precision pair consequently sparsity inverse encode selection determine edge sparsity interest norm sense recent entry well scale et al challenging observe random need complement second least rank consequently latent graphical cast involve insight negative log proxy rank method program attractive guarantee incoherence guarantee recover sign support associate optimum recovery probability challenge potential decomposition see author overcome sparse hand quite relate result concrete incoherence selection assumption return precision norm significantly small small discrepancy dimension raise requirement possibly rank direction develop theoretic exploit model impose nonzero nonzero scale incoherence impose ensure identifiability particular intrinsic graph recover determinant neighborhood whereas difference demonstrate aspect incoherence relaxation cardinality incoherence intrinsic population decomposition identifiability could weak notion concrete begin pair imagine perturb l identifiable decomposition sparse relax requirement long proportional guarantee matrix q observation form sparse robust enforce incoherence way radius low ratio fp matrix range achieve e mass precisely plus sparse low limit thereby also nuclear estimating involve arise possibly partial accordingly seem use error bound induce second component scaling
technical difficulty spirit sample estimator find condition expansion though early involve correction necessary begin lemma couple rewrite yield claim combine expansion sample I eq see equality analogous lemma claim focus use schwarz order may entirely analogous result lemma eq sample algebraic combine obtain desire eq run sgd onto sharp rate iterate choose bind assumption eq application triangle inequality give apply h old conjugate choice markov lemma turn data point algebra eq q taylor lagrange q since old jensen iii bound inductive q define shorthand imply inductive defining indeed inductive hypothesis bound instead analogue bind statement conclude hold prove strongly conclusion argue locally gradient minimizer similar analysis optimality globally chapter note strict inequality inequality inequality divide complete appendix two moment combine proof lemma variant separable banach say matrix independent distribute apply jensen inequality see upper involve ik argument definition complete application success event joint hold side except remainder joint event occur notational derive recursive p shorthand long drop define remain jj th definition schwarz n equality indeed product similarly apply schwarz second apply linearity indeed proof final lemma completely final finish exactly reasoning remainder early zhang zhang em ex em engineering computer department university california berkeley berkeley usa berkeley communication statistical setting involve scale sample evenly perform average show mse decay guarantee appropriate decay amount parallelization attain square decay expense potentially slow mse provide investigate scale efficiently solve prediction chinese search engine logistic distribute optimization subsampling procedure define solving scale centralized minimization among infeasible keep study distribute empirical recent distribute scale survey paper therein contain relevant purely optimization explicit benefit arise statistical computational family computer must high distribute estimation limited synchronization associate simple term size machine machine certain extremely communication failure synchronization yield essentially good knowledge however rigorously generally provide sharp showing rate naive provide error decay match centralize access likelihood statistical programming attain scale bad contribution appropriate form level evenly among processor computer instead return processor estimate correct estimate decay match centralize gold first small empirical procedure section normal model relative baseline split datum investigate sensitivity amount favorable gold access sample experiment search engine click experiment enough involve sample dimension storage essential resampling give substantial naive averaging begin real integrable estimate population quantity population minimization impose parameter risk instantaneous classical estimator deal space compact convex radius addition amount curvature term differentiable exist denote semidefinite condition require method consistently sample sample evenly processor collection processor objective notation describe processor local empirical minimizer involve subsampling subset uniformly replacement local processor compute minimizer average compute standard subsampling roughly estimator argue understand condition sense centralize risk euclidean differentiable subgradient define derivative relation indicator true comparison assumption regularity function constant continuous meaning require insight analysis type smoothness condition method work necessity illustrate necessity indicator w v population strongly second unique given establishe necessity give problem logistic long rank suitable provide associate independent averaged estimate population square easy step I vector establish decay pre depend grow gradient interpretable upper loose original bind vector multiplication perform type regularity condition chapter neighborhood le addition parametric theorem eq linearity trace except bind obtain rate without calculate calculate attain inspection proof expense reduce make even note introduction certainly expect unbiased reduce variance sense distributional estimator behave average reduce desirable bias introduce difficulty note contrast classical asymptotic finite explicit square lastly tie distribution machine relatively processor close lipschitz continuity concern motivate development introduce subsample smoothness condition euclidean smooth third derivative strong mean eq constant easy also non g covariate finite moment establish bound q bind eliminate elimination subsampling subsampling minimize averaging affect select paragraph condition compare grow polynomially hand dimension local per limitation intuitive low curvature population mean risk effect per machine course allow grow total cross model leave open question compute multiple reduce opposed replacement minimizer need order minimizer sketch argument minimizer achieve statistical error provide argument argument analogous iteration thereby obtain view minimizer output triangle elementary bind iteration obtain condition minimizer share convergence minimizer one require perform particular subsection descent descent yield minimizer radius local convexity descent local strong enjoy excellent especially baseline evident degradation much high grow somewhat convexity condition satisfy comparison distinguish gap parallelization figure show grow result suffer proportional see machine expense loss square error cc machine regression cc machine estimator develop intuition benefit drawback describe remark yield aim mis experiment sample begin feature index method offer note error mean error setting subsample correction plot square curve square sample accord regression least even oracle estimate roughly agree somewhat benefit increase case choose reasonably experiment compare mis specify generate specifically dimensional choose close mis improve number machine r description token appear gender gender user word token appear title age user position page ad occurrence ad query ad title click click click ad search predict search engine click business section study search com retrieval book search present user response ad dataset user transform regressor datum description mean title bag encoding encoding assign possible title corresponding index set age interval per feature interval fall bin entry corresponding value unknown categorical also indicate combination dimension incorporate goal predict click negative loss ridge regularization strong suggest practice regularization parameter click split sgd baseline pass entire evaluate square hold specifically five fold use study compute computer consequently full pass pass sgd rough baseline attain sequential pass sdca enjoy rate hold loss error versus subsampling plot well proxy substantial even split give performance pass enjoy gradient since pass gradient give minimax ranking may wish direct prediction end area curve auc auc bipartite broadly show parallel split sampling explore split plot hold set versus subsampling ratio split com mis specify appear inference challenge solving grow grow fast speed storage capability computer performance oracle able access interesting remain problem datum far generally communication study environment informative concentration thank point mistake statement relate feedback fellowship facebook office smoothness present attain processor inspection odd subtract normally binomial calculation somewhat eq allow control moment argument local objective processor event begin inequality relate bit algebra give eq q average statistically remainder intuitively convex begin hold guarantee closeness rough event behave population close guarantee ball u guarantee three follow lemma previously rely relate minimizer chapter care locally appendix careful expansion via initial
justify layer generative model provably close optimal justified provide encoder fine training deep model highlight rich model contrary much hard look possibility test auto complete rbm create outperform state art stack rbms layer future layer selection algorithm low extraction part display color color display color remark architectures generative propose optimistic proxy interpret auto generative stack rbms improve highlight importance hide rich generative mistake comment tag comment tag deep architecture layer network object help representation consume require expert difficulty whole network procedure justification fall somewhat short expectation cite log add reflect validity wise deep new criterion condition optimize deep derive ever intractable complex simpler wise optimistic train subsequent training successful relation restrict boltzmann auto generative hide generative scheme auto approach deep synthetic real auto auto stack auto architecture much rich latent rich inference model basic architecture traditional optimizing model probability generative learning estimate deep architecture distribution latent variable separate parameter deep recursively interested interest observe layer quickly present frequently architecture stack rbms leibl optimum obvious tackle parameter impractical able deep architecture wise bottom distribution variable reproduce recursively replace surrogate objective architecture stack rbms rbm maximize ignore moreover approximation layer improve follow question parameter latter convenient layer reduce hyper search aim layer optimistic section stack rbms auto p ascent space deep ascent require influence perform display training successively train conditional variable bottom part tractable infer bottom optimistic assumption cf provide go realize value obtain enough guarantee suppose trained think color think new top train reproduce perfectly optimal used difference optimum kullback leibler strongly suggest original recursively top fine tuning fail global layer wise training distribution priori version optimization subproblem solve sequentially layer train successful auto keep layer indeed layer layer unchanged suggest rich conditional contrary practice auto expressive prop solve optimization part generative rich reach use overfitte come family relevance representation learn irrelevant experimental evaluate bottom layer whole able produce max likelihood bottom subsequent may converge max display really redundant redundant coincide color remove really redundant really statement show low coincide theorem marginal thus bottom one optimistic assess bottom optimistic display ok ok consequently maxima argument particular writing may help incorporate crucially rely practical optimize architecture already optimize possible probability see optimize conditional represent proof bottom latent assume layer use display ok ok whole take proposition display remove train reproduce perfectly case reproduce perfectly prop propose perfect color ok prop bind prop perfect give reproduce keep fails reproduce global optimum importantly optimize heuristic use may relevance compare optimistic want dp hold reverse optimum express leibl performance difference kullback leibler divergence appear ok nice argument color ok nice precisely loss respect abuse understand obtain log sum I exactly characterize conclude display ok ok ok check ok stack rbms stack boltzmann rbms stack rbms wise rbm learn target layer distribution generative top top bottom rbms bias bottom rbms never train maximize generative generative weight tie first layer different future comparison ascent ascent generative layer ignore tie rbm use top rbm initialization rbm new deep still tie different low layer associate upper account likelihood might actually test see suggest tie part optimality guarantee deep auto credible stack rbms train stack train backpropagation layer reproduce auto p auto training layer error encoder train model auto encoder manifold scope possibility learn top rbm generative concern theoretical stack deep model auto rbms expansion likelihood keep sense stack maximize follow commonly sigmoid give let keep op perspective consider intermediate wish f g training backpropagation raw optimize particular conclude statement keep justified approximation justification underlie situation perform choose form could possibility explore impose wise stack lead tie layer determine mutually follow sense maximize training wise try train stack rbms wise hide rbm know optimal maximize encoder weight criterion rbms coincide tie retain stack rbm train approximate rbm distribution rbm stack rbm tie weight encoder see full rbm clear extent criterion yield optimization criterion train consistent though seem unlikely layer match perfectly nonetheless wise consistency rbm layer likelihood layer fine tuning layer case target distribution non require upper deep model layer incur upper able subsequent consequently layer respect procedure fine backpropagation confirm expect recover principle limit auto encoder layer wise deal issue local yes ok transform yes ok yes except include different term version qp maximum sufficient w vanish variation quantity display let take quantity maximize since ok calculus variation ok variation statement proposition general ok optimum ok optimum prove incorporation display equivalence critical likelihood point constraint follow critical likelihood constraint log vanish elementary multiplier qp n critical point incorporation display look ok detail ok sure empirically deep make evaluation latent new log yet enough dataset indeed try assess modified auto explore bind wise give future log dataset give reasonable picture sure deep spirit rbms evaluate likelihood train stack rbms cd equal consistently single rbms hypothesis deep learn namely architecture capable represent compactly architecture layer epoch give obvious head number l rbm layer layer hide layer cd rate bp epoch epoch dataset dataset probability baseline model dataset scheme give equal independent pixel bernoulli log rbms rbms confirm deep check deep likelihood rbms distinct test part image lie amount capacity one give display air color train propose bernoulli adjust validation sample concatenation log validation stack rbms confirm check deep validation rbms deep auto layer auto use display keep mind possibility bernoulli bernoulli kind new rbm stack study model focus equal although increase rich layer procedure circumstance optimum deep gradient ascent ideal deep remark increase model could greatly positive exploit modify auto use usual auto hide tie depth stack rbms ordinary rbms auto rbm sake generality backpropagation rbm train likelihood validation set distinct comparison generative find optimum propose approximation really layer reproduce auto study approximation provide comparison stack cd cd general training auto depict figure train adaptation bind backpropagation tie generative weight encoder rbm hyper stack rbms bp learning epochs ann standard initialization training figure backpropagation cross loss maximize auto encoder upper weight tie auto model rbm train size deep auto comparison figure pareto front display dataset distribution describe average validation log pareto generative dataset generative deep auto baseline independent single rbm evidence deep auto low rbms compare optima auto outperform rbms auto stack rbms consistently arguably auto framework universal significantly improve rich auto us rich auto generative exactly value likelihood raise whether indicator approximation w model many though enough unless otherwise turn likelihood estimation various measure maximization procedure globally experiment result actual definition assumption wise ideally idea architecture use layer validity maximization possible learn approximation conditional second training relationship un optimistic tight part reproduce practical generally low situation refer optimization might reproduce perfectly layer introduce could high validation distribution bind ideal definition optimistic final obtain perfect upper either class poorly actual layer upper several effect go affect really predict context architecture hyper selection involve generative intractable sometimes visual rare method train criterion maximize parameter exponential evaluate layer otherwise hide always color hyper evaluate layer always become rbms intractable prevent dataset evaluate summary wise hyper layer upper evaluate sample distribution layer experiment robustness hyper mention representation irrelevant train generative representation lower log assess quantitative discuss rbms compare actual model training checking result instead
weight decrease singleton entire removal increase consist maintain singleton usage clear context reward record object terminate backward low cost remove explain low element currently singleton singleton add backward current object correspond row backward forward execute thus backward pair decrease convergence backward forward share threshold backward factor singleton singleton reward q find record break stop estimate support find break backward new share true element correct interpretation main quantification share state scenario randomness gaussian restrict eigenvalue constant minimum task gradient specify towards magnitude large entry entry magnitude constant bind noiseless yield small fast algorithm fewer backward however need thus range complexity contrary zero time row optimize row distinguished atomic setting fall algorithm weak assumption hold proof b inspire j c see exact never consequence stop previous backward fail go lemma bound complete em divide eq inequality imply converse ii setting stopping comparison suppose share portion support location value entry noise draw substantially interestingly tend phenomenon around stable pick oppose single row single different define rescaled version parameter plot success outperform less know theoretical sharp sp sp model sharp suggest problem design sharp ss I gaussian greedy substantial conjecture sharp threshold size show sharp conjecture precision open average classification error handwritten digit optical handwritten digit handwritten task collection digit different people digit otherwise disjoint report set classification find average distribute task change get error terminate forward fail go entail loss function separable next lemma bound estimate parameter stop parameter optimize rhs stop carefully choose would arrive hence q conclude terminate backward fail next consequence stop j hold stop reach lemma arrive contradiction assumption forward step support begin rhs conclude error support forward identical step assumption lemma immediate backward separable column fix jk entail notice always large provide element rr omit b super sub forward optimizer equation latter term first I j claim c theoretical term sake replace hold provide lemma appendix satisfied sample assumption noisy measurement backward operate distinct object iterative addition removal support exist algorithm complexity interestingly extend greedy structural infer observe accord design column interested infer inference recovery potentially small number feature arise recovery recognize regression range support handwritten character course indicate infer jointly often task learning attempt iteratively drop algorithm
loop generalize predicate atomic predicate try infer loop invariant predicate consider abstract incorrect conjecture form x observe variable share state characterize denote I induction versa induction hypothesis formula I statement induction also x atomic predicate find formulae atomic predicate inductive follow show see characterize loop annotate proposition condition loop incorrect conjecture teacher return abstract otherwise applicable loop annotate loop inductive condition nevertheless pair atomic predicate incorrect assume incorrect satisfie necessary collect atomic predicate predicate inductive corollary mi abstraction abstract section try boolean resolve check weak teacher abstract intend equivalence teacher teacher abstract truth abstract abstract abstraction give another chance refine abstraction inconsistent atomic predicate inductive generate atomic predicate abstract formulae recall inconsistent inconsistent predicate loop generation algorithm predicate annotate formulae equivalence first atomic predicate section initial set loop invariant exception predicate invariant case find predicate learn find contradict abstraction distinguish predicate find loop invariant exception query generate coarse abstraction predicate start predicate number boolean support incremental improve overall threshold atomic predicate abstract equivalence resolution box teacher abstract return conclude compare conjecture approximation atomic predicate fall approximation possibility set predicate sufficient iteration infer solution predicate insufficient predicate arise learn algorithm invariant abstract exceed threshold atomic predicate abstraction refinement intuitively predicate insufficient random abstract exception atomic predicate approximate space solver query example add extract translate annotate loop manually average collect ghz intel cpu r case atomic predicate choose atomic predicate pre loop statement automatic interestingly predicate suffice loop fail due ill predicate infer loop invariant example atomic predicate thank small example take order give abstraction addition preprocesse outperform three case tie atomic predicate generate always item buffer keep buffer item buffer copy item atomic predicate program text express invariant specification successfully loop invariant invariant atomic predicate text generation find loop benchmark atomic predicate predicate atomic predicate program text loop invariant find invariant contain predicate suggest redundant predicate predicate make loop easy follow loop summarize implication summarize true atomic predicate execution range chebyshev loop probability atomic predicate invariant algorithm device figure generation perform predicate atomic loop surely atomic atomic predicate atomic needs query loop generation present technique apply atomic predicate imply text efficiency code report need realistic example implicit predicate text additionally loop quantify quantify gray ac com national version publish support center education science national award dual address generation loop invariant abstraction refinement interpolation program text effectiveness learn way annotated post annotate loop post specify intend behavior annotate loop specification verification tool whether annotated specification loop require intelligence recently automate technique base abstraction atomic predicate annotate atomic predicate teacher able loop construct technique abstraction atomic predicate crucial learning extract atomic predicate text atomic predicate express invariant infer loop invariant heuristic atomic predicate redundant predicate algorithm generate atomic predicate technique atomic predicate interpolation logic formulae formula inconsistent logical symbol occur interpolation order formulae inconsistent many software interpolation atomic predicate refinement interpolation atomic predicate loop algorithm add new predicate execution learn new generation effectiveness efficiency loop atomic predicate loop loop body decrease become loop iteration eventually loop explicitly atomic predicate express establish specification atomic predicate exploit pre loop inconsistent extract predicate x interpolation loop specification introduce technique algorithmic technique quantify loop invariant atomic predicate address free recently technique termination technique loop termination combine algorithmic paper author design transition predicate interesting adapt invariant inference interpolation implementation base refinement software check abstract may abstract logic quantify generate predicate template review learn base loop interpolation generation automatic conclude denote free logic equality inequality rational number formula free write evaluate satisfied formula return inconsistent logical symbol occur formulae third condition make generation symbol observe condition specify loop formulae loop particularly atomic predicate invariant annotated loop atomic predicate enough predicate least one propose algorithmic technique abstraction decision teacher answer abstraction engine combination predicate state software check base section review invariant atomic predicate formulae free loop adopt abstraction relate boolean formulae teacher guide boolean formula inference show level view loop invariant framework loop teacher teacher course loop invariant try answer information program text teacher employ suffice framework traditional formulae boolean interact teacher query abstract query teacher otherwise equivalence teacher exclusive abstract abstraction boolean remain teacher guide abstraction loop free loop annotated formula weak loop membership teacher whether approximation inconsistent simply membership yes randomly approximation loop membership query give query give accurate exploiting well learn orthogonal static analysis answer equivalence indeed pre post solver resolution return find
classify validation classification include major challenge accuracy assign everything total accuracy fail capture diversity vote require vote maximum weight grid finally forest use information differ classifier cell examine neighboring cell predict cell question majority smooth noisy classification second pass weight city demand class reduce overall display incorrectly classify however spatial intra relatively total accuracy com matrix com com tendency consideration leave share remain four display result large account mix use correctly classify sub classifier exclude mixed incorrectly classify onto without affect heavily classification accuracy time classify nature actual differ classification accuracy increase heavily area plausible change finally analysis reveal fundamentally different mobile phone activity heterogeneity cdr potential infer act guide suggest mobile phone activity measure heterogeneity simple broad classification update traditional well reflect activity planning large expand aim relatively low aid fraction city high resolution additional balance interest ground may mobile phone measurement cdr hope location public private resource develop apply novel mit fellowship national research fellowship collection analysis manuscript understand people city crucial planning create currently sensor gps device collect massive amount communication million record information utilize mobile phone relationship population course week cluster mobile phone show mobile phone capable useful database application database depend turn city live location understand individual efficient planning choice influence determine demand maximize popular location maximize access city city office kind usage business office hour relatively different might differ somewhat intend information area note part dedicated relation use human traditionally via survey survey require subject record move day whole week survey method expensive limit give survey thousand capture period fortunately past decade nearly country currently phone purpose understand particular call cdr provide location mobile send obtain cost aggregated area level risk information question arise whether mobile usage measurement advantageous result share usage whereas usage usage monitoring allow shift development work apply aggregated cdr infer dynamic I area city supervise region cdr use classify reasonable normalization application forest mobile human first utilize device mobile phone activity university student regular daily moreover pattern level student upon extent et nearly anonymous mobile phone reveal persistent human human song et mobile mobile phone use al mobile phone link mobile phone mobile phone activity km km activity qualitatively city decompose activity usage et cdr mobile phone location similar qualitative similar technique analyze profile cdr proven detect movement census call across area attempt associate source learn date exist employ traditional mobile phone activity partitioning region region profile active order identify pattern characteristic specific correspond two datum mobile phone region cdr mobile phone strength unlike cdr record location location mobile phone provide accuracy continuously across space locate phone activity count call text cdr roughly home million mobile phone mobile phone activity obtain classification office actual proxy actual impose obstacle study phone phone activity record pair partition phone population rarely influence due transform cell lattice size test coarse level enough mix reduce phone compute hour mobile phone certain analysis give single classification area large heterogeneity densely characteristic reflect census phone activity block display actual frequency grid vast percentage examine mobile phone activity macro city mobile phone activity average cell count classification differ greatly maximum hour cell activitie huge order typical mobile phone normalize unit profile remarkably city fall user also partly phone across normalize activity residual interpret mobile phone activity mobile phone city hour mathematically cell time average pattern average notable relationship residual activity behavior higher visible area early activity level reflect subtle phone hour suggest area city mobile phone link high level treat phone activity proxy spatial people give pattern concentration people work shift behavior visible residual activity volume day affect mobile phone persistent phone hour week display activity top activity plot absolute activity density order city logarithm city strongly dominate usage spatial nevertheless dominate difference different perspective region km spatial activity rich early hour activity locate activity become heavily concentrated later activity area suggest activity residual
refer show well leave simplicity value important fold give fluctuation run transform regression since perform high transform change dramatically advantage well across decrease whether l normalization frequency good range classifier combine performance transform keep refer transform transform project smaller omit generalize dataset omit bayes classifier feature reduction transform run bottom axis refer dimensionality effect lda names drug dictionary protein drug logistic lda count treat lda omit outperform binomial omit count binary incorporate via occurrence weight count feature produce document cross weighting tool run except lda binary occurrence significantly datum feature run transform apart run drug drug also several transformation normalization number principal reach investigation thus enable support drug interaction prediction mining patient record drug drug interaction deal drug mechanism efficacy drug aid kind extract publish clinical database drug inclusion still preliminary area benefit literature mining mechanism significance classifier identify relevant document identify causal mechanism drug drug interaction important linear classifier dimensionality investigate benefit publicly dictionary find distinguish relevant give well alone normalization adjust improve classification linear proper dimensionality large help classification drug medical rate refer incidence include aspect drug g medical record database drug drug detect signal database become methodology scale extraction information domain database ultimately research genomic drug collective database especially relationship complementary conduct independently research automate literature method whose mechanism clinical conjunction previously gap parameter probe ki ic text mining approach may particularly causal behind complement mining patient reporting bias previously show automatic literature work orient toward perspective first relevant contain evidence goal automate report evidence towards integration mining li working develop goal report manually wide logistic regression support machine binomial previously find well mining transformation normalization technique use name describe corpus section deal cover discussion li automatic extraction retrieve article manually classify drug irrelevant article contain one four class study clinical clinical drug initially one extensive well detail request field author title subject mesh field si latter contain code biological entity entry process certain convert less character occurred omit token document occurrence combination run classifier simplify angle version threshold proportion occur proportion document otherwise full version protein include additional account entity occurrence dimensionality linear svm interface cross validate logistic regression interface validate naive bayes naive validation discriminant lda covariance toward shrinkage validation validate shrinkage toward equivalent naive multivariate follow occurrence rise occurrence occurrence inverse document feature term apply document total minimize difference feature normalization projection onto commonly precision
parameter relationship hold transformation involve parametrize present w great deal lead discuss fall multiplication wrong definite movement suboptimal use make rarely thing help deal remove dependency inherent topic precisely remain head around provide review ill analogy widely concept whitening problem connection whitening begin poorly mean datum negative parameter much sensitive change step instead large nearly movement get via slow show illustrative parameterization effect parameterization dominate remove descent path blue section initialize fisher notice descent quickly gradient describe descent converge directly value difference dependency assign metric define choose magnitude representative change result uniquely correct objective well g whiten analogue covariance signal remove dimension remove dimension whiten quick covariance inverse mahalanobis whiten symmetric zero whitening refer rescale axis unit inverse rotation signal orientation unitary transformation linear mahalanobis variable whiten common preprocessing signal processing perform direction find write update gradient illustrate function direction parameterization effect formula specific minimized learn take natural metric even metric accelerate fisher require instead empirical alternative approximate field gradient practically simply ignore work surprisingly greatly square problem compute however gradient convergence application evaluate may find condition eigenvalue eigenvalue tend worse dominate deal subject rather unstable give plug play call robust approximation small behaved ridge regression useful gradient gradient frequently cause gradient pass true space natural
recover equation rise compressed sense paradigm signal admit ambient measurement denote compress possible nonzero entry recover quickly instead relaxation also es stable measurement take alternate minimization weak regard recover incorporate algorithm weight x recently sequence minimization belong update performance iterative call two weight converge update algorithm solve subproblem overview sequence subproblem detail preliminary demonstrate recover signal incomplete scope result leave index complement refer van find efficiently spectral determine l pareto trade least square one initialized point hermitian transpose proceed prove pareto curve continuously solution l rise guarantee root method generate ls expression example pareto root solve weighted lasso subproblem arrive update subproblem l exactly lasso ls find generate contain large vector weight subproblem different iterate lie curve pareto switch r problem coincide signal pareto l solved switch path weight apply support path subproblem note axis one oracle curve support oracle ambient varied still recovery clear far show less signal every subproblem allow improve test compare minimization recover synthetic
ref problem introduce demonstrate building consider decomposable ise spin vertex together publish broad transfer discuss distribution inductive summarize conclude hierarchical work string candidate string accord candidate promising learn solution candidate encode diversity new candidate incorporate use restrict termination global optimum number reach represent solution structure acyclic specify direct conditional specify value variable string represent probability parent use internal variable variable leaf assignment node unconditional test split repeat goodness knowledge metric relevant combination favor simple exponentially description complex probabilistic probabilistic crucial building probabilistic population small population learn experience address basic experience examine bias search instance transfer building focus identify bias structural future analyze build implement one sure collect solve work pair classify describe classified predefine fitness q string may additive typically prefer difficulty subproblem subproblem even subproblem order paper variable create denote path variable q subproblem distance length variable mainly subproblem additive metric correspond locate interact confirm numerous spin dimensional lattice ex describe inspire al run apply process introduce start model split compute dependency split recall quality probabilistic contain population prior statistic ex split normalization constant contribution experiment done know evolutionary three ise spin consider periodic boundary condition unique minimum cover node ratio unique instance map create combine graph nearly regular lattice half refer reader preliminary copy ensure population minimum optimum independent run hill hc incorporate cover vertex cover ref use use define effect mean random instance equally sized subset subset run subset analyze remain round instance small run experiment perform across computer configuration base case always run computational node two therefore cpu execution speedup respect execution run speedup multiplicative factor execution improve bias base execution speedup indicate base percentage speedup addition ability base prior apply run instance examine finally examine combination building delay suggest carry building source multiply due requirement effect ex ex confirm ref examine bias yield obtain minimum vertex improved improvement observe much majority instance substantial instance size instance problem vary argue nearly multiplicative distance cc cpu speedup cc cpu cpu c cc cpu speedup c cpu cccc speedup paper extend optimization derive substantial applicable execute combine technique great several topic central technique linkage
relate orthogonal aspect interpretability human view label visual interpretability closely member semantic interpretability view function projection axis attribute experiment visual interpretability show dataset ask participant aim investigate view aim vary level generic order interpretability overview interpretability label address detail human view dataset group relate automate human expression automate discussion visually interpretable view manual view impractical degree apart visualization lee al exploratory pursuit fisher discriminant method et al search projection dataset evaluate neighbor user et propose two measure centroid base entropy class author measure alignment finding view good participant al separation compare measure suggest combination investigate oppose visual interpretability label axis address interpretability propose simplify coefficient et dimensionality reduction arithmetic assess author contain feature less interpretable however report related interaction investigate human develop human computer complexity develop system mathematical expression visually presentation notion human infer experimental participant expression expression human derive calculus represent automatically experiment without intervention detail design execution along automate visual interpretability complete degree field physics biology account participant ask fill course experience participant mine commonly mine visual table dataset shape affect relationship automate measure diagnostic breast cancer dataset characterize chemical three contain seven kind path uci learn repository eight nine country diagnostic breast cancer order view automate choose visual design assess usefulness extract classification assess mention use measure quality include three visual propose measure user present list automate utilize name section vector index consistency measure attribute experiment value automate ensure choose diverse automate measure create width bin bin frequently range across automate measure upon experiment label build observe class unseen item two category generative generative infer mapping regardless common feature classifier usefulness base method feature view choose common algorithm characteristic decision boundary generate dataset machine tree generate boundary decision boundary characteristic separation near neighbor label weighted measure generally algorithm utilize assess choose create partition algorithm build internal split dataset respect partition decision boundary attribute tree bayes algorithm simplify despite bayes algorithm outperform variety machine technique search separate two class utilize minimal problem item quantify cluster algorithms validity indice interpretability validity aim separation class scale rate five view separation automate interface build rate user calibration pre select display randomized outlier median user comparison automate automate strong good automate relationship human table c support naive k near index histogram measure lda value support bayes lda near histogram naive support consistency lda rmse naive k index machine lda indicate fit rating base automate without consideration view near individual notice long match might member class overall lda human seem extent match automated shape member therefore human derive measure individual ten automate median cast linear automate leave six ten see combine human report composite winner composite measure dataset observation estimation naive exclude favor df interpretability understand user understand mathematical transformation characterize projection automatically intervention use interface connection visualization take experiment start participant interface expression consist five variable logarithm root participant expression expression include analysis expression depth size e l tree avg block time spend write z x log u participant inform mathematical expression possible numerical square power expression display second expression back ask rate easy understand interpret difficult show first calibration participant linearize specifically choose display division fraction create easy expression multiplication record spend rating understand disk manual inspection study response correctness summarize rating median take rate number examine expression rate participant frequently correctly expression incorrectly rate participant pearson participant consistent observe behavior answer incorrectly meaningful rating participant much hard interpret present utilize tree expression compose depth block indicate expression participant rate expression easy rating rating operator total rate expression rate participant c attribute operator total tree avg cast learn mapping interpretability participant leave q predictive rating number block df human long expression interpret block increase relationship human automate measure interpretability exploration interpretability interpretability concern easy looking argue validity machine assess view besides compare automate human indicate single outperform extent linear combination automate correlate well interpretability human original investigate human would expression operator participant rate rate long
form overlap segment incorporate nominal prototype separate approximately extent separation similar fix dictionary basis background approach dct basis dictionary initial column frequency audio separation separation whose amplitude symmetry retain separation identify nmf estimate separation qualitatively propose aim separate source online generally separation domain overlap spectra signal separate note hand learning implicitly basis overcomplete amplitude spectra still furth time experimental investigation blind application domain edu blind source propose sparse recent effort representation key background separation partially source problem domain demonstrate separation separation representation blind separation separate signal comprise superposition source bss arise call audio perhaps know independent ica gaussian separation factorization appropriate factorization nmf sparse source separation mixture separation knowledge source manner aim partially background source novel unknown source describe source tool task domain effort audio processing law device utilize device load encounter device qualitatively low audio periodic approximately load audio separated subsequently classify separation would facilitate accurate remainder work description motivate aforementioned audio conclude remark fix decompose partially motivate audio application comprise underlie continuous time consider small prototype observation base approximation generality local inherent let evenly equivalently length segment goal effort essence entail leverage contribution partial column dictionary column express broad approximation recent describe effort decomposition effort put contribution represent linear separation employ simple variable low approximation particular via optimization square matrix interference cause accuracy svd comprise amplitude noise may numerous traditional setting effort aim denote nuclear relaxation non sum absolute essentially count pca possess representation source imply represent cosine transform suppose sparse may directly approach implicitly priori restrictive dictionary column aim base dictionary accomplish respectively comprise column pursuit pursuit denoise form priori represent background approach approach semi try represent semi blind modify decomposition column learn column form assume express superposition possess estimate estimate base alternate minimization coefficient know type approach comparable pursuit omp iterative fashion outline follow subsection lack make depend initialization strategy datum unknown strategy coefficient essentially learn identify estimate dictionary word update extract repeat iterate demonstrate
video video rank knn transformation fr mmd change problematic capture set knn mmd fr video subject assume mobile phone datum set walk individual feature magnitude raw approximately represent whole gender walk retrieve high precision mid early point early recall reflect flexibility similar large goal find group group variability group provide sensitivity score reflect allow flexibility quantify estimate neighbor neighbor point describe capture distribution distributional distance work include provide give significance score dependent dimension domain statistical gain bootstrapping addition interpretation attractive lastly insight discrepancy area represent hypothesis hypothesis test significance normalize apply hold reject error bind differ result reject even though grow value get decrease need ensure rate alternative similarity reject test may principal rejection area change projection projection aid distinguish define value let mapping let distribution projection expansion follow denote ks inequality apply combine conclude proof far surely projection probability least denote notice let fs source randomization projection equality eq clearly depend q therefore infer whether test type level significance sample unit ps ps corollary assume threshold moreover ks theorem consistency bound projection decay power projection clarity thm suppose I let prove totally presentation space define volume box intersect cover mass ba z discretized version distribution discretization turn discrete lemma relation structure two initial refinement discretization discretization split equal difference discretization refinement norm let variable proof lemma yield recall number discretization lemma combine union combine union least get proof belong element ba ta da exist z add bipartite graph case decrease decrease hand discretization x neighborhood bin result histogram may cardinality large claim discretization useful may remove absolute feasible optimal sufficient solution let constraint fix show exist obey stage know result difference bind solution describe obtain constraint constraint optimize plan discretization k bb cardinality feasibility feasible feasible z ij k leave side problem q whose existence td td infeasible feasible next show solution feasibility equality hold conclude difference discretization refinement substitute assignment inequality element sum hold solution procedure optimal feasible solution inequality lemma hyper size definition spherical point unit sphere let neighbor neighbor union bind cardinality neighborhood match q equation department electrical title proof lemma theorem definition distribution sample much research determine score optimally score hypothesis set come simulation detect example mining vision scenario adaptation da intuitive input yet whether generate work similarity statistical procedure score however score statistical testing similarity equality sample generating test one equality equality transform similarity see work similarity predefine physical expect represent similar distribution apply measurement change people put name discrepancy two distribution distribution difference difference specific figure blind follow put distribution minimize cx wasserstein problem mass function rewrite dx plan mass distribution perturbation area equally costly due function triangle quantify intuition similarity consider similarity tv characterization plan costly measure dirac delta perturbation therefore wasserstein tv explain sensitivity optimize plan allow perturbation aspect contribute problem unify neighborhood point mass neighboring non depict scalar objective plan perturbation plan scale width pt plot coordinate optimization identical account difference optimal integral whole solve appropriate let link link assign edge version program assignment typical problem demand example complexity bipartite cardinality score may test notion mmd capture rkhs distance perfectly equality mmd rbf advance highly dependent clear domain accordingly capture case rate let cardinality disjoint cover n rewrite metric change theorem exploit use dependency box inherent namely curse theorem sphere least identical I theorem converge expectation combine difference exponential distribution intuitively dependency bootstrapping bias present different possibility project appendix material type complementary dissimilarity hypothesis p relax distinction similarity flip role nan equality test supplementary material inference bootstrappe ci bootstrappe approximation resample replacement computation
simulate iid sample replication four sample observe plausibility suffer coverage length final ability plausibility function coefficient goal select suitably collection plausibility immediate profile distribute wrong subscript evaluate singleton zero plausibility coefficient type strategy choose numerical analyze diabetes average pressure six tc possible make much figure select comparison four bayesian diabetes section point relative likelihood unbounded although depend potentially quantity respect integrable denote hellinger define metric constant denote universal assumption precede paragraph constant large omit space last outer center hellinger bound exist na application give plausibility vanish plausibility singleton consistency mass vanish hold extra plausibility region rather abstract comment family ii second control theory transformation transformation chi square converge pointwise support conjecture stochastically large check numerically claim I remark frequentist program inferential framework frequentist plausibility plausibility finite sample justification extension plausibility carlo pearson program frequentist error rate despite strategy exact inferential derive asymptotic normality inaccurate challenge require even preferred arguably perhaps issue claim correspond make fail somewhat numerical method exact frequentist property desirable approach construction frequentist assign plausibility datum plausibility use whenever plausibility plausibility function follow hypothesis confidence plausibility control rate sample large sample justify efficiency plausibility plausibility intuitive implement frequentist result problem generality make method appear previously interval certainly new critical version plausibility paper paper nuisance plausibility interest exact sampling plausibility function structure discuss demonstrate efficiency region test theory plausibility method good practically effect model propose exact bootstrap conclude remark give section unknown say e function indicate reasonably sort square negative minimizer relative choice q often difficulty define function act plausibility claim singleton I plausibility variety problem test plausibility base q intuition plausible outside plausibility test control plausibility construct confidence plausibility plausible value connection show region nominal level sampling plausibility unify minimizer plausibility empty likelihood contain plausibility plausibility precisely outside parameter value plausibility plausibility compare asymptotic ask plausibility region plausibility general neither figure plausibility single portion around plausibility understand complicated plausibility properly explain phenomenon play however suppose convexity concavity chi limit one expect plausibility size sufficiently see besides tool frequentist plausibility potentially deep plausibility function inferential employ plausibility continuity statement make sense stochastically large monotonicity may continuous jump know stochastically term plausibility achieve immediate singleton important estimation singleton continuous stochastically general rest go coverage probability plausibility plausibility frequentist furthermore moreover corollary bootstrap analytical validity suitably function particularly efficiently wrong case mild mean plausibility percentile appropriate chi display region similarity plausibility display asymptotic efficiency conclusion region precise plausibility I rigorous plausibility valid fix size motivation investigation let iid version decrease law probability vanish probability atom continuous plausibility correctly distinguish plausibility far difference strictly tool empirical also require quantile propose derive rare need plausibility remark number unless example herein conservative evaluate plausibility equation approximation interesting depend indeed monte substantial transformation transformation tie together model transformation result group transformation certain depend check loss invariant special result suppose dominating respect hold depend establish multipli immediately success fundamental statistic widely substantially coverage likelihood give numerically mass plausibility plausibility plausibility unity plausibility binomial gray plausibility monte binomial almost indistinguishable coverage plot coverage claim simulation interval particularly yy exponential plausibility normality parametric asymptotic normality bootstrap plausibility interval iid unknown gamma literature e evaluate plausibility illustration present survival time certain plot plausibility show jeffreys along elliptical plausibility elliptical roughly region plausibility coverage mark normality maximum likelihood gray jeffreys bayesian posterior consider covariate probit iy ip distribution coefficient plausibility function illustration real relationship exposure death otherwise show plausibility comparison normality plausibility confidence region indistinguishable likely region region nuisance parameter component case kind easier interpret plausibility nuisance example negative profile replace global maximizer obvious carry plausibility turn marginal plausibility q carlo check nuisance straightforward property theorem carry exactly provide plausibility corollary hold rare write corollary structure interest model transformation namely composite give follow invariant marginal plausibility composite see case iid chi reach weak effect turn suggest plausibility convenient justification provide one different might less choice likelihood sign convergence suggest quantity use conjunction bootstrap scheme limit special particularly illustrative unknown relative profile usual residual monotone squared see plausibility interval efficiency plausibility independent nonparametric empirical probability th show quantile plausibility essentially binomial problem bivariate distribution correlation nuisance calculation correlation transformation gaussian convenient illustration replicate probability plausibility display fisher normality approximate normality approximate sign log work parametric percentile bootstrap digit reasonably suffer plausibility range take last correspond plausibility base straightforward evaluate check nuisance shape I negligible iid size robust fixing reasonable
two sale volume namely sale bipartite graph product degree bipartite sale output prediction predict cross sale co sale aggregated category relevant collect book week user category time sale category category setup type include feature involve descriptor matrix contain past sale vertex growth degree market relevant underlie factor instance cluster proxy useful quantity infer future connection simplicity assume node ii indicator eigenvector contain velocity acceleration history average quantity accounting representation fouri polynomial include task completion link vertex asset future appropriate lead graph completion aim prediction relative wish predict top sale volume component assign label magnitude sale volume specific compete compare baseline ridge address time prediction hypothesis completion dedicate link prediction denoise ignore feature version aim singular uv ta uv order convex standard objective denoise separately emphasize decrease appropriately compete baseline observe rank advance book category sale volume good graph f benefit joint regularization represent relation goal advantage although identify case contain optimal explore greatly method determine within domain key contribution lead thm link predict application node formulate hybrid graph predict joint empirically synthetic real node feature forecast system collaborative market bioinformatic rating activity technique apply multivariate multiple autoregressive develop approach enhance assume encoding dependence graph inference graph latter interesting se part recommender appropriate completion realistic setup evolve matrix adapt account dynamic graph simultaneously vertice network cause forecast sale market reliable user product evolve history expect item sale sale induce corresponding item situation arise financial reflect dependency stock predict stock price dependence show available efficiency multivariate enhance broad scope objective predictor adjacency mention factor create set discuss implementation performance objective simultaneous feature experiment synthetic object interest nonnegative unweighted one time represent vector depend function qx tt goal predict future adjacency introduce descriptor encode matrix value qx ff estimate past th equip n indicate value edge large value prediction vary feature local change reasonable evolution govern unobserved evolve smoothly account choice factor measure adjacency coefficient good indicator popularity refer centrality trend read degree singular adjacency formulation set underlie mechanism rigorous history factor assume partly contain assumption collect feature satisfy assumption rank rank factor growth growth exhibit regularity frobenius regularity close predict consequence stationarity mechanism graph correspond base reflect eigenvalue define regularization sum per allow adapt separate correspond task rkhs static reflect predict term reflect assumption matrix minimizer subproblem simplicity rkhs represent regularity degree q w frobenius work adjacency inter intra degree projection statistic length interestingly improve optimization norm term instead vertice issue discuss learn standard gradient project challenge ensure desire domain happen minimization functional degenerate eigenvectors iff define add define trivial henceforth take slack q schwarz basic eq cauchy q let w choose replace nonsmooth nuclear smooth spirit eq lower bind approach envelope unit ball spectral differentiable case relate multivariate vector center also eq randomly assumption linearity make laplacian non book sale volume item aggregate category test book aim predict sale book predict sale co set zero user define product node co product sale volume book target sale volume user algorithm output prediction predict sale sale category relevant book category week period entry item sale time include adjacency involve past realization feature direct volume sale popularity market measure factor structure cluster proxy infer future assume contain indicator several cluster eigenvector descriptor contain quantity make recent average account fouri prediction could criterion optimization evaluation focus completion link task asset least square well use relative graph completion aim report pattern predict instance item market sale volume component product magnitude sale volume address graph completion one direct compete compare consider baseline x account future graph choice low hypothesis dynamic graph shrinkage dedicated denoise ignore hypothesis obtain singular value uv td uv implement cross objective denoise synthetic lagrange coefficient efficiency minimize objective validation set appropriately outperform compete item nuclear free well understand due high efficiency predict future graph benefit simultaneously time series represent relation investigate feasibility convex greatly method condition global attain work approach problem future methodology far predict link dynamic solution reflect paper formulate learn structure predict benefit prediction evolution challenge collaborative filter market bioinformatics response movie rating stock price statistical technique multivariate either multiple regression develop enhance frequently assume encoding correlation discovery unobserved graph per se occur application recommender tool among evolve completion adapt account dynamic goal simultaneously predict evolve one make effect cause sale market semantic reliable evolve history expect sale sale induce corresponding financial graph reflect dependency stock aim predict price infer enhanced tackle broad regularize risk objective formulation adjacency separately convex synthetic mention factor create set discuss implementation issue objective multivariate simultaneous dependence structure joint relevant synthetic notation paper undirecte weight edge unweighted edge th row represent feature node series series n qx recall operator graph standard computation follow eq w frobenius tensor node inter intra onto specific cluster statistic later choose
method add run noise level require impact quality impractical moderate give approximate produce mechanism account dimensional bias towards pca chain carlo simulations subspace produce variance algorithm privacy understand well lower differentially private show sample differentially require nearly theoretical demonstrate experiment many depend minimal differentially private pca grow several question suggest computational privacy interact place mcmc use implement investigate set guarantee angle develop theoretical general challenge provide notion extend use finally many choose look differential privacy selection differentially private manner learn database survey find survey strong guarantee attack definition privacy syntactic definition attack individual database several differentially approximation perturbation desire computation perturbation add exponential base measure output variety mining task differential paper deal differentially prior calculate decomposition mechanism subspace toward private exponential run datum unclear may implementation differentially private reconstruction provide hold plus additional term depend privacy power calculating differential release transformation preserve privacy measure utility example preserve differential privacy privacy base create privacy reference preserve mining set study singular publication differ privacy release pca actual datum private individual let moment frobenius reduction low rank matrix much dimension schmidt decomposition area science suppose top subspace large refer quality measure maximize result top close top preserve privacy privacy quantify take privacy supremum measurable interpreted numerator take privacy privacy privacy privacy differentially individual private close well individual guarantee privacy relatively perturbation therefore characterize scale differentially modify query private pca private privacy differentially privacy top apply privacy differential correspond eigenvector pca easy differential privacy gaussian necessarily eigenvalue entry output dimensional differential utility sample matrix privacy focus special output mechanism mechanism private expensive computationally implement recently propose gibbs gibbs popular assess burn add change induce copy copy copy sensitivity median smooth sensitivity difficult function private solution problem perturbation require property namely convexity norm pca sample differentially show scaling require despite privacy even algorithm pca challenge pca manifold private differentially fact differentially follow calculation complexity privacy correlation gap hold satisfy pca differentially certain technique q differentially private utility eigenvalue gap show scale private scaling match contrast number certain level favorable bind satisfy notice dependence reduction space dimension suggest well application guarantee differentially private match showing nearly privacy column column step mechanism geometric lemma unit sphere bound probability union spherical complement output good boundary maximize minimize sphere fact bind simplicity q private collection correlation collection differentially private eigenvector computation characteristic polynomial also follow product give expect utility average database differentially private radius disjoint hold inequality norm product database database differentially private basis product eigenvector small coordinate packing sphere span weak bind eigenvalue requirement copy copy moment q construction obtain upper bind eq q follow plug put thus differentially consist copy copy database eigenvalue eigenvector q additional guarantee differentially distance eigenvector complexity illustration tradeoff calculation vector achievable orthogonal first loose still tail set ij j yield rearrange differential formula particularly expression differentially provide true top eigenvector privacy parameter space pseudo construct set matrix matrix algorithm input input use pack minimax utility bind private element top covariance expected lemma show small loose set yield low symmetry q density lemma vector inner euclidean satisfie condition theorem low impose condition yield large dominate set eigenvector produce horizontal axis four correspond plot lead plot figure order correlation logarithmic range experiment limitation particularly subspace present good dimension rich regime favorable little algebra q concern large eq get involve mcmc burn must reach stationary theoretically chain reach must interaction differential privacy rich report choose four domain connection medical instance individual activitie dataset contain mix feature discrete preprocesse dimension respectively normalize row maximum normalize utility account four fairly c implement gibbs sampler gibb widely scientific machine finding mcmc still open area much time weak practical user iteration employ diagnostic empirically chain stationarity provide appropriate burn distribution perform qualitative develop characterization gibbs sampler procedure come execution privacy implementation guarantee note suffer privacy try burn confident performance affect burn run copy trace chain initial uniformly column iteration ht ti data gibbs dataset dataset illustrate behavior plot show datum start column rapidly case number sample sampler thus sampler dataset chain good column fact distribution private try generate utility maximize subspace reflect subspace although hold practice datum run several randomized result utility size subsample non private nearly illustrate blue utility subspace well produce subspace utility par choose exception produce furthermore run burn suggest differentially result immediately average bar dash blue utility green indistinguishable utility projection respectively randomness bar standard mean pca algorithm dash line indistinguishable datum onto datum summarize svm private predict majority label classification half onto base use repeat projection projection dimensional algorithm well random projection accuracy compare private pca subspace classification application understand utility subspace point draw
matter select epoch bit datum hashing substantially loading present accuracy sgd datum hash achieve accuracy present loading table datum bit hashing loading dominate achieve accuracy h epoch transfer gpu result make bottleneck similarly dramatically resource requirement online due reduction loading epoch use hash reduce incoming various query classification etc communication hashing approximate similarity bit hashing critical must bit application especially bit preprocessing compute permutation require concern parallelization scheme substantially load beneficial g page hash increase online major hashing substantially reduce amount memory batch become bit improvement hash dramatically epoch significant often epoch reach impossible space study bit hash g u hash family hash technique similarity context matching redundancy file management spam etc substantial bit regression binary task enable massive expensive show million disk gb format crucial apply bit mainly focus view set q permutation practice store storage prohibitive low bit convenience theorem permutation hash vector new point consist expand binary run day inherently example discuss distinct beneficial hashing inner linear naturally suitable application big impact operate linear time linear package dataset regularize linear solve eq regression solve penalty elaborate major issue bit hashing scale bit hash step costly value permutation large concern bit hashing accord classification appear signature signature page still hardware improvement signature reflect expensive preprocesse change data user testing affect new incoming process signature computation graphical offer compare current parallelism access gpu many especially benefit gpu characteristic suit gpu main parallelism hashing typically hash minima speed applicable hashing mention online become increasingly web loading avoid overhead store large memory architecture demonstrate hashing also beneficial substantially load dominate training cost especially epoch overall hashing serve scheme text storage researcher develop loading e loading long g million hash basically matlab verify permutation order alone impractical resort simple hash universal hashing function randomness hash study report hashing reasonably hash reduction unless fairly limited memory beneficial reliably replace massive hash large storage difficulty perfect common standard universal u number choose increase randomness universal u hash storage store hash hash gram gram gram binary feature iterate section describe graphic processor sketch far hash suit implementation purpose recent primarily high cost offer increase computation graphic limit datum memory suitable consist number cycle processor execute locality consecutive access access gpu bit read set disk gpu compute minima gpu hash minima minima back process transfer block memory main within gpu consecutive gpu internal oppose access perform time access parallelism inherent gpu limited capacity even impractical store crucial simple hash function operation expensive trick simplicity note hash odd much integer evaluating compute p else integer evaluate bit trick v summarize datasets gb study pairwise combination product product expand dataset testing l gpu platform processor unit memory bandwidth gb intel processor overhead cpu implementation break load memory use function mod hash bit convert operation bit operation note report mod table preprocesse u loading take magnitude u hash hashing substantially operation avoid million feature permutation actually algebra hash storage impractical gpu processing demonstrate second second fold observe version gpu implementation fold cpu gpu preprocessing substantially load achieve preprocessing become h loading gpu gpu bit figure hash operation separate overhead spend memory gpu minima back memory leave panel adopt batch gpu reading disk gpu appropriately affect gpu report range cost affect reasonably e may practitioner spend gpu speed vary significantly spend order spend process gpu key cpu implementation resort hash hash practitioner validate hash impact hashing bit addition provide another bit hash hash experiment long hash estimate fully random permutation section list sparse dataset gb million choose conduct cpu ghz gb ram windows u scheme svm interval experiment repeat experiment whenever page experimental reporting validate conclusion accuracy average case correspond practice scheme slightly simple u hashing scheme appear well domain whereas hashing show variance provide hashing practitioner please randomness accuracy affect fully implementation hash meaning hashing early limit expand training experiment figure time exceed merely accuracy accuracy accuracy achieve accuracy next hashing algorithm influential hashing bit hashing conduct two large gb use hash train training gb svm logistic solid dash curve bit hash range accuracy value hash substantially storage substantially storage bit hashing compare bit plot bit select h hashing bit hashing see even bit hashing course
image average improvement almost summarize compare result bm average berkeley varied step bm train also outperform bm mlp mlp medium outperform art level good recent tend equally raise inherent denoise db outperform respectively capability patch noisy assign patch bound structure synthetic denoise bm already close rich geometric rich estimate expect yet db mlp rich assumption quality db db db db db bm db db db db db db mlp db htbp db bm db db db bm db value dependent db bm derive perform patch patch tight version theoretically achievable use db bm db bm bind test train bm summarize table noisy version image bm result note quite db relatively small agreement report much bm achieve par achieve bm mlp achieve visually improvement patch patch unable tight bound large size reduce clean bm bm bm use patch step effectively support patch see estimate bound step mean db patch bm difference achievable bm suggest quality achievable patch suggest bound lie db achieve noise level remain bm furthermore patch suffer return particularly become patch observation area figure statement denoise return available increasingly suggest patch denoise mostly flat area mlp area denoise corrupt independent imaging corrupt shoot denoise assume transform imaging handle db db db however difficult transform specifically design effectively denoise simulated noise effort adapt procedure general yield good image whereas also violate poor adjacent canonical mlp train column correlate image corrupt high bit result remove filter filter boundary bm vary db train million outperform middle problem except assume position pixel pixel truth filter db mlp quantization filtering transform dct transform thresholded bm value hyper procedure table mlp match patch omit match plain mlp achieve mlp plain mlp mlp mlp outperform mlp plain mlp large house approximately plain mlp db repeat find house far superior matching mlp ccccc image bm mlp bm db db db db db db db couple db db db db db hill db db house db db db db db db db htbp mlp plain average dataset large improvement dataset mlp db match plain mlp average htbp image berkeley dataset mlp match plain db image lot mlp db mlp explain fact patch blind surface average achieve mlp plain mlp plain surface block repeat matching allow outperform matching plain still achieve average matching learn feed architecture toolbox cpu add noise mlp run produce start load denoise noisy clean image perform I noisy take contain step achieve denoise test image image level perform house outperform bm approximately bm bm method repeat house advantage method low still db bm level exceed bound cluster addition suggest room improvement bm see approach image bayesian patch result superior assume medium yet method great improvement estimate patch infinite size progress toward reach approach reach half gain bm agree room improvement complex see merely achieve result quantization poisson two case seem competitive state art improve little complicated long useful kind addition plain vision clear block encounter worth specific knowledge engineering denoise mlp take approximately cpu gpu bm fast hour cpu architecture layer patch surprisingly explanation operating often box behavior understand extent bm db db advantage sometimes bad art happen lot regular attempt successful ask result combine denoise good plain mlp image outperform achieve theoretical toward gap approach extensively study poisson excellent well combine certain training inner mechanism layer denoise poisson clean count corrupt accord capture digital usually white procedure noisy image approach smooth part noisy rely wavelet conceptually image patch bm successful rely patch elsewhere bm similar image noisy patch noisy patch rely natural achieve bm procedure purely free noisy clean instance noise clean map complexity practice image noisy patch clean patch give patch patch affect quality denoise patch clean potential noisy patch patch invertible almost denoise noise clean explanation least denoise patch clean easily create patch patch denoise choice influence complicated high whereas difficulty capacity usually require model patch model bad potentially denoise possible state art perceptron mlp map patch onto capacity mlp mean sufficiently unit choose patch contain recover free agreement finding enough high capacity modern graphic base plain improve thorough poisson discuss strength work limit two hard limit make step reach third achieve thorough experiment remove extensively image diverse denoise aim smoothing class exploit patch statistic set berkeley exploit image involve wavelet coefficient patch learn neural network belong image image category type cnns various task digit traffic sign cnns exhibit design compare plain result amount layer potentially cnn think cnns kind report strong layer wavelet attempt incorporate auto auto neural pair add art result performance rather representation noise auto become standard deep apply patch learn however learn level correspond overview result base natural image art obtain assumption rely pure define denoise problem noisy patch patch different dataset mlp nonlinear via mlp hidden layer mlp operate wise mlp hidden layer layer mlp hide I hidden layer could imagine potentially train noisy image remove patch patch white feed noisy mlp represent patch mlp parameter update map patch clean patch error maximize backpropagation transform close pixel multiply weight use normalize uniform eq neurons output combine trick ensure part reach division divide modify rate discuss layer hide capacity layer need approximate function provide sufficient neuron compactly would require practice unit noisy overlap normalize patch subtract dividing multiply add place counterpart overlap region weight window patch third slide window patch decrease factor able approximately slow bm intensive operation mlp multiplication operation suit efficiently implement mlp gpu used factor core cpu speed factor allow image unlikely identical amount effectively could image exploit channel publication imagenet imagenet hierarchy image category transform grey scale define six algorithm set select select imagenet set use become diverse lot contain structure choose thorough well compete choose five image performance method choice train extract noise resemble mlp form detail result follow dictionary noisy noisy approximate gaussian novel image sometimes bm state denoise bm prior rather fact similarity rely block patch state exploit use approach bm also excellent choose bm usually least result achieve art introduction learning approach rely bm bm ccccc mlp db db db db db db db db db couple db db f db db db hill db house db db db db db db db db db db ccc bm mlp db clean bm mlp mlp denoise mlp architecture detail mlp denoise image clearly inferior bm house contain ideally suit like bm adapt noisy outperform image suit type note every image cc htbp approach bm five image imagenet dataset db berkeley db perhaps berkeley imagenet plausible imagenet contain regularity bm outperform bm dataset db berkeley db highlight row create mlp complicated structure mlp advantage bm comparison db improvement berkeley dataset db small improvement db initial train outperform art denoise consistent tend bm contain perform patch frequency blind frequency structure match bm patch image art noise level low high extremely describe architecture patch various level ccccc mlp db db db db db db db db couple db db db db db hill db db db house db db db db db db db db
goal good construct graph bp walk addition knowledge exception restrict class learn complete overview mention discuss present precede deal road scale real specification notice without adjustment propagation lead relate bethe heuristic new one organize review mainly base perturbation expansion develop refine ise explores refine expansion coupling expansion dual propagation complementary particularly graph give give gradient reduce tune field sake adapt standard induce solution inverse comparison make section consider spin historical mean resp maximum impose field multiplier associate entropy minimizer log likelihood lead ij note entropy distance empirical ise mean evaluate make transform impose expectation duality e yield solution connect matrix relation gibbs small coupling expansion correspond contribution expansion successive letting subscript expectation successive derivative eq expansion quantity eq convention approximation invert coupling require need mean field inverse vanishing coupling bethe inverse parametrization q relating give ise model neighbor notation neighbor give explicitly invert response result expression albeit suitable discuss ise appear hand side existence check directly give relation non vanish invert full equation nevertheless restrict way correspond construction simply bp equation heuristic proceed lead coincide introduction correspond ise temperature refer state physic introduced originally auto ise sum outer spin inference interpret axis introduce pattern axis coefficient highest plain bp bethe modal coincide attractive basically temperature topology four one factor conditionally tree yield recurrence notice tree tree possibly order term elementary building block coefficient notation generalize point relate basic eq along intersect case distinguish align intersect vertex corresponding lead leave aside introduce name continuous aim construct identify conditionally covariance variate precision matrix pair selection empirical derive easy conduct statistical goal discover conditionally penalize solve eq determinant regularization orient since positive definite thus definite construct norm know element cardinality matrix intuitive infeasible solve norm penalty practice use continuous solve exact optimum guarantee enough firstly square name lasso penalty encourage variable far basic norm allow entry weight constrain penalty diagonal avoid unnecessary introduce structure exact scalar see penalty play envelope norm convexity convex penalty sound also underlie corresponding regularization perform linearly increase result w issue penalty element suitable assumption bethe approximate check sparse either bethe direct graph belief fix belief satisfy constraint meaningful propagation unique allow approximated procedure amount select mutual find span link indirect invert potentially visible correlation indicate well weak assume equation iteratively propagation justify assertion like pairwise variable joint link optimally multiply loop good mutual read good maximum weight relevant classical inference dependence tree start want add produce corresponding since derivative attain correction log sort link quantity current yield distant target contingency table estimation mrf compare bit rapidly inaccurate link incremental propose construct indeed factor read pair precision reference form complete zero add variation give therefore determinant add modify way iff definite one ij ij individually deduce give finally fulfil degenerate desirable remove link desire remove factor assume precede likelihood read rearrange property observe compatibility able use strict connectivity precise condition validity sufficient definite aa modulus wish increase modify link assume sufficient impose express determinant ji ji simply impose track remove wise elementary link strong constraint easy compatible present remain equip define graph correspond link compatible wish repeat reach restrict several intermediate update covariance link coefficient select highest compatible respectively modification repeat convergence one absence simply walk satisfied implicitly slope want backtrack allow converge connectivity adapt dynamically adapt let carry toward connectivity concave pareto front boltzmann trivial point order yield order either bethe give explicitly empirically following consider individual expectation let ise fulfil ise correction implicitly set reference start bethe coupling model expansion expand assume follow follow free response bethe point gibbs energy moment bethe bethe coefficient log give quadratic contain hessian likelihood parameter else parameter associate convex find solve resort mutual partition decide global perturbation form discuss soon ultimately max parameter information read q interpretation direction exact fisher computed way edge link connect group link bethe approximation links information define coupling field consequence drop implicitly maintain impose reference suppose single temperature practice correlation capture belief modal average intra precede follow bethe equation proper counterpart inter cluster single intra cluster former bethe co formula yield superposition higher superposition along indicate natural field case obtain spin variable temperature regime coupling use correspond term spin configuration path closed express last run subgraph empty denote edge loop propagation correction generalize loop subgraph reduce loop cycle span loop convention edge superposition figure therefore weight dual primal lattice duality allow express j ise spin ht loop cycle combination case via disjoint cycle basis still basis link cycle link cycle read loop level constitute systematic take cycle general link cycle factor interaction cycle involve expect strong pairwise dual pairwise graph wise dual response read read arise part involve order precede solve eq contain restrict basic cycle cycle might break rapidly cycle start sized remain pass restrict exist cycle actually sign possibly proceed ordinary belief define corresponding message message eq update obtain cycle result belief function belief interact connect complete shown figure bipartite graph indeed case propose literature decomposition linear cycle far determination concerned cycle propagation precede suitable relevant proper might propagation weight possibly approximate graph edge share cycle dual denote edge dual factor cycle replace product formula read interest generalize expression weight message message factor read adapt expression generalize concern function unchanged generalize degree factor combinatorial burden come back write separate odd eq cycle extend cycle interaction concern lead derive share modify read factor represent dual factor order distinguish send cycle send cycle come weight dual partition read q measure cycle extend fix response ht extend care dual coupling see figure cycle cycle take last pseudo variable measure consider node absence free read involved example possible take formula give latter use joint present context exact weight form determination limitation possibly expression arbitrarily limit present doubly lagrange programming convex split introduce augment minimize function obtain proper initialization optimize jointly respect usually optimize alternatively give lagrangian successive result scale augment lagrangian log determinant programming inverse problem challenge differentiable make furthermore determinant feasible approximation definite determinant term give continuous domain speed procedure hereafter symmetric matrix last iteration auxiliary variable decompose mainly determinant part separate penalty single descent version penalize attack challenge cause exact make name approximate calculate explicitly order optimality unbiased ol estimator square order taylor penalty intrinsic relation explain perform definite impose accomplished perform calculate order optimality eigenvalue optimization clear guarantee contrast definite control margin increasingly iteration make converge gradually reduce margin definite sparsity constraint geometrically predefine upper optimizing reduce gradually simultaneously penalize square regression besides analytical compact functional clear intrinsic robust superior penalization make taylor expansion penalty within good initialization taylor read eq scalar constant solve w j wise weight zero configuration influence self pruning entry magnitude free unnecessary robust induce outli non magnitude design robust stable correlation fact weight expansion robust robustness entry compare step condition therefore cubic define entry cubic rapidly within besides decomposition multipli penalty pareto curve optimization different procedure converge validate penalization norm conditional jointly use cost search optimum avoid vector complexity improve efficiency solution learn name work compress equality penalty penalty design satisfie restrict isometry restrict isometry sometimes necessary norm likely inverse ise ab width cd various rmse plot finite zero b solution assess find actual center determine bp bethe propagation bethe precise field show use method conjunction bp bp bp co beliefs conjunction inversion yield absence field less local bp belief case field vanish bp compatibility approach propose build link schema mrf norm link axis version constraint explain compatibility marginally little maximum span schema bethe costly schema difficult validity bethe control instead favorable compute show datum produce simulator inference correspond traffic consist link extract large albeit simplified network paris link snapshot give link gaussian heavy tail take individually cdf center study actually mrf task display various one solution comparable greedy full pareto figure level
reconstruction e reconstruction solve clearly interestingly estimate almost generalized variance set correlated error generalization immediately multivariate dimension still residual addition improve incorporate nuisance parameter post quantification posteriori application robust formulation obtain portion derive formulation term model parametric posteriori formulation constitute portion student density give smoothly heavy near gaussian also kalman smoothing meta work fix show inverse treat nuisance parameter score estimate freedom residual q dimensional solve routine box source place vertical consist pick pick motivate inversion wave model receiver simply velocity ray velocity arrive ray geometry perturbation interest velocity ray operator transform medical imaging figure source inversion invert primary grid frequency output stage know avoid minima result starting recover important square approach design recover optimization reference cite reduce project project design arrive computed evaluating need update example represent weight inversion might possible find many involve primary significant calibration great inverse dynamic quantification overall optimization square carefully name idea entire corollary immediate ability modify fit nuisance practice c inversion variance dataset gauss cg calibration source bfgs match development present alternative derivation covariance interesting note estimate freedom fix match mass library knowledge degree freedom taylor inversion freedom view engineering discovery collaborative grant research project b cc residual allow home ignore cc lemma corollary nuisance direct primary parameter strategy projection nonlinear square idea broad ml map freedom nuisance large unconstrained inverse variety include optimal design improvement primary compatible minor modification primary parameter secondary degree freedom many example separable square extensively year class parametrize form class major insight reduce long note fast approach descent instance inverse provide detail include degree freedom automatic calibration robust paper section theory nuisance variance freedom formulation important detail see variance student formulation illustrate contaminate automatic illustrate image dependent weight application discuss conclusion easily rather work reduce follow adapt continuously differentiable continuously differentiable function gx differentiable condition exist smoothness hypothesis gx suggest natural design minimize whole consider method specifically derivative gauss type bfgs bfgs allow exploit project modify enter computation next constrain additional constraint normal minimizer x g first unconstraine systematically minimize problem avoid modify provide work formulate framework via forward reflect discrepancy variance affect true come source parametric student measurement freedom variance unknown nuisance estimate propose remainder provide source student estimate important drug approach gauss numerical importance parameter error share eq q modeling operator ml
concept name answer decide deterministic hybrid kb kb dl extend dl safe dl overcome limit dl safe keep scheme still semantic stable semantic predicate interpret predicate reduce analogously answer reduce statement show answer boolean problem currently expressive logic intersection lp lp inferential mechanism induction prominent however distinguish concept individual presence depend classify orthogonal dimension discrimination clause unit clause aim induce classification positive finding explain observation definite clause cover entail interpretation hypothesis cover concept traditionally view partially ii searching clause clause clause partial pair clause usefulness among chapter definite clause standardize apart program substitution c order syntactic admit structured hypothesis refinement function compute generalization search refinement refinement couple bounding clause iii concern syntactic clause search link link link link occur link link occur occur tight scheme attract attention build upon comparative attempt reader closely integrate arise many sort language dl refinement chapter discover frequent pattern dl safe rule rule focus discriminant induction interpretation hypothesis non head play coverage relation one interpretation hybrid generality relation hypothesis generalize procedure coverage relation relation processing enable rule hypothesis represent clause link range clause language name check constrain resolution coverage relation interpretation query answer oppose precisely variant mining pattern conceptual order description pattern unary refinement rule framework hypothesis inspire operator top bottom analogously differently assume coverage generality query answer reformulate show value dl induce dl head flexible integration dl cl safe evolution l kb kb rule hypothesis language recursive rule target interpretation interpretation description extension answer answer ref yes see concept change business trend usage consistent evolution preserve consistency distinguish conceptual specification representation consider conceptual due fact become consider role fact know need together already available crucial requirement follow definition induce concept role kb building rule cover part name kb adapt upon p mm mm r concern generation rule p predicate l admissible restriction head guarantee satisfied concept rule belong represent hypothesis contain tuple occur tuple incomplete reference hand need equip mapping example extended background theory note reduce answering kb reformulate cover nm reference example cover nm satisfy cover cover cover definition choose deal adapt characterization result generality reasoning derive standardize apart kb substitution say denote exist ground substitution variable immediately answer easily prove also quasi refinement top language build disjoint u intuitively semantic fulfil specialized note refinement rule apply mm produce mean refinement h h e onto previous definition rule classifier loop sequential covering learn remove covered rule space perform iteration increase instance view hypothesis begin specific general cover positive inner perform fine grain determine search conjunction hill begin rule promise step consider choose cover degree iff iff increase degree favor confidence cover degree take classified concept require deal unlabele outer start refine specialized cover hypothesis stop cover assume cover rule pose researcher cl community area acquisition bottleneck severe open demand task rule cost address relational framework match art research rule viewpoint suggestion framework implement intend specific section augment expressive thank evolution role operation kb comparative framework review common trivial semantic clause framework framework deal differently expressive dl cl language recursive integration scheme head requirement hypothesis take account course may turn case nature intend impact process rule differ greatly finally consider loose theoretical devoted concern expressive hybrid framework dl many web specify w work group theorem di e mail complement refer rule onto dl language support relational within logic programming database demand engineering automate though inductive machine programming relational adapt onto sake specific onto relational definition dl concept inductive logic programming rule language rule representation read sentence rule field logic lp database play architecture format activity perspective integration source scalable manner core standard standard extension unify expressive follow extension logic monotonic around top aside extend indeed rule mean frequently inference constraint discover new fact expressive see nm likely hybrid system integrate rule interest chapter shall specific language integration rule basis kb constitute vocabulary relate simple demand project produce approximately hour hour rule feasible partial task ml introduction even guarantee correct critical relational rule ml seem particularly promise reason ml lp recognize major approach distinguish form prior logical theory induction chapter proposal learn relational proposal difficulty program atom correct query answer ground query logical consequence answer denote consider advanced handle semantic program presence inherently conclusion alternative accept semantic extension semantic integration research system two deal kb reasoning motivation develop namely crucial semantic loose semantic kb cl share particular dl cl hybrid kb semantic requirement know respect property hybrid
regime graphical relaxation method separate model undirecte observe equivalence dag member technique conjunction standard refer norm matrix form row column position zero th row cardinality vector element diagonal product observe circle minimum cm draw fill black draw observed name observe name joint distribution know de imply exchangeable conditionally see illustration modeling latent occur total topic view generation realization represent topic word draw topic topic occurring draw encoding moreover allow basis framework variable moment recover natural degeneracy topic full degeneracy hope distinguish topic become moment word hope word unchanged correspondingly identifiable appropriately hope canonical follow transformation unchanged sign degeneracy second moment col identifiability col provide term condition topic proceed key establish expansion form otherwise sd ns degree term close necessary weak multiplicative property last generic assume mild perturb independently fix variable prove expansion word describe column column exhaustive search provide state need j represent word child row gap permutation word least column n iv correlation column differ definition set arbitrary section essentially novel traditionally lp inequality often fast projection iterative thresholding accord learn topic hide degeneracy assumption one bayesian model include topic topic bayesian incorporate causal direct acyclic graphs applicability intelligence social sciences structural economic parameterize specifically ss dag dag hide hide dag topic generality hide variance respectively moment variable convenient triple inner consider dag variable skewness identifiable prove skewness chi topic dag independent similar ica also dirichlet allocation lda suitably adjust need impose topic since directly proof skewness third moment observable column topological find pair uv sphere let sake direction vector power iteration describe orthonormal repeat framework combine statistic suffice moment I e accomplish word denote multi tensor hide dag observe provide remark oracle topological ordering need topological dag node result order dag suppose topological ordering induce dag node permutation dag identifiable framework identifiability topic expansion framework recall topic linearity exchangeable degenerate view word unlike topic single observe view random n uncorrelated model hide uncorrelated independent furthermore matrix non let matrix setting application sound source assume topic presence remove linear relationship linear structural progress identifiability paper contribution discuss remove noise fix third looking represent rank extract noise observe decomposition partition return c ab da fix return defer low diagonal diagonal low part I I u ib partition full show fixed let thin orthonormal column fix choose uniformly random put row submatrix fix full partitioning least denoise e recover noise matrix appropriate rank procedure section expansion generic coefficient matrix consist layer formally linear level within level node concern model specifically suppose induced condition hierarchical level induce model satisfie observe e entire node arbitrary relationship theorem section view third present proof emphasize validity general bring require reliably concrete example hierarchical model node require validate consider contain coefficient represent among construct accord gaussian I bernoulli entry indicate make expansion assume positive sake simplicity consider gap gap keep entry unchanged complexity algorithm similar select include chi equally denote chi slight variant make robust essentially er er present find partitioning row coefficient observe moment lead partition estimate coefficient partition due finite necessarily realization realization scaling estimating coefficient return remove ambiguity ambiguity since precision recall characterize support retrieve true edge retrieve summarize result ccccc recall recall sample around line far learn node hide level induce level relationship node observe triangular entry normal coefficient matrix describe relationship experiment gap maximum maximum row previous uniformly family chi describe summarize cccc recall acknowledgement david discussion support award nsf award award fa award nf complete microsoft research observe lead contradiction combination follow multiple linearly different canonical sign aa mm scale row n iv proof equivalent variable l observe aim notice inequality solution optimality claim b z z z twice older moreover put strict contradict optimality scale th ready hold least loop choose linearly hence let ai k lemma decompose part diagonal since pair part decompose low svd kn key determine singular singular sphere q rotation sphere wu ai demonstrate theorem expansion return column permutation inverse ai ordering dag direct triangular topological proceed column get way correspond dag exist exactly zero triangular version row row correspond parent remove node dag equivalently eliminate different triangular topological denote form associate hide write lemma decompose rank argument proof column inverse notice recover moment permutation column entire identifiable within contain fully submatrix ready hold every henceforth observe fully clear subset correspond submatrix nan space conclude let ic dense one dense vector consequently complete vector entry diagonal separately recover leave put correspond definition topological order topological order hide topological triangular matrix entry zero triangular require invertible invertible invertible identity distinct I ib ab thin indicator variable ie I moreover least complete proof chernoff independent topic pairwise correlation iterative column permutation let return denote l w l pt plus pt minus plus corollary proposition numerous statistic estimate broad topic linear order condition identifiability weak expansion constraint direct variable approach topic latent study recognize incorporate model latent datum effect far predict concept human affect abstract belief organize incorporate model incorporate incorporate set causal acyclic dag applicability intelligence sciences economics datum discovery hidden estimation dag suffer model criterion observe third dag roughly speak property influence identifiability dag probabilistic correspond observed model allocation topic dirichlet approach lda low order correlation g topic establish identifiability provable e word topic determine arbitrary impose expansion constraint word support different topic guarantee identifiability topic moment model identifiability degenerate bipartite graph impose word neighboring require intuitively number contrast note subset topic degeneracy identifiability necessary identifiability model degree identifiability expansion word span establish word strong expansion call parametric degeneracy learn topic matrix moment obey gaussian characterize topic however impose modeling via incorporate propagation show sparse graph topic model spectral propose correlate linear discuss rely blind deconvolution ica assume general impose linear view component cross moment topic diagonal prove relax condition strong incoherence condition set column full column decompose employ decomposition apply novel third order connectivity focus correctness plug address difficult reliably technique involve precise deal I matrix tensor future study propose year overview recently provable overview approach theoretical
visualize type cluster low cluster ht discuss later eigenvector could expensive algorithm appear first summarize experimentally second cluster modern business like company company replace present method separate object near method aim square intra distance contain index denote vector locally chapter repeat set point mean cluster toward true perform shape incorrectly perform apparent search pdf point datum end entry similarity induce complete object partition wish identify objective cut way guarantee simply minimize function cut perhaps trivial cut divide sum association normalization one bias pdf minimize np originally due relaxation eigenvector similarity matrix define n degree argue small relaxed assigning index eigenvector eigenvector one formal ht eigenvector eigenvector second small eigenvalue index computationally computation general drawback use method denote close neighbor spectral intel core ghz mb cache gb run intel core machine core mb cache run core mb cache gb run dataset run half tangent spectral accurate dataset cubic practical obstacle approximation develop laplacian popular cluster small representative spectral small remain matrix small submatrix basis nystr om remainder article discuss display numerical conclude finding fundamental behind approximate approach rely algorithmic step ensure sense wish compare computed spectral remainder course magnitude perturbation validate context laplacian perturb eigenvector respectively eq angle perturb projection analyze tradeoff efficiency theoretically correctly spectral mis certain nd eigenvectors perturbation eq yield perturbation et al major spectral preprocessing construct large identifies representative seem representative perform dataset close mean correspondence correspondence recover cluster representative step remain cost runtime course cubic choose enough quantify precisely tradeoff accuracy theory utilize mis mis fast norm demonstrate mis cluster control perturbation laplacian incur cluster small initially cluster name representative assign original dataset cluster closeness dms run cluster representative assignment find neighbor assign cluster majority many way representative spectral clustering utilize identify representative sample centroid coincide fast spectral randomly dataset reference assign proportional column accurate reasonably size cluster subsample similarity submatrix submatrix approximate motivation behind part wish similarity matrix represent relationship column compute nystr om eigenvector unfortunately eigenvector property cluster semidefinite eigenvector efficiently eigenvectors eigenvectors nystr om algorithm nm compute spectral similarity tuning guarantee fact yield worse empirically manually via nystr om nystr exact necessary eigenvector runtime spectral nystr om submatrix entire similarity aim select budget store remain entry enforce approximation index give accord efficiently budget budget denote large perturb laplacian respectively eigenvalue relation factor either eigenvector cluster note preserve closeness budget result run set clearly aim convention percentage centroid utilize experiment machine vary specification dataset machine experiment intel ghz machines mb gb intel core ghz mb cache gb tangent intel ghz machine mb cache gb value nystr tune table display depict rate fast decrease small representative likely portion get dataset spectral gaussian dataset assume perform nystr om fail sample budget perform bad error budget budget algorithm small nystr om sample size change increase substantially consist low might nystr om well problem cluster l om time half depict om ideal small time trend effective c nystr om error display cluster budget inaccurate identify l c nystr om error analog surprising result note perfectly large l c budget nystr om c range figure tangent depict exact second hour second approximation nystr om ht demonstrate dataset l budget om segmentation identification application company contain list age year company child child etc distinguish company stay company resource appropriately effort maintain expense analyze solve spectral method high difficult quantify long overcome challenge utilize historical teacher appropriate center education survey teacher survey entirely take former teacher take teacher survey question survey break interval status code child rating six recognize never child none school teacher cluster yield likely proportion tail cluster easily either teacher valuable teacher age give vary away old child close age tend term work apply yield mix leave child mix teacher contain run spectral express label cluster consist tailed support similar status l status cluster cc work predict group express child cluster yielded tail less first go display characteristic bad consider finding similar child child recommend neighbor bandwidth neighbor cluster neighbor near neighbor newly find cluster neighbor group neighbor reflect appear teacher statistically likely child
write whenever compute play role probability easy status q put together eq expression arrive immediately expression symmetric denote mn ip sg I mp mp mp j mp mp intuitively possess concerned really probability individual individual proposition proposition cause considerable confusion match match indeed come far chance focused quantity unique q odd similarly ratio evidence odd ratio differ case replace evidence database people determination really individual interested therefore odd retrieve see odd probability effectiveness select individual pe ps comprise individual come union contain separately odd size database odd effectiveness increase odd database large give match explain intuitively add many people unlikely match individual add database unique actually unique match decrease database match increase cf likelihood match increase great small database specify probability cf odd long match database always increase contain match unique match increase unique unique match decrease illustrate obtain example choose cast example provide one type perform database allele estimate allele obtain account discussion choice contain actual seem seven frequency reasonable start homogeneous frequency posterior equal effect heterogeneous population dna equally probably member member rise profile ratio ratio odd odd odd belong take relative everything else condition inverse odd get posterior odd belong careful whether take posterior serious finally odd calculate x illustrate obtain probability profile uncertainty parameter varied keep assume consequence know region country relatively past difficult give belong choice compare would ignore namely call naive fix summarize p pg notice greater naive considerable ss x ss sg much however course naive likely ss decrease grow considerable profile strongly probability let strongly choice population population whole exceed naive pg uniform situation three whose priori sc I meaning probability approximately set biased search note section odd decrease probable negligible also procedure become less raise odd unique population pc read monotonically go derive possess population exclude status since solution population situation quite suppose may individual base estimate database odd odd match six match obtain thus dna match search likely match happen ten double search match almost expand database put contain dna contain probability odd unique database minimal odd true database match rapidly grow slowly grow odd pc database match decrease idea database course database yield match database theorem thm thm example give account heterogeneous population correlation effect conduct return conditioning importance wants quantify weight evidence likelihood apply might relevant ratio prior discuss apply identification weight evidence rule couple trial people california match description estimate million match description return abstraction follow terminology population quite extensively detailed population alternatively effect stop generalization paper albeit somewhat article knowledge appear apart duality paper distinction purely mathematical view viewpoint elaborate issue text focus transform evidence posterior odd evidence reason likelihood view quantity unable allow odd already datum evidence background likelihood ratio regard background interested prove point background information issue invariant choice hypothesis purely view concentrate thing early suppose involve probability article case classical expert rest seem ratio prefer one address soon ratio correlation outline review biased protocol type dependencie strongly frequency homogeneous population addition ratio deal find match database include conclusion conclusion recognize text hope specify measure characteristic value characteristic number primarily interested often state call odd similar evidence use evidence way namely side odd odd background information via odd odd transform odd suppose evidence prior evidence evidence classical case uniformly regard member equally probably base however allow way independence frequency incorporate outcome sake ratio equal easily follow really matter viewpoint take likelihood prior clear uniform prior acceptable start subsection express expectation si k p n kp kp distribution e write posterior inverse intuitively equally expect therefore se understand way replacement careful supposed subsection situation suppose repeatedly find population detail question informally else explain else express characteristic count evidence belong characteristic knowing seem unlikely answer depend system distinction strength ideally strength evidence draw conclusion strength discuss expert look apparent likelihood prior probability seem generally admissible justify suffer irrespective must distinction account need expert pointing conceptually property distinction evidence point scenario find dna profile sense point expert procedure g profile could report reduce plausible alternative frequent prior whereas see would determine immediately clear expert knowledge difference variant prefer population reflect justify come focus candidate influence frequency population involve ip analogue check analogue probability ps ps eq weight one value matter whether interpret evidence preferable conditioning average factor weight distribution assume take independent variable whenever odd imply note denominator readily eq inverse concentrate classical population likely happen select extreme follow np also want variance equality remark
verification assignment trial assignment exist alg problem clause indeed include clause exactly keep assignment satisfie also exist one always decrease stop never evaluate algorithm initially assignment oracle trial computation query assignment every time compare solve algorithm formula fix algorithm query assignment medical input diagnostic ask find hidden help find relax requirement cluster formula satisfy within hard finding separates learn randomized trial satisfy assignment formula formula formula input try answer formula success algorithm smaller lower bind trial bind need trial eliminate process clause adapt decide clause show proof slight least satisfied drop half return combine chernoff clause clause assignment satisfy therefore principle clause force subset assignment cut small turn clause case family formula clause verification maintain triple contain triple formula randomized algorithm carefully clause divide clause assignment block clause block enforce assignment block block clause clause index randomized query satisfy part reveal position block two block denote verification imply candidate remove element also obtain query satisfying continue namely least triple either assignment unique thus output yet possibility formula claim unique namely assignment satisfy satisfie b ix ix initially query divide block set formula clause exactly clause clause tuple block satisfying clause assignment reduce give group precisely give group table multiplication report long open structure group version constraint unknown group verification apply restrict group natural attempt solve keep propose consistent whenever add restriction three rule avoid triple computation polynomial trial arise oracle unknown triple exploit table surprisingly hardness stand runtime solve time group hardness shown employ hamiltonian hamiltonian cycle size hamiltonian cycle hardness hamiltonian achievable cyclic shift label hamiltonian denote group cyclic generator translate cycle immediately arise time hamiltonian cycle thus construct multiplication around employ come interaction verification answer require idea efficiently something simulator correctness correct hamiltonian cycle wrong hamiltonian control long care code vertex ask hamiltonian hamiltonian arbitrary way hamiltonian first cycle pick arbitrary cyclic shift group define multiplication module cyclic basically run mapping identify verification table simulator verification correct find cycle suppose cycle shift cycle group multiplication make verification oracle simulate answer answer hamiltonian x hamiltonian cycle several define input hamiltonian time define run trial whenever simulate verification raise design verification oracle really map roughly return outer statement eq emphasize multiplication group next vertex hamiltonian cycle roughly non whole return output output clique cost take query time non claim time immediate give language easily strong include elementary completeness inclusion formula next algorithm statement completeness perfect completeness protocol round complete basically mention paragraph problem come query protocol public coin protocol deterministic send know pair graph currently oracle suppose follow answer polynomial verification decide check answer output perfect payoff utility probability mix equilibrium admit mixed nash polynomial player equilibria number nash normal version function mixed strategy equilibrium player violate trial fast find centralize quite investigated focus dynamic nash game nash equilibrium ellipsoid see matrix nash equilibrium verification oracle return separate claim ellipsoid problem ellipsoid handle perturbation positive volume explicitly employ develop gr formal theorem totally payoff matrix oracle see ellipsoid problem separation consider game respect utility verification nash unknown verification nash matrix without verification return player nash equilibrium entry serve separate gr ellipsoid solve problem separation whether single separation oracle argue strong oracle thank oracle solving identify two utility find find nash comment volume ellipsoid volume handle applicable apply find contain fortunately interestingly verification fit briefly feasible verification ask cover entire center ellipsoid violate verification establish ellipsoid remarkable ellipsoid know violate oracle suffice continue ellipsoid must system solution positive ellipsoid get polynomial conclude dimensional lie extra constraint reduce solve hyperplane solve gr ellipsoid ellipsoid must thin direction short parallel ellipsoid hyperplane hyperplane round number rational number size hyperplane although claim apply game approach obtain note hope nash potential game strategy game algorithm however game admit graphical nash polynomial even aware game efficiently achieve remain verification internet equilibrium identify similar game competitive equilibrium market omit share agent basically connect function classic cost function special case bound rational present numerator denominator violate set precisely input additive unique solution core implication able verification determine trial core sharing inequality cs core subset violated handle unknown get strong oracle describe apply ellipsoid conclude version u propose verification trial instance integer integer always large unique partition solution query oracle big size derive query set sum subset know bad query investigate input lack level polynomially viewpoint meanwhile leave gap examine obvious specific hidden phenomenon work try computational complexity general framework systematic complexity trial tradeoff complexity note argument entropy directly allow pair comparison polynomial note would probably also polynomial deterministic polynomially deterministic complexity polynomial computation verification consider verification ask problem acknowledgment grateful comment point cm thm fact chen zhang nature thing motivated economic computer investigation model search problem goal find adaptively confirm valid solution return violate trial complexity summarize follow despite little provide efficient nash unknown version input considerably theory ellipsoid strong complexity substantially input exist graph simulator computationally lack difficulty algebraic structure exhibit serve supplement computation computer particular focus understand power limit central computing explicitly game usually explicitly circumstance player payoff explore new environment know alone instance business company lack customer sided preference precise example job market job market know precisely like information job company mistake systematic fitting organization event diagnostic serious search collection e way otherwise reaction severe everything composition diagnostic largely effectively could try dna long identify diagnostic input lack input database acquisition extensively speak proceed adaptively accomplished otherwise object goal one consideration trial also approach economic adopt adjust self converge individual involve position management encourage enhance conduct clinical side effect observe analyze feedback future diagnostic ingredient trial employ error propose future formally algorithmic perspective aim investigate broad input investigate lack viewpoint trial question input numerous arise variety application study nash multi search assignment formula individual preference side addition motivate early broad space domain solution define constraint polynomial exponential infinite intensive theoretical artificial intelligence research exploration practical paper input input contain preference list list task namely propose chemical trial candidate one violate return return make complexity study drug verification carry thus truly reveal violate exact constraint I player motivate surprisingly assumption verification efficient verification newly historical return difficulty compare input overall verification oracle invoke either oracle unit unknown input input computational complexity complexity class verification oracle occur extra translate omit become query need standard latter assume free verification query deterministic trial complexity u investigate trial rigorous hardness trial note diagnostic development expense motivate goal protocol trial time complexity lack deferred section equilibrium side preference satisfy also ellipsoid target contain input point core contain volume applicable unknown sophisticated work turn verification crucially use nash structure medical student practical significance complexity trial clause min principle insufficient query instance distinguish employ characterize partial case height correct bind speed examine another whose completely somewhat surprising knowing violate admit input add extra difficulty version considerably version whether hardness u group know computation oracle polynomial compare multiplication multiplication table difficulty trial phenomenon hardness classic complete hamiltonian cycle hamiltonian cycle existence completeness group find cycle verification know issue crucial know design verification perfectly question however favorable incorrect hamiltonian may already use find hamiltonian cycle certain low variable efficient u completely resort nash equilibrium I existence yield illustrate complexity lack imply distinct specific level elimination exist learn combinatorial structure shrink space e explore relation space design unknown environment ellipsoid model model strong connection fundamental difference essence identify unknown object concept sometimes quite attempt solution aforementioned development diagnostic exact input learn case may impossible learn relax requirement formula assignment input finding contrast exact specific application largely available scenario verification situation orient advantage supplement exist context object unknown verification detailed relationship relevant concept task survey give collection row concept row query column answer element map equivalence query cast membership equivalence query framework search class let solution query thus propose solution hide membership gain extra violate necessarily efficient satisfie become equivalence propose correspond set induce table know e need put possible constraint would identity column position identify merely row satisfie hide difference problem however unique lie concept measure exponential learning oracle instance unlabele set iterative give interact fundamentally oracle correct answer label constraint achieve accuracy prediction unlabele use label instance objective pac probably correct pac domain instance concept sample complexity objective approximate concept classic pac although allow rather solution many learning reinforcement reference therein discussion level sample different trial solution address environment exploration explore entire planning objective desire path specific location either space reader refer topic feature oppose robot planning look difficult scenario underlie geometric structure ellipsoid trial search g separate return ellipsoid elegant polynomial expression trial model much pure perspective ellipsoid trial indeed herein crucially ellipsoid base trial time complexity ellipsoid calculation u program make function query input allow view extension traditional query query instance proof interact although clearly would sided market complete preference simplicity exposition list prefer contain namely match label match computed acceptance list propose candidate solution verification oracle pair reveal first underlie result unknown another set unknown discover order indeed verification return otherwise return sort complexity u actually u solve use run make meanwhile solve u trial upper fast order distinction standard comparison technique straightforwardly carry turn measure allow height sufficient degree randomize relation set whenever element denote uniformly among linear extension order thus form order
learn sometimes partial processing device additional shot useful characterization hamiltonian hamiltonian estimating post process reference therein outcome measurement broad carlo bayesian design wide also measurement adaptive quantum idea utilize quantum dynamical improve hamiltonian suitable parameterization hamiltonian estimating purpose hamiltonian expect high concept hamiltonian organize formalism bayesian discuss rao carlo section discuss numerical key element quantum theory statistic interest suppose ultimately achieve encode batch control process offline collect processing choose control past experiment formalize various way natural future control via remainder notation subscript expectation experimental usefulness quantify yield motivated optimize two choice discuss detail outcome choice subsequent utility scientific maximize value distribution utility minimize minimize protocol overview show feedback iteration block background set collect produce evaluate experiment design interest analyze particular rao incur methodology design average explicitly minimize bayes case risk mse minimize maximize bayes question achievable variant experiment standard loss rao bind singular singular fisher information arise unfortunately assumption meet avoid integration iterative algorithm various interest mean rao give give space specifically space carlo back wide scientific application name monte particle importance filter smc behind bayes update rule difficult evaluation costly issue approximate approximate calculate ensure always simplify arbitrarily particle every feed support particle carry distribution without generality initial make particle approximation accord particle location covariance particle weight particle function particle location update unnormalized normalize weight return monte require careful effort introduce error limit first weight effectively fewer use amount resource eventually techniques generally adaptively change location simple choose location shall effective ratio threshold resample explicitly behind incorporate randomness volume parameter apply particle thus particle spread formally location come particle sample perturbation resample particle resample use suggest covariance q involve particle particle location update perturb particle address regard thing since algorithm tailor say something issue simulation quantum interact thus call wish minimize time involve high particle appear utility reduce reduce ideal since routine call average idea formalize particle location semi utility dc find weight scalar u dd np h dp dd particle location particle element introduce sort combine complete adaptively design smc conjugate conjugate nan optimizer evaluate choice specify choice experiment control variable experiment optimize I pick experiment highest well carry provide able task maximize region algorithm describe quantum design point volume lie run record record credible record region speak credible confidence credible obtain kind region explore credible estimate amenable turn particular region compute region least intuition function region large current smc state achieve property choose geometric hull minimum satisfy collect reasonable approximately normally accord information describe mass inside parameter transform variate invert generally normally posterior satisfying particle good estimate covariance matrix dimensional normal meet cdf actual rao use hamiltonian proceed generalization address consistent experiment hyperparameter describe hamiltonian hyperparameter integrate consider realization denote hamiltonian avoid subtle conceptual difference hyperparameter way yield convert estimate drawback likelihood function need ensemble experimental take large may require error approach straight gaussian dynamical hyperparameter develop region hyperparameter obtain region attention assess fluctuation performance especially unknown develop understand either discuss significantly figure build intuition movement note experiment sec describe center likelihood dot red randomly blue measurement monte particle see line indicate indicate initial rao error indicate curve numerical precision choose experimental occur averaged randomly choose figure error see remain optimization whereas sufficient yielded optimally particle smc scale know previous contain guess per high performance introduce new guess choose guess little heuristic reduce possible pick show small approximation cause relatively large value take guess shall provide benefit trial region set surprising confidence vary skewed provide uninformative force informative room search informative guess heuristic heuristic poor guess optimize also median square tend trial randomly bad optimization median substantial contribution true describe simplify make approximately justify yet find consider appear model distribution expect within volume use within choose standard deviation case approximate provide favor region improve collect experiment insufficient experiment model mass averaged guess heuristic choose guess normal challenge calculation distribution use figure absence improve even characterize much contrast qualitative continue increase implie guess heuristic unlikely informative fix landscape sufficiently find mse guess suggest landscape local optimization guess find substantially substantially subtle mse estimate mse three quantity suggest also mse expect optimize model section particle without optimization dash trial single initial guess use experiment indicate solid experiment average trial guess trial region trial guess solid guess trial datum indicate region unknown initial indicate dash line correspond local solid line trial optimize average trial error curve estimate measurement sequential numerical dotted particle mean solid numerically variance quadratic loss region algorithm presence hamiltonian hyperparameter parameter volume mass hyperparameter recall admit provide hamiltonian score gaussian probability lie agreement within estimate probability yield mass unlike ratio tail non tail perhaps surprisingly interest calculate compare experiment grow see infer hyperparameter systematically estimate bias vanish conclude hyperparameter unknown particle another assess infer hamiltonian learn call make previous minimized choosing show sophisticated reduce experimental classical tradeoff increasingly quantum grow quantum particle cost asymptotically perform experiment computational rather experimental time relative heuristic number call call second computer lead computational hz utility lose rate approximately comparable need guess yield experiment although cause cause intersect curve surface seem indicate expensive heuristic may get well percentile strategy continue improved regime consider choose mse suggest approximation ratio become stable require em choose incur show right percentile great show require trivially little communication additionally improve circumstance quantum simulation expensive graphical processing unit field gate array latter device processing provide applie learn parameter collect infer hamiltonian processing store recover even advantage hamiltonian parameter
also show extended perspective noise threshold solve subproblem minimizer coincide minimizer question v point use newton solve appropriately strategy evaluate simply detailed function linearity violate nonetheless solves estimate fairly quickly approximation experiment iteration weight verify computational iteration find framework toolbox purpose velocity perturbation grid frequency hz composite lead procedure outline run multiply dividing therefore normalize source reconstruction reference show source signature outline paper depict normalize depict source convergence propose nuisance regularize problem source imaging draw projection nuisance exist solver experiment demonstrate wavelet successfully figure wavelet figure nonetheless lot effort devote develop solve difficult come always pick solve several iteration exactly obtain c red blue source weight show processing recover give auxiliary signature unfortunately practice signal technique present novel methodology auxiliary propose projection contribution combine projection deconvolution imaging image sparse prove inverse sparse decay data measurement compressive obtain guarantee general inverse guarantee regularization particularly useful linearize several placing reflect array datum represent fouri record series frequency operator record velocity measurement typically unknown wavelet bilinear perturbation frame
aggregation yield energy however agreement energy aggregate together interpolation agreement aggregate coarse hard fine variable influence mm estimate generate relatively label locally temperature perform random variable determine set coarse interpolation agreement coarse select agreement coarse sequentially start add yet small result coarse index fine leave throughout energy compute interpolation construct pyramid consecutive level less energy solution coarse fine fine serve initialization step interpolation refinement coarse energy aware interpolation role multiscale make interpolation coarse sense discrete gauss multiscale multiscale cluster energy compare performance energy result close well show percent energy close pair hard percent baseline art b energy multiscale synthetic energy connect grid label unary term ij pair wise energy optimize see result energy contrast average coarse fine optimization energy single cluster matching frame co minimization adaptively adjust energy regular grid energy compare discrete multiscale tailor relaxation improve art challenge energy energy also algebraic derivation find acknowledgment remark discussion thank rgb rgb rgb rgb rgb art come contrast energy suggest multiscale derive algebraic us pair interpolation principled algebraic propose aware efficiently multiscale landscape energy coarse fine optimization energy improvement method energy weight edge discrete preserve assign label preserve energy energie large dd preserve energy yield energy poor contrast scale energy suggest derivation novel yield substitution vector yield scale wish map fine assignment fine approximated coarse assignment write coarse parameterize fine
good strategy half iterate across method weight iterate outperform also optimize objective continuous present arguably analyze trick use technique last half average propose epoch gd geometrically sized average scheme whereas scheme general iterate iterate scheme therein strongly size finite like update see fw hz w lipschitz respect svm setup regularity b obtain show prove induction subgradient w z w first base hypothesis yield w complete plus minus plus minus pc team sup paris france mm lemma technique weight iterate easy compare exist strong constant follow scenario project subgradient precisely field measurable subgradient unique support pair identically fw w lipschitz unconstraine w show appendix use method x standard proof w f thus w fw fw line sum inequality similar term stay sum q weight averaging scheme implement average uniform average cm bottom view colour empirical benchmark set website causality website website dense standardize regularization although sensitive whole strategy average iterate iterate average since average w weight
sec low glasso mse ff thm act number term frobenius estimate mse guide regularization relate publication li mse consider compare li order weak obtain tight outline report introduce notation throughout report introduce limit mse ff present superiority glasso sparse kronecker present empirically I j define tp f symmetric let submatrix j fs set psd definite ds close simply close cone nc constants number kronecker gaussian likelihood belong minimization definite unique iterative block coordinate descent develop zero well log jointly motivate flip adapt q p flip compute sparsity imply flip lead factor proportion definite satisfy nj k glasso computational glasso section provide prove formulate objective assume argument assume definite subproblem consider eq subproblem follow maximize similar nj I subproblem converge define converge see consider set fold product optimization without vector q k reader appendix interest positive automatically take care assume descent mm descent logarithmic barrier limit minimize minima assume f descent theorem converge objective flip ff achieve optimal statistical thm establish propose improve mse absolute b ff flip flip condition accurate covariance kronecker minimal error ff three ff quickly specify fast computational ff kronecker ff simultaneously achieve ht reflect visually ff consistency establish compression constitute satisfy nk f offer strict generalize thm exploit kronecker structure significantly low estimation ff kronecker need thm inverse hold I inverse three iteration although thm matrix hold ball blue ff rate ff glasso note glasso algorithm would yield inferior ht magnitude constant plot reflect error minimal depict fig region mse go grow rate control grow accurate ff glasso naive ht establish deviation n pf pf w establish chernoff diagonal tight dependence convergence use lemma respect norm section validate rate section iteration penalize step glasso algorithm glasso monte simulation use positive definite base enyi square assessment frobenius estimate calculate trial regularization parameter select nf constant front regularization experimentally performance show perturbation figure compare precision outperform glasso ff ff algorithm suffer outperform ff regime since kronecker htp kronecker lasso ff flip glasso fig lasso flip glasso dense similar see nonzero matrix glasso rmse expect glasso ff ff suffer sample regime outperform ff regime exploit kronecker db reduction covariance ff htp factor matrix nonzero place place ff account sparsity structure ff ff estimate fair comparison ff precision db matrix ff rmse instead ff db reduction reduction ff approximately significant htp center around rmse ff ff bar around ff ff remark reduce address large mse curve intersection ff mse match mse well rate case thm ff illustrate predict top curve htp identity thm shown predict convergent ff kronecker glasso exploit kronecker structure sparsity derive glasso ff validate ff exploit kronecker increase thresholded ff account kronecker thank deviation report let triangle eq sum argument I j equality achieve minimax invoke kkt dual use observe ij evident equivalent verify saddle formulation duality trivially follow positive glasso simple combine kronecker product establish optimize definite penalize flip objective kronecker low primal mle bound sample convergence monotonically iterate glasso kronecker glasso review definition subdifferential reflect geometry proper saddle lemma continuously continuous subset j jx jx bound e convex block hold j ij optimal primal j hold exercise c sum define k imply important ii apply every j convergent ji j j subsequence continuity yield j j show j j critical rule maxima rigorously exist maintain without generality contradict maxima maxima point minimum j contradiction saddle exist nonempty descent jj strict descent induction iteration fm symmetric j denote operator define sequence permutation e k j add I formula method bind u f good infinity constant k tail difference tail bound markov property p lp f inside ty lt elementary mt note mu mu u mu plug tail taylor mt mt proceeding conclude np random n nc follow modification thm due present iteration flip assumption growth sample matrix intermediate max define c probability let cauchy schwarz inequality expand c together union absolute kronecker product precision ff
universit paris de paris optimize extension quantization dissimilarity method refinement non numerous need dissimilarity dissimilarity observation datum neighbor usual hierarchical technique suffer three difficulty firstly classical linkage global quality naive implementation scale large greedy technique suboptimal order efficient prototype dissimilarity technique linkage quantization dissimilarity come cluster suboptimal equip di di di partition measure thank size see favor cluster intermediate arise twice dissimilarity merging increase linkage generalize naive proceed merge small linkage denote compose partition c recurrence devote agglomerative old complexity demonstrate author benchmark linkage previous efficiency close pair linkage naive maintain queue cluster candidate store sort cluster true neighbor cluster small candidate low merge queue long near neighbor computation search remain cluster calculation fast important drawback minima hierarchical heuristic previous section single refinement operate time operate level intermediate move individual fact local minima heuristic dense item happen individual move measure avoid error operate dendrogram multiply level upon consider size analyst give idea apply consider move entire update implementation heuristic give near graph dissimilarity linkage approach hierarchical proposal dissimilarity rely quantization dissimilarity phase method dissimilarity cat database g relational standard naive implementation describe start configuration choose approximately computer implementation cluster cat solid dash cat relational correspond configuration hierarchical quantization hierarchical bottom multi refinement achieve refinement
function bayesian key posterior arbitrary criterion unobserved process gp gp unobserved point normal posterior selection criterion selection optimize area area promise maximum expect improvement ei successful example focus sequentially selection reveal mild leverage distinct phase phase exploitation bandit exploitation introduce select eliminate maximizer hence try algorithm point close benchmark outperform ei art ei whether exploration boost ei help observe ei exploration eliminate region ei select motivate introduce insight aspect experimental section motivate key ei ei hence ei represent deviation pdf respectively observe evaluation variance ei widely study concern balance ei concern though asymptotic ei condition ei try information potentially request lot optimum experiment attempt literature vary degree briefly discuss definition ei researcher make ei replace much success show begin make little difference ei surrogate objective try improve exploit decrease area exploration method change setup impossible proposal exploration ei ei method budget select ei follow ei sample investigation understand ei ei explore necessary introduce section reveal never help ei since regret monotonically hence skewness c literature empirical investigation ei know operate achieve ei answer able exploration exploitation ei especially query budget exploitation ei exploration observe output r leave region product lipschitz hope infinitely strong optimize change radius point sphere assumption violate satisfie hoeffde replace high try volume boundary might case sphere sphere overlap point select pick sphere pseudo next value might negative happens far prevent sure mean exploration need sure affect e pick small start gradually side limited sample implement volume point inside point fall previously newly explore newly explore deterministic direct direct optimize lipschitz inside might slow inaccurate describe exploitation gain optimal suppose explored find would whose small small enough mean similarly replace entail code maximum constant observe output qx r nothing algorithm exploitation gp unknown ei rely lipschitz exploration gp optimize scale deal exploitation phase case complex enough budget width fit budget take width exploitation ei achieve role lipschitz hence budget achieve certain chance sphere well choose true safe derivative long prevent become certain function look cc cell v ij constant x ei scenario different six synthetic function mathematical shown use real contour benchmark output cell adjust output output cell sample benchmark maximize production ph growth medium benchmark except sake case budget exploitation ei ei ei first like different ei ei sample budget ei version ei ei instead take infinity behavior posterior ei light interested used exploitation ei exploration help ei set ei measure benchmark estimate ei benchmark exploration ei benchmark ei consistently benchmark ei ei slightly performance due ei width also note ei improvement ei ei selection em cell like exploitation exploration fail use ei summarize follow regret
probability broad plausible sampling total span curve amplitude use gamma prior scale time I gamma decay duration long assign gaussian equal former calculate loo cv likelihood robust prior nm light curve likelihood value second row use em describe apply describe model deviation uncertainty loo cv look first difference loo cv construction low two model inspection pdfs partition peak amplitude partition relatively broad amplitude phase peak half partition broad inspection extra offset offset reflect posterior mean zero confident relative evidence non formally true term explain reduce loo large relative stochastic explain stochastic shape deviation essentially bar top large happen model component likelihood deviation accurately miss whether require light receive low star variability light reveal variability occur hour star rest integrate light spatially otherwise rotation period observable variability magnitude light curve year apart curve previously hereafter hour long curve use nan mean calculate curve light long widely context reference therein term observe e reject nan accept implicit final generally per se tell important adequate data substitute assessment compare five use although negligible finite duration integration minute section top approximate delta calculation analytic calculation calculate process l nm column curve e ten variability inconsistent test reject p metric sometimes correctly limit know motivate converse indicate fitting identify good curve difference offset need surprising light eight light loo significantly gaussian variability error difference high model therefore none light light light discriminate turn confident seven significantly bad discount interesting light year apart explanation behaviour different coverage interpret forget summarize base loo likelihood curve much model exception equally plausible ten light curve describe model require without seem light curve next remain seven clear winner three seven explain datum four light curve plausible large gaussian variance light curve well bar something plausible multiple stage loo event sample pdfs per examine posterior curve associate nm curve line loo sensitive change loo cv report arrive loo cv likelihood well likely likely examine prior unity change something loo evidence observe behaviour light prior factor ten nonetheless remain issue always light setting cf prior canonical article probabilistic method temporal accommodate stochastic variation many series axis temporal offer total error average partition average sensitive pdfs evidence something confirm take series sample theoretical recurrence derive elsewhere main purpose exposition comprehensive series analysis elsewhere present demonstrate main curve nan variability fluctuation deviation bar well might interpret confidence theoretical curve stochastic probable long term trend periodic posterior early significant curve describe stochastic component component comment remain plausible model exist explain focus practical problem inclusion loo sensitivity comment interested answer question explain prediction probabilistic modelling logical introduce comparison principle measurement signal uncertainty generally write could combine variation top
nb learn significantly beta usage positively use mark nb diverse responsible example large mean confirm beta beta nb across usage might helpful model alone experiment direct yield result zero well explain beta ibp gamma nb hdp viewpoint fix natural count viewpoint result confirm importance together examine adjustment variation potentially fit tb share evident nb compare infer smooth document topic number value convenient variety negative binomial nb process count group propose process completely borel set oppose hierarchical hdp disjoint borel discover compound representation mixture derivation demonstrate nb share nb dispersion hdp structural computational advantage distinct sharing mechanism connection result show dispersion acknowledgment program summation independent bernoulli become sm poisson vocabulary topic th topic crf hdp model crf restaurant table restaurant infeasible sample approximately result enough sample implementation convenience parameter experiment result obviously lower hold thus augmentation one allow infer dispersion eq hdp special gamma nb nb gibbs construct express gamma express thm thm electrical computer university nc develop augmentation binomial disjoint count fundamental derive efficient gibbs unique nb sharing model exist showing infer nb interest count binomial nb nb originally model hdp substantial suffer nb mixture distinct united fitting construction neither explore inference concern nb dispersion infer metropolis hasting paper fail connection nb hdp compare perform joint simple construct computation nb process count property inference construct gamma nb analyze hdp structural advantage hdp nb distinct select completely random modeling typical nb dispersion introduce nb distinct united poisson denote borel give group partition jointly variable measure expression poisson independent count model measurable conditioning normalize prior continuous process dp gamma representation construct equal dispersion assumption count model construction poisson gamma proportion treat motivate layer nb compound mass pmf r p rp rp investigate numerous scientific nb augment p scale compound r p logarithmic abuse add place lemma since gamma conjugacy assume use repeatedly suggest nb process within conditional analytical pmf table customer chinese restaurant concentration chinese restaurant count pmf nr poisson compound key count modeling examine understand fundamental nb nonparametric bayesian number compound l pl pmf summation random w sm sm sm r generate compound lead compound sharing nb dispersion nb count gamma gap represent poisson gamma gamma three inference nb conjugacy ga x ta ga lemma base atom discrete gamma conjugacy exchangeable conditioning equivalence gamma denote hdp normalize hdp hdp gamma nb theoretically disjoint practically nb exploit conjugacy latent inference hdp alternative construction crf stick analytical posterior nontrivial simply crf apply method practical base biased gamma whether shape govern continuous equation nb variation gamma define evy cp dp cb construct jk construction nb nb dispersion nb nb share beta process mark variable thus process k x j jk jk conjugate tractable zero nb process introduce bernoulli process b jk jk jk link inference tractable link ibp beta spike nb parametric show variety differ dispersion share deeply model imply typical group word six differently construct process factor nb tuning document automatically learn topic strength document close gibbs inference even lda topic model nb count framework set hdp normalization hdp construct whose nb introduce process process gamma focused topic ibp compound dp nevertheless likelihood improve allow well nb explore share hdp allow hdp share beta nb explore probability group improve binomial mark distinction update construction nb process table poisson th loading sum score link nonnegative factorization factor proportion count mixture jk corpus typical relate nb hdp hdp nb crf hdp nb modeling variety crf fair comparison implement base atom nb lda nb parameter proportion topic toolbox last sample nb topic measure vocabulary smoothing location ji ji v uci edu toolbox restrict occur corpus term randomly learn dependent term f
relate fitness agent evolve capability learn large lead add bit agent create next accumulate generation median evaluation generation half q small take fitness evaluation cost capability small thesis evolutionary start pre encode gain evolutionary machine learn beyond learn boundary evolutionary possibility ai development ai recursively ai span thereby intelligence evolution cost learn principle valuable suggestion access vast people build present thesis goal achieve complex evolutionary create require thesis efficient go learn blind search environment give inefficient capability work evolutionary learn goal complex need order discuss limitation imply machine study supervised pass phase example field example propagation machine probably computational discover pattern achieve machine study unsupervise learn structure study unknown machine receive environment external agent henceforth goal sensor body achieve goal environment capability specification thesis present goal capability identical machine processing goal agent run platform process incoming control alphabet agent agent modify denote number start memory emphasize use expression size finite use memory arbitrary thesis state achieve run platform control agent body capability state physical pair physical body platform let agent achieve agent achieve goal sequence path energy environmental time possibly agent repeat randomness amount execute achieve thesis limit computational period result dependence thesis present theoretically speed physical computing device energy achieve know achieve phrase agent repeat trial identical start agent achieve learn effectively even achieve start well cost vary repeat identical agent effectively effectively achieve trial agent goal body unbounded agent would large environmental environment integer say grow unbounded phrase short effectively multiple discuss allow arbitrary agent must environment call henceforth able achieve goal string evolutionary bring high fitness landscape phrase efficiently define learn evolutionary information evolutionary change pre calculate fitness benefit target change environment physical case probability success description string exponentially principle state agent string denote agent would inefficient change miss learn value efficiently learn extend agent state efficiently learn g environmental efficient evolutionary cost agent difference ham efficiently principle bit agent zero us denote median
markov chain type delay draw mean model run aid clarity combine rapidly peak decay slowly production peak monotonically know possible reach unlikely rare logic ii plot equation iterate random trace need space value sufficient reliable initial convergence parameter path quality course clearly subsequent note rare assume variance true suggest importance sampling use model address probabilistic checking construct rather representation smc intractable exhaustive smc offer pose monte simulation reduce rare construction exploration importance provide optimum significant applicable kind space system biology methodology importance sampling e probability markov restrict logic precise sampling automatic description syntactic describe produce robustness chapter adapt improved exponential growth state give confidence rare challenge objective circumstance level establish maintain check find motivated simple contrast low avoid transition magnitude improvement simulation efficacy methodology reliability engineering provide prediction behaviour complex system increasingly become compact system becoming increasingly create correspondingly correctly user expect high ensure correctness system testing outcome highlight discover incorporate sophisticated despite testing cause fact unlikely cover mode remain series test check formal system logic scenario boolean answer checking quantify numerical alternatively offer accurate explore state apply protocol precision satisfy property interest exponential limit applicability universe circumstance system impossible compositional reasoning reduction size heterogeneity infeasible also successful match precision quantitative logic ever solve simulation approach become increasingly availability hardware check simplicity checking execution trace individually whether variable advanced technique combine whether satisfie know confidence tight checking offer quantify complex system evidence check checking establish statistical checking cope large key challenge cpu trace cpu core grids purpose processor production despite model necessary rare unlikely problem base since probability rare property temporal rare state rare mathematical applicability work iterative refinement must produce trace present arise syntactic constitute low comparison unique optimum highlight area future error rare bounding formulae chernoff hoeffding property become error useful monte unbounded increase mean importance simulate rare simulating monte originally material sampling good sampling I must simulation must paths use term zero importance sampling fact satisfy estimator meet necessarily address ii scheme fall broad state state dependent sampling individually transition probability former greatest infeasible tractable kind potentially affect differently function indicate path property trace monte carlo random absolutely eq ratio importance goal rare result importance rare property frequently optimal condition rare produces trace rare lead check specification explicitly specify class multiply rate absolutely markov latter density particular vector rewrite rewrite function cross alternatively kullback effective direct regard optimum importance cross entropy directly importance construct successively estimate eq f function eq scalar purpose occur ratio form substitute substitute partially almost hessian k nd note furthermore need positively hessian semi fact unique optimum newton quasi newton force hessian individual sum parameter individual transition equation imply circumstance improve useful side hence formulae base unbiased biased optimum equation previous scalar work reduce distance successive distribution explicitly converge trace establish heuristic trial effective iterate suitable observed increase obviously rate help along failure property express temporal finding system trace successive trace trace satisfy
graph complete even graph complete consequently transition generate operator contain chain state complete time markov markov completed complete irreducible satisfying equation irreducible exist stationary reversible irreducible lem moreover chain operator score search equivalence class direction validity chain operator design reversible easy stationary distribution subset class discuss property need reversible explain markov markov direct chain markov class complete denote property reversible irreducible perfect show transition two complete complete operator operator confusion apply accord complete probability completed complete complete operator denote obtain property operator complete direct transform direct sampling operator uniformly generate uniformly random transition denote property right sure irreducible operator complete sequentially irreducible irreducible operator complete result reversible pair complete q define probability reversible stationary irreducible stationary complete sm irreducible reversible unique stationary construct concrete perfect carry equivalence complete set describe interest compute literature equivalence equivalently direct formula approximate define equivalence proportion number chain probability ergodic complete set perfect markov show equation operator sparsity run reversible equivalence edge complete perfect operator construction complete say complete perfect operator complete efficiently perfect operator operator completed complete six edge six addition guarantee operator introduce sure reversible notation distinct neighbor consist vertex child represent operator complete number six operator occur undirected yx direct edge occur complete subgraph x z cx z xy constraint five operator complete type operator lemma condition define perfect operator perfect set dd key without reversible supplementary material proof precede toy supplementary operator irreducible next accelerated version generate base perfect sketch show step explain subsequent e construct randomly operator complete set operator chain dominate implement time obtaining provide accelerate hundred fast arrange implement discuss complexity acceleration speed step construct algorithm include possible operator check condition te te tp accord tx vx tx vx tx x check result complete getting complete number avoid result complete material three classical whether clique path undirecte two vertex check trivial small complete condition whether undirected path vertex check source look undirected path within accelerate accelerate time number edge structure subgraph construct step equivalently possible less operator operator condition consume path depth know maxima evenly divide consequently fortunately complete experiment less approximate implement experiment cubic notice obtain irreducible reversible markov checking accelerated generate irreducible chain idea operator accelerate complete type edge reach bind easily possible edge version tm nt operator e te determine check set check replacement operator back operator uniformly step complete operator irreducible reversible algorithm provide om estimator sampling approximate normal distribution let variable distribute describe replace define show accelerate nearly roughly conduct experiment reversible equivalence estimation propose true efficient estimation quickly length increase complete constraint edge direct ii maximum measure vertex around ten iii chain grow know assumption latent selection variable inference observational need infer undirected complete underlie complete equivalence causal infer direction markov proportion direct component equivalence sparsity difference study asymptotic estimator pp markov equivalence get true size markov size calculate size chain chain calculate result table close standard deviation second chain ccc implement propose method without acceleration run computer ghz take hour ghz proportion markov size worth estimate neighbor concept characterize equivalence characterization proof essential acyclic occur lemma validity use validity validity condition type two valid equivalently hold contain operator equivalently hold adjacent valid equivalently dd clique undirected path adjacent equivalently undirected path contain operator prove operator reversible irreducible theorem validity valid complete contain complete equation complete adjacent pass vertex vertice parent reversible parent give reversible parent complete chain undirecte skeleton direct structure structure edge edge direct parent parent parent structure acyclic contain cycle contain acyclic child must include parent denote condition rv path contain path thus contain get acyclic complete complete complete except complete operator completed complete contain complete thus show complete result theorem need show define operator reversible operator reversible operator reversible operator operator denote reversible operator six several complete adjacent know graph direct complete vertex common strongly b parent denote result become undirected adjacent contradiction either occur hold must occur lemma occur iterate get direct graph circle stop step eventually lemma complete result extended let set complete whether denote obtain cycle cycle contradiction chain cycle cycle two clique cycle length great undirected cycle chain occur induce subgraph direct induce induced subgraph edge strongly remove ii modify cycle must contain vertex adjacent lemma partially cycle would induce like contradiction undirecte must undirected path vertex undirecte suppose subgraph occur induce subgraph subgraph contradiction yield direct direct complete clique vertex clique hold vertex length subgraph operator satisfy complete result complete clearly hold valid undirected occur occur adjacent occur configuration adjacent adjacent undirected condition valid lemma occur occur possible adjacent adjacent path must adjacent yield contradiction clique valid definition modify show equivalently skeleton common child structure exist structure structure parent clearly parent direct occur hold introduce lemma vertex direct short let partially direct edge undirected occur direct path complete adjacent short must obtained complete direct figure involve thus direct strongly yield contraction lemma proof undirected become parent partially direct child parent contradiction partially direct short partially direct denote direct undirected path direct occur shortest adjacent get exist cycle must partially subgraph structure clique parent parent neighbor vertice parent parent parent modify complete equivalently skeleton skeleton must condition occur structure must child clearly structure occur hold skeleton remain parent direct occur undirected thus occur occur undirected undirected undirected case occur adjacent neighbor become parent parent parent parent condition hold parent must parent otherwise occur also edge hold undirected path undirecte change get proof lemma complete undirected complete lemma contain edge direct exist least vertex parent whose direct find exist go since acyclic finite end parent need reversible reversible operator complete procedure step lemma undirected edge new skeleton subgraph skeleton repeat procedure completed direct whose parent complete empty structure complete edge complete whose skeleton subgraph skeleton repeat remove complete subgraph skeleton contain undirected repeatedly sequence finally partly department berkeley wu visit berkeley grateful manuscript co associate comment suggestion lemma corollary program cb nsf grant dms dms dms nf agreement research school science center economic education education microsoft popular tool represent system statistically say belong equivalent understand vertex design reversible irreducible propose operator markov stationary closed construct operator class introduce operator accelerate provide sparse acyclic vertex experimentally markov equivalence dag edge subgraph subgraph linearly base direct acyclic graphs dag denote dependent scientific bioinformatics business causality dag encode dag observational equivalence represent dag encode dependency equivalence uniquely represent complete acyclic complete short direct undirected correspondence completed call maximally orient modeling task discover understanding equivalence important dag searching complete search base dag bayesian edge direction need active hard set markov complete approximate markov equivalence define dag pe na proportion contain graph vertex year popular tool introduce restriction edge paper markov equivalence operator transition stationary interest class pt section short representation equivalence edge denote edge call partially least edge direct undirecte direct undirected vertex dag direct call acyclic direct cycle consist dag probability conditional imply joint read encode skeleton arbitrary regardless characterization markov class dag skeleton among dag preserve involve consequently uniquely dag
number moment solve efficiently cast decomposition demonstrate theoretic barrier precisely one mixture gaussian separate statistical exponentially imply grow component work bad degeneracy work finally algebraic develop model component analysis ica ica handle mixture model issue section describe gaussian main observable observable small eigenvalue covariance denote coordinate special spherical eigenvector covariance well k w strictly strict separation fact imply k observe vanish h hold crucially derivation claim relationship simple moment give random I recover ks km distinct therefore eigenvalue matrix geometric zero eigenvector sign claim regard plug derive restrict share spherical complexity problem mix worth inverse accuracy moment population central theorem bind exist theorem singular pick satisfie q easy accuracy either w become finally also efficient orthogonal tensor extract parameter robust recommend practical noise zero multivariate gaussian arbitrary straightforward recover definition open handling estimator axis align mild require moment partition conditional x kt describe recover component spherical rotation let x td invariance gaussian spherical conditional x kx dd almost split three view dimension guarantee somewhat conjecture sufficient share similarity independent unobserved signal independent generalize mixture value spherical spectral decomposition approach describe theorem completeness ica non eigenvalue geometric eigenvector choose surely deterministic algebraic recent sample previous moment mixture consider view setting long conditionally view moment map work view suffice estimator signal spherical unnecessary contribute correlation separate view degeneracy virtue tractable previous tie additional largely unnecessary drawback prevent automatic satisfie lemma condition provide availability deal separation randomization matrix denote large norm third eq analogue ny u essentially implement two moment w nx furthermore define moment moment half sample half th eigenvalue average w w w e w w w tt w recall basic special structure moment subsection w proportion inequality bernstein inequality inequality follow per moment third separately first third throughout thin orthonormal freedom thus q schwarz tail term combine handle check handle per ir w shorthand old empirical definite per observe I I I triangle g r r claim recall orthonormal singular analogously u assumption definite column uv uv second u third observe u whiten iw third claim u positive first third first claim w w kt wu w wu write e w e orthogonality I structure first third claim let sphere I take e kt eigenvalue eigenvalue arrange increase ii trial I sign simplify notation sort increase last therefore radius eigenvalue contain inequality simplify permutation range range term guarantee wu since indeed fourth use satisfy lemma exist sample lemma define therefore condition least furthermore inequality eq tail use freedom let vector random inequality term summation zero odd time odd non negative odd integer step standard normal follow cover cover cardinality exist u fact j require example remark microsoft computationally statistically mixture mean moment efficient mixture position connection relate study widely important covariance matrix probabilistic spherical gaussians let component define probability variable multivariate coordinate vector isotropic far k h hz observable mixture indicate component mention accurately recover component mix work efficiently degeneracy component span strictly entrie estimator plug moment moment empirical
norm get reach expansion point last appear discussion expression high initial discuss situation converge simulated initially anomalous introduce distortion moment rx certain moment author optimize implement implement significantly despite fast dimension standard code dimensionality addition add dimension dimension add transform give power situation grateful discussion transform classical problem encounter research importance year come intend suppose include possible recognize purpose trend discovery anomalous idea covariance generalization among widely hyperspectral recent year appear generalization tucker decomposition generalization rx normalize spread convenient go normalize well example slightly gram expansion present completely solve moment practically case exist compute step value choose normalize unity performing obtain suppose pdf unity would normalize moment normalize eventually discover restrict moment third note moment multivariate tensor moment tensor hard treat tensor lack analog procedure despite existence tucker decomposition kind normalize highly nonlinear expect third quadratic logic rx sense rx determine reproduce second carry normalize distribution transformation argument distribution possesse moment moment implement spread space lift want assign point certain satisfy possess requirement lift completely compute coordinate requirement overall normalization fig initial abuse lift lift rotation particular orthogonal rotation third tensor rotation minimize recall still freedom define lift ambiguity coordinate independent amount dimension describe follow condition generalize I normalization normalize rx space third onto compute follow use tucker order employ modification algorithm rotation matrix dimension moment run first whereas carry onto orthogonal rotation orthogonality tensor rotation mix dimension rotation upper corner corner projection
retain section fit logistic involve group mcp scad tend point global local minima mcp scad logistic likelihood refer concern certainly terminate explain write pseudo response descent affect updating remain spherical update equation present mcp generalize family typically unbounded iteration guarantee possess nevertheless well test penalize poisson study extensively obtain either criterion vary minimum beyond strictly convex continuously fact mcp scad present residual previous value next value warm comment regardless choose computationally intensive step calculation operation pass operation fit depend number depend broadly speak factor nonzero require group fix consequently spend end compare publicly package fit increasingly datum package package appear mcp scad tend tend lasso mcp scad mcp scad lasso worth note package problem purpose choose attention entire implementation terminate prevent present scad require compare scad illustrate mcp methodology allow flexible modeling association involve categorical variable term predictor covariate whose true study cross regularization scad fix group default recommendation suggest group case section rmse fit root prediction minus irreducible gaussian mcp coefficient normal vary towards allow enter occur model four amount theoretically gray regularization similarly increase mcp theoretically group perform poorly group mcp scad small model fast lasso far many scad nearly selection thus two behave similarly mcp seem somewhat well panel rmse oracle know advance mean zero coefficient variable oracle nonzero nonzero old cause genetic multiple probe reliable probe influence gene cubic group mcp describe table norm probe group group group symbol mcp scad nr c select scad fairly shrink nearly mcp select probe plot mcp group scad top probe aspect standard rest observation responsible qualitatively gene mcp select gene fit explain scad group correlate gene advantage possibly correlate somewhat predictive versus approach mcp parsimonious group lasso single potentially measure add note mcp fashion aspect adjust mcp make recall special mcp parsimonious considerably course flexibility tuning parameter important parameter group age consist control may disease section represent snp carry snp cross misclassification snps remark improvement misclassification mcp slightly superior group group mcp scad considerably parsimonious without compare snps select correlate cause pathway mcp scad powerful grouped selection lack lack publicly software development prove implementation package acknowledgment grant grant dms dms author thank genetic association anonymous helpful remark lead considerable proposition convexity mcp convex objective group group mcp differentiable everywhere strict infimum algebra quantity consequence updating consist along square loss continuously differentiable thereby establish point stationary note guarantee unique suppose group algorithm possess limit involve regularize strictly function scad mcp proceeding previous infimum proof proposition claim stationary differentiable exist segment inequality matrix algorithm minimize convergence note property stay compact possesse point apply conclude stationary furthermore also eq eq abstract attractive possess grouping group propose problem group nonconvex mcp also extend selection scad group mcp modeling explanatory group categorical flexible analyst scientific understanding take grouping gain variable become build early group covariate lin lasso lasso suffer drawback namely change magnitude biased tend coefficient smoothly scad minimax penalty mcp simultaneous asymptotic imply identity truly coefficient group scad group mcp scad mcp largely due lack publicly algorithm fit publish article scad fit estimate computational provide combine large make inefficient fit scad demonstrate superior approximation upon demonstrate coordinate scad mcp efficient show extend scad mcp fast publicly potential group mcp group explanatory overlap linear portion design form regression let member matrix covariate covariate think discrepancy encourage prevent penalty index regularization tradeoff collection regularity often outcome wish include selection penalization regression loss take logistic binomial loss variable ignore originally orthonormal unclear apply orthonormal explore author group norm demonstrate variable uniformly powerful article term penalize greatly reduce computational lasso straightforward extension group mcp importantly accomplish generality solution penalization software singular group diagonal eigenvector follow j may transform original compute note procedure rank eigenvector denote invertible possess rank transform back original transformation experience model case naturally member identical coefficient article orthonormal group orthogonal consequently subsequent section norm explore penalty group normalization group coefficient group describe estimator originally propose minimize behind penalty euclidean thereby encourage variable select residual large presence thereby normalize across group roughly group optimization group obtain lin generalization describe algorithm clearly illustrate descent algorithm employ coordinate descent considerably presentation fit scad mcp following section impact square center ol solution path orthonormal note subdifferential q square solution design exclude implication equation multivariate optimize problem lie direction thereby enable solution determine furthermore determine abuse definition vector act amount less way thresholde soft single descent penalize replace essentially modify replace updating refer recently update execution loop update refer consist expression repeatedly residual efficiency clear complicated arithmetic group update step lasso complicated view lasso lead clear penalty focus smoothly deviation penalty mcp penalty motivation penaltie p mcp scad originally penaltie group substitute discuss refer regression scad mcp behind understand consider mcp scad begin rate penalization relax aim lasso among coefficient mcp penalization scad move mcp penalty plot penalty middle panel derivative estimate light identity differentiable behind consider least solution simple note mcp scad toward depend mcp scad scale plotted illustrate essential describe mcp thresholding mcp shrinkage hence scad involve scad modify thresholding solution limit thresholding multivariate mcp scad
convolutional formulation color multi modal feature connection map map reconstruction compute inverse connect layer feature unsupervise b perform unsupervised stage complete stage order next stage absolute normalization sampling operation wise feature normalization pooling contrast one exact map index experiment average size feature extract predictor hierarchical x sample I ig I I I ds display since invertible use local contrast normalization figure form perform stochastic datum map constitute convnet classifier component stage nd stage feature traffic sign house number object detection multi traffic house number strictly feed take layer subsample branching produces extract contrary output transformation pooling subsampling convert subsample uv keep uv channel value normalization subsample uv absolute subsampling produce extraction feature map randomly connection mapping break transform value normalization sample fed stage magnitude great improvement detection minimal gain number classification less bootstrapping typically setting multiple extract answer add bootstrappe answer image bootstrapping pass I e total accepted overlap bounding box box overlap modify replace contain avoid instead modify bootstrapping include body part positive bootstrapping detect big window reliably however demonstrate multi stage convnet convnet convnet f stage convnet ms convnet ms windows width context pixel correspond image supplementary fair majority train train table system represent figure along rate show contribution convnet convnet combination convnet ms compare supervise without multi stage exhibit compare convnet reach competitive error rate result publish system clarity plot show convnet datasets datasets convnet competitive datasets convnet sensitive work convnet likely large auc mean less convnet dot line reasonable plot report convnet c fix auc auc pixel pixel auc perform among highlight auc positives convnet convnet ms competitive multiple measure measure report supplementary contrary popular level use extended feature performance benefit art publicly available improved future resolution successfully real hardware combine highly optimize graphic performance roc exp scale roc scale medium oppose auc discrete auc convnet auc exp get convnet ms exp exp roc scale roc exp report dataset curve software available dataset ranking convnet f continuous observe convnet rank oppose rank convnet ms convnet benchmark c c call auc large auc near medium auc auc auc car highlight bold row auc percentage curve plot positive small auc great false positive em auc discrete auc fix medium discrete auc partial auc heavy discrete auc discrete table except auc detection list successful deep vision report convolutional model stage skip integrate unsupervised convolutional stage surveillance variety pose variation classifier forest hand good mid level combine help sort multi feature little demonstrate usefulness unsupervise end technique stack stack stack decoder detector system also learn complicated corner convnet contribution convnet consistently competitive result major use train relatively end fine tune integrate additionally connection enable stage detector detection great enable real without entirely avoid introduce applicable gpu focus feature show algorithm produce successful supervise many find adequate datum design domain alternative unsupervise recently generic recognition train deep achieved actually follow layer machine image algorithm output location bottom row tend aggregate horizontal towards bottom extraction feature perform filter generic parametrize input gradually abstract convolutional code predictor contain code unsupervised multi train architecture follow tune many field code overcomplete sparse coding
distribution balance explore distribution distribution focus get look wish promise reject early main sample increase thing arm armed approach speed mention datum collect predictor generalization attempt step start configuration loop page learn first pointwise estimate configuration accordingly drop line test sequential applying algorithm conceptual overview depict additionally package name via contain learner cs p ip n break np p accord precede top tb sr tb calculate indicator robust trace filter sequential drop stop case complete configuration information whether top configuration rely test store pointwise good performance compare pointwise performance subset remain order sort regard want find configuration similar behavior remain current behind performance build configuration small relatively small available versa apply directly individual exploit pooling outlier affected overall testing affect effect helpful overall good perform configuration look outcome configuration subsequently pointwise test experiment significant difference configuration essence test whether show significant difference case behave differently term performance yet estimator check fine rest pointwise row yield whether significantly submatrix significance configuration increment one test index since indicate configuration configuration must configuration one configuration different configuration remain incremental thus correction actual calculation incremental e add increase speed improvement first collect show step procedure iff amongst configuration step generate step trace summarize learn trace record fashion whether robust binary random configuration top transform error scale top scheme overall represent binary top configuration course want estimate observe behavior binomial mean win configuration variable originally context quality assessment production process biological stop focus approach give deal potential win sequential observe bernoulli want denote accord success bernoulli variable significance acceptance control user meta test hypothesis likelihood meta geometric representation binary sum accept red stay red another fix since sequential eliminate allow evidence configuration want eliminate configuration acceptance mark use approximation expect take real success winner solve equip check trace detailed statistically exceed decision top violated transform winner configuration thereby behavior top approach would change success probability violate accommodate act drop control choose configuration remain stable drop configuration adapt sequential potential switch independence implication please sequential contain necessary need implement algorithm early configuration past submatrix trace apply test submatrix illustrate configuration configuration mark red one top configuration configuration mark gray drop picture configuration keep go heterogeneous behavior configuration mix red perform black early stop tb win pick step configuration determine way course restrict optimal increase model pick configuration suggestion significance suggest usual significance winner suggest setup fast really sure accept winner procedure exploit regime stable regime time future research win probability feed subset capable deal certain investigate suitable subset first property test employ come process give top respectively step drop deferred consequence drop guide stable regime configuration see used guide control observe drop configuration discriminate configuration win adjust act probability step drop insufficient notion learner show real performance configuration behave reasonable strong extensive experimental speed small impact compare heuristic configuration certain inside exhibit act step global win configuration configuration look bad configuration mark superiority global win enough marked winning process process tb visual step execution win configuration straight zero fast end express recurrence recurrence intersection lie decision boundary number split definition configuration line zero construction diagonal reach straight one actual recurrence decision number plus current reach option therefore recurrence path recurrence prove global configuration suppose win trace winner eq calculate configuration correspond outcome binomial probability path point survival path lemma sum conclude early indeed go maximal probability mass maximally spread decision early tb size fit dotted line bind success probability depict observe fast increase step nearly converge optimum rectangular grid impose fluctuation overall small impact balance conservative configuration control dropping impact overall probability analysis assume happen change win lie occur often datum set drop rate point simulate switch configuration change success choose constant change behavior switch turn winner tb indicate relative configuration plot show specific setting switch configuration two false positive similarly theoretical confirm study change variant test remove depend configuration result range later change occur false increase negative significantly low even room even drop configuration box come situation optimal cross amount process explore grid time adjust constraint model evaluation budget available step parameter lead near available budget idea configuration rough estimate total need parameter configuration configuration drop need calculation configuration drop consumption depict correspond difficulty hand need calculate maximal number use fix bind configuration sample appendix really ensure kernel name incorporate without training get regime cross stable essential configuration aware could learner yet world often might nevertheless analyze misspecification indicator configuration show sense grid happen mask normally configuration chance certain redundancy configuration redundancy similarity underlie redundancy theoretically new way configuration notion interesting incorporate function test dimension configuration scheme explain deal fact clear configuration potential preliminary learning evaluation version least dependency overlap set dependency evaluation form behavior throughout property yield task tailor cross choice support gaussian leave right task pose severe stop suboptimal intrinsic task use range devise eq plot point time parameter contain configuration interpretation apart mse learner validation speed gain encode misclassification difference mse measurement impact experiment difference mse well normal low noise normal suboptimal configuration increase tendency increase yield distribution use seem picture speed gain svm show high seem direct consequence inner perform speed gain svm configuration tendency drop rate steady increase kept spread high speed gain transition algorithm intrinsic algorithm choose suboptimal configuration cross gain range direct intrinsic show overall variance much leave datum correct configuration configuration setting extract configuration correct much demonstrate investigate choice benchmark repository entry classification panel difference square error never exceed show although pick problem mse classification task pick suboptimal error always relatively learner impact correspond combination high cross validation performance datum cross speed gain diverse vary picture speed higher task seem solve fast task trace keep regression ratio bank svm bank svm svm top configuration crucial test recall iterative testing pair compare section estimation test compare instance replace test non version result point replace classification rank benchmark get reliable statistic report mse look pair runtime procedure significantly configuration pointwise time bank bank block svm repeat time get sound report mse cross annotated thing negative indicating find configuration impact across always pick perform impact runtime simple evaluation huge choice configuration impact choose setting time outperform like validation combination superior heuristic trade speed simple robust configuration speed cross promise candidate model tb aspect potential module look configuration capture point stop depict top step solely configuration early stop act global determine stop focus landscape module look configuration essential evaluation show behavior drop benchmark easily square use information datum utilize fit less yet similarity come affect alternative calculate sequential configuration accord configuration behavior compare configuration strict drop outli behavior configuration residual residual check calculate raw outli similar specific configuration one see nature configuration obviously speed drop roughly observe datum conservative outlier keep low base ratio impact speed shift keep configuration conclusion fold section modular construction exchange turn extremely devise instance outli flexibility multi structure flexibility comparison observer increase expense set compare accuracy outlier compare conservative sequence hypothesis accept contrary accept neither decision exceed prove test potentially leave lead development kind close algorithm full cubic tb speed close compare conservative alternative increase step rapid close sequential tb indicate negative switch configuration setup behavior start enough turn winner reveal come price drop advantage procedure identify clearly configuration promise candidate large size advantage heuristic effect systematic understand statistically argue one error fast insight lead introduction sequential configuration time without combine bandit problem furthermore get size setting future research performance converge parameter tend infinity discuss minimization refer book therein interpret set ng gx I consequently order vc feed node activation vc dimension rkhs induce converge condition ridge optimization rkh attain regularization hypothesis discussion care correct scaling parameter datum package name version whole intrinsic dimensionality execute run early stop follow trace error remain configuration win configuration tool cast notation deal matrix configuration ccccc configuration configuration kind act use determine whether set configuration perform configuration behave compare trace remain test task test test configuration equally effectiveness configuration least significantly calculate denote total reject degree freedom long entry suffice exact calculate explicitly rank tie configuration position point tie reject denote degree significance section treatment sequential publication prove equation eq yield decision test minimal therefore maximal construction graphical intersection step configuration set learner observe proportion configuration parameter drop small give fact obvious solve substitute rt sake large mild condition power sum mention fulfil since furthermore constraint note condition assume term monotone decrease asymptotic behavior start negative grouping value sum negative finding consume propose candidate method computation power procedure cross validation testing cross standard tune supervise
give voxel voxel exactly voxel contain voxel treat write indicate canonical voxel spherical often voxel voxel size voxel contain show voxel particular identify response pattern extend single voxel voxel place voxel voxel voxel less voxel contain voxel vice versa contain contain multiple voxel voxel simultaneously less voxel voxel show absolute single voxel increase radius radius center one radius voxel spherical distance voxel two maximally distant voxel contain radius namely follow voxel contain voxel radius contain irrespective voxel frequency radius group let since voxel radius radius since one radius contain hold theorems radius voxel adjacent voxel lemma radius within voxel imply least contain consequently voxel establish voxel bias influence detect particular radius increase monotonically radius every voxel informative contain radius since necessarily adjacent logic center voxel within fully assumption center voxel radius radius increase voxel multivariate difference simulation concrete intuition implication ease single slice direction voxel voxel radius systematically take value mm mm contain voxel voxel voxel voxel first voxel voxel scatter text location voxel slice voxel exist voxel exhibit difference condition identify recall voxel contain signal voxel center signal voxel simulate response illustrate response voxel response voxel requirement draw position voxel figure voxel voxel decomposition testing implemented evaluate accurately membership use support machine leave loo cross panel map across slice radius go right expand high cluster accuracy reference cluster information show accuracy voxel slice dot horizontal obtain thresholded circle square separate voxel voxel separation radius mm voxel radius contain simulate manner section voxel map center voxel voxel weak difference organization voxel observe cluster several voxel accuracy red voxel map correspond voxel less separation map map radius mm top map voxel voxel voxel horizontal voxel voxel evident smoothing grow radius furthermore increase htbp panel voxel voxel separation voxel separation panel plot profile information map center horizontal slice voxel voxel voxel dot upper indicate motivate produce difference consistent theorem informative identify manner previous demonstrate effect inherent monotonic increase informative irrespective actual task voxel group monotonic size plausible informative recently study map remarkably map whole radius establish previous ask arise chance suitably question cover voxel brain cubic volume voxel direction voxel contain voxel signal voxel central voxel would informative informative single voxel informative true htbp informative voxel volume proxy spherical volume side voxel contain sphere would simplification cover volume voxel divide volume cube voxel voxel voxel voxel minimum number volume cover take cube voxel voxel rapid size would mm total spaced voxel spherical volume state single relevant voxel voxel voxel enable distinguish due presence random map draw actual voxel informative would information irrelevant informative counting inclusion identify task relevant voxel combine across require informative voxel could classifier compute voxel response assign dependent specific machine assumption appropriate allow technique make analytical alternate interpretation sensitivity count number informative sensitivity trivial explanation geometric property underlie organization sophisticated optimistic assume sensitive satisfying support office collaborative contract w nf lemma popular pattern analysis construct spherical voxel brain despite map challenge challenge size signature issue formally examine geometric mapping geometric spatial produce radius informative brain complete multivariate property capability pattern fmri cognitive provide systematically value cognitive voxel relevance criterion basis inference neural cognitive technical technical motivation mapping concern region response cognitive might voxels relevant though individual conventional univariate restrict voxel explicit concern simple enable sophisticated information irrespective whether informative response univariate unit voxel spherical neighborhood voxel response group base statistic center voxel map central employ dy soon bl kx fm pf sf ph jt vb gd popularity pose map single voxel map mapping protocol irrespective voxel information coarse location unique central voxel voxel voxel neighborhood property map ask reliably pattern signature study qualitative require interpretation map intuition voxel systematically continuous location voxel neighborhood priori informative virtue sharing group signature exactly voxel neighborhood deduce informative signature formal prove voxel signature information brain increase multivariate pattern importantly voxel brain voxel define define voxel voxel convenience treat voxel voxel principal direction assume path connect voxel voxel simplify intend emphasize principle ignore special boundary truncate white voxel latter agnostic surface ad chen abstraction subset voxel voxel voxel indexing indexing voxel voxel radius indexing conjunction structure voxel member convenience write result q clarity restrict entity tb voxel gray vary depend dark gray statistic compute map back correspond information map text indexing voxel difference experimental generally measure whether contain specific compute bl statistic response informative corresponding overall radius geometric voxel voxel informative decomposition virtue regularity sample since choose shape unlikely voxel necessarily relevant response alternatively accurately dependent combine voxel contain task make procedure restrict common procedure geometric voxel unless state symbol requirement sound requirement
quantify cardinality technical make analogous wang ensure neighborhood ef logarithm small meet provide theorem corollary provide risk also ideal yield fast illustrate theory k verify ef ei fx verify positive assume e multiclass estimation compare baseline classification tree multiclass wu et quantile method z pairwise optimize employ tuning refined burden example probability distance measure compute set p x denote cardinality avoid computing correction comparison next covariate freedom covariate example generate covariate testing size example different get large become difficult example additional covariance example error standard superior numerical free relatively cut boundary get intensive satisfactory multiclass apply dataset publicly university california repository http uci ml length width observation white attribute range score make focus score white white attribute measurement different class scenario misclassification set predict p x q replication table evident small except performance due burden propose free conditional estimate construct distributional assumption efficient propose establish experiment simulate example demonstrate advantage regard naturally density technical py pr kx k k x ib ib inequality bound bound let x x p mp mp k mb k establish mp version li p pr pr pr k pr multiclass estimation discriminate logistic regression distributional formulate cumulative function convert exponentially number class study relatively interval multiclass estimation tune statistical machine class probability strength information hazard multiclass discrete know liu zhang wu opposed class estimating probability distributional fisher homogeneous baseline assume ratio generally verify distributional violate gain popularity practitioner classification tree sensitive suffer instability wang propose model estimation conditional binary classifier various weight weight contain obtain multiclass attempt wu lin develop couple multiclass estimating wu zhang liu extend wang liu search vertex contain require intensive computational binary classification vertex exponentially propose multiclass via conditional cumulative series quantile compare free burden importantly establish show highly illustration nonparametric li liu reproduce rkhs induce specify associated li liu rf computational remark asymptotic behavior estimation space employ grid along direction whereas complexity wu efficient cross become inconsistent order suboptimal wu liu wang wu finally performance choice appropriately multiclass indicate dependency tuning estimate conditional k comparative loss eq indicator model wang liu searching distance candidate dependent
finite rate model independent eigenvalue detect dependent result would eigenvalue select also population counterpart strategy determine eigenvector accurate decay polynomially accurate estimation showing section stochastic covariance trajectory discrete corrupted noise study however context perfectly trajectory without bootstrap select sample eigenfunction counterpart parametric observe trajectory establish instance fix eigenvalue eigenfunction bridge gap covariance spectra decay brownian motion connection jump spectrum operator eigenvalue latter evaluate observation instrumental analysis already denote induced let j section sample approximation eigenfunction eigenvector turn develop eigen present eigenvector theoretical operator euclidean square also hold multiplicative constant result sub vector constant x x additional assumption zero constant moment interest meet sub assumption vector distribution sub constant jx provide sharp bound term frobenius operator sharp either state constant specifically proposition depend matrix accurately offer level assess satisfie eq satisfie furthermore continue singular establish independently unbounded employ rest include good rate bound logarithmic derivation norm low frobenius derive order theorem keep minimax rank bound estimator frobenius operator matrix little gain thresholding shrinking would performance simulation conduct situation theorem relative motivate introduction class covariance introduction study low population operator estimation matrix effective rank also appropriately rate sample operator frobenius note guarantee large less specifically small imply growth induce bind make bind non result offer satisfy probability least high q scale norm irrespective show decide believe small upper discuss consistent separate separate relative q notational clutter constant appear eq recall quantity need regarded jump estimate condition make index quantity threshold note implicitly thus well order high index enough large exist call jump spectrum relative triplet order concrete threshold notation constant let hold eq fine jump minimal noise threshold estimate pr specialized assumption spectrum show condition appear naturally brownian specific assumption belong irrespective assumption differ analysis minimal jump note differ satisfy notice index exist immediate exists imply theoretical spectrum occur large noise level follow theorem assumption technical holds exist iii prior illustrate spectral population theorem detect offer dependent threshold manner suppose analysis trajectory ks xt discretize trajectory assumption continuous positive theorem decrease eigenfunction orthonormal moreover hold brownian motion mapping distribution j q scale covariance trajectory focus employ densely smooth estimator appropriate defer detail estimating jump operator accurate illustrate analysis address eigenvalue order need form evaluate whose us spectra close show moreover eigenvector vector evaluate eigenfunction establish modification feature plot apply slight abuse denote eigenvector eigenvector denote assumption fix point hold positive give appendix whereas immediate consequence crucially eigenfunction finite dimensional decaying automatically scale rank jump occur theorem detect jump size spectra construction detect suppose process technical index regime employ translation noise employ recall motion detect covariance apart existence term quantify hold upper suppose assumption directly evaluate eigenvalue eigenvector subject recall reasoning eigenvalue separation analogue hold define immediate identical theorem estimate keep usage brownian motion reasoning thresholde accuracy eigenvector brownian eigenvector relative proof generic u u generality immediately justify orthonormal argument employ transform matrix evaluate either frobenius sub eq let q need non random equality hold independent gaussian constant u sub f combine proposition combine proposition average material need prove proposition next subsection begin instrumental proof exponential use use exist furthermore straightforward last let inequality prove article proposition frobenius smooth change inequality prove let eq let ix n z expectation article proposition adapt completeness result value zero tn special therein matrix singular frobenius f xx xx present proposition sequence independent exist two psd semi definite integer markov ax ax ax sketch note operator therefore inequality appropriately prove inequality sec section easy proposition k k proposition sec proof notice integral kt finite integral equality easily derivation constant otherwise small nd hence upper q hence ii ir eq ij ij c lead q first invoke derivation eigenvector give complete mc full rank acknowledgement david grant dms foundation dms foundation national institute imaging ns national institute stroke proposition class reduce effective class decay spectrum measure complexity define class reduce estimator perform detailed empirical necessarily goal sample eigenvalue population counterpart plot level provide checking eigenvector analysis population polynomially decay extend operator polynomially decay spectra estimation receive amount attention last largely motivated covariance consistent regime understand decade whenever instance seminal rest effective hope year depend type sparsity entry wise wise diagonal decay shown adapt see many matrix
function term regret balance wide range composite special case proper binary necessary sufficient let strongly direction let c proper strongly thus characterization loss regular proper strongly loss tool establish function recall follow tie p note follow rank estimate completeness let f strictly link yy p p p regret several composite property loss exponential exponential proper loss concave strongly proper logistic loss proper c regular eq concave composite apply square hinge loss square probability predict coincide proper associated composite learn apply appropriately obtain loss pt yy yy pt canonical exponential pt spherical yy function regularity condition loss accord spherical proper composite essence exploit proper proper link scoring denote marginal distance rank risk treat show certain rank plug ranking inspire xt adapt rank risk condition version proper yy take noted restriction give obtain hand exponent bipartite rank composite term composite loss include widely characterization proper loss elsewhere concern necessity strong loss term concavity bayes concavity regularity spherical loss upper regret pairwise via natural pairwise standard adaboost standard logistic hope establish study future discussion thank lee conference mining hold ann work support fellowship department science observe trivially x similarly instance negative scoring mis instance equivalently area dominant framework pairwise classification bound et al show regret rank term balance logistic exponential loss bound bipartite broad proper composite simple proper hide balancing variety strongly logistic loss tight condition cl problem arise variety range retrieval drug discovery last several variety ranking rank problem label negative goal minimize mis area algorithmic theoretical indeed enjoy since bipartite suitable follow summary nevertheless algorithm adaboost logistic svms exponential loss yield loss surprising effectively also far quantify score surrogate loss show bipartite ranking exponential loss build analysis bipartite analysis exponential balanced loss nature bound bipartite rank proper composite proper hide balancing variety proper exponential square squared special case set bipartite instance problem definition background composite reduction ranking characterize loss bipartite term together brief open question background binary proper loss assign specifically scoring simply receive tie break simply bound score binary function belong briefly notion regret proper composite give prediction binary yy x concave background material material simplify compare include important start estimation space say minimizer unique proper proper basic strictly strictly decrease proper binary regular proper risk loss strictly several way attribute twice result proper strictly strict convexity equivalently give third contain principle helpful loss proper loss direction concave strictly conversely assume strictly proper strictly concave notion composition say invertible loss widely note composite describe composite form form etc characterize composite bipartite note bipartite review result build pairwise distribution x scoring r
pair verify therefore derive I gaussian matrix universal exist e eq inequality implementation alternate solve admm lie express q admm consist denote solution admm pre constant specifically least eqs via efficiently eq admits add solve iteration intermediate iteration x f problem via system linear equation obtain verify admit rv rr admit analytical plus minus theorem section lemma regression scheme assume assume matrix admit program regularize accelerate gradient multiplier develop efficient key admm scheme optimal program performance composite estimate predictive observation receive area machine estimation assumption set effective base use trace become popular tool trace encourage singular dense application predictive function lead interpretable rank cardinality encourage sparsity motivate use trace induce structure investigate extensively recent develop program recovery trace establish trace norm study trace norm minimization collaborative filter convex paper multiple function basis set function assume coefficient formulate trace norm employ composite regularization induce coefficient simultaneous low structure incoherent propose solve multiplier also component conduct convex basic property lemma assumption basis prescribe regularization conduct effectiveness propose trace sum singular sum prescribe dy set th practice basis sparse low induce sparsity couple coherent low parameter cross use sparse rank globally many technique alternate multiplier consider accelerate multiplier non develop algorithm admm attract deal scheme th search intermediate specifie increment commonly refer operator efficient efficiently dual min verify duality substitute dual convex coordinate cd compute globally solution optimize two fix optimize rate fast iteration optimize via compute unit summarize optimal eq therefore w x f material assumption derive performance trace norm formulation restrict eq less restrictive condition denominator one rsc assumption matrix imply condition trace bind x dy entry eq lemma material last similarly substitute eqs set inequality choose refine bind proportional refined n note tight evaluate benchmark set study admm implementation material trace multi label comparison macro micro classification benchmark business health yahoo multi experiment experimental stop value determine double validation I business health auc micro compete algorithm benchmark set business result demonstrate effectiveness multi outperform benchmark high data numerical study convergence observe admm much slow focus observe trend admm convergence stop change stop attain curve successive curve present plot figure converge fast left plot plot dual alternate algorithm operator similarly alternate successive curve figure efficiency predictive specify coefficient matrix sparse structure formulate problem
monotonically monotonic monotonically fast monotonic unbounded fast propose theorem strictly set sample generalization development rv depend ease point within back fundamental simulation boundary since also contour rectangular depict rejection whether need analytical expression boundary necessary know relationship describe contour enough rs obtain dot dash line solid transform htb standard region standard pdf condition function also increase limit eqs entail follow since rewrite form monotonic rewrite recall hence fast region generated bound monotonic verify vanish vanish condition condition indeed density relationship rv pdf although unnormalized study variable monotonic q denote rv see distribute sequel sample density method obtain connect look region pdf uniformly recall eq write indicate extension theorem simulation htb density pdf point distribute area depict draw equation connect rv density use formalize rv monotonic transform transformation point choose region distribute moreover devoted method transform variable describe investigate connection recall define eq simplicity treatment eq lack decreasing also decrease since recall therefore trivial express increase consideration develop monotonic pdf unbounded rewrite q finally increase monotonic b non monotonic locate mode locate consideration case transform variable eq moreover region bound section remark namely satisfy apply transformation random pdf give rv coincide area transform rejection unbounded b display region depict proposition show inverse du hx px provide rv moreover distribute generalize inverse moreover rv area obtain proposition extend approach whereas generate extension fundamental fundamental link pdfs clearly decrease exactly rectangular circle triangle useful rectangular allow infer function indeed rv consider instance inversion rectangular rv unnormalized rv region p yy v inverting obtain completely inequality rectangle second end contain assume alternative area region second second formulation technique sample eq replace vertical formulation choose pdf summarize relationships pdf dash line set consideration remark assume decrease also multiply obtain area p far minimal equivalent rectangular rectangular region simplicity consider decrease localize mode mode transformation distribute pdf decreasing vertical rv indicate choose later obtaining pdf write rewrite q exactly strictly relax condition need relax study possibility formulation indeed know apply rv apply rv rv transformation htb correspond second slight depict increase way follow combine assume section observation distribute sequel suitable unbounded scheme uniformly acceptance rectangle htb region region axis accept second possibility I uniformly eq illustrate htb uniformly boundary show combination rejection specifically completely pdf fundamental simulation relationship random whereas realization rv pdf function obtain need achieve bound transformed pdf see density monotonic transformation pdfs monotonic piece moreover rv pdf namely rv work see section design deal pdfs consideration aspect rectangular partially science innovation project ref program ref transformation rv uniformly write pdf rv jacobian inverse namely substitute yield integrate marginal pdf rv calculations trivial turn pdf integrated pdf express term rewrite integrate rv transformation rv u gx consideration easily infer light strictly pdf uniform yield expression proportional pdf light decrease decrease draw constant define take important possible symmetric axis axis decreasing rewrite moreover jointly transformation clearly come back decrease define summarize expression rv express decrease target eq easily observe indeed area p htb unnormalized target yy depict solid area derivative cdf calculate use notation cumulative generalize iid ratio rs rejection tr representation generalize density symbol rv distribute unnormalized unnormalized normalize pdf target rv rv meaning unnormalized rv alternatively frequently indicate constant indicate constant rv q close close closed side rv value arbitrary unnormalized rv instead unnormalized instead pdf commonly start type interval consider associate imply lebesgue length sample rs algorithm refer pdf constant distribute sequel rv rv rv pdfs half gaussians inverse target rv pdf give imply target pdfs unnormalized target invert proper unnormalized however rv rv multiplying normalization hence perform different whereas region region lebesgue identical support inverse pdf give region unnormalize generalize lebesgue inverse imply common support paper never invertible use convert rv rv rv rv indicate rv rv target rv transform target rv invertible pdf compactly rv great obtaining convert inverse rv unbounded rv inverse target rv obtain tr invertible transformation inverse rv case write simulation realization transform rv tr rv invertible compactly transform rv strictly q region inside sample must unnormalized target pdf unnormalized pdf pdf formally draw target pdf formally unnormalized target differentiable use inside uniformly define use unnormalized pdf eq let note become pair distribute inside case due simulation tr rv derivative vertical vertical I denote vertical vertical vertical I point vertical department communications la es circuit engineer de km mail david es transform rejection generalize ratio monotonic pdf equivalently transform rejection target classical showing completely monotonic concentrate monotonic pdfs decompose monotonically case transformation inverse handle generalized ratio technique carlo mc often nuclear sequentially smc filter use chain core efficiently question remain rejection exact inversion approximation ratio present efficient monotonic discuss technique case section appendix consider technique often monotonic unimodal pdfs monotonic denote pdf unnormalized pdf able feasibility drawing extend monotonic multidimensional moreover relationship see vertical rejection another discard sample favorable rs occur pdf bound domain density proposal calculate bind ratio find easy task scenario sophisticated rejection high etc unbounded pdf critical region area region straightforward theoretical sampling transformation make transformation rv unnormalized draw convert sample invert obviously restriction unnormalized target pdf furthermore suitable function similar cumulative cdf transform rv achieve imply draw sample sometimes technique ensure v pdf interest us sample draw uniformly unfortunately tail fulfil consequently pdf uniformly inside region u introduce separately explore far goal relationship approach prove region pdf moreover introduce consider monotonic unnormalized pdf transform rv unnormalized pdf provide monotonic pdf monotonic pdfs technique unbounded important consideration notation fundamental discuss rejection technique background paper provide focus possible sampling introduce ratio consideration section monotonic pdfs devote unbounded consideration conclusion sequel unnormalize pdfs mean integrate whole domain consider normalize pdf denote unnormalize furthermore unnormalized monotonic pdfs pdfs pdf unnormalized multiplying variable instead attain inverse pdf inverse pdf usually perform also obtain target rv issue clearly half pdf identify unnormalized pdf normalize logarithm proper normalize unbounded pdf similarly unbounded kp clearly target important remark normalization unnormalize generality attain formulate pdfs although approach pdfs proportional even carlo rejection etc base result sequel density constant equivalent inside region area whereas play auxiliary many monte theorem jointly discard univariate marginally unnormalized depict unnormalized method density apply sampling htb unnormalized target inverse monotonic unimodal density extend pdfs unnormalized target pdf inverse unnormalized describe illustrate alternatively eq depict way generate draw interval e figure always unnormalized inverse non monotonic interpretation generalized pdf straightforward distribute fundamental first coordinate unnormalized whereas pdf able draw second yx generate inside inside e draw e uniformly monotonic proper monotonic obtain feasibility unnormalize pdf unnormalized pdf practical variable since pdf draw pdf inside accord method form random decrease unnormalized random unnormalized pdf call alternatively relationship technique clearly apply rejection rejection rs unnormalized know unnormalized proposal pdf work accept discard repeat step pdf rs remark connection fundamental inside accept whenever fall inside repeat step rejection green target pdf inside locate define eq rejection complement q rs pdf happen dark accept belong red region happen point indicate dark circle discard express condition belong green I red demonstrate equivalence description rs whereas red indicate rejection region locate define sample draw pdf dark green dark red circle fundamental rejection sampler candidate lebesgue tight make crucial performance bound pdf bind convert problem unnormalize easy rejection improve scheme technique unbounded proposal proposal close illustrate pdf support pdf bounded pdf show figure section devote deal problematic transform inside region transform tr htb unbounde finite last proposal simple scenario several author line invertible rv define inside b figure conceptually inside unbounded unbounded target pdf support rv pdf rv different suitable rv support unbounded appropriate rv unbounded unbounded finite support imply infinite unbounded discuss always generate distribute hand invertible apply rv rv unnormalized sample distribute result rv unnormalized target pdf section apply rv sequel monotonic either function inside I inside domain rv imply inside interest rv unnormalized make g monotonic pdfs unbounded pdfs rv sampling technique obtain literature rejection also call close unbounded consider monotonic transformation rv unnormalized unnormalize idea draw invert drawing choose adequate unbounded transformation close look notice term assume unbounded monotonic vertical imply limit situation monotonic unnormalized target transformation increase decrease htb monotonically monotonically corresponding transformation vertical conclusion unnormalized transformation dx f limit tend alternatively order support bi interval extend infinity support towards infinity whether
variable n property imply independent vanish zero tensor tensor constant vanish gaussian variance first variable ica assume latent random via additive assume last assumption serve remove ambiguity otherwise convenience first orthogonal term whiten sign noisy main additive affect covariance whitening could orthogonal without whereas scale follow matrix main clarity presentation constant time division sample nn simple component independent length method already ica modify reasonably way whiten popular include high ignore class additive gaussian validity fourth second ica presence statement restrict exist list maxima th canonical list minima sphere canonical course coordinate maximize restrict ignore rescale tr summarize whiten without mostly algorithm provide moment observation recall fourth operation proceed lead worth first multiplication take role describe transform primarily equivalently b bb idea quasi whiten noiseless quasi whiten matrix construction quasi whitening know multi equation essence however several demonstrate first cauchy variable recursively I arise limit summation generate certainly intersect fraction finally tool proceed use binomial odd dominant particular term come q get follow chebyshev lk j lc choose apply tensor whitening throughout quasi whiten canonical act independent remain demonstrate cosine orthogonality section bound suffice demonstrate fast shall statistic denote max I give define get investigation propagation tensor use growth tensor operation place goal portion useful norm result theorem split precede lemma use propagate tensor bind demonstrate angular correct sample later n I follow equation show ad lemma apply equation theorem apply yield restriction except restriction meet suffice basis canonical approximate whiten canonical stay apply rr yy give equation case q complete thank discussion nsf grant abstract give quick emphasize spherical remove notation consistent abstract proof reference final bf define eq engineering university science engineer party people different room recover mathematically sample whose random ica address propose presence necessarily additive distribution independent routine property tensor ica get provably presence propagation blind set room simultaneously superposition produce variable superposition linear transformation give recover need name range address extensive literature processing community applicability speech various biological comprehensive introduction book ica remarkable fact reach consider correspond maxima orthonormal whose element compute fourth independent apply pca transform principal direction transformation rescaling absolute sphere independent recover direction maxima sufficient sample coordinate paper separation contaminate underlie signal valid analysis still minor address signal polynomial compatible invariant pca special case underlie method base pca sample compatible year blind signal separation ica concentrate provide run analysis analysis al address learn equivalent ica subspace blind
package develop originally integrate crucial ability way arbitrarily development organize calculation facilitate flow sect class implement inference sect sect sect begin terminology commonly measure pdf multiple unity estimate constrain particle uncertain shape nuisance principal build concept density integral handle collection pdf handle easily multiplication representation fitting generation study inspection great modularity principal factor start field distribution background constrain observe event factor number event specify nuisance specification specification idea provide obvious element prior frequentist calculation pseudo limit complete option carlo iteration test conceptual school data limit frequency various outcome broadly permit probability typically interpret frequentist require inference obey brief method available refer detail example test credible class nuisance credible return depend return ability lie within specify retrieve find value side convention value implement likelihood numerator denominator absolute regularity condition demonstrate shape deviation two sided invariance ratio valid physics community program asymptotically variate perform hypothesis characterize systematic systematic uncertainty hypothesis implement estimate interval return object dimensional contour hypothesis visualize interval newly develop inspection profile relate hypothesis give inversion achieve function probability characterize typically nuisance product parameter unity credible interval calculation nuisance remove marginalization integrate perform current interest use posterior algorithm root numerical integration include integration metropolis number step construct markov default proposal mode invariant class visualize input bayesian implement release interface development integrate confidence guarantee fully specify probability implement derive return concrete specification order define observation add desire introduction another decide assume asymptotic mc nuisance nuisance profile nuisance toy carlo nuisance point parameter consequently accord inside class user specify use give statistic obtain toy helpful application importance implement frequentist approach hypothesis frequentist method marginalization nuisance technique often refer define alternate hypothesis along order exclude choose statistic possible outcome functional form statistic monte approximate lie cl uncertainty marginalization distribution generate toy net presence uncertainty hypothesis hypothesis nuisance result consists associate computationally permit merge vary signal side implement test finding level interval limit condition define upper alternate drive project would useful without object file object file complex multiple easy via save file fashion convenience publication newly develop utility permit intuitive string create store allow pdf one create observable parameter class two prefer aim c physics specific single file describe line fact need would everything addition add include comment easy combination channel collection class template base analysis require knowledge instead file model user histogram template event channel efficiency affect systematic uncertainty gamma log systematic channel identical across channel list mention implement technique theorem incorporate systematic pdf combination
context erm costly must optimization paper cs propose provable guarantee secondly aware distinction regard estimating paper principle know accomplish pursuit directly propose derive dimension extend soft section randomize essential estimating minimax finally simulation write norm product b present bp important motivation well result describe draw suitable p constant fundamental condition apply signal additional aim must invariant signal determine determine large hold eq notable feature xx level little measurement recover structure derive technique area deal cs area seem fully usually role technique set randomized approach estimate generate vector characteristic function form tv refer example cauchy write stable set noiseless well study univariate however corrupt show moderately affect impact noise choice control measurement leave first generate measurement cauchy cauchy variance define confidence noise gaussian coverage width confidence vary take measurement mild standard algorithm basis pursuit expect perform lastly growth well assume hold depend dimension sparsity fix confirm simulation lastly small order high follow taylor naturally extend recover year many researcher refer paper application theoretical role sensitive perturbation matrix straightforward attention quantity recovery order explicit section robust motivated quantity quantity elsewhere less follow randomness noiseless deterministic estimating noiseless sx validate consequence decay setting image dependence relative error grow moderately lastly reconstruct consider set alternatively implementation pursuit illustrate coordinate plot agreement three indicate take signal accurate parameter figure average sx according always right middle signal profile profile choose curve probability reasonably reconstruction bp instance choice plot plot reflect theorem implication hand prove implication vector define fact q define measurement version fact imply detail find chi freedom follow delta prove limit statement way depend limit allow conclude relate define consequently may eq derive term complete give asymptotic bound relation almost omit reason theorem analogous essential difficult case idea measurement indistinguishable yet theorem let satisfy old column orthonormal let standard realization define amount show event equivalent accomplished variance gaussian respect fact obtain portion version inequality long j w verify finish make expectation finally definition prove need begin first optimize side enough take infimum without third enough replace supremum two px ax aim eq construct bayes also prior define two put mass rule return ax third california berkeley theory compress parameter unknown due aspect depend know drive practical concern unstable value coordinate paper sparsity sharp rely little width dependence naturally estimate guarantee randomize essential accomplish deterministic sense cs linear q measurement small signal draw ill pose reliably pn draw line commonly algorithm play issue recognize theory relatively quantify conceptual hand estimate assumption sparsity simple estimate emphasize aspect depend know several valuable address assumption likewise check active area recognition image natural device theoretic accurate reliably recover measurement deal sequentially case determine recover take additional may discuss enough characteristic rip guarantee closely rip grow body devote whether treat already yet meaningful consistent tuning orthogonal pursuit omp appropriate examine omp estimate reduce restrict ensure coordinate decrease vector one dyadic red dyadic color code triangle bottom axis indicate number despite cs practical unstable particular equal description effective
count source vice versa criterion define degree relatively external directional zero alternative cut however concentrate explore asymmetric component propose preserve directional divide accord connectivity distinction terminal develop identify directional community sim use node direct laplacian graph define degree matrix remark graph laplacian strong connectivity community di sim nod two way k leave sim stochastically receiver co relaxed sim discover huge direct pair terminal require number scalable search community time community division propose regularize balance lead cluster observation bi adjacency svd find good directional community reason two first directional community balanced division graph division expensive second global np hard undirected regard size addition define us parameter determine trade svd minimization proposition directional community vector result relaxation membership replace interestingly induce penalty sparsity induce helps introduce directional component structure adjacency undirected know component laplacian connect component relationship extend directional length represent terminal directional denote obtain replace singular direct directional singular span moreover directional follow directional community subgraph strict directional q directional directional directional community svd use multiplication hard thresholde local solution find similar exploiting linearity definition h thresholding treat maximize proposition constant tie determine sort entry sequentially large small meet hard direct solution svd search alternatively function monotonically local detail initialize l similarity algorithm laplacian step detect node principal find regularize svd tight community solve combine inspire induce penalty type penalty elastic sparsity control non size discrete report significantly regularize solution net svd local thresholding regularize svd fix become seek time entry search verify search method number define maximize follow threshold regularize svd replace search threshold algorithm find describe appendix scalable elastic regularize extract next extraction repeatedly sequentially advantage identify world web link exploit property devise community regularize vector initialize community repeatedly net value magnitude vector different source modification future level propose among community achieve change obtain change regularize svd smoothly level initial contiguous change huge fraction name elastic vector initialize initialization specify vector level pick discover community search community stop early reach simple stop value time minimum previously detect candidate pre l desire stop search rule burden keep early stop rule tight community direct apply idea modularity first elastic net penalty identify submatrix zero say edge typically network directional edge call procedure take sparse singular singular singular submatrix require directional component keep whereas detect community membership source edge concern regard initialization driving motivation scalability massive investigate computational requirement specification edge know drop potentially community relatively explore locally regularize smoothly believe tackle massive modern setting hand algorithms di include generate benchmark generate directional part consist node asymmetric community terminal node generate law size terminal control aspect community community degree degree information criterion measure configuration simulation result repetition reference directional valid practical emphasize directional community general accuracy di sim community big community repetition error directional community three directional community cm di di sim well di sim give accuracy even strong community accuracy algorithm community however di seem community experiment detect directional community contrast direct citation form science cs paper manually category represent major field computer sub remove self citation symmetric average degree terminal un run candidate cover node sparsity take en take value decrease help community minute minute discover en cover part yet cite comparison consistent perform average degree highlight asymmetric directional community di sim partition require algorithm detect di sim citation arrange arrange remain row citation sim membership column arrange terminal part remain node end internal appear diagonal meanwhile dot dot internal membership sim adjacency separate partition community detect reveal citation yield community reveal correspondence di sim treat manually citation information validate quality category paper paper intelligence value list paper major paper theory database hardware computer system program directional l db ec ir net os correspondence detect community manually assign significantly low community large significant major operating os intelligence ai major strongly connect big community find show field embed field example community relate ai category ai community meet expectation regard manual detect paper part within manual direction massive million scalable detection search community social highly asymmetric micro website china direct contain zero symmetric hour hour algorithm run six ghz gb check quality community detect large community small directional community additionally verify small scatter scatter plot directional community community look similarity node community community asymmetric community popular terminal source highlight directional community integrate role critical characterize community notion community capable svd sequentially identify linearly meanwhile small directional community enable asymmetric direct real matrix multiplication parallel acknowledgement grant dms dr dr dr wang discussion present search find directional find dc article proof negative edge notational convenience cs c adjacency remove row without denote node node whose zero back vector direct terminal undirected connect connect undirected dc examine connectivity connectivity connect terminal connect member node node g node member contradict node directional dc directional component obtain step apply eigenvalue matrix normalize define row say equal component graph span k indicator th component eigenvalue follow eigenvalue l eigenvalue eigenvalue q value principal eigenvalue vector break entry q find exclusive span expression component submatrix directional directional generate corollary submatrix equal include representation submatrix selecting row row principal value eq principal tell enough include directional component row column start direction place entry belong span span include include directional membership vector obtain set row decompose maximal connect within principal singular community connect connect contradict maximize directional component subgraph subgraph prove cs q condition community panel figure size randomly subset submatrix laplacian principal singular sub directional include external add three left panel adjacency directional scatter submatrix laplacian direct directional component matrix external plot graph derive direct external edge right panel pair submatrix component mark intercept maximize small directional describe directional directional encountered connect small directional community directional exact directional node limit small illustrate external merge together pair pair principal value directional external slope still directional approximately regularize embed directional entry objective write notice monotonically keep go keep thresholde large absolute resembles express objective multipli tucker solution boundary happen unless zero satisfy kkt satisfie threshold become satisfie z setting solution obtain z come inequality come way obtain approximate increase monotonicity z increase find lemma z initialize n second already therefore quadratic quadratic formula know di sim singular mean singular vector within centroid distance result directional community separate partition match terminal large common use implementation available default except option repetition v terminal value range determine community size k grid linear community early continue community reach leave citation network four manual category di sim table paper manually assign category h summary citation source terminal node stand c ht l ai db ec ir net os ht paper directional en category ec ir os ht source di sim
way difficult square covariance model slice easy step size tuning size lead computation change metropolis sampler slow paper slice latent highly correlate slice sampler metropolis sampler simple log model dominate covariance gp update cholesky decomposition th latent require decomposition almost costly cholesky decomposition slightly complicated consist gps hyperparameter covariance change secondary unchanged change unchanged change require major discuss table std operation operation operation sampler latent distribution q parameter great triangular cholesky prior standard normal transition reversible leave proposal ratio sampling obtain new update hyperparameter slice secondary hyperparameter metropolis tuning scheme highly sample use possible notice hence component update update model residual gaussian distribution gaussian uniformly true contaminate residual residual covariate method dataset u randomly seed dataset observation drop sample burn predictive distribution compute mse probability response hyperparameter mcmc give gp gp notice gaussian small residual winner giving numerical sampler latent metropolis slice sampler metropolis update sampler computation list adjust sampler efficient slice width adjust deviation acceptance around get c slice efficiency modify five run run bottom trace slice middle initial iteration sampler adjust time plot modify least mix appear metropolis roughly conclude mcmc fast reach equilibrium hyperparameter series main gp elliptical slice observation elliptical wang department university department residual arise propose serve since change derivative unobserved covariate dependent input residual model synthetic deal handle carlo addition residual non introduction gaussian gp text perhaps flexible function degree smoothness additive covariance function regression residual though many variance depend residual include unobserved variance change residual gp std method synthetic section association covariate pair length gaussian could specify zero reflect negative covariance hyperparameter covariance covariance suitably choose constant sometimes residual parameter length covariate lead ard use exponential covariance observe prior logarithm hyperparameter compute integrate let variance gp model expansion q denote derivative second e depend gaussian residual variance ignore consider chi illustrate translate purpose curve axis produce plot observe change look big around notice residual strong quantity although exist see trick produce exist unobserve often gp treatment response secondary logarithm noise residual variance gp prior
sample distribution accomplish call moment measure covariance major impact compact representation certain work euclidean space embedding space discrepancy mmd measure independence hilbert schmidt criterion characteristic rkh despite test question discussion context independence use definition covariance page confirm integrable show rkh dependence measure precisely formal function translation variant kernel two sample demonstrate distance maximum arise consequence arise derive arise kernel sensitive second obtain characteristic arise quite via bootstrap perform distance discrepancy know probability hellinger divergence family measure metric wasserstein metric mmd equality energy show mmd establish total discrepancy divergence equivalence implication practitioner use test member broad alternative one second readily general topological space indeed readily structure euclidean text string structure negative distance negative review mean mmd hilbert schmidt independence correspondence definite mmd give estimate quantity investigate variety conference publication technical proof omit independence type nonempty triangle arise also nonempty say terminology satisfy say function page page hilbert map second type take satisfy square root obeys assume topological borel measure finite sign borel measure borel condition ensure expectation twice von notion generalize exist ready see sufficient expectation namely need come remark energy whereby covariance random distance covariance testing measure dependence weight characteristic joint particular expectation generalization let space energy distance moment integral view characterize independence xy satisfy additional term discussion suggest closely whether xy question corollary understand reproduce space mmd schmidt begin reproduce hilbert z reproduce rkh page rkhs value say notion map embedding finite borel measure chapter let rkhs kf alternatively paper unbounded finite sign restrict later consider finite clearly theorem borel probability measure introduce notion borel hilbert distance embedding q square mmd useful characteristic characteristic characteristic entire iff characteristic alternative see mmd employ variable et topological space q product xy hilbert schmidt joint follow show hilbert operator xy see cauchy schwarz x dp pt dp dp xx yy dp yy identify tensor k p negative type section definite proving equivalence mmd equivalence definite closely adapted nonempty z consequence valid convenience induced kernel say pt brevity drop induce abuse kernel strictly would suffice center distance express type z z proposition valid kernel center covariance fractional brownian page metric type embedding relation kernel hilbert kernel define whenever say generate clear condition equivalent require shift kernel f inner product proposition special obtain pp positive definite illustrate kernel characteristic introduce definition clearly test literature consistency notion relate coincide proposition interpret space w generate n z z clear jensen z z note finite z able sufficient w condition namely generate mmd marginal moment approach approach two independence energy generate kernel generate generate denote page link rkh base mmd rise possibly merely metric induced isometry z z simplicity bound endow note finite moment otherwise energy expectation infinite satisfied finite discrepancy addition must equal invoke asymptotic maximum xy yy dx x dx dx xy use form marginal result use embedding metric addition immediate existence covariance strong marginal p k p hilbert schmidt relation energy covariance remark space x xy xy distance naturally analogy moment appendix distance term review interpretation measure first include completeness characteristic covariance choice nonnegative borel clear correspondence weight integrable continuous invariant test majority obtain testing translation invariant kernel rkhs extend topological unclear two sample characteristic function independence satisfy term review establish interpretation kernel k q quantity exactly immediate negative checking check appropriate borel embed terminology embedding rkhs hilbert characteristic embedding turn distinguish strong kernel characterize equality informally distinguish outline empirical distance mmd define moment generate suffice establish test strong z I statistic recall generate distance key role characterize nan degenerate associate page l p desire class operator hilbert schmidt rkh trace requirement hilbert schmidt kernel pn precisely hilbert case independence xx center analogous chi square product p p class operator remark xy k p asymptotic type quantile distribution yield bootstrap operator extensively compute gram aggregated matrix concatenation center show square estimation threshold bootstrap provide indistinguishable performance obtain empirical page establish converge sum chi present page work design attempt besides bootstrap propose test distance bind independent remark sensitive gram associate kernel assess performance rkh test kernel synthetic investigate kind two differ mean dimension benchmark one univariate hard plot bootstrap indistinguishable outperform test far conservative average I list test pt also kernel set median
allow discrimination character thus remove alphabet contain character character contain character implication development text regular modern form character word chinese text exploit capability recognize regular visual computer instance approach capture denoise model regular wavelet image go one explicitly pattern major pattern position ideally probabilistic generative straight salient appropriately character regular share character recovery corrupt document task robust heavily corrupt page character yes course yes amount self corrupt relatively character character entirely unknown aim optical character character method character generative patch corrupt text static mechanism discriminate character representation contrast allow varying pattern account introduce character discrimination representation truncate probabilistic pixel position patch feature thought patch denote mix patch choose uniform entire patch shape pattern model latent mask mask draw bernoulli mask patch outside assign pm illustration graphical definition background require feature document different background observe appropriately probability histogram model individually histogram b channel compute across patch include potentially nevertheless compute give pattern entry pattern position cyclic positions equation generative histogram show patch class choose pattern mask variable class eqn pattern mask translate combined eqn parameter likelihood frequently parameter em optimize summation joint rule pointwise multiplication computationally computation combination latent simplify observe decompose posterior individual binary q denominator efficiently make principle complexity pattern patch size hundred thousand still exceed improve efficiency therefore pattern position class apply suit factored variational approximation zero approximate point patch class obtain define function position scoring selection give feature pattern define high mask space position mask approximation reliably consider area large value proportional pc py experiment find simultaneously relatively feature low character document evaluate curve histogram blue course heat map generate rgb patch five different character choose colored character mask fig background mixture modal apply class observe blue fig compare variance mask course show converge organize pattern g matrix color individually successfully experiment generate apply document display character b manually corrupt line create resolution scan document automatically cut scan patch interval instead filter process structure patch entry third pixel array cut segment pattern position way increase large character take visualization course represent character mean represent character algorithm pattern pattern document character mask b much character high class patch representation e representation character illustration character detection character document detect identify document leave patch character patch assign match report match identify type character define template result match match match match binary mask mask mask probability mask maximally reliable maximally pattern match instance position close reliable equal zero one scale match quality patch assign input match poor match perfect match use representation character match remove corrupt globally good bound store fig use good reconstruct document match match quality use match visible patch onto document illustrate procedure visible patch match prevent reconstruct one match character reconstruct reconstruction reconstruct character bound competition find reconstruct terminate match accept sufficient successfully document perfectly reconstruction character character character supplement however reconstruction part error supplement random manual supplement decrease unchanged example supplement pose partly extend segmentation processing comparison result baseline apply standard experiment document recognize character essentially corruption cause correctly performance document non character fig respectively approach fp character evidence character improvement training label however fair intend versa generate different
expectation gradient let formulate call equilibrium solution nash j nash equilibrium dependent nash assume realize payoff unknown node equilibrium node keep equilibria different optima bad maximizer capture price suboptimal stability whole system j mention clarity decision state whole uncertain payoff stochastic call robust nash point game state difficult state definition equilibrium find observe amplitude perturbation signal represent payoff instant update perturbation node get realization dynamic environment time repeat sure vanish learn appropriate initialize perform observe learn subsection discretize present payoff function clarity try function rate limitation fine vanishing coincide ode contribution step vanish discrete uniformly existence maximizer j hessian dominant imply hessian hessian integrable ode system trajectory differential equation ode recent development find limit write rewrite jk law zero martingale rate solution ode kt te lt vanish solution ode ode window specify define helpful discrete ode ode need chapter important result ode assumption converge trajectory order ode convergence constant however notion weak vanish learning sort almost sure time ode isolated stability gap sketch theorem theorem first vanishe exponentially bound amplitude ode exponentially independent nash equilibrium payoff deviation nash profile euclidean nash equilibrium close equilibrium nash precision payoff nash equilibrium nash constant nash hold ode reach equilibrium corollary follow ode inequality hold inequality together result constant depend player stochastic ode state dependent payoff assume ergodic drift ode ode almost sure surely stochastic ergodic wireless illustrate contribution interference channel receiver show receiver receiver back payoff correspond receiver interference onto payoff interference wireless channel payoff receiver contain power receive local payoff ascent htb general application utility time power time gain receiver draw dash black thick minimum height em thick fill black scale node black fill sum em black bend right east color thick bend north bend east black em thick bend north north thick bend east black thick bend west thick bend right east node west interference numerically run payoff analytically nash choose analytically represent function transmission structure payoff h jj please payoff follow j assume identically variance assume mean follow wireless tune slow fast tradeoff discussion omit initialized interference bandwidth dotted plot converge htb type payoff bit bit communication channel physical interference source expression receiver receiver numerical advanced etc without channel interference particularly scenario reward tune global optimizer system optimal respective satisfied scalar vector able equal nash minima state sample ode error close numerical generic wireless illustration convergence step size work introduce payoff wireless area amplitude vanish subgradient consider include et local focus nash equilibrium deterministic scalar scalar action wireless example consider user reward surely j follow lipschitz continuity jt j rate step j l j j proof l jensen r r r complete compact need shall remark maximum I impossible imply continuously differentiable gradient absolutely integrable detail condition please q use imply finally check condition martingale k k surely martingale integrable realization value j j j bound show l z j j z sure ode interpolation get line chapter equation vanishe length helpful obtain compact te lt lc te adaptation dt r j trajectory trajectory ode time converge vanish start process concatenation interpolation k gradient payoff martingale ft k mt f j ds exist k weak vanish follow form form invertible combination gain invertible gain almost invertible control transaction liu stochastic application mathematical introduction basic theory algorithm asynchronous optimization automatic transaction subgradient journal sensor rd international sensor incremental subgradient optimization mit mobile american conference noise application mobile equilibria mobile sensor automatic dynamical viewpoint http www liu nash equilibrium general payoff nash person sciences equilibria game economic study pp method pp j pp mathematics sciences york university continuous wireless network wireless communications international pp conference focus proof receive european grant agreement rs france france fr attention system equilibrium distribute become application equilibrium extend payoff develop algorithm nash trajectory define vanish provide conduct stochastic payoff distribute consist node model distribute node interact another payoff node system reward change node k contain action interact reward interact existence access reward maximized scenario interactive game iterative nash draw thick
dataset denote sample miss set least denote corresponding contain miss element variable keep mind mm mm I sub inverse keep instance currently represent sub part context directly miss gaussian maximization compute probability clarity iteration compute eq gaussians fill equal expectation observe assume denote equal q covariance result imputation value sake clarity miss miss regularization diagonal extend high cost burden somewhat nearby analyze computational cost em present require inversion operation pattern miss operation clear miss real missing since lin suffer fast pattern number miss barrier problem motivate reduce cost variable quantity determine another pattern miss observe pattern differ nearby miss compute miss consecutive want cost cholesky show pattern another two optimal minimize result summarize argue fast numerically stable cholesky low triangular triangular compute pattern question update pattern find remove row variable variable find row column add dimension minimize remove remove consecutive miss pattern find order discuss em computation conditional rely cholesky need order advantage inverse partition covariance part matrix conditional partition inverse go go miss miss follow inverse miss sub per em miss small dominate left remove add xy yy yx yy term remove miss update order suppose order order pattern compute cholesky miss pattern miss fast present parent miss pattern matrix visit miss pattern tree fashion thus span note constrain miss update simplicity use number true cholesky varie add dimension argue try sophisticated decrease virtual observed update set find al np scheme provide huge stability computation incremental accumulate incremental link depth may grow bad case never lead significantly exact solve interest summarize section sketch em pattern span pattern span miss pattern initialize covariance ignore increase expensive matrix computation naive em mixture gaussians handwritten optimize gaussian miss remove pixel e mixture rest sample select namely fraction principal keep random initialization ensure computer ghz architecture ghz memory algebra library realize train row grey imputation pixel capture sensible value uci repository validation behave systematically normalizing compare compare imputation expectation value learn imputation value neighbor alternatively obvious way nearest presence miss allow neighborhood base original obtain neighbor one report predictor feed discriminant optimize feedforward descent hide among weight among initial among ridge hyper decay kernel bandwidth dot three imputation mechanism regression imputation gaussian global imputation near imputation try keep set seem reliable miss value go neighbor illustrate generative mixture learn even mixture regressor greatly improve supervised htbp database financial learn company number preprocesse obtain contain record dimension purpose comparison validation test discriminant value imputation mean ridge discriminant ridge imputation imputation mechanism report statistically paper training mixture context imputation computation update span hybrid gaussian discriminant model significant imputation datum matrix problematic learning
condition relation statement addition statement statement establish outer pd solving problem suitable accumulation moreover accumulation minimizer suppose fx compact statement accumulation problem condition bound order affine minimizer view choice together boundedness accumulation exist recall index bound subsequence ki k j statement indeed let imply know use claim contradiction pass subsequence otherwise divide side take limit relation sufficiently since fact identity subsequence accumulation subsequence definition immediately lead relation hold subsection method establish reformulate associate quadratic ready pd let define arbitrary apply subproblem solve l yx k k satisfy step termination subsection method mention enhance solve subproblem b establish omit solve accordingly replace sake brevity omit point affine let also relation definition addition affine together sequence accumulation saddle function convex minimizer follow accumulation moreover observe bound together implie hence observation q limit continuity addition subsequence equality relation limit saddle statement due establish problem q strongly rely subproblem addition reformulate outer suitable generate nonconvex accumulation generate pd fx bound accumulation hold accumulation first convex statement statement let rest proof conclusion statement theorem conduct performance section apply sense code method matlab www perform matlab b intel cpu ghz gb ram subsection subsection vision bioinformatics neural processing see logistic sparse logistic integer solve aim therefore suitably solve effort project terminate termination criterion pd apply set termination pd next conduct pd compare quality approximate first solver parameter default first pd small medium data uci repository tumor internet magnitude discard apply bind pd sparse outcome predict predict detail column column ratio column cardinality average logistic second pd report six solution generally achieve low sparsity c pd c pd method three different size second equal manner sample outcome sample normal draw apply solve five let approximate pd result approximate pd slow logistic demonstrate quality approximate pd surprising c c pd c c pd subsection inverse real recognition see conditionally sparse covariance selection formulate integer follow eq subsection let clearly thus suitably pd eq computation pd lies give slightly pd solve subproblem arise step q solution v tc arise address termination pd outer termination accuracy random next conduct pd routine full eigenvalue symmetric routine parameter first experiment compare generate manner describe particular generate prescribe letting pseudo draw uniform covariance positive empty apply four pd covariance define pd instance four objective normalize give six respectively observe fast instance quality loss pd c time pd c pd c time randomly nonzero diagonal equal sample covariance ij j share diagonal pd method apply pd observe pd completely pattern solution see pd large pd performance pd method widely pre describe entry pd sparse set modify normalize name fourth loss pd last pd pd outperform normalize entropy summary experiment c datum pd pd solve problem noise cs formulate approach solve see apply pd thus pd suitably effort pd method subproblem arise address initialization termination pd obtain matlab set penalty termination criteria pd solve random obtain solver experiment observation let value apply pd solve adopt criterion mean error successfully detail recover cpu column two respectively observe pd large also second experiment except randomly orthonormal computational also pd c pd cs formulate integer control popular approximate solve study subsection form pd subsection suitably pd solving subproblem unconstraine solve conjugate method termination pd randomly initial outer termination criterion pd set associated conduct solution algorithm find solver increase accordingly pd mention warm start approximate pd resp apply plot detail residual average accumulate cpu cardinality second graph observe residual pd almost substantially method conduct generate orthonormal plot pd slow trade general minimization coordinate pd nonconvex accumulation compress sparse generally lagrangian simply replace pd augment lagrangian pd provide recover arbitrarily denote easy recover regularization relative fairly true remark paper problem optimality problem pd method solve subproblem solve accumulation sequence generate pd satisfie optimality condition nonconvex accumulation minimizer saddle moreover accumulation subproblem performance pd apply inverse compressed computational generally minimization compress sparse concerned formulate idea recently selection discover independence popular approach inverse covariance likelihood propose promise minimize logistic formulate solution continuously entry solve iterative square arise deal application compressed relaxation replace fail penalty decomposition problem accumulation sequence first convex show accumulation minimizer point saddle subproblem apply logistic inverse sensing demonstrate outperform organize minimization x solution
dimension formula density challenge even evaluate numerically stable introduction overview nest copulas tackle first copula family evaluation density three nest copula briefly address convenience defer copula dimensional later copula copulas copula responsible dependency several package way follow nest copula copula possibly argument bernstein transform sufficient proper copula completely monotone u frank see generator laplace property copula family family allow example power nest copula reference copula copula possibly condition frank fulfil generator belong proper composition completely generator eq integral compute theoretically especially complexity theoretically copulas child challenge leave generator compute integrate solve di polynomial prove back fa di fa formula q q come frequently denote denote recurrence align leave kx ns ks nx sign inner generator copulas fa di generators generators time derivative cauchy appear correctly allow compute expectation derivative solve monotone align leave coefficient cauchy align coefficient evaluate challenge log evaluate adjusted derivative quantitie part child copula exist theorem nest copula turn generator transformation transformation generator generator form proposition generator theorem generator note u slight copula outer inner generator copula concern former concern plug simplify term power q furth eq nest copula type formula plug kt simplify generator nest copula type follow family generator calculation show hence argument part generator derivative kt ks derive nest component generator generator inner generator coefficient polynomial derivative inner frank generator one inner generator appropriately shift part copula show hold copulas root copula copula refer copula case plug provide clear part basically piece one density numerical typically compute logarithm logarithm face density briefly nest sign term theorem st double typically correspond computed implementation package use kb precise package l two nest copula equal graphical insight copula copula copula dimension degenerate copula st product accordingly display plot copula sample minima display child plot easy copula behave see likelihood copula surprise marginal copula package directly fit nest copula life set figure nest copula htbp u section analogous nest copula briefly address working turn convenient consider level convenient denote copula root copula cd section transform scale grow south level style mm style mm level style level draw mm style draw style dot thick child child child child child node mm mm mm mm mm mm mm mm mm mm replace thus denote respect key challenge align find integrate challenge di formula suitable block di yield special strong lead partial moreover exist function position slightly l q summation write solve similarly eq pattern nest copula previous heuristic useful copulas trees representation branch tree branch branch polynomial polynomial vector equation term stand polynomial apply lb q appear author would like thank feedback align x l n
open since vanish branch use denote lebesgue denote gauss study prove side parallel axis characterize limit modern terminology absolutely continuous measure rectangle equal circle identify point thus random variable way result rectangle side axis deduce also measurable borel b bb main interested q informally negative observe factor agree characteristic variable write independence integral product q true elementary representation split variable put interval interval follow half definition use polynomial use integral follow euler ib proposition entire convergent proposition since product suitable even nonnegative recurrence expand power desired define verify reverse get recurrence simplify recurrence recurrence polynomial writing recurrence integer recurrence convergence series number determine also integer recurrence coefficient positivity correspond sum substitute recurrence sequence zero recurrence give recurrence appear satisfie sequence remarkable identity number polynomial number omit limitation l assume result remark integral representation expect confirm corollary make lemma multiply last integral representation proof proposition real thus dominate dominate uniformity q consistent analytic extensive numerical ib let immediate corollary q order numerically compute sum express compute inversion analytic half need illustrate technique fail fact origin fortunately sufficiently large corollary zero disk practice choose somewhat large proposition evaluate return evaluate equation arithmetic varied dynamically example sum alternative compute advance recurrence evaluation expense number depend interval recall sum value therefore may van root root constant measure detail omit q recall result evaluate function make periodic period function dt characteristic quadrature stepsize measure finite get simple odd computation eqn similarly course analogue proposition evaluate constant computation give difficulty extend motivation analytic independently implementation recurrence directly agreement statistical error carlo region latter van place sum show slow illustrated slowly computation critical suggest plausible seem strong imply pt precise consequently generalise cover dirichlet character euler product would sum modulus would course nevertheless sign mathematical sciences institute national van van pt explicit characteristic accurate numerical practical expression obtain accurate gram positive real sense critical assume
program traditional programming within calculate asynchronous parallelism boolean boolean logical fuzzy logic use form value explore collective behaviour delay representation competitive exploit brief give behaviour detail present require exploitation memory dynamical continuous discusse section use continuous production present discrete action final dynamical array cell lattice set update parallelism extensively genetic network exist extension connection influence body explore value differential wherein logical investigate fuzzy background fuzzy logic lead logical fuzzy life generate toward fuzzy behaviour form herein base connect include read write connection execute parallelism cycle finite machine programming graph structure evolution recently traditional symbolic expressive power represent purely rule direct token stack label string language token translate similarly program node label subsequently stack token specify matching condition action graph stack string token non form fuzzy representation radial condition hybrid system explore base contribution determine extent action node ga date rbf explore hybrid originally introduce aspect genetic since area quick typically boolean generalization machine dynamic affect behaviour wherein behaviour order consist regime around trajectory neither per analytical reach closely describe must play game wherein mutation change connectivity boolean typical high fitness increase approach enable heterogeneity type feedforward value value great rapid change regime occur change update step asynchronous possible asynchronous form wherein suggest realistic underlying discrete network certainly development fashion asynchronous device less towards asynchronous tolerance particularly delay insensitive scheme may hardware asynchronous see asynchronous feature significantly thereby aid evolve exhibit behaviour equilibrium asynchronous change asynchronous network significantly lower expect asynchronous lack complexity interaction transform reduce state asynchronous loose term take alphabet string reward unit predict fitness symbol environment condition input receive match logical match generate classifier environmental covering match current action present action represent prediction make weighted propose covering match current relevant alternating employ wherein exploration conduct otherwise exploitation occur compose execute environment feedback receive update ga ga great ga parent fitness produce usually mutation payoff error fitness average parent include parent update ga oppose step problem loop instance maintain weight input extra maintain environmental classifier weight reflect prediction rule correction control correction subsequently prediction enable accurate payoff piecewise constant boolean fast optimality see asynchronous rule node initially many connection input connection assign way current update etc cover apply l table connection node node input cycle typically implement cycle overall occur explore eight action extra node enable logical regardless join exploit within neural decide last update cycle node would randomly cycle status output create match mutation maintain mutation mutation locally evolve search mutation hand evolutionary algorithm mutation truth represent string connectivity maintain string form entry evolve string mutation adapt evolution connectivity map mutation initially uniform pass mutation rate within truth connection new connect either add node remove occur rule increment initially scheme whenever mutation mutation short term buffer input buffer past clear require present duration define pattern require length distinguish position temporal whereas memory state index maintain internal instead explore soft operation within explore hypothesis inherent content due asynchronous maintain input output cycle significant adjust computing addition short number network place cycle receive environmental input place trajectory different locally therefore environmental fall affect environment require resolve see environment binary bit furthermore task simply short start mechanism employ whereby goal f cell action environment contain non require action solve optimally internal state performance memory linkage recurrent direct np capable represent twice figure figure bit classifier figure mutation rate also rapidly around trial performance also rapid subsequently fewer adjacent increase complexity would three memory resolve evolve bit situation thus large memory optimally bit achieve step prior classifier internal increase observe trial slow inherent selection policy activity unlike number population converge mutation rate reach lowest htbp fuzzy logical fuzzy thus generalize continuous generalize fuzzy topology able dynamic network cycle give set time update q randomly fuzzy logical decide input logical membership membership update simultaneously fuzzy logic min commonly fuzzy mention potentially leave appropriate htbp fuzzy min asynchronous scheme fuzzy benefit figure result experiment increase change state cycle value less cycle due tendency fuzzy I increase initial change fall respective asynchronous update scheme change asynchronous converge around whereas aspect use adopt create assign connection input first message connection assign random cycle population initially empty cover generate rule match regardless join neural fashion binary whereby calculate represent integer reference operation upon input see fuzzy function list integer represent mutation rate evolution select fuzzy within rule along connectivity wherein agent within attempt short locate corner otherwise action would outside move whereby trial agent choose north south east west step state combine
always suggest function eq reference point reference point exist candidate contain generation however unnecessary density reference candidate proposal jointly importance procedure figure candidate work important consist depend need importance use pdf walk distance vary possibility delay rejection trade thank comment manuscript work partially project ref ref process ref ref try classical hasting choose set extension upon flexibility balance scientific implementation family core monte sample powerful markov carlo generate stationary coincide density pdf typically able mh try metropolis weight chain select advantage portion decrease acceptance domain carlo good performance interact adaptive basic modify rule analytic specify extension reversible candidate result asymptotic strategy consideration weight differently delay augment auxiliary stress remark upon flexibility rule within mix building pdfs present construction acceptance scheme theoretically possible design drawing algorithm affect detailed organize explain balance acceptance rule introduce comparison draw mh possible draw pdf movement balance correlate good proposal density whereas author mix together extended drawing candidate proposal analytic arbitrarily function draw proportional state chain scalar treatment draw independent pdfs therefore sample algorithm joint draw calculate weight normalize draw let satisfie detailed condition eq proposal express use choice finally scheme fair generate candidate moreover equation simplicity maintain sequel multiple mh approach weight acceptance indeed choose k finally fulfil e equation separately mh similar function provide summarize draw proposal pdfs normalize draw otherwise probability go back markov method generate condition write delta previous one exchangeable write total candidate correspond integral recall definition eqs expression symmetric vary assume generic balance suitable acceptance separately yx instance I possibilitie non function reference q obtain acceptance need possible previous consideration design reference point author consider drawback scheme since draw balance avoid sample draw calculate draw compute difference position balance show seem evident great say jump increase surprisingly decrease get bad consideration explain reference special fulfil kernel express candidate show reference integral obtain eq rewrite straightforward vary proposal pdfs choose therefore case necessary draw discuss depict pdfs proportional pdf proportional simple instance fig acceptance importance numerical compare pdfs drawing acceptance average markov chain table rw rw one proposal pdf proposal pdfs proposal pdfs table walk proposal densitie independent proposal important proposal weight observation extremely design good density scheme heavy call evy normalizing tail evy goal normalize constant via carlo chain apply three try choose suitable two proposal rw heavy feature average summarize c std rw consider p note possible combination function section different candidate acceptance movement next acceptance illustrate use great acceptance try almost invariant try acceptance rate correlation acceptance rewrite indeed obtain power increase complicated dimension mh provide pdf pdfs shape density shaped kind compare shape pdf performance sampler show htb face remain sampler iteration remain repeat standard mh sample markov draw apply walk proposal pdf I sampler mh standard deviation proposal table provide acceptance probability state average jump etc correlation rate c try mode jump rate table clearly mh since grow decrease jump rate increase acceptance obviously jump movement acceptance mode jump standard scaling candidate since movement namely evident vanishe grow great jump mode quickly face target clearly explore large portion sample study flexibility introduce analytic density draw propose acceptance provide example need reference satisfy balance theoretical infer observation consideration standard mh pdf candidate quickly consideration computational suitable apply kind target unbounded tail moreover advantage mh scheme black box algorithm simulation try independently choice proposal proposal
sx collection take uniform reduce computation notice calculation sx sx cardinality change one permutation state sx formula sx take q case ii site dash shape draw draw circle dash dash dash subset note connected neighborhood site measure probability x compute joint x treat later show herein covariance match choice example neighbor match covariance thick match circle shape draw circle draw shape shape thick thick thick herein field construct satisfie x g ray imaging herein space neighborhood analogue formulate size qx radius n nm I list pixel image pixel I restriction ensures directly generate verify abstract connect natural application meanwhile j site example necessarily albeit necessarily follow mm right dash dash dash dash dash dash dash dash verify connect abstract enough illustrate setup I cardinality denote I I simulate I nearby site x k I uniform random I notational convenience valid site compute x generation target generate automate public computer apart prevent intended abuse automate agent appear character digit easy difficult wu propose name specify marginal pixel embed quantity conditional formula real yet generate markov end generate include detect coin flip unit coin tail classify coin flip either person generate coin coin flip sequence count approximate total period expect way behaviour average covariance flip follow flip simplified algorithm coin flip head analogue wu problem hide still forest surveillance equip capture camera gap correlate either correlation forest observation overhead pixel small area generation use forest target function filter resort specific generate regularity right consistency permutation guarantee follow sufficient regularity singleton proposition neighbor property property site site neighbor site zero type relate marginal singleton valid condition constraint therefore proper equivalent appendix side sx verify tx x sx require singleton simple corollary condition apply also necessary assume neighbor probability x satisfie lemma give form assumption I n repeat quick field definition fact ix ix permutation neighbor rule hence simulated order simulate matter site site site neighbor sufficient permutation give u u use u thing u let I ix adjust component u I triple integer formulate equation triple contain n hereafter write cyclic site neighbor solution canonical permutation n pmf x sufficient permutation n necessity h g satisfy distinct solution canonical linear I without guarantee I prove generator I permutation x point albeit n site neighbor condition joint x imply I ix loose degenerate structure equation canonical structure exist dimensional j k distinct construct satisfy multiplication triple apply n similar say l u separately u u u k l j equation verify also one summarize nonzero solution nonzero formulate satisfied investigation guarantee verify contrary obtain equality n site neighbor hold u application site neighbor site ix I x I let site determine condition I permutation x follow follow x form pmf directly lem necessary formula uncorrelated uncorrelated statement follow immediately formula assume x produce field remarkable concept extend system site ss uncorrelated independent regardless rest state exactly field definition field condition sequence I site formula permutation choose site I compute I treat recursively repeat compute u x simulate value x generation field x make site gain new permutation extra condition actually field site markov field uncorrelated n prescribe dynamically take guarantee site behave every much treat employ reason site multiplication rule property break concern multiplication big neighborhood alternative h nonempty proper pick otherwise subset I site h consist site nonempty since induction ib general nonempty exclusive subset j mb connect nonempty way choose nonempty b start site b pick within b stop repeat neighbor b procedure must stop nonempty exclusive replace third repeat paragraph mb mutually exclusive exclusive b component contain site henceforth nonempty first exclusive I choose site proposition distribution probability ease subscript fourth fix mass therefore q follow section corollary section section herein new quick rely pass convergent method mass site base quite maintain site successfully object limitation impose covariance property field markov derive simulate field field phenomenon use statistical mechanic application field tv image image functional imaging web extraction etc reader researcher synthesis classification optical optical character match recognition recognition video surveillance sparse language chinese interested inverse motion precisely formulate mathematical certain various estimator need simulate simulation smoothing technique chapter technique random field typical certainly exceed modern computer simulate field resort impose discrete field need gibbs sampler briefly speak realization site local configuration sequentially update hundred produce deal propose new incorporate give function site covariance nearby mind configuration discrete simulation far correlate carlo simulation impractical establish covariance field maintain ease proposition site portion random sequentially feasible precisely simulate conditional portion construction actually construct pass dramatically random object field generate condition base singleton particular investigate permutation property case want covariance pair site form auxiliary collection ensure field regardless order component really setup simulate unknown portion discrete valid multiplication rule configuration probability formula use kolmogorov permutation condition condition kolmogorov extension existence vector satisfy n condition site site setup I portion x way k consistency mass list follow permutation g I k h kx x happen x order
integrate plane away derivative vanishing know result likely near useful mnist dataset intersection present behavior embed cause integration disk case intersection involve subtle manifold near neighbor onto onto decompose exactly denote ball center tf follow eqn eqn eqn projection du derivation fourth relax result second eqn third replace replace intuition behind derivation slightly whose projection follow mm derivative intersection e boundary intersection point point boundary boundary effect manifold produce proof smooth nonempty boundary smooth continuous rt angle normal cumulative edge limit close move normal boundary equivalent effect manifold intersect co actually regard piece manifold together advantage view allow intersection continuous case manifold boundary corollary let smooth manifold boundary f eigenfunction intersection continuous within piece manifold useful probability constant inequality appendix asymptotic arbitrary change singular within rescale seem counter intuitive require inside estimate differential operator play likely subtle case implication result place put converge necessary derivative exist impact numerical sample infinite near x see mean eigenfunction laplacian automatically boundary experimental direction hence directional limit surprising reason imagine eigenfunction laplacian near singular structure preserve intrinsic distance edge laplace numerical phenomenon give implication eigenfunction clear investigation eigenfunction situation analogous manifold neighborhood least minimize quantity orthogonality contribution bound department computer science usa laplacian construct considerable exist interest argue boundary aspect geometry realistic boundary process bounding manifold intersect smoothly near boundary manifold together direction laplacian near manifold phenomenon somewhat interior laplacian laplace exhibit implication volume difference large contribution behavior distinct comprehensive understanding toward lead significant deal non key year become intuition surface situation riemannian reflect boundary basic arguably intersection manifold come happen right computer graphic edge surface whenever phase transition boundary occur bound constrain range motion ambient negative intensity paper laplacian near inside gibbs operator different scaling behavior ignore globally despite relatively locate contribute fine believe comprehensive understanding complex design gain acceptance task include supervise mathematical many construct datum laplace heat obtained study infinite reason edge type somewhat related development considerable recent aim algebraic term guarantee special manifold recent interest reconstruct topological manifold line relate provide singular appropriately scale section bandwidth reference infinity appropriate differential manifold exhibit near approximated scaling counter relative inside eigenfunction number application subtle proof smooth answer brief boundary boundary limit eigenfunction smooth one paper intersection intersection seem effect eigenfunction straightforwardly walk normalize graph discuss popular derivative expression involve boundary condition restrict manifold simplify exposition intrinsic boundary smoothness way associate intersect interior point exactly smooth intersection type exposition intersection happen interior edge j technical reason singular manifold moreover intersection tangent tangent piece union notice manifold smooth kf ic px sample build vertex assign q gaussian euclidean analysis weight unnormalize laplacian laplacian primarily involve bandwidth limit various analysis regular see version point limit bandwidth analyze
uniformly cdf univariate cdf note univariate density write express satisfie j f jx sample I estimation function inverse covariance matrix transform light natural candidate large dimensional relative alternative use tradeoff choosing estimate sample standard estimator jk replace respective normal transform truncation tune amenable main covariance entry theory jk jk positive constant detail give show frobenius j nonzero precision precision probability particular rate rate graphical however recent equally flexible approach normality restrict tree nonparametric lebesgue measure identically sample range f bivariate density univariate family forest forest density univariate density log minimize entropy forest satisfy forest hand constant forest connect node find span setting way stage add connect yet visit stop edge replace plug bivariate univariate liu edge degree automatic need adopt split set marginal carry evaluation j univariate estimate cube numerically evaluate kernel point choose th approximated make concern care need truncation use ensure liu span k cycle k j k f full overfitte edge induce liu forest validate f evaluate solely hold hold forest construct estimate carry iterate edge obtain compute maximum span hold estimate property type density old exponent error kernel yield kernel density kernel choice family density access liu omit technical main hold notation follow fact increase increase rate excess still note estimation state order value long integrate slow technique correct work investigate achieve minimax protein study gene analysis assume know sample discussion pre gene pathway treat glasso graph wide parameter suggest lead different regularization compare path regularization evenly space similar subtle nonzero compare value column regularization glasso graph close fit glasso vice gene typically large consequence regularization lasso step gaussian sparsity obtain glasso well surprising univariate unable held likelihood hold log likelihood glasso maxima inferior kernel figure graph graphical automatically hold clearly big co yahoo amazon com collect stock yahoo finance finance yahoo daily price trading denote stock day consider replicate series share compare stock classification stock stock stock stock stock stock stock stock stock show correspond stock stock clustered stock interact indeed example two yahoo target forest graph glasso forest color glasso span show edge estimate glasso forest glasso node good graph regularization edge neighbor remain node point density full sensitive exploit estimate graph stock make dimensional density cluster outlier describe forest stock result display difference common estimate glasso difference forest graph draw hard conclusion effectiveness independence relation fully surprisingly graphical high piece relate spline decompose one identifiable dimension still tractable see markov function equip penalize induce solving require dimensional efficiently log analyze another conduct markov factorize clique neighboring clique exact expensive strategy structure global theoretical paper undirected graphical least simplify tractable consider efficiency rely function distribution acyclic bivariate together clearly possibility nonparametric potential work function accurately support method estimate selection approach discrete normalize strategy variable framework conditioning collection cart go cart estimate conditionally model idea build partition cart classification leaf glasso go cart reduce vary variable explanatory variable simplify straightforward mixture forest read assume joint distribution assume forest forest forest extend handle extension graphical approximation yet experience show incorrect estimate wrong graphical draw finite already essentially clique number discuss flexible nonparametric use estimation graph construct graphical discrete alphabet log linear clique parametric distributional yet alternative particularly build two allow arbitrary distributional restriction copula extension kernel graph forest nonparametric expense two
become active stepsize value small cause constraint become entry support small critical sign accordingly homotopy jump one critical value updating solution homotopy come compute element homotopy homotopy close multiplication experiment propose homotopy algorithm update homotopy cost homotopy solve minimization adaptively every homotopy step accord homotopy add element shrink homotopy act homotopy attempt desire adaptively shrink homotopy path sequence homotopy homotopy value encourage active reduce fast achieve inactive weight change direction homotopy predefine summary solve homotopy problem adjust initialize want homotopy index active inactive follow select fast initialize correspond denote weight select select rate reduce form solution homotopy inactive equal value active illustrate various homotopy choice block length accord setup sec homotopy way homotopy size become small rest select select homotopy select toward methodology optimality support toward along straight move direction maintain respective subtract alternatively inactive become maintain optimality suggest maintain weight value priori scheme adjust setting stepsize cause support select inactive large set repeat update direction stepsize updating homotopy termination main homotopy system compute multiplication element homotopy update rank cost every step multiplication homotopy homotopy performance h accuracy high quality signal cost comparable h quickly expense h exist solver present solver fast sufficiently warm start old warm outperform h recover type random accord generate signal wavelet transform block toolbox interval disjoint region add uniformly transform piecewise signal piecewise signal haar present result haar transform nonzero number wavelet element signal length signal signal jump select amplitude divide three add transform generate jumps transform present fig decrease near generate entry select entry reconstruct algorithm describe example block haar perturb sparse wavelet measurement although weight accord make instead rule good manner homotopy http edu homotopy reproduce weight follow priori select adaptively denote previous homotopy shrink large dense norm proxy support every multiplication identify change support rank update use inversion update every accord outline iteration homotopy solve equivalent iteration weight fix main involve update inversion alternate problem detail solver package use solve old warm solver tolerance order use shrinkage see h iteratively solve use use old warm termination tolerance accommodate weight solve constrain initial weight warm default optimality multiplication summarize select use update matlab implementation reconstruct db reconstruct performance h recover simulate procedure record figure figure signal signal plot first weight superior performance plot row display although iterative h solution h top solution five iteration top row haar wavelet transform randomly perturb wavelet signal measure gaussian gaussian reweighte unweighted perform reweighted row iteration row top multiplication fig fig signal application plot count initial depict since present count first compare matrix product least application count appear count count count third count count iteration count iteration h appear block signal row utilize row iteration runtime display nevertheless runtime small third first reweighte unweighted five reweighte row reweighte third row iteration reweighte unweighte iteration third row brief observe recover well homotopy problem computational iterative scheme iterative warm homotopy efficiently algorithm iteration quickly expense sec homotopy adaptively homotopy method iterative term computational homotopy embed thresholde iterative solve shrinkage generate solution previous determine stepsize entry respect soft thresholding shrinkage shrinkage algorithm iteration adaptively accord solution offer parameter value code method gray compressive measurement setup model generate sparse wavelet wavelet noise entry three presence db snr original peak application experiment third reconstruct warm employ last reconstruct solve code value observe modification cost simple scheme exist enhance scenario detailed regard scope east image north south east north west rectangle south east north west rectangle original inside update reconstruct count average experiment north south east north image south east west rectangle symmetric extension inside box reconstruct image count grant foundation often solve many replace common update two homotopy reweighte present change homotopy replace one second propose algorithm solve weight adaptively select signal algorithm along homotopy change support single homotopy compare solver method yield reconstruction quality fundamental recover measurement observe signal noise obey incoherence program solution term keep commonly pursuit denoise yield enhance weight adjust coefficient original nonzero location elsewhere since location nonzero priori critical perform iteratively solution next appropriate value use compute iteration arise solve solver homotopy homotopy homotopy change linear homotopy path homotopy cost homotopy standard homotopy trace support homotopy respective follow homotopy present weight homotopy update solution change view strict elsewhere solution maintain optimality must keep direction violate indicate add cause
localize multiscale conclude brief graph oppose graph prove particularly visualization numerous science area expansion transform definition enable important classical ne separate signal application way connect thresholded physical vertex describe especially supervised connect distance e signal represent unweighted one associate connect original graph combinatorial laplacian element weight edge laplacian difference neighborhood eigenvector real eigenvalue consider assume fourier expansion function eigenfunction expansion eigenvector laplacian classical analysis carry frequency slowly whereas far associate eigenfunction rapidly provide notion connect eigenvector vary I eigenvector associate rapidly value demonstrate eigenvector eigenvector signal vertex vertex b eigenvector sensor positive come zero non normalize laplacian sensor graph latter laplacian associated laplacian fourier give equivalently different vertex domain useful spectral domain refer signal heat analogously analog fourier decay closely approximated fourier coefficient e way b road represent blue positive negative come vertex signal domain graph domain plot take emphasize geometry incorporate differentiable manifold definition differential operator operate calculus mathematical precision intrinsic examine normalize normalize manifold vertex value note quadratic norm neighboring vertex large return laplacian tell also minimizer laplacian smooth interpretation laplacian carry notion frequency connectivity laplacian transform eigenvector box demonstrate content depend mm signal vertex edge show signal bottom spectral domain smoothness graph signal structure least respect intrinsic see visually ii laplacian low frequency fourier transform popular option define graph partition every connect eigenvector n spectrum also carry high eigenvalue unlike normalize laplacian connect may graphs vertex vertex walk converge go infinity walk asymmetric laplacian identity laplacian due detail basis clear normalize nice contain fact eigenvalue useful extend dc review develop multiscale transform filter start extend filter localize vertex domain classical frequency combination complex transform domain correspond convolution domain notion also generalize theory write filtering discrete version many arise solution variational problem discrete reference relation discrete filter arise differential application image processing mesh example particular application mm uncorrelated additive wish enforce clean view example noisy pass x pixel connect horizontal vertical set neighboring pixel similarity noisy weight take perform diffusion smoothing display row image comprise filter smooth area image classical smooth laplacian via filter signal domain signal neighborhood constant vertex domain relate graph filtering filter also interpret domain yet short distance vertex number comprise connect filter polynomial filter relate filter vertex definition convolution one define product graph convolution multiplication graph translation define change generalize translation delta center weak way convolution center vertex remark act define translate apply second normalizing ensure preserve control localization vertex decay distance increase property translate heat translation operator operator graph eigenvector h translate c heat figure rely frequency content represent eq q define generalized laplacian graph discrete nature however graph domain localize analog set instead transform setting assume unlike generalized generalize require entire diffusion operator example discussion wavelets intuitively apply different power heat describe flow heat proportional weight signal heat vertex heat vertex variable entry far apart small e justification structure increase nice illustration heat notation q initialize dyadic power heat diffusion interpret original show k e kernel dyadic power dyadic power importance wavelet section multiscale transforms signal graph require successively intrinsic geometric connectivity spectral sparsity transform give fine reduce vertex preserve process separate closely reduce assigning connect new additional first often refer special bipartite connect subset bipartite notion bipartite technique algebra mention lee weights walk transition greedy recursive repeatedly part sign subgraph minimize connect generally bipartite graph subset large refer reader therein review literature connection domain closely extend concept signal graph show class reconstructed reduce localize transform design wavelet analyze computer wavelet wavelet graph dependent wavelet tree graph wavelet channel wavelet filter fouri design filter domain wavelet transform thus exploit transform graph signal graph locality spread principle trade signal open recent spread define center vertex interpret mass pmf signal geodesic signal affect purpose example trade exist graph wavelet normalize laplacian one wavelet reconstruction expansion resolution decomposition remainder design simple example broadly type vertex domain design transform base graph vertex localize transform filter vertex node transform transform wavelet wavelet transform unweighted graph compute node neighborhood weight neighborhood outside guarantee wavelet function localize location minimum wavelet transform wavelet originally signal approach set odd node compute neighbor prediction neighboring wavelet balance hierarchical tree partition define level modify graph wavelet base eigenvector wavelet quadrature mirror idea graph construct diffusion wavelet operator basis level schmidt scheme translate version discuss spectral bank synthesis filter prototype graph bipartite intuition present design wavelet short center constant wavelet across vertex wavelet vertex depend exactly wavelet interval take wavelet graph wavelet function vertex wavelet low pass filter wavelet generalize pass satisfy q transform spatial two transform classical invariant necessarily wavelet spread transform scale wavelet scale spatial transform change location take average different graph regular wavelet wavelet axis axis wavelet localize localize spatially wavelet understand spatial signal graph htb ability graph wavelet transform piecewise signal unweighted road color represent well kernel design toolbox scale coefficient respectively magnitude concentrate mostly near pass signal fourier transform f f piecewise smooth severe unweighted scale wavelet cluster area review way generalize elementary set core processing localize multiscale transform generalized operator section multiscale transform review process fairly frequency extend transform surprising property moreover thus quite challenge ahead briefly mention extension method paper incorporate structure way yet construction affect localize transform signal mention clear graph domain short diffusion analyze transform operator useful high adjoint scale confirm fast transform computational signal transform involve computation require eigenvector normalize graph area approximate operator open include fast fourier transform datum deep class wavelet see open field signal localization generalize design transform may heart localization vertex appropriate notion non unlike classical euclidean
subject reconstruct robust face b represents argue solve hence necessary effect consider aim well sparsity block minimization pose tradeoff counting hard work develop tractable relaxation group replace relaxation mix derive entry main understanding relaxation give surrogate however relaxation surrogate depend raise whether preferable relaxation minimization question duality derive relaxation introduce derive lagrangian original relaxation minimization problem importantly lagrangian respectively program gap correspond lagrangian relaxation minimization provide relaxation sparsity lagrangian minimization optimal primal objective relaxation lagrangian generalize lagrangian value choose value describe introduce sparsity otherwise block contain non zero express introduce matrix entry belong otherwise definition reformulate enforce aforementioned original equation constraint g ensure indicator entry lagrangian lagrangian lx unbounded coefficient consider entry lagrangian dual simplified program make going eliminate min operator verify lagrangian give notice going value relaxed real relaxed real constraint verify wise lagrangian hard primal duality gap lagrangian dual dual lagrangian primal bound dual problem np duality gap primal moreover increase conservative primal unchanged duality may analyze effect conservative estimate dropping drop box see conservative reduce entry sparsity lagrangian entry problem importantly solve wise solving precisely lagrangian minimization solution desire conclude e duality problem bind substitute group minimization lagrangian wise conservative relaxation entire surrogate sparsity interesting choice sparsity norm consider norm non block minimization mx unclear finite minimize entry mixed datum face minimization top edge indicate percentile mark box potential explore corollary experiment experiment analyze bound explain tight duality reduce lower grow linearly function trend loose increase nonetheless ability solution pre condition norm minimization able obtain duality hence error image model image system minimize dominant optimization special problem conservative evaluate ar manually align subject contribute un training un image whose great present group primal mixed sparsity entry get consider assign block subject classification minor notice image classification state art correct correct un present minimization allow interpret several relaxation np hard primal lagrangian equivalent derive relaxation sparsity ability per contrast either condition perfect relaxation hard verify importantly condition pre correctness hope prove important instance verification specifically explore tight
regression mle ie adversary solve first square attack attack estimate column dimensional column c polynomially apply produces scale high recover pt pt theorem corollary conjecture attack wide range attack far want release attribute database relate code gender partly arise medical attack record contingency study like classifier risk minimization include transform linear format amenable algorithm surprising solve nonlinear formulation attack distributional consider statistic attribute relate boolean global sensitive record vast statistic formal privacy guarantee security attack seminal rigorous study tradeoff utility access information require roughly datum little recently design private algorithm typical evaluate seek notion privacy bound differential rule notion privacy reconstruct attack sensitive want sensitive relate gender partly arise medical database individual sensitive assume database concatenation construct attack private consistently estimate attribute change error randomize mechanism attribute adversary attack run instead simply reconstruction class natural reconstruction attack construct namely typical minimizing least decode lp attack answer per attack query answer simple adversary coefficient component seem database counting query database inner attribute hash example adversary ask sum database hash statistic unlikely publication paper attack attack appear analyze release marginal contingency marginal distribution subset per attribute private attack fraction arbitrarily remain noise generalize recently attack greatly expand applicability attack setting specifically reconstruction attack release give record boolean multilinear representation table error estimator broad estimator record estimator linear release attack statistic like obtain highly nonlinear like release sensitive attack distributional assumption statistic statistic attribute total boolean subset size submatrix entry evaluate statistic degenerate boolean add per entry attribute mild condition case bound base square lp decode handle fraction entry arbitrarily insight hope broadly release record fact linear know arbitrary value depend public affine variety obviously linear attack classifier see observation optimization every record function must lead equation form first technique add mention attack depend attack give draw underlie analyze geometry arise attack relate product refer matrix show product matrix entry row product matrix satisfy introduce square lp noisy degenerate boolean release attack either lp analysis analyze matrix arise associated clarity discuss attack estimator hamming distance letter vector denote row concatenation eigenvalue arrange value z ns subscript non early often dependence variety processing estimate output summarize sequel entry regression entire attack let orthogonal orthogonal define use sa affect norm entry map som privacy decode name stem lp attack slow attack considerably strong attack geometry euclidean say cauchy schwarz hold say decode bind operation bind describe entry lp upper euclidean nd time bound non degenerate boolean standard background represent boolean multilinear polynomial represent fx x degenerate multilinear non degenerate degenerate constitute degenerate depth evaluate database entry allow repeat let restrict index attribute degenerate attack section degenerate mechanism add entry attack achieve privacy every database attack privacy run transpose reduce database reconstruction boolean degenerate boolean kf boolean polynomial multilinear multilinear multilinear represent degree degree multilinear strictly less aid function jk correspond change position row affect singular row matrix tt j ki define row correspond follow contain vector approximation noisy adversary try I adversary set adversary know goal iterate logarithm part depend mechanism non run outline equation database analyze need follow constant satisfie exponentially probability show exponentially recover noise translate decode solve slightly equation euclidean euclidean make k decode attack matrix appendix notation f decomposition establish definition matrix define attack part boolean depend database add attack decode outline establish section multilinear polynomial high repeating least exponentially euclidean adversary recover attribute analysis bit translate bind reconstruction attack rely geometric property certain matrix conjunction build upon summarize establish product consist row product kt j product dimension order uniformly uniformly random call iterate induction main establishe x matrix independent dimension nc random wise appear independent random refer multilinear step behind simple multilinear multilinear represent representation multilinear multilinear ki ji I ji variable multilinear p repeat coordinate get ni ji ji j hold q dd multilinear constant depend x j triangle last ki thus equation note alone inequality singular independent therefore exist order theorem schwarz right theorem estimator k assign differentiable nm estimator estimator similar natural may
acquire uniform uninformative commonly problem informative open causality consider constraint belief first equivalence maximal constraint network incorporate path markov chain mcmc neither deal dependent incoherent briefly bn dag graph connect equation p iv call drop index equation ignore triple vertex b essential equivalent skeleton reverse connect set imply dependent take logarithm method k log ignore maximization become pair path connect important devise path mutually exclusive knowledge v single belief table path bn cause associate belief causal relation prior mass probability avoid pair belief incoherent bottom induce belief part dag configuration belief describe computation configuration determine p graph dag equation stem mention computing factor preference assign score first employ appropriate uninformative prior belief uninformative uniform prior property factor count intuitive reason everything else prior graph desirable drop score place satisfy dag solve good dags certain count dag grow exponentially option number specifically avoid expensive method configuration never may even variable estimate arbitrary order huge dag upon developed exploit configuration variable become hold effectively path become z yu yu yu u sufficiently negligible formally clear contain node keep add possibility create appear path pair yx yy u belief appearing appear j node configuration influence negligible path directly affect compute graph show variable nod dag random factorize estimate number dag recommend laplace kl sample provide determine approximate node size path r belief sample dags laplace correction divergence measure distance claim belief node relatively l dag cost time memory set relatively approximation infeasible number run small step complete dag constant measure divergence average small reason relatively small variable approximate compute value jj marginal equal belief belief path semantic detect discuss specification impose eqs satisfy axiom belief coherent incoherent equation typical uninformative certain configuration prior preference uninformative prefer information minimize formulate optimization problem efficiently iterative proportional fitting case incoherent equal marginal belief coherent belief ignore seek marginal close input incoherent belief amount measure call provide conduct reasonable comparison belief belief value bottom close belief show configuration score configuration z g yx cost solve dominate practice solve efficiently memory belief obvious would way upon seem however uninformative respect result also u I large path belief practice detect however belief give preference relation example correct suggest belief incoherent probability axiom implicitly reduce belief example px px belief thus imply definition c configuration simple configuration common case method space dag dag hill dag search extend dag configuration closure store closure compute visit fast complex graph trivial dags total overhead straight dynamically closure closure query reduce advantage extra belief improvement belief operator also change configuration application allow swap belief equivalence result describe proportional repeat randomly number dataset dataset scoring score dag uninformative prior equivalent sample ess result plot true network orient high prior perhaps notice informative skeleton tend association tends run correct uniform expect belief identify true effect exactly belief prior role generate belief component appear coherent incoherent path overlap contain result total consider large path randomly assign true remain probability uninformative way remain repeat component incoherent dag search ess start uninformative informative prior swap operator case informative structural hamming introduce markov dag use correctly orient varied path belief varied within incoherent dataset incoherent belief belief similar incoherent small uninformative informative reason score prior ignore usually consider maxima swap happen whole counter intuitive increase size suggest ess parameter reason behavior
j gaussian pdfs I f read respectively intractable message occur message also vector eq pass I pass nx ni code equivalent decode belief proportional gaussian reduce bp rule output gaussian identity specifically bp gaussian message coefficient update manner rest mf f split tractable exploit correlation representation mf message n f compute belief I nx n code part back mf channel mf dirac basically include message pass scheduling start correspond message correspond algorithm pass variable message factor node back message pass end decision belief wireless parameter carlo simulation derive coefficient encounter ep instability ep require multiplication pdfs division pdfs message channel compute versus combine mf bp receiver employ pilot receiver algorithm show pilot essentially bp mf soft channel notice snr value would account bp mf receiver active evenly spaced pilot pilot convolutional coherence channel decode message pass bp include particular derive four message apply employ consider computational stability conclude receiver bp mf em variant effective receiver project support project contract mobile national advanced foundation process wireless mobile project grant grant within national research university university technology iterative receiver introduce inference belief bp field mf include propagation embed bp mf modification consider algorithm probabilistic four pass receiver numerical wireless receiver mf candidate algorithm design advanced receiver requirement motivate successful application decode devoted receiver module design soft well inference prove useful tool detection belief decode probabilistic bp mf usually pass mf bp propagation mf pdfs refer ep belief pdfs exponential attempt unified region energy hybrid message pass combine bp decoding purpose either framework ep receiver wireless ensure belief obtain bp expression reduce message computation bp mf similar member specific exponential belief ep modify message mf belief constrain dirac delta mf maximize unconstrained graphical representation establish baseline derivation message receiver represent r alphabet symbol j alphabet finally aggregate symbol x send channel output vector h channel inter
reward draw rely tu result mab suggest order time quantify performance empirical need currently investigation answer regard chernoff mean sequence x il continuous initially propose lemma large deviation follow ex event notice finally apply combination beginning conclude occurrence anomaly tt deal reward provide since inequality decrease negative write sufficient prove analyze positive ht g term step subsection tu upper occurrence anomaly play equal anomaly appendix mild include end introduce armed bandit mab learning paradigm popular sense allocation reward channel name utility index highest associate sample collect arm confidence assumption sequential face exploration choice decision maker herein focus maker discrete value reward quantifie maker behave problem multi armed mab problem exploitation mab value arm utility past quantify interest index index ucb index provide optimistic ensure select decision maker build policy selecting bias deal remain challenge tackle matter inspire aforementioned multiplicative additive expression main contribution paper analyze multiplicative make available arm collect choose optimistic consider low non restrictive rest refer suggest policy outline independence reward suppose instant select arm instant shall refer seek regret expect playing refer play instant instant present contribution multiplicative index sample machine ex scaling factor index round last lead index machine index play adopt convention history index analysis focus determine sub consistency armed say good policy matter ensure expect expression remark suboptimal upper expect regret k draw prove
control feature channel kullback skew pp feature spatial rate gps sequence tc york rank variance analysis b inversion overlap base method mm toward understanding b g mm team statistical http www co l aid interpretation validation accumulation measure r mm e prototype f h gps mm comparison rank pt proof definition correlation high de email mat cl di universit associate recent nonparametric estimation form area probability nonparametric kernel us frequency spatially nonparametric estimation short nonparametric assessment occur achieve km al show region appear patch locate part secondary et et probability poor area et heterogeneity appear indicate potential great well aim work area area require choice methodology area identification region improvement adopt available science e social sciences standard book focus mainly distance multivariate variable multivariate associate density assume probability option adopt datum california link correspond identify retain parametric construction association cluster mode formulation concentrate kernel type conceptually normal function symmetric outcome parameter simple different cluster separate low refer describe briefly continuous unknown concerned denote position observe form move range cause move accordingly regard increment cluster early vary idea additional specification language r nonparametric adopt outcome bandwidth step toward bandwidth influential vary quite notion build geometry line sequence segment path include give step span admissible mode determine end early allocate sample mode subset core belong aggregated distribution represent density core allocation q option depend whether core represent refer diagnostic technique representation object spatially allocate cluster allocate cluster distance adequate method explicit diagnostic play allocate allocate refer index maximum partition diagnostic index time correct figure tree diagnostic lack surprising give proximity visible variant take exactly examine implication exploration log quantity meaningful outcome adopt quantity measure adopt produce association event examine way regular option evaluate observed choice computation grid region lead non rectangular grid comprise area cover event last refer figure scatter label drop explain early htb selection grid overall provide provide evidence consideration cluster association last focus refer form point feature grey cluster associate involve produce city locate maximum blue locate gray figure association increments htb diagnostic us rank testing classical whether assume population normal tie value make use produce measurement outcome provide group assume nan exist difference non level repeat correction significance regard htb min green gray red cluster represent red gray city low practically green estimation day involve result necessary number day moment great cluster versus index incorporation activity mainly mechanism process event magnitude magnitude comparable magnitude region occur km sharp apparent
domain disagreement source error network discovery publication view reference proposition universit france universit france da arise differ generate source label imply task seek vote da da seek take trade majority kullback leibler capacity majority structural source mm target task mp source ia tr low label domain infeasible hypothesis domain david follow marginal quantify equation capability guarantee usual vc trade traditionally vote hypothesis consist lead generalization essence order error pac expectation error major empirical sg kl domain pm restrict distribution specialized bayesian weight precisely spherical centered expected gibbs give prior kl theory learn weight express pac contribution pac da structural marginal h h suggest instead focus nevertheless derive follow propose trade error classifier p r joint minimize da da pac pac refer independence rewrite expectation g restrict classifier posterior loss center result pac linear need develop rely equation da consequently minimize side gradient evaluate
sufficient necessary omp recently show recovery necessary drawback stand require propose easier read dictionary inner product atom definition ols imply sufficient ol wang sense mention necessary omp ol fail satisfied ols observation vector necessarily omp ol select atom omp numerical simpler strong base coherence necessary ol atom follow refer stand linearly dimension index submatrix column index value x dimension omp vector must say omp ols understand procedure search ol ol select omp pick atom correlation residual sequel different formulation however confusion straightforward omp ol throughout use omp notation whose linear combination eq atom hereafter ensure success use follow recovery succeed weak sense point exist end support number iteration omp omp include ols present perform delay wrong decision failure exact derivation condition narrow algorithm specific condition whereas select iteration constitute necessary condition omp omp recover constitute bad omp ol drawback evaluation carry operation ensuring support easier mainly distinguish type restrict coherence condition exact step omp prove omp succeed tight dictionary term omp select wrong atom us virtue theorem result ol derive sufficient coherence dictionary condition ensure success iteration also wang show dictionary recover summarize succeed exist wrong first coherence success select atom term iteration exist term select iteration atom specifically resp sufficient omp resp ols last iteration differ omp ol omp derive upper restricted isometry project ol connection coherence section prove proof omp sufficient state depend state derive therefore prove theorem characterize appear implementation generalize isometry rip project project rip rip rip asymmetric restrict isometry asymmetric isometry note since many property isometry remain proposition depend coherence dictionary rip result report ready right side proposition calculate indeed full imply submatrix full finally consequence coherence project coherence project apply useful function coherence report state ol combination result meet recently linear combination representation exist dictionary context select bad condition atom first exhibit scenario atom representation result bad condition dictionary dictionary elsewhere play gram eq unitary check distinct eigenvector sort corner zero eigenvalue span proceeding define concept subset subset say subset ol dictionary reach omp exist two representation disjoint dictionary exhibit result input apply representation project respective select q success complete read coherence wang corresponding bad cardinality dedicate condition since evaluate pair mutual coherence cumulative intermediate mutual coherence inner dictionary atom involve type evolve investigate contribution interesting proof proposition technical denote eigenvalue eigenvalue express lemma proposition hold first norm see fourth rip q inequality inequality relationship recursive obviously decomposition moreover full family turn unit norm jj q appendix proof denote change confusion prove technical recall read take statement dictionary reach satisfied ols virtue latter verify dictionary result without atom hereafter remain matrix arbitrary recursive construction omp successively select iteration selection yield
equivalently batch conceptually one location adjustment distance discrimination batch hyperplane approach assume variation observe variation batch factor addition subsequently poor factor correlated batch effect contain remove large process center normalizing deviation array know remove effect dataset large multivariate biological contribution remove variation factor good factor context difficulty arise regression term turn ordinary least square square form deal difficult dedicate random term way adapt dedicated major difficulty covariance amount expression gene way replicate array variation involve control study different know cause help variation third subject allow explicit variation model address unobserved random first factor equivalently know discusse improve section performance method gender use evaluate finish discussion na ab b f removal another fix non effect address factor influence sake clarity unsupervise control affect interest negative gene gene intuitive two w reach singular svd singular value back I maximize plug use step expression simple call simply project datum orthogonal column regression maximize plug algorithm correct direction expect orthogonal factor estimator sensitive variation supervise remove include naive present lead smooth correction estimate unobserved model introduce random normal j know estimator version appropriate correction scaling amount signal direction posterior formulation serve highlight relationship solve close form unsupervised product make convex unless third discrete optimization problem finally discuss difficulty typical cluster center constrain dictionary norm constrain penalize could equation solve dedicate generalize purpose use dedicated discuss fix idea denote matrix spherical convex iterate minimize classical simply solve general fix row couple close iteratively solve observe empirically fix objective sensitive instead us expense correction w hand side minimization hand conditional side appendix identity right side need unsupervised clustering suggest observe unobserved still close involve dedicate numerically general various estimator amount poorly thing use propose available maximize unsupervise generally possible maximize step optimization ridge step unsupervise maximum yield naive hand side joint fixing naive however similar naive setting maximize procedure naive ridge benefit naive obvious replace ordinary dot regular variation interest project sample remove lot center naive projecting remove come lot gene amount result spherical identify naive account information variation along right panel take ridge remove signal uv control remove provide remove leave association summarize fix naive joint iterate procedure alternative arbitrarily equivalent row covariance supervise maximum incorporate encode shrink positively towards enforce separation additional posteriori detailed deal noise gene would lead penalty shrink regular loss assume encode need covariance usual square regression option feasible least ordinary square estimate covariance matrix mention try ml discuss procedure case naive model regression consider supervise show lead supervise estimate correlate canonical column correlation association gene lose direction variation estimation benefit direction estimator first obtaining replicate two say platform profile replicate influence variation replicate replicate difference replicate row gene control fashion new control assume noise maximize likelihood singular svd large value plug regression require constitute unsupervise discuss first gene first variation control difference replicate gene happen influence solely gene association whereas difference replicate affected replicate variation thing control influence simplify describe c develop w asymptotic behavior wishart gene regardless control dot depend gene large enough estimate ignore implement rw da easy db replicate factor remove estimate replicate perfectly replicate two c c leave unchanged combine replicate term amount covariance pair replicate combine lead factor already observe common ordinary compute covariance residual iterate correct amount could factor datum use mean center procedure yield lead know expression affect lead whereas effect may quadratic allow non correction fit affect evaluate correction microarray expression unsupervise typically explain datum away gender centering remove gender remove choice cluster kept replicate correction solid non iterative similar replicate perfect clustering gender correction use lead variation observe replicate signal dotted figure iterative iterative panel replicate method replicate separate shrink clustering send lead high pair gene green line correction control gene result reasonably separation space leave panel naive replicate advantage come naive replicate correction good cause variation even probably well gender gene influence gender particular gene little interest dot green base correction plus error try improvement non iterative two lead well correction rely improvement replicate gray estimator also repeat experiment center replicate correction filter gene since replicate replicate something vary identify array effect performance array measure berkeley laboratory array filter base variability within merge identify restricted neural study three across identify core study allow leave one aside recover correct mean setting first build keep array array replicate potentially replicate select present correction platform completely platform replicate correction good correction perform design full platform orthogonal expect correction easy platform cluster platform centering replicate iteration table show recall deferred appendix datum maximal presence figure platform full reason center platform well remove platform remove replicate available center replicate platform improve regular center presence disadvantage amount platform whereas possibly benchmark becomes center give extremely form replicate naive replicate full platform principal fourth principal remove two platform effect remove fourth platform remove two component lead perfect contain platform allow used presence full reasonably gender one remove platform lead naive design cause value first represent strong batch effect iterate ridge far improves replicate base correction gender quality correction give respectively retain consistently absence correct show come correction remove platform effect gender correction explain design naive replicate performance remove enough platform correction remove array replicate performance correction lead explain behavior estimate remove correction check variance qualitative gene expression assess drug three combination test total sample gene array measure array array array array experiment retain drug suppose effective early time lead array restrict gene replicate platform replicate gene convert design obvious purpose array influence drug numerous variation array array drug hard gender one drug may gender clustering behave h center naive replicate correction display try partition cluster principal center platform array platform like gender cluster drug see figure lead improvement organization drug still clean replicate well performance run different lead clustering objective error clustering correction well organization fail correct replicate performance platform small remove platform drug iterative poor naive replicate replicate procedure work change far gene little allow introduce three benchmark iterative gene performance control gene gene cm cm gender control replicate combined gene dataset overall affect control gender factor rely heavily correction one replicate replicate introduce combine less solely gene even gender replicate variation replicate robust affect gene gender dataset good gene figure interestingly gender control ii control gene gender eigen eigenvector gender eigen gene small control less gender good gene available help remove variation replicate control individual replicate amplitude correction propose unobserved factor interest dataset affect introduce idea negative control affect replicate affected replicate gene control cluster interest affect replicate remove replicate variation sound platform use know explain factor factor addition replicate suggest available remove variation scenario variation remove interest actually rank technique give benchmark difficult technique variation never benefit appropriate surrogate hyperparameter could stand cancer dt author chen discussion r nx n plus additional orthogonal orthogonal regression denote particular equivalent regular ol identifiable normal map equation plug equality carry lead california berkeley usa bioinformatics large contaminate batch variation account analyze spurious association variation account correlate unobserve former carefully gene replicate expression generally remove without compare state correction last microarray gene help like study several involve several
activate frequency task individual individual pattern individual eeg exponential generative eq gamma design basis design test test kind model simple speed guarantee variational gap nmf typical inverse gaussian gives likelihood expand approximated jensen second term low give optimize inference variational parameter eeg group performance hyper common comprise eeg record duration task letter preprocesse raw eeg psd eeg channel psd frequency bin constitute subject desirable intra variability separate kind side propose basis basis share subject whereas common basis pattern individual additionally competition indicate able concentrate well subject expect concentrated region subject concentrate individual basis column achieve comparable often time perform basis subject show capture well limitation show individual present generative analyzing find common class subject subject variability seem capture pattern seem less size limitation individual variability well pattern advanced institute south model eeg eeg variational art competition eeg multivariate electrical potential among neuron high temporal brain brain interface application mobile eeg svm factorization nmf nmf determine useful spectral eeg testing pilot pattern reflect intra variability generative give
remain probabilistic difficulty straightforward persistent divergence minima fail present greedy wise overcome layer rescale seem importantly necessity difficult organization influence organization like would influence layer layer wise simultaneously influence layer deep boltzmann fall vector hide add difference norm principle previously form rbm proportion application somewhat explicitly activation encourage hide effectiveness boltzmann stochastic unit group unit dependency unit interaction specify encode visible visible visible visible connection connection parametrization boltzmann function ensure general rest evaluate q restrict interaction rbm restriction rbm possess distribution unit rbm readily posterior immediately rbm extend imply however still resort much comparison boltzmann typically maximize likelihood accomplish gradient ascent rbm full persistent divergence mcmc update unit filter term vast layer filter successfully detector characteristic mnist superior lower prevent poor regularizer encourage unit layer empirically layer boltzmann contribution demonstrate simultaneously layer well layer future would many simultaneously
limit correction correction monte carlo number replicate non zero interval correct nominal practice however corollary precision function bias univariate parameter multivariate adjust impact nuisance control consistency mean size theorem however asymptotically quality estimate analysis moderate gold great scope provide adjust credible potential credible implement amount exchange alternative would assume correction make minimal grateful use matlab abc financial research provide university south write l x first g l ii l x equality first order taylor represent simultaneous confidence fan method nominal probability distributional frequentist estimate interval correction wide double improve coverage keyword interval correction approximate mm intend credible many problem credible asymptotically sample many problem composite adjusting generate simulated parameter statistic derive draw distribution pseudo credible determined frequentist adjust interval similarity double involve demand demand chain obtain interval estimate confidence base bootstrap bootstrap expansion quantile estimate bootstrap function iterate produce correct interval bootstrap bootstrap coverage double adjust bootstrap compute autoregressive replication interval bootstrap fail interval procedure credible intuitive coverage frequentist reflect posterior distribution calibrate software technique bayesian describe give theoretical relate involve know narrow conclude comment tail interval parameter seek write interval compute quantile abuse notation bayesian case interval bound estimate produce approximate coverage theoretical result come simulate simulate replicate value good estimate population amount large consequently example occur whose see frequentist setting likelihood bayesian posterior case less accurate limit suppose denote th c correct coverage interval write th observe estimate coverage replicate simulated limit manner value l simulate g provide correction nominal asymptotically step obtain confidence interval step upper limit x random example example assume suppose location normal estimator illustration interval coverage amount usual frequentist confidence x z interval x c display correction confidence interval replicate illustrate correct limit correction adjustment horizontal qualitatively limit irrespective correction choose result arise change location adjustment correct location correct plot correct limit leave plot plot interval cp retain property however monte adjustment systematic interval adjustment correct coverage retain replicate second row adjust interval correct coverage confidence cb double bootstrap bootstrap double db correction bootstrap cb correction provide broadly comparable calculation package compute nominal e correction different specification level auxiliary specification comparison bootstrap correction possess structure record broadly double whereas procedure code broadly comparable interval complex composite technique composite likelihood bias narrow approximate abc modelling mechanism behind model abc analyse parameter quantile parameter interval nominal coverage pairwise specifically composite specify bivariate spatial location provide normally specifically score vector ordinary set max spatial standard composite likelihood obtain analytic estimate readily employ narrow compare composite matrix spatial model stable degree covariance spatial model extra marginal parameter cccc coverage nominal confidence interval replicate analysis standard correspond correction procedure correction adjustment clearly coverage base hessian take modify achieve setting algebraic representation available moderate computational ccc hc follow procedure abc likelihood estimate density control approximation require credible analysis daily exchange return type model quantile function control location use particle generation correction draw dependent quantile table credible kernel
model mixture gaussian kernel admit store save store representation second nonlinear distribution clarity key reveal important loss fx f ix instead iy fx risk fx I compute universal simplification preserve distribution nevertheless simplify fx fx variance lipschitz lipschitz second fx fy variable around behave continuous fx risk turn linear train svm x gx gx important differ need furthermore gx consequently virtue kernel place distribution place rbf large bandwidth generative define via example idea surrogate kernel also k kernel previously treat probability restrict robust instance svm incorporate maximization margin cone generalize svm reflect classify relate specifie distribution center miss explicitly involved nonlinear comparable svm expect extensive quadratic qp learning baseline base apply firstly fundamental binary problem gaussian virtual distribution rbf boundary region density region heavy treat implicitly incorporate trained choose nevertheless secondly different combination embed two class wishart freedom consist classify digit digit digit digit initial virtual aforementioned virtual third original example exclude virtual virtual reduce pca dataset svm rbf kernel depict equivalence svm virtual term gb illustrate benefit scene bag represent patch vocabulary codebook encode occurrence detect codebook image generate base framework result four room office therefore focus separate image training codebook training firstly randomly detect local patch detection dim collection cluster local patch define center represent histogram sift svm nonlinear use ic sift vector rbf adopt ic parameter perform choose adopt pairwise voting benefit understanding employ method elegant high natural image propose kernel probability rkh simple nonlinear choose suitable place illustrate benefit pool compare pool example world km thank discussion virtue proposition bound j result loss equality consequently minimizer kx distribution lipschitz function continuous follow accord fm x similarly yield complete bound gx equivalent gx svm minimize gx minimize admit complete proof lemma system rely learn distribution represent embedding hilbert straightforward fashion machine analyse insight svms base insight svm place experimental real world demonstrate effectiveness learn collection problem arguably distribution reason preferable uncertain application gene microarray source reduce unfortunately feasibility microarray cope uncertainty amount array across may give abundance throughput amount challenge space besides potentially incorporate collective point learn distribution kernel generalize closely kernel leibler kl semi object semi recently introduce jensen shannon application specifically domain make first prove theorem regularization kernel probability measure input location particularly reduce flexible svm give iy setting preserve permit hilbert rkhs rkhs endow reproduce preserve see embed associate kx kx second solution combination minimize risk admit indicate contribute usual regularization recover case mx regularization limit infinitely minimize fm x something optimize g cc endowed topology borel algebra
optimization project comparable q probability notation simple full multiplication let margin problem use eq eqn achieve svm clear eqn thus optimality eqn right rewrite w opt opt follow get geometric margin accuracy eqn lemma proof project radius construction minimum ball space denote radius point b b b b b b b b radius exist b project minimal radius radius clearly similar theorem prove construction use ready bind finally regression solve regression rs matlab version run varied run refer fast medium partition ten fold cross estimate repeat ten ten fold dataset cross randomness construction random ten ten matrix classification experiment report random projection need random projection svms run square square average ten choice matrix evaluation namely collection document dataset genome diversity datum correspond classification report use solver synthetic dimension rs rs family use four matrix validation experiment ten family synthetic trial normalize w ij generate family contain family dataset experiment three family grow projection small svms figure running projection svms nearly projection obvious combine svms dimensionality instance tp datum family consist document datum comprehensive maintain matrix binary collect document word column diverse set project versus full rs four table second quantity ten ten cross validation experiment ten random projection set reason large less accuracy present text task set contain feature generate contain point task ten multi vector default setting binary class project datum observe close training training ratio project except zero predict take small dataset datum tp rs data c rs rs rs data pca summarize center compute known pca error principal performance rank pca retain experimental matlab pca keep matrix principal matrix refer run experiment entire denote pca set experiment sample dimensional though projection svms sensitive projection svm greatly like therefore random rank standard quite varied svms randomize hadamard see projection svms svms full compute pca projection compute bottleneck svms rank svms second former pca principal second principal desire sensitive take long matrix dimension pca c project pca dataset average matrix pca second depend indicate deviation ten ten experimental world namely gene convert label classification general full consist image center leave etc regression set time decrease increase c project indicate correlation mean standard ten fold ten four consist cancer cell line ten correlation influence square dimension rs rs show depend explanation projection useful handle predict achieve approximation radius ball relative excellent generalization value svms feature despite full rank improve random method well dense rs random projection method work medium expect excellent indicate experiment benefit projection combine time zero oppose svm compare popular dimensionality see random fast pca extend approximately low open investigation vector point construct soft develop prove margin preserve relative ensuring comparable margin extensive design programming short international conference intelligence short detail svm fast support advanced project air force laboratory fa nsf nsf address computer science institute edu cs mathematical department com respective form hyperplane maximize distance hyperplane separate soft dual lagrangian formulation unknown lagrange part measure classifier ball consist hyperplane width error monotonic soft insensitive dual insensitive loss formulate multiplier preserve subspace preserve I margin subspace preserve hyperplane comparable vc svm regression b n contain contain singular spectral n lagrange multiplier determine solve eqn imply separate hyperplane e appear vector hyperplane generalization svm worth lagrange multiplier determine solve eqn entry separate hyperplane give point support reduction transformation reduce regression svm present construction fast hadamard take address propose ground breaking construction datum depend classification regression memory serve theoretical relative generalization preserve relative randomized hadamard running apply randomize hadamard transform construct need transform run need equal random randomized hadamard three transform randomize hadamard eqn margin dimensional radius row state construction preserve relate technique insensitive respectively practice show problem random importantly comparable manifold regardless margin result dramatically improve need projection run separable show margin separable running projection margin preserve show error free preserve binary meet angle angle well angle inner angle preserve two result additive whereas relative big margin small moreover analyze separable analyze non weight relate analyze claim regularize weight approximately preserve approximation hash kernel datum random projection increase datum sparsity show hash
invariant markov addition first part find ensure mix marginal continuous f omit polynomially ergodic hence basic begin observation hx hx borel similarly proof theorem suppose note corollary eq let transition eq necessary rs principle hasting denote right side jump proposal condition establish accept jump q unnormalized density together suggest choice piece sense choice middle course depend walk normally qx trial error deviation remark statistics california edu school university statistics college university edu quantile establish asymptotic construction asymptotically valid estimator recommendation practitioner quantile absolutely notice bayesian quantile typically monte simulation report notion approximate produce estimate method practitioner rigorously increase reliability inference mcmc entail simulate gx gx valuable carlo finding assess mixing much weak constant serial difficult interval student quantile interval assess reliability explicitly implement batch subsampling simulation rs require previously begin brief require chain section bm sbm rs illustrate conclude recommendation essential preliminary material borel algebra ergodic irreducible recurrent mean geometrically equivalent characterization chain ergodic measure establish gx strong number omit assess approximate sampling monte scenario markov refinement description chain proof markov polynomially order consider monte sense expectation find satisfie q bound work area practically ergodic case easy q marginally student degree variable sampler update metropolis marginal invariant markov chain estimate marginal ensure satisfy every v example replication absolute median distribution carlo appendix polynomially need method substitute consider estimate study chain process estimator estimator familiar corollary continuous consistently mean bm n iy n q putting standard estimating quantile error substantial amount use bootstrap b experience method computationally focus block subsample order subsample sbm note avoid give normal simulate augment markov apply derive alternative estimator recall give simulate otherwise thing apparent represent use simulate residual height simulate pt receive considerable gibbs employ establish practically frequently easy draw chain v hx split straightforward markov geometrically polynomially ergodic exist sequel develop base gx set exist consider polynomially ergodic require follow polynomially exactly need recognize variance moreover consistently investigate finite sbm rs example conduct common replication order interest heavy tailed say normalize slowly vary quantile freedom walk jump proposal draw proposal kernel finite tune scale order autocorrelation result result acceptance varied tail bottom sd length interval rate first agreement nominal median quantile average sbm great case couple appear bm variability agreement slightly well rs show half empirical coverage least huge variability standard tailed bm interval slightly slightly wide demonstrating length sd jump proposal ccc metropolis bm sbm rs bm sbm bm sbm rs ccc distribution bm sbm method bm sbm rs distribution bm sbm report concerned occurrence indicator disease normal bayesian credible distribution geometrically ergodic attention coverage estimate da quantile truth rs example setting replication simulation table bm sbm probability nominal investigate nominal level rs nominal mean bm notice across rs average bm coverage probability ccc ccc bm mm rs sbm mm bm rs sbm know analyze consist average player first player represent reasonable hierarchical specifically flat gamma proper gibbs simulating markov implement interested transform study correspond bm rs draw
ls likelihood nlp normal like example measure label negative example via confusion give table tp correctly false fp proportion incorrectly classify positive rate precision let effort false positive negative worst thus address positive negative actual actual preferable us detail measure optimize case maximize recall choose depend much want combine recall criterion harmonic prevent class q experiment write true maximize positive tend positive f positive f summarize classifier play evaluation binary optimize hyperparameter validation situation separate loo situation next estimate otherwise one value estimate auc applicable probabilistic classifier sense auc however compute estimate smooth nonlinear want gp define smoothed indicator sigmoid make criterion hyperparameter loo combine loo predict positive rewrite q smoothed fashion quantity loo estimate smoothed loo show useful use version classification classifier becoming optimize hyperparameter early gp trading compare maximize minimize overall loo define criteria loo loo cv various criteria cumulative write refer help value optimize approximation compute loo ep use full gradient calculation hyperparameter expectation type optimization gradient implicit site iterative ep cv hyperparameter ep loo cv gradient sequence mean covariance derivative site work use call ep cv optimize f optimization proceed dataset smoothed monotonically case exhibit behavior smooth difference smoothed base take depend problem increase true case car dataset clearly trend case right th always optimization cv depends ard give problem equation method storage complexity recall discuss loo cv relation derive field binary gps originally physics loo loo cv ep loo cv count indicator rough loo error ep ard hyperparameter work classifier feature associate sequential update outcome though optimize evidence prevent overfitte loo choose count work gp classifier optimize include ard directly various loo include gradient loo cv sake treat interpretation imply approach give pass sigmoid loo loo gp treat hyperparameter ill optimize l within ep optimize ep ep optimize ease nlp table l predictive maximization ep optimize ease fm ep optimize method diabetes ep diabetes heart method ep diabetes c dataset car diabetes dataset fm nlp car diabetes conduct summary experiment ep routine matlab code http www code hyperparameter hyperparameter benchmark dataset web project benchmark summarize first available web code conduct correspond post hoc multiple dataset mean point partition dataset performance get rank significantly nan hypothesis rank nlp ep ls statistic reject nan nan reject post hoc pairwise reveal ep ls post hoc significant ep set rank method l method improve test significance detect significant well method significance thus l table nlp performance l inferior performance kind see nlp score breast diabete observe nlp close compare dataset dataset heart difficulty nlp increase score wrong classify example ep score l ml ep estimate relatively poor result poor nlp hyperparameter variance value except difficult dataset apart pattern seem car multi classification consider example treat dataset web edu machine partition class ep cv second nlp smoothed criterion step describe early hyperparameter observe smoothed breast cancer diabetes remain width analysis give method believe arise example carry significance use rank significance nan hoc reveal nlp method smooth f nlp rank observe conduct sign statistically significance demonstrate usefulness smoothed consider gaussian loo cv optimization loo criterion smooth hyperparameter specifically apart nlp usefulness measure useful handle distribution propagation ep expression useful ep cv real benchmark nlp nlp demonstrate smoothed performance ep excellent gp classifier loo optimization nlp predictive py nlp give n py loo predictive loo ep analytical eqn derivative predictive distribution ease z nz give approximation ep approximation ep k avoid inversion note entrie k classifier loo cross criteria practical loo predictive nlp weight useful unlike loo approximate loo distribution expectation propagation conduct several benchmark criterion optimize nlp approach measure f excellent choice criterion cross predictive error precision integrate require analytically various expression two important function integrate within approximation integrate hybrid hyperparameter gibb integrate hmc integrate hyperparameter variable integrate hyperparameter choose focus address model loo cv predictive nlp classifier use integrate hyperparameter maximize laplace ep method integrate optimize information em ep approach loo cv address problem loo rough hyperparameter ep cavity distribution loo select automatic determination ard hyperparameter loo least nlp classifier measure likelihood measure measure like useful image understand example work hyperparameter literature regularize weighted loo curve auc criterion handle regression optimize f general loo experimental compose target pair true label gps model prior covariance k across dimension automatic ard covariance ard latent binary value label w py hyperparameter characterize assumption latent
approach select analyse outcome break avoid bayes amount good actual structural thought principled method classifier bayes underlie set almost certainly extent error true term currently consider domain learn construct theory say provide enough create appear accurate classifier valuable still whether drop heuristic relate decompose inherent variability arise bias considerable insight performance regression adapt bias adopt considerable definition bias negative variance bias variance quantity guess set guess around cost cardinality term bayes specific number reflect classifier form decision boundary elliptical never classifier use study make drop remain sum inherent achievable note many assume hereafter adopt bias accurate value statistical model unbounded distribution integer potential iid underlie practice evaluate training success predict component raise methodology give example partially rely distinguish go misclassifie misclassifie type decomposition use preliminary examine sample argue fundamentally partly instability derive result show stability design degree essence fold cv ensure classified time sample please evaluation purpose ten characteristic naive nn bag adaboost ensemble adaboost bag decision treat entity decomposition classifier cart cart knn naive bayes implement library use require datum artificial five visual inspection problem create image good bad act generator datum inspection cd label operator label inspection describe augment descriptor add total different range cardinality derive artificial cd derive image cd label uci repository observation regression estimate error make unseen pool independent dependent procedure measure simple independent quality prediction close give way process single iterate clarity variance combine show predictive combine change extremely model various field learn sufficient provide achievable height height height scatter sample marker indicate axis correspondence plot constitute combination vertical point predict build word show observe final majority former apparent considerable plot total lie final marker size fall fairly evenly thus bias seem end scale line would consideration take taylor expansion value dataset size single visual inspection fall line bias error account proportion perhaps expansion even variance size even follow intuitively absence training region probability major point simple quantity take far try model confirm original bb model increase determination bias relate either use sum final variance well show correspondingly examine build combination able unseen dataset evaluate cart logistic knn library default final make without build classifier seven dataset see estimate predict near actual poor bias inaccurate original pointing decompose show coefficient determination initial unseen classifier segment f perform pair relative I bias size marker logarithmic different scale somewhat visually trend towards cause trend error decomposition apparent trend show height width different algorithm concern represent mean variance algorithm fairly survey variance ambiguity page negative handle knowledge ensemble take classifier however approach treat entire classifier definition next predictor limit behaviour potentially universe limited future finite predict value complete treat ensemble entity boost create ensemble train set train create training predict fy subset train incorrectly fy possibly item training look depend follow well fold counterpart state refine take average turn attention class far constrain output decision partitioning contribute bias correct x fx contribute give ensemble bias provide dependent furthermore always non quantity constitute strict rate treat entity section build predict attain ensemble classifier require run fold time computationally heterogeneous ensemble pool build model relate bias different training treat regression evidence training decay bayes error decomposition experiment bind derive minimum achievable feed fit predict use feature space cd artificial cart decision tree bayes decision combine discount cross prediction item datum clarity hereafter increase use fit form prediction well coefficient low relate value sample situation size predict dominate observe use predict lower also train perform e across form achievable also different ensemble estimate analogous model deviation law change discount constant constant modelling ensemble ensemble make sample build illustrate cd artificial show build entire accurately ensemble would first show secondary standard deviation curve artificial observe fall standard error overlap regression nevertheless standard predict entire depict investigate early scenario ability approach although component progress way training increase qualitatively similar across different combination create statistical technique examine relationship perhaps surprisingly provide accurately behaviour different component confirm prediction finding validate range dataset model also examine reliably new range slightly datum component suggest complicated depend final bias covariance available function ensemble straightforward methodology ensemble achievable ensemble law error direction combine previous theoretical finding suggest worth law period predict useful major work definition bias investigate whether definition accuracy european contract author view machine system topic establish predict set cost lead accuracy algorithm training composition observe decompose bias although differently behaviour hypothesis take create limited predict full result range analyse various prediction case predict unseen widely understand theoretically empirically e g unseen example validation average fitness run evaluate repeat user change increase effectively achievable thus accuracy acceptable come insight successful place management sufficiently order learn pose develop user management example store come however failure stem descriptor whole must user costly classify field diagnostic inspection medical frequently sufficient store necessary process hoc collecting database significant quickly accurately predict go successful importantly viewpoint valuable save paper future give still learn final achievable focus descriptor limited later predict relationship train theoretic loose motivation investigate approach reliably observe hypothesis use source
simultaneous spread recent tail exponential separately conventional reduce blind paper consider alternative appropriate impose image empirical bayes blind deconvolution exploit blind tune bayesian estimation image decomposition perturb nominal specifie log bayes relate wise gaussian basis tune strategy reconstruction expensive propose estimate precisely nominal perturbation nominal relative truncate reduction improvement full markov simulation comparison quantify advantage real cover bayesian propose result unknown positive recovered reconstruct measurement convolution mean observation kernel model imaging device vertical configuration perturbation nominal experimental field mass probe etc approximation derive error true direct nonlinear direct expansion deviation deviation know basis coefficient influence denote notation rewrite convolution sparsity measurement noise mass single w conditional prior interesting enforce natural furthermore ensure positivity uniformly define e improper informative jeffreys noise equivalent informative jeffreys variable associate hierarchical hyperparameter prior definition hyperparameter express conjugacy integrate hyperparameter posterior yield section present generate describe metropolis distribute sampling simultaneously result consistently output subsection noise pixel parameter pdf distribute w conditional therefore sample achieve posterior stand component stand truncate generation require generation truncate appendix enforce condition image summarize order detailed procedure appendix algorithm transformation costly strategy propose gamma result deconvolution algorithm blind deconvolution nominal mathematical mcmc sec mmse ensemble average burn period sample maximize h amplitude external slice field perform describe image propose assume physical physical tuned table radius mis specify fig image basis span subspace set carefully tune produce ellipsoid specify center nominal pca residual allow axis basis determine empirically fig eigenfunction perturbation basis depict estimate coefficient show agreement correspond eigenfunction true blind show blind nominal hereafter accord appear outperform preserve h quantitative trial six fig histogram error criterion indicate alternate minimization reconstruction notably unknown blind method reveal investigate criterion stage blind comparison fig direct comparison several reconstruction initialize fair comparison modification variation tv suggest estimate poor deconvolution sharp iterative approach fine adapt method term present use basis image prior apply suggest show fig fig blind deconvolution motion incorporate penalization poorly trivial poor due image blind camera motion shrinkage effectively bayes several second method fig modification extension pixel add voxel scalability benefit impractical slow additive paragraph basis obtain select use domain axis grid fine along high bayes interpolation semi blind presence experimental synthetic pass noise semi blind reconstruction initial blind experimental neither ground truth image fig nominal estimate estimate validate difference gibbs give residual nearly experimental blind show algorithm experimental intensity variance estimate region p measure indicator voxel provide confidence significance reconstruction bayesian reconstruction identifiability common deconvolution ambiguity prove reasonable solution trivial restriction estimate reasonably solution beyond scope image reconstruction extend blind transform assign exponential laplacian distribution use pixel hyperparameter image replace reflect nominal metropolis adaptive evaluate uncertainty demonstrate blind blind several criterion method include enforce eigenfunction use algorithm selective future acknowledge dr provide thank comment paper generate exploit describe require without evaluate decomposition procedure coordinate coordinate vector I iw accord I moreover recursively evaluate follow overall expect pixel quantile previously exact require quantile tw computationally especially restrict burn period save selective sampling iteration compare mcmc blind fix blind need careful selection step
correct intervention supervision establish unsupervised shrink transformation variance feasible instability question arise overcome supervision mathematically possible question impose force pde form function convergence supervision assumption pde supervise cluster converge general pde decompose part pde satisfy pde green green satisfy pde replace normalize proper density fact original function pde supervision correct assertion theorem adaptive current historical density leave shrink algorithm design deal law stable due particle speed supervision interesting future aim fill critical gap mining application lack stability employ physics partial absence supervision cluster correct unless transform normal lyapunov anti nature cluster without proper supervision preferred ensure incorporate formulation mean cluster pt corollary convergent anti shift wang wu york analytic equation propose mathematical law successive underlie stability algorithm show supervised shift possibility correct transform multivariate convergent adopt supervision mechanism dynamic set accord cluster set study normally partition small homogeneous excellent review cluster emphasis wu discussion many cluster optimize strength partition merge et determination cluster challenge problem cluster shape boundary difficult issue specify ease normal introduce year algorithm dynamic algorithm treat include wang zero gradually towards cluster law category cluster arguably shift receive flexible generalize operation idea find transform toward variation wang design bias center movement resemble tracking center employ robust recognition excellent image analysis segmentation practitioner apply area underlie understood initial transformation chen point shift point statement also dynamic intuitively behavior major difficulty function transformation setting dynamical dynamic evolution impose shift framework useful criterion reliability general identifie suggest improvement consideration successive transformation general law characterize evolution form dynamic pde analytic differential broad converge normal density anti diffusion lyapunov increase anti uniquely instability phenomenon convergence happen nature shift place supervision dynamic potentially supervision rest follow presents establish brief view sample clustering mean consequently view particle field law require cluster pattern self organization reaction order mean algorithm framework form dimensional density function element surface element denote side minus gauss take inside component derivation discussion law associate differential form supervise influence certain impose supervision lead law framework many dynamic process keep partition process give unsupervised approach functional gradient derivative variant method often govern law move less adjusted interest mathematically show belong category category formulation wang et al center locate wang show force trajectory seek mode movement function long area combine eqn correspond one anti diffusion move region anti unique apply reaction transformation detailed derivation fourier transformation f dx fx fx dx f also follow convergence I uniquely evolution deterministic status unique path lead lead naturally normal respect time converge speed location eq conclusion occur spatial variation time contraction prove result cluster center generalized anti dynamic shrink boundary fourier I fourier side eq transformation integration simplify way analytic function
severe sampling rate ml fit tb plot posterior credible dash square dotted posterior credible indicate small observation level naive lead severe set focus energy interval uniformly define short point restrictive exponential solve energy set constrain energy frequency ern function derive interval appropriate derive uniform length reasonable prior bound obvious fitting naive mis effectively way noise value expression variance level processing rna sequencing apply g variance multiple replicate approximate low total fitting cause bad length synthetic ex tb ex ex tb ex avoid reasonably sec create data sampling equally space fit unconstrained fitting result scale fraction drop gp coincide square point space calculate square true mse large likelihood indicator mse likelihood indicate opposite frequency instance small fit large four present support together model drastically fig clearly length make model sensible bound c htb bin time series use version scenario unconstraine fix processing differential model tf set show moderate fitting vary drive especially fit propose example tb slightly model tf gene discover target without model fit list demonstrating bound bound improve gp instance rigorous bind sensible length exponential mat ern spectrum reconstruct justify collect place prior help fit synthetic usual fit functional freedom plausible essentially model highly constrain ode usually one incorporate delay severe cause case length method base ingredient model short biology carefully justify prior future beyond ode suitable suitably lead fitting failure mode institute technology technique multiple independently gene severe model depend avoid gp focus energy frequency fitting avoid informative prior allow reliably automatically instance illustrate real process gps widely apply nature differently size easily gps work gps gaussian identically widely gps learning square
sufficient data region manifold question remain simple exist system number point estimator answer question develop algorithm lemma unless rearrange j rx x rx j dx multiplying vanish multiply eq whenever apply lemma j j would schmidt many comment manuscript version mm corollary message mml link compression close kolmogorov mml np common approximation strict minimum mml calculate apply rule boundary rule heuristic dimensional calculate dimensional cut estimator prove result certain defer section example address issue idea briefly estimator exponential family open subset pdf bayesian pdf give elsewhere make technical moment n code x code length constant lattice variable estimator cut estimator probability fx parametrization standard result family differentiable page eq say mass say exist discuss informally construction family determine general transform simply ni ni ia ai vanish cut point therefore solve give corresponding derivative denominator step let lemma since never simple numerically behave method j j equation minima estimator point increase global minimum combine though say surprising satisfied estimator make agreement exist boundary seem symmetric g cut odd minus positive place reverse code length number cut code cut point cut estimator due fact difference hence impact minimum large idea minima cut point continuity guarantee corollary exponential exponential though exponential distribution parameterize pdf pareto moment additionally prior case gamma local minima outside range extremely small briefly discuss simple cause cx rx appropriately term many section behave right side numerically
dedicate error message send pilot user vector bit datum symbol ki k complex alphabet pilot draw alphabet aggregate channel ki interference lk h receiver li sample interference ratio receiver consider briefly unify pass combine mf arbitrary aa group factor set factorization index argument similarly index mf approximate pz belief combine mf framework graph distribute collect exchange augment depend could exchange symbol code bit focus share construct augmented pdf scenario replace variable original keeping pdf read next define set factor pdfs f lk lk lk lk lk lk lk lx account form set factor stand code operation lm lm bit probabilistic graph depict structure receiver subgraph factor link receiver process belief pilot symbol channel channel user external iteratively pass overhead adjust depict connect bit receiver make arbitrary factor subsection mf focus mf statistic q define mf therefore belief lk lk lk lk lk lk message lk lk lk lk lk lk obtain select pdf l pdfs non bp eq message l send bp message binary lm lm lm mn c lc lk input decode output equality therefore receive receiver obtain processing message pass main stage receiver obtain initial iterative pilot position include pilot successively repeat li li successively time decode perform bp part initialize equal bit receiver receive compute receiver use successively repeat process decode define denote number stage iteration take go step take belief lm lm lm consider noise interference e table evaluate monte snr almost achieve exchange receiver benefit improve weight precision present decode vice number number pilot evenly spaced symbol user receiver indice db pass design distribute receiver interference wireless unify combine bp mf jointly perform power sharing show remarkable compare system performance share could design schedule exchange approach accommodate value become
document illustrate inference use involve generate class statistical class elaborate step generate generate coherent define extended inter co dissimilarity article similarity extend article extend union article co article article dissimilarity create proximity extend dissimilarity article distance common vocabulary text text mathematically text vocabulary frequency vocabulary frequency construct article give contain inter agglomerative cluster proximity agglomerative amazon amazon user amazon crowdsource amazon difficult use computer human survey question user survey retrieval confirm article success user survey theory member article worker agree member bundle overall agreement independent semantic c comparative ask worker two extend aim comparative check commonly semantic employ worker two bundle appropriate name ask point name diverse bundle result comparative table contain overall user co similarity similarity h c result promise believe structure well conduct towards effectiveness semantic extend year keyword etc automatic scientific article characteristic interesting digital retrieval system properly organize scientific retrieval etc generating article article expensive report automatic extraction article scientific article extraction dirichlet lda make hierarchical agglomerative technique validate semantic make use crowdsource amazon amazon carry setup retrieval effective popular work relevant dedicate services scientific google mention retrieve article proper input query modeling result article google indexing rank page group result coherent task study generalize effective precise definition closeness pair notion closeness closeness similarity dissimilarity closeness item scheme start create bundle scientific second agglomerative inter similarity article validate article document suggest lda base dimensionality reduction important consider independent
dd explain maps hypercube hypercube employ choose angle insensitive exponential leave dd excellent concern normal dd well study dd perform well outli alternative family speed quantify time shift location concern variate normal various consist training base phase distribution classify computation time processor I physical exhibit computation deviation setting dd phase classify calculation computation grow slow datum hull thus depth outside point fast c concern section uci repository contrast dd well know literature detailed description point train number dataset table table diabetes diabetes table synthetic performance term dd respective benchmark datum apply treat several procedure c nn dm dd dd table bad classifier case presence treatment procedure point mh point correspondingly treat random fact calculate mh comparison randomly obviously assignment subsequent benchmark random mahalanobis author classifier contrast md par handle neighbor rule either mahalanobis moment alternatively estimate maximum employ moment nn restrict five rule treat remain exhibit dd depth classifier bottom deviation dd md md md md pd pd ss ss ss classifier nn specify size train test dd classifier class small form constitute testing sample see dd except handle method depend mahalanobis moment perform perform treat mahalanobis euclidean c dd c nn mahalanobis md tr present dd dd perform however depend dd properly robust mahalanobis distance perform handle good machine solver classification use dd depend svm dd two optimally systematically range correspond range interval small arise benchmark fair parameter four employ sequel diabete follow similarly choose big dd calculated use radial basis interface leave employ result twice preprocesse dd inspection plot l c mahalanobis opt time train test pre technique dd bad svm optimally optimize considerably dd svm varied report table classify svm many computational burden phase dd regard take second determine approximate parameter diabete similarly classification completely dd transform variate depth space transformation first fast testing setting compete classifier rather nonparametric approach good alternative certain family consider perform binary knn svm tune dd derive operate elliptical nevertheless dd rather behavior robustness point hypercube insensitive outlier hull must treat classify assign dd classify properly maximum depth rule arise vary specify even classifier mostly large dd dd skewed participant robust discussion suggestion pt plus ex definition corollary acknowledgement classify completely dimensional efficient discrimination depth depth classification perform discuss dd real new rate fast alpha dd recognition misclassification theory recently new employ depth liu point assign label depth centrality indicate employ supervised classification liu al depth transformation analysis dd multivariate regard way classify maximum depth recently li dd depth representation differ notion extension depth depth employ depth represent literature answer ad combinatorial span rather calculate mahalanobi use calculate reflect form mahalanobis depth remain still insensitive depth procedure heavy depth calculate li classification problem plot separate unit validation class point rule firstly classification restrict secondly often assign ad employ calculate excellent concrete application robustness issue outli space operate efficiently space like depth employ dd plot class depth class point classification plus whole dd simulate dimension employ fast multivariate classify introduce transform depth first discussion depth present theoretical dd elliptical mirror extensive simulation comparison benchmark section svm restriction invariant convex level vanishing sometimes impose survey restriction special notion classified class depth mapping mention indicate closeness represent closeness translate closeness e classify depth rule rule al construct degree separate depth several mahalanobi one map origin depth correctly ignore principal allow classify proportion origin classify near neighbor method properly mahalanobis mahalanobis mahalanobis depth depth properly beyond hull sequel either polynomial yield basic include terminate result point object classify accord vector procedure replace long feature leave separate assign order dd transfer set depth properly probability include invariant way matrix measure restriction depth detail cite survey nonparametric depth interest nonparametric spherical elliptical distribution play parametric uniformly sphere random support operate elliptical simple insight elliptical elliptical ellipsoid unimodal elliptical strictly increase claim hold spherical surface also ball around strictly hull unimodal elliptical mixing probability increase strictly prove include projection limit dd depth estimator particularly projection mahalanobi hyperplane mirror respect half independent dd line plot rule correspond requirement satisfied mirror unimodal mirror symmetry dd plot well let unimodal elliptical dd dd common affine transformation dd pc explore classifier procedure dd dd quantify simplify supervise two
fact dual dpp operator solution enhance dpp rule able inactive dpp speedup gain one generally need commonly use cross validation involve lasso consume address dpp briefly inactive idea screen effective discard inactive art dpp rule strong discard strong version version apply rule unclear sequential rest organize dpp idea screening rule set inactive exist screening rule efficiency solver enhance high set screen solver conclude remark dpp rule first geometric briefly safe property basic version dpp exploit geometric lasso dpp enhanced dpp outperform dpp inactive feature form contain derivation dual variable notational convenience recall view kkt potentially make inactive problem apply identify inactive inspire safe first region relax eq find screen rule detect inactive view accurate inactive screening geometric interpretation dual look notational convenience problem space projection mathematically set solution indeed nice illustrated lead enough immediately interior p conclude naturally exist well rest section derivation screening divide contain optimal find step develop geometric dual illustrate fundamentally important dpp screen dpp discard inactive relie operator follow screening contain able notice imply another ingredient feasible nonempty closed operator nonempty convenience convex hilbert projection lasso problem center ball center radius develop dpp upper different inside notation screen rule express easily solve plug bind statement applicable lasso please dpp safe safe st dpp respectively lead estimate commonly approach cross involve idea dpp rule rule corollary inactive reduce problem theorem inactive feature repeat sequential dpp rule sequential dpp lasso sequence k know corollary dpp dpp discard inactive small assume large illustration basic dpp dpp rule easily paper propose screen dpp correspond dpp present careful operator step fundamentally develop dpp imply accurate result screening rule discard inactive dpp operator dpp discard inactive propose approach accurate estimation idea improve dual develop enhanced dpp section screening rule screen rule inside nonempty close hilbert converse nonempty hilbert projection ray projection ray e operator please define dual lemma case technical view easy let define q ray direction lemma inequality distance hold conclude view statement b note reduce briefly review technical convex nonempty cone elegant nonempty closed subset hilbert nonempty close convex subset view key prove statement need x easy arbitrary follow ready view clearly need please note easily complete proof complete theorem inside center develop inactive optimal omit rule improvement parameter screening estimation effective discard inactive dpp rule properly rule discard inactive dpp estimation dpp onto nonempty close nonempty hilbert inequality converse true projection parameter value complete trivial vice accurate dual dpp dpp half know theorem screening easy see discard inactive dpp dual solution see screening rule inactive dpp result screening enhance discard inactive several able inactive screening rule subsequent generalization evaluation develop rule three step optimal lasso recall expand side rearrange follow hand side bound ball follow omit similar easy radius discard improvement dpp family dpp e dpp data b operator screen rule lasso applicable lasso kkt eq generally inactive group coefficient therefore find contain relaxed follow screen word also solution result effective screen discard interpretation let projection onto feasible close notice projection operator theorem rule moreover case lasso specific group problem e subsequent dual nonempty please refer operator theorem develop lasso problem know definition easy immediately need check see eq plug moreover result easy lasso lasso give follow indeed follow immediately easy q inequality due discard parameter measure rejection speedup please safe discard section safe strong version safe notice safe safe strong safe unclear exist sequential favor sequential lasso assume I feature aware safe point group compare basic version safe rule version safe briefly speak version rule make rejection ratio safe along spaced length safe normalize safe structure section cancer face mnist handwritten gene every yield image per object trial randomly image description face mnist handwritten digit report ratio basic version safe five six cancer cancer mnist face provide pointed strength screening stem reason usually method aforementioned counterpart section compare version strong screen real datum perform sparse synthetic datum I generate truth generate rule equally spaced trial screen safe fig ratio safe discard inactive speedup provide screen active nonzero ensure correctness screening discard guarantee run strong twice fig present rule observe pattern variation em c safe strong solver safe solver report combine four section safe strong six real equally spaced datum breast cancer cancer face mnist handwritten digit house rejection ratio safe report solver screen h breast cancer contain tumor gene matrix set microarray containing contain testing response one performance setting image datum mnist digit solver safe solver solver safe strong without solver column second comparable safe rule inactive speedup strong check kkt correctness screen speedup gain size e breast cancer set gain slightly high mnist strong observe speedup e breast gain mnist speedup gain magnitude screening need hour lasso need screen especially strong rule safe discard introduction experiment solver least angle regression lar experiment setting report run lar solve two substantial speedup gain low see discard inactive see em lar breast cancer mnist solver without combine second screen evaluate strong number group entry generate denote size discard group behind solution accurate notice average discard inactive group robust c solver solver strong report screening screen report time second screen far demonstrate effectiveness solver improve solver respectively screen operator screen inactive moreover dpp dpp discard inactive dpp family inactive extensive demonstrate mention dpp rule solver speedup tool plan idea formulation consist structured penalty detailed derivation dual formulation assume completeness constraint therefore problem new dual variable get eq lagrangian primal variable q objective subgradient attain optimum equivalent v optimum denote rewrite q everything dual kkt affine condition problem duality denote primal variable lagrangian kkt exist eq q eq group problem become dual eq eq follow denote eq split problem subproblems smooth let satisfy clearly due get conclude function eq everything
problem variety instance time distinguish metric dense instance assumption stability solve polynomial improve previously primary infeasible solve instance practice realistic fuzzy solve vast completely formal yet interest meaningful way instance efficient partition every many practitioner opinion set find intuition solid study continuous two criterion optimal remain perturbation distinguish another function candidate solution cut partition cut perturb distinguished cut least sum weight degree notion machine correctly metric notion stability stability perturbation modify algorithm distinguish locally stable substantially result regular graph stable instance weak impractical assumption stability find terminology complete graph symmetric bipartite explanatory separate edge separate resp side cut wu weight cut abuse etc minimal average degree potentially follow dictionary useful stand external internal stable locally switch vertex definition intuition optimal concrete definition iff maximal cut locally necessarily resp hold instance unique bipartite stability quite every instance hand numerous cut tend easy stable cut computational perspective hard even arbitrarily stability whereas efficiently decide stable decide cut stable simplify stable instance unique maximal switch decrease define distinguish maximal cut local distinction namely distinction vs cut distinguished highly distinguished stable side cut slightly motivate notion expand expand expand however expand see distinguished distinction expansion derive instance metric dense hard adapt correctly instance algorithm vertex test seed h cut prove part metric dense preserve local instance metric metric instance practically applicable polynomial seek practical reasonable locally good analyze computational significance method simplex solve almost polynomial instance problem improve admit cluster plant instance edge resp resp add edge drop certain consequence plant instance efficiently go adversary modify input improve optimal take forward solve instance dense analysis dense stable cut distribute c mx x x cut polynomially yield randomized instance locally cut cut uniformly stable cut instance vertex map instance prove cut cut map stability cut cut locally cut stable cut map cut construction dense instance let instance consider let easy v v locally metric locally stable cut stable cut stability l z r locally cut either ball l local stability proposition assumption claim stable found consider ball sense show metric stable neither ball even express ball distance identify denote finally locally stable w n conclusion prove psd orthogonal semi cut cut cut ba psd thus show bipartite cut w cut w laplacian iid prove distinguished
consistency satisfied choose exactly part need satisfied important oracle e part existence tight concave satisfy proper choice parameter choose cross validation cross regression also serve paper work study modify observe moderately fall analyze data bic dimensional real high cluster analysis membership adjust rand classification agreement agreement isotropic run member family choose get fail dimension choose associate ari confirm capture bic high dimension simulate model ari ari ccc bic parsimonious behaviour select ari correctly select bic component setting herein one careful modification take preserve asymptotic accounting simulate consistency asymptotic computational convexity prefer concave course problem penalty stay initial mode penalty adaptive lead oracle shall author wish numerous collaborative research science aid collaborative award innovation derive derivation become law large number hold taylor expansion derivative penalize likelihood within p integral laplace apply large usual bic second number penalty mean first line conclude therefore conclude part part time n order tend true decompose efficacy family base clustering depend parsimonious tendency drastically high merge impossible clustering introduce overcome extension factor match model likelihood multivariate gaussian assume fitting decide review mixture focus gaussian review work cluster discriminant mixture eigen mixture actually subset parsimonious family component mixture parsimonious select factor amongst salient family like covariance linear every select mixture away maximum bic theoretically justified number author bic consistently choose observation bic drawback approximation influenced parameter correlation cause laplace addition method prior version mode parameter avoid flat prior fitting case perhaps minimize residual constraint coefficient thus interpretable model behaviour provide penalize gaussian follow model select parameter condition especially mixture though selection criterion note regard author tune estimate extend bic high bic proportional instead consistent interestingly author penalize maximize conventional mainly applicable mixture model author point infeasible possible herein upon mathematical clustering modify suited limitation criterion work property inconsistent ideal criterion analytically work herein propose penalize bic selection likelihood maximize maximize penalize element use penalty penalty scad penalty satisfy asymptotically hard scad penalty prefer lasso easy criterion bic paper estimation property simulate exhibit result compare well property g maximize go function origin locally successive suppose penalty element marginal mle maximization stage estimate variable show membership belong component first stage likelihood z ig complex analytic
I auxiliary representation refer problem mean correlation I modification plausibility outperform drive asymptotically I variance benchmark observable take interest I auxiliary space probability together characterize random inference sampling I tie fundamental quantity predict conditioning invert close measurable serve support sort admissible predictive set admissible set mention admissible unique primarily combine I unique empty predict observe admissible specify eq assertion alternative tend inferential belief convenient report I start singleton I distribution singleton subtle interpretation probability observe parameter original one regard despite start ultimately make explain inference hand datum probability explain valid without common dominating measurable sufficient condition simple ignore let shape I function step singleton simple shall characterize draw finally plausibility assertion singleton q plausibility gamma assertion plausibility plausibility function belief I belief I I false relatively show validity admissible I valid plausibility formulation validity theorem help objective belief interpret unlike example common frequency validity validity output plausibility plausibility plausibility construct frequentist procedure interpretation example collection credible region nothing plausibility auxiliary variable easy however rarely admit scalar association vector first look seem challenge reduce desirable first step reduce efficient auxiliary price auxiliary shall look normal independent step predictive auxiliary singleton assertion c plausibility eq alternative note trying predict uncertainty step predictive singleton plausibility inference base plot simulate sample validity stochastically tend mean wide plausibility I predictive preferred difference efficiency necessary dimensional quantile plausibility correspond remainder give efficiency reduce function since conditioning inference statistic consideration I whereby simultaneous information dimension overall gain efficiency unobserve characteristic need predict unobserve observe help predict unobserve information accumulate characteristic least mostly define relate unobserved auxiliary familiar relate familiar formal decompose distribution take refer understand remark dimension often dimension sort variable reduction advantage I aspect predictive ability section construction I simplify associate fix unobserved see get belief use plausibility depend predict auxiliary conditional association ask loss suppose baseline admit decomposition set association proof say baseline sense obtain former bad baseline I however random auxiliary efficiency construct valid prevent make case validity set setup random valid I efficient baseline I condition decomposition fit conditional representation insight conditional I level sort reduction focus dimension dimension variable conditional case reduction end section appear important I reduce far provide see connection aggregate across sense solution generic distribution induce determine I hard belief set I predictive I consistent posterior I possible develop conditional partial example available incorporate slight previous remark application conditional extend validity result I context obstacle handle collection contain either predictive extension admissible time remark general allow validity identify depend since holds translate statement plausibility plausibility desirable property I conditional plausibility tx tx mind meaningful probability aspect focus relevant subset though conditional validity validity random set tx tx I corollary specify dependence observe could happen depend find conditional definition family obtain probably course problem similarly consideration desirable exponential family condition association method something fact nice check differential minimal invariance etc experience familiar thing satisfactory parameter case intuition insensitive baseline association clear partial e correspond choice solution derivative association formal existence solution powerful tool far track step base student degree freedom somewhat I consideration suggest likelihood observe I build theoretical justification use plausibility assertion plausibility illustration obtain center plausibility I asymptotic normality likelihood credible simulation summarize indistinguishable favor plausibility interval guarantee coverage normality prior alternative naive indistinguishable arguably drive suggest seem suppose independent sample namely available goal come application maximum conditioning recommend consideration reduction step efficiency gain employ differential sx sx reveal conditioning differential familiar classical statistic jointly statistic modify simplify q complete p plausibility readily evaluate illustration display plausibility function derive ignore plausibility sample conditional I naive conditional opposite variability reflect plausibility conditional I plausibility gray gray curve I scale association dimensional sufficient independent familiar default square plausibility singleton assertion evaluate carlo illustration simulated figure display jeffreys prior display asymptotic I plausibility besides guarantee coverage capture elliptical large jeffreys estimator I plausibility I baseline association exist may distribution correlation natural towards independent baseline reduction free decomposition requirement meet I alternative elaborate via type conditioning regard justify localization describe separability extend relax direction start fix section pair association distribution suppose compute point set like produce plausibility refer local validity property local certain theorem local conditional I depend suppose validity localization imply conditional plausibility confirm simulation plausibility plausibility depend place I structural I assertion development solve normal fix construct I shall modify I real vanishe take expression evaluate differential give expression eq denote conditional plausibility local plausibility nominal coverage illustration consider plausibility table conditional frequentist due summarize credible jeffreys message interval interval average moderate large hope well jeffreys local I result along length interval normal respectively I review jeffreys bayes consider number replication treatment unbalanced interest component vector stack useful variability come source nuisance eliminate mixed function fix covariate handle marginalization setup case variance component consider elsewhere orthogonal diagonal fall correspond minimal equation unbalanced model plus sample parameter employ connection note sum familiar characteristic logarithmic since vanish satisfy eq orthogonal association define association chi square numerically random detail I elsewhere brief simulate group size moderate box I plausibility region give walk random contour indicate plausibility region shape roughly consistent work plausibility contour I develop auxiliary strategy simultaneously goal argue remark section fisher case condition auxiliary dimension give satisfactory reduction addition section even predictive conditional standard I set case dimension identify auxiliary predict validity fact could locality assertion singleton way one develop conditional focus thus extend latter extension special case idea association necessarily formulation idea focus framework g regular family I describe herein difficulty new tool come conditioning need dimensional nuisance sort deal marginalization point acknowledgment partially national foundation grant dms dms dms author helpful suggestion associate association r hx hx hx hx distribution hx define distribution hx hx x belief conditional issue fix admissible section like theorem parameter next characterize right turn imply eq supremum prove proof since free tx take supremum validity unbalanced give rescale one assume relatively simple partly fact whether boundary none I full sided credible mean unknown weight framework marginalization present consider association observable something describe look I I pde relationship derivative identify derivative respect simple calculation therefore solution association association omit rewrite obstacle case negative association state word efficient fortunately modification assertion I side suggest get effectively ignore interest care need plausibility size perform test indistinguishable sign something validity locally choose maintain expect arguably standard conditional I gold suggest lose focus choose one function one one function assume assume observe likelihood familiar many exponential family decompose association generate equivalently fit assume independence determine carry summarize factor baseline decompose directly minimal sufficient replace base require result dimension basic basic conditional claim full may statistic great equation technique see baseline association decomposition association determine take independent work analogue mapping factor association I work shall subsection group onto admit association correspond invariant work maximal tie vanish therefore solution equation consequently association free take statistic summarize mapping group decomposition invariant consider association group residual invariant take step proceed baseline association unit maximal I induce distribution expression transformation maximum likelihood quantity maximal condition determine formula general suppose familiar basically find basic insensitive association construction fully depend solution determine decomposition summarize give association one differentiable differential pde system
follow way estimate write explicitly euler c prove rejection state case vx vx c similarly prove upper mala ball require act lyapunov mala kernel suppose assumption observe hold indeed definite explicit exist eq upper q f assertion satisfy chain initial estimate construct lyapunov choose jensen stop process note provide assertion length combine metropolis finally deterministic fix lemma hx h acceptance proposition fix since h h assertion acceptance proof assertion theorem metropolis hasting wasserstein induce distance hence r borel coupling measure assertion w dr nh nh dr nh nh acknowledgment discussion prop prop prop prop prop remark prop prop adjust langevin mala hastings contraction wasserstein mala euler probability density reference measure regular depend contraction rate langevin hasting chain result fact mala proposal metropolis hasting state attract grow show particular product measure acceptance algorithm rwm metropolis langevin mala converge go zero case diffusion scaling maximize speed limit diffusion optimal acceptance rwm mala target sufficiently w point corresponding perturbation measure euler rwm mala replace semi implicit euler proposal metropolis realize infinite arise chain langevin langevin well convergence wasserstein quantify strictly diffusion might expect metropolis hasting chain heuristic wrong sense huge quantify cf chain remarkable exception well work chain uniform measure concave quantify wasserstein euler precise implicit euler require restrictive proposal diffusion result geometric ergodicity particular establish equilibrium wasserstein metropolis hasting chain log concavity context measure space develop thesis fact hasting wasserstein quantify rejection implicit euler proposal bind corresponding ball proof bound measure density normal decompose measurable constant absolutely degenerate absolutely continuous form important cf brownian bound law process abc consider wiener evy number interpolation path dyadic fix let consist component expansion image brownian distribution sampling return absolutely transition strictly let kernel markov monte integration choose accept homogeneous markov transition eq q metropolis hasting reversible hasting standard prove hasting wasserstein look adequate small satisfie straightforward balance case simple kernel give growth detailed balance diffusion solve langevin equation adjoint dense cf algorithm corresponding mh obtain sde mala mala proposal euler discretization detail view euler discretization step euler scheme langevin time kernel stationary cf scheme implicit part substitute semi implicit mala langevin proposal kernel give proposition correction vanish goes fix sufficiently polynomially constant approximation surely finite measure dimension path directional assume q minus norm correspond convenient come end path norm one cf path restrict ball hold constant depend r euler analogue bounds mh assumption exist polynomial r k r r proposition semi implicit euler proposal upper respectively acceptance euler fy determine depend depend euler make state proposition norm e section bound hold kernel wasserstein borel algebra marginal coupling variable define joint derive bound generally couple mala setting coupling mala suppose exist constant moment depend explicitly stationary wasserstein mala chain satisfied explicitly estimate stein step mala require provide minimal step hold polynomial error satisfied radius small take metric borel infimum marginal distance measure bound yx yx qx yx couple markovian coupling markov kernel markovian coupling couple markov apply equilibrium markovian let markov transition initial respectively couple since chain fact stay wasserstein finite assume markovian coupling distribution metropolis product yx yx dy acceptance transition chain variable yx yx yx u yx u yx x markovian coupling hasting coupling accept vice versa mh kernel triangle inequality obtain equality indeed x always simplify diameter x proposal efficiently estimate old hasting q hold wasserstein necessarily close minus consider constant hx dt xy x continuous sufficient guarantee equivalent q derivative consequence mr mr derive mh rejection probability mala probability w r semi euler state euler proposal step size h h vx vx constant vx vx vx similarly implicit euler proposal obtain hx vx h h h sx vx h vx vx hx vx vx upper compute mala rejection vx upper bound rd vx vx ty dt expansion ty vx dt x vx ds ds averaging equation vx vx ty dt ds sg ds ds dt vx vx vx bind semi implicit euler explicit proposal one vanish general valid bind proof vx vx vx vx vx vx vx vx h hx hx hx n p consequence inequality follow old combine order proof hx vx hx vx l hx hx vx proposition semi euler proposal lemma polynomial semi euler proposal pt c proposal
analyse simulation million histogram figure display marginal semi automatic solid represent obtain parameter spherical reproduce correctly reproduce posterior unable reduce spherical marginal posterior regression use semi model semi close mcmc albeit far overhead appear broadly marginal obtain wide especially would construction compare simple bayesian direct argue method toolbox density challenge reliability flexibility enhance focus give estimate al indicate approach automatic expert mixture expert density histogram draw marginal solid density posterior density summary statistic middle semi row quantile plot estimate solid mcmc approach histogram posterior bottom mm mm corollary approximate computation regression mm fan abstract mm approximate difficult calculate active topic current abc common strategy several advantage first easy approximation reference analyse third sensitivity several prior need approximate likelihood build adaptation extend reference frequentist via keyword copula estimation regression density interest initially biological science wide range abc simulate involve simulated weighting accordingly receive see weight practically match impractical post abc posterior g dimension replace statistic reader summary alternative abc construct direct prior summary simulate attractive way firstly posterior secondly likelihood purely analyse maximum researcher familiar inference prior prior bayesian credible base contour bayes posterior perturbation prior expression useful reference prior approximate intractable wave consider approximation approach generalise summary covariance chain mcmc choice summary relate multivariate normal simulate informative statistic appropriate localization summary statistic help available respect auxiliary marginal posterior estimate inference still process pointwise estimate stand functional could serve goodness fit diagnostic abc build recent flexible estimation easy estimating adapt multivariate precisely individually implement density intractable function connection conclude goal obtain intractable datum draw carlo summary determine must necessarily convenient pilot broadly density region define consider flexible marginal mixture expert regression mix vary express component fast parsimonious fit propose copulas margin cumulative density denote normal multivariate normal sample model normal mean conditional component density note exactly another albeit additional experience acceptable approximation enforce marginal summary simple primary flexible construct greatly selection summary statistic orthogonality transformation ij pt ki ik sample derive combine final frequentist provide estimate posterior multivariate informative conditioning mixture provide good parametric technique alternatively condition approximation normal distribution estimating likelihood provide discuss performance normal difficult marginal estimator normal joint benefit simple normal great flexibility copula transform flexible expert wide vector draw range easy linearly reduction via pilot moderately abc adjustment focus production clean require particle observe consist measurement inclusion spherical univariate generalise pareto scale take equally spaced statistic quantile highly instead region density square euclidean large moderately parameter manner methodology summary statistic addition perform fit expert adopt variational number spherical work summary statistic relationship automatic observe display transformation greatly simplify model likely estimate spherical evaluate
belong category simplify problem normalize choose multi represent situation label dimensional ht width compare four train number instance portion instance unlabeled one portion gradually step report micro deviation real world explore regardless pool mml well confidence lack usefulness iteratively learn spectral embed connect high dimensional dataset uci benchmark mml three standard minimal understand also mml two mml mml simply feature mml mml implementation happen rbf j training zero mean square latent model view variance overfitte gradient limit mml decide success consist select table extract color word histogram feature outperform type thus desirable descriptor learner description color color histogram successful descriptor sift accurately image suffer carry ignore perform learn sift preprocesse datum construct color color range visual histogram vocabulary cluster sift description image visual codebook bag task name vs task vs vs vs vs vs ball vs machine vs video predefine use set deviation see mml compete statistically level view confirm transfer knowledge description discriminative performance sift dramatically ball vs reliable sift name sift significantly color necessity multi transfer difficult user mml place multiple view normalization ht c svm sift svm mml mml mml task task task namely census datum record diabetes patient sensitive signal obtain various angle single emission image choose six algorithm view dataset disjoint uci benchmark summarize c feature diabetes heart mml error rate six uci benchmark trial table outperform view technique evaluate mml statistically performance propose framework view demonstrate simultaneous learning feature description one confirm conclusion ht sift mml mml mml diabetes heart conclude mml advantageous multiple description instance multiple task relate mining area multiple attribute investigate together total learn learner use simultaneously one study task dirichlet although many world usefulness practice propose measure success benefit task reject comprise learn together multiple space disagreement branch disagreement classifier semidefinite program conjunction regularization reformulate problem standard implementation applicable although suffer theoretically rarely hold real machine kernel scheme sometimes induce mml require view nonlinear various generally unsupervise traditional classification exclusive multi correlate specialized capture usefulness dramatically large annotation hypergraph concatenation attempt dimension label learn utilize motivated unlabeled datum various learning attract amount interest introduce field propose classify structure unlabeled induce label propose unlabele encourage many propose demonstrate unlabeled survey mining move towards demand supervision view low paper framework handle task view setting simultaneously solve miss identify label apart reliable rate explicit success challenge make purpose use beyond complete supervision dimensional view possibly address problem separately mml optimization reduction set design handle multi many information view combine reconstruction formulation mechanism near neighbor fundamental question task multi present quantify view mml outperform many state semi supervise reduction challenge datum well suit high dimensional datum separately supervise benefit challenge formulation efficiently interpretable reduction mml consider experiment independently collect people shall project face supervise semi supervise fig show five denote people indicate associated face stand green blue people surprising make label marginally infer simultaneously dimension reduction produce belong person associate aggregate gradually image apart width width width subsequent learn technique environment successful elegant framework structural well mechanism view clearly label view examine structure reconstruction create many view set show place single suboptimal raise difficult view fundamentally indicator issue multi multi view supervise construct show mention learn simultaneously quantification successful examine paper section perform structure determine experimental particularly promising discuss issue experimental compete experiment compare compete usefulness learn monotonically improve view significantly benchmark work additional conference extension framework setting facilitate labeling propose quantify view claim approach iterative simple prevent compare mml mml place equivalent mml present early aim simultaneously inference attempt discover structure datum exploiting instance feature abuse refer function explore semi partially formulate function eq near measure near neighbor determine bias advantageous immediately enforce label label matrix minimize function express label multiple label gender involve contain unlabeled problem task relate multi task finite binary classifying contain label otherwise point task description aim asymmetric represent avoid self reinforcement diagonal degree th vector table task classify task label instance instance degree node regularizer vector unlabele identity carry related partially task intuition nearby tend adopt nearby learn encode propagate write something enforce sparsity formulate description classify task third penalty share tuning description decide control section guide error view partially chance less ignore favor class introduce matrix regularizer balance label calculate denote multiplication definition satisfy matrix propagation diffusion balance even label framework task multi learn mml benefit paper experimentally multi multi multi discover aspect description hand weight partially mechanism enable take knowledge mml handle reformulate online learning specifie discuss issue linear fashion mml kind fusion enable detailed important contribution determine obtain zero alternate current iteration predict small value tolerance alternatively make confident treat prior cost confident large derivative confident locate decide find entry entry newly regularizer label apply take advantage complementary carry multiple description optima alternating show recover covariance multipli specifically prediction reduction processing step visualize work datum weight completely intrinsic relation step unnecessary desire vector q reduce vector avoid solution eigen eq recover eigenvector discard eigenvector eigenvector minimize multi apply many problem solve avoid essential approach transfer view empirically usefulness mml perform propagation mechanism instance step build upon carry learn task view wise sum use walk quantify success multi propagation cp positive step propagate matrix interpret diagonal report transfer task occur view value imply view successful occur view algorithm measure world gene dataset collect accuracy value normalize dna synthesis task two color diagonal naturally view different result quantify trial curve mml black proportional relation approximately parameterization lead well thus high outline believe easy motivate multi improve exploit contain obtain highly irrelevant intuitively view suppose description hand multiple description view still knowledge moderate fit task fashion view task source task view weight encode respectively view indicate task l c k c md k j k uk show propose weight avoid propagation sensitive controller label avoid naturally fit preferred choose contrary fit preferred parameter wish greatly minor change parameter empirically respect pick highly dna task micro balance task choose box ignore definition histogram influence view see surface relatively though open question
obtain identity prove constraint display minimum end solution image denoise enhance remove differ example path suppose ij represent record gray across true level denoise serve reconstructed achieve replace isotropic penalty penalty focus path follow convex loss many image poisson ray denoise circumstance square view vector penalty sufficiently reduce goal quantitative notable advance include site summarize recent progress reality iteration tune path attractive option recover minimize linearly dependent violate constraint equation uniquely difficulty four fourth well redundancy neighbor pixel illustrate corner pixel transform segment along path path path accelerate constrain build differ homotopy path introduce effectively handle tracking explore knowledge first make method enjoy simplicity rich numerical matlab regardless slow exist optimization insight predictor interact quadratic piecewise permit denoise suffer side problem reliable separate local current solution development rely strict path denoise role enforce work expand list grouping straightforward remove restriction exact method equally necessarily unclear one behind modern interior broad issue researcher corollary theorem grant gm kl grant hz great stress tend constrain penalty recover penalty examine strategy consistent constant trace constant thus start solution programming piecewise take jump piecewise operate differential segment segment projection onto quadratic programming semidefinite mechanic denoise demonstrate path penalty decrease optimization penalty program ordinary differential constrain regularization device problem penalty operate barrier strength barrier method gradually barrier gradually send either solution optimization barrier approach program application control method central reliably quickly minimum nonetheless penalty augment lagrangian potentially competitive method penalty avoid problem disadvantage penalty quadratic penalty lack objective argue path unconstrained solution penalty increase along constraint path boundary advance lagrange multiplier special programming path one entire segment theory homotopy year exploration method goal assess path follow constrain performance method probably later practically orient paper experience rich numerical matlab include solver review exact derive path particular attention constraint algorithm elaborate demonstrate particular denoise problem apply mathematics penalty discuss function subject affine equality constraint differentiable differential partial transpose second partial constraint meaningful lagrangian l capture lagrangian satisfy nonnegative complementary one take choice create favorable circumstance minimize twice differentiable lagrange rule local minimum inequality program constraint condition impose quadratic minimum prove quadratic minimize surrogate complicated increase lagrange unknown application follow property surrogate increase convex likewise finally assertion generally finite strictly combination strictly two point strictly strict inequality multiply b third restriction result direction sequence scalar tend e n obvious section devise path strategy start program regularize problem primary find solution path mind area path order characterize minimize one strictly minimize subdifferential equation rule calculus book know path uniqueness continuity uniqueness existence subtle solution continuous linearly near convex consider fix point inequality solution fail exist tend subsequence take limit e e unique verification claim defer say remain subdifferential smooth path solve equation ode path algorithm segment determine sake simplicity begin occur plan attack lagrange constraint multiplier satisfy g constraint lagrange multiplier unique consequence continuity hand side observation stationarity constraint equation write equation theorem require dependent equality left implicit importantly implicit following summarize finding strictly coefficient occur subdifferential solution lagrange multiplier satisfy differential segment become active constraint boundary subdifferential determine segment inactive occur move keep constraint active constraint occur inactive comment move continue simplify considerably penalize affine vanish segment satisfy previous highlight generality constraint surrogate along differential two row inverse amount scale inverse sequentially organized operator computational every burden plus compute slow algorithm problem problem invert suppose full satisfy active furthermore f equation computational side improvement cost balanced fortunately practice solution constraint leave sequentially ode loss number regression ode interested reader refer book versus employ euler update ode euler inaccurate connect amount replace easy probably ode package ode intend mechanic comparison comparison feasible point program design project convex algorithm consider toy project close close h path start movement path toward c ccc toward origin toward path c toward axis plot vector point ode solver evaluate derivative converge square nonnegative principle useful modeling nonnegative observational article estimation statistical matrix nonnegative entry imaging range pixel rank minimization nontrivial enjoy property guarantee converge even exhibit alternating solve separate denote column correspond separate path algorithm solve subproblem end problem straightforwardly project typical next path roughly unconstrained programming example minimize affine amount positive semidefinite path equal bivariate radius start unconstraine along circle end ode solver evaluate time programming contour contour branch stand application chemical equilibrium structural mechanic digit circuit process subject geometric leave equivalent fractional power instance constraint programming
must satisfy primal feasibility iv dual feasibility stationary candidate primal plug primal vanish ii hold last two meet give rise decomposable simple strictly sub admit eliminate respectively establishe initial lagrange n redundant identity plug multipli become identity sequel move across lagrangian update obtain I identity penalty upon nk follow similar lead complete algorithm convergent recursion upon imply consensus statement proposition go define limit upon iteration comprise tucker r limit arrive n nk thresholding yield follow ii iv f add iv readily check identity obtain last role optimal equivalent b cm cm signal advance wireless communication edu superposition plus recovery completion rank minimization nuclear surrogate count minimization centralize processing separability overcome limitation characterization nuclear adopt yet minimize alternate multiplier reduce message pass among interestingly attain centralized counterpart regardless initialization outline highlight impact include traffic anomaly rf wireless centralized sparsity nuclear norm low spc considerably measurement observe miss one introduce set unchanged compactly fit square error entry pseudo g describe unfortunately rank norm np nuclear surrogate close accordingly rank control convex appeal special instance attain traffic exactly absence compression term robust available special offer completion stable recovery completion early effort work assume jointly process solve challenge interest wireless preference share fashion raise central processor represent isolated point failure scalability driving motivate distribute propose medium singular aforementioned reason motivate minimization recently though applicable topology also distribute absolute centralized decentralize sparse base small corrupt spatially distribute measurement task separable nuclear norm amenable minimization nuclear separable convex cost multiplier ad complexity per pass single interestingly convergence centralize initialization propose algorithmic four namely traffic volume anomalie ii robust matrix completion iv cognitive cr networks centralized benchmark defer bold letter letter operator hadamard product cardinality scalar matrix matrix top return inverse entry semidefinite trace inner mi mn n adopted consider capable computation message neighbor abstract g monitor internet cr communications network model link agent agent henceforth agent eventually outline incomplete corrupted n l r develop regularize processing locally exhibit centralized estimator keep overhead inter agent communication neighborhood facilitate requirement distribute argue value internet traffic analysis origin intrinsic addition small reduce adopt p obtain convex bilinear lt efficiently handle amenable non coupling address minimum bilinear leveraging provide obtain adopt frobenius enough one globally p ensure less stationary optimum interestingly show relatively globally optimum point globally sufficiently becomes violate sufficiently expect long addition variance certainly decompose couple global variable cf auxiliary estimate utilize minimization become later variable norm cost regularization cf additional nothing importantly understand estimate coincide even consensus impose whole connect desire consensus constraint eventually eliminate tackle constrain minimization problem associate lagrange splitting associate additional positive split notational n constraint n fashion multiplier ad adopt ad iterative lagrangian well parallel tackle task g entail comprise iteration implement coordinate variable lagrangian treat ad generally cycle group ad special section sufficient subgradient iteration implementation augment lagrangian decomposable separability come group agent turn highly simplify four specifically initialize algorithm present l thresholding entry denote r k k k f n k f simplification redundant inherently redundant auxiliary multiplier keep track redundant multiplier communication burden stem program carry agent neighbor communication matrix efficiently observe encourage algorithmic accommodate penalty maintain agent complexity n k one need unconstraine strictly quadratic admit anomaly term operation n n k n b flow invert inversion need addition reduce inversion employ n rank computational ridge large element operational main burden communication identical incur traffic measurement allow datum analysis science g site web surveillance massive volume datum result low dimensional nominal ls pca remarkable presence large related reduce sensor flow see corrupt local low agent want form special discussion agent estimating row constraint n per variable one primal variable close minimize soft neighbor n c n l matrix cost neighbor localization wireless recommender give noise observe entry corrupt rely processing aim complete form exploit nuclear norm whereby feed programming rank brief operator operator linear symmetric characterize introduce whose equal otherwise n general component remain agent suffice primal iteration obtain property subproblem admit solution receive neighbor I k l n agent obtain inversion typically acquire prescribed computational iteration iteration transmission overhead small value observe regression l retain consistency however fail parsimonious adopt regularization agent reason available distribute fashion absence sense cognitive cr whereby sense across frequency map enable spectrum spatially expansion sense form narrow spectrum locate operational variable selection motivate collect location amount solve protocol accomplish accordingly readily update multiplier select fold validation numerical convergence convergence follow find penalty lasso k n k n k r c k convergence agent consider realization agent randomly place distance prescribe flow bilinear draw collect internet internet refer flow record internet assess agent flow end flow collect operation internet comprise flow control condition interval respectively short flow attain experimentally fast convergence evolution agent depict representative metric interest error compare n left agent accelerate collect per processing centralize right depict evolution level distribute centralized counterpart give flow measurement begin internet though management protocol trace flow superposition clean anomalous truth precise exhibit dominant singular receiver characteristic roc highlight anomaly low rate e accurately consistently flow illustrate estimate anomaly flow aid centralize obtain use ad agent pca assume apparent converge
show impose hierarchy section notion hierarchy remove though build remark removal technique column overlap favorable discuss role hierarchy aspect sec primary straightforwardly logistic material modification primary predicting base concentration size c six classify measure top method sparsity pick good level incorporate predictive bottom provide indicate next effect constraint unbiased degree freedom freedom valuable primarily prefer tie advantage characterize optimality tucker effect hierarchy simple take function jk residual predictor linear kkt thresholding c write toward examine statement strong method strong hierarchical loss satisfy sx sx supplementary correspond weak insight prop reveal overall form identical certain effect shrinkage certain constraint loose weak condition become involve lasso th loose match coefficient hierarchy would naturally I dual increase weak identical loose show hierarchy hold show particular jk j jx j analogous get exact establish study elastic term modification ensures note suppose absolutely drop hierarchy weak jk jk absolutely quadratic freedom refer dimension space fit measure notion procedure see thorough discussion produce fit degree freedom ridge depend pattern constraint tight evaluation p replicate strong along estimate df circular bar estimate plot datum therefore useful unbiased amount pt difficult quantity make j j jk pair df strong hierarchical fit main force constraint accommodate interaction pay effect make sense could advantageous make nonzero visually indistinguishable considerable fitting overview restriction discuss iteratively consider remove add consider elimination enforce recent version selection first without include hierarchy post modify lar hierarchy procedure interaction come viewpoint interaction hierarchical selection indicate whether coefficient normal get write similar series modify add inequality enforce hierarchy sense approach method make describe composite penalty broad penalty group achieve hierarchical put induce interaction framework structured literature note remark close vanish rewrite penalty although class involve sum norm combine involve main study advantage restrict true generating consider truth hierarchical case element ii iii submatrix ratio snr snr effectiveness refer sequence add variable choose forward modification modification add give main approach lasso pt method tune select note simulation know avoid biased favor simulation panel color effect respectively solid forward respectively jk prediction assume hierarchy well rest receive succeed panel predict truth expect winner situation surely anti identify main get interaction variable hierarchy even interaction constitute truth present well able interaction inferior scenario hierarchy dominate relevant interaction identify interaction since spurious hierarchy pair favor scenario ii iii well correctly far sensitive joint act contrast select effect regard six mutation drug drug location six site occur focus weak lasso addition standard effect dependence result average six case abc achieve well test error concern rmse dominate several main appear option fast solver rely amount iteratively coordinate minimum convex specifically hierarchy symmetry couple mean get separable block discuss weak discuss lasso solution descent elastic tx r involve lasso gradient convex generalized gradient descent suitably step since replace state descent guarantee fact look differentiable subproblem p solution elastic net remark observation solve along sequence large warm start value symmetry tie method multiplier admm widely applicable apart subproblem problem three separate part serve result practice affect admm hierarchy symmetry involve result update throughout algorithm solve version admm call replace argument propose lasso strong closely tie exploit admit characterization impose hierarchical implication demand hierarchy introduce distinction measure variable hierarchical interaction measurement consume provide implementation weak method gaussian gene prove solution ease equivalently notational write pt change strictly objective amount possible treatment lagrangian variable kkt primal dual pair ty l necessarily term follow equivalent ty result pt characterization assumption freedom write l inequality describe ten row pt ptc ptc follow proof strong hierarchy property include elastic fy pt solve design lemma part vector show imply zero except consider te automatically specify contain begin show union already jk jk j leave begin tu tu u jk jk jk jk jk jk jk u np u u u u jj j ie ie pe np e j set long lebesgue weak hierarchy nearly section jk jk establish weak hierarchy fit kkt get q satisfie ty write continue fy satisfies optimality ty fy h iy fy lf since lb fy fy fy hold plug leave argue ny ny prove nearly situation piecewise degree bind estimate interpretable linearly precisely linearly r span row clearly lie span r jk add show row p p give drop kkt condition subgradient function involve stationarity involve via dual step knot max useful associate fellowship dms dms health contract add set convex interaction interaction include marginally precise characterization hierarchy hierarchy degree reveal fitting hierarchy constraint distinguish nonzero raw prediction hierarchy focus closely tie concern available empirical insufficient outcome medical diagnosis occurrence confident patient either situation I status additive moderate measured variable idea develop order predictor eq refer part interaction summation consequence notational interaction take throughout everything carry restriction provide net training select main estimate practice among interaction effect restriction name call jk argue hierarchy sensible reason special position origin must include hierarchy position origin regression must go take nonzero model strongly norm
reproduce kernel hilbert rkhs prescribe hilbert function functional rkhs possesse reproduce characterized property reproduce kernel uniquely determine rkh thus rkh reproduce kernel emphasize hilbert function vanish everywhere wiener wiener eq depend reconstruction approximation application consider reconstruction error norm borel banach space borel measurable function throughout continuous reproduce property equip borel optimal performance reconstruction eq concerned point shall try remove end algorithm choice finally minimize minimal interpolation point reconstruction reconstruction reproduce kernel determine observation hope another distinct reproduce positive strictly definite furthermore eq reproduce every equation get result orthogonal projection prove corollary minimize quantity calculation tell complicated form discuss space reproduce radial norm define reproduce kernel must monotone measure minimizer minimal mid point unlike nonlinearity exceed example two elementary let careful elementary measuring seem natural dominate multivariate hilbert save computation effort consideration restrict subspace minimize eigenfunction eigenvalue principal machine course point span eigenfunction minimization computing eigenfunction exist transform purpose span infinite algebra consist subset measurable f approximate transform question determine satisfie eigenfunction functional operator moreover imply bound ball weakly word shall j h tu equation imply lebesgue turning theorem orthonormal maximize part transform eigenfunction eigenvalue return positive borel continuous explicit form orthonormal span kx give stand eigenfunction operator different orthonormal eigenfunction eigenvalue k h le ix practically concern cardinality rank notation eigenfunction costly shall eigenfunction efficiently standard follow shall first eigenfunction eigenfunction assume minimizer problem wish close subspace bound projection onto subspace supremum identity kt p kt k kt kt accord lemma distance projection onto accordingly shall show section singular exist span use subspace yield explicitly characterization equation rewrite u kf h u nc k k nu kf get equation introduce matrix matrix eq get hold h h k lemma give problem solve search sampling remark favorable orthonormal search candidate point efficiently occur inverse computation illustrate point reconstruct system throughout kernel matrix therefore experiment consider q two use spaced kernel target sample reconstruct error calculate present plot pair comparison follow lebesgue sampling mark star equally mark eps standard marked eps approximation large situation point perform space comparable error equally sample usage sampling bring relative superior equally spaced lebesgue optimal marked star equally marked improvement figure error mark star eps relative instance space obtain spaced improvement experiment dimensional lebesgue mark spaced eps mark circle star rather lie could equally space example lebesgue distribution spaced eps marked star eps pair relative close equally spaced point consequence error comparable present equally space mark equally spaced point algorithm eps eq mark marked star error instance improvement experiment superior experiment algorithm one uniform support equally spaced point mark circle eps mean improvement figure mark circle mark eps pair except outlier improve conclude sampling yield significantly equally space true corollary thm
assumption meaningful way fix ap minimum iff th differ exist minimizer constant prevent generalize assumption trivially minimum frobenius unique th large differ identity establish estimator next satisfied solution trivial assumption remain require frobenius unique norm objective demonstrate see solved replace invariant norm consider frobenius unique optimal resp spectral objective minimum generalize far rank ex equivalent constrained version take admissible well admissible even solution tune constant desirable optimally usually replace suitable norm counterpart norm strong j sd problem couple considerably invariant norm definition invariant constrain norm nearly prefer constrain norm adapt version moreover suitable constant yield theorem obviously remain different transform trivially iv thin svd norm trace recover thresholding correctness proof characterization subdifferential trace avoid deep form trace thresholding solution hadamard subspace next corollary let thin either form classic point hence let obtain form way derive pp classic result recover look verify recover weak surprisingly appear new subspace clustering consider set subspace estimate ill pose subspace solve subspace cluster widely though refer reader survey successfully membership reconstruction obtain minimize instead independence think turn proved thought recall either point come show unique note shape sim know surprising aspect noiseless nothing frobenius close practice corrupted consider discrepancy choice norm norm typical regularizer frobenius consider program solve multiplier matter choose appropriately one thin svd function remark heuristic computer vision literature provide justification show optimization purpose intuitively amount moderate sim singular noise sim shall shrinkage interaction discrete might instability preferable thin call shrinkage shape interaction purpose also call shape measure regularizer equivalent due equality remark therefore matter result solution sim close derive latter subspace variant lrr lrr trace case performance directly except sim tune sim affinity apply segment cluster misclassifie point follow subspace construct randomly noiseless achieve expect level sim noise lrr range probably discrepancy lrr start trace norm regularizer regularizer sim form single lrr generally require converge svd cost lrr order slow threshold method level sim motion segmentation subspace regard manually independent fair comparison apply though result sim contain obeys subspace row obtain sequence good regularization constant well close art select individually clear lrr significantly slow method include sim lrr lrr provide interesting regularize apply cluster result subspace clustering first body use norm large fan case frobenius three norm fan fan follow fact invariant norm easily remove clear extend iff th large minimizer proof fan value lemma admissible feasible admissible k suitable integer definition note complete thin projection let orthogonal ap k unitary apparent need consider k frobenius minimal choice simplify uniqueness j sd unitary invariance norm diagonal argue objective due xu c follow use everywhere zero solution unitary invariance ba easily verify invariant restrict singular matrix main body norm see value guarantee frobenius step need minimizer also optimality conclude frobenius former principal reduce datum characterize close solution model obtain insight promise typically subspace manifold sample provably correct identifying characterize likely union subspace would subset unfortunately normally membership subspace therefore course difficulty segmentation
construct stick construction paper process allow al deal introduce background completely sampling introduce technique poisson introduce section dependency operator section poisson process random measure completely measure poisson process measure measure along kind work slice describe illustration basic domain process countable infinite set dropping line picture class admit summation part deterministic purely atom jump atom restrict pure jump jump measurable measurable kind construct process take poisson surely number denote call poisson product evy construct measure measure construction completely mean arbitrary disjoint measurable space finite finite always jump bound get point finite without loss generality evy represent probability l evy used concentration parameter sampling logic sampling jump later go evy mt base corresponding go l l evy deal evy generating jump homogeneous note important name exponent exponent guarantee jump lead interpret functional evy equation jump sampling block jump discuss normalize unnormalized construct normalize increase additive term random increment prove distribution also normalize measure take measure survey kind transformation evy use mass usually sample hyper jump depend gamma process construct shoot cox process evy form use normalise l factor normalise normalise parameter evy similar different induced mass parameterize law compare dp cluster create total familiar stochastic special limiting normalize generalized gamma process process normalize process n stable gamma formula l l regularize incomplete mathematical library evaluated necessarily trick eliminate term relative mass variable follow replace introduce variable idea variable future slice method behind deal jump mixture I belong slice auxiliary introduce description follow mixture auxiliary return otherwise actually compound mean expression give integral turn incomplete gamma function large size component slice variable density density evy consider sampling go eq shape separately conditional jump interval sampling introduce dependency operation review transform construct dependent completely superposition poisson superposition poisson superposition superposition poisson subsampling sampling select poisson acceptance transition poisson independently operation describe move integrable measure aa poisson superposition give superposition subsampling give transform new operation generalize superposition measure subsample subsampling define transition atom base superposition subsampling point superposition subsample underlie posterior version condition version recursive simplify marginal weight various integral introducing prior make carry result upper incomplete integral recursion otherwise term may unstable poisson process well ease major issue poisson generalise discount vary concentration might significant computational burden develop simplify marginal variable predictive posterior also present adapt normalise u jointly jointly via transform obtain adapt integrate slice bind jump observe jump value follow n k j k hx k unique count jump index match conditional hierarchical reasoning jump jump retain apply covariance property laplace exponent case dependency involve significantly mass long jump normalize however correlation normalize q dp correspond upper gamma superposition underlie l evy dependency via evy dependency transition measure ba disjoint operator appropriately covariance superposition subsample straightforward extension posterior superposition superposition nx xu jump subsampling subsampling evy q subsampling evy give dependency operation composition operator two operation operation acceptance superposition superposition constant composition transition superposition subsampling apply operation lastly operation form subsample acceptance subsampling point superposition operation rule q remain top remove lemma ready theorem three equivalence subsampling point poisson dependent measure evy evy denote transition sampling subsample posterior subsampling completely bernoulli acceptance nj j terminology pz pz z department communication digital research centre variable rearrange evy formula since something something hold result lemma allocation indicator slice jump auxiliary q introduce jump threshold evy evy proof come posterior proposition simplify yield concentration posterior datum distribution equivalent surely proof lemma equation expand expansion power incomplete expand recursion k n therefore chain expansion hold bt arrive expand inside definition denominator kn u posterior discard theorem proof come end prior see true simplify mass introduce make change tu
much star systematic surface star good although dispersion bias black dot star show namely respectively value indicate datum h model magnitude range g ap average start bp begin decrease detector plot figure introduction improve accuracy magnitude ap range reality represent upper ap star upper assign star ap value equal datum training strictly distribution figure fit limited star prior g star star explain high prior datum half residual whole much star may star vs interval posterior posterior pdfs star circle circle green dot axis ap model ap estimation magnitude range star ever pdf generally width important distinction precision residual top panel project precision red difference low rp spectrum precision accuracy say figure pdf eight star parameter reduce star early ap also ap estimation produce uncertainty pdf bad simply see g magnitude fractional error star distant build star high completeness drop surprisingly low contamination svm true star range ccccc select completeness select contamination select contamination summary star performance inferior science give nothing estimate extreme recognize investigate parameter datum algorithm system star close investigation result select sample star issue ap high star snr something find despite biased e fig less sensitive suffer different svm perform produce regular svm motivated forward fit grid ap table small well ap small mr less bias always ap test naturally report ap uncertainty characterize use also introduce considerably however magnitude possible ap considerably slow work accelerate feasible star observe year allow post release expect include rp spectra available website estimation introduction processing g star emission star spectrum star present star room improve grid improvement grid need star make spectral wide estimate preliminary alpha abundance difficult want fix use improvement modification apparent acknowledgment thank thought ap something support grant well contract innovation obtain source yield star scientific reveal formation analysis rp spectrum three different algorithm effective star wide range account diagram improve estimation broad star spectrum absolute always large amount estimate star across priori performance magnitude estimate well star method survey fundamental star diagram detail accurate census star ever course position proper star source visual addition five also radial velocity star magnitude datum detail statistically part composition formation evolution anonymous star equip together distribution target nm resolution main name bp rp galaxy star physical reliable central understand galaxy fundamental property star age composition star primarily temperature surface estimate ideally star absolute magnitude apparent magnitude development describe refer e bp system estimation approximately object vast module galaxy module classify emission line star module spectra radial velocity star rp also much achieve noise ratio cluster exist try network kernel quite successful many magnitude confusion observe broad partly algorithm take parameter compose separately logic behind collective rarely article confirm work experience survey svm bp rp spectrum uncertainty complete star estimate report final inference account ultimately make user e library observation process place learn datum quality behaviour evolve l environment article effort correctly galaxy noise datum complicate indeed part develop library simulate differ algorithm describe science available version analyse additional prediction notation summarize unit unit ga empirically fit apparent band k logarithm rp arbitrary comprise literature description far algorithm code originally perform implicitly transform nonlinear original normally tolerance svm scale tune search training process identify star tolerance define determine newly star present work discrete magnitude train near build separate ap model iterative initial phase multidimensional forward model newton ap generate observe whereas spectral band ap full hereafter report early bp rp essentially forward infer divided weak thin spline model regular addition covariance fit probability measure star contain information band define hereafter basic spectrum pm g g later pdf ap ap advantage method pdf estimate take account uncertainty magnitude ap possible include compare ap spectrum band gaussians likewise depend bp rp include estimate four predict everything iteratively jacobian find model spectrum resemble spectrum determine apparent consume forward datum even give training although lot observation attempt sufficiently dense broad ap number result heterogeneity limit estimation grid rp spectra arbitrary final grid datum calibrate determine via resolution ground spectra increase vertical band spectra fix lower vertical convert rp spectra simulator library broad spread resolution line response detector like overlap medium band bp cover nm range spectra blue resolution instance difference middle difficulty degeneracy using spectra due see reason former strong star difficult quantify plot bp source course primarily time nominal source noise source accommodate random error systematic currently band high star actual assume magnitude would apply deviation range around colour star star median large nearby star black rectangular value regular spectra distinct grid interval rectangular range step calculate grid locally high resolution I physical combination star really rare star star initial diagram source colour reflect population expect period formation typical white point discrepancy star train test order call grid ap speak ap accuracy strongly investigate datum grid star star star magnitude magnitude set distribution representative obvious varie train svm test set processing mix star magnitude star magnitude send svm near magnitude kind matching adopt train datum ap star data different ap grid sensitive rp spectrum individually normalize divide spectrum remove apparent three reason could arbitrary star select initial assign library construct affect use spectra limit grid may infer rely self consistency gm place derive library star constant make certain star star star star star star f star star star star star star star star star star star star ap svm star star star star star star star star star star star star star star star star indicate dot green dash model solid accurate residual ap true ap summarize strongly plot various ap table permit quick summarize ap star range four spectral type range star star star star star star statistic star residual possible square outlier summarize g systematic mean mr reader appear almost high show residual skew systematic accurately star star prominent spectra explicit line bp rp still third p look bottom show leave star symbol method circle vertical scale panel dependence error significantly bin star phenomenon grid weak effect consequence model rather increase nonetheless temperature star star star systematic part think correct unfortunately rarely plot residual residual use see green line red line ap range plot degeneracy top mean star open circle green closed circle dash vertical type scale figure encourage science star course law large turn somewhat star mr total error residual imply degeneracy svm accommodate observe degeneracy lack sensitive affected try solve
make conjugacy simple exist decrease spend space disjoint measure beta gamma process evy product evy represent tell exchangeable describe mix exchangeable distribution exchangeable binary matrix infinite collection bernoulli result exchangeable matrix beta distribution cluster estimation example distribution dirichlet normalize stable directly hazard pp auxiliary space random evy theory herein evy evy measure atom mapping give q family measure collection bernoulli value atom control dependence measure mapping vary form simplify take unimodal dispersion interpret atom location dispersion unimodal multiple location marginal indicator evy form smooth intuitive measure nearby covariate use measure evy unchanged conjugacy put original allow specific normalize collection normalize completely dependent construct large dependent normalize sub covariate marginally random probability measure remainder leave section describe two hierarchical base simple covariate dependency choose unimodal away covariate dependent model example decay atom correspond model go contribute covariate unimodal arbitrary might random transform sigmoid every value consider construction covariate dependent process homogeneous evy choose location kernel location covariate location fix auxiliary decompose gamma choose resulting imply valid pointwise construct covariate variable weight point select weight atom dependent corresponding give superposition generative sample covariate simplify gaussian ibp combine instance combine k model decompose text finite vocabulary simple topic latent draw word specific extended way topic topic drift allow topic time topic topic localize formulate poisson usage beta topic localize force parametric model activation allow modal disadvantage force treat twitter sense address happen use popularity vocabulary let dx dd db dp distribution present atom rate global topic baseline wang auxiliary dx xt kt expectation exactly form beta programming language inference poisson process derivation easier simply mark poisson poisson beta normalize construct x normalize process snp dirichlet exist covariate incorporate snp model could benefit scheme correspond well space thus naive poorly merge metropolis snp possibility explicitly yield scalable efficient implementation easier employ paper location allow write independent independent inference sampling grow covariate increase design inference represent originally tailor allow kernel manner step let g dependent probability evy ta dp unfortunately easily space however treat mixture show measure marginal pick framework addition represent poisson poisson define process normalize biased fit markovian dependency modeling consider probit dependent latent prior capture likelihood intend highlight ease dependency gain covariate yield ny n observe truncate beta beta ref probit carry gibbs truncate computational case atom converge conjugate update auxiliary iff sample joint distribution scheme lastly sample derive conjugacy supplement covariate dependent synthetic bag item covariate line eight pixel varying imply result location usage location use eight perform describe learn scale kernel depict learn dimensionality probability depict sometimes feature explain evaluate un dependent ibp collection transform gaussian test country observe country process country deviation compare exchangeable feature beta beta obtain rmse exchangeable indicate incorporate modeling result surprising use flexible structure add covariate conjugacy increase cost exchangeable bp run hour day account run simple latent dirichlet allocation ng document word distribution draw word basic way drift topic usage time apply topic usage time assumption allow localize formulate factor assume document observe dependence adjacent instance topic usage beta localize topic unimodal non parametric extend dirichlet allow topic disadvantage force quantity allow sense twitter document arrive address denote document vocabulary word vocabulary use model popularity pn x beta topic effectively learn poisson decompose pn x pn multinomial eq denote topic generate document baseline define control realization topic particular model gibbs sample supplement dependent qualitatively union dataset full text cover year break document result document keep occurrence corpus quantile comparable result topic vice versa qualitative evaluation hold hold word supplement static version binomial see obtain superior show incorporate version much
geometric convex half determine representation setup system study theorem statement share common let follow equivalent setup big class rise hyperplane ask every answer exist hyperplane family answer chapter involve undirecte unlike concept demonstrate demonstrate henceforth undirecte orientation enable orientation orientation else orient way direct graph undirected subgraph follow subgraphs undirected cycle cycle orientation orient orientation mean direction cycle sort linear remain orientation cyclic follow see strict observed cycle yield direct cycle yield yield cyclic subgraph vc dimension vc dimension vc dual vc cardinality strongly easily equality number component g ds se statement implication position orientation agree obtain orientation edge connect agree path particular vertex lie orientation theorem tree short path equal subset separate e ask closed intersection hard demonstrate system example sort subsection lemma get intersection system class intersection theorem theorem convention master thesis dr dr dr vc start numerous application rich elaborate discover researcher set definition give system trace al first strongly characterize strongly al discover characterization equivalence characterization phenomenon different discover field pure mathematic connection hope group enhance functional geometry system extend maximum geometry find new definition differ known duality transpose phenomenon duality via translate claim claim unary notation literature operator preserve property maximum work duality demonstrate stem boolean algebra moreover reveal concern sort reach presentation relate theory sort operator operator simple weaker weak naturally geometry combinatorial orient introduce convex study prove certain concern distinct direct subgraph advantage demonstrate application work study finite thus structure standard clear prefer system call system mean agree cube cube equivalence equivalence relation agree cover follow statement sx merging agree say eq definition identical except highlight duality observe example straightforward calculus statement similarity former replace strongly natural variant partition statement se section statement se first statement strong duality certain english text proof transform text lemma proof symbol pair pair sometimes dual true true valid phenomenon easy verify valid chapter strong system dimension dimension denote generalization accumulate work independently several discusse prove trivial useful let equality remark true prove conclusion consider assume hold equality f lemma let object pair restriction associate system induction pick restriction enough lemma induction dimension easy remains accomplish gives namely demonstrate duality transformation nature transformation theorem hard find claim dual claim case dual present proof transformation take care suggest follow equivalent prove every otherwise pick restriction lemma whose valid wrong system follow statement eq q lemma finish property mention serve prefer pair restriction se se se divide part conclude first valid system se euclidean discuss se maximum se seem possess property theorem cube accord terminology refer pre pre clearly translate restrict follow normalizing cube mapping naturally ns nf system system restriction maximum example operation complement satisfy boolean follow variant unary operator way system operator lemma array order row sort sort preserve sort fact easily array phrase member appear immediate following let sort follow author let se henceforth mean study define regard boolean operator symbol maximum se statement et case system verify lemma easy cube conclusion operator begin permutation intersection appear begin sequence permutation sequence lemma immediately imply theorem picture leave boolean next straightforward lemma every hold yy characterization follow system two system verify hold every cube eq easy q x term sequence member system definition object vx distinct replace every vx lemma desire finish proof system argument seq chapter introduce natural via concept discuss basic operator second generalization theorem regard form system partition correspond cube informally depend surprisingly operator become formal later extend cube cube cube original terminology extend definition operator x call follow hold reader might simplicity note cube x thus determine behaviour operator let henceforth extend x compute locality operator meaningful satisfy sequence boolean meaningful composition operator theorem prove regard remain true obtain meaningful operator similarly operator let cube follow concern cube theorem refine shift statement equivalent hyperplane euclidean half determine hyperplane cut chapter h section follow hyperplane independent addition note plane first amount intersection demonstrate h cell discuss generalization system question give line side htb linear hyperplane usual simply translate nonzero vector note normal hyperplane half associate orient hyperplane e
ern poisson parent cluster cluster sample distribution independently inside disk similar mat ern generate poisson process parent intensity replace parent sample poisson contrast mat ern position point isotropic center deviation potential scenario problem regularity clutter clutter consider regular density unit apart denote generalize incorporate control interaction exhibit regularity computational spatial environment extend inside case sampling clutter generate clutter density top toward bottom target locate window result mat ern whose spatial pattern realization desire six sample realization instance poisson use point average however point clutter discard realization benefit traversal clutter source clutter spatial clutter note rejection achieve actually point conditional poisson point distribution window however crucial observation interaction unconditional one purpose illustrate clutter environment choose point point realization average exactly clutter goal place true obstacle clutter probabilistic disk disk boundary limit disk sampling window poisson obstacle disk window fact obstacle line perhaps traversal obstacle disk know second perhaps operational point view obstacle disk center intensity inside respective total obstacle disk clutter window linear window corner w shape corner coordinate obstacle clutter name p name resp corner shape four corner top six corner point wise leave corner w ten corner point w shift unit axis obstacle sampling window shape place traversal target place along straight horizontal obstacle relatively detecting assess impact obstacle window condition specific point fact many window realization obstacle sampling center window figure obstacle point realization obstacle within clutter ccc disk center denote circle obstacle disk solid show field walks take ard total traversal walk avoid traversal happen traversal length long walk outcome traversal unit b gray reflect color indicate experimental treatment clutter obstacle window treatment level statistically significant traversal length window clutter treatment combination run consist pattern clutter ard walk runtime average simulation core processor speed clutter realization clutter exclude variability clutter realization clutter clutter clutter treatment combination total clutter realization obstacle combination obstacle monte realization sample obstacle obstacle background clutter type ip ern distribution distribution convenience presentation obstacle sometimes label v stand obstacle window corner coordinate obstacle short notation window window shape window obstacle p l l obstacle mention early precision obstacle obstacle combination traversal traversal clutter obstacle obstacle level hc obstacle realization consecutive level highly perhaps positively obstacle linear shape shaped distance magnitude take correlation account traversal difference treatment possibly traditionally repeat situation subject realization clutter type obstacle window obstacle number combination need obstacle clutter assumption among add repeat homogeneity covariance repeat correlation repeat besides plot residual present distribution satisfied setup clutter obstacle number obstacle level dependent try compete structure addition compound symmetry much benefit compare repeat test increase measure setup usual pilot study repeat clutter obstacle setup clutter realization repeat clutter realization setup clutter clutter realization obstacle traversal obstacle stand degree numerator denominator ii analyze obstacle obstacle factor repeat level usual setup clutter obstacle set setup variance autoregressive compound cs cs variance treatment covariance treatment cs setup imply usual structure combination unique ar assume distant obstacle obstacle factor treatment measurement treatment combination autoregressive experimental detail autoregressive heterogeneous heterogeneous combination obstacle var structure comparison var structure selection criterion criterion overall section traversal clutter obstacle form give perform obstacle clutter investigate interaction treatment combination interaction treatment obstacle ignore clutter consider obstacle clutter neither obstacle form clutter mean trend length obstacle clutter obstacle type different clutter type likewise traversal length obstacle form average obstacle mat ern cluster tend shorter traversal regular clutter tend long traversal length clutter tend traversal clutter traversal obstacle average order p shape shaped obstacle I obstacle obstacle interaction obstacle type obstacle number obstacle obstacle trend plot figure significantly reasonable traversal length level compare obstacle p shape form traversal length obstacle shape obstacle form trend increase peak shaped length concave disk obstacle clutter decide boundary often reduce traversal length avoid cost concave obstacle occur shape obstacle traversal length occur obstacle form traversal occur form obstacle short traversal obstacle obstacle ignore clutter type clutter plot compare value clutter traversal tend increase obstacle increase length obstacle traversal traversal length mat ern clutter type traversal length present short p p occur hc initial overall compare length pair profile plot discussion defer pt pt treatment w p hc ern w v hc hc shape v hc hc hc w hc hc hc hc investigate test obstacle type obstacle background clutter background clutter significant interaction obstacle number obstacle obstacle test effect obstacle form clutter shape form length increase shape obstacle concave trend obstacle clutter v shape length traversal length combination present traversal length traversal occur treatment type traversal traversal length obstacle traversal occur shape obstacle short traversal length occur form traversal length occur traversal treatment short traversal occur shape traversal obstacle traversal length obstacle traversal occur shape obstacle traversal occur p traversal length traversal occur obstacle form occur shaped obstacle obstacle number short traversal obstacle form short occur w traversal traversal obstacle traversal length shape obstacle obstacle short traversal treatment traversal short length occur obstacle form occur obstacle short traversal shape short traversal length obstacle traversal length occur shaped obstacle form short traversal traversal treatment traversal length obstacle traversal shaped obstacle form obstacle obstacle clutter type obstacle type obstacle obstacle clutter obstacle p obstacle obstacle clutter level test effect obstacle level traversal length different type clutter obstacle level obstacle obstacle obstacle clutter type obstacle type clutter type obstacle interaction clutter level clutter type level obstacle interaction obstacle type clutter compare main obstacle type clutter traversal length clutter obstacle interaction clutter obstacle reasonable interaction obstacle obstacle clutter effect obstacle type obstacle number obstacle clutter traversal length different obstacle obstacle level clutter traversal length clutter type obstacle type obstacle traversal length obstacle short treatment hc traversal length tend increase obstacle short traversal clutter type traversal occur mat ern clutter obstacle clutter obstacle traversal length tend shorter cluster clutter treatment length v treatment shape obstacle form short traversal length occur ern clutter shape short traversal length mat clutter type obstacle type traversal length clutter traversal length clutter sort shape form occur length hc treatment type linear obstacle short traversal length occur mat ern shape short traversal length clutter shape obstacle form short traversal obstacle traversal occur clutter traversal linear although clutter tend traversal mat ern poisson shape obstacle traversal length sort p short obstacle traversal length clutter p shape obstacle form short clutter short traversal length mat ern clutter shape obstacle short traversal occur clutter obstacle traversal length mat ern shape obstacle traversal length shape obstacle form traversal shaped obstacle type except clutter mean linear shape clutter mean traversal sort order desirable length reach clutter know clutter type actual clutter clutter type determine obstacle traversal refer well henceforth good type clutter shape length clutter form overall shape ern clutter shape pt clutter shape v overall shape p p shape obstacle obstacle table traversal length clutter background var obstacle form shaped obstacle form treatment obstacle obstacle type compound symmetry un ar autoregressive perform clutter poisson clutter mat clutter clutter clutter assume var mat ern aic clutter cs agree clutter var good yield aic likelihood ratio significant follow pick parameter difference type freedom criterion aic likelihood compound symmetry autoregressive ar autoregressive un small aic clutter un ar pt clutter un ern clutter type cs ar type un ar clutter type cs un pt un ar ptc pt clutter clutter ptc pt traversal mark bold type traversal length significant intersect vertical indicate clutter describe shape shape obstacle shape w shape shaped recommend v obstacle obstacle slight advantage otherwise low cost recommend length obstacle type recommend clutter shape obstacle obstacle recommend length decrease shaped shaped obstacle length v shape well significantly restriction slight employ shape obstacle form recommend well decrease shape obstacle form shape obstacle recommend l l v v mat ern v cross clutter obstacle mat ern clutter traversal occur obstacle pattern clutter mat ern obstacle obstacle obstacle scheme localization considerable scientific engineering community cite operational wherein spectral examine location potential algorithm call appear disk shape clutter coordinate scale shift clutter disk center inside take disk simulation environment inspire ard data traversal circle see cc inspection clutter clutter consider instead look realization clutter concentrate away investigate place use obstacle place scheme long traversal length actual obstacle obstacle traversal length traversal occur obstacle traversal length obstacle traversal length placing scheme compare realization replication traversal clutter realization use compound symmetry length obstacle obstacle length obstacle traversal significantly obstacle obstacle shape traversal length obstacle correspond mean traversal length shape obstacle obstacle w obstacle significantly different shape shaped form shape obstacle form shape range insight clutter pattern type combination obstacle obstacle number stand traversal obstacle type obstacle interaction obstacle type obstacle obstacle level trend plot traversal type number level obstacle form instead obstacle c linear shape w shape p trend traversal length shape obstacle form traversal length obstacle number shape traversal trend window shortest v shaped window length occur occur traversal length occur shape obstacle traversal occur p w treatment length v investigate obstacle obstacle obstacle well profile obstacle obstacle obstacle form obstacle stand shape w obstacle traversal length obstacle obstacle obstacle level length test obstacle obstacle find obstacle obstacle level trend significantly reasonable effect obstacle obstacle interaction obstacle type obstacle figure significantly compare effect level traversal length obstacle traversal length treatment present short traversal length tend increase obstacle short occur type obstacle length length obstacle decrease mean traversal slightly however reach short occur type shape pattern decrease traversal length stay stable short occur type length occur traversal stay exhibit concave concave type obstacle obstacle level traversal present traversal treatment short traversal occur w treatment occur type l treatment short w length occur treatment occur treatment occur treatment length occur treatment type length occur obstacle traversal v shaped obstacle furthermore traversal length concave trend overall obstacle multiple obstacle obstacle traversal length obstacle overall shape w shape traversal length obstacle obstacle difference family plot notice linear obstacle form shape shape furthermore shaped shaped obstacle wherein place clutter traversal theoretic specific obstacle place know clutter exact clutter investigate pattern clutter explore number traversal clutter extensive world system setup three measure treatment factor obstacle traversal choose furthermore measure flexibility different correlation homogeneous poisson mat ern hard core point total obstacle sample clutter shape extensive analysis cluster traversal long short obstacle tend reach increase traversal concave trend traversal increase disk optimum avoid window become obstacle obstacle form short occur p obstacle traversal length shape shaped obstacle tend especially large clutter obstacle traversal v shaped traversal obstacle obstacle form close start occur mat ern clutter clutter obstacle traversal tend linear obstacle get among traversal long window small linear obstacle window close obstacle e well shape window close conclusion valid different clutter obstacle window mark cost clutter type nonetheless environment clutter actual real world clutter clutter obstacle scheme world conclusion obstacle length obstacle form close moderate obstacle traversal occur shaped form point follow provide brief discussion several inherent know distribution clutter whereas clutter update disk mark accordingly assign mark fit overall clutter assign conjunction framework incorporation clutter account clutter mark sample obstacle mark obstacle disk center poisson obstacle window overlap argue ideal strategy sensible true obstacle disk reason choose poisson obstacle disk obstacle marks disk true obstacle information actual status crucial ard currently mark dependency instead poisson give disadvantage leave ard dependency obstacle disk center ard currently art presence capability heuristic guarantee yet challenge computational traversal attribute short ard algorithm benefit investigation ard leave future extended type obstacle problem know background place place oppose randomly inside likely term know sensor technology specific area obstacle disk location obstacle disk challenge would false traversal valuable comment suggestion flow article matlab code university obstacle wherein traversal traversal disk shape disk obstacle disk obstacle place traversal clutter traversal theoretic know clutter exact clutter traversal repeat obstacle obstacle place clutter spatial combination clutter becomes cluster short hand follow case datum application vision problem problem introduce graph theoretic considerable attention xu article consider modified version sized need start target disk shape refer henceforth brevity place another clutter clutter piece etc place clutter traversal disk disk obstacle traversal option disk obstacle disk disk clutter execute cost overall traversal assume obstacle location obstacle clutter disk en minimize total traversal give clutter maximize traversal markov computable show nonetheless efficient rd efficient heuristic provably possible complete spatial true clutter broad detect spectral field characteristic point distribution knowledge placing maximize traversal article study comprehensive limited investigation obstacle scheme clutter type various insight obstacle traversal address two question obstacle maximize might optimal way place clutter type analysis measure variance setup measure treatment clutter disk center homogeneous process ern clustered experiment obstacle process clutter window shape cost radius obstacle clutter efficiency adjacency discretization adaptation rd lattice rest manuscript organize formally ard clutter I clutter simulation clutter clutter obstacle place disk obstacle place obstacle equip assign traversal disk disk open fix radius short arc stand complement agent option cost pass
even accept accept unit steady sample score sample high track graph proceed accept consistent graph iteration generate new node e change prior effectively characterize relationship causal relationship pair prior existence scoring follow equation edge multiply influence graph thus large probability graph sample meet interface user unlikely edge bias convenient iff iff satisfy around empirically impact mention follow cubic plot clear gpu refer gpu figure gpu implements part remain part algorithm handle cpu cpu gpu score gpu gpu blocks grid access share global gpu gpu architecture architecture provide cache modify stream sm contain core feature core integer per memory interface support gb sm call sm unit execute gpu implement assign evenly block compare number assign set evenly block local score compare maximal score need assign parent predict local score convert subset indexing problem index number regular consider limit subset index subset subset index subset indexing gpu functions recursive parent parent composed choose update combination give without count straightforward combination one see begin get obtain large combination base parent set gpu store global storing suppose ram gpu ram perform gpu operating maximal implementation gpu scoring speedup score implementation gpu different gpu implementation significant detailed gpu implementation together acceleration rate list acceleration acceleration switch gpu gpu per gpu speedup sec gpu apply cell alarm c preprocesse runtime runtime graph sec gpu sec gpu sec gpu cpu mention gpu base implementation still runtime gpu runtime node part subroutine primary preprocessing runtime runtime runtime parent second parent second partial parent second also compare implementation generate parent implementation generate implementation possible consume runtime acceleration almost generate speedup time empirically accuracy receiver characteristic roc introduce measure roc true rate fp rate fraction positive give observe negative point accurate try curve close corner highly accurate small figure leave generate knowledge edge point assign interface remove without knowledge third add generating assigning remove add prior knowledge use generate fourth become generate h realistic contain noise state flip realistic show roc good acceptable noise tolerance network bn gpu traditional gpu speedup iteration gpu accelerate bn fold demonstrate highly one propose method scoring add enhance accuracy score evenly gpu implementation take parallelism scoring accelerate part work accelerate preprocesse gpu like thank wu author underlie gene pathway combinatorial nature carlo combination still mcmc purpose processor purpose graphic processing implement novel assign acceleration serial use incorporate prior component scoring search able inference big current gpu mcmc priors bayesian causal relationship direct work characterize relationship matching graph use score still accurate maintaining require datum part prior enhance accuracy significantly search add aim knowledge another integrate bn novel easily add existence edge nevertheless intensive demand hardware field gate array unit gpu gpu highly exploit parallelism novel gpu gpu background learn bayesian performance conclude paper set via acyclic node direct causal represent conditionally independent parent point condition variable write node influence parent instance figure parent determine figure common model model bayesian variable expression expression gene mechanism continuous structure experimental type observational observe experimental individually observational require sample complete due super network huge sampling explore small sampling topological dag node dag parent example topological node possible graph table list due reduce combination compare sampler advantage order learning aim explain graph prior q serve complex hyperparameter dirichlet refer parent refer maximal parent plot preprocesse well accept strategy randomly select order order subroutine section start preprocesse include initialization generation score part heavily score use accord consume need score parent store hash node parent later parent need strategy fold speedup experimental node parent change change comparison well
reference marginal able recover statistic observe isolated fix marginal recover tree w reference conditionally provide lemma invertible observable exact matrix version eq perturbation uniformly relate eq rr lemma r lemma apply previous node event component bound fix least ba pd give bind u define k provide statistic g h r inner ai di matrix uniformly simplify pairwise liu consider consider convention choose g improve improve fix substitute result bind mixture relate mutual relate entropy ix hx recall function bound mutual decomposition bound py iy liu see tree liu previous exact denote neighboring follow upon approximate separation function path vertex convenience py markov limit parameter remove claim correspond singular great subgraph fact similarly thus v simplicity isolate define b base u invertible u root close regime proof imply absolute distortion observable desire version slightly bound correspond induce pa aa large xx u u real k ki ia real eigenvalue r k ki unit column eigenvalue distinct j satisfie I I invertible property conjecture claim observation microsoft research new microsoft mixture discrete mixture variable markov propose mixture mixture vertex observe mixture mixture prove correctly maximum complexity scale locally correlation decay offer represent structural quantitative relationship represent natural vision financial amenable inference propagation recent computational requirement see brief simultaneously analyze model think fix set depend applicability change influence recent provable variety paper combine incorporate relationship parsimonious perform propagation graphical model base expectation maximization scale poorly dimension issue guarantee graphical offer include mixture graph product distribution section restrictive since direct significant incorporate mixture aim mixture offer tradeoff fit inferential tree since inference reduce mixture graphical tractable observe learning model graphical base stage correspond union propose union independence efficient node hold mixture sample marginal mixture hide component develop mixture observe conditionally variable adapt triplet suitable estimate marginal mixture degeneracy tree estimate component marginals liu liu span mutual recover component mixture complexity scale pair union component graph extend family include tree graph graph local significantly scope mixture depend proof correctness analysis careful use spectral success setting incorporate local dimension principle node another limitation however impose learn distribution know singular rank least hard another variable latent however mixture model observe variable well provable trivial product significantly advance scope selection outline variety recently focus deal estimation mixture separate recently recovery separation constraint complexity component call complexity scale polynomially impose outline method employ discrete product general triplet pairwise thus applicable selection seminal finding model establish span mainly efficient provable g graphical approach classify convex mixture mostly work tree mixture direct acyclic term use asymptotic decomposable recently learn condition think accord realization setting require component order test distinguish roughly impose although overlap moreover allow mixture allow variable latent estimation mixture operate significantly graphical finite cardinality say pmf local markov include say global property disjoint p property positivity condition henceforth model respect markov positivity markov iff clique z clique clique serve normalize otherwise allow potential discrete graphical observe variable mixture vector mix draw variate unknown know draw decompose draw stage graph accomplish series test special give graph respective mixture via spectral liu mixture extension v sample u satisfy consideration order property union yet respect observe union property recall node together perform test group graph crucial depend relax neighbor instance marginally independent value approximate separation set separation finite recover graph union component markov result graphical graphical output least test graphical work base alternative addition previous procedure graph component graph graphical propose test later recently product r separate variable independent py w form set proceed consider triplet full denote matrix u u estimate eigenvalue v h py I prove recover mixing learn perform triplet remain triplet hide variable triplet share node triplet order triplet decomposition learn order adapt simple observation consider three node removal line w g configuration successful estimate fix describe triplet obstacle triplet decomposition product previously fix triplet additional graphical conditioning unchanged h imply hold operate triplet generalization distribution require place lemma pair triplet dimension see marginal scale see addition follow success subset probability exist node isolate e matrix uniformly manifold r impose also learn require alignment label decomposition discuss component marginal marginal align estimate various characterize provide mixture component least appendix variable op p scaling note special case distribution well approximation estimate marginal mixture impose standard degeneracy existence unique approximation liu exact iy iy connect number n condition mutual liu model represent replace span denote error likely make exponent liu correctly mutual within liu structure marginal spectral succeed recover complexity recover tree liu discuss op approximation graphical technique previously product provable method establish polynomial complexity variable guarantee mixture continuous mixture acknowledgement author uci award fa pairwise reference isolated estimate algorithm carry triplet neighborhood separate fix upon spectral marginal liu
drive ergodicity volatility volatility stock return asymmetric volatility effect ex current commonly investigate leverage stress leverage exhibit asymmetric considerable conditional specification capture asymmetric exponential manner vary family volatility framework volatility allow volatility volatility process originally develop white al framework innovation price basic stationary uncorrelated gaussian time taylor asymmetric model suggest capture return asymmetric behaviour absolute incorporate li model autoregressive depend sign volatility excess residual basic normal skew distribution skewness heavy specification skewness return volatility initially interpret return future return decrease develop martingale property heavy tailed leverage use volatility generalise financial fact include excess volatility sn skew st non specification account leverage asymmetric central hereafter asymmetric hereafter asymmetric hereafter sn finally asymmetric skew hereafter st excess skewness multivariate instantaneous however technique inefficient base specification multivariate estimation volatility set solution normality hasting algorithm provide reliable mcmc associate importance ergodicity essential geometrically addition rapid ergodicity existence limit monte geometric applicability theorem aim ergodicity general incorporate et nonlinear measurable mutually identically distribute observed trend volatility skewness nonparametric series finite identically random mutually adopt strictly irreducible observation past function compact set iid restrict follow assumption iid variance moreover fourth conclude assumption process chain
minimizer inactive active e minimizer inactive duality non whose computed threshold follow operation unfortunately case directly characterize relation equation give median finding treat unknown approach guess say great method design triple q optimal follow lasso projection result supplement satisfy monotonicity employ description modify optimal restricted accelerate first observation element less secondly decomposable involve routine approach compute thus defer supplement brief advantage explain interpretable feature group interpretable nonzero group advantageous especially interpretation vanish incorporate prior amount large may simplify certain permit tuning counterpart note although model propose ideal theoretically moreover computation efficient perform selection conduct pc cpu memory operate dc programming accelerate propose nonconvex evaluating algorithm projection direction adapt formulation supplement generate radius ball fair record run admm terminate iteration stand admm summarize average algorithm replication ccccc admm next demonstrate toward end toolbox generating report distance euclidean convex unique termination relative difference consecutive less terminate k accurate problem great burden one method yield accurate projection one evident follow I standard partition truth possess nonzero group enhance nonzero accord test lasso choose validation conduct choose parameter metric error estimator capability recover structure table well well group glasso eeg genetic eeg electrical fluctuation place technology widely diagnosis genetic encode eeg electrical activity region utilize potential stability place hz therefore naturally divided group lasso adapt validation select lasso logarithmic sample scale select set table although underlie reveal fact accuracy lasso group selection theoretical analyze dc accelerate efficacy propose validate synthetic investigate extend modal multi task promise direction pt pt minus supplementary material sparse group nonconvex inequality likelihood ratio g last towards c pn constant j multinomial theorem j j j c g ig utilize suppose differentiable derivative differentiable addition gradient proof appendix infimum complete restrict c vs get gradient linearization define l x couple augment utilize minimization optimal possesse amount solve projection multiplier summarize note value implementation whenever fast u w x intersection carry projection alternatively square projection p cm plus minus engineering evolutionary institute university demonstrate parameter paradigm motivated application contribution statistically nonconvex sparse reconstruct oracle parameter efficient propose nonconvex compare synthetic achieve high past decade sparse extensively investigate statistical property possess certain explore feature propose ultimately detect site justify sparse encourage feature simultaneously computation suboptimal study nonconvex truncate potential superior formulation preferable theoretical nonconvex ideally wish follow condition define hellinger density dominate b space conclude regard global
point score dataset rapidly c next base area roc curve auc true rate false positive time change strategy peak regard specifically alarm true alarm remove alarm filter change point peak gradually illustrate curve random seed describe auc good test significance level tend perform value plot specific consistently well accord unchanged moreover reason medium evaluate dataset activity sense evaluate consecutive singular eigenvector reasonable evaluate target sequences subspace column span series value ar auxiliary ar ar change score log ar criterion descriptor signal set median distance heuristic kernel indicate outlier activity human activity collect change segment behavior arbitrarily decide orientation axis series change score plot score trend behavior change time change recognize regard roc curve describe tend next national institute environment speech offer segmentation manually annotate annotation final evaluation experiment signal score still clear speech roc curve auc significantly outperform si std c ar std apply twitter twitter interface track popularity keyword occur twitter frequency perform change detection hope correlation change keyword change evaluation wikipedia entry world change occurrence development news platform soon reach point score peak visit stock one year low change spike cut peak formulate two consecutive comprehensive art ratio key block various analytically possess optimal numerical novel divergence paradigm recently possess parametric convergence artificial real world dataset activity speech demonstrate promise density two observe margin reason decide model segment affect hyper discover challenge investigate show however advantage density translate performance detection clear intuition score point even attempt dimensionality change practical usefulness compare expensive validation procedure analytic method iterative detailed comparison improve focused point represent testing provide threshold determine recent often consume setup bottleneck recent report world event show twitter challenge along include interest acknowledgement sl program ms support project mm mm mm change series objective point discover lie behind time series sample method divergence accurately efficiently ratio human twitter message propose method keyword change kernel change series call attract researcher community decade depend delay point target application immediate response robot reaction period accurate detection certain delay change genetic analysis demonstrate change compare interval move typical alarm becoming significantly outli attract pre trajectory past interval dissimilarity subspace span column model lead successful detect south explain rely auto parametric accurate know density vice versa know density density estimation ratio develop g report outperform change point ratio promise advance fold idea square notable advantage analytically achieve optimal robustness experiment base change second improve change employ ratio ratio unbounde basic smooth bound possess convergence plain imply base change compare rest describe review artificial human activity sense twitter conclusion future formulate subsequence subsequence represent transpose treat series sample information incorporate start let segment strategy certain plausibility point specifically dissimilarity likely point change rest review section algorithm kl kernel cross divergence minimize ignore irrelevant second term constraint purpose constraint come non negativity density unique global solution gradient achieve estimator divergence kl estimator apply recently call pe different ratio fit squared specifically minimize constant substituting state approximate average parameter element easy confirm estimator give pe construct pe divergence divergence low conjugate pe notable possess stability robust experimentally compare infinity convergence govern sup overcome q p relative reduce plain tends smooth one confirm plain ratio unbounded contribute use divergence eq separately approximate parameter square estimate ratio approximate expectation density
bind perform confidence choosing choose os mix chain diameter chapter useful periodic taking generalize periodic transition give obtain distinct state converge respective mix aggregate state period theorem latter adapt period try reach arm cycle periodic force learner deal overall claim horizon state space arm exploration determine thus execute episode many sample action hold rs show mdp plausible mdps lemma argument assume diameter first state repeat ks ks sect k ks state term ks rs ks ks ks get color let ps split cf reward contribution pair concern second show episode analogously episode sect obtain evaluate sum episode bound k sequel ac evolve learner action know arm show index necessarily decide arm produces accumulate follow traditionally reward produce probability reward high setting measured extension set govern trivial set often natural transition name bandit bandit cognitive e bandit channel available sense arm simple sense also example illustration much arm two remarkably notably well difficult non bandit regret measure respect bandit take arm regret arm obtain useful compare regret allow regret optimally nontrivial depend diameter bias vector express obtain well good diameter try turn far regret bound problem latter consider armed hoc optimal policy regret unclear exploration exploitation finally bound exp hardness reward arbitrarily chain mdp horizon arm state diameter eliminate however chain arm low improve arm irreducible state know discuss periodic deal space know neither transition probability reward choose observe receive transition matrix learner compete know mean observe minimize average select selecting set eq somewhat next nature demonstrate immediate arm index arm bandit close armed state cc typical look average optimal policy otherwise close natural interpretation cognitive either device channel want channel typical policy reward stay immediate armed bernoulli reward know arm obtain arm switch arm reward choose arm observe reward arm sometimes appeal form maintain policy sample arm index seem independent seem evaluate independently work I case ucb motivate two arm index policy markov also intuitive index suboptimal index right start dash reward reward r observe optimal choose l give give reward probability subsequently arm miss reward l clearly behave consider bound eliminate periodic bad detail arm total depend arm clear intuitively wrong cost mdp step chain evolve diameter mdp recall number time step mdp j consider distinct action mdp markov power transition iff arm mdp correspondingly consist regret measure optimal mdp problem structural close arm tb observe initial episode let action visit prior number time color visit estimate plausible mdps colors mdp value optimistic tr could aggregate aggregation replace reward probability transition state whenever transition probability aggregate helpful parameter know aggregate guarantee mdps colored generalize structural structured set color function ps aa identity mdp aggregate structured mdps modification confidence interval plausible episode optimistic average colored action respectively adapt episode termination basically episode end color color tb confidence execute colored parameter mdp paragraph structure I j j
even specific inequality root commonly subsample behave uniformly provide quantile quantile q understand two side consider replace respectively differ hold pointwise notion way suitable well ensure compatible sufficient closeness ensure closeness discussion closeness metric ensure closeness quantile heavily relate closeness metric coverage usual argument asymptotic subsampling rely show tend construction precisely subsample use construct consistent construct contrast confidence pointwise sense less infinitely often likewise satisfying desirable analogous reason test fail finite course point nontrivial confidence sufficiently rich reason restrict test instance weak also relate complicated setting shrinkage ill develop independently discussion page whether asymptotic relationship subsample result verification moreover converse requirement relationship fail nominal coverage statement fail essentially bootstrap asymptotic univariate cumulative arise autoregressive subsampling inequality statistic discussion statistic uniform asymptotic validity differ subsample fail pointwise valid multiple testing despite area appear asymptotically valid proof result theorem supplementary material result uniform law number aforementione value valid describe subsample integer tend satisfy th generally toward feasible explain remark confidence tend infinity statement true iii bx nx nx bx bx nx deduce strong hold argument validity tend infeasible without feasible region case test nan hypothesis construct apply part conclusion worth though state root applicable especially hypothesis next feasible estimator root nx integer tend satisfy theorem replace tend root q sequence normalize limit integer statement theorem root computationally show obtain result correction factor assume establishe converse tend infinity bx nx p bx bx nx pt let variable root goal approach lie necessary require random true iii nx nx nx n hold satisfying deduce replace replace verify argument pointwise bootstrap construct consistent distribution q metric compatible yield word measure metric lemma whenever nevertheless coverage fail provide equal whenever event p n similar establishe iii kolmogorov coverage proceeding frequently correlation usual diagonal standardized expression variable reflects center normalize deviation nonparametric construct rectangular region root eq form suppose positive integer tending replace suitable restriction generalize root real suppose replace moreover continuously theorem need distribute proof requirement nonparametric consider root differ one sense hold fail bx nx central lem limit hand generality corollary define replace inequality generality hypothesis sequence p problem recently receive consistent level sense tending argument theorem establish test multiple hypothesis distribution sequence hypothesis control accord eq eq stop reject integer possible extend analysis straightforward nan empirical random one confidence natural true replace variable denote natural suppose eq q integer tend remain nonparametric construct example counterpart let generalize way generalize continuous z moment inequality propose testing alternative level adjust quasi follow understand dimensional matrix construct refine moment illustrative simple theorem distribution root generalize straightforward fashion statistic theorem way construct behave large versus hypothesis q bootstrap counterpart extend sided testing cumulative
development dna dependent specification domain bind bind bind activity mkl anti rna mkl mkl mkl protein bind activity protein bind bind dna activity bind nucleotide bind dna direct activity activity bind bind bind bind nucleotide bind dna transition interaction synthesis dna dna dna break end cell dna synthesis dna nucleotide process cell cycle dna cycle environmental indicator rate ie distribution inverse hierarchical conditional number initialize hyperparameter universit de france france environment act large number weak effect way identify correlation environmental integrate signature adaptation algorithm mix probabilistic unobserved genetic variation infer background structure evidence factor number development gradient adaptation selection role fundamental biology intensity environment effect e adaptive cause evolve trait advantage environmental involve achieve wide dna specific signature selection population spatially adaptive detect compare among marker genomic alternative way investigate signature especially beneficial identify environmental population quantitative trait selective act individual phenotype reflect population evidence detect environmental background genomic variation basis environmental adaptation gene drift correction consider association environmental positive study allele account covariance assess adaptation allele environmental explain neutral build face need identify neutral genomic imply lack reject correlation environmental genetic effect factor background structure perform extend recent statistical approach deal thousand rapid control population autocorrelation estimate detect adaptation human allele genomic size simplicity nucleotide allele environmental could environmental association environmental background level response vector deviation prior zero variance gaussian prior environmental variable model factor separate neutral variation explain environmental factorization estimate identify score loading pca factorization probabilistic factorization analyze genetic environmental factorization row decompose data decomposition vector minimize loading simulation choice application estimate population genetic essentially estimation score loading gibbs sampler product speed scale snp addition algorithm environmental retain score absolute preliminary experiment define equation find effect quickly individual fold recover latent factor alternate use deterministic gibbs check consideration incorporate population proportion principal study approach program start individual q score equal identity consider equivalent equivalent vector depend empirical point main conservative principal component exist individual series generate replicate generative test set report distribution conservative moderate number test produce false association represent high genetic generate replicate error model rank generate regression first matrix environmental effect check similar distinct initialization report quantile absolute error square indicate pc regression hide absolute algorithms pc quantile error shift fold poor performance increase series method lm glm pc model enable ms ten adjacent include individual test implement find run compute genetic neutral lm factor range test lm test produce result latent pc value much find lead heat stress genomic dna individual population refer genome diversity centre genome cell panel array filter remove include file weather year period temperature maximum month variable score factor additional total snps great disease trait association rs associate significantly correlate notable disease synthesis activation production gene heart involve rbm development base factorization unify effect environmental genetic environmental allele include genetic environmental genetic specie environmental statistical gene adaptation development summary environmental association detect species genome however implementation example result positive association mix regression estimate suggest utilize separate neutral variation selection phenotype marker neutral sometimes distinction explain extremely difficult neutral background use actually correlation environmental allele frequency genetic much fast analyze computational program outcome take validation computationally theory select find lead conservative estimate still restrict approximately suggest cluster specie test directly fine motivated trade accuracy future development develop numerical base algorithm confirm discover link activity heat stress association gradient example identify gene color associate contain snps height synthesis disease snps involve heart brain confirm correlate evolutionary example result support soft ever genetic throughput sequencing model contribute toolbox landscape new environment association source code program available web work support nsf lm cccc fp lm glm evolve specific response stress heat temperature stress protein disease response stress bind probable bind dna dna cm rs trait color vs disease black red cancer h disease rs rs rs rs rs aggregation cr rs rs pr height height rs brain p cl heat protein cl dna heat bind probable protein alpha cl alpha sf cl nuclear bind protein group cl bind protein domain contain dna h temperature temperature month temperature month temperature temperature bp bind binding
som extend som correspondence approach consist standard computation extend som dissimilarity drawback increase propose however increase stay algorithm relies express som make kernel space version batch kernel design handle string dissimilarity combination classical version som dissimilarity approach name idea som organization describe arbitrary input dissimilarity non grid neighborhood prototype randomly random combination matrix iterate assign close prototype accord distance carry straightforwardly computable distance step accord equation dissimilarity batch call batch relational som introduce long suppose som line som close prototype non position decrease neighborhood whole neighborhood neuron vanish relational som som basis som possess several drawback organization bad visualization unbalanced batch online scan significantly batch summarize som som illustration uniform batch relational som version relational som identical initialization available batch relational som iterations grid organization iteration initialization som converge organization iteration initialization organize minute batch minute processor ram relational som presents application line relational dataset deal numerical surface categorical first relational som simulate roll performance figure simulated distance run different algorithm uniformly first geodesic algorithm perform map unfold som som dissimilarity geodesic important unfolding square relational som project roll separate moreover roll completely rather grid heavily test rectangular grid result clearly batch som grid line som median som som use dna dna comprise molecular bioinformatic aim identify biological assigning specie accord publish dna hand specie discover species neighbor construct dendrogram large become rapidly sensitive unsupervised learn dna although use helpful cluster sequence hundred thousand site specific distance dissimilarity euclidean dissimilarity median som test amongst median som provide encourage main drawback organization highly among allow empty existence group acknowledge som mixing allow detect main specie labeling unsupervise address issue distance nearest cell distance neighbor vertice two distance vertex cell specie diversity radius proportional cluster use graph book us amazon com edge co book accord political orientation neutral extract relational som short dissimilarity figure provide network display direct color classify simplified represent density second organization graph group connect whereas
pair canonical transform fouri relation signal q unit norm u triangle q q follow supremum spin recover restrict isometry relevant full incoherence specify operator easily basis retain spin true spin sparse let mutual spin recover component spin conceptually separate mixture incoherent basis sparsity quantity minimization similar tight therefore spin yield careful spin manifold incoherent basis direction spin case dictionary comprise manifold powerful flexible conceptual image ensemble consider ensemble vary nonlinear manifold include acoustic vary frequency represent disk black disk image solid variable pose six correspond location orientation manifold manifold class construct preserve geometry randomize operator rip constant volume space ambient isometry projection operator reconstruct compressive disk component generalize signal manifold fix template define image denote unknown problem equivalently compressive demonstrate image demonstrate spin consideration recovery signal canonical spin provide incoherent rip manifold shift element corrupt spike spin projection match template simply well assume spin observe spin measurement constitute ambient figure measurement vs reconstruction plot db spin far corollary relationship interest hybrid signal rigorously analyze projection incoherent manifold recovery pair signal spin geometric criterion two disjoint incoherent operator isometry manifold computational spin present demonstrate utility spin thorough study spin future gradient count relate step isometry constant preliminary size give result choice size thresholding scenario rarely ambient concept approximate projection rigorously demonstrate spin indicate spin inaccurate clarity brevity focus attention signal belong spin conceptually sum spin require manifold incoherent operator isometry component manifold spin component matrix reconstruct affine rank attract attention key rank matrix incoherent two manifold matrix vice phenomenon quite challenging need lemma approximation cs nsf n nf nf thank valuable early manuscript department electrical computer engineering signal signal manifold ambient incoherent spin order project method nature recovery measurement spin provably recover manifold incoherent operator certain restrict isometry spin low dimensional match exceed art observation problem instance limited possess effort instance advance separation compressive affine principal write differentiable manifold give linear measurement measurement objective signal also numerous arise identifiability simple measurement noiseless operator observe fundamentally ill pose manifold unique signal situation operator possess nontrivial particularly ambient issue order two issue nonconvex non numerical descent successfully convex optimization design linear type signal prior successive incoherent manifold spin nonconvex possibility spin provably recover true require incoherent restricted rip formally statement recovery incoherent satisfie rip observe b b k b propose iteration z manifold onto component manifold play stability manifold stable operator detail spin arithmetic essence extensively hard projection spin spin project algorithm pursuit generalize mixture nonlinear manifold spin component manifold spin exhibit strong comparable art despite nonlinear nonconvex nature certain special sophisticated higher strong stability guarantee pass amp lagrangian recovery appeal spin conceptual simplicity plus generalize nonlinear manifold manifold paper convention vector quantity appear unless product interested manifold informally manifold signal applicable identify capture signal locally continuous possibly signal manifold riemannian manifold example define include signal excellent refer analysis core orthogonality isometry measurement availability incoherence element generalization normalize simply incoherent manifold direction incoherence manifold orthogonal inequality direct decompose strict uniqueness b b incoherence unnormalize last arithmetic henceforth gm impossible unless em direct sum lie incoherent manifold incoherent b rearrange desire address restrict isometry manifold matrix isometry property rip belong generalization approximation sense matrix rip traditionally generalize manifold isometry certain manifold construction rip dimension ambient discuss section define informally closeness term euclidean nonconvex manifold ease exposition henceforth signal operator crucial signal precision arithmetic uniquely define nonconvex manifold admit operator example signal length canonical subspace projection approximation efficiently simple thresholding describe onto incoherent spin view generalization recovery key spin formulate measurement b return demonstrate spin possesse uniform comparable approximation broad signal theoretical describe spin incoherent measurement isometry manifold observe noisy measurement spin moderately sized explicit spin output z spin true iteration positive paper informally recovery fine precision spin availability operator approximate multiple mechanism projection spin z note implication signal isometry constant use spin spin iterative recover manifold match guarantee automatically mild condition unique full spin proof analyze sparse define error spin incur th current iteration z k k b z k k b z take adjoint inequality
ti remark scale laplacian definition ti desire focus convergence obtain establish consensus agent average dynamic obtain boundedness iterate agent successive refinement I correspond centralized action elsewhere tc tn ni j jt j satisfy q construction contradiction argument action pair evolve scale version obtain construction conclude action event imply conclude hold often together establish action inequality derive q hypothesis lemma suffice following proceed instant induction hypothesis eq property induction obtain q establishe reverse direction hypothesis establish learning establish update sequence agent iterate merge establish reach consensus evolve rewrite whose adapt evolve quantity adapt nk kk state asymptotic average iterate see agent innovation take generate pair trajectory depict solid centralized factor uniformly agent distribute reach consensus readily reasonably importantly centralized asymptotically limit centralized trend equivalence asymptotic convergence loss centralize attribute consensus enable agent essentially track investigate reinforcement setup sensor building entity compute differently environmental setup collaborative competitive learn stationary discount focus approach processing mean consensus strategy assumption communication formulation action set mixed stochastic markovian state general applicable broad memory low dimensional state indicate centralize argue per distribute scheme asymptotically asymptotic consensus convergence state impose action approach simulate centralize practically motivate concern perfectly observable agent instead act responsible proposition result htb department engineering pa mail edu consider multi agent process agent differently instantaneous cost control distribute reinforcement setup state consist horizon discount propose distribute agent sparse cost agent yield network agent analytical interactive dynamic uncertain markov control influence random instantaneous incur process collaborative agent network minimize horizon discount instance resemble control environmental dynamic spatial temperature sensor application building reference minimize measure sense desire important scope agent correspond social entity dynamic interest pattern policy controller shape economic growth scope formulation limit scenario management agent motivate learn instance valuable practically involve information include agent agent instantaneous cost bellman generate sequential trajectory rely trajectory implement control direct various exploration technique would cost instantaneous stage cost locally agent cost central resource bit communication medium fully distribute agent computation extensive multi survey formulation range stochastic call investigate viewpoint formulation somewhat setup optimization unique stage local agent cost key additional instantaneous realization stage global cost observable specifically instant instantaneous cost whereas often formulation decentralize stage cost time comparable often decentralize controller emphasize require literature locally cost mutual neighborhood communication pre decentralize specifically perfectly access contrast decentralize controller control action perfectly agent limitation agent simultaneously receive instantaneous potential consensus communication one cost appropriately communication consensus one agent functional inter exact instantaneous appropriately potential suitably design optimal performance network agent consensus control minimal generic lead scale markovian analysis technique broad class problem viewpoint centralize explicit consensus distribute scenario problem network literature optimization scenario broadly network goal static objective agent aware view extension environmental dynamic model static agent obtain minimize contrast scenario formulation transition state sequentially cost rest set sequel learning setup formulate present propose formalize inter agent communication intermediate distribute present learn section centralize section finally conclude component wise denote use correspond cone denote denote definite matrix denote vector zero symbol use dimension operator standard euclidean matrix denote existence object correspond respectively inequality object interpret surely state inter agent communication denote agent communication pair exist loop graph li pair edge matrix diagonal definition positive laplacian matrix order eigenvalue eigenvalue algebraic connectivity value detail control markov denote generic note bold symbol govern satisfy agent often reduce former control proper augmentation apply satisfie evolve dependent modify policy horizon discount factor global agent mdp concern horizon associate provide centralized programming denote bellman readily see contraction imply start obtain form basis classical successive value know reinforcement method lack information bellman class method stochastic call action often recover trajectory oppose rely implement direct adaptive offline simulate response far trajectory time dependency due relax context rely centralized instantaneous cost centralized obtaining expectation instantaneous cost agent location feasible due agent medium motivate learn collaborative local communication scheme agent learn counterpart state process agent learn scheme base stage formalize agent impose requirement characterize locally conditional require condition random adapt formal trajectory control markov accordance obvious measurable give instant sequel characterize agent locally message formalize message obtain possibly agent slot algebra agent network available local agent information reward consist locally reward information agent view require across formalism readily join I e instant induce moreover inclusion strict usually agent communication inter strict global explain fundamental exchange run lead wide eventually obtain accurate successful involve communication inter generate neighborhood satisfy distributional failure failure link failure spatially network failure time wireless motivate interference wireless failure connect stay capture assume capture broad class asynchronous random asynchronous protocol analyze fall hand event communication link exchange system instantaneous action eventually instant adopt show pair note agent simulate often impose note centralized condition u n collaborative distribute weight reflect sequence update instant refer transition reach realize slot stay maintain adapt serve weight w identification eq send process update agreement consensus innovation incorporation sense result algorithm trading impose sequence follow consensus innovation potential persistent sequence guarantee innovation weight consensus dominate innovation comment sequence associate innovation potential consensus innovation condition far effect randomness local asymptotic mixed scale
conclusion value optimisation scatter indicate probably case mean optimisation remain r orthogonality display bad negligible surprising design orthogonality criterion design exhibit term ml design slightly orthogonal orthogonality criterion unique w design criterion slightly restriction orthogonality restriction unclear design good orthogonality lead orthogonality improvement ml design slightly bad good improved combination visually plot nearest choose design worst minimal bad property crucial projective sa negligible impact response parameter omit surface onto original none project projective call situation small corresponding simulation hypercube hence projective property measure quality projective property minimum project domain use redundant design present onto histogram represent redundant easier redundant list cn show superiority redundant design bad decision number call strategy fine add coarse grain batch one produce one shot fine grain sequential possibility generation sample sufficient advantage non orthogonality course optimisation may display coarse grain sequentially produce one design shoot orthogonality property searching hypercube latter test sort optimisation orthogonality purely factorial design integration grid generation bad orthogonality last group employ modelling place suppose error sensitivity hard confidence broad design equal level discrete new sequential randomness free point figure plot distance aim show strategy start design preserve column row present design within sensitivity although bad generating show importance monotonic relate impact evaluate engineering majority monotonic quality sampling monotonic consider discrete discrete plot together correspond full design involve mutual correlation design store difference parameter plot per associated apply design accord restriction model multiply cc cc ml cn max max max max cc cc c cn max max max mean overall sa ml optimal design term variance free slightly bad design free suffer large average compare variant classify hand criterion contrary evaluate suffer sensitivity aim analytical devote illustrative benchmark optimisation represent ten bar area bar benchmark continuous discrete together cross area discrete p material lb loading concern bar material loading give thank symmetry bar group hence variable cm p material lb modulus f model maximal restriction restriction automatically specify design feasible bar bar design h anneal iteration correlation compare use consist statistic parameter independently maximal correlation multiply list ccc cc max max bar bar sa structural model cn ten bar design predict three significantly improve well ten bar criterion design balanced able among bar obviously parameter eight hence bad eight comparison ease mutual usage base sa analytical structural conclusion cn present aspect make optimisation tendency design subsequent analytical structural error difficulty optimisation nature tendency even restriction bad design suffer high variance criterion regard optimisation advantage simple computation good orthogonality prediction criterion generally restriction design ml design orthogonality sa sa bar projective criterion concern optimisation besides drawback lie necessity bayesian ml criterion common winner comparison course consume growth computational exploration task basic investigate input simulation e point design call aim available suitable sensitivity orthogonality hypercube sensitivity sa tool investigate essential part response individual system present contribution particularly aim able reveal nonlinear monotonic relationship output sa expensive exhaustive perform relate contribution present review several generation generation aspect discrete continuous possibility continuous domain beyond review literature include method generation present difficulty arise devote mutual compare projective improvement section present assessment design usage sa respectively conclude criterion assess preferred preferred need evaluation admissible input value orthogonality necessary assess parameter may preferable orthogonality nevertheless base employ one propose potential distance probably know minimal two maximal minimal use quadrature normalise dimension value space discrepancy consume ml work field regression determinant dispersion inversion negative point order become eq design illustration fix four corner domain coordinate corner value colour instance optimisation criterion idea high response add overcome problem modification imply add b need keep mind add preserve c two define manual two orthogonality cn commonly design center scale define eigenvalue generator discrepancy design review design heuristic annealing employ design concern briefly possibility optimisation particular sake clarity present four five four corner evaluate last shape way whole row grey colour colour corresponding ii detailed surface accord domain imply several concern optimisation optimisation space optimisation situation present criterion poorly another negative demonstrate criterion cn extreme corner pointing character cn evaluate corner undesirable smoothness appear optimisation criterion much anneal feature concern r corner optimal design well test quality ml
result college political book case correct block variant employ monte result obtain obtain block degree actor node connect cast member reflect actor always something impossible modularity besides block temporal line actor correlate datum enables fully scale thank point correction american b universit module length principle seek prescribed block structure maximum block simplicity yield efficient monte carlo application network actor bipartite module problem literature systems blockmodel attention drastically majority maximization derive structure inference blockmodel communities unknown infer predicate choice amount blockmodel ref generalize accommodate arbitrarily arbitrary community structure penalty monte arbitrary structure blockmodel compose block node twice far impose direct analogously becoming fix degree respectively ensemble entropy entropy blockmodel ensemble correct belong node edge analogous material overview block maximize network compatible henceforth entropy number block otherwise matrix principle necessary blockmodel eq upper necessary observer one datum loop simply multiply number take derive ref b without restriction must exactly restriction show correct direct replace equation infer unit width dl mm mi vs benchmark width mi vs pdf prescribe together infer vertical line mark c true detail pp impose diagonal structure otherwise original common express pp value criterion useful albeit recover discard serve structure mi vs pdf partition pp different grey line ref difference impose detect convex global give easy even prescribe partition block possess impose criterion ref blockmodel variant show recover fast infer hence compute resolution modularity interpretation simply modularity entirely avoid properly modify modularity knowledge improve infer structure eqs variant closed width unit mm width profile b blockmodel american network correction describe political book infer red match infer correct blockmodel circle block actor
underlie number tend hypothesis indicate l evy ad tends clearly indicate propose skewness difference fusion device analyze fluctuation phenomenon mode h mode rapid drop device operate device investigation physics diagnostic tool characteristic potential probe array behavior mode like transition reveal radial three typical fluctuation radial observe obvious amplitude position set constitute consecutive examine part evy examine tail set whereas examine datum datum statistic depict follow confidence consider tend whereas value confirm one propose indicate analysis evy ad hypothesis distribution examine skewness parameter corresponding interval add possible prove normality active research determine develop able close show law essential identify alpha simulate stable distribution work might detect evy develop procedure assess stable fluctuation measure fluctuation concern observe evy fluctuation measure stay evy index detect stable sufficiently heavy tailed pdfs low turn qualitative change important scope check device well discuss alpha l evy available law clearly combine visual fourth check method furthermore density transition occur transition gaussian statistic stable alpha stable evy stable central gaussian independent reason evy stable naturally appear determine characterize evy index stable law slowly decay law asymptotic process general four l skewness location deviation comprehensive alpha stable evy distribution quantity evy exactly motion arrival find evy central brownian walk jump heavy tailed review evy light medium evy evy field stochastic electrical engineering biology economic lot analysis evy parameter l evy stable quantile quantile evy stable law consideration evy fast come hill log log focus assume parametric whole section v address issue evy index shape pdf gaussian log scale hill evy example fig ref especially biology contain evy belief analysis inspection l test relatively rarely physical biological effectiveness application analyze fluctuation device detect l evy device sec pdfs simulate evy close sure almost visible plot moment simulate demonstrate evy outline l evy demonstrate one large number theoretical evy index number evy visible empirical totally skewed evy distribution symmetric asymptotic evy contrast difference pdfs demonstrate technique evy htb algorithm recognize evy detail statistical test goodness check idea check probable reaching analytical cumulative piecewise zero jump point usually measure either supremum quadratic well kolmogorov ks distances analytical incorporate define suitable ad test kolmogorov ks exhibit poor sensitivity ad fit tail crucial stable law testing propose ks ad goodness test match normal skewness converge skewness deviation skewness l evy stable implement various package g first stable general structured whereas favor outcome statistic reject ks ad test propose
certain diffusion chain marginal information imply rest part introduce unsupervised couple dx np xx unsupervise unsupervised form principled labeling build I partition misclassification learn train c text method search partition minimum scheme correspond labeling mention rather study misclassification misclassification error scheme misclassification error quality estimate average generalization cs conditional density prior let measurable far old underlie isotropic bandwidth one assumption prove kernel density uniformly converge bandwidth assumption probability kx dx indicator almost sure n n min f note indicate unsupervised connection nn hard give nn cost function neighbourhood q misclassification misclassification give show nn facilitate introduce cover rp r misclassification nn I em nz l x x r l eq x moreover n h h f iff cx approach nn far asymptotic misclassification soft lm lm lx n rhs nn c plug kernel density suppose eq sure e cover point z g r dd rhs suppose g min g generalize almost min kx dx vc function envelope exist open min b min g vc obtain classifier lm em class indicator bandwidth satisfy old h old follow equality convolution h follow verify misclassifie almost volume cluster boundary region separate low density separation cluster prove slight proof plug weighted lemma bandwidth rhs follow rf pi lm plug volume boundary pi ds vertex lie edge determine induced volume misclassifie plug suggest plug normalize minimum solve computing laplacian hx forward laplace backward operator moreover forward operator operator normalize capture riemannian consistent similarity reflect geometric volume misclassifie classification practical method misclassification near plug connection error plug volume cluster similarity classifier close relationship remark edu classification unsupervise manner exist misclassification error except classifier misclassification error popular near classifier unsupervise bound similarity prove recover type map close classification normalize laplacian similarity misclassification induce volume misclassifie classifier laplace set representative include dissimilarity cluster identify cluster complex lie low parametric learn classification learn max manner far
token rgb rgb rgb rgb rgb relations consist ignore computing tool graph high system signature relation positively propose spatial secondly force interpretation mechanic devise embed multidimensional rely assumption dependent capture short property dimensionality ability preserve local produce maintain meaningful neighborhood gradient trajectory sensor result superiority propose embed various reduction extract complex set popularity decade back principal component classical multidimensional recently local fisher discriminant discriminant modern enable capability reduction technique meet limitation structure hyperspectral acquisition object pose challenge set employ variance unfold embed subject good point laplacian draw correspondence laplacian laplace heat devise geometrically construct sample high widely method extend property classical nonlinear embedding extension dimensionality probabilistic widely separate lead embed distribution embed stochastic neighbor graph capture embed include distance non kullback leibler divergence nonlinear dimensionality compression visualization reduction represent manifold comparative study manifold hyperspectral reduction community problem tendency embed result increase discriminative boundary form within develop reduction visualization incorporate benefit unify framework nonlinear dimensionality form high predefine draw interpretation suggest design dependent novel embed enforce pairwise range force towards dominate force dominate range environment neighbor field importance together similar act barrier generate force map reveal meaningful structure similarity construct second local kernel signature space image characterize channel pose challenge visually exhibit overlap signature structure review force formulation formulation multidimensional field framework connection popular nonlinear field unbounded setup various force interpretation relate study biology discover pattern force analogous physical force early propose interaction maintain stable developed discuss later power law individual strong short compare parameter constant inter law artificial system carry complex rely fact basis rule aggregation short planning biology control study seek address relate maintain stable obstacle grow importance force field area remain concerned draw collective behavior devise predefine formation manifold give force field loop element property compression extension pair force embed vertex map particle motion edge force law govern model configuration map pairwise map heart mechanic embed present imagine particle move velocity centroid denote position move aware position correspond force edge graph individual centroid inform dependent force field interaction velocity describe symmetric interaction th map map attract attract whereas represent superposition define interactive insight section force dominate dominate unique balance manner barrier force embed function follow ij ij odd symmetric origin u ij artificial term whereas term establish alignment interaction force act along potential explicitly instead neighbor superposition field enable cope change rewrite reflect force map along occur around define motion potential neighborhood let describe dimensional map map embed adapt platform reduction many nonlinear visualization though field preprocesse building visualization task ask illustration popular neighbor le stochastic neighbor benefit various functional combine insight create embed preserving assume neighbor presentation focus version weight use binary method ensure effective neighbor low gaussian pair weight map proceed minimize kullback distortion neighborhood distribution dimensional neighborhood far demonstrate superiority include locally optimization unstable lead experimentally attain meaningful expansion dimensional work reveal term nonlinearity force field form vertex space attractive force force interpretation field magnitude dominate cause establish force describe stochastic neighbor well model modification lead improvement student embed kullback divergence correspond expand cost dynamic force field term vertex map attractive force square law large force describe formation establish force short field spherical variant embedding assume surface probable value similarly desirable unfold many spatially drive probable control map implicit unit yield motion contrary describe manifold turn define sphere neighborhood configuration magnitude unbounde pairwise distance also make spherical apart force field I unbounde inverse distance exhibit capability split unbounded configuration unstable technique present proximity embed dimensional embedding obtain idea minimize magnitude obtaining whereas uncertainty uniqueness embed constraint compute solve eigenvector problem connect field identify artificial whose motion map incorporate cauchy interpretation step insight force field energy component involve choose incorporate neighborhood high technique equation however area hyperspectral suitable parametric neighborhood extension pixel tend drive therefore fully address image remove smooth enhance result feature improve technique partial differential equation hyperspectral demonstrate significant improvement devise pixel graph spectral artificial unbounded vertex potential suppose force map I potential corresponding force short range distance force dominant observation model formation pair whereas set exist table force function exhibit behavior c n ne ce nz nz objective nonlinear require stability learn meta rate adaptation embed even fast minimum configuration description termination configuration rate optimization world initialize parameter neighborhood randomly distribution new u iterative optimization equation equilibrium distance motion guarantee minimum configuration invariant visualization consider energy negative motion graph continue decrease invariance principle configuration converge conclude result embed force field property discuss termination embed embed neighborhood neighborhood weight dimensional optimal embedding establish another comparison enforce map coordinate learn nine natural acquire delta signature remove water overlap band band classification carry hyperspectral acquire national visible sensor resolution water remove leave class signature subtle visualization low coordinate scene north area west west consist tree image resolution gray select testing euclidean compute classification class scene hyperspectral comprise pixel locate six original scene california cover comprise water discard band green min heavy center c water c short water evaluation le take neighborhood obtain principal component reduce construct neighborhood value ensure neither unstable observation observation implement gaussian determined generate graph principal establish spherical magnitude set norm termination solution algorithm iteration terminate embed map admits namely pick lead le pick eigenvector dynamic serve describe gradient field map minimum configuration trace affect cost corresponding scheme field present hyperspectral class interior water field formation high random change cause pair map smooth optimization iterate design dependent range force generate magnitude contrast initialize seed display trajectory direction field indicate severe optimization point negative direction gradient water demonstrate gradient point reduction demonstrate performance problem manifold tendency onto characteristic increase class representation effort visualization complex figure reference le superior map method compute coordinate seem separate seem spread imply tight demonstrating class separate effect significant le produce interpretation image reference embed embed construct spatially embed maintain disjoint truth boundary appear existence water class little ten class presence le little meaningful interpretation embedding show figure generate embed information disjoint neighborhood establish neighborhood interactive field generate relation many maintain embed straightforward denote low euclidean respectively model local representation highlight coordinate continue frobenius turn preserve force tendency map small separate property useful suitable task close include entail establish automated image object class label determine nearest angle cover dimensional carry pixel several visualization table trend indicate outperform enable accuracy observe mean ks classify label belong percentage provide interpretation consistent seem lead accuracy however lowest achieve embed representation separate achieve ability make label distance force field optimal metric k near always class term separate term formulate map classify trend provide representation low embed achieve visualization correspond category tree spectral signature mix subtle graph incorporate boost category advantage pairwise embedding separable suffer algorithmic optimization nonlinear objective misclassification error embed dimension reduction rely call embed estimate neighborhood low allow regular automate property acknowledge open research deep separate study include figure channel choose low drop display nn classification le ks class c c ks embed demonstrate nonlinear category resolution technique framework valuable couple dimensionality reduction
illustrated drive formulate comprise markovian suppose originally access e heavy drug widely several health development understand comprehensive branching markovian alternative base constitute strategy assess reach generate random traditionally associate existence priori learn specific characteristic population research factor account propose population collect participant seed participant ask ask participant drive quantity associate association within tend categorical contingency value row status status pearson contingency check observe naive ai ai ai ia ai size individual however account structure friend population author confidence interval coverage confidence build naive characteristic drive sample characteristic risk incorporate risk risk logit characteristic individual contact know latent markov network individual hyper parameter diagonal prior region laplace de et al comprehensive population comprise comprehensive criterion ii another age old participant study long comprise equilibrium reach start seven seed start slow six seed eight seven participant month study treat seed subsequent deal local instead besides heterogeneity pool weight former single city comprise country evident structural secondary structural drug progress another carry confirm finding much probably social difference large take underlying reason choose site unfortunately participant major open scene structural drug attempt researcher health worker knowledge parametrize study contingency table pearson independence test nan hypothesis suggest dependence analogously contingency table participant pearson test nan suggest evidence therefore structure population provide use confidence evidence obtain function logistic regression logistic effect well agree although coefficient interpret odd participant three ci receive material participant odd school odd ci participant college education participant odd ci summarize suggest participant reduction remove connection lead dependency factor relate odd ci participant time odd participant live live participant time infection
component unit equation repeat run burn apply experience behave augment produce sample unobserve turn enable etc remark specify metropolis hasting gaussian independent exponential element draw otherwise save element algorithm posterior initialize expand run produce augment posterior unobserved produce predictive full unit size step complete pmf hence sample level package front include parametrize population unit overall mathematical size naive dispersion tail option package focus enable degree improper prior size binomial poisson log effectively common typically thin tail prior sample proportion sample size size simple prior infinite moment flexible beta proportion translate discrete assigning uniform size mapping close easily numerically median amenable used section modeling design add great see amenable pg pg design proportional condition adaptive full equation worker drive hard reach
subsequence hence x k x statement em suppose modulus hold whenever let eq imply immediately let suppose obtain consider convex constant dual cone cone lagrangian lagrange multipli lagrange multipli q pair relation eq lagrange multipli box constrain convex hard aim study throughout function gradient solve thresholde eq subproblem close solution study proceed subsequently follow establish operator subsequently lx lx lx x follow show magnitude component method suppose define second relation argument conclusion lemma em minimizer moreover let local minimizer problem continuous moreover hence minimizer addition know minimizer establish proceed strongly modulus follow change local number sum monotonicity hence ii first arbitrarily depend monotonicity yield convexity apply obtain one q hence contradiction satisfy next establish upper summing project apply fact obtain l fx fx imply iteration establish local solution method apply perturbation modulus finding define strongly arbitrarily method number satisfying pt satisfy observe hence local solution use conservative improve performance dynamically result variant solve subproblem satisfied go go go end em method need number finite termination satisfy outer iteration similar argument derive together method local moreover final similar argument replacing imply convex programming relaxation minimizer penalty dynamically point smooth convex assumption lagrange multiplier feasible exist observe let approximate approximate variant quadratic moreover suitably assumption strongly convex establish apply find minimizer give define accord replace modulus let find local minimizer c generate local minimizer lagrange multiplier inequality assumption local find convex convex variant strongly perturbation associate modulus moreover clearly establish replacing lagrange generate find minimizer theorem iteration generate approximate eq together approximate em conservative practical update dynamically result present proceed project variant method sequence multiplier problem go show satisfy iteration method apply loss variant follow lagrange multiplier inequality apply know imply em inner terminate outer iteration accumulation minimizer accumulation local satisfie exist subsequence subsequence necessary addition pass subsequence necessary upon take limit know hence local cone programming propose box constrain programming sequence converge minimizer solution solve regularize programming apply quadratic establish approximate local dynamically show accumulation extend work solve author thank suggestion substantially remark assumption support discovery author leave department engineering university thank visit thresholde constrain converge local solution method cone apply relaxation approximate local propose penalty dynamically accumulation minimizer pt key iterative
procedure sense flexibility knot hope initial spline procedure knot enough least hand fan li xx basis may variance q able operation operation classic xx square estimate truncate important scad estimate estimation criterion suggest generalized cross fan li let eq q hence modify generalize validation criterion factor due lot basis select adaptively fan fu predictor estimation criterion also aic bayesian schwarz suggest penalize spline estimation criterion wu model classic strong result rao wu rao wu agree lot wu rao wu procedure well two slightly smoothing spline spatial initial knot two vary knot method insensitive though cause knot depict figure knot select non spline frequency example histogram scad every scad scad tune tune knots eps eps width examine examine impact factor simplicity spline method sensitive factor factor method rao wu rao knot may affect cm knot cccc cccc knot cccc knot factor multivariate chapter model outline suppose th define impose univariate knot spline non penalize concave penalty replace release impose concave univariate additive concave still make accurately select dimension corollary section true cm true cm plus spline find non spline optimal knot insensitive knot method simulation compare smoothing e concave spline concave knot much attention attract penalize spline smoothing spline regression simplify knot spline penalize spline penalty quadratic penalty penalize spline van propose regard variation kind regression spline theory penalize chapter smoothness position knot firstly variable adaptively optimal knot lot work direction adaptive backward approach establish spline remain estimate procedure insensitive avoid involve high dimension like spline efficiency penalize optimal knot selection still penalize little spline spline easily multivariate estimation penalize penalty avoid select knot knot simultaneously insensitive knot enable study chapter penalize compare procedure fan li non concave penalize traditional property high spline extend linear basic idea concave penalize take penalty threshold spline basis coefficient knot penalize avoid procedure effect shrinkage fan select claim spline sensitive knot knot spline green approach trade flexibility regularize insensitive knot trade flexibility control regularize property validation validation green chapter spline various chapter optimize likelihood spline chapter newton fan convex spline penalize spline simulation discussion nonparametric regression knot q jump knot dimension truncate power basis hence classic power spline truncate spline minimizer weight function scale sake interpretability penalty fan three bias result automatically reduce model instability whereby continuous fan li satisfy continuity three principle useful spline generally penalize spline cause number knot penalty produce rough penalty smoothing smoothing reduce knot adaptively keep smooth spline knot fan li smoothly non follow fan scad scad discussion refer chapter
subsequent yield perfect classification bic value datum model true c c group plot value ht line yield analysis purpose consist species datum consist original measured age purpose evaluate clustering consider ari display cluster cluster cccc est est cccc est clustering approach parsimonious describe package parsimonious mixture gaussian distribution estimate via package classify correctly two misclassifie select group color factor evaluate bic bic among six characterize remain four top choose covariate hence scatter presence observe red species green indicate versus support loading though informative unique constraint loading ensure uniquely load fit result huge computational burden approximation useful reduce accordance formulae inverse inverse partition utilize start term ig variable explain section follow load iii isotropic th term g k g g yield g attain k g accord n g k simplify g g estimate loading solve manner without consider last update update g g computed matrix update g g g compute accord get moreover estimate matrix p load unique datum example attention matrix weighted constitute vector constitute response explanatory assume impose variance introduce alternate expectation maximization algorithm maximum estimation parameter family artificial model direct mixture model component sub group describe cluster base mixture group ideal modelling framework manner mixture regression model joint marginal well approach gaussian mode parsimonious model gaussian density tool purpose applicability space remain challenge issue gaussian lead matrix gaussian matrix principle mixture develop work start family component parsimonious clustering paradigm linear basic maximization address aspect discuss evaluation artificial suggestion step introduction next suppose partition weight mixture usually mean vector conditional become linear leading see assume independent triplet distribution importantly mathematical become factor recall conditionally generic shall herein extend across whether isotropic parsimonious variance loading variance isotropic unconstraine unconstraine unconstraine unconstraine unconstraine unconstraine unconstrained constrain unconstraine unconstrained unconstrained constrain unconstraine unconstraine unconstraine unconstrained constrain unconstraine unconstrained constrain unconstraine unconstrained constrain constrain unconstrained constrain unconstrained cccc four letter whether impose constraint covariance variance letter apply load isotropic observation unlabele scenario draw know small proportion ig indicator observation notational prefer cluster cf classification substitute dynamic section specification miss stage generic incomplete ig z ig otherwise ig say easy consist cycle cm correspond partition iterate section illustrate cycle detail family give miss n eq st complete current ig ig k g j step maximization ig miss factor conditionally ig cycle st calculate follow z ig ig give maximize eq computing guess st carry set g g g start standard select natural opinion parsimonious procedure group multinomial first constrain cccc second start membership initialize ig initialize initialization continue accord scheme display cm cm cm decomposition detail value eigenvalue th element eigenvector acceleration acceleration iteration l k analysis stop l analysis herein characterize relevant classification integrate complete
vc q unlabeled request return er em specify adopt likely case aside indicate multiplying factor improvement know occur agree prior regard slight depend generally equal dependence undesirable application easily follow factor whether always achieve tight access value case sometimes specifically condition plug corollary bounding summation arrive theorem sketch include calibrate ba log appropriate unlabeled label request er examine asymptotic unlabeled sample log log log request indicate theorem multiply bound second interesting indicate surrogate access prefer use somewhat surprising ignore case calibrate loss indicate size ever competitive method optimize reflect indicate extent otherwise complexity result dependence typically directly log log log tight dependence apply fix almost everywhere survey class marginal fact density condition exhibit scenario particular choice pf w f p f p p f pf know plug probability sufficiently budget universal contrast k log indicate say active assumption method turn certain represent xy q xy condition result van calibrate q erm er analogously brevity condition calibrate unlabele request function factor indicate multiplying case strong distinguish case corollary reasoning proof universal calibrate satisfie finitely argument appropriate request return er sufficient multiplying particularly case indicate erm slightly specifically f j j xy specifically corollary reasoning analogously follow universal constant classification calibrate satisfied let label request probability er sometimes theorem small erm include vc vc major analogously brevity leave f hx logarithmic derive erm achieve vc excess rate vc major contain special define subgraph envelope hull vc conjunction adaboost hull vc van note vc hull envelope vc hull envelope function derive vc hull calibrate xy condition theorem satisfy least corollary derive analogously erm active since analysis significantly convenient offer unify algorithm inefficient study relax loss class vc van uniform classic jensen next f pf satisfie satisfie combine monotonicity easy proof would little algebra take suffice make right side suffice define monotonicity j xy suffice plug complete proof appropriate universal calculus reveal u u j choice j u u require condition definition strictly j summation let sum n log n convention note appropriately success least note therefore imply state sketch follow analogously integer log ba substitution place theorem bit calculus choose identical remain show satisfy necessarily toward end note expression sum n appropriately constant sketch analogously theorem modification u j follow definition constant proof theorem proof constant u j reasoning include proof lead prove vc subgraph application condition throughout adopt notational except apply variant specify specify chernoff law event determine determined completeness define sf inductive claim trivially inductive step take event mf furthermore inductive eq mh mf fact monotonicity xy xy universal side definition e mc brevity let right complete inductive prove hold j er furthermore sm chernoff least side suffice complete note discussion expense logarithmic next arbitrarily even prove exist every mm serve base inductive fix v universal f g mf follow derivation localize risk least sf v claim h v implication simply eq v sf principle induction establish j q appropriate specify definition prove every I serve inductive take inductive exist I sf j chernoff bind law least xy imply j j xy xy imply j jj hand induction hand fact e summation sum right inequality take suffice satisfied imply er note union great theorem complete sequential design supervise algorithm sequentially request select instance pool unlabeled label request label work use active specifically surrogate label classifier return give active extent passive additional passive accurate go subtle topic namely certainly active develop computationally condition e say many yet reach level practitioner turn heuristic practical attempt problem common perform convexity within execution guarantee loss formulation adaboost indeed come understanding use surrogate learning actually primarily surrogate sequential design learn large pool unlabele sequentially request pool active learning produce small achieve already specifically minimize make produce relatively passive surrogate bounding via inequality error technique zhang active optimize guarantee passive might seem possibility surrogate way less direct help may guarantee even surrogate insight truly identify optimize location focus request elsewhere construct active optimize surrogate increasingly number request achieve bound analogous passive minimizing find passive extent tool develop thesis conjunction localize rademacher work surrogate relevant zhang develop excess surrogate risk conclusion insight study plug rule loss value smoothness study analogous learn obtain remarkably obtain method work method optimal term surrogate loss loss strong generally lie zhang extent active active develop several maintain plausible candidate request make mistake label request technique number mistake derive request method surrogate behave nearly previously study analyze request achieve excess risk base complexity surrogate method result determine excess risk excess evaluating bound label request method immediately number request excess constructing algorithm make generally active passive method instead propose surrogate optimize increasingly space thereby strong space g denote usual simplify event measurable van discuss xy py xx xy ph certain conditional follow hx h hx hx hx contexts function write equivalently f variable refer arbitrary protocol initially unlabele may select observe another request protocol conditionally independent ni ki throughout primarily interested satisfy condition discuss statement convenient xy interested optimize characterize z represent xx subject error necessarily minimizer reasonable calibrate though convenient infimum actually z necessarily instance interested modify natural handle general substitute risk f xy xx p rely combine stand formal assumption loss choose function surrogate necessarily note significantly restrict care surrogate add essence analysis relation h however leave open important learn passive generalization significantly h r h xy xy h related define ph g h ph g ph radius radius xy h r transform guarantee excess surrogate abstract transformation define xy calculate context also arbitrarily subject subtle relationship excess hold calibrate argument calibrate satisfie context fix r f though modification replace family originally convexity quantify specifically condition calibrate take discuss many calibrate loss specify context machine optimize exponential vector machine term hinge hinge condition require loss learn estimator binary classifier quadratic loss quadratic exhibit gx iy subsequence reason g g h r passive serve derivation motivate bound excess erm mu erm h h return think kind erm erm h h minimizer erm erm point function erm erm erm mu erm follow quantity q interested quantity however toward define follow take h quantity variant mh claim hold roughly within factor follow localize sample quantity completeness xy erm h typical calculation derive sometimes envelope function proceed p entropy tend complexity calculate much van explicit conjunction relax p relaxation source slack however case condition tight convenient explicit later section make benefit allow abstract enough capture specific measure conjunction toward definition h p satisfied instance make see explicit example quantity quantity p p p mf h p crucial h p p pf invert bounding express abstract h xy xy perhaps simplest make surrogate know basic comment potential drawback context optimize number achieve excess specifically modify fix xy er mention several optimize loss many learn excess surrogate passive specifically problem z equivalently state constraint x f slight imply minimax optimal active achieve specifically positive decrease convex twice variety calibrate loss neighborhood know loss satisfy produce provable passive general describe request implication improvement active sufficient merely surrogate produce algorithm interest interested help computational maintain surrogate propose relaxation unlabele budget mm request label v behind rate informative therefore focus label request uncertainty update step empirical sampling maintain shrink region practice maintain implicitly keep track step define step check one likewise find problem function convex long efficiently quantity present internal previously define appeal interpret way compare empirical risk conditional compare empirical risk original sometimes problematic computational challenge considerable class continue derivation description relaxation case mild dependence noise condition represent main request er briefly idea rough outline maintain also round latter upon reach index r theorem algorithm request label label request index chernoff statement indice label budget formal keep minor request indicate often achieve certain type class vanishe calculate alternative calculation measurable immediately fix u theorem label request er modify way analyze analogously convenient interpretation analogous restriction modification constant event argument argument way lead interesting possibility update update occur point update could choose factor modification restriction allow additionally check another update substitute abstract result
style conference wide http www file pdf contain illustrate various satisfy rectangle paper title bold horizontal rule top bottom thick title page start page author name list leave name address follow author side side please pay regard figure level word level first proper line apply regardless within text order end section style format style acceptable consistently blind publish use paper widely e review anonymous author name citation anonymous text bottom appear horizontal clean dark place figure figure may clean hand table appear title line title proper terminal terminal body cell aspect style file modify width font perhaps please page file letter letter file cause year file file reader available box machine generate file acceptable please http www pdf generating file figure otherwise please pdf letter ps ps check pdf file shape implement shape try figure file program eps eps simple clean figure achieve window microsoft pdf office http www microsoft ed save office os via pdf click drop box save window via ps file create computer file http www com pc take ps click click advanced font font outline select click ok file ps create pdf file ps file option file contain embed ask margin come figure always specify figure width eps graphic graphic graphic bundle http www graphics ps properly please acknowledgment acknowledgment reference acknowledgment level citation consistent reduce font reference long cite reference mit book realistic neural neural journal assumption laboratory mit institute institute random constant direction reconstruction manifold reconstruction bound higher previously k novel tool motivate suitable one close low proved practice well amenable theoretical lead work set variety semi embed ambient approximate suitable sense geometry manifold typically support survey quality approximate measure reference therein minimize associated encoding crucially work focus important widely k algorithm see approximated extension induce collection possibly result manifold resemble locally tangent mean support provide k mean algorithm thus combination fact novel develop tool rest begin discuss reconstruction mean present space endow compact drawn respect goal learn approximate reconstruction reconstruction easy respect word increase mind give belong effectively constraint negative space typically definition space setting continuous distribution euclidean domain lebesgue size see equation induce subscript virtue minimize set mass locally number though closeness global randomize global minimum respect randomization collection affine analogously k mean aim minimize minimizer relaxation flat often cluster interpretation rewrite sum pairwise distance consider k mean quantization error interestingly coincide precise sense problem quantization distribution excellent generating support point analogously euclidean decomposition effectively produce euclidean problem quantization manifold define constant approximation support perspective interesting dictionary interested dictionary associate reconstruct maximally reference crucially quantity interest question characterize piecewise affine manifold manifold bundle sec k yet currently mean approximate sense increase ultimately interested understand quantization approximate minimize thus population infinite increase derive rate technical depend order hilbert study keep report train small increase experiment think concentrate around analyze factor derivation somewhat surprising trade error order entirely might expect reconstruction really result could tight tight remain e derive whether bound tight importantly aware point exact distortion justify heuristic choosing perform show trade indeed regime trade easily e setup unit sphere nearly orthogonal clearly indeed origin complex setting direction embed hilbert complexity rate arise solution generalize derive rate result theoretical either widely past involve throughout follow smooth metric absolutely density consider access solution grow equation vanish convergence compute depend px dc ed k empirical equation may bind follow ns ns consider total difference measure tend become large note discrete variable probabilistic performance choose complexity explicit error novel quantization quite transmission distribution finite moment measure absolutely therefore unit cube clearly make formula result sequel manifold know appendix let absolutely replace clear restricting mean equation error uniformly widely recent year particular sn restrictive condition moment mean whose solution practice mean original relaxation provide closeness global equation practical approximation proof result let choose equation argument multiplicative incur third multiply equation probability respect prove next section problem begin introduce uniform expect affine combine result expect smooth metric affine space positive second grow smooth hilbert manifold embed separable constant eq dx b begin begin upper rademacher subspace projection onto independent unit schmidt inner lemma finally desire consider simple analysis metric tangent absolute value fundamental tangent measure choose dd intermediate clear everywhere associate since absolutely therefore minimize among fourth quantization metric definition match geodesic follow adapt characterize quantization riemannian packing result must diagram triangle follow minimize region diameter measured distance tangent begin establish tangent manifold geodesic appear give neighborhood tangent plane geodesic form clear dominate point satisfy equation qx imply collection neighborhood hold consider point equation lemma contain neighborhood clearly compact admit lebesgue every diameter set since hold inside need kf theorem absolutely definition definition mention exposition absolutely
eigenvalue correspond instead operation rank prior diagonal consist corresponding eigenvector mode determinant determinant column one diagonal determinant inversion lemma draw posterior covariance basis rewrite fast low triangular cholesky drawing form avoid optimize induce require choose location induce expressive automatically location input challenge gp preserve small input trivial overfitte kronecker product separable respect test reduce suitable density covariate target region space contribution contain associate th slice grid similarly consist th slice contain grid non distribution laplace factorize total slice grid denote consider closely derivation present issue become computationally dense grid large limit form mcmc limit measure posterior use inspection factor initial one la approximation remove burn density estimate grid size second examine approximation logistic simulate la mcmc dirichlet gaussians mixture variance assume mixture allow component gibbs compare advanced gp default mixture model exclude using would toolbox test mat ern skew datum grid truncate mixture truncate less sensitive measure time estimation sample student la grid size take core ghz multiplication take option second second comparison simulate student x compute estimate la illustrate low show la take mcmc minute test set student mixture gaussian measure divergence grid times grid size equation reduce exclude kl la performance sense large prior density density set demonstrate scheme estimate dr mcmc dr dr difference dr illustrate directly la fourth process la close art laplace avoid posterior la grid take time fine paper cubic demonstrate predictor laplace approximation speed variational bounding approximation ep laplace variational binary however poisson slightly laplace process well ep could performance la diagonal quadrature free ep way gp preliminary turn slow improve correction correction latent correction could la mcmc plot toolbox matlab acknowledgment author like thank de importance sampling anonymous acknowledge resource provide ki department prior model analytically inference grid integrate type sufficiently interactive reduced speed transformation gaussian modelling density parameterize smoothness control covariance challenge intractable construct focus ensure normalization describe property establish grid leibl infinite approximation consideration bound extended unbounded interval transform integrated metropolis set latter automatically location analytic make fast fine laplace grid quick practical focus way relate literature posterior involve mode penalize maximum consider marginal enable posteriori map covariance alternatively use derive moment process approximation evaluate likelihood hyperparameter grid newton mcmc guarantee guarantee hyperparameter dominate covariance matrix straightforward computational interactive figure speed stationary covariance avoid dense grid exploit reduce exact prior grid exponentially suitable reduce computation impractical normalizing term avoid elaborate rejection conditioning algorithm place make space easy however scale construction quick combine idea integrate value slow tailor la rejection importance hyperparameter result additional inference several experiment independently focus unknown maximize pd limit delta locate belief unknown obtain realistic equal employ logistic density transform unconstrained smooth estimate density covariance gp gaussian widely use smoothness scale decrease magnitude combine placing gp polynomial lead density tail go eventually basis demonstrate illustrative mixture gaussian latent fourth show hyperparameter value exponential collect center prior latent vector associate contain basis regression fix tail towards negativity posterior zero rejection discretization belong th denote count prior bayes due non approximate integrate implementation laplace gp section efficient mode computation likelihood far obtain multiplication reduce gp brief mcmc resemble laplace approximation intensity estimation expansion approximation lead full diag element matrix implementation form avoid use multiclass positive maximum mode stability inversion preferable conjugate evenly lead toeplitz covariance embed enable toeplitz speed multiplication frequency domain multiplication convolution single embed matrix multiplication fast two form la ii hyperparameter marginal determinant evaluate intensity evaluation exploit low intensity require evaluation marginal newton typically fast implementation default reduce compute respect compute derivative magnitude informative freedom prior equal addition estimate integration composite design la scheme joint predictive monte
propose fix minimal complexity dominate whose bind atom assign alignment protein share variation similarity compare variation atom score refer respectively sup sup invariant translation semi valid k deal metric area auc author bind classification map problem decide auc nevertheless provide supplementary metric rmse determination coefficient suited paper nest validation rmse bind outer fold validation fold union fold test since average square nest denote fold affinity coefficient error always return eq therefore predictor give decrease report sign rank database protein along protein sequence bind energy force field diversity structural protein domain database computational structural dissimilarity database structural protein bank refer unique base bind energy modification approach bind datum bind bind affinity fold minimize overlap fold avoid fold since predefine fold nevertheless conduct method contain dr binding per allele transform ic learn allele purpose useful pseudo highly position potentially contact specificity author method use pseudo compose dr respect chain consist also conduct well design quantitative recently pls networks ann biological activity gp find protocol thus tend cluster use split select small dataset kernel attempt predict bind consider major challenge consequence la protein yield secondary structure validate try prediction protein protein assign great protein alignment aspect protein interaction interaction surface protein bind protein share bind motivated sup binding sophisticated rbf vary kernel motivation design gs descriptor sup sup sup bs bs gs gs l table bind affinity protein well gs simple bs sup sup bind sup sup benchmark atom provide relevant figure illustration sup sup absolute binding maintain bind drug discovery ultimately serve drug rational gs ability build perfect predictor quality responsible ultimately achievable biological dataset method experiment predict bind energy specific bind train gs validation validation predefine fold provide predictor generate three common metric root square rmse roc auc rmse auc find see significantly inferior dataset rmse cc c c drb drb drb drb drb drb drb drb drb drb drb h ia far potential gs author cross allele training use determine bind allele bind specificity use beta chain experiment sequence yield slightly suboptimal gs allele obtain allele assess auc supplementary appendix rmse individually yield allele show outperform value ic calculation require predict author indicate globally cc drb drb drb drb drb drb drb drb drb drb drb drb average cross result extend rbf gs kernel database gs use gs likely rbf kernel table rbf method gs rbf kernel ridge slight support hyperparameter tune insensitive require short gs outperform rbf datum limit consider method gs l design pseudo bind gs elegant eight kernel ridge bind kernel first capable accurately bind target ii bind state task well quantitative affinity database benchmark would substantial tool community predictor still major machine hard computer expand author gs ad conduct provide biological insight supervision final manuscript acknowledgement foundation innovation sciences university work nature pr grant mm say alphabet symmetric convolution string alphabet ai lx I finite alphabet semi ff string note line rewrite string string contain put real fact l suppose calculate roc curve auc problem transform case aim distinguish achieve allow require predict bind bind affinity threshold convert technique calculate auc convert bind energy bind bind value range latter auc drb drb drb drb drb drb drb drb drb drb ia ia calculate explain discriminate drb drb drb drb drb drb drb drb drb drb drb cm cm theorem protein physical protein template choice protein interesting display activity drug drug furthermore base possibly affinity method potential accelerate low drug discovery select potential biological validation specialize sequence bind incorporate chemical generalize eight radial basis programming computation ridge bind propose kernel relevant good prediction bind affinity specific major benchmark dataset quantitative model benchmark dataset conclusion benchmark p art predict protein bind apply predict reliable bind improve model interaction pathway lastly research accelerate drug vast majority protein protein biological process protein protein interaction control interference chemical discovery novel pathway agent interaction surface secondary essential bind interaction secondary furthermore activity drug drug serve bind region specific process target identify utility throughput screen million test biological however chemical library generate false negative challenge reasonable accelerate provide increase aa recognition complex capable affinity response bind affinity protein model pathway high binding bind affinity predictor affinity huge set candidate stochastic traditional classification bind bind bind quantitative obstacle database bind protein family bind protein machine biology database contain bind great success simpler bind significantly allele target reasonable require allele accuracy multi reasonable bind allele allele know propose capable multi prediction kernel bind information protein propose gs kernel generalizes currently weight also inverse elimination also protein one predict bind energy vector bind energy protein consequently equation thus kernel extremely successful decade choice good subsection design choose protein biology protein homology detection kernel design small like exploit exploit unify manner call pair string string string position encode vector component encodes possibly similar encoding string length encode comparison gs contribution rapidly position control contribution differ measure square vector gs sum comparison parameter distance dirac delta spectrum rbf radial rbf rbf string string great string weight kernel gs eight list free approach gs free measure accuracy elaborate
netflix contain million user movie book music recommendation graph employ similarity movie movie build movie wikipedia user whether certain movie netflix retain music book wikipedia set netflix contain wikipedia record movie shorthand recommendation collaborative filter netflix partition part movie identical part rating movie density serve source world source book movie set target different item construct netflix share movie netflix extract share movie netflix rating rating netflix knowledge movie task item totally task music source movie utilize wikipedia record graph movie perform rating netflix movie predict rating rating c c dataset target selective selective pmf c simulate selective utilize baseline pmf show work non adopt technique recent prove uniformly weighted utilize domain baseline selective framework performance result collaborative task target pmf fail give especially help selective source domain prediction selective transfer outperform selective truly helpful issue selective observe selective significant netflix source world case handle source factor affect table find movie much improvement simulated entity movie rating netflix try system recommendation domain target rare source domain user graph knowledge movie domain heterogeneous utilize source target domain selective level domain selective globally noisy contain inconsistent observe transfer level transfer eliminate irrelevant source highly target even variance weight fix one adjust adjust order fix prediction learner movie base reflect training topic training suffer overfitte selective selective number latent topic compare fast go slightly large decrease obvious clearly overfitte fine grain knowledge selective selective learning etc pmf etc selective transfer relate collaborative summarize knowledge work learn ever grain consistency source collaborative filtering recommender gain probabilistic machine introduce concept selective transfer well overfitte utilize domain build application domain work transfer collaborative involve manifold alignment jointly build focus auxiliary recommender distinguish user preference align framework transfer knowledge target domain recently researcher propose via weight source individual boost base well limit task systematically filtering besides perform selective transfer variance empirical would introduce useful source embed criterion boost transfer source domain target experimental world selective transfer method long robust overfitte pt filter user historical user many model prediction recently several work knowledge manually part target novel criterion empirical cf setting consequently boost perform selective several state selective improve rate real world recommendation collaborative cross recommendation recommendation attempt recommend movie tv book page recommendation aim rating item historical user preference record system although item often usually small available rating extremely sparse service limit recent via collaborative filtering source domain transfer previous trust similar world cross music web site rating music international rating web site rating good rating inaccurate music site obviously international site data tackle cf challenge source work adopt cf target item get truly word inconsistent dominate happen domain give careful user expert criterion far variance domain like preference transfer observation consistency source implementation criterion selective collaborative instance help build contribution follow domain filter influence factor novel selective transfer collaborative extension boost issue cf base implementation probability semantic solve various wish prediction regular recommendation illustration user task observe recommendation example contain adopt commonly collaborative either appear source domain derivation target domain item co collaborative base probabilistic analysis filter probabilistic integrate selective transfer learn observe careful reader generative base introduce topic item rating mean variance mixture estimation minimize extend cross domain transfer filter knowledge knowledge source domain assume user source world approach jointly model domain target relative could domain motivated domain expectation statistical estimate derivation tb weight boost initialize generate minimize weak fitness weight source update hypothesis target domain propose selective novel empirical selective mutual like transfer knowledge record reflect preference record empirical variance factor boost care mis predict instance domain
opposite estimate separately necessary construction propose auto probit actor network probit maximization employ treat bayesian description estimation actor type question vector preference covariate coefficient term responsible covariance structure structure mechanism exist basis tie describe act structural membership mutually exclusive effect correspond compete structure group actor embed term model describes account integrate unobserved matrix pose significant since treat description regime autocorrelation first estimate expect unobserve complete calculate expectation respect replace estimate possible analytically estimator mode see produce approximation find mode expectation degenerate solution bayesian observe choice decide unobserved chain generate series distribution summarize appendix choice truncate unit distribution multivariate follow gamma distribution language mechanism ensure generate please use close choose prior assess algorithm ideal pre distribution temporal narrow show temporal consideration available suggest high effective sample high posterior car compare original consist decision mid car otherwise car price interest preference among network measure location live explanatory actor information age education information car car actor location construct home code thus structure membership contrast separate coefficient show explanatory network term equivalence calculate simple distance structural equivalence undirecte weighted adjacency distance number individual otherwise node inverse plus positive equivalence element adjacency connection respectively addition avoid denominator equivalence leave right deviation interval thing second structural get increase first autocorrelation equivalence variance autocorrelation investigate phone around call back phone take soon back automatically phone company raw phone record month period phone information age user pruning algorithm imply equal relationship call include several explanatory gender phone connection structural assume phone call party eventually adopt technology call drastically normalize number element structural adjacency obvious mechanism impact relate relationship trace autocorrelation coefficient second autocorrelation structural variance term log show autocorrelation opposite model auto actor specify fit e hierarchical find solution solution use validate quantile accurate structural recover variability vary number observe capture affect get get transform side decompose likelihood parameter get estimate expect first analytical equation first right q analytical generate interest follow truncate distribution identity generate n matrix corresponding bs generate metropolis increment one get software develop greatly chance validate return estimation bayesian software implementation history program quantile validate simulation base generate software code true credible interval bayesian distribution quantile value series software test number draw posterior estimate quantile draw software true software quantile replication expect simulation run want distribution mcmc simulate replication validate hierarchical generate network generate keep draw parameter count software correctly write randomly estimate replication pool quantile sorted show replication roughly confirm mcmc draw burn listed figure autocorrelation autocorrelation term equivalence autocorrelation plot right bottom coefficient variable red randomly show valid th randomly n show red otherwise bivariate apparent come htbp r invertible finite coefficient matrix represent weight rise characteristics one obstacle measure closeness type actually decision difficult conceptually address probit behavior regime auto sensitivity impact nature interest include govern connection explain topology make network embed topic technology past network hundred people collection enable access technology handle scope aspect complexity individual decision product technology solely attribute age education though due mechanism handling indeed development associate unobserved share tendency ability predict person beyond characteristic produce correlate member phenomenon autocorrelation outcome actor distinct network family outcome explanatory term term whose specified interest autocorrelation estimate autocorrelation accommodate actor friend research autocorrelation
uniformly iteration chain stop randomly sample square error respectively correlation fig depict function function increase close goal bivariate target use mh black box simplicity gaussian set stop pdfs form pdf mh pdf parameters gaussian location weight gaussian quickly remain mh modal multi target build extend proposal update gaussian lemma corollary carlo widely communication introduce modal density vector previously two chain monte mixture hasting chain problem scientific digital communication learn etc mcmc normalize simple proposal produce mcmc hasting mh drawback mh method high sample mean extremely depend discrepancy target target several extension propose reduce burn among mh adaptive proposal technique usually parameter capability technique least unfortunately problem adaptation proposal lose chain mh carefully recursive empirical sample covariance whereas jump algorithm mixture obtain fully mh complicated iteration introduce mh gaussians use recursive formula adapt mh multi modal target always improve mh organized mh describe box usage finally conclude multi modal density mh covariance variate vector approach fully adaptive mh update since adaptation adaptation stop use recursive formula initialization initial n n identity gaussian pdfs accept otherwise set parameter find index add parameter denote er set update definite pdf mixture expression update technique show sensitive information initial box way gaussians great dimension select cover support diagonal big value train period parameter big great numerical suggest could great gaussian generate adaptation mh irrelevant vanish therefore cost control adaptation gaussian locate target discard important refined improve toy univariate target pdf clearly pdfs proposal uniformly respectively I stop
rule db I refer adaptive evaluate two mean excess ii exist distribute adaptive filtering subsection special rule propose fitness filtering fitness I equivalent rule fitness selection fitness give fitness fitness c degree n ik jk j jk metropolis hasting correspond fitness utility define hasting rule fitness definition exist summarize fitness definition graphical distribute fitness ii filter topology node error aware weight instantaneous denote local exchange instantaneous neighbor fitness note fitness form use I update propose directly adapt environment adjust accordingly base instantaneous mse aware power implement performance lower verify noise poor principal adaptive signal whole enhance analyze dynamic expression regressor graphical game structure eventually model reveal good signal node concept signal probability filter probability show center node node center possible steady variance steady adopt strategy information noise interact one instant nod instant user specific node neighbor expression intuition adopt beneficial adopt node well therefore utility follow sufficiently calculate accord compare therefore hold conclude signal whole filter adopt percentage adjacent similar either imply follow dynamic I subsection dynamic understand update include strategy fitness q fitness normalize include fitness fitness fitness among neighbor node fitness strategy probability replace strategy meanwhile I select show among phenomenon fitness q fitness percentage meanwhile taylor weak go parameter dot dynamic variation within limited player derive strength limit long biology help close strategy combine besides reflect node signal negative signal good closely good subsection begin equilibrium rate selection e rapidly converge accord therefore characterize focus diffusion characterize regular degree suppose edge probability good signal db I equivalent undirected adaptive filter characterize good connection node update update diffusion I update rule diffusion diffusion consider adopt favorable state even node use ess game theory ess ensure another obvious good favorable ess check whether stable ess follow distribute network characterize ess node node equally respectively scenario majority adopt fraction ess evolutionary stable left hand ess adaptive graph incomplete graph ess incomplete graph characterize regular degree ess strategy approximate rp rp u u I since three compare adaptive ess network leave respectively regressor simulation average independent conduct kind aware power propose aware exponential aware update use projection first comparison variance node hasting kind variance propose well adaptive perform degree algorithm relative degree steady kind adaptive filtering algorithm steady besides performance steady performance six convergence algorithm fair network compare variance degradation significant hasting since rely less information clearly power achieve relative variance similar adaptive power advantage notice performance fitness function b b iv simulation generate degree type trace regressor set node I iii update either reach reach calculate run reach simulated theoretical gap approximation derivation good along noise variance ess I rule node solid represent setting network node strategy denote color use process strategy propose exist adaptive algorithm game node always unlike bottom theoretic provide important distribute offer unify design future give substitute define increment case approximate finally adopt update backward kolmogorov satisfie follow differential weak approximate bad connection substitute theorem expression initial term state whether increase decrease percentage shrink favorable receive engineering ph high visit group electrical computer china dr currently department ray liu post interest game theory wireless social dr distinguished award national fellowship distinguish receive technology china college associate electrical engineering college currently origin wireless communication information interest social signal dr chen receive fellowship chinese award university distinguish fellowship mention school research award ray name distinguished teacher college technology lead research broad communication cognitive communication security green communication technology include signal processing award distinguish year award award award school cite author process vice journal advance signal processing chen ray liu processing estimation network distribute filtering algorithm regardless nature characteristic study theoretic distribute graphical evolutionary formulation node regard selection framework two evolutionary game adaptive network stable strategy verify adaptive game recently adaptive filtering interest wireless hoc localization distribute sense cognitive classical centralized distribute robust node network fusion diffusion process adaptive filter instant satisfie follow vector measurement covariance white independent combine incremental incremental allow node neighbor adaptive summarize project mobile traditional mainly design rule topology statistical degree relative variance incorporate node intuitively sort adaptive exist offer combination unified reveal rule fundamental answer essence parameter similarly evolutionary game general bottom focus framework fundamental summarize evolutionary game node regard local combination different neighbor regard evolutionary exist special case propose aware distribute filter complexity evolutionary theory analyze derive diffusion information good work detail adaptive filtering problem graphical game information section iv section simulation vi finally always adopt ess player rational take equilibrium strategy ess evolutionary game utility matrix whose versus population fitness fitness fisher ess eq fisher population sufficiently strategy strategy process update adaptive evolutionary adaptive classical evolutionary consider complete scenario player location incomplete evolutionary game structure player relationship valid replace
albeit claim formalize algorithm condition online stability play adversary step online regret condition assume lipschitz equality regret picture regard follow let adversary regret stable regret intuitively proceed average batch loss unstable strategy stability condition formally new set batch batch end average batch process picture lipschitz lipschitz element stability pt prove bound last batch maximally regret stability regret condition forward regret fairly general convexity major simplify exist technical brief generic sketch stability exploit lipschitz regularizer l non bound optimality regularizer divergence evaluate update minimize would commonly obtain iterative sublinear dual although solve successful maintain intermediate sparsity however impossible every behaviour exact give regularization strongly add approximate eq stability q triangle successive iterate eq know couple stability write simplification use rhs bound bound sublinear stability light crucial concept relate ability minimize online remarkably simplify extent arbitrary stability regret arise solve approximately compare offer per like perturbation stability iid unfortunately gap exist stability class existence concept batch set think major empirical risk concept online stability way fundamentally batch learn ex mm property definition conjecture axiom claim ex ex ex ex microsoft stability implication bound introduce novel look bound forward also bound general non stability regret online restrict regret regret simple exist tight approximate version follow stability ability recognize connection stability generalization fair say stability adversarial show minimizer erm apart insight stability potentially help design algorithm example setting generalization concept erm online adversarial online sequential two player adversary loss player adversary player control stability derive critical area privacy connection iid online set erm serve hypothesis sufficient analyze erm set unfortunately canonical scheme make connection stability difficulty study connection regret concept sense end stability essentially leave also uniform alone guarantee example move bound force end forward incur move adversary move fundamental online bound regret regret equivalent regret forward like stress general convexity contrast equivalence illustrate learn follow regularize mirror demonstrate arguably simple approximate version solve important practice precision provide version framework section usefulness analyze exist online recall bregman find notion key behind online strictly interior nonempty bregman rr convex modulus convexity refer strongly present characterize optima particular bold letter denote dot product refer arbitrary norm unless compact norm describe play point suffer measure player perform compare move know move advance goal minimize regardless sequence online linear game set introduction description stability set last md type online progress connection unlike extremely generic function prove exist family fact connection consider algorithm iid iid fashion define new notion provide connect move sample general follow choose minimize play surprisingly simple achieve adversary play add follow give strongly norm tradeoff another describe bregman divergence update unconstrained minimizer call mirror try minimize current loss similar general interesting note special mirror descent gradient descent look fundamentally spectrum algorithm mirror corresponding counterpart
successful build bayesian linear datum exact prohibitive present variable though use novel provide efficient implementation lin cut plane implementation cutting couple heuristic remainder organize dependent section metric scoring decomposable compete bic minimum description order decomposable find empty go permutation define know program operation solve dynamic see variable minimize arise treat however ability bayesian may fact equally htb permutation list history dependent receive mathematic community decade list city find cycle city city np develop overview reader popular pick removal add replacement underlie city asymmetric replace opt opt step note increasingly extend naturally edge add randomly acceptance rejection replacement opt opt implementation find ignore history opt opt implementation due integrate history dependent part effort improve approach dataset publicly uci repository census several attribute work week country education status attribute unfortunately miss value individual discard datum break capital continuous attribute number possible bayesian greedy hill h education capital hour week country learn hill capture study work week education status hour work week hour simple arithmetic accurately input order random variable predict significantly comparison ratio check predict individual less square expression p case find approach predict apply ni machine repository consist try answer relate state discrete clean dataset entire entry entry highly fig dependency air air temperature surface air temperature find dependency surface dependency seem failed quantity quantity suggest c air discretized prediction concentrate wind eqn wind wind predict learn hill new structure order expert space hill technique structure lin hill
allocation simple hierarchical flexible discover topic use efficiently massive set encode variable factorize estimate hide want many prominent strategy monte carlo variational sampling posterior run chain equilibrium collect distribution set parameter member close posterior inference optimization variational kind describe researcher speed tailor specific correctness optimization optimization maximum objective optimization provably optimum particularly term independently set subsample amenable datum estimate subsample subsample independently expectation detail idea attractive point variational implement repeat require repeatedly set analyze graphical probabilistic mid seminal statistical parallel originally publish lead mixture understand lead automated allow write inference review early whose set much wider amenable coordinate field approximate service alternative chain monte despite popularity focus develop mcmc strong guarantee gradient langevin advantage develop variational variational derive variational algorithm probabilistic massive set divide part apply hide within review hide variational ascent algorithm objective gradient coordinate stochastic technique variational estimate variational objective repeatedly subsample build inference datum nx correspond global global may involve observation collection vector partly random keep global local hide variable distinction dependency local conditionally hide q notation refer frequently arise endowed gaussian global mixture variance local hide complete conditional hide parameter shorthand convention conditional complete conditional global complete determine local e equation conditional relationship imply specific conditional hyperparameter dimension equation exponential parameter derive stochastic discussion conjugacy family conditional contain learn bayesian latent allocation kalman variant factorial hierarchical probit analysis factorization nonparametric keep use explore prediction future list modern turn field inference root strategy variational optimization introduce family variable index free find member close interest closeness divergence use distribution call field variational variational field ascent inference kullback leibler kl maximize marginal negative hide jensen fy fy q term entropy variational depend variable family try member maximize solve member close kl divergence pz depend simple family govern global local factorization variational family distribution lead ascent optimal without conditional equation likewise substitute mean advantage q respect expectation computational gradient variational ascent ascent hold variational conditional coordinate coordinate eq involve third constant quantity depend hidden quantity depend equation derive likelihood separate leave substitute form identity simplify derive set global natural complete holding optimize play thank local depend identical depend th computational scalable section coordinate inference iterate update global guarantee optimum computing expectation direct graphical conditional update tractable many aside field inference expectation maximization across global reveal begin reflect complete must analyze expect something parameter variational inference detail component variational classical finally global old intermediate improve global calculation arise coordinate gradient account traditional discuss riemannian distribution classical function gradient ascent equation away direction reasonable direction space might set parameterized dissimilarity univariate indistinguishable euclidean distribution overlap reflect distance dissimilarity distribution parameterize transformation kl method q ascent ascent space kl complicate riemannian define transformation euclidean nearby matrix plug equation ignore term derivative identity variational gradient mixture address variational respect fisher metric fisher fisher reveal follow analogous go closely coordinate consider variational subtract classical natural coordinate reasoning around importantly easy compute classical parameter many compute ascent inefficient point stochastic inference use fit parameter form follow variational conditional natural easy discuss noisy gradient decrease algorithm optima complex write subsample optimum overview optimization overview equal type optimize realization iteration convex whose resulting suggest fisher replace stochastic noisy gradient function result figure current optimize variational variational ascent repeat weighted intermediate ascent variational variational maximize write return local optimum parameter local natural hold reason jacobian stochastic optimize maximized subsampling decompose term local choose variational natural gradient objective noisy suppose objective thus natural replicate compute replicate natural whole consider gradient noisy optimize comprise gradient noisy size update parameter sample appropriately compute global datum intermediate weight step control old information iteration delay rate way satisfy iterative optimum describe basic inference algorithm stability bayes method estimation sample stochastic optimization datum finally gradient gradient gradient expectation global use expense help optima converge basis datum see per may want hyperparameter maximize empirical maximize fit figure increase variational maximization optimize currently scalable derive variational lda nonparametric counterpart hierarchical dirichlet process use latent encode model collection aid extend apply many word arise share proportion represent document exhibit health document business business topic depend assignment dimensional q lda collection document variational collection complete conditional derive ascent form coordinate natural variational inference topic iterate document local coordinate ascent routine iterate update assignment proportion update complete conditional equation back parameterization assignment expectation proportion document variational depend document batch inefficient collection document topic complete analyze document topic initialize randomly schedule appropriately document variational global dirichlet variational per topic multinomial follow phase optimal iterate equation batch document use variational intermediate equation contain replicate document next iteration topic inference assign topic corpus document per lda root scale variational fast batch variational inference reasonably size subset lda collection document limitation lda researcher cross practical nonparametric nonparametric variant lda hdp membership text hdp assume collection document posterior hide determine topic need describe hdp flexible unseen broadly inference bayesian us model mixed membership model grow expand prohibitive search specific structure validation use nonparametric topic scalable correlation section dirichlet breaking hdp topic stochastic variational place context place flexible prior distribution draw datum draw flexible potentially mass variety review see dirichlet parameterize non negative atom closeness scale small place atom look spread around atom draw representation gamma marginalization chinese restaurant stick break stick explicitly define draw discrete atom q atom stick stick breaking use infinite recall following combine infinite imagine stick call aside break proportion aside stick result stick length collection realize return stick th combine g draw place mass draw tend describe formally see dp stick break intuition stick break atom stick encourage draw atom break location tend later break location dp dp dirichlet vocabulary topic draw dp collection proportion construct hdp stick break construction corpus construction construction chinese restaurant alternative stick breaking mention hdp generative topic corpus breaking proportion document break proportion draw draw corpus define probability topic break b draw document stick length membership unknown advance unbounde however posterior advantage complete conditional exponential family separate global global variable break document break stochastic hdp begin indicator interaction level indicator proportion th account index variable indicator conditional topic sum document allocate statistic keep index conditional break proportion stick break di di conditional distribution generative family main parametric model optimize variational breaking allow level breaking proportion topic enough topic necessarily topic truncate variational topic easily correct stick truncation large corpus expect exhibit subset conditional update variational batch global intermediate sample local stochastic hdp level breaking proportion beta optimize conditional multinomial di di di di k di z w hdp summary initialize set set size sample uniformly intermediate illustrate collection variational inference fast well per hdp model corpora variational well batch subset study effectiveness stochastic allocation lda hdp compare collection investigate mini stochastic traditional document collection vocabulary stop rare nature span year vocabulary york times document span vocabulary wikipedia wikipedia processing observe word vocabulary collection aside document fitness set fit corpus topic topic predictive vocabulary assign divide hold word disjoint approximate imply distribution avoid compare bound form evaluation dirichlet parameter topic use variational topic proportion approximation depend metric evaluate hold hdp via stick breaking truncation stochastic inference introduce schedule equation control old early although stationary speed size minibatch hdp hdp concentration equal level truncation unique hdp inference large model give large number lda datum modeling hdp stay robust overfitting overfitte fitness hdp lda regardless corpus proportion give hdp certain priori likely appear topic l lda batch document sensitive consistently stochastic hold batch slow preferred hdp hold varied batch enough turn sensitivity hdp present fix explore rate mini figure corpora ten big sensitive corpus sensitive rate hold size fix varied rate prefer inference hold varied big variational massive stochastic variational objective noisy arises repeatedly subsample datum illustrate two latent dirichlet dirichlet model stochastic collection million document generalize many setting develop way apply membership blockmodel communities social non uniformly adjust stochastic mcmc update modeling update direction conjugate exponential expressive rich use expense mathematical expand probabilistic tool example capture topic change present develop non conjugate scaled optimization develop update another recent advance close fully factorize posterior collapse variational trading simple another structure variational relax field well arise connect inference sample uniformly
cause assumption incorrect time nonlinear instantaneous model assume direct suppose extension regard time series substantial causality structure finite broad class exist provide assumption causality wrong causal conclusion simulate iid try structure acyclic graph joint say constraint causal dag reconstruct graph g recover equivalence distinguish different vx jointly independent drawing element acyclic yx xx stand lead extension post nonlinear additive discrete bivariate provide section principle time value full contain series vertice full address problem causal summary causality causality article cause past help predict translate phrase help multivariate assume var causality g tx ix tm ta tx residual follow significance p p causality nonlinear chen bivariate noise test whether degree correspond short relaxed already say infer causal structure independence effect linear extend ts instantaneous effect relationship describe suffer causality intrinsic effect predict causality instantaneous summary still identifiable instantaneous causality conditioning causality structure fail exp bad conditional test desirable partially perform violate conclusion lemma fitting checking residual check implicitly simple although causality series extension causality furthermore testing suffice section describe identifiability consider pdf pmf say set tn tt node appear require acyclic assume identifiable model hold assume kx exhibit acyclic full identifiable condition regard feasible result stationarity ergodicity var nf I I rp ip f ip linear special causal hard drop iid ts fail causality find output dag estimate principle output correspond know intractable section concentrate series without feedback loop additive recover dag modify regard independence atomic user input tuning causal occur independence cause causal relevant structure independence test scientific discovery relationship rather hypothesis lead causal reject independence number useful develop heuristic size generate code mat experiment lag x see figure g infer large contain wrong causality draw circle draw circle circle circle circle circle draw dash draw x z z z z gaussian effect length causality structure lin lin ts nonlinear instantaneous simulate n z ground causality thus causality wrong answer wrong gp specific nonlinearity causality ts wrong r dag lin correct simulate tn figure show linear mainly gp sample one effect gp accurate answer due causality ts answer causality show bad causal conclusion linear experiment opposite tn correctly identify causal discovery length tn ta line discovery dag wrong causality ts answer circle dash b inner sep draw circle draw w instantaneous effect make fit true ts causality lin correctly wrong lead false old duration next old interval whereas causality g causality experiment temperature minute expect ts causality infer remain insufficient temperature cause time result diameter ts probably relationship price decide whereas lag store exp experiment give wrong decide causal benefit framework causality compare substantial practical ability multivariate discover structure complex model preprocesse trend case instantaneous feedback loop fit force although promise lie scope present conference satisfy variable parent noise crucial series believe hold technical full
prove well product start arithmetic bind proven note subsequent q value definite write nonnegative product form positive complete hermitian software find decomposition sum hermitian construct discover proposition arithmetic necessary considerably inequality actually arithmetic semidefinite reader comprehensive concern technique specialized inequality vary arithmetic mean inequality positive list arithmetic definite derive resembles mean mean nonlinear map tuple geodesic flow riemannian however average matrix arithmetic consequence scalar sum arithmetic geometric matrix mutually high also product satisfy arithmetic contrast without replacement large replacement collection k unit follow dominate operator inequality upper final order property deterministic product construct frame odd identity inner factor harmonic frame cast validity conjecture frame k additionally arithmetic mean harmonic treat use effectively fouri appropriately group coefficient generate combinatorial prove broad expectation independent identically symmetric explore identity hold variance theorem would infinite matrix analyze let without arithmetic since iid power low final contribution random demonstrate replacement record fourth entry verify calculation rr rr way lower use linearity couple v u j since distinct odd zero must write apply since index equal expression expectation prove commonly detail extensive fourth moment surely replacement grow scale exponential replacement expectation square wishart matrix wishart demonstrate kk also analysis problem look sample independently depend expand since concave hermitian jensen replacement reasonably mild condition subgaussian moment inequality sampling least mean similar reveal replacement sampling bad numerical evidence replacement randomize article learn completeness example demonstrate six comparison randomize define third row row generate haar sphere replacement rate column harmonic frame running row run haar I piece open conjecture demonstrate frame quickly sort combinatorial exploit arise employ prove beyond conjecture conjecture conjecture frame list version certainly inequality argument follow could analyze arithmetic matrix product incremental increment reach seem suppose minimize amount minimum path simple within success extend result incremental descent coordinate modify randomization tool jensen replacement nonlinear acknowledgement suggestion support award n nsf award cr support air research laboratory contract fa nsf award award research google finding recommendation express necessarily view include wishart geometric self loop complete arithmetic fix lower matrix angle slightly goal fourier nn recurrence pattern computation kind odd term induction fix follow write remove assume place permutation choose appear permutation element occur factor complete case care cyclic unity harmonic shorthand compute use geometric invariant frame tell since sum fix conclude remainder alone claim equal unity root generate eq equality unity union line line unique yx inspection else hence possible series eq lemma conjecture subsection subsection height pt pt sciences university randomize base decision pool progress theoretically replacement focus least formulate expect convergence replacement demonstrate inequality hold many well provide gap discrepancy explore prove consequence descent keyword definite randomized nesterov free replacement sampling quite
increase factor per equation application health proposition necessary accurately give scale disease would surveys health status health care measure make incidence observe compact continuous smoothness side question term approach inverse exponential although approximated affine interpolation quite well appear quite line inverse pose generalize publish relation incidence three death former theoretical example disease treat incidence death model model incidence disease prove death sometimes refer normal people death transition intensity henceforth denote incidence intensity age duration book duration follow differential equation describe system intensity figure system look relatively heavy easy imply eq population although population merely value could ode importance incidence age patient uniquely respectively clearly force incidence uniquely qualitatively quantitative infer cause force ode age profile know directly incidence cause allow study incidence study need recently inverse ill pose article organize generalize allow central partial pde ode case inverse pde disease section finally death henceforth age assume duration additionally proportion introduce ct partially hold incidence hence important overall negative disease stay result death absence necessary affected initial cauchy hand pde year solve problem follow show disease show disease address age time want age arise course incidence change cause formulate straight problem design period birth death disease place uniform equation denote positive term approximate life incidence death constant time simulation piece integer year birth diagnosis person ill age person three correspond identification counter birth year year decide non person year ill compete risk simulation des accomplish cumulative inversion decide death occur first case age disease death get death age exactly people year transform diagram extraction person event assume course advantageous context try measure incidence furthermore later age incidence addition show age incidence dash red blue comparison solid initial affine numerically ode ht visually agreement curve actually age specific predict percent point age disease measure incidence much direct function know profile assume incidence information assume incidence express product limit stem incidence non low
interaction original prediction evaluate membership protein unweighted interaction result prediction table measure analysis unweighte substantially primarily real value pair protein affect wide edge statistical protein transform network generate much result network unable observation original benchmark edge network class show perform original transform measure perform almost metric produce high increase able large major well neighborhood configuration two protein protein mean auc auc decrease c auc class auc change increase increase decrease auc decrease experiment protein original confidence utility content protein extensive investigation explain subsection result investigation protein even connect original versa configuration consequence transform improvement measure attribute change network part focus prominent change identify change scatter plot figure remain transform produce indicate tendency network focus assign accurate reliability interaction lead substantial difference transform include well formula denominator numerator assign protein protein degree network bottom corner low numerator high high indeed high network retain original connected neighbor point corner figure denominator high high numerator equation node original form neighbor configuration lead natural structure drop responsible study case subsection similarity filter interaction protein spurious interaction number connect benefit analyze difficult methodology edge original give noisy analysis study extent noisy version measure average auc noisy worse encouraging able dot even network interestingly function robustness transform original serve important improvement drop protein operation address protein study able prediction advantage measure discussion quantify protein measure advantage value reliability score able substantially improve prediction continuous version produce change original especially structural factor likely spurious introduce related protein researcher protein direction extension validation functional direction would measure perform characteristic presence weight edge since measure combine well support nsf grant fellowship school confidence measure mean max auc classes auc class decrease auc decrease h c c auc max classes auc auc auc apply text interaction cover protein bp member none network outperform metric capable extract rich refined protein interaction change quantify introduce transformation examine dropping measure record drop transform eliminate lead elimination examine comprehensive procedure two prune original transform collect non original edge network respectively respectively figure show small percentage edge drop drop noisy indicate indeed effective valid interaction percentage drop transform network result major original process change namely global coherence edge drop add overall avg avg study examine change base transformation influence transform edge three original drop keep approximately average share protein analysis along trend observe although small fraction retain drop add drop coherent edge add variation type determine functional answer last connect protein transform encouraging note transform transform network coherence network match preserve original drop coherent coherent although add coherent fraction transform small transformed institute multiscale department genomic new york ny usa science university mn usa mail abstract protein interaction study complex biological despite rich face quality challenge similarity form address measure interaction graph convert interaction transform corresponding effectiveness estimate transform original find transform original particularly reveal improvement measure link biological address disease protein module function protein since protein tend highly base protein despite rich embed interaction challenge affect prominent primarily positive interaction presence network affect another interaction interaction lack completeness mainly specific protein annotation biological insight gain datum present positive major challenge rich study local interaction issue interaction traditional approach connect however addition direct association protein association idea two protein direct interaction protein similarity network module module discover original fs similarity protein show performance utilize measure context handle protein interaction protein robust protein likely benefit performance measure context understand interaction firstly comparison difficult furthermore measure functional module discovery set use comparison hard attempt fill extensive comparative context unweighted interaction follow systematic graph network measure estimate compare quality original biological several transform accurate obtain contribution change due investigation ability measure noisy important well prediction effective novel association performance efficacy handle interaction protein detail assess protein source microarray since focus complementary combination accurate material evaluation protocol annotation interaction set include interaction protein unweighte study include database go analyse vote member protein interaction similarity define use notation direct similarity one measure two coefficient follow assume form unweighted incorporate presence interaction propose probabilistic measure significance neighborhood configuration two name chance binomial non unable take however significance measure protein section attempt fs functional fs measure common neighborhood protein interaction unweighted measure refer avg uv avg protein network protein least one protein score reliable assume protein direct separate similarity protein probability neighborhood conditional similar generalization name interaction unweighted weighted version note protein topological overlap co expression network association common neighborhood unweighted straightforward respectively numerator denominator inclusion desirable sensible contribute generalization zhang co network measure transform protein hc demonstrate application originally design
dynamic statistic process switch nonlinear continuous allow analogue biological us perform rescale dynamical describe across force position model offline discriminate learn approach change derive base switch propose monte constant diffusion dynamic approximate dynamic maintain present detail switch representation implement winner multidimensional sigmoid strength external behaviour winner implement determine centre determine identity dynamic implicitly one one lyapunov stable provide remain switching element lie interpolation observation parameter set dynamical system ensure stable fix switching logistic sigmoid choose increase logistic maintain evolve dynamic behaviour point point switch external switch movement represent gain describe learn dynamical limit improves govern implement motion around cycle position implement strength thus position dynamical implement position combination normalise circular von cycle evenly ad hoc consider learn position phase suggest allow actual model assume equation w covariance diag process switch induce switching provide flexibility reliably datum aim infer switch filtering solve optimally filter dynamical linear dynamical resort extend kalman filter trade version suggest filter integrate sigma transform sigma factor prescribe wiener standard I choice scalar correspond manually select facilitate switch choose moderate discrimination performance otherwise likely switch choice experiment describe synthetic human motion filter long datum possible infer fraction response trial identity choose typical motion capture dot connect illustration purpose show c line error grey switch output determine switch true green period begin high b noisy infer state noisy reliably inference dot trajectory plane form overlap depict position shift thus dot dynamically hide period dot dynamic generate uncertainty unless otherwise trial simulate switch current external dynamic online base dynamic variable start switch fix trajectory trial trial switch impulse quickly switch typical movement sequence increase step plane introduce noise average twice time prescribe fig nevertheless still movement gain free value performance cf half cycle response input r true walks walk noise walk noise motion trial identify correct dark blue red second switching dynamic motion example aim principle capture similar switching present motion walk important walk remove capture marker data dot see frame walk span dot highly walk principle walk capture normalise mean trajectory close normalise walk four normalise another represent infer stand normalise original marker position marker challenge test model original generate walk switch introduce jump total walk uncertainty trial dimensionality true achieve response addition add gaussian noise marker deviation equal marker position yet standard give furthermore misspecification uncertainty model nonlinear kalman dynamic process online hide synthetic plane light represent switch achieve stable dynamic process environment switching perform misspecification represent goal motion present dynamic cycle successful
take variation tail balance negative motivation loss net negative yield balance hill establish cdf normality characterize model equivalent cdf essentially statistic hill substantial considerable value represent issue tail reference square rmse enjoy estimator fortunately bias reduce exploit censor couple variation define equation eq reduce quantile establish author define eq replace get estimator establish normality carry section conclude remark section appendix optimal fraction function quite function determine known order unfortunately second regular asymptotic estimator variation specify rate constant sign near extract fraction asymptotically k fr cdf quantile taylor identify example fr view realistic show indeed assumption asymptotic follow give term normality thing normality meet need major approximate brownian integer define corollary sample ccccc ks ccccc ks ccccc ks ccccc ucb r r c r r c r r panel panel eps shape eps population parameter horizontal represent size new rmse moreover panel case tail show range tail panel pass normality test panel probability cdf brownian amongst state order eq cdf formula make representation statistic rewrite show theorem use application calculus us behavior integral elementary k rewrite eq make expansion observe may q n use approximation dt show remain converge ks complete theorem normality us integral formula simplify let verify first rewrite nc nc easy eq let show recently eq nc combine normality element analogue calculus finally calculation bias tailed bias high newly confidence easily estimator perform mm mm remark mathematics quantile introduce normality goodness square interval reduction extreme value heavy distribution hill I negative v tail infinity tail index e cdf tail model large price return etc construction rewrite quantile
namely modify summation start inequality density parameter equation determination coincide generalize modified formulae equation formulae introduction q definition approach prior signal read nonzero appearance denominator formula background interpret really flat lie conditional provide determined formula coincide naive eq drawback interval unity background contradict full lie infinity must unity drawback correspond bayes inequality inequality read derive coincide satisfy evident equality look q frequentist eq result modify frequentist show support grant modify frequentist definition namely pn definition prior propose modify physics determination observe due parameter probability determine lie except evident popular option interval inside confidence number event determine value determination identity determine parameter lie modify frequentist equivalent
present phenomena stochastic interaction interaction particular community naturally label chemical may movie collaborative rating email main contribution generalization describe identifiable context label block transition propagation threshold label conjecture validate belief propagation detection draw address label reconstruction regime et spectra grow high et al belief determine regime initially al tree simple census threshold survey understanding threshold still coincide et al community contrast detection label explicitly consider follow node split block namely block node refer belong node relate one observe draw al relative type infeasible feasible degree make un several support conjecture couple see edge consider follow starting node consider branching characteristic birth child poisson distribution poisson parent draw everything else consider depth denote subtree root together bayes entail recursion uniform leave uniform constitute robustness belief ratio less belief root belief I variable point say insensitive ready denote average branching tree quantity insensitive prove implication un label infeasible insensitive classifying correctly nod conjecture technical branching process parent branching endow I moment sn sum path let er transform er determine behaviour deviation weight branch sure similarly branch q expectation summation read large summation borel lemma finitely thus branch indeed nd positive sure tend tend infinity lower bind minimize desire linearization formula read absolute suitably label l weight derive laplace point supremum necessarily convex duality case equivalently equivalent strictly sensitivity perturbation perturbation parent child type parent distribute tree non get exactly type vertice child ij type vertex throughout child opposite label correctly reconstruct clearly infinite growth realization distribution adapting infinite label reconstruct type th level derive bind effective electrical upper maximum background notion follow child bl ij r r ba tend infinity hence computation ij also uv uv g eps ab cm eps numerically label stochastic model symmetric I among node type type reconstruction value validate conjecture great lead characterize success reconstruction use introduce denote assignment community setup assign take overlap metric range assign randomly label graph belief base vary value seed line curve attribute fig accordingly belief propagation label correspond curve shift fail threshold overlap achieve analysis context label conjecture symmetric community affect rich community potentially characterize namely sensitivity coincide label tree main conjecture characterize theorem
span span proof find light qr iterate triangular obtain qr give define kb tu u tu notational simplicity break lemma state complete proof satisfy rip norm th spectral rip matrix lemma u k v show recover use measurement prove observe constitute sense satisfy approach incoherence done exhibit similarity sense incoherence perturb completion iterate step partitioning set prove establish subspace w h incoherence see sense alternate enough incoherent maintain guarantee lemma define also iterate pt u start alternate use proof general let q similarity update sense essentially special incoherent induction distance incoherent factor step incoherence incoherence incoherence step incoherence w multiply constant similarly multiply q assume argument prove lemma prove stress incoherence also incoherent incoherent directly run sample total large step stagewise one x recover nm significantly information modify call stage stage solve stage recover top step k also k noisy sense goal I formally present proof base f hold th step initial stage ok show initial stage formalize proof f satisfied iterate iv alternate minimization empirically appeal solve main motivation result theoretical would aspect result matrix rip incoherence algorithm iterate computationally fast optima high statistical minimization rank problem perturb perturbation extent rip incoherence demonstrate rigorous result minimization pca light aspect initialization easy iterate show succeed initial orthogonal subspace suggest iterate preferable stagewise adaptation modification alternate fact mostly definition microsoft com edu university mail edu alternate widely applicable method efficient form major win entry netflix step convex paper theoretical performance minimization completion problem pose tractable certain minimization succeed compare exist minimization guarantee particular find empirical represent bi form reason crucial several practical application recommender several estimating million matrix load memory hard disk optimize recommendation may rating million hundred thousand store rating final rank efficiently optimize modeling impose constraint matrix impose pca look due bi finding minimize result bi linearity correspondingly popular overall solve efficiently usage date almost theoretical work motivated practice completion complete observe popularity primary come user become appeal provide fast parametrization refer measurement recover back study completion entry completion element measurement without sense ill moreover svd involve manifold need manifold intensive exponential several robust additional relaxation rigorous minimization establish recovery norm assume problem recover give additive scalability problem completion provide scalable respective exist albeit drawback rip theorem level decrease base observation rip hold alternate view analyze perturb see detail proof rank span orthonormal basis space depend span column distance subspace decreases iterate angle distance ready correctness inequality follow rip follow
dimensionality improve furthermore signal less crucial hyperspectral scene fractional linearly mixed simplex abundance rely library usually laboratory term dictionary library different library observe constrained observe spectral infer signature implicit fractional I hyperspectral implement simultaneously linear baseline describe identification class technique fractional devote devote sparse contextual mathematical summarize obtain experiment plausible distinct negligible partition illustrated spectrum approximate linear mixture fractional measurement denote fractional abundance g give fractional name fractional area th fractional abundance vector sum abundance abundance researcher sometimes abundance account consider error signature cover hull green red vertex correspond simplex mix green identify simplex exploit algorithm adopt direction show fig noise vertex correspond hand side pixel contain material spectral datum middle infer fitting yet seminal work several hand correspond near resort adopt suitable linear fractional signature tend condition sensitive yield characterize expect value snr snr sd couple ratio snr must acceptable significantly corrupted fig mix fractional simplex db united survey fractional well column band remove noise matrix code read snr indicate signal subspace snr singular decay high signature nevertheless big shape slowly band assume good low linear vector yield gain snr usually advantageous necessary operate subspace require extraction suggest exploit objective minimize sum projection empirical maximize additive power power mathematically sequence transform first transform snr sd identity differ optical develop laboratory dimensionality extraction module identify pixel variability scene collect pixel angle metric create identification signal criterion minimum come mind criterion adopt approach turn develop pearson theory method determine hyperspectral base detector build sample term whiten infer estimate select mix dimensional linear dimensional subject topology method estimate example analysis manifold linear projection projection pursuit also refer orthonormal orthogonal projection spectral onto replacing project storage snr gain direct consequence explain latter assume covariance power project denote power signal subspace estimate side noisy spectra mixing interval fractional db project datum noisy project spectra identify db colored additive db top noisy scatter eigen blue dot final although projection often remove percentage change value eigen possible attack contextual leave I filter bm structure scatter plot noisy dot green dot eigen image bottom figure simplicity notation project original product simplification reality variability treat primarily invariance shape fairly signature affect positive factor pixel instead entire spectra pixel observe simplex rather illustrated transformation match reality mapping transformation impossible identify affine orthogonal onto dark accord set projection spectral angle affine choice issue illustrate fig publicly hyperspectral cube united hyperspectral line band hand side angle high angle vector usually area figure side project vector discard project vector pp volume approach pure pixel assume mean spectral vector simplex enable efficient view strong pixel probably often use hyperspectral conceptual mean pixel dimensionality snr random correspond vector pixel one fact simplex pixel find define large iterative implement vertex project orthogonal span already determine iterate simplex algorithm iteratively grow simplex finding vertex angle cone cone represent spectral vector start identify make cone making angle terminate tolerance maximize cyclic fashion quite concern project subspace span determine consider complete datum nonnegative abundance lattice value lattice operation construct max spectral contain appropriate use notion al seek mix volume simplex define pixel result fig simplex minimum datum true mixing simplex project onto simplex usually consider recall volume convex hull give origin simplex seminal identify apply projection iteratively simplex minimize simplex v identification lagrangian introduce allow positivity relevance depict perturbation true lead dash original positivity project onto theoretically conditioning aim solve linearly stand number follow maximize minimize b approach constraint hard optimize convex lagrange computationally aim hard positivity cyclic linear pure pixel version effect chance act soft constraint fractional chance volume mix alternate involve solver factorization solve original set minimize square simplex volume nmf implement alternate programming latter robustness h algorithm aim solve distance simplex vertex volume regularizer ambient non exactly recently ice implement alternate ice regularizer former minimization square nmf ice ice incorporate proportion prior large initialization proportion proportion zero discard ice sense datum correct independent source dependent constrain solution meaningful range pose adopt posterior abundance assume priori bayes paradigm stand probability observation model unknown popular joint matrix clear ice nmf classify minimize play consists conversely assign abundance respectively convenient ensure inherent observation mean gaussian autoregressive iterative respect abundance linear distribution interest conjugate physical informative joint posterior close estimate design remain impossible chain propose generate sample parameter mainly unknown chemical spectral mix distribution assign abundance efficient operational respectively instead estimate spectra hyperspectral estimate identify dimension bayesian approach depict toy illustrative compose pure fulfilled maximize volume correctly contrary statistical hypothesis star independent uniform admissible abundance equivalent seem span characteristic detailed paragraph simplex suggest recover conduct choose volume simplex implementation bayesian rely identically gaussian lead matrix vector note colored covariance handle many low band projection subspace largely spectral abundance compute em estimate ica attack assumption fractional hyperspectral fractional infer mix mixture dirichlet density enforce constraint abundance cyclic infer augment signature investigate interesting normal compositional stage stage respective gaussian second stage pixel abundance multinomial dirichlet abundance set quite consume joint maximum estimate joint sampler inherently theoretically correct speed dominate fig nmf algorithm generate library fractional mixture dirichlet mode dirichlet randomly initialize reflect situation region scene tune correspond scenario poorly contrary yield useful hand side dominate dominate image circle identify although really sure true reasonable conclude example fig bayesian refer link spectral observe signature linear pure signature advance ground amount optimal signature library mixed pixel scene combinatorial efficient regularizer mixed grow availability library area strong angle basis pursuit pursuit matching select say available amount find library pixel scene fractional library regression system satisfy linearly unique least noiseless pursuit omp replace norm term approach pursuit constrain denoise term multiplier interpretable perhaps totally fractional reconstruct solving provide incoherent sparse signature tend limit impose hyperspectral point simplex add sum depend optimization convert constrained square feasibility problem case hyperspectral library admit sufficiently sparse unique act induce regularizer point rarely reason noise run experiment hyperspectral angle spectral signature abundance dirichlet yield beyond reach evidence impact coherence minimum signature introduce optimal dirichlet mixture stand abundance respectively degradation perfect perfect curve challenge db useful notice plot material snr value correct selection material crucially availability hyperspectral library acquisition library consume library consideration adapt vice way library directly information idea term signal processing reference al attack modify exist learn representation learn signature scene approximate material criterion generally hyperspectral implement spectral usually algebraic positivity fitting term likelihood direct ignore contextual great hyperspectral hyperspectral contain image pixel process individually hyperspectral cube take advantage follow integration contextual information guide abundance step spatial attempt spatial pixel abundance dependency particularly adapt image abundance pixel partition abundance markov field labeling variable conditionally pixel individually generalize region must choose extension base nonparametric markov several exploit spatial information designing criterion addition classical positivity include structure autocorrelation abundance measure vary smoothly information incorporate within include account automatic extraction spatial singular value decomposition use spectral extraction preprocesse vector scene intend preprocesse extraction mention exploit contextual assume deconvolution variation regularizer spatial enhance nonnegative factorization replace regularizer neighboring abundance limitation regression signature add variation term individual band follow collaborative add pixel decade research spatial resolution environment signature material spatial instantaneous imaging hyperspectral resolution component development recent development paper development hyperspectral cover mixing versus nonlinear identification integration describe physical many algorithm high activity limit many address manuscript however combine provide snapshot challenge hyperspectral field include signal modeling algebra regard trend image general particularly application monitor tracking g type chemical contamination application response decision near different heavily consider spectral hardware architecture gate graphic unit implementation together discovery correct method gibbs sampler advance multi development offer possibility hyperspectral piece accurately process practical yet several mention proper distribution become structured library area process become cover different area investigate software quantitative perform statistical acknowledge green team make hyperspectral united states publicly library signature acknowledge center available community superior mail lx image instantaneous thousand channel higher refer high spectral resolution identification require identify material multiple spectra measure material pixel material signature ill pose condition variability many search tractable describe contribute potential small spatial spectral resolution monitoring region spectrum band range record light locate illustrate measure organize form cube plane correspond acquire band vector acquire spectral hyperspectral refer process pixel spectra image collection spectral signature per material present represent material hyperspectral spectra grow suppose percentage cover want care distinguish signature remove interested present scene obviously application percentage scene pixel states mixture author laboratory set calibration hyperspectral accurate linear material cross area highly dominate dark object inaccurate amount accurate estimate contribution material pixel regardless large hyperspectral ten year currently
optimisation modal slice specify energy boltzmann draw slice identify global mode project methodology include multi mode dominate boltzmann mcmc pose well functional optimisation mode boltzmann potential exploit slice develop chain mode avoid associated optimisation modal methodology optimisation literature build seminal hasting mh boltzmann optimisation additive functional dimension boltzmann distribution analysis mcmc local generally require mixing slice remarkably mix slice case energy mcmc popular simulation evolutionary mcmc liu al lee lee gray particle solely energy focus stochastic level wang algorithm rest describe base optimisation sampler conclude problem minima min optimisation multi modal minima simulation simulate exploit optimisation find energy z partition clearly minima mode simulation explicit knowledge boltzmann limiting case boltzmann distribution interest limit mode boltzmann minima original extend simulate draw recurrence start ergodic estimate modal however dynamic equilibrium distribution issue chain insight slice back chain converge high project lower describe development slice suppose wish high un density let auxiliary slice normalise provide slice define gibbs sampler hence draw wang slice product suppose auxiliary slice x u conditional I interest additive boltzmann slice extend slice pick set classic optimization inside long narrow flat get minimum need choose last nonlinear mcmc use collapse van conditional conditional invert slice sensitivity burn minima local maximum boltzmann quadratic slice augmentation slice inequality follow combine argue similar fashion constraint conditional sensitivity slice long chain level equal contour underlie function region distribution slice invertible set result conditional truncate normal slice figure show burn boltzmann mode straightforward domain conditional auxiliary slice uniformly run gibbs conditional defines sampler sensitivity namely slice boltzmann efficient optimisation slice sampling modal flexible enough additive multiplicative fx x minimum straightforward example quadratic minimum immediately identify simulation simulate anneal mixed optimisation sampling alternative ny liu wang application analysis york annealing b york simulate annealing york university thompson anneal lee optimisation categorical input sensitivity importance variant statistical mechanic chen sampler liu computation local liu slice sampler van partially collapse sampler dynamic problem certain comment et g g geometric
I jx seem tree definite exist asynchronous algorithm plug root r dependent computation leave two theory laboratory f ed electrical university ct belief multivariate gaussian equivalently minimum guarantee convergence fail converge quadratic function failure mode understand via graph parameterize tree remain positive demonstrate iterative quadratic gauss always converge work study reweighte minimize equivalent positive covariance definite gaussian belief propagation message pass variance provide variance sum area dominant covariance scale tree condition necessary definite computation tree definite matrix arbitrary tree occur dominant loading load matrix subroutine produce feedback produce repeat decentralized amount load feedback quick provably convergent min algorithm convergent sum diffusion belief reweighte propagation coordinate maximization pass guarantee correctness plausible search convergent message first graph cover combinatorial characterization characterization allow conclude convergent message pass correctness correct assignment outside walk investigate reweighted belief algorithms undesirable typically answer diagonal may definite although similar loading parameter reweighte variance definite converge exist parameter extend idea general convex outline reweighted generalization cover characterize examine variance reweighte algorithm quadratic examine reweighte min sum finally section summarize main sum factor proof derive min minimizing factorize self edge minimize factorization bipartite graph correspond omit single reduce graph figure typically omit clarity message pass execution min pass back pass jx neighbor difference understanding update correct central min arbitrary variable affects locate constant avoid numerical issue may execution think vector index edge pass value message typical assumption choose message order min marginal message set belief approximate ix j ix min solution construct equivalently uniquely locally global contain arbitrary graphical min version message pass min sum need belief correctness convergence algorithm graph single demonstrate tree min depth tree root length backtrack walk backtrack successively vertice node root recursively leaf neighbor copy node operate copy potential construction represent circle minimum none edge potential subscript message time receive message add one message tree terminate lemma belief algorithm correspond message initial example computation view change update belief small say algorithm converge real objective always equation theorem min guarantee guarantee alternative message pass suffer drawback effort convergent message pass result algorithm obtain standard choose call weight problem choice surprisingly weight guarantee correctness cause incorrect solution message message analogously x c ix ij jx x vector message belief message factorization correspond marginal belief exercise draw scale node label node node right label right edge x x edge x node label factor self adjacent variable reduce computation tree produce pass order however pass multiply potential tree multiply summarize message pass time tree computation leaf computation create connect node tree root contain backtrack length start length computation computation root multiply branch multiply concrete associate weight correspond pass new potential along standard node belief computation tree min generality tx tx tx tx pairwise factorization tx ix reweighte though appropriate reweighte step update parameterize quadratic eq update valid message analysis hold asynchronous belief locally always minimum assignment locally belief tree converge minimum completeness proof lemma locally minimize quadratic consequence definite reweighted solve dominant convergence computation denote entry graph cover iterative message pass address great strength two precise neighborhood copy cover study relation local cover pair cover cover original connected graph cover finite cover graph cover cover universal cover associate potential base sequel specify object objective pass reweighted sum distinguish graph identical node node message receive send message send copy local pass assignment copy lift belief define belief variable lift assignment assignment factor entry function nk identity quadratic notably critical cover critical similarly definition hx critical point vector fix hx lemma critical eigenvector eigenvector lift problem point cover cc negative eigenvalue unfortunately via minimization may figure cover iterative reweighted base reweighte reweighte converge correct assignment correspond cover characterize equivalent definite definite implication proof use detail consequence walk intuitive explanation condition message min theorem tree product correctness correct positive walk belief minimize early belief lift cover locally belief message reader reweighte message pass whenever semidefinite correctness belief edge example produce pass scheme quadratic correctness correct original dominant dominant may eventually possess non positive happen infinity course correct understand affect reweighte reweighted remain course belief belief determine begin consider ij valid ji c ji follow j induction hypothesis exhibit weak suppose computation c require tree positive sufficient convergence variance computation positive see ensure element force computation cause behave almost choice computation generate exploit computation scale dominant case message result suppose trivially satisfy c ij induction symmetric diagonal tree ta tc ij eigenvalue tree bound way long monotonic decrease remain definite limit belief definite estimate converge belief must minimize see message pass order update perform order message asynchronous asynchronous computation extend asynchronous asynchronous allow update cover via vector order variable message copy belong already bipartite kronecker double disjoint algorithm asynchronous kronecker double cover asynchronous alternate message label label node label asynchronous two specifically double early analysis kronecker double bad might solution reasoning message pass iterative minimization definite equivalent system study elimination cholesky system gauss show use algorithm positive minimize cyclic gauss produce iterate gauss symmetric ordering follow variable analog perform n algorithm produce cycle gauss gauss semidefinite matrix strictly diagonal gauss exist immediately semidefinite matrix converge semidefinite gauss extension may point correspond system system first construct j ki ij ji ij converge
pattern recognition tag machine learning supervise hard ml pg feature extraction work supervise learning include pg popular learn engine select technique pca vector like pca recommend deal want ml pg library provide pg library feature extraction mechanism proof library file library file level library transform file internal implement list default ml pg library perform library first question need increase library reason pg librarie lemma library introduce illustrate pg user small library pg list run table efficient inefficient lemma prove pg interface machine pg table group proof agree one possible lemma fundamental natural operation split keep frequency click name split description show mode pg library cluster current may technology aid interactive case cluster library pg interface two figure incomplete development library lemma ml pg pattern side figure pg statistic take extraction machine interface suggestion ml pg vector pg discover series sum odd different correlation fact sum current suggestion pg brief description flexibility modularity pg come free light backward pg implement translate form cluster name external concern external tool system external even section highlight ml light handle handle pg convenient environment divide learn input object example high serve may cluster example happen certain find run determine ml pg handling program cluster pg generate handle output statistic program three frequency explain cluster interactive mining may proof pg choice search pattern refine extreme avoid value produce library find big value cluster determine experimentally ml mainly learn heuristic optimal cluster pg interactive interactive consideration size library auxiliary cluster user ml stand produce cc c c frequency library frequency parameter lemma table pg lemma proof n induction note desirable ml ten suggestion relate lemma ml see figure proof finish n notice goal inductive rule imply way suggest pattern high similar lemma potentially pg analyse output pg actually double criterion close cluster range distant indicating indicate probably assign wrong whose ignore pg fix interface access effect proof property provide therefore frequency trivial initially frequency trivially pg time frequency cluster discard low significant pg item frequency serve proof hand acceptable differ recognition ml pg allow user vary threshold parameter show come experience library line modular approach pg wide choice keep ml pg different highlight c c matlab gaussian indicate pg frequency threshold cluster lemma cluster irrespective choose lemma result effect level ml pg library value frequency parameter proof split smaller notably see simplification cluster figure parameter discard typical usually produce big big split interactive cluster form choose goal example highlight form belong cluster highlight form become frequency increase increase effect increase demonstrate mining library goal level bring level focus example whereas focus relate finish pg ml pg transform file format consequence ml pg file node whose child lemma belong file machine process way depend pg convert list pair contain name list goal dependent cluster search list current pg high frequency display overview automate interactive proof background introduction general interface probably different technique generic planning interactive theorem interface recommender mining implement could method prove termination implement theory discovery mechanism derive program symbolic tailor certain library certain proof properly deal library hand statistical machine library problem discover ml belong category successful provide efficient integrate statement relevant result successful library since proof infeasible receive tackle library send syntactic predict whether proof library hierarchy figure approach rectangle naive anchor anchor north white difference tool handle grow pg achieve extraction compact feature ml pg also library interact stage library big well tool aim increase goal user feature extract order formula proof vector sparse svms bayesian mean gaussian big interface library present pg machine interface prove statistical result non highlight pg flexible proof ml pg interaction learn help analyse interpret avoid pg benefit knowledge determine frequency frequency easily extend modular tool regard level statistical design allow environment learning mode addition library notation level plan feature extraction method plain pg detect pattern extraction implement pg way first consider five integrated pg whereas proof varied patch partially proof library spread library tackle close lemma pg pg combine easy mechanism implement search pg pattern find move symbolic family ml plan integrate machine help tool track fail machine strategy experience could also discover various include interaction extraction mechanism interaction pg cluster library think server user useful especially program purpose pg name notation grateful anonymous comment suggestion individual research interactive prove fm project participant rgb rgb thm lemma thm remark definition remark figure pg extension allow goal structure library interactive write matlab pg automate matlab result pg interactive development interactive prove decade prove solver become increasingly efficient type environment conceptual mainly advance enforce employ art offer user generate lot challenge inherently especially output order external order proof suggest valid mention improve perspective statistical automate interactive interactive successful mainly improve issue statistical challenge rich finding proof structure apply notion regard traditionally proof nature aspect recognition statistical interactive challenge consume challenging find understand guide interactive interested include fail guide tool experiment interactive lack interactive mining extraction proof semi automate inherently interactive user important range among experiment build upon maintain strong interface trend statistical interpret result uniform environment matlab underlie programming language learn interface source interface wide variety hand translation use view meet technology improve user behaviour feed back proof primary accomplish become interface rectangle control anchor north fill white white rectangle build machine matlab pay attention address vision call pg ml pg manual pg interactive relate proof development goal tree challenge issue pg automatically aspect machine length challenge ml pg must enable range machine suitable assume want amount pre processing pg question ml pg automatically connect interface collect analyse interpret output stage challenge backward machine tool proof less demanding seek statistical arise backward finally ml interact relation proof detect library even challenge aspect development interactive potential recognition aspect benchmark statistical pattern recognition prove pg go useful option experience survey work extension possible run kind statistical help expect ml pg library lemma list notably simplification goal induction goal trivial library list user pg problem previously pg lemma library namely old give table go proof operation lemma shape statistical goal trivial goal n mining shape lemma pattern case general similar pattern also may apparent formula type abstraction automatic interactive column table pattern sequence apparent always serve important ml pg sensitive effect one several proof step immediate transformation proof single throughout dimensional array shown allow induction trivial goal top lemma feature feature correlation n successive proof step consider goal rewrite move prop rewrite prop prop move prop move rewrite prop l prop table odd pg name stand big index big seq odd odd add big library highlight bold contrary odd pg learn patterns style chosen ml pg feature associate level extraction worth level base application difference concrete concrete plain package implement impose pg plain row represent almost ml currently consist extract close correlation trivial trivial row table encode property plain type argument hypothesis none pg goal apply extraction bold table lemma odd none prop l l prop move prop prop move move prop bold sum odd pg big n fact big add add library ml pg feature table flow use odd tree move move rewrite prop move rewrite big argument inductive name add add big lemma library column
find ccc ccc bernoulli horizon c min bt learn policy bold report policy consider training horizon already bernoulli prove gaussian make tuning sometimes happen horizon learn policy systematically outperform policy numerical nature policy extremely hard interpret understand relate symbolic policy box enable performance symbolic interpretable strategy interpretability performance tradeoff several field worth equivalence strategy resp symbolic horizon policy good training horizon prove well policy propose paper knowledge exploration test armed bandit horizon policy outperform publish robustness wrong highlighted evaluate armed bandit opinion direction improve policy overfitte occur call technique along idea could identify candidate behave good policy certain expect aim certainly relevant policy alternatively see adapt well bound perhaps identify policy give small expect regret good performance armed exploitation scheme principle investigation study successful ac exploitation many bandit formalize canonical form knowledge lack approach incorporate address class e propose step I target ii strategy performance parameterize symbolic appropriate bernoulli various play meta learn e outperform strategy greedy robustness learn strategy truncate many science artificial intelligence finance step reward machine response characterize distribution rational risk neutral play expected variance reward reward distribution decide play goal consist try use current decide play effort essence difficult impose play theoretical armed focus design provably asymptotic assume reward distribution bound play confidence reward arm round high strategy index typically involve performance usually report manually tune share similarity problem trial simulation good policy play armed exploit bandit armed number training consist search e yield performance tune hyper index policy within much broad prior two compose learn generate symbolic formula empirically play horizon fully significantly wide previously policy careful test learn truncate distribution idea armed policy policy identification pure exploration formally armed index policy state symbolic report armed policy arm bandit play process arm play refer optimal denote bandit minimize maximize ideally tb kb bandit bandit policy arm value response arm play play machine subsequent policy score k arm tie break random bandit parameter note sum give exploration playing rely prior e bandit many desire strategy exploit knowledge family whose policy fully give play horizon since cumulative regret value perform training make hypothesis optimization solve section consider strategy define parametric family candidate feature rely describe feature define product operator policy history play combination feature rich e strategy propose possibility history perform four root logarithm inverse arm multiplied way give combination degree resp policy parameterization change bandit episode global optimization yet rely region sample solution repeat population model probabilistic candidate train problem sample candidate probabilistic adopt prove quite policy gaussian policy literature three sub propose small formula build upon formula close formula advantage formally formula unary unary atomic variable constant provide cardinality four elementary operator contain absolute opposite figure formula occur k policy index formula discuss symbolic parameterization good policy multiple time propose partitioned equivalence formula equivalence typically far trivial specific rule perform involve advanced static believe difficult solution consist formula return formula random formally follow respective realization ff f lead cause division logarithm discard minimal minimal cardinality formula naive finding implement inefficient contain formula relatively bad relatively formula formalize multi associate arm select episode index policy reward quantity episode arm reward try formula identify bandit arm arm multi armed index pure exploration bandit work bandit algorithm reward instance depend time play far select bandit clearly correspond bandit rely armed technique problem hundred benefit compare previously propose policy case exploitation strategy test policy learn former policy policy default hyper tune scenario priori parameter care issue bandit problem bandit
role increase amount behavioral naturally minimize contribution definition iteratively membership snapshot nmf matrix max membership role structural evolve role less important active inactive role currently probability shift occur role future structural represent analyze behavioral role basic center star drift evolves analyze dynamic role mass shift role active expect role would denote list max snapshot node feature iteratively membership give nmf membership consecutive inactive take role time role active role inactive whereas lot role role increase dynamic refer evolution structural structural behavioral node behavioral drastically communication stay structure therefore structural behavior center star incoming bridge connect network consistently behavior drastically role notion decrease trend g role versus versus home besides track detect change occur similarity role membership across node membership detect anomaly minute r ex dynamic analyze ip trace dynamic consist million node real citation propose model quadratic unable model investigate sized node unable realistic million exploratory approach node minute second dataset million approach take linearity method apply intel core ghz gb framework also trivially behavioral role parallelization make attractive pattern naturally apply stream fashion monitoring error demonstrate utility track individual dynamic result indicate behavior entire important communication domain explore dynamic specific behavioral pose particular type seek question characteristic learn behavioral role day change day behavioral role normal another node change lot trend evolve behavior across role drift axis importance structural dynamic ip communication role structural property role behavioral axis first analyze last dynamic university email email university email account trace edge across hour activity feature result behavioral trace address communication communication begin minute interpretation dynamic role show interpretation role role whereas role combination structural characteristic major evolutionary particular consistently property also cluster slight role many interesting certain role cycle trend structural behavior node additionally nod pattern activity become active inactive represent white structural world email network structural pattern significant trend pattern use role interpretation stable role period surprising membership day email communication role period represent inconsistent inconsistent behavior anomalous activity two day role node dominate role individual role clearly node identify dynamical dynamic email communication ip trace often dominate role nevertheless network dynamic nod human behavioral fluctuation day lot scalable vary completely capable handling handle minute capture particular topology coefficient biological miss perhaps evolve property may long pattern traditional connectivity pattern understand frequent actually instead fully dynamic paper imagine type select difficult connectivity understand pattern make manual impossible knowledge biological network representative truly property also compute edge manually tune interpretability extensive manually costly time inaccurate novel pattern furthermore pattern stationary system could completely automatic capture main disadvantage approach however analytical tool pattern widely plan sophisticated analyze connectivity nevertheless time anomaly goal detect node link anomalous interpretation pattern anomaly pattern anomaly fail capture novel explore therefore representative generalize evolve capable capture anomaly suitable attack may plan detect take g entire network change future structural dynamic track level connectivity require edge whole node use basis sophisticated future transition apply task plan nod structural dynamic acknowledgment laboratory contract de ac nsf number air office scientific national science engineering fellowship reproduce purpose conclusion herein interpret necessarily policy either imply nsf network biological behavioral role represent connectivity parametric automatically learn role behavioral center star bridge connect community novel arbitrary track particular dynamic non membership perhaps indicate anomaly trend among network pattern evolve time overall effectiveness tracking dynamic network dynamic representation building tool fast explore database biological network nod attribute dynamical change considerably induce pattern automate far observe network necessarily considerably usually significant continuously automatic identify track tracking analyze structural fast completely automatic manner approach capture linear heart build sophisticated tool dynamic individual activity instance ip ip want behavioral role allow characterize behavior detect begin dynamic behavior feature time track membership dependencie role track behavioral role behavioral role role connectivity pattern role similar network recursively automatically therefore node share analyze dynamic dynamic whole dynamic present evolve importance pattern entirely example behavioral dynamic initial privacy twitter mobile device etc fluctuation change entirely contrast dynamic structural structural behavioral instance hour serve work email activity behavior star large incoming connect community follow explore serve tool structural specify pattern explore perhaps suitable world network clearly nod network dynamic pattern communication agree human intuition increase predict temporal work mine one importance node work exploratory discover structural automatically therefore social communication biological applicable domain structural variety pattern become connect network period structural dynamic learn diversity adjust behavioral sequence graph graph generator feature behavioral internet topology generator topology generator internet learn facebook role analyze google govern e network
feature within conclusion induction accord signal without suppose relation take set iterate x note combine l q choose obtain signal eq induction hold index magnitude u n n last use decay eq contradiction definition matching pursuit natural extension omp atom per omp add atom orthogonal matching pursuit rip show rip absolute within slowly decay particular recover slowly within pursuit omp sparse commonly compress sense q sense sampling omp isometry rip recover compress eq q sparse eq I analysis omp primarily direction coherence rip analyze rip condition improve rip rip omp recover uniformly omp allow omp iteration omp atom least square direction rip omp sparse type algorithm base omp orthogonal pursuit sp variant extension omp matching pursuit omp select omp atom procedure omp omp study name iteration index feature z r x x rip recover iteration main give rip recover signal recover numerical us conjecture recover slowly decay within rip recover set case answer suppose matrix rip initial recover zhang omp make result extend paper additive strictly liu rip rip depend theorem answer question interest find matrix whose see rip make rip one comparison performance give parameter sample set distribution subset generate sparse omp repeat succeed record illustrate omp omp theoretical whereas right depict decay sparse naturally want signal less experiment decay run step decay omp firstly definition decay without decay rip decay omp improve rip decay omp iteration right rip rip lemma denote vector index generalize jt similarly inequalitie q part u cauchy schwarz imply n eq space furthermore imply similarly since x fourth follow accord na n noting combine last rip since p imply u nu indice u u h u x n lemma eq put arrive lemma result
red width pt south near color line south node start color red width south north regularization apparent certain present correspond top regularization worth series contiguous traffic describe accommodate dependent since flow satisfactory dl technique complexity suit monitoring wide count measurement sample collect purpose phase link slot assess entry link count tune choose show reconstruction comparison fix diffusion entropy penalize l solve ls regularizer encourage traffic volume stochastically dl base compete approach furthermore decrease remarkably actual traffic monitoring delay connect path delay operator assessment plan diagnosis challenge primarily grow delay measure partial current delay p tool environmental krige scheme delay measurement introduce therein topology delay build idea capable spatio put kalman filter variation delay topology krige predictor path comprise due contribution link model delay collect depend traffic temporal periodic well drive delay spatially path model link sec spatio generally refer random propagation channel spatio entail treat ls find namely recently diffusion wavelet delay correlation network operation minute delay network bottom wavelet bottom right delay several around change delay map summarize operational spatio temporal early efficiently past measurement towards kalman employ yield equation term gain final give covariance delay yield noise kalman allow prediction well second namely measured turn design monotonic greedy algorithm readily increase framework admit problem formulation capable measure art tracking path selection heuristic note parameter adapt include delay collect month ip square left observe far improve traffic anomaly across flow traffic engineering traffic physical cardinality operational traffic flow associate specific adopt mean multiple connect pair path discrete horizon otherwise estimate topology also physical traffic temporal traffic addition periodic respectively thus validate temporal flow since normalize rapidly flow experience due failure behavior g attack aim service amount account anomalous f traffic change difficulty anomalie anomalous spike flow traffic interference flow stem link measurement operational reality traffic indirect traffic measurement tuple e link measurement compact q argument keep unchanged flow traffic term nominal level anomaly occur short time possibly flows suppose anomalous instant anomaly traffic row give level measurement critical monitoring aim anomaly argue sparsity property instrumental network th flow one addition affect magnitude event examine link anomalous flow subsequently contingency heart seminal anomaly absence miss component analysis decompose anomalous model residual dominant nominal span dominant anomalous orthogonal operational phase exceed amenable priori traffic effectively rank alternative bilinear leverage provide obtain non bilinear term r partition table adopt frobenius optimality could considerably may point globally however globally violate unless implicitly cost inside introduce auxiliary copy per copy yield connect impose neighborhood agree framework regularize put adopt alternate direction multiplier admm unconstrained quadratic refine subspace thresholding anomaly map exchange estimate directly connect overhead problem far admm offer literature empirical especially convex favorable linearly bi potentially extensive indeed problem attain consensus optimality centralized monitoring computer addition change miss challenge identification effect load balancing cause e anomaly slowly table dynamic partially general change load traffic completely link live low subspace may true update frequent network scale acquisition monitoring task datum network reveal week thus safe lie exploit spatio anomaly top previous acquire recursively historical naturally place possible counterpart exponentially weight l distant anomaly environment formulation coincide provably convergent base anomaly traffic critical nesterov acceleration anomalous flow robust rank subspace corrupt incomplete namely incremental descent subspace projection tracking miss problem issue relate model instance outline sense physical wireless cr well completion collect link level protocol anomaly standard flow leverage low solve aim monitoring cm end internet application file prefer establish connection close resource quantify choice delay unfortunately pairwise infeasible measurement distance pair matrix strong delay multiple intuitively correlation nearby node belong overlap common link low distribute requirement network motivate factorization algorithm require symmetric avoid overhead leverage observe traffic reference collect observe total network miss typically column internet structure good delay typically imputation interest sec either path delay observe spatial cr rf fashion global power psd capture channel gain cg propagation medium per frequency map enable identification band localization tracking activity reader treatment introduce build psd frequency spatially frequencie unknown virtual spatial grid cast problem vector control narrow band nature operational due nonzero correspond map active motivate lasso basis sparse ls cope matrix arise inaccurate channel estimator capture propagation psd band plan include bottom span band estimate right recover total center utilize transmission level location dynamic dynamically heterogeneous key full yet delay well accurate identification partial corrupted look scale sp collaborative monitoring objective management include adapt environment operate ii development scalable tool track purpose management iii ensure robustness face attack develop adaptation design categorization cm cm cm edu communication specialize transmission system device become orient heterogeneous service streaming thank advance speed capacity demand experience entail high failure attack research vision enable robust operation management dynamically evolve landscape human intervention analytical relevance sp monitoring sp construct scalable tailor heterogeneous network service internet device volume day wireless connectivity dynamic rely mobile united capability make device cognitive cr ensure service operator comprehensive landscape key management risk analysis security come challenge traffic volume interest instantaneous network management capacity plan traffic count readily acquire management instance link approach series square complexity internet traffic effort toward network monitoring delay great interest service affect experience number origin impractical even base inherent trait task operational paradigm current entail central monitoring interaction prefer consideration couple additional limitation first resource heavy tend network enough separate reference therein parsimonious descriptor monitor management state incorporate traffic volume delay activity intend traditional signature increasingly raw dimensional anomaly reliable manner diagnosis statistical instrumental maintain experience dynamic environment ensure network security monitor management complex complementary online construction map network anomaly flow recent advance statistical processing sp kalman dynamical minimization matrix semi optimization multiplier sp dynamic monitoring enable health lead enhanced robustness deal incomplete measurement domain measurement stationary traffic delay krige internet comprise node flow count collect incomplete due hardware matrix synchronization count simple measurement processing software rely interpolation series accuracy miss entry series none true real stationarity contiguous individually origin traffic flow relate term carry difficult link count goal estimate flow approach flow gaussian logit ultimately mechanism regularizer link count contiguous time miss historical measurement leverage structural regularity semi dl approach dl dictionaries representation mean capture success compressive cs signal lead advance audio motivated link count linear column dictionary admit predefine dictionary fourier respectively exhibit traffic
france lot attention devoted kind scheme combine tackle elegant quadratic program come theory low misclassification evidence naturally fusion add preserve improve behavior fusion real fusion rank combine multimodal issue lot research effort see survey fusion input bring source semantic audio event modality correspond generally different view classical commonly tf bag texture sift spatio temporal descriptor able discriminate scheme generally fusion merge unimodal classification heterogeneous sometimes score available classification fusion fusion outperform significantly tend give often refer stack extra multimodal fusion stack associate classical look combination usually weight majority vote cost machine assess diversity adaboost multimodal fusion method adaboost introduce weak another risk completely fully classifier since output propose majority vote score account majority aim fusion able positive additional ranking pairwise conclude empirically section vote pac dimension I real value otherwise aim lead majority vote risk majority auto moreover consider opposite restrictive represent quasi elegant trick usual core majority vote n I moment margin generalization propose minimize minimize risk vote account quasi find minimize uniformity weighted majority vote pac rate vote appear classifier fusion modality randomly subset I achieve low risk application document retrieval relate vote adaptation improve measure real value ss j sm example score map example top behind preference positive method measure notion order preserve multiclass hinge order positive j h j h force hinge previous eq hinge quasi term hinge consider additional slack abuse highlight difference simply drawback make hard propose negative force positive score vote lead process cv lead map empirically extension fusion stacking implement base goal example independently draw keep test objective good benchmark
robust asymptotic design phrase asymptotic optimality sequential minus identically distribute I vector density degenerate testing versus common sequential select measurable terminal rule stop accept test control test sequential test maximal type sequential hypothesis simple probability seminal statistic parameter see uniformly asymptotically replace statistic f apart moreover approximate often overcome adaptive observation approach initially sequential lead delayed estimator bayesian cost density q integer hypothesis two serve replace subset mention statistic ratio always easily discretization hypothesis loss efficiency advantage many application second wide present area detect signal sensor take density resp present sequential parametrize stop call weighted generalize ratio ratio replace sense attain asymmetric channel jx asymptotic setup want answer select far select correction addition kullback leibler accumulate attain open favorable prior take observation expansion operate select lead induce especially channel expansion essentially design remainder paper preliminary asymptotic operating test whereas establish optimality section compare specification test use quantify kl order divergence situation attain situation every trivial coincide throughout paper time respectively transform connect number number quantity important great testing moment z size say emphasis parametrize arbitrary follow also role asymptotic operating characteristic section expect size rely decomposition hold weight eq follow precise slowly change page decomposition random walk imply allow random walk approximation theory specifically define vector covariance matrix follow lemma prove b distribution apply mu asymptotic additionally define argument lead along positive finally inequality first remain similar way clear b correct approximation expect sample size rest kk mt necessarily moment moreover select belong theorem bayesian upon resp resp define h h I sum ct sd sd integrate stop problem sequential structure seminal state sequential follow specifically mi mi cb rely order threshold theorem go write simply mi similarly threshold establish sequential test thus threshold clear right suffice q negligible specifically sequential fix doe inequality due recall cm ct second rearrange eq inequality become similar asymptotic select strongly believe way away zero allow asymmetric integrable thus word kl accumulate follow accumulate independent depend additionally select pd pd imply consequently kl follow additionally select pd threshold select pd follow weight scenario test density quantify similarly q uncertainty alternative require testing probability eq side expression determine cardinality error every I ie choice would favorable particular prior rank sense I assign relatively ratio prior setup assume embed simplicity resp performance channel use set channel strong present channel high strength check accuracy compare realistic setup table emphasis signal set choose threshold whereas c three column compare two experiment level whose go indicate particular since also sharp type probability h c remain figure probability dash line triangle circle see asymptotic moreover identical present case probability scale dash line asymptotic whereas refer triangle resp circle resp work detail simple composite discrete asymptotically appropriate optimality divergence favorable simulation experiment
number consequently expression proof di nonparametric prior derive letter sample science page institute mathematical california de poisson stable ann cm secondary appear parameter rely approximation central unconditional approximate estimation specie inverse prior introduction exchangeable infinite exchangeable integer exchangeable function backward identify arise extreme namely conditional evy stable stable q partition exchangeable gibbs correspond specie al typically lie diversity specie unknown specie et al assume realization exchangeable belong predictive conditioning stand huge amount nonparametric implementation mix stable obtain explicitly mostly explicit alternative generalize mixing hence exponent rewrite change combination burden implicit prior additionally predictive specie ratio term increase look therefore possibility gibbs coefficient rely non central number approximation posterior rely apply finding approximate gamma approximation easy task aim generalize number combinatorial coefficient term hence number correspond partition sum generalize number block exchangeable gibbs arise poisson eq generalize number limit stable partition almost limit example proposition agree approximation enough notice pd h polynomial stable correspond q introduction predictive bayesian nonparametric species weight specie number specie exchangeable namely number et central generalize number convolution replicate previous section look exploit central follow approximation hold central number central rewrite gamma exchangeable gibbs stable proposition drive generality box new block additionally et eq poisson square new specie conditionally basic observe intermediate even formula poisson generalize large size would approximate generalize idea reduction report formula nonparametric prior exponentially letter diversity gibbs partition proceed conference computation complex bayesian prior st neutral species exchangeable triangle unify generalize control reinforcement non nonparametric
numerical paths ridge concept implementation work purpose rule example manifold circle although circle example aware cloud find pt web clutter difficult reach point four phenomenon bias homotopy apart piece suggest great require reach intersection corner violate far component ridge converge relate currently investigate question first establish procedure adapt mode theory work allow corner intersection develop extension sequentially remove shift investigate adapt investigate run result purpose lemma recall hessian onto q let denote dimensional normal enough induce distribution density lebesgue measure bt da define quantity bx x lemma p bx p x xx z xx x e w dt finally z ok x ok part loss coordinate span span xu z x ok b xu h l result turn claim dx dx calculation order restrict change dt similarly acknowledgement suggestion improve thank suggest simplified theorem support nsf national grant dms air grant fa problem ridge cloud density manifold numerical accurate many problem intrinsic low existence cluster inference research commonly unfortunately manifold even assumption homogeneous meaning attain offer define set essential underlie open space estimate density case density dimensional manifold finding cloud much like goal hyper ideally ridge focus point cloud definition ridge define hessian projection local maximizer direction hessian yet think analogy hessian hessian space direction show circle smooth ridge coincide convolution fact bias uniform ridge indeed mode main well estimate polynomial estimate paper many propose make definition concentrate near ridge provide understand hyper ridge kernel paper informally together ridge hausdorff distance define hausdorff boundary ridge ridge treat practical density instead assume estimate finite algorithm accordingly extension call ridge apply dimensional ridge cluster manifold relate reference therein field work include paper visualization exploratory issue history global locally definition relate appear literature also structure sensitive method paper ridge rather use suit study generally vast shape representative throughout value may section hyper section start point cloud five derivative density ridge far subset convolution common assumption logarithmic valuable ridge deconvolution datum draw additional clutter point noisy consist thought rely hessian q eigenvalue eigenvalue write column write vx vx vx space call local normal span l lx tangent space flow line curve flow intuition pass toward unlike path toward path project algorithm discrete step interpolation approximate flow map respect flow gx ridge curve satisfy first critical critical one regular point call definition ridge thus lie dimensional ridge record main denote radius exist jk space ridge intuition mode consistently derivative instead positive analogous imply direction size ridge background version symmetric eigenvalue eigenvector correspond similarly symmetric square symmetric q examine show convenient gradient path parameterization often subscript tf derivative write formula derivative arrange derivative integral unique unique path condition suffice show show differentiable rule whose th near establish ridge formalize notion function drop along move away ridge want emphasize point immediate ds ridge side hence l l g eq l il sg l l g l sl l sl sl sl se ds l h follow etc fx hold r decomposition small clearly enough gx gx gx lx lx lx gx gx estimate hide manifold bandwidth ridge define assume second derivative vanish boundary kernel density usual result law result prove error prove use law g jx kx kx h h assume q suffice purpose theorem next section let role take tend rate without boundary twice usual way chart away add subscript want ridge surrogate show subset neighborhood property lose corner gaussian fact let surrogate ok k mr would hausdorff near neighborhood manifold gaussian paper mixture see nearby mode highly mode think fact mixture refer ridge function around several surrogate neighborhood whereas preliminary
row signal model observe zero support far average signal denote theorem support notation dependency recovery together constant capture tradeoff understand quantity mild linearly indicate benefit new meanwhile characterize role limit measurement cause drastically special signal identical merely however properly certain condition support wireless communication block contribute non theorem instantaneous capacity conduct random block diversity representative trial trial matrix choose noiseless successful trial study intra algorithm size block block zero entry model coefficient assume range value adaptively intra completely ignore result clearly show correlation exploit exploit performance unchanged phenomenon correlation exploit correlation perform exploit correlation degradation conduct noisy experiment treat time column increase start column nonzero zeros nonzero ar snr treat exploit smoothness short smoothing inter l true variance learn vary concatenation multiple measurement vector exploit correlation compare correlation non review intra model intra measurement framework incorporate level limit application involve endowed exploiting improve performance nsf grant correlate phenomenon exploit intra exploit intra exploit correlation properly exploit essentially operate operate exist reweighte estimate previous iteration algorithm let suggest strategy intra block correlation iterative reweighted assume variant framework framework regularization extensive computer performance two framework ability high exist domain break bottleneck wireless make correlation latter inter become properly exploit family performance degradation use approach iterative reweighted type detail lemma work among value entry intra vector correlation sparse inter correlation context measurement model sparse present incorporate limit recovery discuss impact intra inter limit attention development measurement mn measurement many application structure uniformly dependency application exploit view concatenation block block model attention application general nonzero motivated appear detail problem typical vector row impose mentioned structure correlate challenge consider case call modeling zero algorithm correlation intra intra block basic assume multivariate distribution tend due relevance thus block capture correlation principal scalar estimate type maximum procedure likelihood hyperparameter exploit well partition th block consist issue structure stack thus kronecker overall th capture inter capture time natural vary assume stationary situation abundance option adaptive kalman give signal vary algorithm localization event stationary segment series certain duration model multivariate signal certain statistical signal deterministic assume dynamical
number ever case unique ji ji recall active concept derivation supplementary material share among corpus ibp exchangeability current allow write factor histogram concept unique concept follow et proposal material sampling share e particular exchangeability p ji ji ji ji straightforward mm induce nest compare concept learn hdp hdp collection number distribution vocabulary corpus record health care mark two year either house statement day select document correspond active measure removal name entity noun token appear leave vocabulary augment corpus take sentence co occurrence project approximately sampler choose comparison select probable sample introduction concept find coherent focused hdp word evident histogram fewer illustrative incorporate induce call drug benefit order quantitative conduct user idea cloud hdp choose pair coherent ask ten vocabulary salient generate remove word question identity participant hdp correspond advantage model example coherent concept hdp prefer concept hdp topic preference subtle understand american participant cloud consider participant claim follow closely understand preference study illustrate birth death proposal ibp concept birth call draw unique concept force birth proposal j respectively birth otherwise maintain hasting rewrite definition plug simplify propose acceptance replace therefore value sample proportional concept aim concept specify distribute gamma construct purpose work directly dirichlet maintain infinite multinomial concept assignment frequency conjugacy multinomial abuse notation utility dirichlet along draw j imply random variable ji document observation conjugacy conditional distribution corpus semantic derive material simplify follow document ji ji follow sample ji ji ji ji f denote dimensionality semantic normal wishart matrix covariance respectively diagonal independently write likelihood analytically likelihood standard normal wishart conjugacy yield unbounded restrict attention noise although concept equivalent simplify across document presence indicate absence gray identifiability illustrate generate underlie infer definition concept reconstruct additional vocabulary recover concept filter user participant sure health care participant ten question result favorable vote hdp ask many participant prefer treat sample tie loss binomial binomial test initialize sampler semantic feature concept add remove hyperparameter term representative concept concept sparse structure associate infer inherently concept standard hdp direct feature demonstrate utility representation nearly member researcher million scientific article try read information ir evolve move search remain representation ultimately atomic consequence task set document representation must expressive fine grain name mr distinct name entity th united may end coarse grain representation topic problem individual estimate interest concept seek appropriate represent computer vision community concept model model interpretation concept incorporation ir system vocabulary occur underlie semantic concept differently document set address property rely strict concept uncertainty encourage instance say assume document create refer topic inform undesirable option expressive inherent specifically concept nest beta vocabulary benefit source example vector sentence information text lose bag formulation additionally vector information word feedback weight word assignment concept corpus encode semantic similarity membership mm model topic vocabulary traditional example top hierarchical hdp salient use health care package additionally concept obtain approach contribution use beta concept incorporate concept formation mcmc include study record showing efficacy approach wish unbounded concept probabilistic prior mixture cluster concept coin particular attribute particular flip coin assign concept formally know coin bias concept document exploit know write base mass finite expect realization formally beta define realization poisson rate base improper distribution measure contain prior active conjugacy predictive assignment concentration ibp ibp draw bernoulli infinitely concept document customer select customer share concept absence concept level document lead ibp customer concept word happen customer previous select come vocabulary ji z z z x k nx mm coin bias process draw level beta process assignment word concept indicate concept idea discuss note nest weakly combination entity encourage concept component close correspond sec avoid incorporate sensitivity way maintain exchangeability computational document collection word concept entail identify concept indicate ji binary nest beta feature assume discrete come vocabulary presence actual text document multinomial distribution hadamard product word multinomial distribution importance base provide issue sec concept contain word address semantic describe cf illustration fundamental semantic identifiability explicitly vocabulary allow help guide representation provide efficiency much want concept co occurrence count g sentence basis work study distributional know company keep semantic sec semantic often occur semantic explain concept consider correspond semantic likewise column per simultaneously car value feature concept weight j ji set inclusion responsible meaning incorporate independence
write equality bound operator order v x vs therefore lemma prove claim follow multiplicative lemma establish eigenvector perturbation eigenvalue k ki ia ki eigenvalue aa notational ia k c triangle inequality give cauchy schwarz follow third norm bound eigenvalue simultaneously subsequently column k matrix obtain column matrix invertible satisfies lemma claim define I inequality spectral r rearrange claim multiplicative j v project vector vector distribute obtain bind proof establish gaussian clear statistic true magnitude simplify norm use property eq rule verify mixture satisfy nk note handle idea coordinate dimension remain check matrix km ks hadamard large axis roughly speak non vertical block satisfied partitioning coordinate induce roughly row value call partitioning view preserve distribution view similar cca require separation contrast put least matrix selecting row index together union submatrix e ie chernoff let u eq require concentration multi clear conditionally recall independent therefore effectively bound statistic subgaussian definite standard theory decomposition decomposition invertible invertible n fact together see case matrix l x show mathematical identify result document identical case simple satisfie rise triple differ lemma theorem fundamental treat current practice search scale broad dimensional include gaussians gaussian model method rigorous simplicity offer tool apply machine treating generate identify important include mixture maximum em drawback practitioner alternative maximum approach method back pearson order several equal adapt variety intuitive guarantee mild high dimensional determine moment accurately broad class high dimensional model result estimator implement algebra eigenvalue decomposition estimate low involve explicit indirect variable determine view hmm present future view axis align gaussians coordinate non redundant mild work relevant due variant offer em mixture vast thorough find text advance science decade comprise work learn subsequent provably gaussians roughly deviation gaussian separation expect application spectral projection enhance separation develop grow dependence show related thought modern literature method technique decomposition pearson root computational present notable exception situation evolution mixture polynomially mixture component success transition matrix work remain space approach mixture hmms component dimensional second moment know discrete moment demonstrate moment moment method problem availability exploit mixture separate view mixture separation direction projection availability view remove separation entirely gaussians study degeneracy view text video linguistic datum paragraph predictor semantic example warm simpler wise wise exploit work view method moment mixture document explicit algorithm correctness efficiency view mixture gaussians hmm appendix inner integer context suppose partition document word independently multinomial topic corpus vocabulary document short accord multinomial vector draw independently accord specified encoding become conditional depict degeneracy document prevent distribution whose tensor triple identification coordinate expectation view di view involve parameter triple via operator assume matrix invertible invertible probability distinct every topic exactly geometric observable xx algebraic transform vector observe notable consequence perform transformation xu mb xt eigenvalue equal non zero probability every exactly observable suggest approach base relate focus estimate estimate handle estimator context general frequency triple arise moment concern triple v matrix moment tensor lemma straightforward generalization matrix invertible observable kb scale vector nevertheless carry possible reconstruct kb eigenvector whose entry system element th polynomial observable template decomposition crucial distinction polynomial involve involve lemma suggest estimation b present average independent copy similarly matrix orthonormal corresponding vector pick invertible km euclidean u ki u view estimator care depend abstract scalar technical splitting discrete technique hold gaussians present exist pick rotation matrix manifold permutation addition multi view framework hmms technique parameterize observe multivariate covariance mixture covariance j various dd j axis align diagonal partition view randomly hold require v column cluster degeneracy show separation efficient comparison minimum recover concern condition slight recover pair matrix finally gaussian still concatenation condition requirement matrix covariance full meet even component markov latent model state form state conditionally consider homogeneous vary describe hide state readily handle handle restriction hmm identify independent give u therefore recover observation matrix recover scale zhang discussion comment david anonymous pointing perturbation appendix also illustrative modify implementation empirical copy frobenius norm euclidean claim let union mu invertible imply let permutation j bound feasibility modify begin third middle end article stop instead single extract eigenvector automatic heuristic leverage score c bit save run
view actually conditional hmm consecutive recover emission spectral view state come hidden organize section short view view state simulated illustrate correctness effectiveness view section short summary view vector target variable take nlp view word topic mention property word connect dimensional response view view view htbp pair view linearity assumption lot fall example widely nlp hmm observation gaussian edge edge h often word english vocabulary lot easy internet getting lot human observation lead briefly go though simplicity mean since view refer conclusion view identity introduction cca svd correlation canonical theorem claim aspect view together drop uncorrelated group predictor hilbert predictor onto assumption span covariance span span projection onto therefore optimal predictor dimensional cca regard dimensional order view feature uncorrelated help view hide get hide view illustrate content help classify find combine hide new available assume view cca square optimal dimensional optimally everything subspace span predictor predictor canonical random canonical component predictor subspace indirect cca actually cca rotation I variable moreover know cca first last subspace easy svd column reconstruct find follow except tx th tx space lie pick canonical direction I denote moreover q canonical lie direction cca last direction hence satisfy rotation last canonical similarly covariance canonical convenience q full rank column step step rotation matrix compute find optimal dimensional subspace dimension view cca view reduce algorithm reduce cca dimensional huge contain useful information reduction space feature cca normal normal view predict three dimensional three predictor label linear one time rotation trial square learn large square large h second sample size view amount experiment square loss average well sample square loss experiment advantage view label linear decompose view moreover variance due predict amount size different square plot right label view feature become small cca multi view assume carry achieve dimension view need disjoint part act view
let past l l bn remove parent minimal joint parent minimal negativity kl separately parametric generative graphical direct process node might exist causal strictly causal satisfy positivity violate graph unique process process edge equally minimal generative unique causality identify causal influence adopt wiener apart past use formulation work direct name causality condition recall statement causal prediction cumulative loss sequentially predict thus direct influence process process direct iff graph immediately graph case exist arguably bad ambiguity generative graphical direct minimal model equivalent make correspond pair search first help correctness satisfy parent bi bi iff equation parent accurately learn bi ai identify parent require determine satisfie potential parent require exponential direct alternative motivated chain trivial sequentially remove condition generative proof appendix alternative direct graph separate test entail direct condition process condition direct ai ai ai algorithm recover algorithm element line execute parallel number graph dimensional dimensional statistic demonstrate involve recovery next direct algorithm network identify parent pairwise test etc small degree relationship exclusive counter process ai ai ai return proof increment even degree recover minimal dimensional minimal bound degree result graph optimal show direct involve sufficient identify parent show parent intersection parent formally sized subset degree ki ai return return parent trivial algorithm parent information test q question output show modify return validity marginal exact parent divergence form approximation direct direct sum direct along specify degree let follow case ki specify however use exact direct information interval recover point ai reliable ai develop find good estimation confidence practical simultaneous rectangular multidimensional denote ai scenario value ai ai parent error characterize I approximation well given perform poorly different scenario attain minimax regret input b mb b b lb mb lb b return confidence identify robust approximation ai thus construction alphabet plug second parametric jointly result jointly direct process jointly stationary markov finite markov pairwise network jointly direct couple network irreducible notational simplicity indexing start history assumption stationarity definition jointly plug j constant event examine sample characterize order compute empirical direct estimate time sample appear biology field complexity network p estimate mle analogous interior p hessian q pg twice maximizer iii extend parameter vector jacobian entry singular multivariate delta specifically information pair continuously jacobian dependency different direct occur nonetheless joint matrix sample unknown calculate confidence separately calculate sample confidence markov coefficient use trial trial parent non scale magnitude distribution pg identify direct parent child compare parent infer dynamic capture parent estimate least square fit entropy avoid used length number complexity n initially parent resolve except parent parent test parent respectively show algorithm identical degradation dynamic misclassifie edge present computed version setup section degree draw normal stationarity show proportion dynamic increase monotonically percentage edge identify concave peak parent parent parent parent identify identify next plug simulated network binary value node limit bias respectively pick bias round example bias color correspond figure perform almost game game amount use use resample parent replacement mean direct information width normality interval perform algorithm percentage infer game bar random would thus quickly suggest robust utility infer news user online micro covered event middle east analyze precision correctly proportion influence proportion randomly influence depict roc show average algorithm approximately non influence tn baseline user single even parent influence correctly tn variation amongst reflect monotonic decrease precision truth graph c acc alg alg alg alg visually depict axis overall influence precision baseline network significantly research social sciences economic biology widely meaningful interact underlie case reliable demonstrate utility news influence direction future research estimation technique direct information several rich well characterize another graphical estimation assume likewise assumption depend past biological network graphical handle topology beneficial assumption strict causality relevant observe section strictly causal influence influence need causal nonetheless non x linearity iterate expectation divergence joint natural strict causality minimal instead direct parent algorithm correctly recover direct repeat causal causality corollary ensure hold state direct model corollary parent p strictly closely p chain correlation mutual notation correlation condition close lastly identify statistical dependency causal use linearity rule mutual strict causality direct propose causality direct form causality preliminary set causality predictor conditional causality direct far beyond strong causality conditional sequential setting conditionally help thus want strong anchor capture causality causality setting term side help sequential consequently precisely quantify causal predict sequential sequentially form full past specifie know except simplicity mt l help decision combine time simplex hellinger could discount manner prediction measure form causality causality logarithmic cumulative expect cumulative minimal test compare model error linearity expectation focus show condition direct information j continue past give direct hand non negativity use notion reduction tx x predict cumulative loss causal side predictor probability interpret quantify strength sequential demonstrate sequential special axiom alphabet direct I prove causal pc property lc satisfy pairwise property likewise lc lc generative lc invoke extend condition place turn induce direct information contradiction j ig mean I model direct yield non chain chain way bi zero consequently bi I occur equality use chain add zero reverse valid generative parent minimal ai ai ai ai recursion parent set return contain set suppose current inductive base hypothesis trivially inductive hypothesis rule lemma set true holds remove remove algorithm parent show direct involve let process process exclusive note argument three return parent first contradiction parent evaluate necessarily j j ia ai ai execute line case evaluate parent chain parent loose maximal ib j ki bb jj lemma thus appear ba empirical hoeffding entropy translate entropy direct estimate measure error direct event realization hoeffding four realization order pair next correspond denote concentration l evaluate entropy maximize increase bound entropy j x analytic bind attain aa increase maintain probability increase several condition check define information assumption alphabet q evaluate rule move linearity plug proof condition specifically simplify addition normality assumption form sequence follow derivative finite definite iii assumption include matrix th r linearly hold column orthonormal matrix inverse column direct error wise analogous variable denote normalization normalize volume respective integrate normal rectangular specifie corner positive volume eigenvalue volume constant two coefficient repeat form imply grow parent parent maximize bring scenario independently hold rectangular irrelevant optimizing hold identify parent regret apply ks line parent lb j either add side return thank inf j computational science fellowship de er support grant fa fa nsf grant propose represent generative base distribution information causality quantify sequential develop bound use event valid near known certain criterion analogous structure undirected use direct characterize plug direct information confidence point estimate lastly analysis real twitter network influence analyze causality generative direct economic social involve interact agent neuron stock computer expert seek computer infect time activity price traffic work influence agent question whether influence agent want user micro twitter many several news wants identify strong influence decide pay activity want influence reliable compute indirect influence distinguish influence user sure influence influence issue propose graphical agent depict influence show information generalize causality one influence direct connect principle beyond strong causality use independence network nod million graphical prove correctness infer graph degree optimal valid return bound adaptive early instantaneous influence influence identify reliable estimation interval plug empirical estimator propose identify influence activity news account middle knowledge accurately infer influence organize follow discuss make algorithm identify exact estimator demonstrate algorithm contain direct graphical direct communication channel direct
lr test denote likelihood parameter table two model repeat item group implement package item lc restrictive item equivalently dendrogram show lc obviously cut dendrogram several value correspond lr suitable multidimensional lc section software follow speed analysis aggregate record pattern record configuration distinct label index provide configuration among distinct multidimensional trait list datum apply original code response k latent start deterministic start user point parameter lc specify link difficulty item difficulty row equal index item measure correspond dimension latent class estimate vector item array response class lk aic bic index outline two procedure pair perform item dim class item est est multi specification dim specify multidimensional model multi specify multidimensional default multidimensional structure dim obtain est output est lr df lr merge item collapse list sort represent hierarchy dendrogram statistic lk maximum result aggregation np list hierarchical merge height use package application illustrate item consider ordinal illustrate choosing class logit cluster mathematic testing service national progress replicate example aggregate record class pl link display dendrogram figure detailed object height first collapse identify statistic aggregation aggregation concern define group item item item aggregate visualize also type lc former detect global logit link carry lr item response structure response structure item test dim firstly point link dim model freedom test link dim constrain unconstrained degree freedom coherent perform adopt cluster grouping item outline presence threshold item application four response type properly parameterization difficulty difficulty know constrain difficulty p parameterization account logit parameterization basis sake likelihood bic provide est start rs est multi rs vs lk df np df p lk df comparison lk df lk p vs np rs lk lr rs analysis rs fit four response free taking lr parameterization dimensionality accounting criterion indeed select logit link constrain free dimensionality logit logit c free constrain constrain constrain patient suffer mostly first belong severe patient condition outline selection binary logit link difficulty initially class extend trait trait introduction latent detect notable simplification case trait characterized multidimensional treat formulate treat ordinal item different link provide order maximize contribute specific lc base likelihood ratio tool useful nest multidimensional lc availability application collect service implement cluster concern scale ordinal subject class explain type reject favor measure provide package multidimensional class package deal name traditional trait also trait depend type model estimate package discuss selection illustrate concern refer aggregate describe fit keyword trait deriving make trait case trait unfortunately restrictive traditional flexible extension take one trait among main contribution example multidimensional wide see overview topic another advance literature latent characteristic context introduce certain number patient treatment secondly trait maximization way multidimensional characterize assume multidimensional comparison traditional formulate discretized variant class represent mixed ordinal mixture concern mention extension multidimensional lc trait latent trait random subject individual moreover parameterization item diagnostic rather interesting multidimensional base continuous latent trait perform goodness fit parameter require aim lc ordinal package issue trait rely link allow response cumulative local category link function presence possibility possibility discriminate differently different difficulty special level category basis item extension traditional introduce trait procedure aim type parameterization dimension allocation model output package estimating model neither trait among package multidimensional trait propose package application collect service within assessment progress binary homogeneous group trait concern ordinal patient characteristic latent trait follow propose multidimensional lc devoted devoted illustrate implement analysis model specification constraint response item item latent trait latent trait realization discrete latent trait latent trait category express category parameter identify difficulty suitable belong specification three link base local first compare category moreover category previous category model see item coincide global typically trait contrary intermediate level award reach intermediate model model base item may discriminate discriminate item category share degenerate response also observe logistic pl refer parameterization difficulty item unconstraine ii special constrain difficulty category rating parameterization case category item difficulty category combine obtain specification member eqn free rating scale cm item level cm accord multidimensional derive latent trait know traditional multidimensional lc multidimensional lc rating extension equation multidimensional pl multidimensional lc due independence contain measure trait denote identifiability constraint latent trait element moreover free item rating identifiability lc obtain probability free parameter item depend type parametrization difficulty class probability equal ability however difficulty unconstraine parameterization parameterization unconstrained ht difficulty free express category extension item response zero zero zero element equal indicate concern way moreover logit link logit vector suitable find ability single taking difficulty explain unconstrained column constrain constraint accordingly suitable column scale parameterization remove constrain accordingly remove deal likelihood deal aspect mainly concern observe formulate propose may express free
discretized spline rw smoothing rw rw rw derive solve stochastic sde resolution spline equally good computational aim extend rw carefully smooth sde representation explicitly exist method adaptive smoothing convenient understand continuous satisfactory adaptive intuition build correspondingly rw knot spline equivalent improper sde dt wiener refer sde ft du improper ft e sensible straightforwardly carry method unfortunately prior intensive dense sde note provide good overall product denote choose element ft let common choice piecewise linear h jt j field interior determine continuously finite substitute linear dt weak sde neighboring basis give modification dense computationally expensive show change replace give straightforward result cubic smoothing spline expansion approximate show approximation depend enough function describe actually sde cubic derivative however aware global function theoretical sde spline flexibility straightforward spline control smooth present sde formulation sde td ft dt smoothing function compare global spline second smoothness spatially smooth spline ny ft dt piecewise derive reproduce hilbert computationally intensive reproduce dense case achieve dt leave side proof side th h discretization boundary affect leave corner h h h note involve sde dt instantaneous quick decrease adopt formulation minimize ny ft dt also dt write define appendix mean h h n n take smoothing assume differentiable proper smoothing intuitive sum previous link sde dt use dt dt leave right remain variable full approximation numerically integrate laplace work hyperparameter case use reduce rank vary sharp peak highly gaussian add true internal knot add example spaced noise ft adaptive spline estimate ordinary smoothing spline square eq sde fit second sde knot knots median square table slightly estimate slowly smooth significantly peak spatially different yield sde offer inference mcmc number well inferential knot capture efficacy software reason space error acceleration draw minimum illustrative concentrate general gamma regard shape scale heavy sharp peak spline adaptive technique cauchy fit credible cauchy cauchy adaptive sde mcmc performance give drop back model attractive adaptive smoothing smooth drop compare cauchy offer credible wide capture opinion cauchy error well overall cauchy paper unify smoothing spline base smoothing spline sde easily adapt element smoothing spline demonstrate effectiveness application proof use adaptive sde hand entry dt tt tt dt th tt dt dt dt dt tt dt I tt I I nonzero entry happen dt letting write dt dt sde linear system jt nonzero tt dt tt tt tt tt similarly dt dt I previous I dt tt nonzero dt dt n dt entry approximately boundary diagonal matrix define spline popular evidence support partly elegant firstly become prohibitive number roughly secondly amount smoothing function class adaptive smoothing certain differential drive smoothness markovian make study example demonstrate keyword adaptive monte spline partly support partly elegant mathematical I I function index spline degree smooth sum derivative cubic smoothing frequentist view reproduce hilbert bayesian yield partially improper restrict spline become computationally obstacle stem smooth underlie spline set typically impose point knot spline unfortunately mark penalize p spline require spline note distinguished basis penalty penalty compete truncate ridge penalty spline gain statistic implementation formulation
confusion dictionary advantage sc report qualitative sc independent hand advantage indicate sc outperform sparsity become transfer employ pixel handwritten character digit capital follow experimental instance possibility train task learn target task approach dictionary task set vs task case instance character per tune hyperparameter create process describe task associate train target run multiclass accuracy task character character learn possibility learn experiment assume digit compare show dictionary miss pixel per image employ pixel explore code widely process justification insight bound hilbert depend quantity intrinsic datum code promising improvement compete would valuable vector could arise structured norm lasso g family regularizer mind qr k encourage code group code divide dictionary hilbert task express acknowledgment part grant ep international joint dictionary combination transfer principled bound generalization setting real dataset advantage grouping extension code approximated space quantity principle bound numerical real dense decade development linear guarantee performance reference therein atom dictionary free principled choice exist representation underlie benefit individual justification new task automatically task environment task contain learner task demonstrate give single concern subject considerable attention idea meta dictionary support predictor atom dictionary sufficiently task dictionary together justify analogy sparse coding unsupervised replace observable correspond combine coding list incomplete transfer therein common perform generalize present first coding provide learn learn present connection present practical also address problem learn demonstrate utility set learning apply thereby hilbert organized section present bound numerical experiment conclude turn technical exposition notation way hilbert integer dimensional dictionary canonical regard code implement assumption set could readily ti ti value tm tm represent predict label lipschitz value atom vector minimum section present experiment generative model standard distribution well minimization always specify rule assign I propose frobenius use employ use place depend bind denote redundant eliminate restrict learn learn setting interpret context task training ti input upon return minimize th task error minimal analogous risk bind input state implication appear average square suppose surface dominant extreme datum bad sense little dominate dictionary atom could bind noiseless generative number minor increase suppose task generate model list dictionary extend disjoint correct total relate cluster sl ball radius bind sl apart sl outperform utilize correspondingly task sparse superior theory employ crucial end regularization scheme frobenius place quality may poorly perform set environment context transfer risk simply draw sample test read loss incur predictor linear adjoint observation task draw simplicity fix sample large independent datum retain dictionary environment quantify compare well minimize achievable result ts implication interpretation remark imply behaviour dominant denominator excess theorem possibility task possibility appendix risk task say term plausible discuss example linear model vector input assume uniform sphere haar measure task fully ball draw combination square reduce code quantity choice order bad code experimentally section propose code multi rr base line versus
stays correct apply mean th line c stay graphic
graph assignment run side part propagation offer would belief allow recall comprise additive multiplicative satisfying multiplication identity multiplication multiplication requirement meet usual arithmetic identity operation backward property preserve rise useful call next far carefully allow exponentially follow four tuple property weight multiplication observe component multiplication leave result fourth component thus belief propagation second perform need perform message important remarkably suitable rise tractable normalize marginal efficiency belief configuration graphical generalize inference sufficient normalize time pass compute diversity q assignment pass run compute though note pass perform second second order significantly modern processor tuple use follow operation easy computation scalar normalize marginal structure alternative marginal structure single assignment ask assignment efficiently message pass form therefore computable marginal simply separate runtime single factor run pass obtain everything compute require make sampling practical familiar marginal probability run take pass marginal offer efficient sampling show naive iterate conditional marginal efficient integrate part return belief propagation assignment singleton create new message assignment since see unnormalize assignment example assignment go handle maintain assignment hold naive algorithm run might expensive run extend input factor run arrange belief propagation pass proceed backward receive message conversely message message backward pass independently execution factor variable belief suffice compute part send full unconstrained forward right assign forward leave message likewise send variable unchanged backward computed illustration mr thick dash yshift yshift cm row sep sep minimum circle draw minimum cm draw minimum minimum cm circle minimum minimum circle f minimum cm minimum edge edge edge bend bend bend bend leave bend bend right swap b bend bend bend bend right swap b bend swap bend right swap swap message need circle assignment propagation loop essence complete forward backward pass linear message assignment send hold imagine root subtree root distance draw circle child node child node child c child child child child thick fill subtree anchor sep subtree north east include disjoint immediate tree neighbor weight node message neighbor incoming message message assignment subgraph claim message neighbor belief propagation assignment go proceed part sample incoming message set current message message take account neighbor claim visit visit know walk entirely assignment hypothesis neighbor know assignment conditional complete walk start proceed depth message thus procedure second traversal fixing assignment final piece replicate dpp number eigenvector first asymptotically dot tb b leave paper per total similarity cite paper importance weight cosine dot normalize tf define eq document filter remove common appear rare appear filter word score stationary thresholded cosine plus term centrality score design diversity cosine zero imply correspondingly dot feature dimension behavior sample offer type examine illustration cover topic apart visually salient locally control discover multiple tc thing explanation mobile database mobile server organization handle mobile wireless challenge management length subset project tf centroid word high paper two news comprise york times collect part corpus article article corpus list number discard article deviation six month period article article cosine article six cosine furthermore go enforce document feature feature control make one model maximize half project report provide resolution neighborhood article annotated reflect position article publish digital read likely online fig visualization article article entire million place figure depth exceed modern instead upon display visually ranging leave sample color compare baseline method task month period sized slice similarity cluster basis article slice slice average cosine coherence repeat clustering exhibit article successive slice naturally align attempt smooth publicly fit set slice topic topic randomly produce baseline obvious distinction always span nearly period select one dpp tend continuous news explicit relation collection risk health raise ability home though ill world choice meet may message name salient single breast disease patient benefit heart breast patient health black breast patient news topic model half salient word quantitative evaluation generate baseline news summary summary approximately daily news summary france corpus distinguished tag summary cover news dataset gold summary month tf evaluate obtain vector cosine percent hyperparameter metric validation describe automatic variant six produce summary cccc sim summarie bold high verify bootstrapping rating average score average article identify bold distinction baseline former orient choosing approach orient article distinction see online complete task ask first article totally single clear rate average column rate coherent mean poorly since time slice ask implicitly ensure enter display article ask thought remove improve coherent average right task tf advance second include baseline fitting projection matrix sampling core summary mean offer possibility wide practical application suffer correlation arise conclude briefly mention question shannon dpp q strongly currently useful quantity remain future asymmetric encode dpp dpp inference constraint set might learn dpp quality approximately use parse translation corollary theorem conjecture chapter section proposition conjecture probabilistic arise physics traditional model hard negative efficient marginalization conditioning focus extension learn community apply informative summary sentence pose automatically important news modeling learn analyze discover process lead space goal make introduction combinatorial impractical tree approximate interaction intractable offer arise physics theory elegant correlation offer marginalization inference extensively rise deep new aim focus extension dpp fix result database dpp see correlate make inclusion item strength correlation similarity item co occur assign item dpp prefer distinct query focus salient diversity place work diverse information retrieval unlike diversity context kind interpretation query topic alternatively item exhibit quantum particle dpp dpp serve metric weak large pose dpp final application summarie news choose diverse sentence diversity image return google pose task improve human image incorporate bias toward non overlap news task automatically extract news balance intra coherence inter background along modeling extension theoretical result aim enable material begin introduction tailor interest focus discrete simplify proof description efficient describe decomposition dpp fundamental tradeoff quality diversity compare expressive representation show still provably requirement efficient quality diversity news allow item practical expressive dpp dpp experimentally structure number doubly diversity inference pass structured toy pose process process quantum anti effect year give matrix attract mathematic make formal combinatorial probabilistic accept survey overview relevant process pattern interval record output characterize spike spike tend occur might tend spread correlation finite point without extend continuous simple distinction document simply measure empty unnecessary nonnegative eigenvalue use sufficient dpp marginal observation singleton element dpp diagonal element tend co demonstrate think measurement element unlikely perfectly similar conversely independently element co occur correlation draw dpp sample process set plane relate spread independently paper focus world receive much section briefly example remarkable whole technical uniformly say location less process previous probably thus likely position decrease independent symmetric walk integer walk let walk begin position fact intersect dpp walk intersect far apart edge span spanning distribute edge begin orient arbitrarily nearby likely uniformly demonstrate assign zero cardinality particular case minus spanning issue subset dpp hermitian draw entry normal unitary dpp shape square half square color gray suppose known subset b cover vertical subset distribute dpp data symmetric whereas atomic every semidefinite eigenvalue less normalization follow zero everywhere else trivially hold cardinality less give split partition write column whose inductive term give shorthand write write measure diversity prefer q reasoning marginal intuition l kernel obtain zero except eq rescale conversely dpp l ensemble invert inverse dpp inverse equivalent give restriction reason nonzero rare cardinality dpp limit ensemble describe offer alternative l ensemble atomic offer appeal furthermore need effort interpretation always square intuitively describe measure use product say volume view span magnitude probability similarity increase intuition verify probable vector orthogonal span volume feature define else magnitude feature appear multiply span volume decompose kernel direction magnitude section dpp distribute dpp kernel dpp kernel may seem complement encourage dpp assign marginal probability hold constant define take dpp element bernoulli immediate generally expect plot ensemble include infinite appropriate uncertain image situation may cardinality front ten desire goal achieve advantage realization review associate estimate large interactive memory gb already partition write matrix well go multiplication use practically computer interactive storage remain extreme marginal set item compute probability see probability configuration conditional dpp ensemble marginalization dominant inversion gauss elimination kernel sample latter practical working complete ten dpp row element obtain simply drop element also dpp one diagonal entry zero everywhere else inversion appear combine dpp appear closed formula allow arbitrary marginal conditional dpp note dpp marginal partial eq q dpp expensive dominant inversion although appearance inversion conditioning expensive marginalization dpp theorem tb input orthonormal loop phase eigenvector produce second previously cardinality random express elementary phase elementary dpp mix elementary dpp phase dpp elementary dpp kernel elementary orthonormal subspace span due elementary arbitrary expand vanish dpp arbitrary definition hand expression reverse marginal agree fix draw dpp whenever almost theorem eigenvector loop select elementary dpp mixture loop sample geometric introduce span span generality span height formula projection onto proceed iteration begin update basis probability choose exactly argument identically select generate visualize influence dpp grid together initially successive select shift avoid choose dpp unit fig particle circle color dpp probability offer cluster eigenvector eigenvalue strength cluster perform choose select identify accounting overlap expensive operation schmidt expensive cardinality potentially save often bottleneck require interactive large minute instance eigenvector since choose kernel multiple desire recently sample significantly sometimes refer posteriori hardness closely entropy determinant covariance find find principal covariance submatrix maximum dpp mode semidefinite input dpp mode hard dpp reduce cover element decide cover constant intersect semidefinite reduction require time value must suppose base product column correspond cover orthogonal optimal dpp hard dpp hard even branch mode heuristic set find optimal hour modern computer order application greedy algorithm achieve approximation poor may nonetheless whenever yield return maximize submodular subset discuss kind monotone approximately maximize polynomial search grow variety way show rise wide involve diverse item physical diversity take sort class probabilistic particularly appeal machine section briefly wide world process perhaps right poisson item come coin setting mean process disjoint generalize least likely since poisson real world address various modification poisson introduce correlation construction make induce embedded euclidean remove radius item space apart ii mat ern design keep begin uniformly small remove item remove lead dynamically early radius inference ern computationally moment compute iii process expensive ern enforce radius choose issue mat ern spirit surfaces cell surface arrive random bind remain like mat ern process core pack much achievable restrict find initially select ern intuitive seem plausible analytically provide framework item energy course without constraint assumption instance common interaction argument lie local term markov markov pairwise process potential piecewise radius otherwise cluster continuous integrable discount general core typical pairwise depend control diversity call union center little item diverse fall twice radius decompose individual similarly however item might appear item interact potential mrf element discuss expressive possibility mrfs simply mrfs intractable even normalizing process carlo approximations nonetheless moment rarely dpp entry sum permutation dpp submatrix appearance index natural generalization rise special case irreducible theoretic character partition element determinant single describe opposite dpp diverse recent paper property detail furthermore process induce computationally matrix work class efficiently computable leverage perform inference beyond computing determinant replace count determinant modeling advantage third originally index operate set additional modeling preserve setting however randomness study estimate practice finite draw poor concern set generator randomness ensure appear even speaking set diverse evenly origin point discrepancy define discrepancy box set cover unit cube deterministic uniformity property unit q discrepancy lead quasi contrast monte stochastic discrepancy offer uniformity generate efficiently seem plausible uniformity characteristic set tool work generating prediction learn possibility point process deep exactly characterize theoretical promise exhibit make intuitive computationally variety fundamental serve extension later intuitive unary decomposition splitting quality comparative expressive power show qualitatively characteristic despite advantage impose super dpp show dpp inference simple projection dramatically bound consider finally formula ensemble analytically ratio concern interpretability practitioner understand dpp kernel totally similarity primary practical want diversity balance preference diversity dpp gram quality term diversity decompose dpp arbitrary high dimensional entry think goodness item sign similarity main potentially simplify independently diversity diversity diversity model choose quality tend quality item quality diverse fail quality combine balanced return determinant volume span magnitude previous decompose objective diversity go diversity intuitive setup exist turn dpp geometry square volume span probability item increase probability contain model offer already task like essentially practical make advantage global negative traditional elaborate expressive compare take representative random field mrf whose purpose mrf variable value denote bold assignment set encode dependence might fact tend co mrfs field parameterize depend mrf clique nonnegative clique constant mrf think characteristic define independent neighbor converse clique mrfs offer intuitive interact encode independence unique clique limit clique mrfs large clique potential clique node potential mrfs clique unbounde inference inference also hard problem likewise probabilistic hard approximate constant mrfs identify mrfs submodular tractable mrfs depend clique encourage take formally least depend pairwise mrf eliminate ij parameterization sometimes visible boltzmann mrf whenever mrfs encourage potential necessary build mrfs diversity opposite potential graph indeed potential cycle inference failure improve algorithm wide informally approximate effectively potential outline mrfs familiar potential directly relationship similarly mrfs come expressive dpp relate binary negative negative correlation model dpp mrf difference individually semidefinite kernel mrf actually mrf induce virtue disagreement see prevent dpp form dpp therefore obey imply guarantee semidefinite trying intuition mrfs dimension difference equivalent correlation start apparent dpp mrf unnormalize per potential unnormalize single see dpp node mrf item dpp ccccc mrfs potential show subset edge mrf dpp set visualize manifold slice four slice dpp slices mrf origin appear gray surface rise qualitatively similar surface mrfs describe anti anti impossible improve constrain slice constraint away mrf slice plot express grey dpp blue mrf surface primarily similarity dpp distant item rather far look distant induce mrfs constrain model data mrf exclude item generally depend express say item naturally restriction inference rely kernel inversion may efficient construction semidefinite diversity become database furthermore identical q thus dl apply quite fact include normalization marginalization sampling dpp normalization equal determinant eigenvalue nonzero dual dpp marginalization course require time q dot product marginal give arbitrary pairwise marginal obtain determinant implement represent orthonormal relationship tb select iv seen allow setting effective case also diversity language reduce maintain dpp random extremely nonetheless distance classic randomly onto dimension preserve volume set point connection volume projection dimensionality dramatically maintain result projection dimension sample probability least span project effectiveness projection restrict portion restriction application relatively formally seem bit since imply conditioning dpp apply work dpp dpp quality dimensional condition dpp project projection say dpp volume conditional must adapt reason low sign definition inequality directly follow combine representation exponentially expand expression nice volume normalization denominator simple matrix dimension analytically derivative likelihood plug identity formula advantage use diagonal positive semidefinite everywhere else identity matrix expression determinant way complement dpp similar asymmetric offer appeal geometric inference kernel prior knowledge dealing set dpp diversity introduce dpp choose student think student tend merely entire article sentence advance separate great even importantly learn nothing generate summary unseen article time place alternatively sentence appear article single address rarely repeat depend input enable share article denote imply input g sentence article conditional dpp conditional assign every dpp efficiently quality quality diversity receive identically conditional kernel parameterize base prediction unseen input learn log observe optimize dpp learn unlikely dpp estimate ascent bfgs must exist optimize fundamentally begin diversity fix support automatically consist desire even long proper score use feature distinct parameter advance ease go single drop extend parameterization show able negative exp expression concave modeling empirical count dpp assign higher diverse compare example diverse attention overcome bias impose determinant see practice sum instead switch summation eq q marginal item appear dpp count per inversion idea efficiently note need diagonal exploit multiplication entire multiplication unfortunately asymptotically irrelevant still dual fast along line recently currently tb instance demonstrate dpp quality document news summarize news news article selection sentence relevant sentence diverse fit document article preprocesse try dpp summary construct place sentence policy unlikely perform justify automatic like sentence order understand cluster use collection approximately article ap cover sentence word use training summary evaluation summary character space depict set human reference summary performance follow automatic evaluation overlap human reference summary sub simple show well f primary development recall turn removal actual recall setup require unfortunately reference summary high summary summary human simple round human normalize add summary precision reference remove reference count character oracle summaries summary reference automatic competition well reference human probably human could summary optimally nature compare believe summary training score summarie human use diversity stop sentence word document article finally per entry word proportional similarity cosine cosine word thus salient feature score good cosine confident remain fix throughout experiment hand training find list feature cosine distance tf produce series bin boundaries determine bin evenly spaced bin quantile current summary small sentence bin character five global bins position document position plus indicate position appear cluster similarity compute cosine feature salient occur frequently cluster raw ten local eigenvector row cosine raw score global bin count may feature include previously unseen task apply inference posteriori high reason primary metric impose character summary thus goal summary budget length simple tb cluster character limit iy u discuss formal generally nonetheless seem bayes decode automatic speech decode specific loss function realization evaluate since exponentially summary expect efficient inference sample satisfy impose sample character outside inference report show time require processor decode randomize lr summary cluster parallelization bfgs optimization parameter system simple merely since consist news article entirely many identical similarity include advantage dpp take diversity contribution impractical properly nearly long short inference employ baseline document perhaps simple relevance similarity measure sentence logistic regression sentence score sentence optimize sentence add refer submodular implementation rely include method test originally result update version highlight stochastic improvement dpp outperform significant boost logistic inference run produce relative dpp perform report contribution performance drop significant intuitively two essentially centrality important removed assign probability dpp position team tends spread team exactly five likely exist elementary fix however achieve focus equally aspect notion fundamentally serious limitation express dpp uniform cardinality dpp may cardinality dpp series trial characterize certain might look may tend diverse position warm might impose cardinality offer search engine diverse mobile user dpp real world conditioning dpp content size simply expressive limited correlation define time diverse image query dpp subset dpp size dpp concern content dpp obtain dpp set cardinality dpp give semidefinite standard restriction normalization dpp cardinality subtle attempt difficulty likewise valid marginal eigenvalue matrix full easy special item expressive quite dpp condition case depend flexibility practical advantage assign possible since effectively situation size hundred nice know expressive increase expressive whether convenient useful case close seem simplify dpp polynomial polynomial examine characteristic property recall dpp apply dpp elementary last two eigenvector elementary polynomial give every eigenvalue formally compute identity difference essentially summation fast numerically stable rely precise number eigenvalue kn inner iterate elementary symmetric polynomial thus every advance since dpp repeatedly efficient two phase subset eigenvector reject sample begin rejection eigenvector formalize intuition dpp whenever decompose elementary dpp elementary dpp dpp dpp accord mixture like task recursive elementary tb input e l nj desire return loop single index induction hypothesis loop begin suppose added loop occur immediate inductive probability return nothing begin observe since begin inductive otherwise iterate overall dpp generate sampling assume expensive sampling dpp normalization dpp normalizing simplify observe side dpp normalization conditioning marginal polynomial whose thus probability standard derive know singleton use item elementary dpp elementary except marginal scale probability eigenvector compute marginal require polynomial fashion would tree leave correspond represent associate elementary symmetric polynomial auto fill black child draw black node child node draw child draw child fill child child child child node background line pt leave interior node represent leave leave combine leaf constant interior use remove eigenvalue polynomial along leave represent leave root tree take necessary elementary polynomial singleton marginal inclusion dpp dpp exclude offer dpp easy still cardinality cover greedy approximation like demonstrate image motivation run search engine goal image user unfortunately search look city perhaps furthermore user specifically shoot expect search small argue least diverse respect another want maximize search return satisfy maximize want diverse natural course evaluate task amazon establish diverse comprise comparative image prefer use whenever correct set task seem work practice give expert advance optimize idea combination receive loss make mistake logistic alternate step project projection standard algorithm create query retrieve top restrict search pass safe average image city great paris category spread evenly across order compare directly baseline method probability full differ single classification candidate redundant set use sift candidate uniformly instance obviously redundant choose usual order decide actually diverse diversity amazon worker label reason candidate result offer candidate order instance category fig label example five candidate right correct calibration instance belong level inherently difficult keep check five instance four agree end half image form block kernel image kernel normalize ensure assume image equally relevant partly justified fact come actual google search relevant function image variant pixel color space color sort align bin either dimension sift variant toolbox sift descriptor descriptor category combine cluster normalize near cluster descriptor process use five center half kernel create kernel every vector flexibility act diversity increase derive training combination tune query run result candidate dpp kernel apply test kernel optimize mixture minimize technique diverse idea set add round weighted combination relevance detail merely small give prediction dpp good apply attempt learn replace metric substitution optimization non dpp practice optimum easily find provide advantage random split significance bootstrapping regardless outperform achieve decision make number rather human obviously significant measure improvement percentage versus kernel actual use dpp mixture category dpp majority bold fig dpp exhibit significant diverse might sift center sift color color center color center center sift high category try cover image draw whether select satisfie virtual generally simply similarity expect well method high similarity average single virtual whose desire dpp dpp indicate result average perform good virtual mixture select ten subsequent drawn fraction fraction virtual user model gold measure expert combination expert result average virtual dpp job cover result significantly see section offer inference ground exponential naive would imagine linear position obvious move
coincide gaussian probabilitie posterior component I coincide associate generate write r I kx kx aim highlight finite regression wrong although well separate polynomial consist cubic report generate ht name visualize beginning represent base gray dash allow dedicated marginal histogram component scatter display group regression separately generate joint via display ht suppose forget classification regression start parameter confirm value adjust rand index ht contrary shall obtain differently mixture estimate corresponding component mm display ari value regression ari impact possible remain scheme range randomly indicate use discrete set generate replication classify ari replication conditional ari improve remain extension mixture polynomial name polynomial justify carry within em framework bic use membership know polynomial apply artificial datum excellent clustering compare future several extension initially concern characterize outlier finally evaluate theoretically separately conduct indirect acknowledgement helpful comment suggestion di mail present cluster bivariate dependency consider base base framework estimate use use bic integrate complete condition posterior probability membership coincide excellent artificial mixture polynomial weight employ statistical purpose indirect nonparametric see hand application original datum biology mark overview focus direct indirect represent bivariate hereafter vector constitute suppose linear conditional regardless shape mixture regression mixture adopt latter use unfortunately linear capture polynomial regard regression highlight indirect view implicitly affect classification bivariate cluster furthermore elliptical approximated well mixture several like helpful modeling purpose application group elliptical density consist elliptical fitting rarely satisfied become difficult grow propose elliptical represent correctly elliptical create provide alternative difficulty contrary bivariate allow increase applicable purpose easily interpret presence model detail component frame artificial datum suggestion mixture parametric respect associate th component mixture implicitly assume density model constant hereafter denote n paradigm represent way functional component integer degree model become polynomial factorize denote eq free model aim classify unknown membership unlabele general scenario estimate fit adopt unlabele correspond know proportion improve define zero component origin label observe x indicate l polynomial suppose em ml gaussian step detailed th follow unlabele th independently regard maximization subject constraint augment set equation equal yield eq maximization I nz ij qx ij nz nz qx ij nz finally regard maximization algebra ij nz nz ij ij ij nz nz nz nz nz nz ij nz ij nz ij r nz nz nz nz I nz nz ij obtain em respectively say unlabele via map analysis adopt classification algorithm although maximize almost except step one partition em section environment value constitute model specify strategy detail mixture regression maximize log likelihood run draw acceleration use iteration base decision make regard whether reached log k iteration l analyse algorithm k parameter determine optimizer particular provide method option among logarithm quantity purpose consist convenient selection range couple choose couple good exist criterion integrate reference choice mixture bic use classification model mixture selection base bayes eq present component represent component attempt cluster approximate
maximum algebraic contain many bind check feasibility iteration initialize g hence bind terminate analogous consider far observe relatively determine principle every situation theoretical practice analysis lipschitz consistent set however search feasible constant supremum behave alone scenario entirely possible robustness thereby lipschitz control value identify detailed one stand notably concentration hoeffding turn influence rescale cover calculation first numerically second impact find implementation response model exactly implement necessary follow form associate provide representation refer x store compressed form elsewhere convert numerically represent come constant regard add class calculation whether integral overall calculation outer loop outer inner loop apply candidate ever scenario cx impose outer loop describe expand solver interface solver termination size population objective value improvement great consecutive outer loop objective optimizer generate position direction optimizer also constraint algebraic impose outer optimizer necessary scenario cx valid constraint equation evaluate optimizer standard multiplier objective evaluate optimizer fx px approach explicit build object optimizer generate first constraint c underlie discrete solver impose shift coordinate optimizer pass c scenario use impose unlike constraint impose feasibility scenario little discussion scenario load collection subject area interest dominate numerical convergence show bind indeed datum ignore find experiment reduce determine discover decrease compute markov difference confirm run consecutive long mass locate worth significant algorithmic computational result distinct much observation approximate illustrate point discrete cube uncertainty velocity dominant phenomenon natural try calculation always valid greatly find relatively consider automate product dimensional event simplify improvement future show column position column coordinate bottom effectively approach great generalization admissible independence independence moment include use optimally propagate uncertainty hierarchy acyclic output relationship figure g il bound precisely suppose respect value play pair geometrically cone must remain close whereas note confidence interval nature application real easily define admissible scenario admissible also analogue problem induce q measure validity similar uniform norm many application may strong therefore room metric lie distance inverse system system neither even observe uniquely partially assign subset notion continuity space neighbourhood equivalently hausdorff response lipschitz extension define single need acknowledgement us award fc na california institute technology material california institute center science finally anonymous helpful comment thm thm thm thm j quantification convex r universit provide quantification rigorous carry objective pose setting output solution trivially monotonically upon furthermore extreme carry analogue simplex efficient high system suggest optimal maximally finance rigorous uncertainty practical concern uncertainty quantification address cope operation paper thereby presence uncertainty pose scenario input uncertainty uncertainty concern unknown partially distribution infinite reduce type physical know method variant system variability diameter optimally bound bound study extensively discussion lipschitz locate hand smoothness point calculate without evaluation couple concentration produce last optimal datum know motivated show unknown lipschitz approach large calculation optimal see survey historical remark topic structure exploit greatly burden shown propagate bound bound determine propose use offer rigorous motivating easier less terminate number addition extended objective next hence maximally informative induce great change optimization upon assumption interest note hoeffding often reality assumption toy detail example variable consider constitute failure give value failure use set information alone trivial impact piece q lebesgue distribution gx g z information generally improvement possible say measure set unless take evaluating pose lie close contour notably non monotone establish problem treat determination function upper failure treat directly optimally bound failure least consistent still provide non discuss concern redundant general remark section implementation many redundant large quasi lipschitz short differ component lipschitz lipschitz constant argument borel measure kf f index sensitivity bound short diameter diameter place value independent finite independence relaxed control event system complementary provide rigorous computational physical interest determination diameter great failure safe know sufficient truth value know exactly gd question address constant probability failure problem determining evaluate maximally treat bound uncertainty quantification verification method applicable stand physical quantify bound suitably space place unified framework paper question use calibration difficulty good use bayesian area attention condition response globally lipschitz attention globally lipschitz broad old use modify suited requirement inequality constrain hold inequality example constraint suitable constraint must though value isolate elsewhere type pointwise evaluation finitely order upper obtain available state constant space let exist apply old value modulus continuity preserve modulus language metric say subtle hilbert lipschitz banach consider scalar optimization negative lipschitz variable intersection double illustrated note close cone contain empty say four dot dot note feasible lx kx x g ask bind rely denote lipschitz value sense denote gd short g ss sd kx xx gx complete proof although l say may mutually short also upper upper give g solve least kk candidate mutually run constraint mathematically formulation identical differ addition optimality number large metric part arbitrary eq q may necessarily member estimate lx lx gx constrained entail feasible compact extension constant restrict even simple suggest consider fix great l note quantity finite differ defined easily k observation corner cube size constant affine determine everywhere half vanish section spirit emphasis provide diameter failure necessarily short input simply give couple numerically may seem next extreme find search structure simplify structure correlation use remark let cube opposite corner ham cube separable borel probability regular countable dense borel measurable hausdorff separable compact simple inner topology right mild space extreme obtain combination dirac support matter convert product eq indexing upon make easy easily search finite feasible extreme value optimality space eq assertion discrete cube failure final equality contradiction extension short contradiction establish similar omit infinite dimensional value follow write assume distinct redundant bind see p dot location observation grey dot location dot feasible assign error maximization basis possibility q sharp next observation unit failure depend maximizer event sufficiently consider threshold must otherwise observation lipschitz give five case contour plot neither boundary critical conclusion infer attain satisfy unique dotted line various dot dot failure impossible least upper application whole input space partition lie lie lie outside reasonable one obvious formulation objective notion information content calculate entropy make notion optimization constraint determine extreme value behaviour rather solution elimination redundant section notion redundancy
classify beyond replicate reach three curve explain classified entirely level level vary display smooth author exceed usual relate thought status patient differ hour line period rapid within appear increase final period slowly curve greatly reach within positive curve low deal variation age sample provide henceforth distinction class alternate find sense programming know employ attribute machine svms applicable technique classification include handwritten digit recognition operate represent find separate separation hyperplane ideally far concept refer perform accord gap motivate formulation accumulation decade progress learn upon discussion improve observation pair form classify spam filter email spam longitudinal patient identify patient seek hyperplane separate observation contain vector observation lie half hyperplane lie half space find side eq convention break arbitrarily hyperplane exist training set separate hyperplane several identify hyperplane chapter near point hyperplane give iw I margin hyperplane via optimisation note objective positively homogeneous equality refer hyperplane support remove act optimisation space separation consider penalization approach slack variable measure th meet relax slack lead optimisation classify misclassifie penalty observation hyperplane reach increase objective hard correspond linearly separable least must zero agree force remove slack recover original margin two class criterion sometimes desirable reduce contribute separability unnecessary express desirable know encourage encourage interested clearly recover increase interestingly proper goal control huge regularize optimization descent serial randomized performance alternate sensitivity positive specificity already give correctly separate together refer reliable classification svm paper robustness quantify work portion remain provide useful feedback overfitte training training set classify hyperplane otherwise fail specificity gain false positive negative classifier create robustness separable specificity support remove hyperplane upper cross bound negative negative bind positive perspective understand support vector act test sensitivity specificity pseudo sensitivity specificity pseudo consequently specificity equal pseudo sensitivity pseudo specificity optimally test non fraction sensitivity specificity value else know refer number classify train subscript vector machine raw attempt find svm optimal employ nan read hour hour observe equally hour average threshold decide adjusted margin separability hyperplane test oppose calculate recovered svm specificity rather hour read hour maximum margin performance calibrate optimality guarantee latter great distinction two suggest high reliable validation classify differently remove optimally indicate classify precede read read margin svms separability classification error margin measurement specificity sensitivity extremely time see margin take hour perhaps entirely rt svms exclude present figure actually due threshold towards negative clearly contribute interesting error increase remove likely cause stability inclusion dominate train big parent svm dominate freedom greatly remove parent svm remove pseudo inclusion validation accommodate performance figure around hour cause reduce inclusion increase average argue exclude margin still hour margin hour argue inclusion reflect couple remove subsequent accurate svms lot choose hour positive four negative high read svms false negative average svms test take hour confident probably rt hour notable reading hour read hour average winner exception hour choose positive occur fact result penalty increase svm obtain specific sum consideration specificity svms justify training improves arguably adjusting consider far potential test machine observe help distinguish class variety namely extract information difference arise order replicate must wrong meaningful four identical arbitrarily replicate replicate within replicate flat replicate replicate show aggregation distinction lose average sensible three flat show increase flat would closer negative sensible replicate great reading hour replicate replicate increase occur replicate significant plot heuristic maximum increase train length specific validation replicate associate pseudo specificity fit optimally robust full svm suitable replicate weak robustness absolutely relative reading dimension consequently surprising dimension training practically separability seek separability compare plausible overfitte induce separation curve random also general present attempt separate hyperplane confirm separability dimension poor contain simply stack svm stacking vector consequently lose measure within independently seek induction try eliminate essential information discriminate class recall nan take fit curve function least space attribute dimension provide noise unlikely clear curve aspect occur unlikely reveal approximate piecewise major profile approximate stage three connect simplification define gradient location allow choose highly clear curve figure curve decide think derivative approximation sample degree way appropriate improve fit useful keep proxy intercept conclude information piecewise help find curve train svms try poor lower frequently unstable figure display curve fail trend sigmoid function growth positive description plausible follow trend model case sigmoid flexibility accommodate many express identical therefore aspect stack dimension curvature locally regular analysis involve stack dimension stacking assume present extract certain value feature svm training point apart variety difference define central follow stable divide second derivative approximation hour must multiple hour interval measurement minimum gradient time approximation calculate significant noise cause suggest issue partly great certainly appear flat guarantee sufficient preliminary enough sufficient trend noise employ identify trend longitudinal smoothed easy implement contiguous fix longitudinal central odd length enough large capture visually compare smoothed raw smoothed selection sample visual clutter plot first approximation smoothed difference value brief highlight value firstly information gain reflect rise trend derivative zero increase reduce trend maximum rise rather increase appear stop second derivative close trend suggest assume curve perhaps behaviour mean examine vector stack derivative overfitte robustness measure selection svms robustness consistently ccc case omit clarity feature classification assess work appropriate optimisation infeasible imply separability separability dimension stack meaningful appropriately sized ensure separability performance present colour graph tendency see consequently light robustness tend separability overfitte fit explanation apparent interpretation figure another smoothing derivative svms explanation investigate induce observe svm induction actually stack svms long frame instead curve difference suggest real error roughly balance shift positive less toward dominant select feature fit robustness difference either maximum feature inferior measure correlate curve high similarly svm extra positive negative improve robustness inclusion achievable via analysis rt analysis patient age date lp duration procedure plot display figure firstly plot age segregation conclusion consequently additional impact axis additional svm clear apparent profile positive horizontal axis difference prove induction merely see measure distinct patient age old old old patient characteristic correlation exploit characteristic improve probabilistic classifier report case curve time derivative predictor show predictor separability reducing omit rt clinical case rt diagnosis case class need multi class construct separate svm one class classification assigning class give great hyperplane soon apparent reliability type svms maximum separable freedom validation confirm exploitation omit stack end next try derivative information construct difference way meaningful find separability case robustness highly absolutely reliably patient discriminate age date lp procedure far available divide duration disease observe disease patient duration alone influence partially explain profile diagnosis
efficiency feature class therefore concern simple knn network since contextual requirement great show categorization store amount prototype cardinality calculate document centroid measure similarity regard use final hierarchical classification assess cite centroid comparison hierarchical hierarchical big flat method confidence main concern label result boost accuracy obtain tc superior grant pn ii te organization new categorization retrieval mining content spam filter mail categorization web characteristic typically categorization attract field machine mining pattern categorization totally class run hierarchy single category variety categorization take benefit goal massive far hierarchical category structure top upper used concept specific propagation major disadvantage misclassification recover propose error limit classifier per parent node iii classifier benefit node classifier great hierarchy classifier clear strategy method strategy return reliability goal confidence assignment weight goal add related occur accept reject candidate label consider reliability score hierarchy experimental categorization reliability score reliability validate text conclude literature hierarchy root typical child process start proceed class positive c c c separately strategy membership node fact classifier correspond leave unlikely membership avoid force level strategy commonly assume evaluation start go root level base local next classifier permit stop internal hierarchy certain class predict classifier pass phenomenon whenever threshold score node test originally force leaf worth point leaf node combination top prevent stop false negative propose label evaluation try ensure reliability hierarchy assign classifier idea sample option classification process manual account hierarchical parent relationship level high rate weight candidate use formula label generate accept answer classifier threshold rate lead equal false testing apply hierarchy hierarchy accord node select plot histogram truly reliability specify equal rate fa false rejection fr equal regard
identify positive positive run model fact mixture negative solving sentence admit negative sum adjust sentence equation element rank assertion discuss mle us start case function sum let vector sum variety non maximal interior critical global maximum try ran sample polytope converge one eight local lead precisely solution maxima lie negative q discover maxima carefully geometry em case comparison em global project acknowledgment thank comment national science dms dms dms example corollary problem fundamental statistic matrix rank mixture distribution two likelihood numerical geometry point discovery likelihood complementary maximum fundamental typical encounter discrete respectively write thus non closed I sample negative rational projective divide problem find rational point algebraic compute point reliably find maxima among determination bold number small compute symbolic fail beyond author solve use degree finding interested topology algebraic degree sign euler characteristic open suitably ultimately topological table know rank result n x x factor sign euler organize constraint mixture identically numerical geometry practitioner refine computation might theorem proof computation statistical closure fold independent algebraic distinguish equation affine system equation regular point subspace complement inner subspace positive logarithm likelihood critical contain entry solve express orthogonal make exclude strictly singular point exclude degeneracy jacobian defining format jacobian format homogeneous row format similarly write follow column rank jacobian third translate requirement derive formulation elegant geometry serious namely solution little row impose replace first row far sufficient ml get replace let kernel hadamard format linearly column either suffice explain span matrix equal p product invertible indeed equation note redundant far variable concern way reduce row replace critical still simplification lead computational result recall matrix matrix imply provide give u equation see consist many sum simplify column diagonal last entry column nine nine nine equation solution rational nine nine solution generic algebraic ml arise product rr nm homogeneous coefficient expression refinement fact upon variable root exploit various aforementioned table build balance tight computational linear product discuss impose variety independence state equal entry theorem entry multiply similar symmetric multiplied diagonal system consist local column sum hence fact diagonal symmetric arise entry eq equation take system six theorem suggest ml degree symmetric first version later document advance project use numerical algebraic address global numerical significant symbolic equation generic computation fast numerical homotopy critical fix homotopy change methodology wide likelihood treat agree statement homotopy substantial commonly statistic option preprocesse formulation generic first homotopy build discuss notably homotopy build root count option intersect notably essential algebraic preprocesse subsequent solve case processor perform summarize separate processor distribute processor second minute attribute represent alignment dna table take solve subsequent second integer become degenerate preprocesse polynomial appear clearly use small shall preprocesse introduction numerical homotopy book homotopy compute set complex isolated root compute point approximate solution software construct family contain isolate sufficiently track solution path isolate generic count degree generic root know root isolate root track ml root connect along segment general position contain avoid gamma trick trick arc critical issue due choice local write suitable matrix give affine random homotopy quickly critical preprocesse namely computation serial double duality root root homotopy read solve homotopy system arise generic construct table root return homotopy root use implement construct advantageous significantly less degree intersection advantageous intersection build discuss idea solve system intersection build product polynomial variable define system arise fact ic algebraic yield finitely hyperplane isolated point solve univariate hyperplane homotopy compute apply homotopy computation additional confirm trace centroid hyperplane linearly centroid critical matrix linearly test randomly generic theorem hold critical analyze hessian lagrangian polynomial critical maximizer form tangent remainder symmetric critical likelihood maxima local minima six define extension instance coordinate proposition irreducible coordinate hadamard point illustrate distinct complex critical real consider precisely seven six maxima respectively expect real positive number maxima maxima equal real maxima algebraic geometry include ml duality regard ml variety equation verify whose grow refined statement entry equal statement conjecture maximum likelihood duality likelihood preserve reality positivity critical perspective result exactly estimation matrix vice versa refined formulation duality speed complementary trivially ml degree illustrate theorem specific already literature conjecture maximum datum datum follow scale normalize occur illustration ht compute critical express rational remain point fall four symmetry calculus remain distinct pairwise smooth depicted tracking homotopy path function analyze arise member family pairwise arise establishe performing path using solution remain distinct retain critical size q sort point
compute first patch average stage compute eigenvector reconstruct average patch mm snr figure signal different level level moderate noise achieve patch nevertheless quite pc pc different cause much image lose general practice stage eigenvector general level patch contain patch extract dimensionality patch texture extract region patch image low spatial proximity image proximity conversely contain experimental instance patch classify accord compose class extract region texture etc author observe patch distance note proximity texture patch pixel texture patch distance patch gradient patch much texture patch low smooth eigenvector able reconstruct patch reconstruction patch texture eigenvector patch graph texture figure snr eigenvectors patch eigenvectors snr initially reach index adapt incoherent need reconstruction figure eigenvector pass see evaluate denoise gold pass evaluation image periodic texture add image pc pc pc pc level moderate middle follow denoise provide provide k svd display reconstruct original outperform squared visual wavelet yield consistently bad noise svd l reconstruction wavelet svd yield mean smooth texture even stage estimate rely eigenvector random patch smooth organize along low structure patch reconstruct denoise outperform work raise address compute eigenvector zero method implement option propose eigenvalue fast recent indicate method still computation patch clearly coarse processing eigenvector patch reference therein algorithm currently existence extension collection lee patch optical organized construct show lie image sub manifold provide eigenvector similar pc pc pc pc pc pc pc pc pc pc pc perturb analysis perturbation graph laplacian acknowledge perturbation acknowledgment grant dms award eigenfunction computation influence eigenvector diffusion operator perturbation change connection patch organize patch reconstruct demonstrate gold image denoise goal experimentally perturbation laplacian novel eigenvector patch graph image computer universal transform replace perspective indeed recently image shape contour patch image use patch reference taylor dataset patch aggregate eigenvector provide expand patch image intensity obviously smooth graph laplacian yield patch often corrupt eigenvector graph problem become eigenvector result note non eigenvector perturbation addition bound predict image angle translate effect perturbation encode eigenvector image original eigenvector aim understand low laplacian geometry geometry quantify eigenfunction laplace equip experimental understanding robustness laplacian jointly eigenvector experiment extend image location define notion patch block odd integer patch set patch start explore patch ignore imagine intensity dimensional though patch form collect lattice horizontal vertical lattice pair translation coordinate patch discretized argue move object form sub differentiable lack acquisition device smooth provide illustration patch extract structure shape surface encode move cone fig explore different generator cone orientation effect orientation inside encode patch always edge patch manifold review patch interpretation manifold patch denoise construct local point measurement tangent global dataset reconstruct tangent patch coordinate observable remove delay therein estimation coordinate tangent plane plane basis plane therein project noisy onto tangent plane ideas patch ball project tangent manifold orthonormal multiscale recently worker compute construct global diffusion manifold try around comprehensive denoise denoise geodesic manifold replace mean explicitly graph various interpretation idea pde depend local neighborhood graphic community implement discrete version surface therein propose diffusion discrete operator apply diffusion kernel patch compute eigenvector laplacian diffusion eigenvector spectral decomposition similar fouri seminal work perform remove image diffusion estimate noisy expect eigenvector different perturbation eigenvector propose originally think regard vertex structure patch connect construct call patch graph weight patch near neighbor patch encode region domain parameter drop rapidly zero patch noise fully characterize patch weight entry symmetric define laplacian opposite laplacian manifold ball radius around volume ball wrong sign share small similarly eigenvector kn write vertex eigenvector inner product define image left encodes intensity irrespective take horizontal derivative eigenvector fourier analysis intrinsic patch propose eigenvector encode feature texture etc graph patch replace regular lattice image patch reduce single pixel noisy denoise clean set access clean appear circular stability laplacian study experimentally add weight iterative recover eigenvector corrupt noise eq image e graph neighbor vice cause accord set edge near perturb define laplacian expect eigenvector reconstruct clean perturbation eigenvector depend separation predict problem separation small large limitation invariant subspace translate term perturb approximate effect employ coordinate plane orientation laplacian balance texture similar perturb row clean comparison visual observation analysis paragraph appear reasonably eigenvector pc pc pc pc pc mostly capture texture original instance texture fig quantitative perturbation quantitative comparison transform practice research analyse radial angular frequency etc orientation integrate orientation thin sampling two within perturbation individually dyadic eq dyadic eigenvector motivate observation eigenvector dyadic similar fouri inside eigenvector show index scale index radial total contribution group perturb eigenvector energy clean red perturb start perturb energy clean distribution radial create confirm try capture increase h il respectively radial bottom radial obviously suitable denoise topology topological change create removal significantly graph laplacian modification eigenvector verify experimentally entry location topology perturb weight perturbation topology local perturb eigenvector experiment factor preserve guarantee experiment base noise accord neighbor graph perturb constitute previous paragraph word perturb display perturb visually eigenvector radial eigenvector remarkably plane change image spectral geometry encode metric phenomenon pc pc k image pc blue red radial frequency image
simplex certain ideal matrix parametrize family simplex definite perspective package allow collection statement compute vanishing case analysis acyclic compute associate ideal vanishing follow package graph create contain entry normal store hence take direct acyclic know pd vi definite parametrization imply generate vanish satisfy I follow j j j graphical variable explanation independence translate matrix briefly explain construction generate variable whose simplex p primitive quantity create use p let statement translate certain probability follow generator p p p ideal degree three correspond undirected independence local statement direct undirected context statement n capability graphical acyclic satisfy recursive factorization parent map vanish q ordering vertex acyclic vanish graphical binary variable vanish generate ideal vanish ideal path previous ideal independence global statement ideal compute vanish gaussian model mix mixed ideal parametrization triangular direct edge otherwise symmetric zero entry vanish ideal ideal principal ideal generate ideal g problem consist solve graphical mixed example identifiable index contain rational acknowledgment people lin david partially national science institute mathematics anonymous helpful comment suggestion research sp grant fa air force scientific advanced research project ss support foundation dms david foundation thm thm thm thm edu edu edu package undirecte vanishing ideal
three flip reading comment cosine similarity pearson correlation tool preserve put correlate maximally apart transform fall second transformation derive metric distance pearson coefficient cosine use denote already know metric angular equivalently anti object maximally apart correlate anti correlate angular pearson coefficient cosine similarity strength pearson commonly deviation coefficient pearson sample define cosine standard information retrieval cosine angle transformation pearson cosine evident terminology model solely interval henceforth pearson indicate applicable dissimilarity satisfy triangular informally always short distance allow building accelerate skip take distance distinct distance object satisfie commonly state distance distance preserve shall metric namely increase concave refer state exceed essentially mean relate g important ease reference preserve preserve first inequality rewrite fa ba scalar rewrite fa ba ba proof quick determine concave twice concave understand acceleration imply calculus twice preserve transform dissimilarity transformation generic principle ax metric currently distance triple derive embed equal distance vector identity page second hence find preserve obtain trait angular certain interest trait angular preserve counterpart single object use ax ax ax angular ax ax angular distance
monte replicate randomly total accuracy plot via bootstrap resample solid circle data remain curve plus cca get f dot nn result pair dimensional classifying mi consider simultaneously classify test dotted plus curve pair dash indicate incorporate domain via term classification correlation reduce additional regularize cca choose properly trivial noisy keep figure regularize cca result cca regularize counterpart noisy dimension c non c regularize cca cca tf mi tf mi mi setting training relation figure superior cca investigate choice combination class regularize yield setting document embed early regard indexing outline table pair example classifier relation indicate choice class superior cca classification cca tf tf inference relative monotonic give inherent nonetheless characteristic regularize version match reduce apply wikipedia datum domain efficiency result analysis improve regard text document canonical generalization improve finally increase available document improvement performance amount matching work identify embedding multiple enable fusion source canonical generalization investigation cca document canonical efficiency object domain example translate language domain classify representation task ik need common training classify separate different learn represent relation datum consider canonical common cca training investigate near relation datum investigation discuss manifold well setup present conclusion survey kind explore view domain testing language later paper dictionary latent etc translation involve classification translate match whole divide follow space domain learn multidimensional euclidean cca generalize cca common e combine generalize firstly low manifold cca paper focus learn investigate dissimilarity space manifold matching show mapping manifold dissimilarity denote kind fidelity fidelity well dissimilaritie fidelity q measure define multidimensional dissimilarity dissimilarity ik jk ik ik jk ik ik multidimensional mapping column orthonormal requirement similarly imply subject language write differ english article neighborhood article english view case point training document manually topic topic people date thing respectively document class relation total document train classification start dissimilarity matrix dissimilarity dissimilarity graph topology dissimilarity matrix contain dissimilarity wikipedia connect document short document english neighborhood document document connect text dissimilarity ti ik jk ik jk indexing wikipedia description common pick via scale embed dimension joint model fix throughout dissimilarity training document problem different large preserve noise second indicate total c nearest nn assign close
status study disease death consist depict symbol figure incidence scale age duration disease simulate population population accomplish disease model compete birth cumulative failure failure diagnosis death measure birth person know event death disease person failure integral eqs calculate question random variable available store piece date birth person diagnosis person age death person person contract age diagnosis na follow h simulate na file file store person store text file dot file devote analytical relation common measure measure duration disease diagnosis lose life year disease obviously age diagnosis simulation typical question mean diagnosis subject another interesting incidence status typically current status incidence estimating incidence status decade useful integral eqs describe integrate integration give event co system axis age sometimes refer age entry entry death clinical trial diagram second subject without life end concept hand eqs integral failure time associated life choose sure occur year calculate life life field graphic exist calculate intersection volume voxel start calculation parametrization ideally suit approximate integral rule j necessary calculate f voxel grid calculation interpolation voxel j transform would well obtain affine consecutive j similar compute intersection grid accordingly death disease interpolation section present year birth death incidence total contract disease important age death disease year death duration ill compare disease age black several group confidence analytically agree quite article simulate population death consist disease birth subject without subject life diagram locate part life direction change allow modelling disease duration play disease follow
convnet ms convnet accuracy demonstrate clear dataset point inferior pooling pool art stage give slight increase performance unsupervise learn method mean auto run attribute supervision high seem exhibit address problem introduce edu house convolutional learning inspire unlike popular design automatically optimize traditional convnet stage state art dataset improvement analyze stage sf character recognition document handwritten hard like behind human mainly de image illumination recently dataset extract dataset digits input big label contain character digit image template matching way superiority superiority traffic sign challenge accuracy art convnet different pooling implement open cm cm convnet compose stack feature stage contain convolution module module module pooling module convnet lp oppose stage stage oppose h pool pool rich representation compare feature add fine lose improved work work however gain sign likely explanation observation gain texture multi characteristic channel set extra easy train information sampling order yield allow measure put emphasis process contrast channel contrast normalization channel invariance overall convnet feature two convolution filter filter output also global classifier hide unit rate gradient dataset pooling represent report epoch
directly c rest write sign consistent vector n probability lemma write iy eq n w I put ready proposition n utilize c since combine q show lemma w c suffice complete satisfy pn I I p proposition hold probability proposition hold stand theorem ise binary markov apply range scientific engineering infer represent edge equivalent dependency rest graph focus draw network type graphical ise structure equivalent diagonal precision paper inverse covariance appear recent year paper establish many fast glasso recently ise constant make demand spirit fitting assume sample model node study life covariate genetic study role tumor development since interest tumor observe various clinical phenotype tumor mutation status factor include motivated situation model binary incorporate covariate connection covariate mind impose allow select covariate paper way quantitative fit separate subject interpretation covariate concern nearby region covariate ise lead covariate strength subject organize ise establish asymptotic evaluate instability conclude physic q binary z probability sum equal otherwise specify related parameter independent suppose covariate ise jk conditional odd way imply conditionally jk jk jk jk model express vector instead depend eq choice linear parametrization jk logistic regression parametrization straightforward relationship depend among desirable formulation convexity negative log likelihood maximize likelihood spirit maximize give conditional likelihood q network interpretation suggest good sparse encourage propose conditional likelihood regularize regression guarantee estimate compare magnitude initial separate regression jk jk jk jk min conservative sense turn separate often conservative identify detail regularize entire logistic careful rearrange obvious treat separate fit model omit regularize derive fashion regression covariate slight notation drop intercept irrelevant true without additional j j j hold regression exist bind effective covariate rest assumption c q uniqueness jj j j establish proof vary signal strength covariate roc zero estimate zero specificity curve replication sample adjacency randomly exception intercept term generate stop estimation min separate max joint combination first fit augment uninformative total remain zero false curve generally increase particularly dominate joint method tumor play tumor development complicated relationship time vary tumor dna profiles microarray cancer patient advance breast cancer collect another base copy profile dna array copy event bin band sample status cover group patient retain association among event association clinical characteristic include mutation status variable er status variable tumor stage ordinal denote array datum I jj contain phenotype covariate matrix ise covariate fit selection infer stable repeatedly replacement tuning record non pair note correspond interaction covariate measure finally rank list depend heavily primarily pair gene locate likely depend column relate name third record tp status status tumor generate hypothesis experiment molecular mechanism breast pairwise selection tumor tp associate genome instability tumor tumor association interesting association group find breast region several tumor patient tp contribute tp pathway study suggest positive breast cancer previous finding find association tp role tumor gene cancer htbp c main tp status gene gene gene gene p q p p p p p p p q highly node role pathway covariate selection subsample node finally across stability covariate list interestingly rank role breast cancer study region tumor region breast cancer report candidate tumor transform confidence selection change tp status frequency finding lead discovery tumor breast allele survival association role gene
leave aim residual empirical partition test baseline scalar ridge ridge integral ridge evenly weight kernel well combine identity residual enhance use constraint ridge outperform kernel functional response induce lc value simultaneously operator deal regression method predict movement regression performance work mkl mkl collaborative minimizer consider tool guarantee generalize corollary banach suppose proper bounded space attain functional give present convergence infinite reproducing let kx q strictly convex directly strict convexity us eq ridge rkhs converge number subsequence convergence value convergent analogue convergent subsequence space subsequence obtain proposition hilbert operator state subsequence convergence rkh operator reproduce subsequence minimization fix converge subsequence whose minimizer pair minimizer convergent subsequence converge bound de analyse des de hilbert deal case value function considerable learn entity typical include brain interface design precisely movement measure electrical activity give period instant whether move amplitude signal amplitude clear formalize functional point benefit multiple value movement movement fix several value reproduce rkh whose map target work draw rkh value kernel recently reproduce build kernel try scalar kernel seminal effort carry theoretically analyze problem motivation propose framework finite value kernel mkl without difficulty arise space point work coordinate descent multiple although combination kernel value reproduce correspondence positive definite value kernel traditional scalar kernel value cm reproduce hilbert value furthermore hermitian adjoint definite hermitian w rkhs reproduce hilbert reading value kernel lagrange multipliers lagrangian lagrange multiplier banach finite lagrange multiplier suitable infinite constrain lagrange multiplier minimize respect ridge belong rkh similarly scalar convention cast derive admit supplementary material line problem follow block kkt kx present devise scalar mkl coordinate descent closely gauss operator typically analytically iteratively solve initialize non simple system still fix rewrite close optimization operator basic matrix value make detailed supplementary showing generate continuity boundedness argument minimizer scalar space reproduce technical norm cm cm f common build scalar carry set positive operator integral fact operator provide interesting extend structured identity finite kernel encounter kernel infinite space solve value form value kernel operator product analytic solution invert gauss method initial cm cm kernel value inverting matrix gauss iterative iteratively convergence satisfied expression positive operator value variational n jk split role indeed problem vector deal constraint derive multiplier equality constraint kernel easily gauss highlight operator real involve brain interface address related movement focus direct scope prove aim fourth dataset competition grid place depend unknown record band pass hz signal
well screen fs fs often perform combine study draw conclusion relatively fs prefer fs converge yield well simulation cr cr cr sis sis fs sis fs fs cr cr cr sis sis sis fs fs ct dataset frank author ct ct slice histogram describe structure histogram location air body histogram form axis construct manually ct volume know detailed dataset et square fs fs want improve fs fs fs least error number fs fs highly procedure fs fs term classical screening enough word subset call fitting well rule property model initial virtue screen initial estimator screen subset include subset screen hence fs wang well well screen fs addition satisfactory fs modification also fs fs fs choose wang fs fs bring whether rule believe valuable topic national natural china grateful laboratory science variable inequality omit ax ax note v ax ax superior subset hard complete complete unique proposition ft generality pa b c b b c b c boost prediction fitting van de york california school science http uci ml ct images hill company york http corollary mathematics science method screening include residual square rule well fitting regression algorithm propose search subset yield property popular screening screen show competitive high subset combinatorial reduction em orthogonal number candidate variable wide variety scientific become coefficient loss generality denote cardinality often eliminate important exhibit variable model actually sparse fan two estimate sparse screening stage guarantee effectiveness screening possess sure fan fan fan wu fan song li aim screen much relationship fitting square sum square good fitting say interestingly variable pick fitting well fitting screening make fitting tell sum asymptotically size one likely include denote number nonzero component equivalent optimization square algorithm reach sum screen obtain exhaustive branch later moderate subset search infeasible forward optimal solution e basic idea possess e square put fitting screening well call fast real underlying actually desirable include yx x ax pa eigenvalue matrix hold rank column eq x ax make design wu lin identically sure screen fs literature fitting tend include small mt ratio dm theorem indicate superior subset state sure subset point call power see normal nx q minimize subject constraint fs track path flexible minima relatively point final write monotonicity ft proposition fit improvement asymptotic mt p mt k square stop take p correspond subset well asymptotic screening subset corollary connection point sure independence sis sis take similar yu well combine successively whereas keep besides virtue boost paragraph convergence monotonic converge study counter special case stop sum decrease tool wu certain recall theorem effective nonconvex solution neighborhood practice neighborhood provide direction sufficiently know consistent obtain way scad regularity condition van fan like disadvantage least replace fast subset screen ft ft monotonicity converge simulation study design kronecker product wu hadamard configuration cr cr sis
could discrete distribution give dependence serve tool evaluate specification compare parametric nonparametric kolmogorov integral former develop datum integral strong bring uniform guarantee transform nice follow add apply transformation use extension copula test uniform von kolmogorov transform necessary estimation make prove parametric bootstrap correct arise ml idea project transformation ng projection test require projection approach extensive static study base moment comparison parametric nonparametric despite stress situation could provide specification contribution dynamic asymptotic asymptotic bootstrap valid rest introduce specification experiment conclude ty dimensional parameter space cdf use subscript nan correct conditional order would family distribution nonnegative integer fy strictly version strictly increase typical p ff matter result distribution copula marginal invariant realization realization fix realization respective denote property fact u u motivate know either uniformly function von kolmogorov statistics q test lag independence correspond obtain generalization kolmogorov discrete distribution alternative deviation box correlation lag note variable nan correctly specify specification good goodness fit alternatively residual eq normality addition square capture alternative example misspecification well test distribution bootstrap parametric statistic recursively bootstrap repeat percentile prove limit bootstrapping derive simple know fix alternative denote ba r fr f ff smoothness tr linear nan e dynamic probit logit introduction adjust satisfy close variable study effect expansion estimator smoothness continuity uniformly parameter although satisfied estimator know assumption mean account asymptotic expansion study limit eq q cr df define alternative equal nan local assumption hold random around may nonzero stand appear project effect suppose assumption hold v test justify bootstrap procedure e prove triangular array tt x tt similar assumption require hold investigate finite sample exercise autoregressive dynamic specification specification set try hypothesis static dynamic interaction probit check dynamic marginals logit alternative sizes table bootstrap base parameter lag von kolmogorov denote residual normality residual lag additional omit nan probit probit static probit probit probit interaction logit static probit static chi static static logit dynamic probit dynamic probit static chi interaction probit dynamic logit interaction static interaction probit pt traditional almost slightly improve fast rate ks size static logit static probit hand static probit test static well case dynamic misspecification add logit improve add dynamic alternative dynamic probit logit even well high lag account versus lag summarize dynamic misspecification statistic misspecification marginal distinguish statistic possibly test additional noise effect since test residual dynamic misspecification indirect effect power attribute develop introduce still wide check goodness many financial parametric asymptotic test experiment base
covariance know correct formulae krige proof enable interpretation krige key though formulae krige weight valid square integrable good coincide case case ordinary krige write covariance integrable bayesian construction improper coefficient equation use equation take come counter q early regular krige equation diagonal happen enable decompose weighted krige krige formulae incorrect formulae covariance krige would conditional formulae enable avoid co sup e attention weight david support david lot effort computation krige update formulae enable krige variance avoid costly inversion already available addition traditional formulae new formulae formulae sequential correct establish correct expression variance covariance parallel counter first notation recall formulae real krige krige formulae krige formulae argument counter sake simplicity case wiener even krige case stationary kernel initial weight lead krige krige lead expression
robust loss omit design consist c find condition rather strong exact improve yet maintain restrictive tucker kkt moreover gram design say condition meet link q ps compare bring lipschitz respectively one constant square logistic let constant effective arrive furthermore take depend fix keep away stay zero formulate differently imply hold appear appropriate choice estimator error likelihood oracle np definite matrix condition enough say compatibility prediction argument argument population I order compatibility another refer paper selection also small heavily convexity loss unbounde priori bound result loss mixed effect regime pa theorem example section kkt consider theoretical extension result quasi likelihood robust positive high co much literature eq orthogonal lasso extension model least loss therein concern lasso say good oracle state equal true result compatibility neighborhood stability equivalent hard strong latter imply compatibility concern oracle inequality early paper context orthogonal consider along later proof developed remark technique penalty design hinge penalty design compatibility eigenvalue penalty consider generalized possibly compatibility cover case lipschitz compatibility one spirit condition concern oracle model extend beyond paper generalization robust high discuss concern selection modification lasso procedure scad introduce quantile aspect study apart theory description datum case huber loss loss numerically present finding prediction compatibility dependence linear situation large assume assume vector quasi eq quasi describe lipschitz assume quasi robust likelihood section handle let abuse vector f large correspond regularization mean shrinkage expression square e usual call estimator either euclidean quasi present condition selection paper map allow turn sup nonzero denote small call approximate elaborate issue brief outline see active well term coefficient zero refer appropriate sufficient result state facilitate give formulation next robust definition inequality likelihood address similar compatibility strictly depend tuning paper except see condition become involve much sparsity compatibility line connect sparsity alternatively square suppose lasso error jj regularization allow see rather say sparsity compatibility constant stay estimator constant know measure term include say error ahead idea approximation neighborhood compatibility sparsity least estimate immediately bound abuse terminology prediction behavior level sup inequality allow neighborhood link far note quasi measure linearity term choose true constant serve normalization albeit principle say exist exist q exist impose link hold logistic canonical come appear result let use square describe effect choose eq neighborhood arbitrarily reasonable take comparable canonical compatibility eigenvalue condition error compatibility know empirical risk minimization penalty coefficient prediction know true actually generalize truth sparse one sparsity compatibility trade arbitrary thus condition constant regularity
refer matrix factorization visualization interpretation temporal network membership represent binary capability attribute covariate link prediction latent space distinct technique much graph remain I nodes vary characteristic domain single non metric rich evaluate future work may wish well discussion focus base link however link form use trace page visit walk discuss task relational link event type assign stochastic modeling prediction predict presence static frequency occurrence link prune away link facebook noisy add used remove link indeed assign could weight weight link often possible effective discuss link weight link interpretation jointly link construct weight link somewhat assign link thus link link except assign link negative relationship flow link generate continuous instance might word appear link feature view summarize weight feature summarize process collective toward compute much supervised learn weighting link construction exist link represent importance link discuss weight accomplish apply technique simply link perform apply decomposition singular similar technique discuss result unweighted graph unlike design exist graph technique unseen know additional sensible link simple weight aggregate property link aggregate phone communication network case interaction link like aggregated generate weight weight link facebook identify user friend facebook predict link facebook datum participant strength link large network via tree find perform alternative two education level interaction find interaction feature helpful day communication kind feature attribute topological feature communication take normalization send friend prediction predictive accuracy parameterize possible evaluated training approach manually prediction two dataset interaction propose link capture coordinate optimization predict strength strength task directly evaluate strength autocorrelation gender status researcher event occur interaction metric link base separately incoming message impose decay base old metric implicit represents demonstrate predict however basic task weight heavily connect frequently recently alternatively appear summarize link ever link new recently past classifier exponential decay kernel weight yield handle serve incorporate v link relationship facebook create represent enable subsequent classification link label text link unsupervised technique dirichlet analysis traditionally technique use topic document collection document topic form manual reveal sensible concept relation however semantic obvious infer topic link aid topic represent kind technique mind software document lda technique content associate link go impact vary topic run lda one significantly temporal topic describe label essentially bayesian dependency associated art message author message infer make use role email network demonstrate simple technique network label predict label learn relational link prediction page utilize feature anchor possible link link performance separate inference often significant challenge limit consider approach learn entire graph classifier treat link negative create sign nearby link indicate relationship edge interestingly decrease train vs status partially explain ability technique section work text general sign opinion mining sentiment reformulate predict associate kind label design logic extract text yield node represent object produce link analysis example group intend graph two people yes directly cluster people description vote topic implicit link add discover construction link construction many considerably computation feature construction discuss summarize feature describe datum aggregate average common link precisely aggregation collect feature center subgraph target positive sign adjacent link node way aggregation link summarize kind link figure source go link input rather bottom show four subgraph link subgraph link subgraph vary amount display use input link solely link construct value message link feature count might compute form compute aggregate phone call link weight depend topology link coefficient link extent discuss feature link use link nearby link instance p work identify link target link form take label use link feature work graph sign figure feature sign target link path finally relational feature close close feature label node distant new base friend two people node predict construct node consider distinct discover email imply message graph supervise paper certainly add node knowledge base relational existence focus prediction node community characteristic social process kind facebook newly discover node represent node refer latent depend link include many node become link close share impact reduce influence propagate collective exploit node group node side may associate affinity group discover technique create leave alternative may represent unknown dataset potentially simple relational disadvantage allow propagate influence newly large liu large link collective unnecessary good clustering depend discuss assume create originally create link kind discuss relational node discusse relational attribute type algorithm agglomerative algorithm self map since study relational attribute desire link group original finding grouping kind two topology table latent belong value pair value instance detect extend group short path go along link link connect assign correspond reveal remove high relational addition link cluster coefficient describe metric node walk reach node nod euclidean link modular often lead useful simple module find modularity simple identify kind similarity compute every node link remove link place weight intuition vary form particular leave large form cluster original remove link reveal community group identification transform row algorithm interesting matrix variant involve compute original motivation identify cut connect identify node walk overview describe relational metric adjacency describe identify group enable ultimately yield compare ignore also number learn supervise cluster clustering liu metric web node prediction instance represent prominent normally prominent community search secondary treat link represent detect pattern add technique topology graph technique attribute produce consider simple similarity combine single similarity mention algorithm instance weight attribute information iff link control importance attribute attribute similarity entity resolution attribute incorporate hoc set logical probabilistic system logic component primarily node person primarily describe connection people principle approach kind model link instance model belong member link membership belong search state chart hill membership well use estimate news generative sophisticated treat membership belief group membership particular round round soft assignment membership membership demonstrate focus simultaneously multiple g connect also depend group membership group connect group assign cluster area complex facebook might way might status multi node topic form predicate consider distinct individual cluster baseline alternatively node represent real world object case entity cluster representation create link representation yield efficiently section use enable inference collapse super yield logic see interpretation weight feature node seek node labeling seek discrete represent likewise systematically reduction subgraph weighting labeling general use label practice technique tend label rarely nonetheless interpretation substantial actually label vs discuss section represent weight social discover prediction whether attribute topology construct attribute weight contain represent sale sophisticated strategy indexing concept corpus rank connection quantify rank extensively elsewhere ignore structure topology topology search example web conceptually implement systematically compute describe eigenvector algorithm rank identify influential variety centrality local global characterize metric coefficient centrality addition ranking ranking particularly relative ranking short path g addition alternatively formulate technique extend define notion temporal local notion closeness metric understand dynamic accurately identify concern interaction communication people could interaction node node join apply link link weight also instance various approach weight seek vector add anchor high approach probabilistic discuss far relevant transformation recently adversarial air moderate spam web technique topology necessarily kind attribute discuss propagate trust site try spam air widely suffer human site favor assign label label classification labeling consider end facebook predict political desire anomalous detection indicate ever connect estimate label enable subsequently node enable learn accurate labeling final goal label change stack classification label relational classifier relational collective classification cc analyze training inference stack bias estimate single pass cc extend idea generate training simulate classification approach outperform stack cc accomplish aware describe collective done knn logistic simply exploit assign supervise assign content communication traditional abstraction recent technique lda link incorporate link discover describe attribute topology topic specific systematic instance classify likely represent information neighbor construction construction discuss feature input operator discretization automatic relational link node value feature value possible include link node relational describe consist four type relational relational value feature consider relational value relational node might construct dimensionality several feature value thresholding etc compute exist node topological shown count adjacent short pass relational link new kind aggregation mode link value link target could relational feature construction away value adjacent instance value mode alternatively feature number count adjacent feature apply recursively topology combination recursive classification network might compute might hybrid feature topology aspect relational feature potential collective meta depend neighborhood contrast applicability independent occur semi co kind describe summarize case aggregation operator relational input walk topology color indicate walk conjunction input rarely ever discuss aggregation refer return another frequent compute fraction meet c operator may threshold node base complex relational via join compare performance alternative aggregate topology count adjacent link carefully handle temporal aggregation raw define importance recent temporal discusse notion distance modify node closeness traditional domain operator intersection represent presence contain word intersection collective represent union node adjacent value adjacent node e indicate draw relational outperform count potential probabilistic relational without compute specific dependency naturally accommodate vary neighbor sense choose aggregation still different clique link appear later clique enable random co citation link subgraph pattern pattern adjacent produce true exist simple pattern link involve star node link star node three know direct possible argue triangle share common side help avoid degeneracy arise generation subgraph sign positive relationship pattern left compute feature dimensionality reduction kn information original capture accord dimensionality component ica reduction adjacency create explain collective classification relational demonstrate apply pca substantial mention operator elsewhere walk classifier sampling topology easily imagine type discuss technique prediction weight weight computation easily lead refer however feature discuss newly discover create datum similarity graph instance type also discussion feature link text node whereas link relational link aggregation accomplish computation include node target link operator well suit vs select manually prior experience trial error situation automatic non relational study select receive search evaluate summarize search search relational possible specify raw consider friend divide easy operator possibility effort expert step appropriate usually specify operator strategy consider feature evaluation usefulness strategy selection evaluate way determine retain instance candidate evaluate improve immediately feature scoring feature retain decrease new continue correlation mutual strategy bic many frequently must metric summarize automatically describe system search relational tree extension relational involve topology g predict class measure yield value discard remain choose inclusion adjust bias relational sum exhaustive argue tree allow complex alternative single exhaustive operator base logarithmic recursive random add temporal spatial feature exhaustive pre add remain system system candidate view create base predict include recall pr feature continue extend feature add ability dynamically object different demonstrate helpful system develop independently describe impact system weighted formula systematically candidate clause empty testing clause top inefficient proposal clause response markov learner template guide construction approach learn long clause consistent pattern create enable structure parametrize attribute improve justification propose extend clause demonstrate fast well baseline seek learn explain whole political force subsequently clause conjunction appropriate smaller useful logical technique approach logical exhaustive decision feature relational link apply construct relational use refinement equality refinement possible proceeding refinement seem combine specific bic include reduce notion aggregation aggregate relational confidence discovery cd possibility instance clause metric evaluating clause primarily relational instance build transform labeling section transformation transformation instance way avoid labeling might propose perform link labeling entity merging solve task simultaneously notion inter relational relational feature task task compare perform technique across instance link focus recent associate link kind kind input text text document link document link link connect email message list prominent kind input type perform node link discuss document link first associate instance mention discover typical weighting use weight topic discover use represent kind perform seek topic word contrast design lda imply text document likely topic simultaneously semantic extraction discuss text second type add lda prediction use weighting generative document topic multinomial link link directly lda link lda extension link alternative lda replace link link model model bernoulli condition link link lda model document word document document work divide link incoming argue limitation largely overcome incoming much scalable lda likelihood lda link comparable pairwise slow change use approach encode lead introduce topic compare lda argue force topic pairwise link provide accurate suggestion pairwise lda baseline change model type link likely topic consider author create new document outperform lda link text link email message relate protein protein interaction several previously link model author art extend labeling strength topic assign node art link model allow role lda idea specifically share information share model block label link dataset email outperform lda baseline task protein category group model link prediction discover document surprisingly approach demonstrate link prediction discover link connect also learn discover topic discover graph connect topic connect frequently document represent topic add discuss additional relational transformation highlight representation transformation subsequent task whether accomplish goal guide provide ground link particular evaluation rank document last link classification run change surrogate change labeling prediction naturally problem weight relevance direct metric related autocorrelation presence sensible collective success strength autocorrelation social likewise attribute could assess naturally appropriate vary technique different upon ideally transformation execute selection surrogate evaluate retain improve even desire walk section use link final accurate supervision however supervision always helpful final program linear link lead via algorithm ensure particular transformation task remain suitable refer effect relationship e cause cancer challenge causal statistical take circumstance situation provide control discover discover different demonstrate understand media system remain extend broad range relational article task also subgraph frequent informative classify subgraph semi technique classification subgraph weight generation many node subgraph generation attract use prominent product method global potentially attribute classifier graph survey additional statistical prominent dependency structural logic transformation regardless kind statistical subsequently interact kind link connection relevant publish interaction domain multiple incorporate temporal represent examine temporal network specify manner simplification remove transaction efficiency support update comparison need purpose specialized maintain kind survey relational mining alignment involve pattern similarity useful overlap private publicly support preserve privacy transform way minimize information e prevent naive operate replace individual unique adversarial true identity user discover particular adversarial attribute identity user within early sensitive relationship kind describe cluster approach remove address attack adversary able recently study privacy issue goal topology create model generalize describe drastically change oppose make change allow variety analysis perform available resource attack must addition obtain related graph challenge need enable release importance relational article relational present section primary interpretation accomplish simultaneously technique work section highlight task link survey suggest develop instance reformulate traditional link likewise topic discovery use labeling node vice versa much discuss link together range transformation wide range weight decrease challenge technique come social retrieval inductive statistical relational relational resource evaluating change consume challenging understanding affect characteristic interact combination characteristic thank anonymous majority research research laboratory support fellowship make air office scientific engineering fellowship relational relational artificial intelligence transformation relational transformation transformation link label link construction node weight gray em minus width em depth transform statistical relational david department west usa department usa apply research artificial intelligence laboratory code dc usa cs edu david representation become due relational domain article examine issue relational choice relational feature affect capability introduce representation relational domain incorporate link include predict existence systematically motivate detailed compare transform highlight challenge remain research assume identically violate encode business person task involve associate person generally relational describe type link relational internet world scientific citation bioinformatic association communication location mechanism implicit relationship relational address reason relational high characteristic across link instance friend political people inference relational reasoning identify issue heart intelligence decade important example whether feature add high decision ai domain large complex way model performance relational datum use chapter relational relational algorithm relational represent attribute g logic focus address grow application people place likewise relation link content representation choice range aggregation kind path researcher recognize importance separate study examine weighting prediction survey assess relational give link subsequent representation transform feature apply transformation vary upon intend sometimes substantially add node add represent underlie simple increase dotted predict represent predict link link link produce discuss focus subsequent focus examine relational figure pre increase output valuable prediction insight protein survey multiple purpose applicability g suggestion great follow prediction survey many could g side facilitate transformation collective running analysis classification create possibility label useful pre stack wide likewise survey issue knowledge representation dependency link structural logic network briefly focus modify modify feature change logic program node addition program closely analogous logical could logic anomalous transform transformation understand appropriate intuitive representation identify specifically transformation task predict iii estimate iv link consist feature examine necessary comparison organize next relational label weight section consider summarize transform discuss method evaluate transformation challenge conclude datum facebook relational introduce relational aid define type consider facebook www popular social link facebook user gender relationship status school movie preference likewise link formation send link formation one political moderate assume construct label every representation sometimes focus useful pre process subsequent analysis political correlated degree connection type kind relationship instance indicate people join facebook addition facebook weak person often weight notational add component symbol transformation predict might combine measure similarity instance recommender implicitly link two rating movie book case cosine commonly metric link problem transform train approach machine
function lasso estimate test number govern predictor average compute pool together level sensible feature thus represent precision run run problem tend within precise fast high ex contrast affect measure fall specify scale stop stop burden reach affect obviously precise difficult descent setup ill condition system surprising proximal experimentally roughly comparison believe handling explain optimize greedy overall highly competitive accurate much rough dominate agree competitive variable conversely sensitive optimize second method competitive arise difference protocol accuracy stop criterion step occur early slow considerable obtain optimize soon sensible problem figure implementation highly fast improvement rough fista homotopy fast alternative large condition true restrictive fulfil obviously require rough either removal irrelevant relevant recovery high need hundred medium sized generate set determination average correlation medium sufficiently say evaluate compare lasso return various argument default correspond report htbp error mse mean bottom expect bottom obvious well level htbp cccc ms accuracy opt focus performance slow quadratic solver become precision par ten time slow viewpoint sparsity induce optimization viewpoint penalty compute low minimum objective function provide assessment test state art implementation compare mixed possibly ridge penalty lead known net derive viewpoint address non efficient wide accommodate penalty symmetric ridge provide structure elastic net wide range inducing fuse irrelevant bound stem triangular side reach choice vector side slightly proposition state prove define relate penalty infeasible inequality stem lemma trivially conclude define compute hand stem trivially proof generic problem trivially tight gap guess near value complement current vector function much compute gap nan compute meanwhile monitoring could comprise zero optimality violate compute meanwhile easily optimization france et universit france universit france france interpretation elastic net beyond viewpoint unify efficient medium software solve get rough compete aim solve viewpoint provide novel interpretation machine robust framework location affect closely concern show assume penalty assume corrupted suppose response adversarial according induce recover formalism regression simple unconstrained quadratic strategy iterative plain strategy solve approximation medium several computer variant uncertainty thereby detail derivation lasso ridge penalty net general purpose formalism section optimality resolution scheme demonstrate solver exist motivating linearly variable sparse perturbation variable datum consist source affect design corrupt strong though set option lead equation relationship pattern neutral perturbation denote response form sum contribution input input neutral noise discuss support amenable unbounded deviation confidence level neutral confidence noise scenario valid estimation perturbation belong without specify derive robust regression eq robust easily follow robust reveal case corruption minimizer ordinary amenable propose know two follow dual spurious regularity condition express convex define convex finite possible perturbation induce elastic net implement strongly subsequently prescribe magnitude activate group derivation problem suggest unified iterative resolution assess bind gap current solving series small increase involve guess solve respect active currently penalize submatrix comprise index adversarial pick move along descent direction update adversarial check coherent value bad gradient respect choose minimize current picking problem variable active provide behavior net measure computed plot large display hand meanwhile optimization monitoring well subgradient comprise situation violate eq compute meanwhile obviously objective objective trivially tight optimality complete complement propose magnitude gradient solution bind low size duality check gap nan compare performance efficiency assess value computing time return obviously package compare comparison language library calculus library algebra second package bias simulate post genomic signal processing domain predictor exceed dimensional setup active bad subspace thing local govern affect heavily run characteristic reach rather determination conditioning variable coefficient true ill three art coordinate name elastic net admits embed routine approximately differ regard active resolution descent implementation approximate optimality reach end precision warm routine resolution nine situation parameter medium predictor low medium medium high medium level medium medium
sf sum prove together cdf binomial q early sf sf j prove microsoft com microsoft bayesian idea study empirical state provide novel thompson simultaneously open regret prove et novel conceptually beta bandit armed exploration trade inherent study mab clinical trial unknown arrive sequentially decide treatment patient choice reasonable treatment patient time treatment bandit many generalization study basic stochastic multi available bind ucb algorithm discount reward bandit problem randomize regret basic probability arm ts ts idea independently emphasize bayesian free arm see interpret assume bayesian knowledge bound thompson directly comparable ucb attract attention thompson provide technique general setting favorable comparable low display news competitive popular delay delay play delay require decision arm play ts randomize unlikely delay microsoft search ad idea thompson competitive ts namely provide bind bound e bound logarithmic reward problem exist far lower bind pose thompson optimal bound thompson analysis technique conceptually arguably extend distribution beta contextual substantially state ts formally mab give slot machine play yield reward obtain play repeatedly reward play mab decide arm reward goal maximize expect play amount arm formally define introduce notation play regret performance measure guarantee provide thompson reward reward thompson sampling closely extend bandit bandit maintain bayesian convenient prior bernoulli reward useful observe bernoulli trial trial success sampling algorithm initially arm time observe play play accord summarize thompson arm play arm observe else expect first matter state course optimal arm armed stochastic bandit thompson algorithm big bind armed bandit thompson big notation contrast bound dependent bound depend prove regret precise give asymptotically achieve bind bayes ucb quantile low sampling also achieve independent regret ts exist imply suboptimal imply additive dependent appear involve problem bind ucb give theorem follow step essentially condition history arm lemma core decrease arm desire merely introduce early introduce notation denote cdf mass binomial number play time arm threshold bind event event intuitively later hold time step play play reward define probability assume probability leave hand trivially exceed suboptimal hold q last equality event involve hence f f true side inequality trivially suffice exceed suboptimal arm prove inequality give involve essentially chernoff hoeffding appendix observation concentrate mean happen I careful binomial sum play lemma mark observation change play give give eq rt notation absolute constant obtain independent x dx e rt
empirical work example approach practice orthogonal series kde non estimation datum inference population sample solution continuously twice integrable derivative essentially smoothing degenerate integrate hx kx z x base usual spline minimization problem model smoothing spline twice q control fidelity datum converge square typical spline part spline criterion fix remain spline write element penalize square write minimize give suitable contain function unknown different give ref basis nu nu kde sequence cauchy schwarz hand uniqueness inverse fourier hilbert space hilbert space basis fourier smoothness choice hilbert drive series also estimate nice
simple case adaptive directly use multinomial case equally version take hour complete run intel cpu result allow sampler result pdf mode non clear adaptive well n autocorrelation plot acceptance ratio respectively allow one brief interested forward let composed concatenation genetic type go mutation x zero probability mutation mutation diagonal generate number reach reverse sampling forward previous mutation uniform concentrate infer shape equal adopt approximately level proposal walk c take hour smc unit gpu left plot trace sampler result encourage practitioner adaptive ht ht bottom autocorrelation plot acceptance respectively perform inference stop process level auxiliary recent
lead normally load pc usually variable sparse loading vector enhance store loading vector incorporate induce mechanism formulation via induce two norm denote optimization formulation extract loading compute arise modeling induce si cardinality two constraint version stop effect choose enforce encourage sparsity si usage x ax x ax ax ax p x ax ax formulation involve previously formulation version four propose formulation constrain directly cardinality load introduce additional variable
hierarchy document label multiple redundant membership overall percent document content two branch market economic social percent label branch example market market market branch document multiple level topic trading market market branch market therefore completely membership corpus topic table policy account comment c capital issue rating asset production service market market c c competition management management e economic economic price price finance sale finance e e output capacity trade balance trade start lead mm cb lx cross branch name european community ec internal market ec ec policy g ec economic ec ec ec ec ec international world health issue issue interest science weather service market market market market market soft trading market mix cb lx branch mm exclusive topic parent expect usage candidate highly exclusive child child topic rate greatly parent child mean branch consequence although model regularize fit figure differential usage across
survey among simple approach greedy item type sort operation next density second identify many feasible round another fractional version fractional solution fractional ii item solution integer fractional relaxation choose j cost determine high type depict budget arm budget small low still estimate set greedy receive upper limit drop subscript since np hard use near optimal arm confidence arm indicate arm next arm budget repeatedly draw equal reward unbounded current imply converge solution unbounded budget form
store leave correspondingly statistic integral calculation classification propagate step particle change require first update occur new evaluating stay dynamic swap ad leaf computational cost incorporate leave explore benefit ad heuristic discard estimator datum discard full may discard enable operate stream eventually become use massive synthetic illustrate multivariate modern batch x online keep intend represent I budget full comprise fashion leaf ht cm full reveal even nearly full predictive second respectively sized data cost gap grow roughly online stay classification consider uci machine repository classification attribute predictor train fold full fold two stream
size specifically depend difference normalize implementation advance table refer compute gram rewrite n ns require memory small would run instead regularization less amenable regularizers elastic regularizer well feature easily incorporate negativity closely stage center alignment propose selection problem perform however hessian positive non feature suit review relation approach eq joint window
sp ex e e e kernel e e ex e e e ex value benchmark source compare form eq baseline psd ordinal regression sigmoid sp insensitive sp use sparse implement forward greedy surrogate mse accuracy size significantly learn able guarantee sp offer mse accuracy sigmoid superior seem sigmoid dataset method baseline along give
iii form iii theorems ii asymptotically imply asymptotic type iii combine maximum c type ii summarize definition maximum appendix relation show advantage respective error type estimation type supplementary difference type supplementary accurate corollary maximum type performance method g discuss type assignment
approach functional combine spline eigenfunction represent basis eigenfunction predictor project onto function instead incorporate penalty eigenvector penalize estimate arise expansion term function determine empirical structure see scope longitudinal setting allow vary concern uniqueness instability arise lack predictor infinite estimation subspace uniqueness mr spectra plot amplitude involve repeat subject subject collect value predictor specific effect random similar slope longitudinal assume decompose several invariant mr collected stage scalar mr spectra interest association mr evolve time mr show figure transform frequency pattern spectrum spectra
tc event least de know u grid close mapping construct fix set exist close remark corollary thompson heuristic multi bandit problem idea significant demonstrate well however many design thompson sampling contextual multi armed payoff context study version contextual thompson prove efficient theoretic low armed bandit mab exploration trade inherent armed bandit bandit round arm arm make choice arm play dimensional arm reward arm play play round learner aim relate likely good reward
phrase movie movie learn appear near though analysis train gram positive sentiment avoid specific gram movie set use difference energy gram sentiment sentiment explore training pair well yield examine gain energy class bag average energy linear classify document result give notably model bag bag bag science china sciences economics engineering probably slowly retain whether effectively describe rbms softmax unit efficient train rbms
simulation trial variation guide probability three importantly sharp missing value multidimensional simply miss zhang require miss observe equally fit autoregressive compound miss care user split illustrate idea survey health vs health vs fair poor assess separately parent teacher absence code response child teacher child teacher report two response generalize equation simultaneously fit child health fair otherwise page cd intercept status health teacher single parent poor health guide figure split health status split parent child health terminal terminal report behavior teacher interaction parent report teacher report child parent single parent teacher report frequent child teacher report parent reason
calculation inequality obtain substitute prove operator I definition adjoint convergence minimizer regard sequence pt minimizer applicable optimization problem need convert set sequence random feasible problem function converge pointwise prove pointwise convergence distribution theorem converge pr rank perturbation let satisfy function accord limit sufficiently sufficiently nuclear encouraging decay establish transformation directional q accord sense tangent cone set argument unique feasible result explicit characterization n r subdifferential lead together conversely easy kkt necessity n n tend characterization subdifferential equation eq since exist uniqueness proof similar easy b tangent assumption n proof necessary consistency explicit f note r yield desire conversely easy satisfied necessary part theorem due necessity condition consistency proceed theorem multiplier simultaneous know tend exist argument tend complete first rectangular self
finally easily solve substitution part unstable virtue uniquely expectation triple expectation rating applicable raise whether uniquely even proposition expectation value ce c e c invariant trivially q satisfie imply assume conclude complete square prove imply q nn precede fall reliability uniquely range straight line parameter category case estimate popular triple uniquely situation involve typically
iterative maximize scad standard minimize coordinate detailed defer analyze standard literature statistical fan assumption independent satisfy subgaussian subgaussian variable constant need constant regard assumption np satisfie appear due nature establish oracle sense support asymptotic ol estimator ol demonstrate correctly mean
create supervised learning task visualization ad create different yield insight particularly belong group place result low within group clearly experiment propose design line available entire advance line become past snapshot snapshot propose utilize place belong together place position neighbor temporal context previously summarize grouping propose consider three force intra force inter force intra force force force different group force correspond force group subject inter force meta force belong close utilize force direct intra force meta force da tu add virtual virtual group virtual virtual differ square grouping cost propose da static graph scale graph contain temporal apply characteristic dr attempt preserve dr grouping dr pose close space pose version optimize denote group membership point notice group regularization control two grouping grouping group ordinal
character lead total hierarchy conditionally layer close ard layer share correlate ard bottom fig discover common intermediate layer ard set non two subject perform opposite ard weight automatically model latent method plot encode information interact subject constrain space two projection latent dynamic encode top different whole first small dimension ard blue wide red ht abstraction analytic evaluate quality overall hierarchy mnist utility digit set
uniqueness statement slice mode belong cp tensor routine component slice span true equal step randomize span repeat attain non singular principal reach step mode
potentially run distribute accordingly acquire want answer completely fast method utilize distribute extreme update neither appealing bit completeness relate lie minimization widely fast force learn bfgs offer robustness relatively fail situation acquire previously situation old certainly leave increasingly similarly reasonably regret reach error descent combine advantage number bfgs sized batch forget change significantly outline problem interest widely outline algorithm
redundant tree available randomly random forest recursively split tree focus detail framework regularize tree way select splitting belong belong enter need upon gain select criterion tree consist tree tree build relevant regularize simplicity single regularize ensemble indeed form tree ensemble regularization feature need measure minimal markov minimal
derivative reversible distribution let first since reversible satisfie reversible reversible invariant satisfie show upon define proof expect upon draw two geometric irreducible chain number setting indicate lack ergodicity utilize proposal fall could proposal bias towards centre global end define q autoregressive similarly lack r b iy recover target attempt improve markov kernel concentrate markov analogue extension hold irreducible admit invariant mild proposition factorize x follow artificial dp decay suffice x
informally unbiased formal justification taylor bias parametrization like equal adjusted mse inference linear function effect empirical blue suggest blue suggest effect test construct fix simultaneously matrix explicit section accordance approximate scale distribution estimate analogy derivation estimator denominator degree trace estimate derivative detail
irreducible appear add jointly jk jk jk w accept probability q remain variable slice build truncation completeness briefly recall admit stick break slice latent give allocation dirichlet q update augmentation technique w accept acknowledgment thank pr helpful work european intra fellowship preference college european intra european fellowship rank develop nonparametric popular random specify effective simulation model apply preference degree amongst existence preference establish determine subject matter datum consist order list object ranking arise shall top school consist private application chapter provide detailed item assign represent assume item item among item choose select proportional overall partial select many collection unknown sensible item assume possibility item appear I sampler available infinite unobserved group outline section gibbs limit mixture whose directly
rate level nystr om question translate version pca add little overhead rf combined square loss g group pca classification approximate unlabele covariance turn select organize summarize describe elaborate test analytical chebyshev imagenet comparison kernel visual system good performance among histogram kernel propose cross different bin histogram include also hellinger shannon invariant kernel approximations rf nystr om sub operate kernel long slow speed propose good learner embed deep code prove
cluster shape conjecture method one another worth investigate fully derivation proposition statistic relatively derivation test assumption calculate distribution let likelihood simplify could statistic derivation statistic likelihood ratio derivation statistic normal x lead lead proposition collect region notation statistic define note region show case signal consideration region region noise show c straightforward similarly bn stochastically eq boundary
much fast especially dataset quickly depth assessment require acknowledgement work national centre grant perform grant scene allow one video classifier still ensemble computer application image action reality attention aim bring wide generalise package briefly bayesian version list variant discuss
function use basis fourier natural structural move posterior q integer wavelet inverse probability event nc j n ng nb ng nb j later two q correspondingly quasi write log quadratic result map quasi borel integral soon gx relation soon integral hand hence practically expectation relax keep essential establish technical since theoretical frequentist allow use level approximate basic theorem I usual minor primitive iii used identification assume identification conditional reader discussion completeness domain ball substantially relax however analysis inverse continuously invert plausible know value decomposition see wavelet system suitable quantify ill basis quantity ill least l ill pose
type prediction dataset relation type relational incorporate relation fraction type link prediction observe help boost model capture correlation among reveal influence link prediction performance likely vice versa relation impact performance follow paper propose new social introduce specific addition object provide treatment derive conduct dataset relation reveal distinct network evolutionary link scale human association provide paris
intra block observation sensing sense noise lead cost together covariance th semi th block parameter cast specialized variant block exploit structural beneficial block class compound auto etc cost type maximization author
formulate evolve intuitively change human page dynamic available spirit research place graph row probability manuscript utilize walk diagonal degree none column linear system table section rx computation decay time vast idea incorporate update related et vector change new
improve sample period precisely summarize estimator observe difference become indistinguishable significant way length accept fail step therefore adaptive good contrary accept step adaptive note result finding r ccc r r ccc r iii difference average exact l l l approximate estimator conventional histogram limit conventional time available whereas period sampling period bias decrease shown summarize iii h tt accept fail step shown accept big iv exact iii iv result realization linearization euler thin partition guarantee simulation realization two
patient report breast year respectively survival shorter report event table interaction comprehensive network pathway interaction remove pathway r package node undirected edge protein protein type direct interaction package microarray probe accordingly microarray map ignore network approach
follow goal mind tradeoff tracking residual thereby track manifold vary old cost online estimation impractical address cost f form subset sequence previous illustrate dash red union correspond past detail apply sequence residual assume point I detect soon method streaming setting change incorrectly change stream alarm time point detection false long alarm multiscale online subset subset use multiscale multiscale harmonic literature subset parent child roughly cover child tree index subset leaf use index tree notation transpose matrix matrix offset hyperplane origin hyperplane parameter specify ellipsoid capture tree virtual maintain fine leaf instantaneous need explain tree subset use determine
constant fraction weak tight see agnostic approximation agnostic construction agnostic thm thm agnostic regression thm formulation cl pt agnostic agnostic thm thm second deterministic opt opt opt opt nk whose correspond point response table opt agnostic probabilistic fail probabilistic algorithm away event choice sense agnostic success unknown approximation agnostic construct without bad vector fact exist bad
mlp output perfectly detector fact never combination feature input generator generator cause high activation hide neuron maximization base technique normal mean norm initial patch find feature activation maximization strong generator neuron htbp correspondence input pattern discover maximization deep demonstrate correspondence mlp four detector inferior detector mlp single detector visual appearance detector therefore denoise high capacity seem operate principle layer detect noisy patch weighted patch consider see mlp difference mlp previous one output patch somewhat generator figure detector generator patch generator look similar feature detector look seem focus input look detector focus center input explain patch input patch patch correspond center region patch correlation fall pixel outer denoise center activation binary mlp single lie essentially binary dictionary hide zero layer information unit hide entropy train mlp mlp mlp unit layer unit detector generators generators unit low generator high feature detector high entropy feature low latter seem
elaborate proposal transform correspond learn particle strategy however user reasonable indicate resp describe simulated figure modelling determine persistence really characteristic range behaviour properly must understand spectral quantity coefficient belong particular towards consistency size prior independently outside prior fact consistency strong indeed simulation ghz resort language library program author web page smc final approximate posterior correction particle cpu minute variability run top plot see plot reveal rate birth death may successfully particle go axis grow slowly roughly smc sampler step time hour minute satisfactory result box expectation approximate ten run identically histogram bivariate sake top plot repeat run smc versus colour proportional
following provide sampling use construct specify similar except apply pack recall consequently choose equivalently somewhat theorem though emphasize choose loss since pack consequently complicate privacy begin state let accord support proof appendix binomial differential combination channel f mutual information combination channel lemma thus obtain choose q property differential require index establish bound amount underlying bound locally differentially private packing element induce mixture summarize previous paragraph hold locally differentially private interactive place proceed establish establish require formal proof proof corollary identical nearly paragraph uniformly interactive differentially private channel immediate low inequality separation b note le identical argument pack conditional contraction proposition construction le obtain proof outline exhibit exhibit special packing hypercube packing packing usual conditional pairwise separation pack discrepancy theorem hinge loss since distance bind indeed pack hypercube sample long channel locally differentially locally differentially previous paragraph hold choose note universal result apply repeat completely imply bind noting complete low reasoning choose result mirror obtain match second statement mutual follow uniqueness high mutual allow level allow must must must must multiplier apply technique convergence
form constraint q since q computation definition bind acknowledgement part work network p project grateful constructive discussion mm mm lemma group effect france universit abstract huber regression operator satisfactory method huber call elastic coefficient cause ridge ordinary huber enjoy approach compare elastic satisfactory illustration purpose keyword elastic huber oracle property implement http fr subject outlier encounter predictor eventually response heavy error ordinary square ol estimator huber type estimator hand linear analysis sparse enhance fit easy interpretation prefer simple put light relationship covariate tune suffer satisfy nontrivial consistent differently
identify validate current gene credible significance great dataset non constraint experimentally validate lasso routine significance threshold roc various value varied threshold set roc dataset plot show purpose sensitivity specificity reason lasso coefficient use false auc rr bayesian large term auc note meaningful select auc perform constraint infer protein target substantially base apply broad discuss gene detail interact
preserve local structure dimensional manifold propose maintain nmf performance application sample partially g image face kind corruption corruption treat noise estimation nmf kind underlie recent deal partial usually corruption give ahead ignore assume position
robot essential maintain physical coordinate frame sensitive error error derive identification explicitly rewrite feature follow look kronecker notation element dynamic pick form give transform repeat representation convenience discovery system simply interpretable track kalman need approximation plug kalman h gps state collect observation u u x coordinate compute step motion pa simultaneous localization mapping algorithm identification desirable include consistency optima optimization multiple track low requirement track extended transition severe non gaussian posterior encounter finite demonstrate world theoretically justify comparison good attempt estimate location hypothesis filter optimization likelihood
w diagonal see implie theorem residual induction lemma second approximately satisfy assumption let algorithm permutation let projection extract column condition say generality projection reduce first inequality projection orthogonal orthogonal projection last w k actually empty imply extract th ip ik al recover column order compare bind theorem one separable matrix without separable let recall matrix hull add separable et identifie see equal column eq theorem matrix tighter particular extract imply large well bound expensive least et al rather difficult trial separability sense separability satisfied summarize simple growth hyperspectral comparison depend solve program algorithm
application accelerate contribution characterize hardness value direction directional index require explicit representation branch max exact exist technique space shift radial algorithm set posteriori shift applicable shift invariant equivalent input solve study shift embed inner product approximate value follow representation locality sensitive lsh locality sensitive hashing lsh normalize kernel without normalize similarity rigorous technique branch bind restrict runtime suggest preprocesse kernel example kernel require un kernel recommendation normalize inaccurate item
roughly impulse additive provide lead outli bayesian detect describe additive explain illustrate methodology simulate well set concern ip address server department university conclude poisson identically variable
calibration misspecification confidence exponential evy range pricing calibration focus parametric cf therein l evy approach parametrization misspecification activity evy construct confidence calibration neutral measure option price calibration l evy process assume evy mass sd evy component second set mass assume evy fa sd modification improvement square quite fast fourier whereas focus theory realistic sample size jump finite confidence derive confidence activity set sample self decomposable scenario sd well coverage simulation variance introduce use self decomposable option good residual datum calibrate model option behave
company even sale market assess seem interaction estimate finite solve krige stationary field mutually normally spatial however conditional interaction term flow paragraph problem address paper develop spatial though refer location collect measurement aspect sense general differ suppose eq q measure measuring say h ultimately act measure measurement distinction value measure location n paragraph estimate main statistical analysis arise follow spatial h parametrize set pattern pattern assume
estimate terminate whenever may happen design small implementation allow predictor stop convenient complexity ode solver repeatedly evaluate derivative path stop event check derivative semidefinite inactive penalize along optimizer coordinate number segment ode evaluate hessian glm loss take detect inactive thresholding per evaluation optimizer utilize warm fast adaptively choose size simple software ode ode package matlab differential equation user fulfil knot scad mc solve ode partial due scad penalty knot mc ode rarely numerical role lead parameter path usual implication next follow certain avoid procedure aic variant frequently recall bic laplace fisher observation maximum posteriori likelihood plug aic derive operating necessary fall path somewhat pick
improve confidence hc bias wu preferred group mean robust conventional estimator unclear pool variance analysis become economic adjust question ols adjustment rapidly unit standard brief remark actual substantial imbalance cox inference covariate imbalance adjustment perhaps either consequence occurrence randomization population mean member assign biased finite context equation bias square analogy lead tend depend largely importance omit quadratic covariate omit first result determine randomization adjustment suggest focus social illustrative conduct service financial college student except school randomly treatment service third service group
logistic adaboost without attain analysis produce output iterate must focus suffice mean deviation cause issue apply choice apply generalize limitation present boost tree manuscript organization primary force iterate finite break piece core complement albeit hard core direct establishe effectively carry instance significance prove sample quantity risk display simply many bound consistency appear simply guarantee share arbitrary structural tree constraint split meet risk note support variety associate mention related generality sufficient boost work map weak learner function convenience operator abstract convenience search classification instance boost throughout make function notation hinge origin requirement three precede loss arbitrary
go bandit derive ucb chapter bandit nature process markovian three distinct playing effectively ucb randomize call markovian case stochastic adversarial refer survey markovian bandit player forecaster implement bandit performance horizon play arm sequence arm study select receive reward play horizon forecaster forecaster allow averaged definition expectation forecaster pseudo weak since compare expectation regret build arm select arm possibly round forecaster reward independently past reveal forecaster pseudo write seminal bound chapter describe naturally stochastic bandit game quite recognize instance motivate theoretic machine step gain would go forecaster select gain still minimize regret adversary mechanism gain mechanism independent adversary forecaster behaviour adversary instance clearly adversary player randomize pick gain game simulate presence adversary gain depend randomize player round consistently choose gain non connection equilibrium observe theoretic equilibrium behave differently opponent change behaviour interestingly minimization round forecaster choose possibly external randomization adversary gain vector possibly external randomization observe arm adversarial bound forecaster opponent irrespective opponent adversary start since hold opponent randomization forecaster play opponent playing topic david seminal payoff observe opponent move consider game round observe turn equivalent adversarial non adversary simple playing recently propose approximately discover science apparent connection bandit bandit adversarial arm process probability state change markovian fashion reveal player go bandit process compete resource project get resource matrix nature seminal provide optimal efficiently notable special markovian bandit assume state change update reward observe markovian bandit economic different adversarial
especially mean advantage disadvantage soft feature view proximal minimization present chapter demonstrate connection iterate start appropriately descent negative regularization dictionary chapter eq q separable element therefore unconstraine optimum set soft literature slight split encoding obtain proposition appear threshold feature separate compete entity triangle code treat proposition important insight proposition nice code study feature code proposition tell even approximate code even typically see single sufficient show discriminative three question possible encoding lead extent investigate experimentally examine soft feature variant proximal code present chapter fast soft first fista
datum study ability estimation dimensionality large cause rank small direction difficult distinguish projection tend separation panel estimation min rank min max min demonstrate reduction axis align noise contain investigate project onto direction reduction method cluster consecutive within sir implement whenever randomization projection include scale sir evident rich scenario randomization perform regime improvement scenario ridge h tb rand ci rand ci reports ambient panel report bar standard dimension set assumption strongly estimator randomization latter sir randomization pca denote randomize implement regression behind covariate response latent decomposition bs independent noise principal q latent factor
notational convenience estimator corollary give section fast slope take conditional quantile bound previous convention apply address tight restriction rate least nonempty side indeed prop file three criterion investigate simulation singleton xu define likelihood asymmetric eq aic see may analogue bound bic du du integral replace summation monte finite case repetition website quantile correspond spaced point quantile simulation summarize figure table criterion error error show
aic often cross normal glm simplicity drop covariate vector zero denote response normal aic fit square degree freedom stein unbiased take degree angle lasso freedom nuclear normal derive unbiased comparable non orthonormal b convention expression freedom regularize fit freedom several aspect least degree third surely rank degree freedom qp exactly plot well I count equal freedom na I count search number effect freedom theorem limit require existence exist give conduct first regularize covariate second classical regularization regression nuclear lasso covariate penalty power regularization simply correspond penalty coefficient tune instead examine nearly unbiased scad thus
compared show advantage viewpoint convenience exhaustive run exhaustive moreover efficiency deal exhaustive compare choose competition look fig show find feature although set hard specify ht depict follow find specify acceptable find feature discuss third competition well case user know viewpoint rough view aspect optimization analyze aspect first extension concern since require rough form accord cover rough deal neighborhood system rough well might fuzzy example rank directly
generate lsh thresholding thus lsh define hyperplane different opposite hash point match cosine specifically point lsh guarantee within time optimal empirical study lsh efficient hierarchical decomposition vision extension lsh lsh probe lsh propose lsh lsh address limitation locality hashing shift hashing mapping invariant convergence guarantee well relatively base need preserve pairwise distance database
sequence round sequence consider proof eq forecaster share simplex extension forecaster share kullback leibler divergence p generalize theorem bregman kullback bind rewrite summing apply I I universit di sup paris within team classic paris sup paris paris france lemma definition mirror regularizer achieve carefully share equivalent generalize discount capture
polynomially relaxed stage bundle improved complexity preference economic begin modern explanatory utility finitely price observation piecewise concave reveal excellent survey utility finite view come hypothesis length relate give think future decision hypothesis produce polynomially circuit intractable utility reveal preference computational goal produce monotone utility non guarantee agent utility jump continue function efficient polynomial complexity nice work consider reveal set rather reveal bundle select value bundle bundle arbitrary agent bundle pair price bundle
probabilistic long modelling mixture component source dp process I speak interpret whose measure strength
mixture recent advantage parameter significance illustrate real keyword adjust rand index validation significance nonparametric integrate shift plug choice full note seminal concerned unimodal cluster shift cite preserve density crucial significant dimension extend scale digital visualization new problem closely gradient finding application medical imaging sense rigorously analyze one curve embed ascent path gradient concentrate estimation useful tool density arbitrary formally crucial bandwidth multivariate setting bandwidth consist definite bandwidth constrain degree along bandwidth identity mean smoothing coordinate direction note bandwidth validity previously check unconstrained counterpart reason smoothing encounter analysis detailed analysis derivative simple lead substantial become bandwidth selector develop sophisticated drive choice applicability
e r qp fp qp fp qp r qp fp qp e e ball next display project ball varied reveal ii value time approximately large moreover run apparent code scale bar still nontrivial projection relatively currently medium problem htbp group feature separate important less important information task mix usually matrix
unimodal multimodal gaussian unimodal asymmetric multimodal mixture unimodal rotation angle gaussian add extra rotation cause dependent across vary angle dimensionality gap gram component break permutation quantile type hypothesis accept reject experiment rate dimension solid acceptance rate test span dot reject line dot figure statistic kernel dotted line statistic base joint entropy span graph propose notice correction configuration small acceptable type statistic estimator method notice method angle determine hypothesis propose independence gram size therefore gap monotonically phenomenon relate entropy notice smoothing associate slowly characterization resemble quantum
convex bias term regularize absolute obtain set runtime optimization find stochastic gradient descent sgd sgd find runtime favorable sgd approach stop tend begin especially reach fast convergence become interested solution coordinate solve conjugate namely associate optimize dual variable optimal know immediately gap sub optimality version sdca round theoretical understanding gap sdca loss paper smooth equivalent smooth main finding smooth hinge precise several experiment large sdca sgd sdca understanding rate svm basic ascent establish mean achieve
evaluation devise prohibitive typically accord acquisition acquisition lie estimate point estimate apply functional statistical minimizer relate highly hence intractable
pc loading second project one pc loading project attain loading model enforce orthogonality unnecessary task sparse enforce orthogonal pc involve pc include etc spc etc method minimization perform variance objective function sparse equivalently formulate eq separate respect column follow small divide solve construct construct convenience rewrite form dimensional closed form I present equal omit element th equal proof q large element
forward smc f n filter acknowledgement project initialize input thank associate enhanced article un dx x px dx dx p eq semi q q define backward normalize td td dx n n dx group tf dx
recover graphical rigorously method compare maximization step support part nsf award fine piece hereafter refer address graphical
version simple skew simplified advantage expectation calculate e reader different form skew efficient th matrix degree shall fm finite distribution fm component fm generate adopt denote shape respectively sample fm implement admit mle yy framework complete likelihood unknown fm implementation alternate change log use maximize iteration calculation expectation bayes membership component current expectation positive hyperplane evaluate express central recently truncate multivariate correspond
preferable h suffer limit parametric semi parametric model paper possible parametric context derivative rkhs rkhs provably iterative study exploit universal propose safe discard variable prediction selection nonlinear toy towards role differ previous local scheme computationally algorithmic solution general distribution lebesgue differently explore analyze selection view computation consider towards natural improvement possibility consider supervise naturally suggest generally plan differential operator selection beyond lr absence would discussion suggest proof sm discussion science mathematics mit biological institute brain department artificial intelligence laboratory integrate project health child grant national foundation nsf di support usa science technology b foundation x x define thank empirical property quantity proposition meet operator continuous derivative schmidt partial derivative schmidt base hilbert schmidt operator respectively proposition extend henceforth satisfie follow write f problem
exponent brief proof entry length whose specified think compose column later property mapping encoder q mapping produce column compression q style codebook column exponential construction choose column develop automatically decode encoder binary binary indicate zero bit involve share common code two distortion encoder decoder achievable
probability comprise specific together membership estimator identifiable contribution assume correct situation primary possibly circumstance toolbox circumstance usually infer unobserve preference show sufficient contrast conversely reduce component belong impossible sharp region provably accuracy high theoretical grow contrast consider small amount vanish impossible inference however network parameter highlight difference impossible inference come introduce infer justify fit reduce approach deal search common way interaction far apart originally
represent pattern share b evolve share middle model rely evolution vary put alternatively therefore algorithms provide latent might estimation successively follow w em scale mixture follow slide maximize accord reduce convex previous conduct carlo conduct bayesian requirement prevent implement particle employ hasting latent x
time fast discriminative object representation maximize inter object region boost instance drive etc object representation design adaboost robustness appearance cause rotation illumination tracking liu yu present base mechanism boost lead tracking build online weak pixel wise employ object localization instead boost image patch object attract attention line tracking distinguish classifier adaptively discriminative supervise base constructs review cosine signal first dct section dct matrix dct compact dct dct goal dct express discrete digital mutually cosine frequency information dct dct n reflect energy e signal texture discrete discrete dct whose mutually uncorrelated furthermore dct sparse compression tracking dct cosine form write formulate dct decompose dct operation z terminology algebra terminology tensor algebra dct n accordingly dct special dct
take marginal filtering sec mean cross integrate source uncertainty law total expectation expectation take xt rewrite fm se center input predict xt dimension gp across na f ji integration serve test gp tractable choice combination integral signal dimension see compute law
dependency relate number report publication eq play population furthermore incidence plus age profile patient uniquely force incidence uniquely qualitatively forward force
g way connect ht ccc observe inner style circle sep name z scale sep style inner sep name h z h observe circle sep hide circle inner name name name name consistent topology capture recover motivated design way connect join together tree variable relation total taking resolve reconstruct set suffice make method new main contribution approach method practice test make order
berkeley segmentation graph orient generalize boundary detector propose pair connect parameter segment edge segmentation output range simplify optimization perfect code solver tolerance add constraint work describe recursive contour early low bound lp relaxation term potential neighboring relaxation unary potential cycle collection cycle enforce consistency fast house implementation branch remove solution inconsistent namely within component add enforce consistency edge bound comparable tight low give batch cut upper solution histogram comparative
w pn op pn ij v z ni z row lemma z z piece n corollary applying theorem induction mod pick similar mod mod identify identify order may define z follow substituting term lemma I c w p x op x op z cn ki op union bind w c op I op w op term I iw I conclude w pn combine pn argument ix h z pn small h h complete variable argument show ix ix ix I apply I
penalization variance hold penalization penalty equal concentration start analogous particular proof p p eq bernstein depend constant notation definition eq tm lemma eq deduce computation train set compute j risk train mn cost penalization criterion j eq v yield initialize datum unique j iid random value measurable common distribution q provide simulation result analogous illustrate extended version compare setting standard deviation datum drive separately loo cv sample deviation drive consider knowledge l loo cv cv complete test validity without sum mm right provide mn mm setting figure b mn v b b mn b mm theorem corollary definition remark study fold square goal theoretical choosing minimize prove non
show p continue px qx n px qx x exist chain monte mr author mr mr mr mr mr author mr mr avoid mr mr mr mr mr author carlo markov mr mr mr mr algorithms automatic author carlo mr metropolis mr author mr ed sampling chain mr mr mr mr mr mr unbounde mr graphic theorem support advanced research fellowship capital award project markov chain monte marginal chain spectral normalise pseudo chain many hold gap equivalent ergodicity unbounde recover case independent hasting super contour imply asymptotic pseudo marginal accuracy estimator increase interested defined achieve scenario carlo metropolis hasting hasting call proposal implement wise either unbiased nonnegative estimate paper
differ take instability define I cluster factor cluster fair clustering instability cluster item assignment figure depict instability request instability request pattern number instability two median instability bar instability minimized request pattern instability pattern randomly exist division pattern overfitte phase extra structure result instability instability increase realization subset analysis select pattern fit request facebook application methodology request request differ find request technique request facebook application must review rating application facebook facebook application perform instability outcome facebook fit tb entry sort decrease color conditional htb
come contain information correlation region multivariate correlation spatial transition see correlation random prior geometry geometric incorporate pay attention field change bottom layer look layer interface interface field positively interface smooth key consideration remain thing certain transition manifold reasonable interpretation leibl divergence completeness account definition need exposition boltzmann entropy arbitrary positive definite matrix line q nice length curve minimal
non thus correct estimator empirical take well use define numerical artificial dataset mm kde c kde kde denote normal mean kde behave depict sample kde show accurate kde depict next kde distances distance distance kde estimator cause fact kde tend density difference tendency kde base typical reasonably true
nd introduce hidden variable background nd pz w pz bayes pz pz constraint version pz derivative nd pz pz nd nd pz obtain nd pz nd
performance sequential derive process parallelization asynchronous update share latter suited communication compute machine cm paris france paris paris france universit paris paris france
contour see contour action pair state couple fully observable obviously help
draw body demand universal generate demand uniform generate demand distribution go round execution demand time execute execute demand environment universal demand determine output use mae square simulation environment
discuss resolve convergence gradient contrast reduce gaussian smoothing see minimum subsequent less sf sf comparable case sf considerably pair sf achieve achieve sf exhibit sf experiment perform sf trend sf pt c c sf sf sf sf sf nb segment discussion list retrieve essentially choice mean wide previously applicable restrict sf cauchy search similar table also trend small v particular quite may prefer sf nature table relate behavior sf justified analyze gradient ideally tune difficult pair theoretically
isolate adjacency permutation unlabeled node keep word minimum permutation unlabele correspond provide g node count argument supplementary state fraction node ise homogeneous degree scale eq nearly match frequently trade complexity datum fitting criterion schwarz stock return software code ise model depth potential keep randomly potential employ latent graphical relationship word occur latent graphical word constrain keyword select indicate appearance divide equal group purpose financial modeling dependency consist stock list sp dataset performance graphical simplify interesting sophisticated kernel complexity fit well avoid overfitte propose method lda module control neighborhood present range recovery theoretically guarantee h learner criterion threshold threshold thereby output choose value reference dataset experiment stock return choose previously discuss
chemical specie solution specify reaction pair reaction reaction place current number must chemical assume reaction population ax reaction together state reaction specify jump occurrence poisson reaction specify initial exact relatively partial physical discrete parameter property kolmogorov equation master evolution probability master intractable observe process incorporate show markov posterior computationally strong require carlo approximations systematic obtain approximation physical yield fluctuation extensively see applicable fluctuation system limit describe set state dependent master likelihood brownian increment thus limit applicability methodology transformation likelihood less inference linearity non
space unity mode define act scaled unit temporal dimension embed illustrate temporal closely resemble fig embed evident representation mode recover sec embed prominent datum mixing find place temporal mode useful physical available dataset influence factor non markovian initial compatible factor dynamic coherent candidate embed extent describe degenerate mode evolve quadrature instance capture prominent low frequency involve organize situation information separate verify mode periodic frequency window
arbitrary result restrict maximize say try equivalently variance common approach also criteria point disadvantage absence density measure probabilistic model desirable permit comparison may extend model use class density classification posterior membership compute thus appear estimation
exception rand interesting gap believe available sc affect suggest hmms instead automatic annotation music digit estimate annotation auto ms gmm emission long model therefore annotation retrieval account consist provide annotation range acoustic song audio coefficient window audio instantaneous song audio approximately second automatic tag modeling audio tag database tag database processed learn song tag song level ms pool together h algorithm tag tag tag song extract feature song tag song annotate retrieve tag rank tag tag sc write retrieval tag h tag use necessary implement sc approach song ms single hmm poorly hour times annotation yield sc song hmms centers hmms method sc hmm cluster center estimate hmms assign cluster hybrid song hmms hmm form h mixture tag proportional hmms cluster tag audio empirically ram h overlap audio evenly subsample sequence actual song non overlap believe em roughly similar still slow report running song require processor oppose would extremely intensive considerable maintain mixture bag tag feature far consecutive tag tag annotation automatically song precision recall score tag measure tag retrieve positive ranking
learn often large tractable exploit extend recent advance carlo search avoid agent belief mdp episode outcome simulation many future obtain belief innovation considerably representative competitive algorithm consistently algorithm sampling belief avoid bayes expensive simple mdp suit planning domain reinforcement illustrate benefit infinite state challenge intractable generic make unknown mdp mdp sa discount component mdp tuple mdp use unknown distribute observe history belief ph dynamic current inside augment space tuple
generalization cover construct ball center ball center call mass ball small ball disjoint ball concentration ball pr lemma constant complete constant finally question extend density give converge length partition fine chapter suggest many application plug distance estimating present simplify go approximately similarly consider
integer expense adopt anonymous reduction project let r qp projection definition relax p f pp p f pp p opt statement theorem conference title shape total projective include median cluster projective total reduction type greatly early result total sensitivity positively projective projective problem high article shape fitting shape refer b
feasible sequence banach converge belong satisfactory solution sufficient existence feasibility condition banach rate banach satisfactory time advantage important satisfactory see global mu mu j mu mu optimum remarkable distribute banach range aim possible fast rate speedup technique speedup satisfactory reverse choose rate big converge banach learn convex use speedup speedup quasi static lead learning find satisfactory realize context express generic satisfactory payoff generic reverse rate banach iteration target background noise load satisfactory convergence satisfactory field banach reverse speedup summarize banach banach iteration get clearly error iteration satisfactory solution speedup la acceptable tolerance study additive simplify equation per class requirement speed acceleration partially player numerical payoff happen hold observe realize observation
hold background refer appear sequel value possess expectation satisfy provide operator e operator specific play covariance operator definite self adjoint depend process result space banach proposition need cauchy statement conclude fourier eq statement orthonormal denote lebesgue since turn imply fix negative adjoint orthonormal trace v stationarity verify negative adjoint triangular kronecker latter two imply converge continuity adjoint negative get e cauchy schwarz dominate complete g p eigenvalue continuity fact e c observe complex conjugate eigenvalue spectral limit transfer operator sl filter spectral cauchy sequence prove appendix l yy routine argument iii dependent z r h cauchy inequality furthermore deduce sum independence conclude e e de office research support science policy office cm cm
type stage stage transition start state select take unknown receive immediately terminate problem define transition bandit sec radius policy maintain distribution stage observe payoff norm multinomial employ solve augment mdp action additionally act optimistic probability policy maintain dirichlet see distribution reward draw sample mdps parameter individual reason correspondence
relative algorithm propose highly compute additive eq also randomly time simultaneously row square norm row sampling subspace sampling ok expectation intersection dominate svd choose least seek devise mild requirement column optimal selection column aim choose construct achieve minimum constructing well hard polynomial particularly bound
address problem partial fundamental service automate focus fully decentralized page localize community localize partitioning available leverage decentralize execute acquire node need exist exchange among scalable comparable exist collect china topological already direct friend topology user many community contribution formulate new constraint page justify applicability user design community evaluate use scale social organize survey work formulate topology social proposal commonly vector possibly subset community satisfy intra community inter linkage include maximization cut minimization popular combinatorial np hard researcher eigen approach several batch focus detect community pure community researcher try incorporate enhance algorithm combine topology
fails successfully classify feature strictly require correct extend cascade mostly cascade several stage classifier either reject input pass prediction input cascade enforce reject classified input later however structure skewed class imbalance generic detection reject response haar cascade suit expert optimize building cascade weak note heavily stage regression boost work learner gain feature introduce additional learn tree reinforcement
one fill table set encode function user manual call function optima member semi
drop assign define ratio substitute equivalently notice give substitution let leave multiply side mean exact implementation specify hermitian matrix eigenvalue remove normalize trivial automatically step eigenvalue number feasible choose minimize say n eigenvector eigenvalue eigenvector need appropriately combined feasible matrix consequently feasible eigenvector increase make small small eigenvalue definite generalize eigenvalue feasible trivial drop normally great satisfying whenever care toy follow affinity cut constraint want group cluster happen make distinct feasible mean want thus cut plot fig group nod group indice entry optimal relax bias toward unconstrained finding similarly formulation clustering interpret colored way node color assign node cut graph entry interpret relaxed sense minimize interpret
case disease duration duration exposure population risk usually formula go describe risk expense institute diabetes year broadly incidence need determination sometimes duration exposition
triplet approach rather allow counterpart bound convergence rate framework cover establish accordance robustness bound introduce propose assumption specific form stability induce regularizer limitation parallel analysis tight tackle regularizers derivation involve compute limited accommodate robustness tackle triplet constraint illustrate following propose ability metric notion robustness originally notion characterize metric robustness
dataset patient sample pre statistical selection run cross dataset pls pls double check ht cancer cancer cancer normal posterior extremely excellent pls use pls extraction also notice widely g encouraging hold water problem experiment depict select pls pls pls common collect use pls find gene confirm pls gene believe believe patient analyze cancer simulate cancer gene forward expression generate formulae generate identically q correlation simulation facilitate gene gene interaction scale mean value gene x control patient perform classify simulate normal advantage approach know dataset responsible classification rigorously examine gene gene interaction heterogeneity interact gene assume dominant allele allele capture potential mode generate train dataset interest thousand gene greatly microarray nearly gene perspective gene interaction disease discuss different interaction associate computational article search level seem impractical genome point drive far home
approach carlo smc likelihood free therein concerned sde contaminate system dimensional statistic real represent increment standard motion matrix assume regularity directly might conditionally inferential method error e tx tt nj system concept paradigm methodology letter hasting mh regularity convenience q simplify likelihood generate numerically sufficiently accurate approximation scheme dimensional simulate draw stepsize numerical euler sde law gets simplify whereas free methodology attractive simulate function previously mention typical sde trajectory posterior large sample abc useful section difficult generalize multidimensional methodology propose sufficiently close summary tolerance abc simulate states observation sim embed abc mcmc chain sim abc kernel
effect membership discover globally predictive intrinsic property also transfer cluster datum dataset suggest outperform state art behavioral relational become object citation co many relation object discover among relational predict unobserve share common service relational make statistical challenging correlation give various structural relational characteristic imply divide within relation object form dense take social dense circle group relational generate proportion rare largely observation contain relational side
word topic informative assume indicator specify simplex dimension informative know whether incorporate word generative include divide entire vocabulary informative word determine word token step ib di b di di di corpus corpus token dirichlet multinomial prior number token th vocabulary token dot represent token ni simulation study generate third lda
guarantee skeleton times eq operate independent observation write derivative brownian used involve stochastic independent diffusion work give brownian motion volatility stochastic popular modelling financial asset interest financial derivative asset diffusion volatility diffusion sde reflect ease exposition stochastic volatility leverage still paper volatility diffusion denote form pair multiply volatility normalise brownian latent survival survival target aim hazard reflect event diffusion parametric formulation assume positive diffusion motivation consider occurrence event give eq obeys exposition unit although observe hence brownian motion censor refer simulation diffusion volatility survival aim efficiency unit cpu environment carry latent diffusion survival language equilibrium suitable burn period sequence amount serial correlation autocorrelation truncation sampler time burn period match variance obtain independent ess relative sampler mcmc application volatility hybrid univariate trajectory context assess monitoring draw report ess ess reflect percentage draw consist propose brownian experience literature aim rwm mala performance
article review sparse measurement sparsity different channel joint sparse joint tree former utilize utilize channel bind joint practical happen mr image tree naturally b channel physical object nature human tend boundary several connect like forest forest mr form unfortunately exist recover channel separately exist result well htbp previous channel support subtree kind forest kx ki similarly forest coefficient channel zero zero forest forest structure solution intuitively tm tn measurement satisfy
performance experimental construct set site constrain non negative turn I unconstraine p basis vector moment experiment section describe entropy choose inclusion gain kullback posterior inclusion take differ second quantity compute maintain loop hyperparameter construct minimize choose computation inclusion hyperparameter site site alternate estimate parameter conjunction selection validation except summation happen view nlp viewpoint
lag adjust dimensional upper determine instantaneous preliminary notation cm shall instantaneous unconditional test nevertheless remark eq instantaneous hypothesis lemma clear gaussian note expression simplify notation optimal date base one step predictor instantaneous causality non triangular instantaneous causality pp test case instantaneous obvious test standard q block usually probability vary interpret instantaneous causality interesting entail build unconditional consequence variance test spurious stationary general iid consistent
dependent svm ed bm bp ed ed kernels ed cs exhibit among ed rank complicated ad hoc cost cost bb winner rule ed cs ed ed bp svm kernel ed bb svm ci ed bp ed bp bm svm svm predict recently view sensitive function cost unlike separable enforce large slack offer optimality implement rule cs svm sensitive theory metric classifier sensitive cs cost evidence confirm superior method convex r nz conjugate duality induction eq q primal dual regularizer decision I formulation introduce hinge q v conjugate ci hinge bp cs hinge q moreover rgb rgb rgb hinge sensitive draw connection probability cost manner
tendency favor tree surface cut biased cut leave node optimal tree edge parent edge receive weight cut parent cut node entropy map section train independence rest maximally construct classifier well figure two section step feature output network input image multiscale pyramid ideally linear categorization pixel achieve target pixel elementary deviation multiclass location distribution pixel training maximally hard target discard available produce descriptor kl component label assign cost stanford background evaluate semantic dataset public
produce much low leading although gain much going everywhere extreme obtaining take space discrete accept generic set move improve acceptance much possible move slightly much large concrete optimization os illustrate fig one try maximize indicate curve whole easy check immediately due maximum reject refine refinement max max find small predefine threshold usually completely task view range consider
test precisely mean I frequentist good I near distinct assertion interest predictive ratio specifically limit r lebesgue optimal assertion respect sense set plausibility ratio favor iff determine pearson statistic construct free conclude side I classical notion optimality restrict class set light familiar I exactly illustrate powerful unbiased random plausibility optimal poisson choose unbiased integer critical test x construct predictive random depend comparison show plausibility plausibility xx black gray line general set dominate group show mention choose particular simple dimensional equally trade make simplicity optimality currently define simplicity theory function default date satisfactory picture present unknown automatic long run calibration identification auxiliary present I calibration belief mild theory develop help resolve illustrate belief predictive validity convert
cell preserve count choose select cell non cell typical associated current temperature whether accept neighboring drawing uniformly accept less likely make temperature move ingredient schedule general temperature favorable neutral slow method rely one close quantify euclidean projection take p give collection close nonempty intersection alternate convex recursively nn nn odd closed method alternate
measurement signal domain take consideration multiple l encourage domain enhance mean coefficient equal traditionally synthesis linear predefine call outcome operator representative element
sophisticated structure advance decade focus universit paris machine relate raw limitation limited handle description rich relation network etc quite world aspect observation hypothesis justify mathematical learning efficient develop analyze context statistically independent decade numerous survey solution multi perceptron mlp som mlp artificial neural network point consist real bound transfer introduce linearity
deconvolution uncertainty reconstruction deconvolution give measurement statistic measurement generate proportion target aim system target compute call result reasonable parameter weight probability generative integral saddle scale concrete also saddle equation p distance p nm
party classifier htbp acc acc party party communication cost versus size communication cost versus dimensionality protocol baseline htbp protocol protocol well technique perform perform synthetic dataset outperform baseline usually fail yield classifier particular protocol achieve far result particularly support node good partition formulate act constraint svm player natural goal early solve communication efficient distribute take input working space look past streaming pass sublinear work look stream protocol streaming word storage pass prove indicate sublinear input exposition first streaming work let player
calculate clearly experiment cl interestingly consider similar comparable ei speedup observe speedup achieve fully three inspection reveal speedup explain select ei close quantity easy consider two jointly set exactly set expect speedup investigation generally increase go forward change significantly compare lot observation likely meet stage experimental batch batch experiment ei batch practically intractable therefore select
eigenvalue decomposition generality basis contain u obtain projection perturbation iii concluding show triangle q mean final conclusion add stop satisfied inequality schwarz fact two tensor imply stop condition sketch guarantee somewhat form appropriate termination terminate universal condition frobenius condition outline relative stop stop invoke iteration eq consideration condition remainder essentially another simultaneously examine w cast simultaneous linearly independent permutation rescale simple eq chosen give sample parameter include diagonal separate perturbation sample guarantee loose instability contrast mild choice quite simultaneous beneficial uniqueness row requirement uniqueness translate perturbation fact simultaneous simultaneously collection cc find unitary matrix simultaneously interesting question yield improve computational section section tensor learn latent consider statistically efficient dirichlet exploit extract orthogonal symmetric value decomposition compute decomposition efficiently similar analysis establish analogue vector computationally variable decomposition model topic moment statistic prove basic paradigm
finite bin parametric unfold determine experimental unfolding solve priori directly priori remainder paper organize illustrate numerical given solve unfold problem use offset plus pdfs non location width kernel use common generalized vary form parameter unfold find infinitely discretization perform g discretization histogram quantization error follow
valid se generic design estimator proof nonparametric se require exist trivially measured continuous mapping parameter enter square continuous se nonlinear explicitly assumption minimizer least continuity uniqueness law use generic take l individually probability stochastically close nx pr n w nx ps
posterior inferential let parameter mind start I variable take value space measure association together characterize association I I treat unobserved tie quantity turn success framework predictive start measurable index generic collection predictive predictive nest define distribution measure construct admissible restrict predictive support equip three I associate empty predict unobserved associated association specify assertion interest inferential belief e c case often
decrease frequency item primary tumor vote kp j primary tumor kp heart option dense threshold common problem range association fall illustrative example advantageous performance frequency threshold range answer useful fig dataset j resolution frequency key management good imply extensive analysis solver different axis main detailed cp finite logic retrieve cm observation enumeration option significant level impact ability early discovery interestingly preferred expressive iteration remove subset redundancy search search overall good choice search search acceptable frequency threshold multiple search cm medium threshold quickly search light discover search low maximal frequent encoding orient prefer threshold minimal significantly option suggestion adequate medium towards large suggestion indicate frequent option solver frequency threshold constrain resolution tune
value kernel eqs gram operator operator associate vector kronecker operator important towards main value kde generalize take try perform incorporate regularization kde conditional value way take account regression moreover rapidly dimensional know fix operator kde formulation succeed effect explanatory input another take account output scalar function similarity reformulate kernel joint induce see map function joint output value literature recover
carlo universal indeed generate independent correlated mh drawing proposal unfortunately situation limitation drawback rs generate reject adequate pdfs rs analytically ratio pdfs best pdf proposal correlation chain correlate mean converge practice establish burn ensure great mcmc focus extension point propose technique rejection rs acceptance function procedure build proposal exponential inside associate newly reject always refined proposal proposal discard sample decrease cost due unimodal generalization handle target jointly indeed instance technique approach attempt overcome limitation metropolis tackle multimodal proposal remain metropolis accept properly pdf mh initially accept also step determine whether accept rs sample rs support build pdf accept mh support mean pdf procedure point improve never add guarantee pdf multimodal
impact trace extend realistic span detector scan scan specie enter however slight chance pass detector impact mass detector arbitrary thousand acquisition chemical acquisition area normalize round close empirical result poisson poisson variance tie variation detector cumulative distribute conditioning event cumulative normalize reveal satisfactory datum poisson adequate model variable describe detector response dirac delta pmf q account spurious detector eq indicator define function otherwise modify drop log strictly remarkable give follow strictly log concave transformation log likelihood remark variable irrelevant great operational scenario complicate additional appear multiplicative specie output original
nr k nr ar nr k nr nr ar space uncorrelated call component euclidean k kx k apply proof explain covariance define j towards define z j note base type uniformly random definition previous inequality w x degree freedom centrality non conditionally centrality normal straightforwardly fix q ji eq derive conclude j lemma straightforward consequence lemma nz n hold eigenvalue inside circle eigenvalue inside circle event consequence include denote lie contour circle eigenvalue eigenvector j nz n z z lead n j j nz nz nz take contradict straightforward assumption classical us expectation
action restrict situation set action vary drawback mapping state mdp discount function rl policy rl environment problem mdp provide policy improving use simulation approximate form estimate policy repeat properly action action classifier without q binary action carlo mdp optimal generate classifier state label algorithm rl multiclass supervise
result lasso multivariate noise var square lead equal issue choose address literature residual possibility minus simulation paper explicitly implicitly apply method lasso detail simultaneously perform parameter approach simulation simultaneous model var var method ss tendency var phenomenon specify complexity increase square ar coefficient estimate var stage correct ar efficiency two compete lasso method result ss perform among three stage lasso ss model dimensional iid specifie series change three compare three method five metric coefficient ki ki coefficient coefficient k ki ki ki jk first reflect estimate method fold sample comparison stage ar coefficient number lasso ss mean spurious ar compare produce coefficient non coefficient var reflect variance ar ss large stage stage small mse var method notable marginal variability ss
vary seven match select agreement another file record linkage class choose third sample fourth fit mixture probability agreement match likely tend deviation approximately beta sd narrow percent block match status know comparison produce hierarchical necessary tuning prior distribution formulation covariance correlation transform across block hoc suggest logit size logit scale limit range log scale four log hoc follow probability range value repeat agreement constant initially
marginal logistic copula bivariate copula section pair imply expression algorithm simulation bivariate residual limit laplace conditional threshold simulate independently limit set I constant bivariate extreme logistic bivariate simulate preserve stochastic ordering dependence decrease survival simulate observation asymptotically logistic
imply first follow rewrite graph dimension super exponentially choose computationally paper approach distribution graphical presence investigate probability edge identify pair achieve model edge concern super inferential paper likewise reduce basic prior derive mainly property variability section inferential appendix exact computed graph statistical intuitive representation compose describe different graph separation conditional two commonly find undirected direct acyclic graph well determine probabilistic low element adjacent node conditional link arc dag indistinguishable
modify degree currently compute scheme add early iteration add kernel kernel want discuss update kernel suppose effect less negative suggest instead keep operator combination make value version optimal perhaps run end suggestion operator slightly weight enforce low kernel mkl sum polynomial kernel term degree first kernel among weight roughly zero value experiment achieve experiment three uci regularization improvement conclude specific ensure pick choose dimensional along coordinate run dimensional run discretization geometric fashion finally uniform specification give ridge tune kernel low threshold however image dataset finite mkl dataset table
norm understand parameter convex step size take index choose iteration assume choice q neither rate apply query uniformly index mean minimize computing gradient axis estimate gap leave vary aggregation fix vary sample work risk experiment next optimality iteration nf true evaluate unbiased risk see experiment confidence interval iteration attain lie perform experimental microsoft web search microsoft engine retrieval benefit aggregation partial preference ndcg three surrogate aggregation inconsistent fisher consistent recall ndcg prediction vector ndcg increase give query relevance observation set pair score average log preferred structural ndcg addition corollary minimizer square least square logistic surrogate loss previous loss completeness fisher surrogate score loss access score aggregation strategy empirical risk order sample minimize minimize risk horizontal axis aggregation statistic vertical ndcg plot loss regularization goal empirical minimizer aggregate varied extent
interior equation newton first inspire nesterov proximal map accurate robust sense fine furthermore computationally process convex convex smooth gradient proximal shrinkage choice respect computation combination consecutive fista choice proximity p q proximity compute available show error accelerate maintain advantage one fact fista recently criteria proximity penalty admissible minimization scheme special proximity evaluate reduce wise thresholding subsection proximity write compute initial lemma allow efficiently proximity operator consequence computation amount operator onto intersection theoretically show project intersection
amount case great assess statistical behavior frequency well less basic understanding statistical simple giving measurement note part suppose successively unknown process either location location variance call relevant location go definition recall whose
maintain validity plausibility assertion range alternative model unknown inference trivial case function simple produce construct p I side assertion uniformly favor equip determined produce plausibility value side binomial unknown goal nuisance conditioning determine minimal sufficient auxiliary goal scalar leave write testing versus ft predictive plausibility exactly p
forget
pattern precisely either cm enough pattern exactly behavior path first tell inactive associate variable side variable instead close switch segment rigorously regularization rewrite optimize obtain equivalent r solution therefore span necessarily true denote segment
equivalently end g sensor mutual node necessary detail instead analogy network note variable condition process nod weight lemma imply give budget immediately algorithm question budget set lemma exist compute function increase keep mind q bad algorithm prove need g set lemma principle see e chapter state flow flow flow flow unit flow exist imply finish proof flow composition prove program together flow feasible iff affect cut see follow ai ai us minimum non iff negative scale assignment define flow cut include side flow total flow cut side give approximately precise mrfs width graph bind entry degenerate full assume approximation mrf time pass decomposition
integral calculated step di sm fractional moment one psd fractional keep terminology fractional calculus calculus complex keep density cite introduce integral derivative show psd integral valid belong interval integrable generalize function treat paper di eqs path integration locate eqs application previous software calculate apply thus whole eqs fundamental equation choose reconstruct di approximated application
gp recently reformulate showing consider gaussian obtain robustness linear multivariate start generalize polynomial categorical depend heterogeneous effectiveness estimate effectiveness group user take account characteristic stay education framework generalize paper canonical particular belong product show generalize variable useful common simultaneously rather use principle real problem pattern really histogram probability state remainder show likelihood approach introduce
keeping introduce reservoir dynamic network reservoir compose unit toward reservoir reservoir design reservoir network identify external spike traditionally neuron expression precisely neuron behave load activity rate stable reservoir solve linear equation reservoir concept need output dynamical parametric function use system compute reservoir network
precision address challenge new far superior train optimization behind scoring source incur runtime maintain recall labeling number source artificial pattern challenge joint multiple source intractable cost source significant focused er mining community typically precision recall reference discussion er achieve recall combine expert overall comprehensive offer effort label numerous science motivate source suffer label easy matching source master learn heterogeneity explicitly requirement amazon cost privacy sensitive trade negative precision labeling er moderate heterogeneous address transfer algorithm score source adaptively fast er movie major internet engine movie entity
maximize iteration immediately improve locality satisfy formal guarantee illustration comment sampling template guarantee case minor purely uniform policy parametrize empirically respect exponential uniqueness sophisticated give formal denote accumulate depth go uniformly choose auxiliary notation number key result proof outline issue high claim call horizon probability bound statement establish horizon choice branching partly go mdps cumulative eqs mdps period period try lemma constitute prove theorem stem mdp finite horizon parameter step particular induction verify chernoff inequality hold actual chernoff hoeffding directly inside biased stem numerous root
branch tumor frequency tumor generative implement infer frequency infer uncertainty measurement dirichlet major namely tumor derive advantage begin expand subsequent fitness survival subsequently frequency tumor snapshot evolutionary contain multiple major advantageous appear appear back illustrate circumstance highly allele frequency frequency tumor tumor tumor evolution generalize model three frequency panel consistent panel may change take branching simplify population cell discuss estimate frequency sequence copy per consider population allele available copy change specific whole genome sequencing site occur contain cell population greater equal tumor population different branching topological branch circumstance already establish plus
countable sampling apply largely without modification introduce sum pass finite allow particular important follow give duration graphical otherwise conditional indicate range new state whose derive full slice
knowledge reasoning formula valid valid construct place assignment easy claim subsequently boolean give formula logic form variable basis formula threshold list formula semantic logic atomic formula focus formula polynomial atomic tractable everything essentially carry usual formula insight logic offer independently main formula reasoning semantic guarantee high formula true good estimate validity formula know valid whether interesting develop partial assignment whenever draw mask property value assignment process allow know problem hard restrict learn something evaluate way assignment high formula kind certainly essentially true partial assignment evaluate iff false partial evaluate iff true bc regardless formula partial false precisely motivate likewise simplification evaluation restrict partial assignment recursively follow represent evaluate suppose restriction formula state say axiom proof hypothese triple formula system triple hold triple say check impose formal object sake reader aware expectation fulfil application although least preserve classical semantic fall know preserve restriction system system close partial satisfactory subsequence axiom system restriction derivation formula derivation partial step consist formula variable logic sense mean extract case restriction formula ahead typical proof essentially interested version give proof limit tractable limited hypothesis either else limited formula resolution
capital scalar full one subscript column th respectively subscript respectively non vector norm frobenius division recursively optimize objective stop optimize differentiable contain fortunately update optimize alternate optimization variable row except follow term residual wherein look carefully piecewise w l change slope sort wherein sort accordingly remove piece slope increase change sign sign slope w l w slope three range account negativity solution ht summarize successively stop satisfied successfully previous warm accelerate convergence n h r I jj end stop end main spend sentence sort piecewise cost worst piecewise th slope change stop output bad complexity find however scalable optimize equal column th problem negativity constraint deviation presentation much square l especially contaminate outlier keep negativity capability inner r prox dual e e
detect detect equal test favorable detect observation far present five signal result result validate effectiveness feasibility show cause degradation ideal observation time signal scheme wireless eliminate influence apart consider experimental simulation successfully feasibility
result fix want clear infimum apply reasoning plug everything get restrict attention treat x b imply q desire side e take define conclude prove since hold reasoning plug prof follow inclusion since uv u trivial extend take factorization marked replace v u uv eq bind
document sensitive topic toolbox crf hdp dataset word small crf hdp crf become parameter crf hdp base fix optima nb lda document weakly couple result large large ratio usage nb improve tune appropriate parametric nb outperform hdp percentage nb get nb count nb sharp far j kp particularly mean rarely ten infer nb corpus variation compare curve geometric beta nb gamma nb transition smooth gradually kp topic model large allow rarely confirm gamma learn topic beta nb mean positively correlate correlate hdp process viewpoint fix normalization viewpoint count interesting examine view concentration adjustment variation proportion fitting crf hdp nb predict hold model model crf nb crf hdp binomial distribution count shown find crf hdp crf fig indicate across document use
pareto later power behavior various physics biology hence law though first exponent expand include tail decay gaussian pareto measure begin birth shannon distribution lead increase application point theory measure introduce mechanic entropy axiom shannon entropy information
original size easier safe elimination problem life method work especially often interpretability text pca promise corollary electrical computer sciences university california berkeley berkeley provide small number maximize sparse apparent interpretability generally computationally
increase year ever increase theoretical optimality result generalization scale use practitioner deal setting parameter test sgd decrease originally recently asymptotic understand like affect convergence speed researcher make per overview early adapt descent multiplicative approach update fisher estimate simple natural newton combine covariance second approach retain hyper tune another disadvantage start problematic landscape continuously contribution maximally decrease expected formula
art leave adaboost cycle conjecture tie adaboost leave particular computational community emphasis statistical characteristic believe capture essence consistency statistic scenario ml hand round reason round ml rarely amount data essence practical size ml truth often recover care learn perform whether recover ground datum come statistic certainly concept extremely useful ml fundamental play include core cycle cycle also provide often low dimensional also exhibit cyclic randomly dataset adaboost cycle real alone vary technical e matter carry infinite realistic g randomly seek something cycle adaboost margin interest previously maximize adaboost turn come science machine perspective nature study running rate forget rate something mostly function version nothing original optimal adaboost recognize usefulness asymptotic adaboost really adaboost whether adaboost error reasonable know move rate say recognize significant iterative dataset decade asymptotic variant study convergent property often round nature manuscript within focus old learn fundamental central many weak frame adaboost sufficient tool ergodic margin adaboost classifier practically function generalization stable show dash line select log weak base find round pair example input note round error logarithmic growth select suggest real dataset numerical error calculation result open whether adaboost adaboost dynamical invariant dynamical system average adaboost converge ultimately stability hope open htb axis letter dataset round plot originally appear max put forward believe formal heart disease round axis converge letter behavior typically adaboost margin adaboost quality loose explain couple question margin remarkably seem stationary may community adaboost show margin tend yet little significance result ml community care surely learn classifier sense per margin tend limit believe useful dataset implication justification converge consistently formal stable practitioner formal guarantee adaboost classifier generalization similarly unstable behave require establish establish considerably certainly trivial concentrate adaboost generalization important without establish concrete adaboost seem fundamental interesting guess read relevant summary community emphasis deal asymptotic happen round hand asymptotic study question seem asymptotic consider quite iterative ml adaboost article recent literature manuscript hundred year literature thus scope mention likely paper begin
period nontrivial historical highly direct right skewed heavy tailed event attack record even statistically unlikely significantly historical close discuss estimate tail fitting parameter distributional large distinct maxima peak finance several uncertainty upper frequency replace identify define tail severe true uncertainty discard statistically one principle average example aid return finally tail simulate empirical inherent variability likelihood bootstrap extreme technique principled provide tail instance negligible statistically unlikely plausible away likely remain least great mark although mark relaxed incorporate uncertainty principle choose practice large empirical meaningful historical remove describe univariate covariate straightforward denote particular denote event value tail variable tail region binomial variable deterministic event least big bootstrap event integrate drawing sequence bootstrap one event event qx fx event correct jointly
computable minimize gradually around either discrete optimization provide case free energy broad iteration generate determine fp tw fw w fw bioinformatic compute applicable normally objective non provide solution soft guarantee solution convexity objective interesting since bind straightforward apparent analytically case analytically version readily description also make relation
atom residual ss section result fundamental lasso critical motivating may interest stability margin retain perturbation mean inactive atom perturbation stability perturbation dictionary inactive perturbation depend incoherence high contain qp algebra view difficult strategy four problem objective value consequence closeness q duality show complement residual perturb consequence considerably strong satisfy q support atom solution demand former one nevertheless hold considerably theorem theorem behave lead desire optimality solution original perturb govern system aid subsequent lipschitz sparse index induce operate similarly empirical measure associate operate compose return conditioning speak overcomplete set overcomplete individual overcomplete overcomplete error problem implicitly
give algorithm independent output sampler ig pair initially run testing use draw use oracle way phrase sampler prop say interpret mean get sampler sampler oracle rejection take affect correctness probability happen whole thing exposition perhaps basically actual goodness issue issue issue sample even still mention leave real come fix thing assignment tv f hold output simulate requirement precisely index desire tv f claim call conclude linear generation three four count technical giving ingredient recall record approximate count uniform algorithm arbitrary run probability give whose exactly opposed representation parameter output hx algorithm provide al algorithm time query tolerance ingredient proper nn intuition unknown fx satisfy f equivalent note u intuitive henceforth assume e constraint natural first approach program variable turn valuable present note relatively suppose feasible base representation solve simple condition fx union feasible solution satisfie satisfy example feasible along line negative contain constraint prove hence fact course infeasible algorithm problem naive one ellipsoid one intuitively one algorithm one intuition actual invoke stage learner small mistake guarantee terminate small number stage force mistake recall notion learn boolean online unlabele ask identify minimize mistake mistake mistake example essential class class mistake mn current maintain base receive reduce online ellipsoid mistake proceed follow previously basic idea execute learner stage sure solution decide counter output terminate generator learner satisfying assignment output generator continue stage generate upper stage stage terminate online put output b n execute current hypothesis give run ix go current hypothesis satisfy section satisfy condition condition event event call generator successful union bind least successful h ix x condition conditioning certainly p succeed hypothesis observe point give identical point execution fx bind stage complete correctness running involve
minimal definite tu un sufficient assertion observable key ingredient minimal particular triple principal spectra tu q prove thus sufficient show imply strictly negative spectrum unit disk process equivalently state drive autoregressive move average polynomial describe hence l evy process assumption ensure model describe stationary output analogue strictly negative part algebraic degree triple shall inference irrespective identifiable impose strong collection positive definite root ib tu calculus triple rational conjunction realization invertible representation conjunction imply imply contradict theory develop equivalently multivariate process see b apply choose way hold previous specify parametrization continuous impose rank imply observation tu eigenvalue matrix entry number eigenvalue equal observation eigenvalue imply state sequence able impose interior positive hold follow b tu infinitely
large density nn unweighted nn behave near rbf behave smooth insensitive outlier contrast reject minimum cluster size vary unbalanced varying vary proximity partition partition unbalanced inherent cluster balanced partition partition cut cut ratio ratio partition unbalanced unbalanced partition cut unbalanced unbalanced cluster curve cut small cut rbf choose cut unbalanced volume depend account insufficient
scalar problem transform scalar nonlinear operation follow transform similar proximal review equation well equation computational generality key motivation characterize behavior se amp review particular se realization component component appendix precise assume output let denote lipschitz se adaptation sequence value gaussian distribution analysis mild define adaptive remove assumption true conditional thus vector first represent component output output dimension asymptotic state nature constant deterministic constant depend output choice recursively parameter well describe repeat write equation ignore value pre
constant depend lead positively homogeneous ff envelope w w fa fa equal result function corollary relaxation bind penalty illustrate graph penalty blue combinatorial potentially sparsity obviously encode relaxation allow case simple cardinality always norm relaxation w literature fact coincide show detail provide see extend consequence tight necessarily suggest tight characterize extend true couple concept capture intersection relaxation reflect immediate unit f combinatorial envelope q construction decrease envelope extension submodular get tight relaxation f immediate corollary fa always small contradiction satisfie imply consequence construction relaxation figure illustrate value remove formalize illustrate low envelope specify combinatorial range enable answer small large motivation
look put value problem start compare statement follow address embedding asymptotic convergence algorithm space rkh necessary subset measure measure detail moment affect rate assume follow schedule upon follow schedule every exist value n tell achieve logarithmic factor embed map rkh value song al bound thing rkh expectation applicable converge coupling minimal main likely deep detail ensure statement theoretic fulfil subset rkhs last integrable intuition fulfil need sense optimize cb spectral correspond value operator ax l complexity namely essence assumption measure finite yx analogy song need schmidt covariance operator mean give rise although still translate property regard value due two schmidt operator compact general compact operator main rich associated finite fulfil condition q invertible yx schmidt priori unclear fulfil interesting measure rate
dimension array consider hadamard operator easy idea source analytic auto compound toeplitz finally array array however computationally demand delta covariance value repeat array useful statistical recognition compression pattern dimension product original random covariance eigenvalue column call th calculate covariance covariance least one zero kronecker covariance eigenvalue vector r I sample base
repetition correspond method fp tp lasso mcp cv mcp cv mcp cv cv mcp cv cv cv mcp cv mcp cv cv mcp cv cv mcp cv cv different correlation selection tp loss repetition package loss cv mcp cv mcp cv cv cv mcp cv mcp compare paths mcp smooth path stay simulation mcp sparse near exceed newton notice path correction stability newton path coordinate path identical mcp concavity path although path stay optimal easy mcp path concavity get flat short mcp path path lasso whole path see two
combination attribute effective affinity measure effective method agglomerative direct fundamental concept widely receive much attention role data power linkage superiority cluster believe work powerful vision graph partially support research research grant foundation china support introduce team key author thank read li acc detect connect initial cluster n cn v ab ab bn ab sample vector n weakly initial create neighbor cluster initialize near two pair clusters b ab ab bn ab ab
cd table cluster title xlabel ylabel list name black legend pos mark x thick error bar mark x red mark mark solid thick dash explicit cluster title xlabel ylabel run name black white legend pos north west error bar mark mark option dash cd mark table title xlabel ylabel run black legend pos mark thick mark mark bar efficient primarily work well investigation improve correspond graph graph become remain number probability mass write marginal follow poisson function hand define multinomial n algebra furthermore poisson apply multiplicative hoeffding chernoff inequality chapter form
estimator hence condition outperform hand reflect sample change develop high must get ball radius define short connecting bound large symbol generic expression follow need compact density high concentrate near precisely notational drop
converge depend rule convergence burn addition show inference sensitive count consider analyze many clinical trial anti drug precede week base logarithm age treat count successive visit second bound model iv code record visit otherwise ii intercept mod iv poisson intercept slope ij age poorly decide covariate age deviation fit initialization quasi time second take variational especially application parametrization effect center parametrization fit center parametrization fit center give marginal emphasize relevant partially parametrization dash parametrization center general know center automatically choose optimal parametrization improve marginal dash line density variance compete infection information patient seven patient seven total patient arrive variable severe
suggest carry step exist parametrize sample measure must n kx mean point store cast lemma measure close reduce find near reduce find optimal finding non x n always close increase however point sec problem minimize
find coincide jeffreys dx poisson formulae fx jeffreys proportional
split visual instance viewpoint partition truth viewpoint annotation ground category semantic aspect split heuristic obtain bottom category notice huge variation appearance pose camera instance direction different close build slide window detector accommodate diversity amongst recently recent detector et entire vision towards challenge detector perform well state detector discriminative behind root
determine parameter identifiable e calibrate follow probabilistic discrepancy account gaussian process available combination model available small limited example device model physics use device calibration physics multi physics reliability device require solve several accounting calibration computational source view adjust comparison calibration square bayesian inference account various computer uncertainty uncertainty due insufficient bayesian various source discrepancy account calibrate framework effort devote engineering application issue system calibration point interval identifiability use computer calibration common calibration amount aim provide potentially issue network system node observation incorporate facilitate node calibrate account pdfs note focus calibration develop moment paper calibration use calibration
access accordingly respective label instance cluster update variational n class cluster label server eq break q second site separately send server send server site difficult become retrieve label server get present site make
brain space valuable nsf grant grant ns entire margin gray equal product simplify notation assign number iterate fix margin spatio temporal field field x horizon fully mark red extend fully red gray grow stay constant truncate red lie lie furthermore k limit another happen may simple base familiar
literature especially could triple guarantee property sake completeness appendix restriction restriction basis z om g g basic material spline normalization concern lasso think spline knot g ig ig ig ig appear well behave cone dominant penalization statistical eigenvalue restrict originally eigenvalue restrict eigenvalue ig j md play j sparse subset dm dm preliminary proposition statement bind order condition value iii iv concern
ordinal perform trend among alternative independence test statistic asymptotic homogeneity decrease advantage illustrate assume linear force effect snps centre trait hand hypothesis dominant exhibit independence network specific particular dependence expect modern trait molecular part curse gene trait redundant system either processing gene marker study concerned markov literature learn one bayesian include curse markov influence
electrical engineering dc computer fp overview white paper np tree ed group v cluster center f multidimensional survey discrete mathematic structure clustering retrieval analysis management house p massive ultrametric embed journal scientific ultrametric ultrametric analysis ed computer pp fast cluster search concept ultrametric f ultrametric
vector year achieve misclassification method cf crucial user behavior indeed single user compare rating across day week distribution almost use week discuss generative incorporate rating rank approximation section propose binary classification contextual module regularize regression replace composite several aspect investigation confirm claim early importance accounting present allow evidence precise form extremely day context recommendation note music track dataset tend rate recommendation identification three deal classification evaluation rate filter rank rating training
smooth choosing choice theorem pt pt pt band notion coverage guarantee combine idea optimize prediction finite regularity converge minimax fast bandwidth simulate desirable interval automatic want goal produce prediction iid next fall level set observe shall set fall nonparametric prediction regression construct relax usually parametric nonparametric smooth behavior remain construct adapt band sense band investigate study finite desirable guarantee band give weak validity good solution achievable propose sample infinity valid band
th area curve cdf assign vary specificity substituting nonparametric obtain marker eq q auc statistic form compare location normal location statistic possible pair normal assign location normal otherwise average location within subject view statistic auc form range fall sort measurement statistic great statistic compare screen number marker low rather interested sensitivity th marker particular measure
importance top list half life exponential formally associate item list item assume utility user test evaluation ndcg contour plot asymmetric count item count contour line differ pmf equal contour figure user contour line performance largely affect score sensitive svd outperform comment nmf well mention ndcg trend time issue appropriate cf list arbitrarily scenario realistic within since netflix bigger much limit minute severe scenario realistic exceed well perform cf mae time observation apply nmf pmf pmf regularize work constraint svd one color good work pmf pmf exclude
true regular singular condition let constant constant integral outside affect theorem prove general kind make form assume sufficiently small manifold ax infinitely satisfied fx du index one equal mm say use analytic function compact set function recursive enable resolution fix resolution system local log canonical concept algebraic follow definition local say essential equation j k coordinate odd model say odd mm
dx dx taylor series perturbation respect become eq condition reduce lagrange multiplier multipli variation interval
ia paper subject centrality amongst researcher theory experiment among high show analyze summarize nine range empirical supremum euclidean take conservative suggest less known computationally convenient conceptual monotone transformation see parametric centrality arise asymptotic expression avoid matrix
smc abc hmm variability seem slightly allow grow smoothing criterion preferable forward time smc obvious increase associate linearly approximation appear grow less obvious red static throughout datum smoothing smoothing forward smoothing allow dynamic resample smc hmm smc abc hide proposal run burn addition computational cost smc similar computational smc procedure error result observe accuracy estimation smooth use smc update forward exact roughly well forward smoothing smc mechanism observe abc hmm reasonable consider effect evident smoothing particle mix likely
proof hereafter w show lyapunov lyapunov function individual drift condition characterize involve definition individual worth point drift vanish vanish practically situation increment vanish may role accommodate update numerous later scenario exist constant eq establishes set establish strong return weak guarantee existence time return contribution rescale lyapunov respective drift compare scale control define surely jensen concavity identity I deduce obtain iw x iv x iw conclude notice comment lyapunov degree find establish region resp notice concave could lyapunov scenario whereas note practically relevant encountered practice establish strong satisfied scenario dependent drift condition previous abstract add drift
main state subject combination eigenvalue extent fulfil design negativity tailor class comprise design arise paragraph design identify evaluate must condition fulfil mention motivate matrix study noiseless broad class I imply negativity constraint cf apply regard hyperplane consequence hard continue present satisfied design non negativity regression gram lower follow I partition corresponding principal bound equality simplex figure apply matrix entry function spline kernel function traditionally point evenly effectively band I u h mention remarkably deconvolution intensity measure model dirac arise limit device paragraph deconvolution spike train bivariate denoise study design population gram population limit design gram design eigenvector scale uniformly cardinality uniformity consider specific scaling structure structure gram cardinality event well assumption noise show appropriate result remain valid distribution gaussian include cardinality extra depend let negative distribution trial probability cf shift form increase sequence I larger decrease extra arbitrarily quantile virtue various monte generate standard bind active dot quantile frequency solution problem consider whose gram correlate random paragraph design gram possesse involve non design
recover fmri validation considerably experimentally validate problem connection section detail discuss penalization experimental n n mn na mn na normal covariance model simply reflect precision concept estimation selection number zero fit overcome dense matrix precision regularize encourage among term log technique box dual via determinant inequality constraint perform regularizer graphical enforce know assignment unknown assignment change regularization scale free variable reinforcement structure model mrfs heuristic conditional block ise
induce define induce get work subspace imply fix classification column formal cover mb grouping classification classification classification empty inclusion small element element serve add appropriate meaningful x v vb mb set size although classification ambient sketch basis determine whole similarly belong contain element capture definition hyperplane conclude cell nonempty give lemma prove lemma apply lemma would like belong dim perturb lp precisely column unchanged lp slight matrix lp simplify integral constructing
release r permutation estimate permutation calculate calculate assignment eq approximate estimate slight conservative especially small nan alternative increase however conservative slight general large practice covariate otherwise change increase permutation numerator theoretical approximate simple robust permutation truly proceed potential variable basic gaussian matrix nuisance unknown however standardized version space partial mean scale variable interest onto nuisance ready correlation proceed correlation condition covariate however approach nuisance variable adapt original algorithm deal
min redundant eliminate profile extract redundant classifier follow pattern corner white b c I corner thick white b rare amount available association rule pattern class short occurrence specify appear preprocesse perform evaluate scale application real world proposition class possible method overcome short come advantage allow identification correlate mining deal database discovery relationship variable association rule point derive perform deal target occurrence classification tree etc possible effective ignore class devote past evaluate contribution
semi definite definite follow call expression fundamental large also row column claim concentrate prove third positive semi definite resp obtain relationship c deal notable analysis method reveal combination addition describe section section review multivariate method viewpoint give pca xx b w class scatter typical example term typical canonical correlation cca formally cca coordinate maximize either yy take lagrangian obtain xy xx
broad broadly consistent inverse scaling complete star capture correct complete graph contrast inferior take reach roughly star reach iteration threshold line graph star cc value row correspond graph star last case star vertex simulation walk property subgraph sampling bad simulate iteration plot function exactly complete simulation emphasize depend precise location sampling choice appropriately agent protocol quantitative learn crucially agent polynomially fast connectivity play turn relevant primitive point number question protocol achieve depend polynomially appear speed loose several could potentially speed finally develop decentralized handling situation complex learn vary arrival group learn strategy deal situation try action
latter wide classification paper automatically theoretical guarantee method improve probabilistic prediction prediction adaptation regression goal calibrate reporting possibility prediction poor theorem predictor predictor probabilistic diameter reflect uncertainty explore efficiency empirically use loss popular prediction minimax way former version nine set uci repository datum set usually work original interestingly slightly score classifier example simplify seven nine confirm wide study prefer achieve predictive empirical recommend output carry valuable
interesting give figure stock return example entry conditional concentration identical method
ii iii grow infinity maintain constant ucb sense ucb quantity limit assess tend prove interesting item ucb iii technical detailed discussion number prove term term ucb strategy discover request adapt miss simply know reinforcement entail prefer miss ucb simply request expert high upper brief ucb prove find item completely satisfactory main non step obtain interesting item request I good efforts ii subsection independently define every interesting see exactly idea total
run ari similar table performance consider preferable ari small ari std scenario generate add bic within component merge average slight classification former ht ii bic iii simulate triangle rejection point triangle component generate merge within analysis data mixture four model window ari perfect averaging notably perfect nice average merging cluster value bic good section consensus cluster form clustering object clustering heuristic clustering suitable value available http correspond window clustering cluster five set consider average probability one probability approach average apply give outperform consensus
day probably cd year profile change format week inform available totally email see format keyword engineering design h k sciences ai stand profile entry title multiscale de france author van h p management science multi keyword spatial aggregation landscape keyword management risk f profile year observable attribute marginal alternatively empirical frequency occurrence correspondence sort
walk item accord target preference combine trust collaborative recommendation take walk rating propose collaborative walk filtering incorporate social show approach prediction social create graph interpret recommendation similar connect item record unfortunately construct clear author weight capture preference effectively item
infer merely time estimate noise third space burden burden increase great difficulty clearly impractical sake completeness account matrix constraint normalize sample wishart new scaling augment infer turn metropolis hasting possible sample remain component complete cycle suffer case da da half discard burn prior choose fig show auto augment isolated improvement alg improper prior px restrict value depend variance law fig autocorrelation value improper scaling demonstrate da da comprise action prior improper mean px da translation discard px da translation practically da shown figure plot use initialize half fact quite far true finally translation autocorrelation compare da appear beneficial popular
need infer latent membership observe make formulate three infinite finite upon process allocation finite infinite modeling document modeling compare lda dp corpora achieve document baseline extract dimension reflect orthogonality learn scene understand recognition place govern I human potential human pose mixture associate human versa mixture joint pose object govern pose preference explain pose front certain therefore object parameter
model compare previously study ml ise mrf observation generate ml compute intractable ml finally variance estimator abc ml ml gibbs coordinate implement auxiliary tolerance mrfs mrf ml markov whose minute intel core matlab
rely predefine incremental approach bellman successively reduce construct orthogonal basis bellman derive function eigenvector induce gps reward p mdps rl ergodic introduce terminal episode sequence state reward train discount terminal terminal case state episode function short v eq scalar parameter collect ordinary rl observe define term define n unknown capture mdps remainder solely consider e function
minimize update hold generality focus row matrix covariance last variable inversion block involve lead remove term solved coordinate direction vary soft iterate follow column descent block summarize start let solve repeat utilize representation
manner satisfy instant incremental symbol assignment scalar nonnegative satisfy instant node intermediate cycle incorporate represent perform incremental neighboring node intermediate update p retain standard covariance representation filter require consider satisfy every instant node follow incremental kalman start time implementation arrive order facilitate comparison equivalently rewrite degree neighborhood diffusion write observe employ estimator generally employ enhanced performance update adjust enhance discuss convergence kalman filtering detail along diffusion smoothing early mechanism global assumed diffusion involve reference strategy step satisfy leave regard scheme coefficient nonnegative corresponding behavior diffusion noiseless strategy statement condition adjust complex noiseless let individual practice situation network attain chemical physical identifying minimizer would converge pareto solution global individual cost real individual constant generate towards desire global norm gradient restrictive differentiable condition relax allow require norm hessian would exclude unbounded body chapter adaptation update exact use version weight error realization gradient become condition past noise noise variant long grow fast requirement furthermore condition require variance satisfy eq verify relative noise vector let steady state noisy consider assume minimizer data strategy employ zero development diffusion adaptation insight student laboratory http www edu students chen yu lee di tu j chen yu early article ease reference kronecker product compatible kronecker denote replace scale consist combination property kronecker bc ec te ec ec consist node connect node consider edge connect loop exclude self loop set still loop loop path denote consist connect addition integer since denote entry eq term location associate incidence define every connect column display one entry entry deal simply assign index negative high index node incidence consider example column incidence manner observe laplacian incidence row small algebraic connectivity zero subgraphs connectivity nonzero algebraic identity add b zero network separate say laplacian matrix laplacian generally subgraph algebraic connectivity obvious must would diagonal algebraic contradiction converse statement graph connect eigenvector e already b verify individual imply connect entry desire add leave add doubly add arise frequently list right leave doubly leave doubly one add conclude spectral radius unity eigenvalue equal integer strictly strong characterization eigen regular eigenvalue circle eigenvalue correspond eigenvector right eigenvalue entry part follow frobenius regular doubly matrix hold doubly eigenvalue nonnegative definite hermitian b likewise symmetric eigenvalue real nonnegative follow part matrix orthogonal entry therefore end scale justify
refer guess mean document identify appropriate word fundamental task translation ir corpus paper corpus base employ learn feature text word correct manually speech consequently competition consider english lexical task unseen see na I bayes nb near neighbors nn machine svm learn algorithm learn consist nonlinearity model behavior algorithm create simple lot feature input composition manner abstraction
adaboost auc area roc curve adopt diverse sensitive imbalance traditional criterion etc convexity even optimize auc yield hard avoid pairwise surrogate usually adopt g exponential loss hinge square etc surrogate lead improve actually expect surrogate converge risk also call bayes optimize yield limit sample optimization surrogate calibration loss calibration necessary yet insufficient auc inconsistent pairwise surrogate risk whole provide sufficient auc minimizing find exponential auc exponential provide
probability member choice haar wavelet multivariate basis task develop use appendix implement ks cs r differential entry application complete posterior distribution instead posterior object object euclidean cluster uninformative would conclude actor actor situation multidimensional obtain achieve represent actor decrease take value example possibility choose solution probability solution transformation ensure continuous like aforementioned uniqueness show embedding classical return dissimilarity dissimilarity provide rank discussion concern rank least dissimilarity begin scale diagonal entry matrix whose th entry dependence need general remain
demonstrate norm arise section present superior know often inner function denote rademacher depth mapping learner nature protocol round minimize study space set able algorithmic idea game distribution solves henceforth omit understood range moreover partial recursive game eq minimax specify mix recursive appear tool yield array constructive translate minimax interpret take present value serve regularizer online exponential follow relaxation first yet tight upper sequence x admissible minimize relaxation meta however need valid relaxation relaxation tx irrespective adversary hoeffde admissible deterministic eq recognize potential know loss player difference potential extract potential origin author conceptual arise conditional characterize view sequential rademacher shall refer already future whenever rademacher admissible appear prediction arise relaxation conditional rademacher tight rademacher admissible version chernoff inequality tell proof softmax maximal weight realization finite probabilistic concentration deep exponential relaxation optimize dimensional cost schedule loss suppose upper admissible furthermore important absence automatically mirror aim relaxation arise step algebra example
within edge term combine equality let p way twice put everything together label graph transform list neighbor method label line technique scope significantly experimental confirm often beneficial pdf span span linearize traversal arbitrary simplicity assume traversal soon visit get line backtrack produce visit eliminate elimination display bottom show bottom elimination eliminate eliminate adjacent drop directly adjacent right eliminate connect node weight without obtain edge weight gray predict remain neighbor prediction shown initially create together suitably weighted order create starting perform backtrack edge current backtrack one traversal eliminate soon encounter edge iterate elimination happen adjacent eliminate among eliminate let algorithm predict operate metric label di j v v e path I mistake free drop edge define use shorthand point please proof mistake loose ii diameter extremely argument change rescale value make compare label edge concern eliminate contribution g exponentially light mistake refined way equal mistake subset imply add extra mistake mistake label mistake logarithmic advantage high make intuition rule close combine rescale edge insensitive appear
appropriate weight majority margin thank soon motivate material base national health grant foundation grant
baseline algorithm know aside naturally approximate crowd appear accomplished process typically much population characteristic nothing closely crowd predictor let crowd l denote arrive vote number time far consistent quality collect smoothing shrinkage crowd toward valuable whose certain time opposite vote threshold ml ks l loop v v tt new pool high estimate another uniformly pool enable balance exploitation estimate vote
embedding measure restrict measure discuss borel probability satisfy reflect random consider test appear appendix underlie negative describe relation energy theorem prove appendix depend choice link family kernel accord may generate rise possibly subset merely provide triangle schmidt pair jointly hilbert schmidt mean discrepancy distribution kernel mmd k k k xy x x last k show demonstrate link covariance prove xy similar make yield rkh equivalence lead result
natural near nonparametric predictive valid conditioning integrate empirically exceed suggest distinct prediction analogue forecast interesting many risk access netflix root broad cover understanding
seq cn lambda exp exp k sum n exp mu I exp n x col col shape break cat value sim shape n alpha ta sim sim alpha alpha alpha alpha alpha col c col alpha col alpha x x x tx alpha alpha alpha alpha alpha alpha alpha alpha alpha tx n alpha alpha alpha alpha l alpha
directional lp x x selection index plot convenient formula contingency xt concept apply non continuous variance regression multiple logistic discrete dependence coherence trace yy yx regard coherence quantile mid copula em nonparametric relative theory
summation state give definition estimate hmm maximization iterative algorithm hmm cf reference therein incorporate section iterative em hide observation access state partial access happen ease notation ever model labeling framework b order define backward equation variable z actually observe noisy observe backward
careful approach work space query subspace cauchy variable say prove embed sufficiently e polynomial require vs recognition point solver reliable speedup fold rely price computational profile failure quantify use repeat end maintain regression recognition complexity observe improvement example query strategy basic building block database hard subspace problem near neighbor sublinear point randomized locality become sublinear algorithm et lift space near neighbor several hash hyperplane sublinear computer science limitation intrinsic sublinear near hyperplane hypothesis boolean version variant variant difficulty sublinear unlikely well motivate cauchy projection choose cauchy family remain cauchy exploit previous level obvious heavy yield preserve df logarithm certain non distortion result notably distortion incur nevertheless elegant turn
fraction implementation work support scale proposition proposition fact restriction constraint question question remark fs mm grant support nsf grant support david nsf innovation fellowship support dms nsf innovation fellowship david google award grant wu method exploratory corpus inference exist provable practical inefficient inference provable implementation run popular collection without supervision topic model vocabulary token document specific distribution intractable result researcher
differential prove side correspondence equation ode ode exercise calculus solution meaningful obvious type age population true restriction
finitely noisy circuit completion single set observe position find complete circuit circuit entry return average appropriately polynomial per take variance circuit high try decide except contain imagine bayesian explicit yield entry illustration statement circuit graph simple cycle bipartite arbitrary disjoint number disjoint circuit occur moreover simplification elementary become algorithm take least regressor complete circuit observation efficient computation polynomial complete efficiently ht completion denoise entry position observation estimate complete circuit circuit write set weighted mean determined sign circuit locality circuit also reconstruction say complete circuit r solve side suitable estimate plus wise error local completion variance logarithmic single entry observe position observation variance variance estimate return standard error actual reconstruct section finitely closure uniquely associate bipartite corollary bipartite vertex isolate position grind convention isolate care indicate ground set want maximize closure
change point copy interest sort accord position observe emission distribution point segment square share level observation figure level occur change cp maximum segment segment approach location peak dot lrr line change r change curve include second slightly change due bioinformatics genetic pointing phenotype disease detection copy characterization dna genetic treatment hide map likely extension hmm procedure change specify transition improve extension include reversible markov monte carlo account
expect per expect top axis ensure expect gamma probability heterogeneous implicit reference exponential mle leave submatrix adjacency simulate clearly plot except would identical block break adjacency deviation model level htbp regularize maximum likelihood asymptotic membership high population grow slowly stochastic blockmodel edge decay probability decay block stay bound away otherwise induce second implication small size number diagonal grow linearly
classic measurement error regression independent design motivate measurement enhance algorithm scale scale view validate characterization angle selector predictor uncertainty perturbation matrix source theoretical point view interested theoretically argue variance efficacy point validate empirically characterization algorithm lar ds motivated characterization experiment national energy laboratory repeat spectral value domain small predict fraction variance interested mean understanding indicate research algorithm way familiar lar variant tailor parallel evolution
discuss explicit bernoulli digits string represent string sum equal final dag small one large dag recursively sum integer bit integer entire computationally resample integer integer numerator repeat dag nan triangular recursion end remain stay zero connect newly add connect hence bernoulli reject alternatively direct binomial sampling exclude add complete order since integer dag rather equally likely permutation procedure mapping dag uniquely identify edge map simple ignore mapping draw compatible edge node permutation sample total dag detailed number illustrate list integer big far leave small dag divide way dag fact node integer would next arc also least arc node last far bold bold may draw receive arc merely permutation node label draw reconstruct permutation adjacency sample inside line denote draw bold old adjacent matrix permutation label arc next step ensure receive
actual classifier predict difference utilize originally investigate familiar establish ir community correlation three consider crowd score correlation crowd evaluation outperform blind round typically effective outperform accurate though reasonably regard outlier though still fairly base expert crowd approximation additional likely collect future comparative finding consider induce system impact measure school consider classifier share expert traditional evaluating limited expensive classifier scalable yet preserve high evaluation label crowdsource scalability raise serious regard blind investigate label aggregate label output label performance additional crowd direct classifier supervision assess establish rank correlation classifier vs crowdsource expert expert classifier rank accordingly score tie ranking score
entire finally item already herein vertex option option create addition different code gr gender r vertical blue quantile plot score performance nearly diagonal line plot strong test obtain r display item ht compare evident nearly overlap also display code point individual actually attention confirm produce result add gr site format gr omit produce separate plot ht plot dash diagonal situation vertical quantile answering highlight among group distribution slight people toward people large people figure figure note score density item testing efficacy plot presence confirm environment within along well application
f analysis appear remainder author outlier contingency among true table treat couple connection residual test detection find context contingency deal count contingency sample cell count great identification recognize outli contingency outlier goodness test apply algebraic statistic approach towards outli contingency go back outlier minimal set contain enough cell full rank remain although base minimal minimal table derive pattern independence example nevertheless order applicability notion run outlier organize briefly base estimator
contain ranking object conditioning conditioning phase object phase set ranking predict rank neither object rank observe phase four different conventional framework use correspond imputation link know solely exploit link incorporate feature information label rank new setting fix able conditioning object set realized rank encounter learn rank literature retrieval document rank new predicting capture previous use joint query possibly encode typical kind particularly design retrieval bm match task require human expert information always construct perform access possibility representation experiment example addition case object object domain enforce relational symmetry consider explicit one nonlinear kernel know similarity aim object advantage kernel rank kernel building pairwise capture similarity edge predict kronecker generating modelling consider structured define input usefulness kronecker remain challenge require computation processing usage overcome computing training advantage kronecker traditionally solve g reference therein relate link set computational kronecker exhibit solely prediction body g similarity path kernel kronecker sense infer similarity path walk could use related pairwise kronecker kernel domain kronecker application bioinformatics task target knowledge concern
dataset represent informative consideration connectivity unable small maximal algebraic os r enyi graph contain edge result graph laplacian os r enyi algebraic os enyi random connectivity know algebraic use concentration connectivity os enyi inequality hoeffding pm bn pn pn least match side red green os enyi value obtain indicate circle initial take black value figure nearly indeed algebraic connectivity heuristic produce optimal also algebraic connectivity optimal average os enyi graph blue solid finally give os r circle formulate yahoo movie rating rating yahoo movie rating movie nonzero rating movie rate review rate movie review make movie give movie imply comparison complete majority movie receive movie occurrence movie receive ranking discard leave movie review remove remain
discard mean information lose loss also old window detect performance chart detector assumption delay nothing inferior benchmark define misspecification detection set equal delay change order choose method generality knowledge require learn change affect occur detect consider early moderately case contain correspond locate proceed performance chart follow observation know reference choose specify approximation note previously serious control limit
interesting gp constitute constant variance correlation return noise clearly approach finance allow capture effect volatility vast majority financial return capture address bayesian paradigm obtain expression covariance let modeling th consider uniquely describe possibly infinite impose latent gaussian gps rather effect mix introduce function relate q n cx eq latent impose accommodate fact definition cx innovation also impose gamma complete conduct bayesian bayes g prefer variational due well scalability
condition hence ct require essentially limit empirical evaluation apply large comparison size problem subtle motivation basis induce entry gaussian result entire randomized use hold list worst show performance trade compare ct advantage ct base transform relatively implement straightforward theoretical algorithm advantage set ill condition slightly randomly ill heterogeneous leverage illustrate linearly spaced gaussian linearly space way ill due due dd rest low leverage ill condition submatrix snp snp genome description lead submatrix create rgb convert intensity result implement evaluation closely seem use run third cccc e third size superiority condition close worst bind ct cccc conditioning algorithm choose conditioning well condition algorithms test increase decrease expectation algorithm interestingly reasonably reason
sgd machine take draw respect polynomial averaging set besides polynomial decay run iterate report log repetition iteration omit indicate polynomial decay well achieve averaging early amenable significantly average require performance particular simple iterate attain well theoretically also extend sample individual iterate smoothness finally bound
simplex hyperplane empirically demonstrate substantial accuracy exploit refine enforce convex portfolio illustrate degree european future proof sciences de paris corollary definition challenge relaxation lead norm application benefit feature constraint efficient sparse projection onto simplex use
independent monotonically jump infinity constrain physical limitation communication bound drop jump infinity cc cc monotonicity make analytic approximate expression detection allow overfitte operational stage state active risk overfitte high active research successive correlation sense temporal interesting phenomenon correlation mention layer access keep track detail place eliminate sensor probability set similar use geometric certainly com wireless sensor
v innovation conventional r vi innovation order innovation estimator noise innovation innovation maximum likelihood state four simulation interested identification problem innovation introduce noisy convergent innovation filter exact variance also demonstrate approximate distribute decrease innovation linearization filter algorithm performance simulation thin innovation satisfactory innovation innovation estimator maximum discretization decrease innovation innovation much exact innovation less adequate tolerance adaptive order innovation automatic approximation effectiveness innovation identification reduce observation distant easily measurement miss within international centre physics author partial accord tt functions taylor expansion drift matrix function symbol viewpoint formula exponential depend method van alternatively suggest computation moment formula
distribution proportional label frequent quantify company fix size edge software show topic stock microsoft yahoo report reflect clearly potential deal yahoo stock microsoft agreement yahoo difficulty useful impact whole topic stock isolate rate financial major observe company name confirm successfully economic information drug national product deal global regression news topic meaningful among news click market imbalance extract news record sometimes information news record top news exclude blue exclude overall support trust contain topic stock reflect report
use first modular modular notice perform maximization guarantee ensure next increase monotonicity follow submodular reduce iteration maxima modular reach optima modular next value local minima submodular maxima optimum gx tm gx f tm fx j modular optima gx gx j gx gx gx gx gx vx vx ensure tight approximation time submodular question form bi randomize importantly practical greedy bi randomize pick amongst randomize combination lastly note local heuristic submodular entirely step heuristic via procedure submodular modular
denote random belonging minus alternative set hypergraph say acyclic connect acyclic graph acyclic greedy algorithms section use lead exact greedy clique decomposable decomposable graph approximate find target entropy sx sx sum selection consider width bounded parameter problem decomposable graph entropy since clique size clique characterize selection incidence incidence minimize equivalent form decomposable
velocity angle first angular nm angular restrict cm symbol mass eqs solver keep xt discretize continuous however alone keep position unstable introduce complex balance yield physical controller yield result stay inside valid work intend produce meaningful close balancing roll roll measure axis angular velocity control center mass nm dynamic model interval reach angular computed physical front radius vertical mass horizontal distance mass mass mass roll angular velocity keep roll angle restrict roll suppose terminal define terminal state stay matter go forward use keep xt us compute finite possible action discretize cm pt department cs usa school science college ab united continuous agent generalization loop drive transition influence influence early identify salient dynamic act intrinsic reward require external reward state know initially address carlo address prediction iterate induced application exploration model keyword dynamical self title continuous enable
source plot visually tb mm see stability much give e mean confidence mean confidence seed validity inductive region significance proposition inductive
e c dim sp sp cpu generate need hour minute solve minute obvious generate objective value always much hour minute take minute dim latent large expression fast cg thank consensus research national foundation fellowship institute mathematics application part science office proof sense completeness introduce define function rewrite
group diffusion topic originally implementation http www ac research relate topic nine topic nine sample list extraction set web page four set manually page link co undirected composition collaborative internet movie movie recommendation movie focus movie whether office contains link whenever share production company edge production two movie report node know label remain phase use assessment phase labeling run run fold cross validation external fold successive rotation specific fold moreover fold external cross validation fold internal tune classifier
component section recover transformation component component certain distribution ica state certain intrinsic state ica randomize simplex simplex learn presentation ignore ball transform linearly ball measure section ball map sample use linear scaling turn find axis align alignment ica ica somewhat rescale distribution fact even isotropic ica justify algorithm denote routine vector coordinate square isotropic distribute entry p nx n p isotropic ica routine state explicit simplex hull generate scalar obtain approximately separate inverse multiply matrix remove row isotropic v permutation diagonal entry sign correct orientation sample output every let invoke obtain separate work iid coordinate accord isotropic ica
state metropolis acceptance eq eq one flip p momentum
comprise texture color feature edge detector texture original image filter decomposition texture euclidean distance absolute position vary tree fold cross retrieve counting retrieve fold achieve good accuracy category par baseline fail global distance less angle forest incorporate relative position pair implicitly capability speed benchmark learn feature function form position clear question remain
assign give clustering comprehensive apart hamming proposal concern partition ideal goal introduce distance partition difference clustering clustering moment number cluster introduce distance surely distance practice hausdorff distance measure hausdorff compact compact metric try capture proximity set symmetric nan identify hausdorff first rely involved measure seem ideal clustering incorrectly represent significant cluster result clustering two clustering distance permutation metric partition moreover usually combinatorial represent particular comprehensive understand set resemble possibility logical add equally possible well name bottleneck
put wish book decision informative comment via relationship social filter social influential detail filter recommendation know preference user prediction user variety adopt relate item rate user rate item rating collaborative predict rating target certain collaborative filtering prediction recommendation display rating item unfortunately ignore social user take effect consideration unfold epidemic make recommendation start explain model
typically encode continuous space tractable differentiable nothing determine alignment collapse onto remove one refer output recurrent layer layer length sequence encode vector length unit compute iterate input hide matrix bias term rnn application lstm exploit lstm composite vector obvious mean hide gate gate state gate gate
intersection face random independently cone marginal weight mixture recall argument empty inside well similar limit restrictive want interior patch numerator allow isolated intersect patch really closure isolated patch avoid closure entire field case difficult exactly maximum isotropic ec dimensional volume ec high ec ec approximate maximum giving field volume essentially ec light gray region threshold expect ec ec surface diameter two parallel tangent field field necessary intrinsic
hdp whether hmm hmms bayesian hdp hmm hide chain collect transition form integrate sequence generally expense expensive treat prior approach model focus model duration explicit distribution bayesian perspective literature parameter estimate procedure particularly constitute basic underlying formalism generative standard draw duration new probability random duration graphical picture see transition super super state length segment observe symbol super transition sd state specific duration defining must end segment boundary final right censor algorithm modify censor chain similar alpha beta dynamic hmms inference message pass message symbol message hdp symbol message duration segment begin observation duration indicate begin sum future boundary expression constitute contribution segment provide censor survival duration subroutine sampler hmm expressive model
integrate k last line evaluate complex effect jk kk jj jk cholesky j k jj jj jk kk kk k newly start derive task calculate predictive partition predictive group derivation computation calculate bayesian task preliminary give newly task predict procedure variational keep particularly much random effect yield task gp multi task task also model multiple output work derive differ
heavy coin unique minimum occur hence finally proof dimensional walk walk less heavy coin respectively heavy gets let heavy coin respectively expect algorithm follow inequality achieve heavy minimum cost measure future strategy good outcome history modify initialization optimality action major heavy heavy interesting devise coin condition set coin
edge efficient independence graph exploit sample estimator dependence sample finally combine stand rw graph simulation topology facebook require least facebook graph sampling walk measurement important number practical user user month business case user strong report compare facebook different case network scalability reliability obtain architecture system protocol www instant rich content million network track predict medium obtain potentially thus extent estimate population major technique currently context variation drive widely public health
thing buffer expression observed lie equivalence gene total lie neighbourhood equivalence lie gene posterior great gene microarray explain equivalence indicate identify correspond specified profile profile would htbp nm nm nm nm bc nm nm nm nm probability equivalence day day posterior equivalence decrease value gene equivalently day gene equivalent small htb show set requirement evidence particular
dimensional one continuous lemma restrict meet condition thresholding finally backtrack procedure terminate specify duality attain author satisfy work well practice note accomplish cholesky cholesky j u j require section criterion however approximate hessian take step safe
work extend wise regression generalize specify model poisson specify specify conditional conditional regression arrive joint cx procedure mixed categorical regression closely baseline generalization pairwise edge unfortunately incorporate likelihood procedure graphical representation probability absence conditional maximize penalization graphical model pairwise edge either variable conditionally variable two absence edge entire mixed motivate non sparsity penalty relaxation scalar regression irrespective balance way group treat equally fully factorize px py r base
entry customer customer must exist customer covariate latent section construct poisson give poisson operation yield rate allow varie covariate define dependent normalization process dx three poisson construct superposition transition restriction rate superposition superposition poisson countable poisson superposition poisson transition tx tx tt dx measurable associate atom qx dx qx respective associate special treatment partially construction bivariate poisson represent baseline superposition process bivariate take form partially exchangeable additionally distribution hazard lin superposition subsample chain poisson chain dirichlet addition transform use kernel gamma gamma operation atom subsampling affect location size lin generate process wide variant gpu need lin mcmc chinese chen slice drawback construction application ad validation alternative construction drawback use kernel process use wider dependent spatial poisson
filter recommendation accuracy insight recommendation interpretable svd internet service internet market vast maintain customer recommender strategy example suggest base recommender system cf content base item match cf approach netflix user rate preference overall item rate extensive review discussion refer reader recommendation boltzmann broad netflix neighbor gain popularity result netflix perhaps netflix prediction ensemble many take predictive table netflix base label knn error rmse factorization svd table drop rmse achievable combine together also netflix project focus fundamentally one focus propose add value paper extra capable item without cause general user relatively
column separation bss mode lead cp cp useful allow bss notation bold letter bold unfold tensor I pn ny j na denote rao wise hadamard product reader notation q outer b exactly cp ni nj illustration rd outer tensor regard e letting shorthand reason hereafter alternate widely standard update use quite attractive essentially permutation matrix number suffer slow respect effort improve reduce accelerate preliminary tensor tensor transpose tensor accordingly example transpose permutation transpose mode e transpose unfold thank rao replace rao successive vector simultaneously
multiple pre separate background foreground scale sift identify feature use identify sift major stage detection localization orientation assignment invariant generate successively gaussian laplacian filtering localization accurate contrast discard absolute pyramid detect small lie discrete neighboring pixel determinant orientation histogram region histogram gradient magnitude circular window orientation bin descriptor vector sift represent pyramid lx equation calculate determinant equation eliminate orientation sparse sift descriptor smoothing
supervise keyword method v cca v cca cca v structural learning tag protocol truth query retrieve v visual pca reduce nc structural refer deviation query nearest retrieval cca compare plain euclidean distance scale component consistently adopt propose evaluate view tag section performance class baseline feature pca get precision term poorly baseline interested keyword dataset tag vocabulary improve embed project tag coherence image e query also image cca third keyword retrieve retrieve search incorporate view three target version view view nc clusters almost identically automatically supervision list performance cca v replace lower high cca significantly though noisy view help tag view report comparison structural cca v reason discrimination batch batch optimization sgd beyond scope design sgd neither produce tag unlike cca baseline suitable retrieval query show tag note advantage view tag fact cifar imagenet lack tag retrieve tag main ten keyword particular retrieve complex query multiple keyword tag give minor form tag intuitively cca objective influence distortion tag rare one observe effect accurate tag
neighbourhood average cloud dependence function least component simultaneously component dominate one put thought profile measure borel weak compact converge profile equivalent copula indeed homogeneity homogeneity eq consequence intensity integrable indicator computation representation unit find total measure profile vector must law appear max profile vertex asymptotically asymptotic perfect dependence
homogeneous p distribution rl log likelihood hellinger likelihood frobenius example statistic produce arise hellinger appropriate associate entry contingency cross result call calculate accurately always extra principal advantage frobenius frobenius
poisson exchangeable homogeneous random ibp remove family distribution joint marginal later cause sure one imply direct familiar subset truncate restrict contiguous line restrict subset restrict exchangeable distribution exchangeable distribution de representation n example ibp zero per condition latent ibp accord restrict arbitrary infinite trial probability
encoder denoise encoder justify apply denoise well local matching auto perform achieve local derivative target implicit one gaussian location result method base partition function denoise encoder perturbation contraction magnitude applicable extend auto denoise second argument successfully auto encoder small chain local reconstruction jacobian respectively factor function finally empirically verify include experimental criterion auto train
isolate employ whole network pc search comparable summary scalable structure complete information serve meaningful biological protein gene serve central area underlie relationship bring understand graphical acyclic dag flexible proper result integrate model bayesian average grow enumeration impractical overall structure network variable parent network range protein average support logical attempt beyond number thus employ manual yet influence result knowledge domain exploited distinguish closely variable guide partition frequently similar pattern neither practical collect special second quantify bias result lead structure result attempt correlation information recursively network multiple size intra community
parent covariance parameterize mat ern correlation modify function kind spatially vary mahalanobi describe write diag rotation angle ern mat ern deviation geometric rotation angle feasibility vary spatially accord combination determine generic transformation ht l l smoothness rotation angle cumulative standard normal distribution mean see smoothness order determine parameter smoothness solely integrate n full furth probability density variate mean describe jump markov monte gibbs metropolis emphasize
generate accounting perform compare expectation well correspond bias tradeoff smoothing sm sim reduction mse associate hour motivate approximated initialize present detail section assess validity deal series experiment conduct validity trajectory capture experiment epidemic observational epidemic
policy action include indicate index introduce shift prove shift around must ode sufficiently large theorem come addition introduce ode define region rely establish meet strategy derive policy give ode p track solution ode value operation thereby v v n n claim lyapunov sufficiently explicit account hold definition regardless transfer sufficiently direct establishe track solution ode second ode strict lyapunov validate energy propose extensive km km exist macro bss micro file transmission request arrival file size ease process traffic load realistic historical indicate bss e w micro bss transmission operational consumption operational power macro bs micro w channel modify influence fast noise
propose endow working side exhibit linear rate research originally motivated framework computational optimization possibility kernel equivalence popular polynomial since kernel application capability training recently kernel improvement second minor acceptable actual run statistical assess significance conclusion addition confirm article introduce machine treat feature space call instance infer prediction term correct category candidate assess ability correctly map call hypothesis induction problem multiple category possible separately rest train binary svms versus classifier organize direct acyclic frequently performance method decision model space dot embed mapping decision represent feature input precisely computationally avoid belong value pass sign classification label prediction mechanism decision predict well unseen minimize implementation induction principle classification problem address build reliable pattern misclassifie margin pattern attain mechanism full implement regularity margin instance without
contour fit spline covariate correspond credible wide region edge space maxima minima indicate fit spline plot quantile grey grey figure sample band lag apart model expand lag suggest modelling residual identically u average degree freedom require trend element eight effective freedom five sd intercept forecast day day advance forecast store throughout bar year condition triangle slight model indicate provide variance joint daily trend provide also lowest fit goodness fit forecast subsequent analysis univariate daily trend credible mean quantile fit temporal trend vary year day peak around peak period daily peak pm daily trend day trend contain peak trend decrease effect day account autoregressive autocorrelation lag capture model wind wind linearly wind weak wind observable interaction wind speed direction wind angle approximately correspond
constraint mac correspondence prove nest unique mac constrain versa local minimizer exist nonempty kn nest exist nonempty neighborhood mac constrain problem mac nonempty local nest constraint obviously exchange everywhere non strict minimizer max mac formulation correspondence well manifold minimizer mac vertical necessary tucker mac point simplicity exposition special layer analogously r omit kkt nest mac constrain problem equivalent nest necessary condition minimizer minima maxima saddle mac eq kkt problem conversely eqs kkt hence correspondence stationary kkt saddle correspondence minimizer saddle mac first qp penalty sequence qp exact global sequence limit point kkt multipli theorem theorem involve function local minima applicable derivative weight differentiable mac qp nest positive qp find minimizer k multipli noting turn sequence mac
integer proposition ji ji infection denote infection model version generalize vector log likelihood number node please course follow parent cascade infected node pick large remove cascade consideration proceed remain cascade cascade u I result suppose suppose ed least greedy neighborhood attention establish cascade clearly could tailor instead standard information ensemble cascade need approximately low ensemble corollary cascade epidemic ensemble infection collection cascade infection observation say approximately recover graph singleton graph probability randomness generation infection recovery define fail theoretic entropy mutual
notable feature scheme laplacian simple evaluating take rank popular sequential backward search work f x reduce potential nature search different backward search incremental backward never consider likewise feature find feature diagonal place solution tend simplex select reveal non may give sequential incorporate necessarily nest characteristic useful multiple feature subset size indicate feature counterpart optimization correlation statistical measure covariance imply converse capable linear would quadratic dependence give positively pearson schmidt detect dependency require formal hilbert map
nesterov acceleration technique convergence number superior generalize function distinct secondly condition accelerate single consensus useful convergence proof let note bound also sum next ij k substitute omit increment start use convexity x j turn result back eliminate k j reasoning give lemma proceed hold suffice show due second positive
onto nine dimensional computed modeling face versus image last row show projection face explain qualitative subspace order face computed show model generalize decade intractable researcher optimization admit robust focus aspect book comprehensive robust overview major challenge notion quantify recall point arbitrarily place estimator bad quantify recovery stability approach residual early orthogonal consider method hybrid modeling aware tractable aside receive attention square residual book generally computationally tractable data maximum principle often formal subspace fail subspace compute covariance aware review pursuit pp construct direction scale component repeat proposal appear aware provably maximize pp remove offer algorithm provably correctness criterion tailor randomize iterative consist eventually identify project sphere pca recommend method practical guarantee researcher start effective technique variety algorithm guarantee assumption split plus corruption first problem rank norm regularization tradeoff goal recommend norm return appropriate outli contrast formulation possess appropriate discussion
strictly multipli one could dependent multiplier loss monte base use multiplier equivalent usually independence sensible work multipli nan propositions eq define independent copy finally copy result theorem adapt ii conclusion replace somewhat claim proposition obtain monte study detect question single change cross dependence margin alternative happen margin hypothesis combine factor representative question via resample distinguish size per autoregressive exponential autoregressive normal multiplier observation move
reach hour boolean go model minimal find go due perfect main advantage compute minimal allow meanwhile exhibit develop minimal question biological relevance discriminate precise pathway nonetheless exist fit reduce compatible induce perfect strong performance efficiency experiment describe perfect size range minimal model criterion parsimonious fitting observation however mention least
available frame hour intel figure show increase approximate flat setting result spectra atom ccc dictionary self constraint indicate residual code cardinality train dictionary term trade code code omp report result residual norm residual curve identical
state produce bottom actual circle induce uncertainty panel model rapidly whole lead exploration optimal transition episode already counter uncertainty cell approximate function state similar continuous domain gp value separate problem interpolation transition learn advantage situation limitation bellman globally space advanced discretization grid grid curse limit minor concern simplify make transition know conceptual limitation simplify make address acknowledgment take agent artificial intelligence laboratory university research
positive heterogeneous homogeneous step allocate channel access iteration channel step sense outcome round mechanism mechanism introduce fair resource allocation round argue subsection multi assignment job framework latter adequate relate let refer assign unique one resource formalize set maximized cost minimize logical logical logical resource aim interference resource advanced technique interference user performance division division division name agent virtual entity plan decision every coherence worker exploit primary moreover characteristic quantify channel availability ratio sense characterize f resource observe implicit functional relationship relate primary resource state resource among maximize secondary network observe
expert select competitive cifar state art grant amazon web services universit algorithms tune often rule force appeal idea automatic hand automatic within bayesian gp tractable induce gp enable choice parameter impact machine algorithm variable cost duration experiment leverage multiple core previous reach set dirichlet svms convolutional machine rarely regularizer generative
consequence exist comparison mle art analytic centrality centrality seem room remainder centrality main centrality dataset international centrality maximum exist analytic parameter centrality mle bind imply rao conclude remainder line transpose euclidean operator sequence event go grow comparison various analyze centrality algorithm comparative interest assume I n iw ji model ij independent invariant scaling equivalence w outcome equivalence representation onto define equivalent upper distance estimate order object
rise I class standard prior rule correspond denote I divergence use divergence interesting px evaluate divergence marginal decrease practice approach classifier incorporate basic modification follow class capacity feature marginal class easily provide natural q average list dataset denote feature use rank consider classification assign density extension although inefficient require additional I
shape restrict fit onto great progress newton quasi extension successful constraint constraint correspondingly room unified approach smooth attractive onto intersection write unconstrained introduce challenge take denote progress make parameter region example appeal convex distance closely tie c stationarity analytically due projection resort principle rely call subproblem fortunately close euclidean rectangle c set matrix g ball analytic last organize follow place iterative illustrate distance five different set close machine advance present theory algorithmic concluding limitation distance enjoy great convert smooth principle function current combination
unobserve observable also arise distribution see next variable parent rv style thick minimum mm u xshift mm b causal represent arise trial take outcome factor affect interest na I estimator effect biased encode amongst assignment treatment assess effect restriction make observable however finite
c combine conjecture microsoft usa university large introduce upper governed margin hold rich family feature characterize tool hardness property linear classifier use rule effectiveness complexity tight classifier learning focus instance free vc rule learn classifier maximal excess understand positive understand rule restrict require predict upper tight low instance vc cover well upper entire characterize identify distribution focus select classifier margin input treat origin correctly predict specific margin vc class classifier complexity optimal error
ice parameter heat bt output ice ice height base fourth span year ice prescribe ice year configuration important measurement ice measure measure rectangular spatial model ice large ensemble run show ice plot control heat ice mark indicate log ice take location set location index incidence measurement design consideration determine column enkf describe give compare determined measurement standard identity output parameter element behavior mean variance spatial sample covariance spatial correlation take maximizer treating maximize gain shannon minimize course number need compute define write computation relatively system
transform introduce vertex rewrite graphical cavity belief locally normalise require cavity serve find lattice cavity finally qualitatively show qualitatively correct consider cause teacher student lastly bayes gps locally normalise locally normalise error variation see individual vertex globally normalise spread prior change example cavity case teacher cavity gps enable look ahead extend cavity graph student teacher limit curve graph heat normalise exactly kernel give heat kernel method relate lattice discuss section eigenvalue find call consider p simply modify tree limit contribute expand everywhere similarly calculate decay explain understand behave outline previous obtain linearly break suggest look integral integration part lc l cutoff unity state visually cutoff version remove decay expect find numerical suitably normalise plot clear dash may surprising cutoff appear understand intuitively take let evolve diffusion process break diffusion scale diffusion reproduce quantitative scaling adapt walk normalise x mul exp mul mul understanding leave mean function vertex write nx independent
orthogonal consider sufficiently radius display versus oracle initializations curve imply appear fact account agreement prove interestingly would expect left plot radius much dependency coherence initialization potential minima conduct presence noiseless local minimum develop assumption lead also generative spike believe total remain plan formal future appropriate use relaxation technique useful acknowledgement european project fp project appendix statement proof simplify core appropriate problem dimension number possible local neighborhood control version present minimum reference quantity q universal provide c regularization minimum great success decompose contribution concentration surrogate term us generative dictionary noise introduce provide condition find theorem heavily concentrate constitute indeed corollary consequence discuss bind strictly exploit quantity rip noise level parameter define
metric trivially verify adopt argument assume aspect somewhat quantity formulate class function invoke law definition moreover see imply rx exists dominate empirical iid taking every observe suffice verify dx proceed number equivalence cl closure one subsequence md kn subsequence take negative minimal respect nr suffice equation extend originally interest fr value computationally hard fr element variable practice iid realization employ instead infimum necessarily
epoch count static remove historical relationship count forward online single infeasible data sampling base parallel hierarchical receive data processor execute independently parallelization parametric create explicitly master end describe master first post iterate new read joint receive label iterate label update child master receive iterate post master phase master iterate master computation master many new topic begin child happen across child create topic maintain count back master child master sample help quite similar section experiment carry medium aspect model goodness unseen label topic qualitatively trend major estimate insight find trend scalability strength ability million evaluate importance factor able combine factor usefulness factor medium would able perform array medium available truth quantitative
leibler form scale rescaling constant natural introduce uninformative probability measurable absolutely respect respect call x reference entropy sentence sentence countable alphabet sentence entropy equality general conceptually refinement limit always enumeration sentence equivalence require justification entropy sentence monotonicity routine order enumeration limit equality enumeration sentence sn let elementary algebra tail show borel generate borel algebra ni measurable p form martingale z consider z existence show definition measure find measure relative practice scale meet constraint rescaling integral converge measure ix uniquely ix n jx construct end relative choose indicator elsewhere tell function piecewise constant sentence proposition lead follow constrain expression informative contribute define relation indeed factor possible informative measure theoretic first finite convenient sum restrict term elementary expression minimize constrain minimization separately non strictly finite compact lagrange multiplier derive uniquely finite elementary necessary condition probabilistic coherent finite determine probability sentence answer alphabet alphabet probability meet remark informative equation since pairwise proposition finally proposition conversely equation put well suppose valid eq valid finally separate guarantee first sentence sentence valid condition
lda cca versa needs introduce regression cca outperform par p cm name correspondence face contain pose item text instance tag office object amazon department science engineering california assumption train underlie unfortunately instead label exist target improve classification domain method domain vision apply datum make sometimes impossible sometimes transfer leveraging hereafter unseen domain case
field discriminative augment discriminative discriminative value greatly generalization operation next fed perceptron resembles initialize respectively treat constrain remain transpose learn
share chain chain approximation elliptical amount chain take different time may slower one update period share validity fraction spend measure overhead point operator transition preserve indicate round long multiple time need approximation randomness compute algorithm maintain collection advantage core core make use collection seem good collection chain collection burn motivate distribution section convergence amount summarize amount approximation iteration every varied approximation change four work update slice straight point elliptical slice along use step affine invariant make differ elliptical slice perform variety scale adjust member parallel population parallel another parallel involve sample separate chain encourage chain auxiliary hamiltonian burden user combine stack support practical
conventional accuracy subspace two mahalanobis mahalanobis confirm mahalanobis compose class sample therefore small zero accuracy comparison achieve pca mahalanobis kernel perform bad mahalanobis high gaussian kernel increase observed subspace estimate classification signal classify mahalanobis mahalanobis class sensor spectral pixel feature nine define principal increase conventional kernel pca mahalanobis equally accuracie influence accuracy classify report mahalanobis mahalanobis test variance influence compare mahalanobis kernel estimate
prop suppose combination none q prop fix raw distinct candidate bid treat either occur ad ad etc say matter ad ad ad query reflect inductive result p map selection prop maximizing maximize eq since one suffice single decompose ad bid raw I respective ad expect ad since ad negative ci quantity ad hard selection mechanism prop consider sort ad bid show efficiency hard
date consumption distribution variable consumption realization multivariate amount almost customer collect expensive minute total consumption customer storage get estimate profile reasonable select sample population compare compression population conclusion situation rather simple design survey property sampling treat far know nothing framework investigate estimator use apply functional arise domain functional response time index structure section concern mean linearization curve variance without replacement proportional replacement linearize sampling adapt stage thompson population functional element trajectory straight generalization univariate customer company want location interpretation centroid three
say x go arbitrary let tune accept accept explore reject see tune accept proportion accept history article approximation sa sa adaptation condition p tv verify rate convergence therefore aim stochastic common diverse field model branching process advantage discrete apply try
clear record element nonempty recurrence relation convention subset tuple record contain record belong tuple record record classify tuple appropriate record notation product function tuple inefficient problem bipartite record linkage block thereby reliable categorical code quickly record code link discussion link record tuple assign subset agreement link file record decide remain assign record tuple tuple assign see practice tuple classify partial subset record tuple fine equal record subset potentially present file illustrate gray become record match census subset census subset pair link record assign link record linkage classify belong direct implication obtain tuple subset determine
explore misspecification forecasting initialization adequate historical observed must care take training datum forecaster take misspecification base outperform adequate extensive study demonstrate use dynamic forecasting method little system forecast likely poorly demonstrate system surface long svm determine width end dimension correspond set pick grid number training speak actually factor use cross approximately grid geometrically validation pair grid validation performance l one repeat change validation
strategy exhibit tree sphere enough summation preferable expand determined procedure cutting expand appropriate preferable opposite summation less copula aic function never great algorithms construction nine bivariate appropriate ordering copula sphere variable first order I success good evaluation sphere algorithm exhibit well resemble indeed exhibit random cause efficient evaluation conclusion careful copula decomposition respectively copula flexible product copula rich confirm correlate always necessary
rejection bottleneck mention nevertheless balance mention tb allocation sec start procedure cumulative shift shifted assign shift obvious amount allocation unique construct overlap position order explain shift example update cumulative gibbs determine uniformly detailed known good way make negative generation order candidate order position update uniformly symbol nothing satisfy detailed net flow globally autocorrelation method candidate sampler surely explain significantly rejection case minimize create
limitation observable process causality require exclude whole universe impose source entropy face process set able source solely graphical applicable estimation bias effect estimation skewed come positive link material coupling attribute coefficient mit filter influence parent mit strength link regression regard equitability property coupling couple partial correlation approximately analytical mit adapt air temperature anomaly heat towards height km coupling pressure pressure mit panel significance white estimate use separately surface significance average e lag parent spatial lag mit leave panel peak indicate month mi significant lag couple delay difficult
figure discard search find high place slope checking cf gps job provide low upper bound limit vary costly would avoid optimum easy surrogate aid optimization restrict gps enable evaluate optimize reader general surrogate utilize gp confidence ucb deviation gp choose study rely heavily idea surrogate key trivial notation follow set
regard verify log regularize problem estimate regularize yx formulation convex convex desirable section high formulation formulation impose natural dependency interpretation special univariate reduce high regression replace row formulation element penalty arrive regularize maximum linear penalize also study sparse regression unknown univariate variant neighborhood write entry row entry row matrix formulation due quantity regression know multivariate regard generalization graphical precision author suggest estimate selector mean contrast estimator convex framework investigate performance monte employ sub
instance section collection categorization validity light previously point classification deal state compose assign learn infer word assign subset unlabele cope classical consists divide binary decide category naive baseline relevance collection object assign tag due absence another tag motivation look result binary assignment capture improve utilize assumption relevance still categorization year basically transform adapt adaptation contribution algorithm field variation cope intra dependency improvement deal probable category document belong category characterize
demonstrate increase hybrid mining attribute fortunately increase generalization generalization role indicate optimum mark circle define attribute hybrid able researcher recognize importance explain top bottom top description resource instance analyze task enable work manually pure manually carry method thereby task bottom role algorithm role mining different instance candidate role role cover role role select role differ proposal cost role role mining variant paper problem definition discover role upon prior role assignment rule mac focus role select pattern unsupervise hybrid generalize particular connection concept cover approach method solution motivate analyze world usually realistic prior mac one core model highlight role role assignment theoretical comparison sound confidence assignment modify assignment mac user mining prediction compression problem role configuration definition mining inference analyze variant access control robust generalization ability ability role mining convert joint joint depend outcome realization vector individual contribution ik sum pick half factor term sum sum range modified set bit except successively product sum probability two start exclusive event p substituting yield necessary gibbs collect distribution ik ik beta beta euler integral new
beneficial also lasso despite number simply provide support zero prefer might gain predictive include correlated variable drawback elastic q response column view trade ridge depend relative equivalent method observe outperform elastic net measure second handle sum orthogonal trace penalty
loss minimization cast separable many proximal regularizer norm interesting naturally smooth term square typically subgradient norm efficiently common situation machine processing regularizer atomic norm norm decomposition compact translate correspond may choose constraint I differentiable interior gradient converge g bregman always see project start recursion x
computer build south road qx email ac website http com language thought language thought permutation require begin
certain regularity fa fa g tail boundedness mild quasi assumption could regression endow property far adjoint q w g exist sequence bilinear eigenvalue assumption hereafter satisfy sum sup sequence delta convergence enable expression g theoretical derivation underlie assumption know polynomial say marginal view rely ode bound infinity eigenfunction finally use notation fr z ng w fr derivative ds g explicitly write k z g ng sg ds f identity operator develop key asymptotic rigorous theoretical develop present norm exist constant choice process possibly continuity condition proper necessarily state type dominate q
change adaptively subject threshold aggregate eigenvector subject give grow extract projection without invariant subject towards define penalty denominator perspective subject rank aim remove set subject want dimensionality subject interest training determine validate manner type subject discriminative scenario important valuable datum illustrate domain discriminative subspace datum discriminative stationary subject user subject give similarity discriminative non stationary subspace task furthermore common perform subject subject reliably subject present bad affect please amount maximal remove limited subject important advantage avoid utilize vary discriminative subspace
location main area region desirable fine km resolution lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc marginal level limit area km cover see blue set take complement indicate grey panel indicate region certainly ni method pair level avoid uncertainty cover uncertainty contour curve contour lc lc lc lc lc lc lc lc lc lc lc lc lc lc lc leave complement grey contour green show contour trend cover related feedback surface rapid receive much attention period individual occur significance testing field thus find simultaneous trend year prior measurement determine trend restrict spatially vary evaluate distribution choose correspond measurement phenomenon would entire assume measurement one pixel
bivariate nine devoted illustration financial proof code test study framework theory empirical probability assign mass elsewhere measurable evaluate function mean sequence weakly notational weak denote indicator lower recover version state advantage general low may goodness consideration choice half paper van satisfy x x open measurable fr map np asymptotically linear q proposition prove appendix subsection exist brownian limit state proposition class depend family respect proposition
unless provide author forward step show behave effectively sensor datum author expert recurrent neural rnn environments low sensor datum environment process error filter schema rnn also take application view forward optical flow detect forward acquire obstacle free compare expectation novel incremental generate capability purely optical spatial enabling predict locate time phase optical field capability robot follow
evolution traffic study involve affect weather prevent provide traffic flow weather weather direct whose variability quantify impact weather traffic pair datum traffic different weather road variability road condition occurrence traffic study obvious heterogeneity datum weather road conduct nevertheless previous modification enable velocity furthermore enable different weather change conduct therein achieve build select observation traffic different
deviation building intuition amount proportional amount temperature controller control day span day provide use implicitly variation usage due building keep use use notational reason default controller energy usage day controller day estimate day statistically significant estimate characteristic statistically difference enough exclude controller day statistically suggest default mark
gap total obtain proximal sdca one pass general unlike batch gradient accelerate gradient relatively even significant loss convergence prox prox sdca gap nonsmooth advantage duality check criterion discuss convergence nonsmooth nearly
properly horizontal material follow sentence input file english name wider take width page live area environment table location immediately sentence file comment author alignment author enumeration table
generative discriminative pre layer mm method improve positivity encourage amount use finally avoid feature finding applicable cs optimize single global contrast exist scheme heuristic model store information appearance decomposition signal although pool incorporate range demonstrate secondary issue within present detailed machine representation recent feed attempt generative deep belief network compositional mechanism representation give invariance perturbation level large common preferred pooling many hierarchical directly pool region global pooling max max averaging sum joint hierarchy something build layer hold fix optimal joint notably machine
assignment customer generative specify customer customer form index partition table customer restaurant table configuration customer factor notation factorial similarly except numerator factor find numerator customer customer restaurant collect configuration block customer exchangeable construction exchangeable turn particular demonstrate instance bernoulli represent allocation collect successful draw feature allocation latter case feature distinguished frequency order order reason suggest random random achieve maintain across associate assign uniform natural follow unique augmentation order random factor number block nearly equation argument construct size partition case implicitly summation condition long hold argument explicit necessarily one partition exchangeable consistent allocation whose order feature frequency represent identically form appear feature order choose exist argument call assign index block ibp recursively chinese restaurant like crp form customer partition feature crp ibp start choose sample customer yet restaurant recursively choose sample customer customer sample equal customer try customer belong feature allocation
recognition operate genetic evolution generation individual leave system condition genetic programming fusion return system order make decision rejection none equal system low far good accordingly reason use evaluate lowest generate low increment high step far comparison inter intra score roc far mean couple fitness various genetic present evolutionary produce division minimum number return mean mm score score distribute linearly score linearly tree depth em individual depth limited mutation selection fitness inferior
accurately exact would noise noise atom finally code indexing significance grow combine atom consider combine dictionary dictionary learn multiplication matrix interpret dictionary investigate similar finding dictionary reference dictionary allow significantly problem fast goal describe selection formulate submodular canonical derive neighbourhood small alternative submodular formulation mention generalised form author select support non help quadratic objective include boundedness dictionary matrix optimisation practically complexity iterative
fair fair sequential aggregation rule obtain rule grid report fair prior adaptive run weight allocation select sequentially good pair maximal grid eventually construct cover span implement extend grid user start say value test different try impact bounded place grid possible grid table fully character meta limited performance would period allow enough base rule symbol well good single expert p benchmark grey black quantile residual group half move measure residual want come aggregated hour subset aggregation discuss study aggregation meta benchmark expert overall aggregation benchmark performance hour good combination forecast share aggregation slightly bad seem excellent term probably benefit intermediate around intra depict third value residual distribution concentrate benchmark conclude aggregation never prediction expert expert error strongly favor
global order sufficient full initialize cluster dx I poisson isotropic etc initialization mle good give diagram dual cluster al cluster bregman bregman interpretation efficiently weight maximize complete likelihood amount minimize weight ratio mean standard bregman bregman choose q fc b fc seed optimal bregman factor explain triangle bregman bregman interpret remainder taylor consider proximity property zero mixture weight increment combination rise weight al em mixture bregman duality bregman divergence bregman decrease complete converge monotonically decrease expect doubly
positivity temperature large temperature grow mathematical formula energy significant role free define shannon entropy free energy cross instead multivariate define side energy introduce statistical positivity differentiable monotonically increase energy coefficient partial follow reasonable accordance accordance I principle finite accordance data principle mass energy state part interpretation shannon adopt select accord free principle maximum entropy difference arise ground basis shannon entropy energy estimate finite
td td step follow step combine base criterion adapt hmm obtain merging th term merge merging criterion cluster obtain close local iteration hierarchical em choose framework view maximize non informative uniform satisfie integrate approximation bic maximum bic consistently devote classification global
project read air research laboratory contract fa conclusion recommendation express material view u release thm thm corollary remark family notion symmetry probabilistic framework exponential family equivalent class marginal deal variable usefulness general framework variational map lp relaxation lp bind result art function relatively efficient
need reservoir computer good performance operate drawback precede absence reservoir process limit analog bottleneck point modular computing hardware parallel analog possibility back output apply reservoir computing category generation intensity propose drastically optical experimentally reservoir report although reservoir reservoir computer usually neuron typically neighbor linear combination instantaneous coefficient evolution reservoir discrete total reservoir input describe topology favorable dynamic treat
batch devoted therein intermediate definition assumption direct convex strong convexity combine fw fw furthermore fw fw average evolve accord next convex optimization serial begin sequentially repeat recursive typical descent characterize appendix serial algorithm immediately regret standard study distribute discrepancy state
de un ensemble de class la action dans ce ensemble le du age des la la distribution axis les des en dans un en un une de la
accommodate input position obtain term setting enforce benefit become ambiguity improve ready introduce equation denote discrete time policy actor reinforcement algorithm parameterize generic ms valid exploration term actor update adjust choose distribution gradient lack gradient traditional matrix outside type actor update gradient desire hamiltonian parameter algorithm balance system action k r x actor x balance actor raise stability balance lose effect preserve passive definite hamiltonian immediately specification rl guarantee exploration necessary guarantee
coefficient classical separate political alone vote drive political traditional ideal adjusted figure stand become overall traditional adjusted interaction specific depend vote name popularity ideal adjust also assign advantage include popularity extreme increase issue qualitative difference ideal quantitative issue well predictive vote divide vote individual pair fold vote vote evaluate hold vote assign probability hold several ideal label describe study issue ideal rather topic lda topic make issue datum variational mcmc ideal issue remove contain maintain mixture improvement traditional topic change five ideal issue adjust label issue ideal lda adjust lda adjust issue summarize table issue adjust vote permutation vote issue adjust issue adjust ideal demonstrate issue give well fit roll point validate exploratory tool traditional ideal characterize call datum demonstrate approximate collection vote adjust fit discuss several vote party line explain preference like issue model measure log adjust relative formalize define vote log vote adjust likelihood point issue improvement vote log likelihood measure house
whose h breast cancer alm admm alm nonsmooth area splitting method way accelerate distinguish wide applicability problem covariance show code implementation handle prescribe application sample interesting advantage varied definition motivate problem completion cluster robust pca sparse simple simple non advance domain optimization proximal iterative smooth fy ng
click directional click feature mutual mutual information entropy determine user must convert bin span quantile make bin mid cover velocity mid pressure stop direction velocity stop duration direct distance pairwise velocity velocity velocity acc end trajectory large end acc acceleration end line acc phone stroke orientation leave coefficient analysis percentile pressure trajectory informative position adjust position orientation orientation insensitive please rank rank constitute gain complement hold correlate percentile percentile end segment direction end connection understand feature redundant color green encodes color plot highly selection still understand discard thereby speed merely decision distance highly almost thing constitutes ignore trajectory classification due discard orientation end velocity feature feature stroke exhibit motivate stroke framework neighbor rbf kernel various reason work stroke observation close observation classifier merely observation huge limitation
infimum quadratic coordinate descent paper simple newton descent minimizer approximation descent minimizer find line repeat meet algorithm search necessary ensure q equal q reduce optimization gradient xt algorithm imply definite computationally beneficial g middle loop penalty block separable differentiable separable separable twice differentiable quadratic decompose follow symmetry minimizer consideration cyclic j j j last term penalty due part ordinary descent block descent claim follow minimizer descent
gaussian development affect affected reflect copula payment incur significant shape quantify contour sample contour vs scatter plot third eigenvalue eigen combination variation payment incur year ht plot marginal box plot box box posterior distribution present presents copula log observational figure structure copula plot copula copula posterior tail payment incur uniform copula contour mixture copula year incur assumption year posterior surface posterior dependence year incur mean scatter correlation correlation versus payment incur mixture box distribution box plot box marginal posterior box posterior distribution section detail able payment incur give integral analytic distribution incorporation significantly posterior development payment incur clearly shape dependence hierarchical mixture admit analytic augmentation stage sample cumulative payment loss year note base mcmc interest sample complete model dependence construct normal approximation locally factor precision covariance gaussian may copula augmentation alternative distribution total give payment incorporate development precision development good agreement dependence ht posterior predictive box kernel loss estimate mcmc denote depend
theoretical bias causal first ensure estimator relation utilize collapse graph e node parent graph causal diagram parent edge node edge direct applicable selection mark parent collapse transform contain model conceptual population model collapse selection diagram follow diagram selection diagram conceptual identify ii consist causal causal include identifiability surrogate design identifiability causal effect identify rule calculus effect integrate effect allow rule however express estimate identifiable specific instrumental clinical trial identify even causal identifiable form may
across operator nonconvex penalty mc parameter realize model procedure evaluation problem degree freedom degree freedom selection introduce df bic df p df penalty freedom well turn refer improper improper converge slow unstable issue penalty tr tr prevent solution select value whereas generalize bayesian criterion penalty prevent occurrence loading dimensional diag penalize likelihood traditional rotation
discretization state facilitate arbitrary applicable increase exponentially space exhibit kalman gaussian filter try adapt filter assume state represent kalman apply series offer employ linearization density true property approach approximate accuracy depend dynamically linearize linearization mixture induce linearization quantify covariance base moment preserve splitting introduce direction nonlinear linearize linearization linearization applicable formulate sec brief introduction linearization novel splitting derive sec describe component conclude remark time
technical representation briefly estimate gradient main introduce state conclusion prove condition fulfil section present network criterion normally value behaviour analytical formula normally realization carlo function form variable offer potential analytical study begin management scheduling project project activity activity time perform activity perform normally interested describe oriented graph arcs correspond project belong must formulae di f activity describe duration random represent involve completion express
pz chart plot z difficult distribution resort chart chart give mean chart control comparative adopt outli outlier outli number largely shift place put choosing outli take take outli carry simulation order compare
minimize effect run validation analysis different suitable nn classifier network approximate per classifier neural network nn nn bayes class variance measure cell variant choose minimize mse network classifier comprising train due take average discretization ideally marks go play aim aid classify style dimension many recommendation game various base currently promise world rating play especially play similarly go quickly human opponent game auto provide pattern provide experience hope look analysis pattern especially hope insight various shape improve investigate challenge extract game play classification highlight room improvement record discover classification origin game aware previous
rmse pls hard pls pls hard significantly drive pls ols large show rmse pls pls soft slow rmse rmse least pls pls drive promise finite rmse four next sample penalize step drive variant plot approximation close left reveal size informative panel influential plot obviously line coincide demonstrate construct pdf c comparison suggest adopt interval plug obtain specify drive choice rmse suitable confidence first four index pure cc cr rate setting drive interval previous except coverage close nominal proportion coverage rate decrease sensitive example implement penalize study expression gene
graph pure therefore idea good algorithm time label node nod benchmark example respectively g window benchmark form web page web performance construct graph gain pure combination performance naturally site method al label consist similar pair tell construct pick oracle know node consist unlabele unlike percentage construct even connect edge connect label unlabeled take construct take pair filter edge
party I local standard deviation age presentation deviation visual assessment alternatively one parametric region list model gender region g display probability support function age package surface bivariate bottom indicate surface gender region surface probability similar htb party sign change frequently obvious play profile adequate appropriately reflect evident aim precisely group
screen target ht response model expression signature signature prove clinical reliable disease signature coherent signature routine clinical practice establish signature design classification signature association underlie enhance derive signature network validate wide detect model local condition independent predefine classification gene extend approach predefine reveal unknown know related modify applicability datum identification coherent component predefine pathway collection entity complicated gene suggest gene predefine predefine small module signature search set miss purely orient biological bring connect difference relate approach single additionally specific expressive module signature specific
perspective continuous concept remain constant bounding equilibrium denote convexity vanish exhaustive iterate nash equilibria exponential tb sort uk note constrain equilibria fashion observe candidate nash equilibria maximize increase equilibrium nash equilibria joint frequently observe maximize aside constrain equilibrium equilibria linearity constraint thus fit subsection game check whether nash operator account equilibrium make satisfie np empty complement equilibria search regard refine hypercube function possible linearly separable first number tight bound player conclude vc vc neural easily unfortunately weight I find induce experimentally find estimation game equilibria rely mild space likelihood trivial sensible variety base may look would range fact broadly objective want behavior need tend try convex upper find slope upper maximum bind kl bound informative since compare red bound equivalently next maximize equilibrium game sufficiently hypothesis identifiable kl kl kl kl kl kl prove third approach sigmoid equilibria additional ascent maxima gradient ascent hard proportion equilibria implementation regularizer encourage attempt lower control approximation maximize equilibrium minimize equilibrium maximize proportion equilibrium loss far introduce note avoid obtain speak make hinge require minimize develop efficient solving hinge encourage attempt fit loose simplify player independent regularize joint novel output logistic equilibrium kl
identifiable approach correctly probe zero propose keep probe deviation high variance j yield noisy prior expectation probe support probe content element probe affinity estimation probe prefer accommodate potentially probe ij ij maximum j affinity estimate across yield final specification probe level j j weight inverse ij learn far validate comparison preprocesse standard datum ideally batch confirm compare regular moderately sample estimate pearson version batch memory resource sample correspondence batch confirm convergence probe probe convergence later indicate array typically case online file
cutoff law cutoff law alternative fit normal exponential cd cd cd pt branch volume moderate none stay moderate wind speed none none wind speed moderate cutoff city moderate cutoff moderate none rare gene analyze city intensity come similar analyze counterpart intensity consequence choice large slight raise compare furthermore poor power law cutoff heavily contrast fail reject cutoff lose obtain intensity case conclusion direct implication illustrative work branching argue law imply branch forest law distribution branch entire collection branch analyze law cite support claim statistical west result match diameter branch diameter branch prediction instance stay well fit power particular hazard probability worth hazard leave decrease stay heavy tailed investigation covariate predict hazard article principle conclusion coarse power sound make method practitioner variety field behavior datum regardless number field quantity compatible hypothesis tail distribution explanation effort
however sometimes great discard hypothesis general reject logical reasoning evidence almost expect shall may occur let main x x p notice eq evidence build come acceptance hypothesis practitioner become summary frequentist general computing statistic statistic choose depend value give frequentist metric dimension hypothesis arise nested hypothesis new regard logical paper present evidence frequentist belief show final remark evidence discuss refer review evidence manifold evidence measure write notation nan item hold nan least coherent plausibility
I es expectation logarithm unbiased walk precisely know get geometrically rate tw accord unbiased geometrically investigate es contrary geometric chain irreducible borel start addition one precisely suppose
scheme approximate iid uniformly km km proof theorem outline coherence r km rs matrix omit proof would adjustment advantage theorem error achieve small recovery noise imply minimize large error
application consider phenotype interest clinical trait simulate snps phenotype rs rs detect bayesian association study account serial structure variable represent trait characterize phenotype significance strength phenotype unified approach challenge met adopt show variable spike detect weak application genetic diabetes demonstrate relate measure limited phenotype continuous phenotype association marker interest effect phenotype impractical genetic variant advance parallel partially less moderate propose uncertainty however must science engineering ls health lx disease receive genome institute national diabetes disease institute diabetes disease national health diabetes diabetes study support institute health grant
matrix linearity may responsible variant hide specify dirichlet distribution think probability simplex denote distribution q pseudo uniformity pure one one rest allocation represent independently fall represent vocabulary multi exchangeable model random dimension full rank long share exchangeable rich view factorial series hide abuse product factorial make mean shift make linearity furthermore recover transition furth topic underlie hope recover column extreme good recover multivariate see transform invertible indistinguishable statistic issue
entropy entropy uncertainty past transfer right side lemma information follow entropy non conditional entropy equation mutual negative later equation equivalence come go flow state three lemma multivariate
successfully equivalent match discrete namely generally implementation corruption implementation fundamentally get would limitation ability discrete continuous review would interesting generalize besides seem qualitative pattern illustrate arise force mostly parametrization energy experimentally mathematically formulation assess score inconsistent gain constrain chen acknowledge support research use quadratic corruption behave first equation auxiliary rewrite put quantity focus quantity optimum read rewrite term equation taylor expansion expansion inside represent term odd reason expectation vanish px px use geometric expansion q result study asymptotic would would say deal pointwise asymptotic expansion problematic contribute use quadratic corruption rewrite loss index mind consider family taylor involve take expectation substitute taylor expansion express
library initialization seed feasible seed early initialization include slow attempt environment minimize trajectory environment position target initialization trajectory move exploration goal exploration original straight line point goal seed short left configuration configuration environmental configuration library notice straight initialization obstacle therefore difficult initial seed mark seed find also execution blue end initialization generate default order evaluate library free initialization seed fall slow contextual role quickly evaluate trajectory result
rational middle plot entropy rational mean line procedure converge entropy state consider empirical conclusion conjecture rational converge cm cm rational certain moment compare version obtain exponential lead lead direction therefore assess ability outperform algorithm originally another whose entropy suggest fail far efficient moment suggest entropy suggest question well
single post recovery advantage rely subject constraint optimization constraint graphical trace regularize max formulate fairly relaxation relaxation max generally cut norm fact cut negative classic differ several way problem multiple cluster integer sdp number cluster round technique projection typically employ classic relaxation linkage work aware cite focus bad factor guarantee guarantee need cluster hand clear translate affinity enough underlie true
estimation recover element error negligible propagate reconstruction quantify element provide achieve column minus element accuracy instance empirical accuracy simulate node difference clique instance basis oppose accuracy c error reconstruct evaluate term versus network model coefficient bernoulli corresponding network method runtime accuracy comparative degree fit model denote blockmodel mixed membership value summarize comparative consistently second desire
spline basis additive implement discover cf sf fitting cf sf smoothing pf extension name summarize extend integrate apply first illustrate validate rf predictor pf sf spike explain model aic pf sf strongly associate estimate intercept pf sf df spline basis pf cf sf pf partial describe contribution except drop linear pf represent time channel thereby notably coefficient sf space light fit confirm sf end increase discount possible interpretation may fast input sensitive observation fit interaction seem minor deviation improve prediction much somewhat study
conduct multi bandit strategy evenly budget arm identification sr plain successive design arm budget return arm identification idea extension bandit arm figure require propose sake attention permutation gap arm high arm gap cluster
successful obvious variational achieving commonly use discovery rbm energy energy rearrange energy model energy identically comparison reveal energy turn inclusion undirecte direct undirected describe immediate consequence transition modeling tractable training rbm guarantee factorial due away rbms
least negative vertex relaxation equation enjoy prove unchanged every instance think simplex attain regret solution statement begin note look ball pick theorem effectively adapt simplex randomize enjoy since bind adversary since bf randomize simplex bound assume however tend conclude interested bandit barrier bandit play simplex convert armed bandit develop self solve bandit pick adversary bandit pick pick f f conclude step pick bandit simplex use armed note choice output round sampling pick bind bandit simplex observe reality observe unbiased estimate
partition partition affine hyperplane shift hyperplane gap cluster project onto principal trend measure gap adjacent unbalanced size figure two world singular plot depict algorithm gap ht ht division situation two separate ht aside trend sort right singular sort permutation v eq obstacle implementation something gap end gap take separate
fraction close gibbs hmm incorporation framework via link therefore state hmm covariate hmms require six wish wish state disease disease change gibbs sampler employ forward metropolis true hmms covariate describe model use disease covariate monte sensitivity hmm influence process hmms subscript notation disease state observation eq conditionally independent l chain month take missing time example disease give covariate disease deterministic forward fitting model month month covariate influence month covariate covariate month site disease detail description ht dd pc pt pt pt l
state economic subsequence assumption stationary show mix coefficient single process stationarity deal try joint extend ahead set notion function p expectation take therefore may prediction recent I series somewhat subscript within forecast memory forecasting model grow method use make sequence series use say ar obvious meaning memory allow control forecasting loss assumption unbounded loss retain control allow memory allow grow implication considerable training specify tool aic asymptotic class increase uniformly w explanation inverse cf worse note expect supremum event error expect concerned discuss way understand visualize confidence effective eq minimize thereby ensure small outcome sample risk take take movement toward
functional minimizer compute independently functional admit theorem widely also technique machine therein theorem hilbert space rkhs functional exist wise functional coincide section start
mixture consistency value purely arbitrary long speaking degree equal unique neighborhood need converge theoretical unbalanced material block like label truth instead weak assumption label plan far guarantee practice initial option empirically perturbation initialize pseudo empirically topic appendix tail consistency bernoulli let apply result rhs chernoff optimize yield let thank mathematic discussion unbalanced remark support nsf focus group dms grant dms algorithm fitting scale network fail sparse new pseudo range include political perturbation work spectral fail mild condition
framework top standard agent entity receive act distance object agent sensor well ball team learn hand code however learn individually world happen step macro ball must decide hold ball automatically receive ball continue ball leave rl duration episode immediate pass individual call act work consider state vs result large space make benchmark finally approach optimistic use implement try actor transition result outcome pair optimistic learning meaning execute policy changing estimate q optimistic iteration online become available interact paired actor
use approximate nonlinear estimator linear nonlinear control system map infinite act reasonable dynamical nonlinear dynamical hilbert leave lyapunov operator nonlinear thank thank receive international incoming fellowship nsf thm definition remark remark quantity arise nonlinear nonlinear system exist readily extended system success mapped develop computable approximate induce approximate ergodic stochastically force nonlinear easy system study analytically analysis yet necessary analyze dynamical basis previous reduction
subspace invariant decomposition boltzmann machine filter together group cause window filter filter subspace direction invariant unsupervise inherent use representation pooling somewhat datum representation away procedure level exhibit control lose subspace pool invariant sensitivity direction goal extraction many situation return pooling expression subspace subject expression form associated appearance lose feature lose successfully task irrelevant unfortunately ultimately
except greedy notation assume randomized apply state construct individually simply implicit include assumption indicate policy necessary consider policy common approach section focus type make easy theoretically show bind iteration policy optimal upper choosing otherwise highlight advantage bind small limit right instead bad programming bound policy
error line comparison response accordance essentially line oracle except comparison confident comparison active know mention practice fy fx identify request comparison sample opposed ensure intuitively strong ensure equally spaced line difference away bound coin whole exceed probably correct boolean comparison evaluation class possibly error decay strongly convex tight resolve gap due world desirable extend
data decision reliable provide number win bp bp nearest binary method approach lr moreover need seem similarly well bp human bp reconstruction bp statistically significant bp statistically human assessment figure present train via resolution mix super estimation section demonstrate nonparametric image super beta bp assignment activate binary assignment cell distinguish characteristic examine posterior matrix element illustrate upper sensitive sufficiently use equal reconstruction histogram bp batch size vb set algorithm batch present subsampling patch dictionary segment call mini evolution hold natural patch learn dictionary patch represent
without feature see conduct svm hashing convenience slightly regularization svm hash performance achievable reader able easily accuracie linear panel regression panel hashing compare well accuracy permutation e second merely absolute already present permutation hash less original hashing occurrence empty phenomenon bring additional reduce advantage reduction show code code superior coding add new expand desirable algorithm encourage hashing work permutation scheme conduct empty occur sample vector permutation coding work let well accuracy permutation without without matrix sample
precision graph follow keep independent keep number replicate point replicate number link fp false false discovery rr aic bic aic bic well average graphical neighbourhood lasso rr glasso glasso see graph necessary dynamic graph human cell table c aic graph model describe accord scheme summarize characteristic lag constrain temporal lag constrain across self interaction couple lag interaction sparse offer straightforward interpretation I particular part zero priori orient parameter estimate real describe finance consideration determine system big molecular gene structure penalize lack specific consist
lemma lemma lemma complete substitute discrimination power generalization discrimination ratio note moderate characteristic inherently fit evaluation generate population covariance discrimination scatter discrimination scatter since expect generalization bound side figure power repository image segmentation constant class
chain assume singleton must contain
strength range average hamming outperform thresholding post nontrivial sophisticated signal strength investigate half case pattern different experiment report inferior ideally result repetition table suggest outperform various two pt c focus memory accord way tuning parameter u pe pe need take case lasso scad shape mc range vector tuning know scad mc set hamming assume argument comparison report hamming repetition suggest outperform comparable mc gram process cd c scad scad mc signal severe b end except generate appear adjacent opposite ba repetition suggest outperform mc mc signal major signal cd c scad mc scad mc scad p population covariance tune therefore misspecification affect appear adjacent parameter mis report misspecification instead apply tuning comparison tuning ideally experiment insensitive misspecification outperform misspecification adjacent panel adjacent triplet panel pattern panel lasso severe compare pattern signal vary block fraction sign pattern block sign block hamming base repetition report pattern negligible inferior sign outperform theoretic insight
limit goal seek expand eqn simultaneously approximate inexact solver execute dictionary separable kernel denote infimum minimum supplementary write notational mind impose value trace denote cone definite whose sparsity regularizer admit representation broad extensively sparse allow norm norm admits close norm admit optimize specialized solver involve partial involve follow quick bound trace conjugate cg iterative rapid numerically start regularization rademacher complexity give give material scalar concept
get engineering institute science technology corollary stochastic setting optimization nonsmooth linear multiplier nonsmooth closed solution minimize augment directly demonstrate algorithm structural function strongly function
let product notation save covariance uniformly mutually iii q pa ii b pa b ds pa take side lemma ii optimize conclusion sum finite dimensional range weak strongly dependent concavity finite dimensional suit self adjoint dimensional ii eq I ax get
interesting nature mu empty intersection infimum play infimum bottom multiple maximal respectively form mu explicit maximal gamble confusion coincide confusion ex xu yu background yu coordinate xu away yu scale baseline ex coordinate xu yu background xu xu yu yu xu yu cycle intersection xu yu yu xu yu away cycle baseline xu coordinate coordinate yu intersection xu xu yu intersection yu xu yu close second nevertheless top mmd md cp pos dash pos cp pos md md md pos sep white dash mmd mmd relationship set diagram set go top diagram indicate dash line set maximal element move correspond assessment confusion step far move assessment model viewpoint closure typically bottom confusion diagram simplify non trivial assessment correct extensive criterion give intersection closure closure effect generate least dominate assessment dominate assessment structure encounter closure proposition mu mu mu closure apply operator form interest use indicate one agent assessment impossible conservative closure confusion interested turn observation mu dominate counterpart envelope coherent combination dominate assessment closure applicable whenever extreme extreme point set constitute instance style belief priori assumption gamble capture replace small context valid appeal use negative gamble condition one model de section set desirable gamble restriction event issue treat model section assessment share theorem lead natural assessment formally mu assessment combination assessment natural extension gamble interest coherent interpretation accept statement call acceptable therefore axiom status dominate status simplify closed assessment confusion status confusion possible coherent background background ar reject status gamble cone result relation allow representation variant regular east color size column rp xshift ex east west yshift south west rectangle yshift ex
environment figure implement classifier white list order receive validation threshold fix choose false relatively true empty begin black white valuable phase focus address list important whether identical informative derive feature produce correspond white list test arrival reliability mechanism machine machine na I decision bayes contain email randomly select w minute history present heuristic baseline window address threshold ml good ip behind place heuristic interestingly algorithm seem much sensitive compare work produce apply auc score highest whereas low ip configuration achieve comparable performance notice base match discuss continuous email high resource consumption learn white classify incoming email placing handle incoming mode update black activate save computational resource activation amount spam list spam execute batch update execute minute minute minute list update time address update figure superiority
simulation suggest strong property decentralize decentralized detection scheme analyze scheme discuss extension correlate support stochastic observe kt deterministic specify locally space coincide restrict process find time quantify delay formulation bad case conditional expect history word delay history know random cumulative alarm much rich class adopt idea detection delay kullback indeed measure q u optimality imply solve proportional brownian motion index drift exponent case imply detect brownian motion centralize apply decentralized communication constraint account detection center detection sensor center stop measurable transmission
recall immediately single trial illustrate semi reveal lead vanish contribution namely learn indicate fast explanation actually decrease turn original word learning randomly choose integer circles monte panel leave time course trial associate single mind write expect sum provide expression evaluate number trial double vanish expression term dominate decay already scenario interestingly exhibit monotone decrease selection increase fix increasing must increase maximized simulation effect word guarantee
target birth death implement target state association observation quickly resort approximate monte carlo mcmc different offline estimation tracking scenario prefer implement infer target approximation beneficial explore target tracking smc step assignment sequential proposal filter appear previously context da tracking essentially like implement mle technique formulate static contain discussion appendix smc mapping capital small letter density w denote write law distribute joint distribution expectation tracking surveillance noisy observation markov density paper specific hmm linear specify mean covariance x state measurement target surveillance target surveillance evolve transition density element target new target target create mean state superposition newly observation happen point addition
visual image transfer visual new category right available head body four similar texture idea answer support target common paragraph transfer task depend model random forest helpful task give answer question course expect incorporate always learn unseen task concern learner fail lead event happen life friend sometimes word thing occur tb besides learn learn classification symmetric jointly refer combine estimation helpful especially task couple relevant classifier collaborative filtering mention
computed programming computing field binary graph notice factor partition pairwise without form unary factor appropriate without set accordingly due
monotone lemma ne satisfying continuous ne triangle train n n p ne rgb rgb dark lemma conjecture note school electrical engineering department statistic university exponential family density statistic mild result approximate family propose exponential distribution address degeneracy typical family estimate family prominent role estimation family unbiase convex know readily non kernel convenient alternative number vector relative
handle subspace subspace find parameter addition proof agglomerative local affinity manifold local good build dealing subspace sensitive cluster resolve similarity similarity deal assume subspace complexity exponentially dimension subspace hence computational advance sparse low ssc rank lrr low build algorithm handle noise know principle paper study ssc lie union subspace represent combination infinitely express combination point ideally motivate infer overcome cluster choose neighborhood deal pick close solve hard minimization recover extend representation point unlike recovery problem basis basis subspace datum challenge program convex miss class affine incorporate corruption optimization experimental outperform world segmentation fig ssc union linear generalize condition connectivity regularization increase ssc motion cluster paper order last lie sparsity increase lie ssc motivate perfectly subspace generalize deal entry affine linear subspace lie subspace permutation assume refer number address consist find belong
show possible picture class role class corpus role usually frequently co occur role role see appear come noun come appear role relationship see together instance relational simple case blockmodel sbm role heterogeneity complex blockmodel single membership blockmodel
stochastic stability lyapunov deterministic force exhibit upon rigorous yet detail achieved proceed analyze detail discretize make poisson non ignore issue start reason indicate conclusion
uk negative media finding journal analyse see background fundamental usually refer collection reader social network develop content twitter previously present try resolve limitation generalise capacity rate social pattern extraction daily way able work pattern discovery task twitter content location method form second time tool thesis mm chapter fundamental theoretical notion start general research common learn short well bag present field retrieval vector extraction say statement notion learn characteristic experience previous probably make statement primarily human could machine computer definition past year broadly develop program play intelligence without address general machine respect measure experience machine quantify abstract improve performance several learning task one try identify primarily root machine quite year basic category reinforcement learning scenario also target response target trying discover content reinforcement instance discover try mainly unsupervised effort description theory use variable variable case break task independent know value dimension one intercept observation mapping function hold target know ordinary ol residual sum eq intercept derivative equal derive ordinary square ol solution interpretation square affect appendix ol generate predictor something ol equation regression know control dependent shrinkage aim main many correlated correlated shrink coefficient impose variable sum sparse still exist predictor hybrid approach disadvantage discrete quite often reduce ols define refer norm ols ridge close accommodate identity top diagonal way add process norm encourage sparsity offer interpretable maintain solution retrieve know lasso optimisation eq control shrink easily control transform ol nan moderate choice initial regression regression programming estimation efficient way achieve enforce modification coefficient coefficient direction joint coefficient current predictor become zero variable square direction iterate predictor compute residual explore predictor direction predictor variable active correlate predictor continue along least fourth propose drop arrive nonzero value assume predictor perfectly probability sign mean sign correlation relevant irrelevant make weight infer pattern differ fix relaxed bootstrapping categorical recall target represented category label label representation structure primarily later extend regression sort top label classification depend figure media content web high score big risk epidemic actual tree whether epidemic simplified instance middle tree classify yes likewise classified assess manually practice mean supervision order make application text mining computational failure diagnosis medical scientific intelligence construct automatically tree past tool chi automatic detector quick many briefly chapter interpretability self explanatory easily understand handle miss tree rely make create decision np apart optimality instability minor section cart sophisticated program et cart cart capable great established exist software package implement tp four terminal entire specific making similarly previous tree number perform recursively splitting form cart construct binary tree divide divide variable observe use variable create branch core cart subset set subset subset decide optimisation response terminal regression control denote square loss overfitte set decide version assess improve sample accurate replacement original bootstrap consist member appear appear appear bootstrap predictor primary sample element bootstrap deviation predictor aggregate well predictor replicate predict bag average version prediction operation bag bagging improve significant impact unstable predict bagging reduce problem test classification bagging interestingly necessarily tp give respectively unseen variable voting decide aim create dimensionality extraction method manner dimensionality find bioinformatics text speech processing aim effort dimensionality interpretability infer speed learn follow extraction selection specify basic selection analysis correspond primitive divided product indicate perfect anti absence rank important represent research version name modification effort cart instability cart able text stream use main collection material usually ir automate algorithm perform computer section go basic ir apply throughout corpus collection collection gram convert something structure receive form gram document weight equal exist alternatively count occurrence gram treat gram possible semantic random reason usually bag word inverse approximate actual similarly present tf bag count frequency normalise additional reference vocabulary result derive vocabulary
next mean equation call estimator norm concentration time algorithm ensure converge theorem represents summarize setting algorithm experiment synthetic report dataset employ extend draw follow multiply concatenation compose dataset sample c sphere affine concentrate nonlinear nonlinear roll uniformly hypercube sample hypercube band isotropic nonlinear face dataset
protein one incorrect illustrated fig ground problem already absence model
segment enumeration member guess least enumeration relationship may specific either eventually enumeration output hypothesis unbounde finite enumeration initial segment enumeration even code family
machine pattern terminology classification construct score univariate object assign know perfect score great support way measure rule measure complement discussion complicated classifier crucially affect classification tend object tend class receiver
mini achieve neighbor size mini batch lead mae approach outperform art randomly select mae bad demonstrate test surface surface well mae correction structure b minus l scalable collaborative large pp signal decomposition canonical pp em multidimensional pp lee negative matrix factorization regression shrinkage via zhang benefit pp v compressive inf zhang yu automatic
arise g iterate provably consist split sake clarity without generality backward close proximity adjoint operator dependency iterate drop formula allow compute iteratively
come exhibit equivalent normalization say satisfy obtain norm actually norm amount aim ai belief idea express message term belief start rewrite ai ai ai ai conclude whenever belief bp address series viewpoint look message belief
sp finite interval lebesgue c n later become infinity place restriction coefficient approach continuously differentiable th vanish slope allow run zero smoothly correspond since assume hold penalization theorem assertion uniform arbitrarily slow obtain bind inferior property procedure expect subsequent enter estimating equation small rate might b finding demonstrate use scad encounter quantile section implement adaptive sub perform wise might desirable rather keep certain range quantile preferable quantile introduce kind adaptive far disjoint preliminary converge uniformly bound wants set different quantile different interesting present fashion affect problematic assume exist
give robustness independent positive number subset suppose fix consequently apply inequality bind inequality go subset lemma zero two bound identical element nonzero set taylor expansion hold jensen define lemma side sample assignment maximize likelihood close finite text variance bernstein bind finite population assumption locally assumption detail maximize label small lemma place detail note large must rewrite remainder combine f follow additional empirical result standardize student misclassification rate size deviation normalization di
x x give divide sample either direction previous jt subtle recursively r convention recursion sampling achieve x x x use conditional recursion subtree message surprisingly indeed look bp message formula algorithms ex however center approach several firstly sketch hmms natural obvious benefit secondly provide compact parallelism tree
area characterize si tie ti q increase every convex choice divergence symmetric hz hz scale prop tie test likelihood respectively mild work al eight
lipschitz x low bind arise growth closely resemble determine modify proof originally lower active satisfied decision translate rate optimum think around optimum role sequential connection demonstrate complexity active dimension first optimization precisely classification boundary dimensional agree intuition boundary detail find boundary minimizer feedback minimizer continue exponentially passive minimax know upper strong convexity popular theory decay
q equation rewrite order moment parametrization grow curve various value grow grow dash function distribution modify third f wishart come homogeneous area return heterogeneous heterogeneous complex I component respectively block vector follow complex c give covariance multivariate return area wishart use model nm center wishart degree freedom analysis describe order define area different degree generalize characteristic al extremely heterogeneous datum reason tractable harmonic
constant passive marginal concave improve bound provide capacity closely disagreement characterize active immediately imply concrete complexity literature purely agnostic noise active deal concave broader mixture separate deriving might interest match nearly unit nearly concave allow paper label machine draw space also class linear keep notation classifier goal hypothesis xy consider protocol passive label hypothesis polynomial label also access sequence make request label hope active
os os em em em lr lr inf inf inf inf inf inf inf inf upon decision graph analysis em inf inf inf inf inf inf inf inf inf inf os lr em inf inf inf inf inf inf inf inf inf inf inf inf inf inf inf regard statistic pose little difficulty exception present small five surprisingly
phenotype phenotype narrow sense list quantity three trait method agree suggest enough datum half training individual closely different family study split intra individual randomly roughly inter family intra intra share inter covariate effect size root square rmse bad inter family split phenotype summarize rmse compare measure give trait measure snp large poorly trait cd consistently outperform seem bayes perform similarly trend tend improve capture large genetic effect small environmental genetic background tail capture either insight perform illustrate benefit approximation superior performance illustrate benefit fix pre fit prior robust specification show difference method reflect rather fundamental evaluation suffice nonetheless competitive speed implementation effectively effect addition snps effect substantially additional snps compare causal snps scenario order fast speed consistent hybrid behave result considerable speed statistical hybrid allow small balance infer computationally tractable moderately implementation moderately equip modern perform wide range setting accurate trait phenotype consistently outperform two modeling despite burden believe general problem phenotype example control one treat status correction estimate derivation text phenotype
integer adapt state join standard build select matching whose range usual similar fitness place ga execute exploitation high multiply receive input prevent rule l logic fuzzy action continuous adapting es great trial performance apply fuzzy action ga continuous achieve discretized trial continuous give receive action greater achieve minimal none drawback predict enable true reinforcement component evaluation
autocorrelation monte carlo texture three model layer cd except unconstraine model generative deep projection visible usual expectation collect layer quantitative texture sample texture patch within test patch compare quantitative comparison model competitive constrain texture cut texture texture image set zero result frame fed gibbs texture seed texture evaluate index compare truth compare texture method consideration fairly competitive texture multi texture layer explore generative
researcher constant budget concrete example illustrate ball covariate ball r risk often hinge logistic let uniform value complexity set stochastic computation output satisfy see give remain classification increase sequence structure capture order scenario design incoherence selection however inequality selection form exhaustive simple nest oracle assumption complexity arise case may hierarchy time provide classification aspect similar expansion wavelet instance satisfying computationally begin class budget modify quite inequality incur setting whether nest hierarchy across hierarchy recall constant see notation give characterize growth penalty budget coarse presentation result coarse present uniformly amongst budget indeed class attention present tractable look assumption unnecessary hierarchy assumption
interpolation support etc use target orthogonal wiener coefficient coefficient value jump asymptotically variance estimate vector matrix gaussian n section validation predict distribution switch construct gap height major source include four pa pa correspondingly construct use polynomial basis validity validation assess surrogate experiment device device record point combination h comparison experimental aggregate respect pressure bottom correspondingly marker prediction combination graphical comparison pressure systematic pressure mean prediction always experimental experimental combination value calculate dash represent fail rigorous make individual failure four pressure pa failure discuss reflect existence directional parameter requirement illustration device consider calculated pressure failure bayes number failure
acceptable wide positive detect drift positive increase whenever change stream discard classifier decide two drift detector affect artificial detector give detector change occur polynomial stress polynomial require contain adapt control define similarly two ii obvious without fair choose give false example pc lda gauss dataset difficult point algorithm low overhead calculation overhead situation artificial contain benchmark literature point identity draw gaussian classification datum feature uniformly curve time stream detect estimate take gauss gauss denote gauss occur stream stream change
three compute three assess importance influential observation suggest kullback leibl distribution hide focus parameter sensitivity suitable problem speech bioinformatics contribution forward backward triplet extend complexity configuration show outlier anomaly must influential context prove efficient detection appropriate rescale influence
form leave leave identically testing block expect block expression block probable large graph correlation degree begin observe large suggest correction approximate taylor computing hard use let put large expand yield simplify set keep term give plot block model raise challenge globally challenge way consider divide pattern link tool belong impose homogeneous degree block
sufficiently self sufficiently evolve real activity node value memory system accordance fuzzy buffer nod fuzzy yield buffer associate bin weighting since neighbor overlap convenient bin associate neighboring bin imply represent show subsection tuning redundancy spread fashion pick buffer ultimately give correspondence ks implement discretize derivative notational activity consider discretized non entry read respectively value calculate correspondence self evolve discretized discretization derivative discretization distance neighboring turn relative construction far small hoc low requirement closely spaced compute activity process numerical differential numerically evolve differential equation discretization propagate boundary computed discretization propagate derivative induce compute th derivative fact free coarse th discretization construct grain th derivative free grain weighting match discretize limit fuzzy system linearity equation uncorrelate small turn choice node uncorrelated standard representation neighboring generate delta magnitude function signal neighbor remain however node accord snr make node
collection focus applicable model performance causal even relatively infer causal among actual consist interested multiple approach applicable lose output connection replace respective mean aggregate denote important exploit full oppose towards approach design
family avoid need representation derivative univariate polynomial basis function available recurrence polynomial polynomial division difficulty fall robust alternative require evaluation three recurrence preferable conclude analytical mit edu technology become optimization space gain formulate stochastic ii deterministic quasi polynomial model objective conduct partial sample estimator quantify solution robustness experimental crucial development understanding across careful translate financial resource traditional factorial composite largely explore response experimental guide particular parameter inference discrimination extensive endow gaussian extension linearization approximation analytical impractical power nonlinear design perspective approach foundation noisy indirect constraint heterogeneous focus experiment design experiment parameter inference design objective theoretic leibler complicated must use objective available design many approach simultaneous perturbation approximation gradient great broadly involve average latter must invoke deterministic employ gradient involve nest differential box adjoint evaluate prohibitive address surrogate expansion
represent rating set j x exhaustive co dataset result every retrieve close object retrieve position ndcg ndcg base distance th similar object rating ndcg criterion assess repeat generative size grow rating interpretation bar deviation set image ct
minute ghz intel core level level minimal level fine effect gp kernel set variance match gp splitting predictive fig analyze distribute location head spatial per brain field outside ratio typically collect subject describe noun trial concrete fall information single key capturing vary still model trial similarly maintain smoothness trial model trial gps norm mu pp training visual likelihood analysis independently sensor test run independent chain global search discard first burn result hierarchical simulate sec cut point cut point temporal gp optimize hyperparameter predictive condition time straightforwardly
individual claim book test set accuracy compute book source accuracy book set explain well share com tag box car cat child purpose tag tag manually tag annotate user aim error yield follow book precision tag among compare algorithm conduct intel ghz cpu gb physical windows operate compare converge reliability group reliability
allow recursively pyramid label reduction well look perspective reduce especially move degradation principled scale fine scale look interpolation act coarse label notation emphasize type contribution trick form directly exploit move multiscale completely specific application practically wide diverse family energy multiscale effectiveness multiscale interpolation poorly interpolation fail landscape principle aware compute variable variable soft influence fine variable hard aggregation case variable influence coarse label energy pyramid approximate decrease degree exclude solution scale exclude interpolation exclude energy interpret one aggregate assign low however energy aggregating allow interpolation strongly correlation variable estimation
size draw get size draw dependent guarantee hinge divide side allow every bernstein prop apply obtain x lemma appropriate draw draw erm able negative class idea spirit online parameterize convert refinement state let consider properly cover denote properly cover follow bound cover spirit eq eq clearly maximize note need unnormalized dividing use jensen q unnormalize two value unnormalize
indeed risk obviously minimize long find mild require prefer high soft svms remarkable bayes obviously minimizer much achieve finite vc dimension word classification asymptotic asymptotic bound statement approximation generate choice kernel entirely need remark distributional short result consistency hold replace practically reason fast higher enough
localize galaxy arc measure show quadratic root compare length blue peak peak turn closely little rd sequence galaxy green point map scatter shape principal pc branch turn scatter length scatter apply scatter next mostly ht galaxies arc mean root distance curve spatial change equal order arc alone principal along galaxy property along w shape principal branch galaxy eps panel representative galaxy shape galaxy arc increase bar spectrum normalize perform galaxy marked emission line line galaxy spectra evident slope spectra emission line etc increase band reach arc galaxy middle curve red dominate agree st b subtle arc length branch connect turn arc galaxy distinction galaxy galaxy appear st branch principal identify green galaxy galaxy st turn strong second branch nd happen rd branch dominate form green turn galaxy dominate th emission line galaxy st branch line transition interestingly strong reach nd turning become th
dataset proportion algorithm competitive htbp example well mkl scalable kernel scale fit plot per million htbp perform range example run transformation sample approximation reduce fit reduction help kernel time outperform column demand memory improve without technique scale well beyond thousand htbp minute hour computation scalable mkl indicate deal limit direct head comparison make make show achieve time achieve kernel implement
see autoregressive process assume independent normally distribute root lie outside ar quantity consume derive recursion v quite show decision recursively length put variable variance chart independent cf lie let causal ar thus eq consequently x possible calculate quantity recursively b x identically expectation stop put moreover chart point zero continue chart run ratio sequential process random know
selection represent artificial member gene genetic gene marker indicator categorical individual depend address bi distinction goal make orthonormal bi derive result develop selective development theoretical computational bi selection genome association issue least square regression extended attempt relevant definite group local tucker condition ordinary square uniquely define definition orthonormal lin however scale proportional equivalent triangular cholesky criterion become q transformation j x x jx jx standard take gram predictor recommend lasso study idea lasso van de reference therein show consistent variant condition consistency error matrix estimation group sparse zhang property group estimate group formulate considerable progress lasso version
inference lda inference objective different likelihood inferior topic soft topic I cluster cluster document topic document cluster document visualize project vb separate clearly phenomenon meanwhile document focus document explicitly often place one document cluster document separate document explicitly include extremely topic principle trade time sometimes frank wolfe provably decide limit quick reach ap small learned lda reach find case average problem stable
relevance hypercube observe stable stable dy u ingredient function agree relevance hypercube choose relevance hypercube hypercube sized contain relevance stable agree conclude variable structure conservative iff label contain understand restriction relevance hypercube tree restriction label discard far occur conservative conservative branching contain leave branch nothing prove branch branching express formula conservative contain hypercube constant
substitute compute inactive also apply extraction recommend pick candidate sort generate final recommendation htbp mapping match interest obtain recommendation similarity popularity item suppose user specify domain q previously category active inactive mention value possibility preference eq acceptance proportion especially receive observe recommendation preference eq popularity
paris universit paris paris france novel move trajectory build optimization cluster profile study superiority propose give insight cluster traffic become activity basis result serious delay environmental collect information road g rate
heavily illustrate accord rp reliable estimation slowly feature precise estimation simultaneously open massive solution projection rp five demonstrate order ii dimensional parallel fit core architecture include diverse signal advance acknowledgment european union social provide grant research european ec fp conclusion recommendation material
prior coefficient see effect numerical improve rank bottleneck ridge regression dimension inversion cubic runtime approximate polynomial log sense create light number pairwise input even generalization performance uninformative instance suffer perform exist selection review ridge elimination redundant unfortunately run infeasible nevertheless cross rr rr completeness include rr rr half decade book build add cross stop expansion unnormalize normalize feature deviation carry forward expand reason prevent ever million selection exploit inverse update previous cubic fix basis million million parameter default describe predict runtime ridge backward lasso give contrast rr employ cubic essentially pass forward add backward phase specifically construct consist expansion bound candidate forward backward add reduction square rmse remove rmse terminate add finally outer choose phase iterate parameter well processing perceptron output unit input hide single combine single use mlp runtime neural matlab matlab neural package support weight parameter determine
training create instance give w line determine test final sample take classifier experiment discrepancy regular average average average last perceptron figure phenomenon average explain increase answer found consider thus nevertheless still outperform algorithm worth drastically improve previous proof simple notion divergence work motivate specific
algorithm inspire estimation parametric generative window supervise learn responsible attribute permit anomaly univariate per attribute normality score combine yet univariate discuss formal pdf scale draw convenient bag attribute attribute draw classified task model calculate likelihood propose hold x dd anomaly aggregation harmonic normality respect set normality instance aggregate normality score especially computer security cover anomalous example anomalous rule learn responsible wise normality score rbf normalization distance function similarity smoothness attribute normality weight weight technique equation attribute entropy approximate density weight technique range lastly employ leave training use training first normality technique likelihood obtain score accord instance score univariate technique algorithm anomalous respect approximate approximate value need thresholded case identify system anomalous anomaly particularly anomaly measure attack expert
hellinger area analyze model distance tend closely observe law look immediate estimation discuss parameter present experimental intensity employ acquire nominal look scene dark area forest
increase cpu implementation minute roughly simulator output six allow fit use cpu example average cpu hour thus perform table implementation computation gpu parallelization burden large minute suggest complete implementation gp gpu although focus hardware gp ad modify give plausible gpu correlation substantially reduce chance want alternatively
therefore approach may proceeding lemma em let respectively statement unbounded unique extreme unbounded ii linearly inequalitie exist linearly inequality together immediately point unbounded statement extreme g moreover independent total entry satisfy divide rest active active total inequality number active independent due observe observe inequality one case similarly virtue relation similar argument independent inequality inequality combine conclude integer j follow let fy fy ji fy unbounded equal finitely cone follow fact immediately
basis intuitive intuitively segment segment majority trajectory relevant contrary portion trajectory play key formation contain devise adapt tf case trajectory frequency occurrence trajectory rarely visit length trajectory measure segment whole trajectory total trajectory contain segment trajectory attribute road finally compare trajectory cosine trajectory trajectory road depict put emphasis fact common road segment
give high decay significant boost irrespective search good weight learning rate decay able small size high momentum scenario stochastic apart dropout dropout give standard take network belief fine tuning backpropagation dropout dropout unit unit learn rate use constraint impose length incoming hyper epoch propagation decrease take boltzmann machine backpropagation field activation deep whereas dropout major encourage useful rely specific effect feature learn figure dropout simple look like whereas backpropagation difficult discriminative less speech corpus dataset evaluation automatic speech recognition american english read phone speech convert give signal need extract output target open library bank extract ms normalize
highlight model sm interest jointly shall observation four interest middle facebook two facebook name page status common update popularity four user interest mild interest four interest likely contrast interest dominate popular four interest page social page notably four internet topic topic start popular correlate differ topic dominate truth school especially house carry sentiment interest seem parent normally interest high topic contain self converse put facebook interest proportion range whereas intuitively coherent friend topic inter topic rarely interact whole inter weak art topic visual whereas topic political support though interest positively former fine city phrase like star
stem ep message kullback leibler kl divergence old belief refine distribute fashion approximate well message pass suffer divergence approximation force ep converge slow message poor function propose relaxed propagation relaxation kl penalization current relaxation message pass moment constraint datum understand also primal differ equivalent
partial partial feasible hold already hence trivial call feasible indeed apply provide euclidean optimizing denote call lipschitz specify property continuity optimal additionally tight follow px combine continuity basically state optimum moreover convergence preserve please idea give bound projection constant optimize continuity lemma lipschitz al l three lipschitz cauchy place continuous derivative monotone hence I lf lf ff obtain switch section follow inference mrfs polytope formulation optimize dual estimate devote formulation introduce notion
phenomenon strong predict music book movie sense rating preference node contact motivation recommender netflix team win integrated similarity similarity focus paper applicability leave future mf mf integrate solve second prevent value ex ordinal entry round value ordinal rating require matter long practice constraint compare
path distance find probabilistic transform investigate give aim distance special considering recursively discrete group infeasible large focused compute important database efficient query traditional top query score probabilistic formalize unified ranking database graph uncertain edge direct visit edge concatenation visit go node propose generalization path system framework probabilistic link adopt method relationship particular
purpose consider vector response coefficient error belong say generator negative probably sometimes difficult three measure al satisfied generator elliptical density reader exist compare model dt n thus nonnegative elliptical interpreted mixture normal sub class
heterogeneity bottom present execution execution spend decrease detection time execution herein figure method never fast follow note discrimination similar focus table consistently reach look window pixel field band sensor display enhance purpose main dark leave heterogeneous due edge detect five agree dot provide compare eliminate extract information use use assessment aim turn determined heterogeneous region heterogeneous five carlo carry criterion
background foreground success also video video extract relevant feature high vision limited success extract class object image regular texture corner texture speak surface texture projection camera low imagine rank
codebook term trade rate well codebook grow exponentially code superposition code square error computationally efficient codebook construct recently dictionary low enable base distortion code efficient distortion successive trade complexity parameter per encoding choice excess source decay fast decay computationally decode encoding rate source compress distortion letting rate successively asymptotically distortion rate encode excess exponentially successive distortion gap typical realize distortion distortion rate designing rate question give excellent exponent minimum code computationally code source
sampling terminate th stage n nn n w prove lemma consequence equivalent continue n sn justify suffice simplicity virtue q cr r cr prove establish definition finally proof combine continuity eq continuity imply q prove nu n imply law z z prove z contradiction show claim b bind b prove desire coverage lemma stop continue sampling rule continue n
work fire model control h university em lattice series parallel application carlo chapter journal pp model mit existence limit many book assumption however example small limit distribution unique class expand b probability
explore refer denote univariate discuss bernoulli possess dimension construct bernoulli realization py k simplify denote quantity bernoulli correspondingly bernoulli realization interaction order interaction bivariate interaction multivariate vector log formulation bernoulli member bernoulli fundamental link j j among addition natural parameter multivariate bernoulli exponential interaction define determine gaussian independence multivariate univariate bernoulli
follow matrix row diagonal concentrated row ensure topic otherwise identify distinct remain row represent distinguished minimizer topic observe unit perturbation matrix nonnegative one admit rank robust sum decomposition quickly verify factorization construct factorization constraint eq member solve feasible suppose let twice fact sum matrix margin ensure th satisfie ii r trace row identify identify topic simply factorization
variety closure ideal vanish model rigorous anti equivalence affine scheme variety encode property vanish polynomial vanish hilbert finitely many generate ideal follow affine ideal closure closure hyperplane closure cube dense require apply hmms visible state trick visible visible emission let emission trick homogeneous homogeneous homogeneous find cut applicable consecutive process thus linear map write vanish use obtain fast derivation use parametrization parametrization describe moment coordinate generator motivation generator give short algebraic geometry cut show generator costly operation generating find instead make moment moment particular binary algebraic subscript index view probability zero provide
comparison mean coincide please involve statistic carlo programming language page complementary multiply sure problem subtle particularly example subsection black art goodness fit moreover require extra work principal require mean example draw small please although natural draw bin bin root draw bin probably useful determine discrepancy large fluctuation outside physical science exactly observe due due arise necessarily test suppose exactly fluctuation remarkably extensive introduce modern bootstrap test separate arise related homogeneity contingency goodness square log latter three member divergence bin bin probability draw respective bin actual draw fully notation classical pearson statistic g hellinger draw number draw distribution differ carlo draw testing hypothesis parameterize
use involved however extend partially partially clause otherwise permutation variable clause clause respectively map clause color add variable versa color clause occur weight add node edge occur incorporate distinct cv weight connect node depict result colored generate f f state colored colored construct clause class set bound degree specialize color remarkable million set position exist message pass algorithm belief propagation operate belief operate approach contain receive
usage hypothesis year one paper computable namely coherence case group coherence define upper pointing bad coherence early central thesis paper coherence particular address measure satisfy coherence straightforward definition verify finally define ready state result group nc large brief significance
family parametric namely depend copula definition rotation extend overview density refer derive formula copula derivative partial gauss copula definition xx yy xx yy xx f yy f yy present subsection hold general pair discrete paper gamma dispersion claim count model distribute f yy approach appropriate claim zero binomial claim accordingly copula average claim copula truncate claim joint parameter gamma
make incorporate improvement accounting use compare property linear fmri decode superiority fmri fmri supervise decode rank behavioral cognitive assess specificity certain cognitive kind implement fit fmri activation variable brain
depend every rectangle I meaningful special give dependence ready use class thought expansion constraint remove time accomplish induction argument focus removal idea constant involve also verify exactly infinity every exhibit cover
weight end complement adjacency es exactly combinatorial heuristic strength understand theory exactly approximately cut thing correspond dimensional euclidean space tight duality reason span distortion minimize lot third return graph cut originally riemannian manifold parameterize independent node finally spectral method method deep cut connection diffusion see biased compute several procedure effectively metric third prove partition cut return big graph space distortion clearly graph structural finally flow consequence average pair thus spectral flow complementary relax different may flow connection know fitting connection discuss remain fail complementary base path cut guarantee reason approximation partitioning provide good atom understand algorithmic flow implicitly regularize analogous lead nontrivial eigenvector laplacian regularize accomplish evidence spectral filter metric place explicit impose nice quality approximation challenging suited algorithm phenomenon observe work highlight base
contradict minimizer analogy special since minimizer x b z set prove contradiction b contradict x minimizer eq recall definition dual complete sample easily formulation belong subsection remark firstly restrict attention loss widely use function state mx b ij x
disjoint clique compose clique clique tend exponentially follow scalar tend exponentially establish hand lemma subgraph give disjoint subgraph consist decompose repeat follow entry probability tend exponentially exist tend exponentially k provide sketch technical establish satisfy eq vertex disjoint bipartite vertex sample plant scalar establish multiplier assumption construct block block index consist one block equal implie assume semidefinite matrix complementary equivalent require orthogonal rearrange system formula force row orthogonal nonnegative choice exist system apply orthogonal ks semidefinite decompose condition feasible correspond scalar unique optimal subgraph tend remainder consist establish disjoint subgraph tend exponentially construction moreover series nonnegative tend exponentially therefore semidefinite tend decompose
assume cosine become tight prove instrumental incoherence among isometry extent subset column behave like isometry satisfy normalization subspaces two disjoint small satisfie r assume contain nontrivial meaning nonzero rely equation w lt tr u ft rl w rl c lt w lt tr ft measure role condition I e x e row issue define small column row contain identifiability span column simplicity singular say projection th allocate direction need orthonormal row r hold program exactly recover pair upon observe function sparsity increase nonzero element column sufficient indeed note nonzero role sufficiently spread moreover restriction place position affect one multiply side theorem coherence
simplicity ard widely finally numerically b termination criterion difficulty difficult reflect real wide range hyperparameter carefully work well synthetic dataset dimension input draw gp se variance dataset derive simulation relate energy dimensionality datum report dataset concern dimension test split dataset available accuracy prediction set standardized log mean square normalize mse always predict predict ghz core least likelihood routine toolbox code section investigate efficacy iterative investigate utility compare prediction hyperparameter hyperparameter synthetic attempt dd
nonzero singular component less comment context potentially improve propose concentration graphical lasso build selector neighborhood b several neighborhood deal begin neighborhood aspect marginalization latent procedure state differently require preference impose wise global constraint require certain size vanish estimator latent believe selector additional selector
leave repeat effective thus ensure spatially variety l l begin convergence follow series thus membership class primarily old h old sm time differentiable derivative old class sm require condition interval wavelet one fundamentally affect follow sm class class sm coincide provide still correspond function r p define pm r variation unchanged contain meaning whose expansion h class coefficient wavelet often always small height jk axis j argue spatially rate rough wavelet also approximate describe tm sm I correspond f
noise dataset world separable technique metric highly nmf local minima associated initialization issue target easy model keep memory datum distribute scale share distribute tweet factorize less exist lead unlike require near propose approach new identify correspond expand cone ray entire eventually figure next extreme ray pick outside cone project vector call residual separate ray find maximize call intuitively pick cone ray ray h geometric algorithm applicable nmf provide datum separability exactly
graphic section family normalize gibb similar complicated algorithm class central idea monte mean use broad come gibbs finite function deal ise false gibbs graph vast devoted gibbs instance
b q conduct assess estimate hyperparameter fixing advantage p perform label field turn posterior unknown couple assuming assess compare value present different separate finally comparison art report consider gamma model frequently pixel extensively apply gamma paper prior distribution fig density gamma note overlap experiment map single markov chain correctly jointly fix ease interpretation scenario highlight blue propose perfectly table mmse simulation column table display generate simulation scenario report depict realization mrf synthetic fig perfectly
ratio regularization ratio vary neighborhood prune ratio begin decrease become unlike omp perform ratio higher appear loss never drop although stable behaviour low interestingly become low norm stability come cost loss ratio nevertheless test ratio close value effect low ratio penalty elastic net logit omp include illustrate denote estimate surprisingly none algorithm attain small error parameter affect algorithm regularization high provide comparable versus regularization mid ratio outperform logit mid unstable behavior sampling ratio change change dramatically regularization relative compare ordinary test sampling ratio sampling ratio appear logit omp almost sparse logistic indicate
variable computation hope achieve preserve cost sg new call sag randomize incremental combine cost sg sag incorporate gradient sg sag iteration like access recent gradient example possible sg pass problem sag unseen convergence translate testing achieve see training context sg apply focus sag typically cost sg review literature attempt aspect sg despite year sg focus aware sg preserve sg give technical applie sufficiently discuss include depend numerical comparison base sag sg pass datum available accelerate sg review outside scope comment relationship closely sg momentum sg momentum
impractical accord wish improve follow approach specifically two easy obtain bound capture due eq use induction song al operator schmidt reproduce hilbert consistent embed lead parametric calculation admit consequently much allow across example test illustrate general reinforcement challenge work identify integral
test wishart distribution employ statistical homogeneous et derive distance wishart work present stochastic test distribution beyond geometric al use technique intensity
stationary various kind stationary particularly kind model piecewise stationarity receive considerable attention graphical influential source universal possible partition powerful modeling piecewise efficient live code partition result binary stationary provable redundancy guarantee recently parametrize prediction change environment predictor directly compression weight computationally efficient weighting segment distinguish towards contain long
sample bandit I denote bind one let hand union follow chebyshev x h old inequality entail always easy see indeed restrict computation tight factor clearly property dependency term estimator empirical show indeed deviation remark basis approach fix tail obtain
phase synchronization nature review w j specification journal analysis th ed read understanding transform wavelet synchronization essentially physical journal asymptotic partial journal business economic w vector analysis vector journal dynamic testing efficiency equation journal autoregressive var brain ph thesis university eeg base generalized synchronization var york n integration integrate synchronization synchronization u synchronization dynamical physical review economic population couple theoretical physics york chemical york schmidt stationarity journal synchronization introduction series ni bivariate generalization present journal conference david coherence application eeg g physical direction couple interact theoretic physical avoid physical review dynamical concept polynomial synchronization review correction equilibrium synchronization universal university phase drive integrated york study force numerical ed relationship synchronization physical synchronization study physical review synchronization physical coupling interaction physical physics p reversible transition synchronization physical letter explore synchronization population couple detection application review j synchronization integration gr heart national united biological behavior population couple r interaction synchronization power vector synchronization physics report autoregressive time p var shift time thm proposition pc lead powerful decade theory couple lead physics area
hard soft machine repository classify normal variable individual area hard compare table datum point vertex five hypercube roc right tree perform hard mean element independently actually use constitute error sample soft replication another describe constitute error summarize accuracy
fast ei inconsistent performance ei outperform ei simulator al jx p contour global show multiple global easy versus global criterion force follow jump neighbourhood precisely ei try minimize prediction uncertainty consequently large trial estimate global implementation point use ei criterion evaluate median global clear well ei gp spike axis benefit smoothness response suit slow attribute inconsistent ei select ei play role candidate simulator function give narrow dimensional detection would volume e exclude function simulator small recommend leave
bayes determine whether differentially estimate factor almost simultaneously go acknowledgment thank useful discussion innovation university institute systems biology road school foundation road china mail author express researcher conduct biological gene go sense representation commonly use rate g strictly control family rate prefer approach control false discovery
learn pp residual machine partially observable dynamical association intelligence parallel online repository kalman journal engineering search machine point td advance system process pp h gradient temporal thesis j reinforcement robot advance information pp advance pp predict temporal reinforcement learn introduction mit mdps temporal abstraction reinforcement artificial intelligence temporal advance processing system ari gradient r scalable time architecture knowledge interaction agent make prediction model journal artificial intelligence mit artificial intelligence use reinforcement represent wide fact dynamic may
soon project material c web figure acknowledgment work institute disease grant grant grant foundation discovery trial laboratory support ai public grant ai ai thank trust distinct accurately cell crucial system biological study rely protein cell common problem datum identify differentially present bayesian beta cell response strength across subject distribution propose expectation algorithm frequentist include basic change experimental gene single method specificity additional robust misspecification combination dirichlet cell population particularly truly homogeneous individual difference cell lose mixture provide single cell protein development flow since numerous cell
theorem accord q f r compute rewrite edu nystr om base impact approximation concentration integral improve error nystr om frobenius nystr om method approximate large
answer hidden state deal easier avoid optima em usual
iterate solve minimize q solve fix q binary usual erm minimization replace many boost strong weak classifier finite subset classifier operate sigmoid boost boost solve stage find eq express solve keep term stage ease indicate reject term close agree expression go simplify variable reject go stage last stage binary initialize hard classify reject use nominal classifier alternate accord sufficient estimate update state optimize follow way stage pass stage pass convergence input f boost subproblem boost subproblem allow surrogate equation enable follow thm surrogate simply smooth descent learner
generality fail concave vertex boundary remove argument vertex particular edge vertex edge vertex new minimal flat shift put reduce case increase step assume large maximum note ks j irreducible define order locally minimal flat tell consist width local though flat multiple cut graph
use website site static page site web usage methodology define come analyse representative trace usage long request total duration second assume human duration request request exclude come web elimination justify usage rather site filter status fail request percentage
human source color fall two category user whole require neighboring intervention texture segment reference method perform intervention pixel value statistic require case image voting confidence whole treat pixel intervention
hence asymptotic useful many answer find rate convergence scale decay positive write exist write corollary derive type majority analyze small specifically corollary rate configuration height tree short tree recursion decay tree let suppose apply fusion probability similar decay ratio fusion turn characterize total bayesian fusion rule majority easy obtain ratio sense minimize use derive follow immediately tree fusion case bayesian least good fusion consider convergence combination tree agent mention tree branch odd majority test since fusion ratio locally total achieve globally interest decentralize globally tree test globally exponent assume message threshold
qx rule define c also assume hidden version item u q u q x hmm triple inequality eigenvalue go identical hoeffding increase obtain figure utilize internet capture google gram popular token token thin svd vocabulary gram per obtain multiply
coordinate within distance coordinate conclude ne approximate fix joint outcome distribution good vice versa first hence prove intuition p accurate approximate fix proof sized complete prove
class simplex exponential function boost hinge consider half loss constraint assume latter considerably slow computation hinge multiclass svm relaxation require far loss hinge loss simplex code fx v introduce notation fx fx loss introduce certain notion geometry coding derive theorem moreover linearly class loss h consider simplex code decode misclassification dx replace
outcome risk recall formally motivate theoretically introduce continuously parametrize minimax compositional paradigm present evaluation formulation model discussion basic live input typical typically addition apply inductive bias select set present precede goal transfer learn good probability task task measure index couple notably equivalent natural classical argue relevant expect yield supremum simplex supremum assume attain simplex empirical motivate
f fx fx true inequality uniform ratio inequality literature q surely constant v surely take cover c j v apply satisfy j u part lemma e v f f f last relation bound f give h replace error assumption almost surely h v z together apply eq q identity
std std std std std x gp gp rapidly prediction area typically area view region low htb e input datum predict view uncertainty around region prediction area correspond low figure low predict view uncertainty around rapidly area view region region high concentration figure e entire region section view prediction constitute gps neural resource mm ms numerous summarize main trend figure follow figure result reduce may whereas result increase deviation ten fold cross generally block well behavior robustness nn kernel gp could apply attempt obtain simulated annealing quasi bfgs update datum approximate marginal comprise training computational resource optimization consume part attempt code matlab core processor machine core optimization interest across analytical gradient significantly reduce initial attempt attempt time hour
determine conversely vector satisfie solution minima must great estimate improve replace rough consider local minima providing number minima residual
mn optimum parameter conjugate function lemma task appear classification extensively latent consist key frame belong category include scene scene weather scene dimension dataset image include cat dimension real value histogram moment define discriminant latent split cc ibp f show ibp discover learn svm discover latent feature learning model svd weight svd initial finite f score achieve table illustrate use either increase bad ibp randomly run appendix detail fold cross hyperparameter mt comparable nearly much ibp winner ibp dataset interesting discover latent show value per overall category different example discriminative value per category deviation probability like process lead feature inactive semantic discover dataset rank interpretable f feature category f different latent mt real scene uci repository treat label assignment treat task dataset task per dataset education student secondary publicly evaluate various define score student school gender band school percentage student school percentage student school gender setup describe variable feature form student school school student school student school ccc ccc acc acc f micro mt ibp svm acc micro f mt ibp mt mt ibp svm mt
provide cost dl complexity replica method analysis plant solution learnable element sufficiently support encourage dl strategy non convenience simplicity plant constraint consider differ impossible main study critical constitute expect property realize energy unity stand
bound easily permutation function surely let careful reading show quantization term distortion vector quantization penalty see target look already stochastic ensure asymptotically use lyapunov lyapunov stable require strong illustrate adaptive algorithm law number hold sampler penalty robust robustness toy alpha alpha left proposal make sound characterization user show quantization introduce approximation examine direction arise would make post counterpart prohibitive model work concentrate modification probabilistic reversible jump mcmc switch inferential study stable throughout lemma supplementary extensively cover zero r lebesgue furthermore v p iff linear
latter dataset spike train lead spike train segment auxiliary spike huge dataset demand segment train whole load spike train matlab ghz intel processor calculation took take less matlab code spike variant www application spurious gain instant instantaneous matrix train e spike train intra inter dissimilarity limit instant instantaneous cluster spike train train ms dash matrix pairwise four mark green regular top spike association indicate train cluster ms ms eight last ms contain regular real spike bottom instantaneous cluster figure fall overall spike spike train change ms spike train four cluster b reach form four time reflect pairwise past distinguish sensitive th differ considerably fourth spike instant event compare column time fourth contrast regular spike time interval average future interval interval relevant past difference interval spike dissimilarity precede balanced spike reflect precede spike although spike normalization dissimilarity spike train spike show instant average certain interval continuous separate individual interval interval linear superposition matrix whole interval column spike train frequently cluster third spike
co establish recently requirement exchangeability exchangeability imply blockmodel piecewise also provide simplified approximation nonparametric inequalities blockmodel identify practitioner actual blockmodel main statistical enable misspecification effort devote community draw community find flexible nonparametric generative understand exploratory organize inequality base blockmodel fitting quantify collection clustering main statistical theory study recent sec proof technical fitting blockmodel involve partitioning block bipartite graph vertex represent person represent exchangeability exchangeability array permutation permutation finite set column bipartite broad class indeed unlabele adjacency discussion separate exchangeability one require result representation class separately exchangeable binary array array fix generate exchangeable bipartite generate ij yx separately preserve
example average adjustment dimension reduction ridge network aic minimum adjustment alone statistic substantial adjustment majority analysis gain performance obtain adjustment technique partial sometimes rank disadvantage procedure aic bic evaluation potential able optimum example summary implement expensive computationally square adjustment raise benefit expensive form stage comparative likely rank expect outperform dimension production clean may alternative adjustment work naturally usual variance trade neural high analysis original square linear zhang presence cause degradation perform adjustment figure beneficial quite suboptimal third produce improve network weight large accordingly weight avoid overfitte approach produce superior general primarily attribute dependent summary extra determine appropriate regression example complexity relatively parameterized manner primary directly give model estimate statistic determine quantity derive naturally statement make advantage summary example fully dimension substantially analyse considerably curse exist dimension apparent reduction
sparse pca introduce invariant rotation simplify reason clarity relative express ratio general center say find indicator desire test value throughout assume determined see eigenvector candidate behavior statistic nan key assumption relevant arbitrarily hypothesis discriminate spectral lot attention perspective rest section argue moderate dimension discriminate infinity consistency entry large eigenvalue discriminate alternative asymptotically high large behavior hypothesis quite accordance hold reference establish fourth condition almost intrinsic limitation fluctuation discriminate hypothesis make formal spectral moderate phase behavior regime phenomenon phenomenon subsequently qualitatively critical spectrum exhibit eigenvalue
action even moreover many case regret background definition arm mab present suggest mab policy evaluate arm
eigenvalue respectively analysis establish rate shall introduce call drive implement norm loss upper bound risk major establish minimax rate convergence derivation sharp design scalar vector spectral present directional low estimating along direction le method column direction match thus minimax convergence estimate norm mean minimax satisfie view drive ball optimally pattern easily investigate estimator perform paper closely connect grow literature covariance study sparse covariance rate see fan zhang convergence introduce simultaneously
study q recover iterative reweighted also solution np converge minimizer chen derive minimizer hybrid pursuit smoothing propose wang q chen quadratic smoothing trust newton solve nonsmooth special suitable al interior nonconvex minimization problem box suitably minimization problem regularize derive minimizer extend solve provide propose novel lipschitz locally develop solve problem accumulation sequence computable minimization dynamically update outline notation low nonzero stationary minimizer
r f x l x x l k x x r l since x f r r tr end policy relaxation relaxation drop polynomial programming also theoretically relaxation reinforcement generalization computational j min mode rl aim design interact maximize reward signal develop researcher make problem many field finance engineering end several subproblem rl high environment batch rl batch dominant generalizing usually generalize contain area poorly cover property nearby cover low area cover explain policy need system
bn bn k j k use repeatedly kn order iii arbitrary continuity observe sn sn kn p sn sn sn sn kn cn sn sn sn kn increment existence increment initial stopping imply diffusion matter verify second stochastic x diffusion integral wiener value value satisfy assume hold well n x x p db u ba u exist satisfy ds ds cn vx ds p n constant p x mt mt n note du sufficiently exist constant c proof sketch th trial project increment bx n consecutive trial surely dimensional sx jx bound jx c writing sx dx dx sx sx dx dx dx dx
improve cs effective extension demonstrate algorithm provide well original extension cs imaging compress cs powerful theory acquisition recovery attention signal field theory unknown inherently reconstruct lower need exist processing several imaging hyper spectral imaging acquisition sense hardware play role inherent year advance application collect repository also variety specialized solver complexity focus recovery wherein emphasis robustness originally recognize scheme statistically inefficient corruption model corruption bit error transmission pixel buffer image processing work address cs combine exist cs formulation convergent cs formulation solve problem computationally recovery overcome limitation robust cs robust loop thresholding share nesterov alternating
validation nonlinear nonlinear nonlinear generalization principal component pca straight hence describe detect important frequently process complexity pca fitting cause often limit pca sample pca find flexibility little flexibility follow trajectory flexible relevant datum illustrate fig neither
discuss provide supremum increase system result generalized supremum denote convolution basic paper predictor equally space na term variable formula fluctuation estimator cause instability consequence ill pose require regularization exclude value multiply fouri transform smooth function compact satisfy exact assumption analogue express h analogue classical manner reason asymptotically kernel example theorem consist approximation field region approach small high derivative thus bandwidth justification choice useful constructing interval refer deconvolution kernel advantage carry call rapidly corollary sake parameter application
represent expansion factor neural bottleneck depend lead occur fine amenable dp adopt representation broad transfer possibility basis locally transfer careful accommodate change extension change reward change problem mdps observable belief decompose solve solve mdps extend mdp framework could efficient tractable coarse fine vs acknowledgment support grant fa u sub nsf grant dms author acknowledge helpful discussion markov first expectation obtain equation coarse chain transition matrix recall state denote interior state restrict compatible state fact chain mx measurable suitable markov property stopping time probability law right hand third equality equality px sx partitioning bottleneck bottleneck state non follow mdp unique negative transition bottleneck start p enforce whenever define future keep track sx give equation harmonic strong markov h sx process time conditional expect reward ultimately assume stop discount reward x nothing homogeneity prove proof use second discount reward boundary coarse h k ap h h ks ap p ks ap ks r h equation bottleneck potentially bottleneck define another graph induce second reward calculation discount factor derive discount discuss stop well lemma solve define give depend q equality fact compress discount find depend appear side compute compress previously compress vice versa solution discount negative separate square still possibility edu edu mathematics many decision often multiscale together goal infer leverage hierarchical abstraction multiscale repeatedly mdps wherein sub problem mdps representation task within sub globally problem significant aspect multiscale decomposition yield task amenable localize transfer policy potential operator compression illustrative include involve reinforcement transfer leverage sequential hierarchical generally suggest ideally layer abstraction broad occur task divide dramatically collection small computation super divide approach hierarchical subproblem subproblem discovery multiscale fundamentally multiscale planning relate concept multiscale procedure partitioning repeatedly markov contribution consists incorporate within multiscale multiscale geometry reward play prominent role wide regardless accomplish multiscale bottleneck control geometry partition one result hierarchy distinct sub efficiently propose perfectly compressed mdp subset multiscale hierarchy fine compressed transition analytically macro may different original problem scale function fine successively mdp multiscale way computation restrict conditioning time fine scale globally coarse scale condition mdp behind contribute coarse pair coarse scale repeatedly localize modify asynchronous assumption per iteration state converge globally sub systematic scale plan reinforcement domain decompose distinct part part appropriate transfer sufficiently mdps framework transfer proceed match various scale appropriate solve partial exploratory preliminary definition brief overview policy dependent discount describe multiscale mdps concern consideration introduce multiscale domain comment open subsection well definition notation formally mdp see tuple consist action set tensor discount
proportion equilibria always take consideration selection issue performance proof constraint game theoretical formulation sec algorithm state te ne represent network consumption theorems ne te action profile ne meet reach player satisfy consumption minimize generally ne large satisfied
attack adversarial I nx denote dual corresponding depend margin point sequel low letter part submatrix attack whose maximally decrease accuracy attack fix proceed validation section develop affect convex ascent iteratively optimize initial attack attack current vector attack attack gradient hinge differentiable
smoothness choose enforce prior maximize law deviation parameter shape could contain spectra change mean smoothness prior possible discuss specific smoothness write operator form calculation hamiltonian get differ extra term denominator calculate inverse hessian hamiltonian hessian reconstruction use spectra moderate strict show reconstruction much shape strict turn smooth resemble color conjunction terminal option load package graphic graphic macro ltb lt lt lt lt ltb lt lt lt lt bp power spectra smoothness solid power spectrum blue dash reconstruct smoothness dotted reconstruction region dot line sigma hamiltonian dot hamiltonian give translate interval mode visible shown reconstruct hamiltonian sigma component spectra interval spectrum calculate note uncertainty hamiltonian case scale rough uncertainty power spectrum typically example intermediate package color option load package graphic graphic macro ltb lt lt lt lt lt lt lt lt lt bp residual reconstruction realization solid eq
outperform bm visually mae compete ray ever provide information present early stage deep implication per voxel channel cube remove spurious propose bm approximation reasonably intensity provide importance fully transform bregman divergence variable poisson implementation classical optimize square update equation illustration improvement direct simple simulation gap appear form transformation transform refer method algorithm poisson refined transform refined multiscale partitioning bm refined bins bm mention patch patch length dyadic multiscale adapting consider inspection qualitative
general result regret analysis suboptimal constant mention exceed introduce q inequality follow present q sum arm link beta binomial display deal binomial trial draw proposition care ts sequence bernoulli q let notation hoeffde bounded sequence nb rewrite ease
every greedy scheme probability arm apply chernoff hoeffding exist non optimal arm obtain tight eq great mean henceforth rest arm arm accordance cumulative choose well often intuitively scheme ucb current sublinear average time arm exhibit polynomially decrease every ucb outline chernoff
heterogeneity degree diversity rich natural step realistic fact multiplicative model moving effect require rich class property degree correct blockmodel network amenable understand affect property observe variability nontrivial variability summary model summary multiplicative marginal network move statistical setting establish p root function derivative whenever converge approximation lead convergent convergent write side expansion convergent expansion yield asymptotic form induction nk statement recognize power thus prove double index iterate application relationship denote regularize incomplete part establish beta substituting q appeal lemma upper bind integer incomplete gamma gamma apply expansion lagrange expansion bound recall eq integer respectively low gamma q denote
eq linear see flexibility linearity refer linearize efficiently programming ls regression problem control benefit computational extension suitable series naive produce coefficient convex validation albeit intuitive intermediate disadvantage linearize combination slide cross window sequence boundary separate window control performance incur average select set close integer ls illustrate slide window
ir argument lag ir function normal computation obtain dotted line plot option nan innovation alternatively autocorrelation discuss function reduce considerably calculated calculated h prediction benchmark observation bold horizon west evaluate series consider relevant implement test produce forecasting ahead default mention matrix empirical variance name name model illustrate performance
special observation treat second reduce treat near near similar example bias near attain interior point simplicity support want generalization obvious symmetric moreover integrated kernel approach interior close boundary smooth define prevent support upper boundary density next hide score eq notation function
little beyond either efficient normally ef extend ef concave global maxima result choose cauchy application mc adopt adopt approximation establish error affect long carefully suggest parallelization agent alternative rao scalability generally order comment extend preference synthetic world preference experimental suggest approach flexibility world aic observe fit predictive aic criterion p dataset
link affinity develop tie feature belong multiple feature logistic base membership node affinity entry base share derive task
segment basis optimize accord avoid basis full summarize approximation original clearly illustrate htbp get principle thank anonymous constructive comment cluster functional axis france objects analyse spectra
mean stop rule span stop early suppose stop denote denote prove loss element vector identical real also far independence technical corollary procedure substitute result lemma pursuit omp redundant complex complex noise recover representation extend case validity discuss illustration transpose
learn entropy skewness locally embed subroutine version zero denote rotation angle run preserve geometry sample distribution rotation angle object dataset convert object perform detection detect proximity image detect euclidean detect fig successful preserve geometry image pose perform group machine sample unknown nonparametric supervise state several well image show promising estimator estimator operate derive divergence distribution denote euclidean distance th neighbor dimensional unit define estimator q claim estimator apply almost equivalent rx dr theorems point interior eq away zero uniformly similarly away condition consistent state lead unbiased p continuous uniformly let bound
programming path vertex similarly pass allow delay adversarial sampling show depend low eigenvalue form x symmetric example binary solution consider product show associate edge nj nk ni j ni kf ni equation write distribution simulate sub exist might graph path relaxed exploit programming solution
predefine predefine indicate frank wolfe dual predefine bad batch frank wolfe early additional solver several value include average version appendix trick uniformly sometimes widely recommend mention text objective value exclude substantially solver optimal htb pass ylabel false area legend north east header col comma lambda l txt style thick mark mark col sep comma col sep include lambda confidence densely mark repeat header col dataset txt index comma dataset lambda ls txt densely dash thick mark solid header col sep comma lambda index header col comma dataset lambda txt densely dot style mark solid col sep comma lambda txt index header col comma lambda txt color mark mark index sep comma include txt densely style mark index false sep comma include dataset lambda txt scale xlabel pass ylabel primal area style legend legend pos north east header true col comma include dataset lambda ls txt style thick mark mark header true col sep comma lambda product ls header col comma lambda txt densely dot style thick repeat index header col sep header col comma lambda txt color densely dash mark mark option solid true col sep comma lambda txt index header comma dataset lambda txt color densely mark mark header col comma dataset lambda txt header sep comma lambda txt index include lambda txt densely thick mark mark option header col comma xlabel pass ylabel style legend legend pos north east index index header col comma lambda l txt color blue style thick repeat index header sep comma lambda ls txt index header true col comma include lambda densely style thick mark repeat header comma include lambda col dataset confidence txt color densely mark option header col sep comma include lambda header comma lambda txt densely thick solid table header sep comma lambda txt col comma txt green solid header col comma lambda txt color black densely dash mark mark mark option x header false col comma lambda simpler predefine original htb xlabel pass ylabel primal area log pos east col comma lambda product txt solid style mark x header true comma lambda header sep comma lambda ls confidence triangle table x header col sep comma include lambda densely thick repeat mark option solid table x header col comma lambda opt txt col sep comma data lambda confidence txt color densely thick repeat solid table index col sep comma data l txt header true col comma lambda style thick header col comma lambda txt color densely dash style thick repeat header col comma dataset x true sep comma include confidence txt style mark col comma txt table header true col comma solid mark mark table index header col comma dataset header true comma lambda color style mark mark table header sep comma txt ylabel legend header sep comma lambda txt style thick square true sep comma lambda txt table header col comma lambda product ls txt style thick mark mark triangle repeat header col dataset lambda txt densely dash thick mark mark index header
see fix term depend proof lastly optimal estimator contain orthogonality see optimal estimate argument deduce claim imply two rv see orthogonality principle hold course estimate observation result optimal equal optimal estimate estimate optimal principle prove attain expression since respect eq expression nonnegative minimal lower bind bad attain find complete simultaneously auxiliary rv study eq specifically set e x prove set estimator fix result maximize within comprise therefore last establish estimator attain minimax optimal complete rv use orthogonality principle orthogonality equality result orthogonal rv mmse rv form office research office advance national foundation science foundation grant foundation google research technology ac edu
unnormalized univariate optimize factor rather single regression regression computationally statistically factorize implementation need far draw achieve offer far equivalently univariate approximate implementation equally statistic rewrite condition normalize appendix property know approximated easy long give seed monte approximations beneficial variance reduction provide gradient update factor propose assume regression figure probit large empirically gradient algorithm present combine rejection sampler approximation sampler correct proceeding might univariate calculate transform sampler parameter cdf matter probit z might equivalently proceed something entirely mostly although offer insight second approximation option approximate twice differentiable hessian relationship use identity stochastic combine classical square explanatory variable alternative form efficient classical regression explanatory directly equivalently transform explanatory variable principle rewrite invertible may variational thus regression carlo sample case regression give finite functional hold immediately obvious general choice statistically observation implement choice
gd report evaluate kernel chi square matrix create representative group lasso recommend segment scalability group standard present class follow weight perform sift sift sift color sift norm see feature preserve fourier powerful approximation carry scalability extend approximated g square leave multiple learn accurate plan explore alternative basis li
landscape provide inference correct variant well heavy tailed edge vertex illustrate use correct present variation orient throughout denote edge number block block subtle problem model bernoulli belong replace connect result analyze control loop degree correct block multiply impose normalization vertex constraint connect block direct network call direct number edge impose rr ignore constant log parameter degree completely degree fit block generate community perform quite poorly
property degree rich entropy internet entropy internet tend form fully connect mesh clique impose restriction network rich internet network link internet variation node ambiguity rank rank average consider internet http www network evaluation degree link agreement
minimize j pm j tm verify approach normality j sufficient mu mp approach replace bind verify
complement restriction various literature rip analyze l problem slight integer say satisfie restrict property equivalent know convex still possible sparse strongly segment connect moreover characterize smoothness namely much restrict instead global constant vector verify desire inequality key whole convex behave grow situation along homotopy convergence constant key idea gradient homotopy solve regularization decrease reach nesterov adequate warm pg list algorithm make presentation regularization since l optimality condition control reach solve absolute introduction ls type relatively rip turn precisely exist l whenever parameter appropriately condition iterate path argument state convergence sufficiently dominate adequate sparse recovery pick practical purpose recovery closely assume exist rip satisfied
formalize notion line asset comprehensive exposition decentralize vast game biology physic asymptotic structure research begin seminal cover observe e structure extensively measure private surely use error likelihood ratio unbounded private private signal derive error rate learn previous node ratio convergence extreme decision learn ratio author decision tree right unbounded ratio private error study show truth structure review perfect generate sequence test know failure decision model observe certain channel decision node learn bound constant slow agent learn convergence memory error node immediate learn strategy
large scale linearly distribute pass consider stop alarm verify asymptotically optimal detecting pass mp implementation stop establish model exact provide otherwise issue adapt infinite thank property pass protocol neighbor node chain message pass message ji ji n ij readily available message use nk k let pass posterior root initialize message one recursively reverse direction edge repeat reach step base normalize let base n nk ij pt following produce value time contain asymptotic stop share loop divergence run I j
subtract adjust variance output estimate package plot different department university structure unknown propose encode markov much
major interface class define spline final sample determine analytic analytic segment define offset gaussian width test effectiveness classification knn briefly knn one knn zero else filter building codebook match label test near perform codebook label codebook codebook svm good classification dot class fit minimize obviously straight line synthetic poor expand fit nonlinear polynomial transform independent trick apply expand calculate square dot
systematically layer may ultimately center ccc auc error offset center layer configuration offset top level simply discard one generative discriminative wise evolution wise evolution see epoch center stable clearly discriminative inaccurate become eps eps eps eps eps eps eps eps eps eps eps eps eps eps eps eps eps pt
determination asymptotic aforementioned expansion fix symmetric eigenvalue complementary set refer coordinate either second q submatrix index corresponding expression convenience several stage jointly refer stage coordinate define relation select coordinate combine ambiguity choice set ng depend convention correspond large eigenvalue let follow ng c define consistent reference denote establish ba valid observe imply follow generic algebra equality inclusion I n technical argument appropriate large except negligible also note deal involve analyze set n na ab task section expand isolate finally complete section establish c nh g nn useful probability rhs going fix carry important c next q
sufficiently add side add side criterion substitute fx remark remark newton minimize sum nonsmooth proximal newton convergence smooth computed method tailor bioinformatics processing case result nonsmooth proximal proximal newton many bioinformatics loss necessarily penalty regularizer trace describe type convergence interpret gradient curvature method composite art problem relate project constrained type behavior application drop composite function denote nonsmooth part trust region accelerate idea nesterov family iterative implement use statistic processing generalize successively curvature exploit quadratic cost
scale property shall uncorrelate observe eq estimate r package eq q discretize difficult analytic close em optimize iteratively quadratic semi analytically eigenvector nu daily return bottom omit converge minimum rv alphabet remain p p low optima repeat start position select x usefulness find tool
million stochastic model stochastic hdp use break posterior allow word come topic beyond truncation something reasonably use something small node three specialized nonparametric nonparametric process tree greatly quickly center contain relationship briefly document k measure partition child stick break set think capture distribution obtain subtract negative level vector second level second coherent sub result cluster initialize tree variational v randomness equal document compare inference lda hdp give entire versus topic gibbs papers corpora experiment year vocabulary size per three corpus use truncation truncation number increase corpora child stop root share document five fold validation predictive outperform variational also corpora benefit document share big omit present stochastic algorithm million article page somewhat consider vocabulary remove stop
thank program support des de la france grant university bs wang present opinion agent network political opinion party mechanism sub form isolate external agent add opinion influence social political opinion compose numerous locate axis mechanism splitting take account decaying end quasi remain study vote carry keyword opinion political network approach understand complex considerable modeling base political
two capture market bottom formula normal european plays role china european bottom european subject quantile time posterior mean black flexibility vary smoothness characterization dynamic major financial plot regime occur correspondence immediately market show clear international financial agreement financial level end begin capture correlation economic lead rapid correspondence budget recent grow instability period financial save notice peak represent dependence financial change correlation economic peak represent begin possibility prediction new obtain quantitative financial updating present gibbs observation simulation smooth show observe log quantile red conditional trace performance characterization together improvement analyze usa bottom unconditional online ahead use prediction predictive step ahead merely prediction
micro tuple assume role centroid allocate functional micro micro summarize adapt keep choose much window construct allocate allocation centroid micro correspond start window centroid allocation appropriate mean window consider w jt f jt f jt jt b max w w functional alignment
assess adapt swap transition action low adapt adaptation choose change stability filter significantly practical hence well adaptation plot rmse incur action policy expect exp well bad action adapt attain rmse rmse action exp change action dot switch action comparison exp algorithm great improvement ahead decomposition ahead perform remarkably though parallelization show ahead pf even serial
write pruning consider example pos sentence tag real vector classifier output correct pruning value linear pruning find goal minimize proportion answer
appear lipschitz replace note need separable pick exactly immediate consequence one eq establish easily vary procedure arise via let j ji writing ii follow depend doubly remark otherwise reduce arise combination clearly doubly combination establish nice doubly nice regard combination satisfy sense h fx fx fx j fx equip j jx jx h fx j smooth proper let replace j expectation last establish proper uniform monotonic establish minimize upper update hx fx ix establish uniform monotonic call conservative make nice nice sampling thing translate monotonicity objective merely applicable monotonic let uniform let serial parallel monotonic fx h establish remain monotonicity fx fx point nice fx fx fx show adopt expectation happen identity without generality indeed inequality plug ready bootstrapping
consider suppose deterministic variable assume whenever realization general convex slight variant constructing query away always handle depend uniformly rr upper describe may regression instance follow f td remark even set regret since emphasize result formalize plug improve calculation calculation subgradient recall also matrix back eq value require hold consider without term corresponding eq equal assumption distribution bound back eq generality play repeatedly choose
continuity fx fx v k indeed lemma argument give k w k fact recall k z hx hz side also condition optimality immediately em continuity set study inexact subproblem constraint subproblem kkt subproblem satisfy kkt inexact inexact programming
input update accordingly separable conditioning around feasible define q go feasible give column assign let rest j hence diagonal entry instead pick cluster ht rank initialize nx centroid large rw w k index identify identify cluster remain index I I j contradiction observation correspond individually follow must moreover contain normalize equation therefore iteration eventually indice small also
feasible either iteration algorithm empty solution single doubly discuss utilize ensure e discriminant function equal doubly active scaling accordingly adjust remain coordinate calculate straightforward computationally show reduce multiplication give initial identify lagrange correctly classify doubly active scaling necessary place accordingly sample place either c c modification follow contain tn take input e must like costly costly scalar
various illustrate aspect focus specifically co pick sign member differential evaluate baseline knowledge terminate epoch p favorable baseline dual average regularize regularization prox convergence method exploit strong square impose constraint converge trial observe baseline prediction ht ccc eps eps versus iteration tailor baseline remark keep henceforth maintain fair algorithm epoch square measure version epoch length epoch henceforth refer show large epoch quite rapidly phenomenon bad solution decrease rapidly though knowledge epoch slow iteration though exploit practical epoch remain relatively importance decrease ccc eps eps trial versus simple implement optima minimax optimal optima logistic attention idea naturally extend interest study development mirror method lead multi step convergence acknowledgement aa google fellowship sn support yahoo award appendix recall prox lagrangian lagrangian conjugate
pca belong satisfie assumption hold tool control ball agree factor case square fact distribution sharp notation proof dimension compatible define leibler kl measure sharp dimension greatly exceed size lead
relational predict property interpretation entity entity properly categorical mutual automatically interesting property situation occur consist relation job however recognize job necessarily capable correlation g principle language separate job look believe concern permit great job relation binary interpretation entity multiclass interpretation attribute interpretation currently database library predicate specify graph signature auxiliary predicate database read interpretation library except generation c incorporate interface solver set scenario grow task independent interpretation simple learn structured output domain naturally model e distinguish finally distinguish interpretation concrete extended task value biological activity water partition structure activity relationship formulate case handle introduce relation signature trivially sophisticated amongst task collective interpretation illustrate classic page four interpretation relation text r model relation represent classify possibility unary entity possible may subtle perfectly interpretation page atom page part connect category output study literature page independent domain movie movie entity relationship unary partial movie movie movie produce informative occur available algorithm space compute explicitly space explicit advantage deal framework undirected function directly kernel intermediate technique logical relational attribute possibly attribute tuple feature transform relational format kernel representation graph literature kernel
distribution marginal copula complete cumulative approach treat statistic marginal say tool extremely powerful relate test originally far complete description correlation
plot experiment face image image piecewise smooth signal pseudo admissible use normalised mean zero use learn noiseless aware iterate iterate loop iterate plot initial apply arbitrary learn show coefficient sample figure use noise formulation great suggest operator perform explore detail aim experiment denoise use operator operator tv keep corrupted face two setting bottom row visually successfully corrupt smooth copy table get learn instead db initial signal horizontal patch operator bottom operator horizontal line learn synthesis promising experiment comparative framework image previous patch reference overcomplete dct sparsity synthesis omp recovery convex optimum omp run additive
gene contain establish snps gene amongst set pathway subsample snps selection pathway model perform lasso matrix pathway select snp coefficient satisfy denote snps select snps map map pathway snp snps gene rank frequency open programming use module execution nonetheless considerable burden processor implement design increase computational taylor penalty avoid intensive addition identify pathway reduce follow final ensure place efficiency pathway major bottleneck generate entirely fit parallel strategy computer server node parallel server due study time exclude whereas job signature characteristic describe longitudinal voxel variation increase marked patient volume expand average per coefficient slope brain per plot voxel structural time point correct age discriminate ad cn voxel image showing extent voxel able discriminate ad top fig voxel extract final voxel value exceed voxel discriminative ad highlight bottom
considerably error compare average pointwise schmidt four square schmidt optimality projective projective numerical precise expect thank discussion numerical work project develop innovation education introduce let invertible matrix matrix substitute eq obtain introduce minimize eigenvector minimal eigenvalue eqs rao assume order definition probability py finite mix sphere obtain maximization rhs maximize maximal value multiply semidefinite take trace py explain unconditional reason call fisher know divergence everywhere part likelihood condition fisher unconditional divergence unconditional fisher however
combine region properly scope end multiple mode probability proxy expect jump move mean distance chain autocorrelation tend affine prefer autocorrelation practice evaluate short chain hundred principle go large issue twice compute return go particular almost improve acceptance fraction disadvantage large burn initial burn autocorrelation time burn together
smoothness follow extend smoothing stochastic stochastic vary nonsmooth part composite realization max stochastic smoothness parameter observation relate approximated ready algorithm nesterov smooth surrogate original analysis easier identify proper series achieve notation throughout center smoothed line alg apply nonsmooth rhs scalar could arbitrarily hence retain side
small assumption satisfy completeness trivially replace essential guarantee termination propose attempt reasoning restrict checking property incomplete may terminate present section learn intermediate termination teacher implement two check learner make teacher true fail teacher generate execution mapping thus teacher return learner fails return negative execution contribution towards composition enable maintain learner return teacher conclusion real succeed otherwise practical way negative execution terminate algorithm terminate learner use semi analyze reasoning describe partition directly max support distribution distribution total candidate state iteration complexity bad computing checking assumption well address leave work deterministic sample traditional partitioning discovery compositional investigate teacher partition
dx inductive ks euclidean infer inductive affine inductive hypothesis affine hyperplane else exist affine denote lie construction trivially generality assume pass origin thus x nz j rx jx jx dx dx j x terminate hyperplane satisfying follow dx nn hyperplane multiple every must integral actually upper claim theorem remark pt award california research fellowship california berkeley nsf university function degree since exact uniquely specify recently nothing efficient approximately exact approximation representation run integer previous weight also strong conceptually simple structural showing threshold close take take decade field machine theory weight boolean weighted majority game theory shall coefficient paper expectation respect may provide least surprising elegant completely constructive rise rough statement motivation briefly see extensive account motivated electrical engineering early suggest later researcher vote
digit base sequence string successively mention made order generate digit string produce digit string c next pair digit purpose string pass statistical test string familiar digit everything place choose imply equal obtain experimentally frequency frequency hypothesis consecutive order f f f f f f case case table equal frequency table show main confirm generator string statistical test string expression consecutive bit select string expression string fulfil reason
branch evolutionary time comprise substitution rise problem computation exponential poisson linear treatment global poisson computation inference inference model separate inference transform rapid availability sequence inferential cope inference increasingly procedure bottleneck modern molecular dataset issue align evolutionary figure depict evolutionary string string evolve stochastically branch tree computation branch make branch v string substitution star denote nucleotide circle substitution research paper markov string stochastic branch evolutionary tree likelihood far realization generalization broad class finite know dynamic simplification restrict case
draw posterior coupling time eq error test unknown gamma component possibly handle algorithm acceptance point ii probability occur restrict ii accept class mle run unknown prior denote common hyperparameter choose acceptance probability ratio class ii restrict similarly ii accept candidate q inverse gamma prior hyperparameter transition
might high hand panel censor solid embed contour contour full visually curvature family inferential example reduction effectiveness idea approximation maximum important tool manner problem tie understand relationship rigorous numerical understanding describe make asymptotic well exponential consider show level parameterization dimensional plotted implement figure become parallel fact describe linearity system describe geometry simplex extend advantage compute understanding variability htbp determine look model e distinct arise likelihood define hull non concave simplex work concern observe distinct probability lie euclidean projection pre image maximum likelihood completely likelihood likelihood map geometry see simplex space advantage explore exploit simplex exploit give simplex geometry structure
algebra theorem order finite processing achieve range simplex three set contain figure enyi argument well enyi equality first alternatively follow zero jensen whether equality partial enyi distribution equality note old sufficient extend let secondly r enyi divergence jointly order enyi jointly convex argument pair imply strict imply hellinger convexity imply ordinary hellinger hellinger integral imply essentially convex important result theory inequality kullback inequality distribution strong sense enyi ordinary set define simply generalize let q normalize normalizing normalize p bf fa jensen inequality side equal mixture element define eq cover kullback leibler example remain prove evaluate monotone q trivially imply put everything therefore converse dividing
find newton optimality kalman kkt newton method newton operation system scheme addition need extend smooth red dot blue dot smoother green outside axis specify previous kalman smoothing demonstrate efficiency kalman smooth function definition write explicitly follow form gauss subproblem decrease
follow incomplete framework adaptive policy prove explore policy conclude adaptive population successive follow respect uniquely determined assume period infinite exceed upper infeasible constraint redundant model programming randomize maximize expect standard e collection pdf program g basic feasible degenerate correspond correspond basic correspond degenerate degenerate include let optimality
unique causal strength causal additionally satisfy non treatment treatment treatment refer bound identify set causal function value function observable take must lie like offer little understanding develop inequality find violate contain model otherwise treatment see even demonstrating explain determine def seem parameter infinite see distribution satisfy condition begin model look satisfied constitute bound associated optimization
continue good performance experiment model generate flip coin retain coin let retain ratio repetition robust however proportion miss gs continue outperform additive linear flip coin evenly jx j repetition figure reasonably presence nonlinearity gs continue degree moderate say h h idea utilize lasso attempt exploit emphasis rare careful reveal subtle transition properly able exploit phenomenon ji paper framework similar design way technical devise current different gram matrix pick variable weak term challenge signal difficulty heart innovation gs screen overcome challenge objective consequently main use hamming optimality imply vice space corollary must note get arbitrary large gram paper rd state since ratio derivation match penalize optimality gs sure screening screen property gs line random ap iid signal independent design matrix model bernoulli gs really except negligible exceed integer constant p condition claim subgraph sparse ise another interesting model relatively degree generalize test proof simplicity connect subgraph must size connect combine em write short generality start update repeating terminate favorable configuration see arbitrarily follow index consider alternative keeping except perturb know know small minimum type ii hand ii
note discretization value vector piece conceptually appeal remain whether output regard discretization process reason connection measure space additional illustrate move differentially private vector function examine function subset covariance dimensional require family distribution differentially space finally result dimension sample database differentially field finite multivariate distribution demonstrate imply finally note borel privacy statement carry function lie reproducing correspond covariance gaussian simple basic definition first rkh closure combination correspond reproduce
randomization eventually randomization observe adversary end player observe require address adversary coordinate action information semi bandit observe coordinate observe variant optimization rigorously combinatorial repeat adversary choose draw adversary choose bandit measurable si measurable player incur loss observe full coordinate bandit try cumulative online time action possibilitie bandit bandit version adversarial armed communication represent choose suffer delay delay traffic edge observe delay delay
h activation visible reconstruct visible example distance bias visible stochastic cost boltzmann connectivity visible layer figure seek structure activation rbms train divergence idea represent optimally practice via visible hide correlation presentation successive gibbs proxy strategy use early stop rbms stack deep belief
second interest common term beta z beta alpha word document z alpha
large symmetric datum carry formulation implement task fidelity specification information point setup supervise real characterize smoothness fidelity weight datum vertex know understand example fidelity consequence attractive class goal address potential fidelity minimization attempt interface region absence fidelity term trivial diffusion final result well potential interface flow label class require
lemma I lead result summarize case pa pa n n b pa b pa b w n r argue exactly tail add remark section
q choice cca linear lda limitation cluster task object belong widely data retrieval bioinformatic unsupervise learn need sample many sample cluster certainly dissimilarity example human extend location remove similar show incorporate extract face manually face observe profile face individual helpful feature split construct remove common feature let perform cluster object detail reduction extraction consider link bss simulation total ten four column benchmark name mat six standard snr db first extract run latent signal good simulation simulation sir interface sir
prediction although adjust svm effect opinion error pay type svm first meanwhile important try control adjust appropriate control experiment denote radial type stepsize test fix type low prefer coefficient error fact error rate control second error component mainly predict whether
base pac inequality multi idea mix statistical give belief large information essence associate predict specialized appropriate pac bayes characterize deal upon majority classifier put lot enough bind classifier informative majority vote give support closely majority enter play normal different estimator
among list occurrence item finally define model expectation give marginal laplace l evy another application derive observation gamma atom random mutually distribution simple derive conditional form integrate total mass leave derive proceeding nonparametric ranking smoothly change ranking ranking gamma distribute random markov enable
row contain weight allocate smoother matrix name return tractable strength rate use mathematically demanding exist g analysis logit transform xy smooth back transform guarantee provide motivation model relative success transformation depend set interval relevant section eventually rate require substitute use package analyze
guarantee corollary exploiting focus observe phenomenon across notable obeys found diameter satisfy platform graph vertice graph vertex multi indicate multiply edge graph choice multiplier ensure easy cut coarse previous kronecker vertex distinguish cut scale strong explanation demonstrate figure graph triangle subgraph change noise end develop scan suggest feasible graph spectrum develop balanced kronecker graph statistically
q fact u u v u note order prove soon consideration since triangle equation u u conclude along straightforward acknowledgement optimal discussion regard van mm financial engineering university edu di universit di microsoft statistics microsoft com bandit contribution algorithm action instantaneous work show expert basic expert
argue property ds yx discuss ds threshold lag weight predictor conjecture ds lag behavior impose enhance discuss suggest orthonormal adaptive towards perspective double exponentially likelihood coarse lag address frequently nevertheless ridge enhance ridge lag ridge choice generality consider scenario group identical predictor reason obtain desirable bias predictor shrinkage sense follow claim lag
fr size estimate performance report carlo hypercube monte carlo mid numerical analogously replace expression histogram logistic v extreme logistic centre right case estimator high although bivariate exponent constrain estimate change regard well
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model couple area via multinomial age limit population level predictor presence disease health spline specification provide area within age denote unit variable latent area h refer area indicator latent small area pattern unit area weight belong class assume within probability take model multinomial one category status category enter aspect finally mean denote unit include variation explain include intra area covariate prior assume prior specification since mixed effect set fix effect estimate reference item fit describe parsimonious consider assume order class odd become equation
alternative sign integral put infimum attain gx dx right side slope alternative density assume integral sign alternative denote statistic limit see j hz
fully entropy semi supervised precision semi fully knowledge publicly available task consider seed study aid significantly supervision take rating rating adapt competitive sophisticated require jointly text datum competitive sophisticated world dataset star word star sentiment overall size aspect center star weight star sentiment publication identify sentence user aspect rating specifically review sentence maximize score rating setup motivated user prefer summary discuss aspect merely show increase supervision semi supervise improve segmentation high different user variety segmentation user primarily easy classify aspect correlate outperform lda lda incorrectly aspect sort rank cast aspect sake reproduce baseline figure aspect rank worse supervise classifier similar outperform
rl agent apply electrical collect slice model rl observation select step occur challenge state code convert variable encode policy place grid hz domain error monte carlo set size carlo sum rp rp number sample size projection iteration rp error projection result rp avoid include rp number projection interesting though place minimum error size relatively robust
method particularly magnitude truncation example rao particle degenerate report slightly paper space dynamical originally space decade research carlo monte carlo lead substantial challenge algorithm q static arise scenario model via marginalization non markovian particle family inferential construct proposal sampler sequence interested gibbs would sample procedure carry conjugate prior replace step metropolis smoothing difficult address smoothing final weight
apply smooth penalty single share regularization improve rkh term value rkhs multi task solution f k give functional extension substitute task give prior give j j distinct example ff df n furthermore optimizer express zero capture variance connection q data effect model task jx jx assume effect sufficient capture mixture mode like gmm maintain amount add group zero mean predefine kernel relate point z j know kernel relate shift periodic j j focus stand circular dirac periodic period shift mutually mixed model similar prior realistic cluster define function characterize j additionally denote
establish time fact next algorithm computation costly algorithm cost combinatorial np hard cost decentralize wherein decentralized indeed shall see decentralized armed problem abstract communication formulate bound unable computation let regret argue alternative define arm increment counter chernoff variable armed bandit problem per prove suboptimal suboptimal arm event make arm transition jt I time arm play replace chernoff get last expectation perform index allocation allocation update suboptimal allocation arm transition suboptimal bound player pick equation jt jt mab player select suboptimal arm jt
quantity calculate pmf less windows pc ghz dual processor pmf look like equal equation machine round indicate first roughly apply claim pointwise random monotonically decrease pmf support work trial error large reduce find lattice variable dft compute
technique packing packing follow careful formally framework provide consequence intuition conclude practical concern prove low major proof identically distribute pca look mutually uncorrelated coordinate maximal find manifold dimensional orthogonal least spectral orthonormal associate eigenvector must pca replace decomposition sample estimate lead eigenvector eigenvector additional structural necessary estimation notion appeal successfully context value parameter sparsity inherently coordinate coordinate span manifold notion term norm span wise orthonormal belong radius speak sparsity generate orthonormal notion orthonormal column coordinate orthonormal norm orthonormal unlike orthonormal ensure dimensional sub tail intuitively estimate turn
slow convergence qualitatively close additional impose either hold distance hausdorff next qualitatively nonparametric like either satisfy eq regularity equation hold parameterization need simplify follow give population n ij estimate assumption replace lower infimum take oppose notably dependent partial justification like posterior rate allow differ like surprising exactly extreme implication impose pairwise extreme point polytope interesting determine datum level hierarchy minimax optimal estimating support due elementary space hausdorff obtain state unbounded extreme within ambient irrelevant replace thick property corner support hyperplane angle form vertex lemma probably absence direct reference g lemma sequence
reconstruction projection level compressive simulation effectiveness approach compressive variety performance arise edu arise field image digital measurement signal estimate incomplete additionally corrupt constrain tv
convex hull subset hull sample identical solve minimum uncertainty intuitive consider uncertainty decrease sample solve constraint regularization depend regard define compare confirm way derive variable lagrangian lagrange optimality negativity lead constraint conjugate min rigorous min theorem gap label parametrize set let solution point min satisfy kkt hence solution condition nonempty nonempty z svm function know form estimating bias term boundary line segment vector base learn learn elaborate parametrize set way function set svm hull addition infeasible distance conjugate invertible truncate prove x yield though learning method conjugate q kullback weight derive parametrize associated uncertainty
uniformly think might preferable value yes appear find mistake cause inequality something inconsistent check ok yes arise ok phrase appear ji bit appear appear old think relate ok think current ok constraint support term sort restrict rip require effect least square term compare entry draw gaussian iii together q theorem bound regime
uniform proximity location predictor unsupervise bayesian modeling year nonparametric estimation infinite component limit posterior density infer mixture represent model modeling distribution popularity popularity generalization dirichlet process couple segmentation datum
combinatorial relaxation obtain interest automatically select connected subgraph direct acyclic coding encourage tractable formulation choose sequel natural parameter encouraging encourage sparsity function encourage connectivity negligible count cover support thereby encourage regardless framework handle us acyclic node node formally graph link every vertex arc path weight component present next figure cost choice whenever experimentally path start illustrate cost arc arc encourage path connect path code walk connection simplicity appendix follow issue objective question graph thick blue fill size draw fill mm draw fill red thick thick dot bend yshift mm xshift mm right xshift leave yshift mm xshift xshift mm yshift node node edge yshift mm xshift node yshift xshift yshift xshift v v near v xshift bend yshift xshift xshift yshift xshift v xshift yshift node yshift mm xshift yshift edge yshift xshift node yshift v edge xshift node yshift mm edge xshift dag along cost weight state sketch main first key component transform flow flow arc correspond along path equivalent cost look every arc decide
lattice detection ideally rate decrease thereby statistical significance datum confirm claim claim lead researcher publish reporting experiment fully various protocol system mechanic prevent ever quantum relation prevent simultaneously mention possibility read continuous toward critical quantum correlation would unlikely able go make experimental choosing help distant galaxy soon device still certainly subsequent soon possible memory opinion ensure freedom choose setting cascade classical randomness pseudo etc never super make proof ghz done implement claim experiment prove quantum outcome yet exhibit bar follow line local event another quantum mechanic arrange event contradiction probability observable event event experimental design assume exist coin course view super everything ever go world generalize outcome two particle possible mechanic mechanic refer different accord etc measure interested interestingly requires maximally amount quantify notion far formalism maximal quantum isolate example even repeat never like strong
exist positive function deterministic function particularly interested sequence function want say growth martingale mapping sequence stable continuous eq never reasonable rewrite growth estimate martingale growth rate suggest prove density kullback inequality theorem contrary statement satisfy obtain
next term I treat variable lebesgue integrable measurement belief lead convention symbol occur density density also interested equality argument argument argument measurement use tendency mode measure dispersion prior become simple gaussian rest prior equation square index behave dirac delta give eq assume instead general also understand diagonal ji j ir p power distance measurement precision weight general distribution eq choice generic choice possible whose measure decrease rational interest matrix degree practice instead systems note two uniquely discuss usually small avoid integral
nest v blind krige develop weight computer experiment calibration computer model liu bayesian liu dynamic computer k l model computer functional response element model w nd ed york efficient multivariate deterministic function computer combine computer wu f l equation evaluation computationally uncertainty uncertainty confidence bootstrap augment augment regular department nj school ga school institute ga conduct optimize residual krige profile dimensionality functional severe krige regular grid simplify complex gibbs regular kronecker employ efficiently
investigate computer combination carlo review relate well moving bring cost goal physical may least one center sum experimental interest function identical component fix pair basic kind measure natural consist function design correspond independently fractional factorial replicate second estimate involve difference three former great convenience measure integration estimate necessary interaction sum integrate kind integral product reveal internal exhibit moment combination
magnitude lasso note coding loop dictionary computational lasso smooth term top eq use cross validation iteration regularizer example regression marginal datum context incoherent way use bound enforce eq direction represent constraint use lagrangian update q incoherence constraint normalization least square update correspond converge provable yet nevertheless idea speed exist bi representation kernel similarity zero
probabilistic note inequality truly size oracle produce notable produce hold high later selection establish small small possible understand terminology see selector remainder small suboptimal selection successfully inequalitie precisely average convex combination carefully flat simplex define estimator refer theoretical call estimator lead choice choice successfully average irrelevant implement comparison inequality term suffer gaussian perhaps importantly limitation may technique actually inherent exponential consequently since expectation also paper
time large distribution sublinear achieve sublinear time time stochastic problem compact subset linear tail sublinear achieve action order adaptive cognitive exploit primary availability link dynamically communication nearby delay link model hoc
compute probability markov whose implement hasting estimator constant design choice q burn stationarity negligible alternatively period discard another estimate avoid simply essential determine influence concern control take normalize sampler control thereby convergence speed markov fast control control normalize variance ask standard estimator size normalize moment dx fx order close density play role choice let approximation explicitly close emphasis form chain
ab solve nd new york box c york l inference c theoretic interval utilize letter confidence
pair rand agreement index undesirable ari ari take classification ari also happen classification bad simulated site assess draw gaussian illustrate face mixture empty would expect mixture indeed merge classification merge mixture skew accommodate skew appear mixture apparent lin lee analyse anneal start mixture distribution treat remove
consider dynamical train dynamic gps point model synthetic ep ep mae report trajectory mae average kalman smooth ep iterate know iterate backward smoothing ep improve poor criterion uncertainty track period show iterate backward smoothing ep posterior follow indicate blue bar measure ep smoothing small infer track ep require iteration solution mean posterior match ground tb blue gray
throughout section triangle triangle lemma conclude boundary furthermore triangle unique edge triangle edge close origin argue interior ray namely contain note slight modification need involve rescale rescale instance consider strictly correspond intermediate lemma triangle contain vertex lemma exactly two edge triangle triangle consider column simplex problem arise modification issue plane plane nonnegative origin denote nonnegative vertex cone minor invertible nonnegative instance coordinate triangle vector plane nonnegative
orthogonal q bind side matrix copy q bernstein
likely visit visit state visited count unlikely patient day expected visit patient low fairly trivial case probability begin add reach infinity capability elegant transition time computational time dramatically parameterize asymptotic probability interest semi time extension scope applicability many state appear problem state methodology performance element find numerically number may program
recent researcher dl performance denoise merely reconstruction orient fashion explore information representative dl method directly discriminative coefficient discriminative simultaneously classifier discrimination name final track I face dl supervise dl discrimination dl organization classification use
reason obvious estimate company log unlikely large interested reader illustrative demonstrate apply complicated monotonicity constraint concavity constraint application present section illustrate potential section review method derive operator necessarily strictly convex discuss concern various discuss future generalization consider convex twice row partial transpose hessian penalty exact method way constrain optimization problem calculus sufficient coefficient make sense uniqueness continuity vary concern continuity foundation uniqueness unique minimizer gradient linearly coefficient path uniqueness sequence e continuity uniquely solve stationarity therefore continuity check sufficient convex function continuous strictly square solution
analogous article demonstrate effectiveness problem arise processing b show hmc variant mala diffusion space hmc superior demonstrate walk behaviour behave nontrivial acceptance dimension proposal summarize relate viewpoint many inverse assumption flow approximate satisfied image range condition item discretization noise markov mala langevin proposal straightforward fashion rearrange apply term proposal explain measure metropolis well choice pt desirable preserve sde invariant remain standard absolutely continuous carry proposal eq derivative satisfying advantage space scaling limit gap use limit recently hybrid monte concerned distribution conclusion relate proposal central see discretization mesh refinement demonstrate modification describe scale limit proposal variance stable mesh refinement gap idea gap closely resemble dimension via gap mcmc arise dimension demonstrate wide range lead respect field reference designing problem produce insight design
tree inference conduct various propose literature neighbourhood joint mle induce require gibbs popular enjoy mle strength provide credible penalty follow mrf framework offer characterization underlie gaussian prior ideal exhibit well unlike laplace unimodal apply context expectation propagation unfortunately mrf hard feature even intractable cause double nevertheless mcmc explore bayesian spike mrfs devise langevin sampler experiment show actual well
space location counter bit store always address pattern storage string intend different bit center pattern every increase counter retrieval summing address retrieval
necessary exact guarantee low extent relaxed graph broadly might exact guarantee either variant absolute estimate threshold selection analysis incoherence note relate
descent know converge geometric error require sufficient overall conclude naive processor previous involve combination stochastic ease limited requirement case machine approximate descent begin stochastic choose step stepsize project point convergence reader deep alternative convexity assumption require moment globally entire similarly hessian require nonetheless assumption hold common hold domain average gradient base sample result attain bad early circumstance difference th assumption correction base correction significant term arise bias stochastic trivial work trivial machine number split one specify generation choose similarly set correctly mis estimator variate independent experiment loss minimize specifically setting subsample stepsize give performance I split oracle cc machine across simulation square accord oracle line use plot set average number machine grow regression estimate return sample uniformly plot infer vector versus split plot red centralize gold apply batch make
good reach training tuning tune help note tune wise intermediate stage fine apply experiment layer bottom mention rbm train generative fine work practical optimizing simultaneously global hand architecture correspondence easy directly incorporate aspect distribution simple well formalize argument let fix marginal maximize likelihood might define turn know coincide former arbitrary improvement yield incorporation good point transformation fix point incorporation concavity maxima unique fix incorporation display concavity critical maxima invariant distribution concavity maxima incorporation incorporation coincide maximization em justification construct variable inference auto display remove ahead incorporation even generic map close optimum look look search sense simple already respect justification know thus describe arbitrary ahead incorporation though sense make close look look sense already good property respect model know feed maximization em increase assume approximate put optimize particular tuple sum algorithm transform optimize ok recognize em color ok recognize argument algorithm tuple independent eq display argument
infer individually turn depends actually result mode regularize level contribution novel greedy design target structure sparse matrix guarantee performance support subtle oppose sometimes statistically efficient convex programming case huge restrict directly recent via share approach paper regularize encourage conceptually constant multiplicative factor greedy
set symbol inconsistent formulae consist predicate interpolation inconsistent exist generate inconsistent domain p predicate abstract boolean boolean relate formulae abstraction formula map boolean formula map free formula specify boolean canonical normal lemma useful abstraction function predicate cube direction predicate canonical atomic proposition quantify canonical canonical conversely assume proposition canonical observe atomic proposition boolean since formulae query teacher teacher responsible two query ask formula answer query boolean polynomial query size target language basic term assign formula formulae loop sx ss occur positively give annotate loop formula
many classified area fully develop thus city com com com goal algorithm correct actual use incorrectly area mit clearly area usage political lead error may incomplete mistake examine prediction closely display activity cell ii cell incorrectly incorrectly predict activity dominant activity activity pattern find group support phone activity residual cell incorrectly group display algorithm identify share activity characteristic class reality examine potential cdr predict usage demonstrate show potential activity activity dominate life reveal subtle five category may aid resolution capability fine supervise
lda full dimensional reduction classifier dimensionality average describe previously interestingly tends keep principal lda diagonal lda account occurrence feature extracted name entity protein interaction task occurrence entity protein document extraction tool count name identify document feature occurrence feature publicly tool entity chemical name name clinical identify name entity dictionary provide li
prevent scale follow parameterization wish metric identity mahalanobis identity symmetric system step illustrate gradient always problem parameterization white everywhere constant express j many fashion bfgs rather rapidly
average realization experimental outperform recover moreover approach reweighte require dominate basis problem decade outperform find modification spectral subproblem replace subproblem constant estimate
service experiment maintain st resource bioinformatics conclusion recommendation material necessarily reflect foundation mm pt height laboratory mathematics science st laboratory science one university st di di automate practitioner collect statistic probabilistic previous run bias future purpose technique np spin vertex demonstrate even run yield multiplicative inductive experience
validity cluster pairwise distance sum small pairwise across whole value compact class measure separation inter index separation propose define separate experiment introduce report match amongst four study include index base introduce exploratory pursuit classification centroid group indicate structure closeness centroid view place close centroid score contain projection centroid separation histogram sum bin factor indicate separation choice bin size influence score bin dataset participant dataset task direction goodness point color name name ask
contribution row list dct choose well aforementione utilize obtain component even list achievable procedure snr well classical illustrative comparison depict histogram measure block primary motivate audio classify periodic prototype signal px c dct spectral comment broadly audio nmf examine blind
simple find assignment matching assign match illustration see obtain solve problem measure j ij dx amount step distance neighbor average bipartite many testing nonparametric consider equality asymptotic equality dimension minimal span statistic portion near testing wasserstein relation histogram histogram signature center converge choice signature capture wasserstein continuous distribution notion signature manner capture fine tv notion tv distance single similarity remain namely would partition last establish machine et base discrepancy choose mmd rkhs
order asymptotic reference could efficiency computation empirical improvement composite likelihood herein promise alternative bootstrap investigation considerable effort free include interpretation herein imply calibrate meaningful herein sort inferential framework shall elsewhere monte carlo approximations frequentist inferential procedure knowledge computational pay simplicity plausibility evaluation carlo estimate consume parallel cost principle problem nuisance infinite marginalization something propose make inference grateful comment liu anonymous associate relative technique maximal likely property plausibility emphasize establish plausibility consistency plausibility outside property imply chance make incorrect decision plausibility function assertion closure eq thing control sufficiently overcome process handle admit
tt adjacency introduce descriptor encode information matrix dimension prediction f qx ff estimate past row prediction equip norm n element likely edge evolve slowly change assume govern evolve smoothly network factor graph coefficient capture popularity refer centrality trend degree reflect factor set slow evolution factor unobserve rigorous statistical specific history formal contain factor approach study simultaneously dynamic collect feature exhibit regularity adjacency feature keep infer stationarity predictor neighboring graph function assumption notation trace root rank prediction problem square loss loss error different adapt separate x prediction denoise reflect predict assumption author minimizer subproblem presentation addition regularity laplacian
careful finding normalizing framework analyze interest broadly machine acknowledgement author detail manuscript ks would like thank research support ad california technology relatively refinement nf f eq technical basic randomly fix consequence shannon unit inequality packing sphere parameter satisfy eq vector distinct surface equivalently inner product end jj union inequality imply pack work eq product product bound question sometimes refer perfectly noise construction scale theorem theorem ad institute ks science engineering experimental support grant ad institute principal tool contain algorithm operate risk privacy paper differentially private propose new explicitly optimize output differ nearly furthermore illustrate dimensionality tool datum hundred reduce intrinsic dimension discover relationship transform greatly
red bit solid hashing loading average require hash lrr loading ratio course write inform sgd code provide dataset panel accuracy versus panel accuracy improvement sgd nevertheless epoch perhaps approach accuracy bit continue time recently overhead energy order learn truly frequently occur matrix due prohibitive storage signature resource successfully apply learn fairly efficacy become increasingly context search show normally highly document hashing demonstrate permutation bit hashing permutation increase threshold bit find similar address bit hashing task match accuracy random hashing number challenge able hash current gpu massive parallelism impact operation bandwidth
remove characteristics loss clarity quality common enhance compressed shift compression back achieve db compare state mlp noise sa achieve db peak peak bm db mlp corrupt mixed gaussian imaging imaging corrupt observation come distribute free follow take peak bm variance ii denoise bm example design design specifically potentially lead well result lc cccc bm mlp db db db db db db db db db db db corrupt poisson gaussian peak mlp peak example peak divided compare bm pure outperform state art many denoise notably bm patch reference patch denoise exploit patch clean image effect two ask image perform rather bm repeat structure answer train patch image sometimes average previously patch patch noisy patch size output one perform imagine provide patch architecture mlp hidden size training procedure note block neighbors bm choose patch distance threshold maximum bm denoise step merely neighbor directly noisy high employ
belief prediction detail practice state median value eq cdf invertible datum consider one fix reflect none overcome multiply mf bethe modify dependency global connectivity additional link selection calibrate fair ht width width ab comparison mrf stable favor well display approximate one project position set belief ise ideally calibration ise locate center help optimally curve coincide one hide generative compare generate traffic simulator fall gaussian version greedy reach level ise obtain encode full follow poor ise encode paper many case bethe start develop solution gaussian sparse compatible compatible simplify natural ise evaluation bethe regime hence keep compatible certainly basis schedule gradient efficient broad class simple extension local experimentally work support national research grant prop prop prop prop pairwise mrf problem inverse ise somewhat relate inverse mrf belief propagation certain approach bethe field solution pairwise mutual perturbation different way idea start bethe computed bethe expansion choose fundamental cycle basis loop characterize connect propagation pass problem refine regularization
perturbation velocity interpolation may design long evaluation derivative gauss update subproblem cg iteration cg student histogram final corresponding figure clearly inversion home ignore eq fitting modify lead interpret parametrized signature signature formulate index link eq student solve penalty refer velocity depict source percent stage use
relational meta inductive conversely extension elegant relational chapter devoted lp database survey powerful proposal conclude remark future programming root know logic form complex symbol mutually either quantify omit notation implication clause admit call implication implication call body intend quantify definite clause theoretic semantic symbol assign clause syntactic semantic know formalize proof logic inference derive resolution sound clause prove clause body semantic logic program accord semantic logic alternative none correspond view reality lp orient towards clause prominent database clause database treat restriction distinction predicate divide two fact involve clause reason
hide document view require additional assumption single recovery certain rank observe vector propose technique technique describe latent condition technique hierarchical model apply deep fig useful property structure include satisfie rank illustration latent variable tree path illustration effective central include independent chernoff expansion rely tool decomposition area cross moment word satisfy topic find matrix indeed impose natural degeneracy denote column establish column thus prove leverage dictionary counterpart ingredient establish word neighboring satisfy condition identifiability word exhaustive propose recover additional latent linear bayesian moment specifically equation acyclic dag correspond third excess framework reduce analysis spectral hide hide noise variable reduce ica latent observe original term analysis moment decomposition third direction exhibit moment recover expansion extract high depict view hide mixture conditional
alternatively acceptable om adequate tradeoff ht ht l budget nystr time cluster frequently accurate dataset easily cluster nystr om often quickly large dataset fast inaccurate efficiency run apparent face approximately spectral take three reach sample size increase preferable datum addition right use teacher run useful quite teacher cut minute would likely factor fine acknowledgement thank thank pure mathematic lastly nsf corollary theorem separate within cluster cluster utilize spectrum datum solving problem develop summarize application employ cluster likely leave company light cluster applicability
write situation probability publish search sometimes incorrectly problem subtle case q whether odd odd relax variable q necessarily search also selection likely write search uniformly random middle account search odd take course odd make I e last describe odd remove conditioning contain odd equivalently symmetry middle appear odd evidence everything odd situation see correction denominator fact bias case correlate contain write satisfy success independent give rewrite whether large role default population equal q default question weight contrary weight evidence consider background choice prior odd arrive reciprocal careful one want divide computation expert one face
correct come branch statement inner outer non combine claim hamiltonian claim know answer trial find terminate correct care even completeness use serve cycle far output hamiltonian answer finish proof cycle really correct infer return element multiply hamiltonian runtime give problem incorporate bound check accept hamiltonian cycle reject complete remark polynomial create simulator simulator notion interaction simulator cycle generalize invoke invoke simulate conclusion become time oracle verification likely polynomial oracle weak search give undirected ask exist otherwise u two unknown exactly want new chemical know chemical compound either verification pair may define strong piece distinction matter weak hardness result strong graph bind one low try propose knowledge imply bipartite node index vertex empty efficient try propose newly piece add edge decrease initially isolate finally stop trial efficient solve u come address question oracle list oracle look quite nature collection avoid exist whether process wrong follow necessity solve verification oracle solve solve graph clique exist isolated specific verification oracle return violate must polynomial instance clique clique isolate return thus find clique contain look output clique verification oracle returns edge first decision yet return design happen edge already clique contain clique least oracle
increase change drawback avoid analytically marginalization section marginalization analytically tractable converted expectation intermediate hyperparameter covariance slightly single hyperparameter translate estimator hyperparameter peak good derivation estimate mean parameter decrease approach region assess performance number design performance setting build complexity detail reference second generalize exist substantial unknown hamiltonian control evolve internal hamiltonian want experiment hamiltonian record unknown protocol provably protocol slightly generalize decay know treat normal exponentially whereas finite reach go remain convenience unknown generalization fisher rao utilize asymptotic approximation normal bayesian rao way smc numerically estimate way admit similar mean condition intermediate variable point entirely remove leave normal example consider hyperparameter identical formalize fashion dependence
popular particularly useful imaging nuisance must estimate imaging modification nuisance proceed wavelet nuisance parameter idea exist extend present imaging velocity smooth conduct
energie relaxation dd tight far multiscale energy coarse exploration allow avoid energy principled derivation coarse coarse fine aware interpolation multiscale landscape discrete preserve energy energy suggest hoc heuristic coarse
proof become nice note zhang propose weighted decay average averaging scheme whereas iterate paragraph exact proof much tight last
large structure penalize covariance interest many economic many lead parameter greatly exceed address frequently impose lin et impose al kronecker matrix miss optimization flip ff sparse call kronecker glasso dimensional assume whose separable kronecker channel wireless communication receive matrix arise recommendation netflix kronecker product graphical know time community variate normal study matrix normal n variate collaborative filtering kronecker factorization generalize fold measurement likelihood ml estimator ml alternate flip ff model order report know measurement structure graphical glasso glasso estimator idea glasso vary
efficient hierarchy set analyst cut dendrogram adapt heuristic frequently suboptimal construction merging decision lead refinement move cluster simple approach heuristic select switch lead error long move continue quite
clearly step whole exploitation walk entire fairly well regret well exploitation different explore property evaluate budget phase exploitation find maximizer evaluation property try remove many search search close ei distribute normal variable simply point write q point costly optimization find optimizer carefully maintain perfect exploitation area budget leveraging follow exploitation aim
way something frequentist course series method pose periodic fourier period could fitting parametrize however method one pose limited model type take uncertainty limit equally spaced parameter restrictive present von powerful considerably pay hoc suboptimal length cover whereas comparison mathematic relatively allow integral summarize rest paper early claim interesting possible summarize method hereafter arrival
nb explicitly become irrelevant construction treat nuisance interpretation fixing hdp group gamma process prior analytical posterior dr e cr poisson dr dr dr measure g continuous group component distribution express nb discrete marginally equivalently jk ji j n jk insight unite framework direct assignment crf likelihood would analytical normalize gamma model finite hdp may discard useful begin nb share nb dispersion alternative share construction distinct nb dependent scale thus draw
estimate value million measure evaluation unit capability year environment average human year simultaneous agent agent generation get local generation evolution divide start million efficiently start zero environmental year use evolutionary possibly upon capable small learn large efficient learning start
make ht extent system motivate feature moderately large numerical method model chain comprise respectively component fail evolution begin fail rate model investigation component failure express logic u failure show dash solid line highlight effect circle relative experience decay division step converge parameter type solid line make less agree failure fact include intermediate plot path close optimum course agree well exclude estimate variance efficacy
set markov ratio complete complete proportion direct number number report material length acc acceleration proportion mark solid box st rd direct complete box proportion near ratio near box box indicate rp box box respectively plot window window relationship number distribution maximum chain complete window approximately show chain increase solid circle chain complete size component complete box box accelerate algorithm accelerated accelerate fast estimate estimate four edge number size distribution complete line point window difference panel display second display maximum number structure three via plot accelerate window difference window top bottom component maximum component generate chain suggest speed approach nearly complete via study discrete show estimator chain second pp algorithm equivalence via proportion edge cumulative show reversible irreducible class markov equivalence equivalence class reveal undirected even equivalence thousand important equivalence class appropriately markov nearly equivalence widely calculate vertex potentially complete vertex complete vertex appendix approach proof definition notation contain cycle every cycle equal possesse occur reversible corresponding parent
alternatively span mean number markov use view estimator estimation computationally insight thank regard straightforward show assumption hessian diagonal noise alternatively orthonormal eigenvector multiply aa scale preserve product include special case mixture axis covariance align coordinate group view matrix kt theorem satisfy incoherence th large diagonal span basis coherence property partition formal gaussian apply coordinate partitioning orthogonal spherical multiplying cause coherence rank uniformly orthogonal
triangle become symmetric anomalous normalized distribution circular norm describe datum lift add dimensional quadratic initial firstly author neither aware make vanish describe section situation analytic although converge rotation
rmse theoretically covariate variable flexible equation f effect outcome maximum domain thus six visually effect nonzero represent b model group certainly lasso achieve parsimonious model mcp scad group mcp recall true carry design involve nucleotide snps version snp depend minor effect phenotype study mechanism yet detect important snps snps subject phenotype dominant effect include version variable ignore group snp simulation mcp false broad conclusion reach previous group variable either account grouping grouping ad hoc fashion mcp scad term mcp produce scad group gene expression genetic association determine snps associated fix mcp scad select validation analogously evaluate cross validate briefly microarray gene week
horizontal dataset variation scale ratio determine recognition accuracy equal extract yield unsupervised initial bootstrapping htb exp scale roc convnet report positive sort top bottom area measure bootstrapping phase sample scale desirable box file publish website version effort accurate evaluation curve discrete point instead compute curve sum area interpolation curve annotation dataset miss add modify code report fix auc yield supplementary material impact modification avoid
evaluation ease collect notational fold lp ng gx nc pp take speed validation reliable small subset setting motivate understand effect make estimate output ip overall error expect finite possible configuration approach predictor consider predictor parameter ignore minima test location test cross systematic configuration performance predictor configuration condition converge appendix counter neighbor prediction compare say section configuration full configuration hand suboptimal parameter correspond introduce fine grain see b indicate solid dash dash drive force approximation sure quickly uniform concrete see uniform convergence convergence minima typical train support subset training parameter sake simplicity see minimum rather one drive force still converge helpful continue discussion concept empirical denote error would difference risk class result coarse reach dash assume stay small extensively study theory know link dimension dimension hilbert mean small configuration complexity error error increase tend true become choose complex complexity ignore fast unfortunately make tight approximation realistic paper prove future work mechanism fast plausible look concrete example instead possible ensure sequential analysis learn body mostly focus evaluation already available reduce
upper going increase red colored center cluster panel expand cluster cluster voxel increase figure voxel diameter consistent accuracy exhibit degradation profile would smoothing multiple voxel exhibit experimental neither voxel contain informative alone informative accord voxel voxel voxel task response voxel take figure randomly deviation dot line voxel response dot voxel mean indicate vertical voxel condition correlate voxel denote voxel condition simulate response voxel uncorrelate response voxel simulation signal form response distinguish weakly response htbp scatter response voxel gray dot voxel spatial location voxel indicate single voxel separation either voxel text response voxel dot
organized computational tuning parameter example application devote present method multiclass computational multiclass conditional cumulative distribution furthermore regression function th conditional quantile define value various al chen add follow increase importantly k x estimation estimating specifically regression method ng li liu
meet assumption jump investigate appropriate threshold estimate population discuss eigenvalue theorem particularly informative combine corollary either term order rate still kk eigenvalue requirement take jump derive unnecessary would consistent eigenvalue jump occur identify dependent close fix assumption n eigenvector proposition depend gap eigenvalue recall denote counterpart select delta follow result reasoning make applied spectrum retain close counterpart estimate gap consecutive eigenvalue let define need hold depend constant usage plot outline eigenvalue retain separation population theorem eigenvector suggest small furthermore assumption plot
composite regularity satisfy eqs strongly r follow therefore proper loss canonical link composite comprise strictly factor correspond link multiplied function eqs satisfy proper composite desirable property convexity reader note loss canonical note link scale link appropriately proper canonical link r algebra proper strongly alternatively start scale proper define link verify strongly composite therefore purpose canonical
existence right value l duality subgradient subdifferential large value li li li leave singular completes give interpret projection ball admits follow multiplication ascent superior multiplier implementation detail admm implementation material estimation sparse low building block optimization dy eq respectively universal operator lemma material previous random value
mode localize sign indeed since consider always see general equivalent deduce rv consider must connection inverse analyze monotonic pdfs recall generic pdfs pdf lebesgue inverse version generate sample point accord note strictly clearly unimodal yield unbounded minimum pdf segment figure depend monotonic piece monotonic part increase mode pdf unimodal decrease combine inequality depend interpret localize switching region observe define write rv lebesgue subset unimodal localize generalized write target application unimodal loss region axis region describe observe figure b rv pdf region density compose rv simulate generate accord accept distribute distribute consider pdfs region represent note boundary eqs finally represent axes switching axis picture domain pdf partition form disjoint continuous even generate illustrate recall q hence note form monotonic piece region g introduce region target transformation bivariate relationship transform region also interpret pdfs piece monotonic transform region distribute rv transformation rv another must bound technique pdfs observation refer unbounded decrease vertical first pdf unbounded produce obtained evaluate general pdfs rv situation increase pdfs random pdf pdf high order
entry simplify property last come independence univariate rearrange factorization quasi whiten apply lemma give apply yield positive number diagonal positive root bb ti tb whitening b need fourth invariant distinct several tensor index fourth statistic statistic let I ic multiplicative imply estimate directly purpose analysis approach estimate fourth lemma
keep fix user statistical leave namely statistical recommendation method different statistical still discussion conference acknowledgement member statistic idea thank organization rich useful conference present progress team collection software tool implement modeling implement map within overview hypothesis testing estimation discuss development member root team exist extend root tool analysis establish readily tool compare
relaxation relationship analogous sharp invariant inequality iff eigenvalue stable eigenvalue separate set semidefinite exploit measurement next collect n refined measurement detail retain construct parallel valuable width depend signal ratio order construct rip since select ask randomization essential lead think game nature namely nature choose maximize choose freedom fix value take bad sx aim noiseless measurement rule large always choice state
design capture range grid en early corollary dms one link incorporate direction link reveal regard terminal connect adjacency exploit svd propose linearly efficient huge network benchmark network problem effectively interaction refer insight one interesting search module typically characterize link connect within practical direct wide web citation depth community community restrictive structure assume member violate main source citation present among paper computer citation restriction make symmetric community reveals completely turn allow different role detail htb comparison community asymmetric citation network implicitly express asymmetric social asymmetric community context web wikipedia drive play formulate pair community node connectivity quality directional community aspect capable directional aspect detection scalability huge network requirement regularize value edge community direct community propose theoretical link denote vertex source point community exist importance terminal community treat role community edge concept community ideal situation component undirecte
covariate random give give I constant real residual reason sometimes nearly covariance add diagonal change secondly produce add away gp covariance ard stationary hence achieve instead hyperparameter denote multivariate covariance prediction response depend response though variance residual taylor
reproduce kernel view measure choose design test strategy tool exploratory appropriate domain family arise chi square describe extend pearson straightforward schmidt clear whenever whenever invariant kernel broad dependence generalize exploratory take structured domain remark notion universal space say universal dense endowed universal laplacian universal due kernel universal direct universal also characteristic universal characteristic equivalent mean include kernel characteristic learn kernel notion translation acknowledgment b acknowledge carry university equally lemma institute mathematics provide statistic test covariance statistic maximum mmd reproduce space rkh establish energy compute mmd exactly
letter constrain would self natural several page character page character page alphabet letter exclude character would page consider character english page execute procedure improve character character linguistic predict context probabilistic generative sophisticated side recognize pattern patch simultaneously image invariance sift character handle add dramatically font separate clean text document heavily corrupt far improve sentence foundation early education research grant institute strongly manual remove letter character representation supervision character
circle node edge thin black em em sum em em em bend right east r west color black thick bend north thick bend r west north north thick bend east west color black em bend north em cl interact environment gradient optimum case concavity payoff gradient ascent ascent along extend j component ascent k knowledge state iii payoff state action equilibrium access unknown ascent feedback reward advance player project asynchronous differentiable sensor interested reader survey subgradient convex present application mobile greatly
many missing focus gaussians covariance expectation suit component non parametric e window put training make maximization training miss provide imputation keep em miss variable always like mention sensible absence gaussians attractive low analytical sampling gaussian train learn least gaussian use imputation observation generative distribution useful regard help discriminant reach accuracy contribution em novel inspection provide
subsection throughout develop method conduct experiment test inverse present conclude remark symbol dimensional respectively nonnegative maximization operate real cardinality number nonzero denote arrange entry likewise submatrix closed let normal tangent real denote identity positive semidefinite resp definite cone resp vector denote diagonal matrix element study order nonconvex necessary optimality minimizer follow condition assumption also local minimizer assume follow hold define minimizer condition problem nonconvex convex point fx fx x affine feasible exist local virtue know minimizer point minimizer second sufficient solution develop minimization show feasible denote follow definition imply conclusion
recently gain since copula investigate copula inner generator appear nest tractable copula arbitrary dimension family nest copula generator derivative inference interest able one nest copula monotone q completely monotone generator denote follow copula refer intuitive example joint say measure association furthermore construction explicit copula censor complicate argument possibly copula desirable parameter goodness transform see copula
also hence denote half consider real half obtain contribute part take simplify ib laplace infinitely zero term expansion zero infinitely real immediate expansion give real consider show eq constant ib xu xu x reduce ask
around compact showing provide average static mutation trial converge increase htb example update asynchronous parallelism topology recursive perform adaptive open scheme allow topology meet problem discrete ensemble asynchronous reinforcement evolve retrieve memory randomly update network match cycle environmental control network self complexity introduce dynamic necessary exploit solve mechanism adaptive adjust memory limit input topology term increase maximum content scheme boolean fuzzy logical collective ensemble asynchronous solve input reinforcement value grid previously research explore general complex problem wherein additional beneficial system investigation discrete dynamical asynchronous boolean represent discrete asynchronous fuzzy logic possible adaptive open number test
exception draw generate depend apply candidate deviation exactly mh summarize probability movement correlation number rw rw rw rw correlation mh rw rw without indicator indicate correlation point pdf true shape grow draw reference totally whereas outperform mh suggest generate generation reference always scheme independent proposal pdfs consider gaussian random walk try weight acceptance weight importance weight yield small produce standard mh drawn try target density independent pdfs try scheme
site addition simulate general site extend site neighbor site covariance herein flexible algorithm generate site field random site field proposition remainder next summarize simulation sufficient regularity singleton proposition effective generating proposition appendix recall notation state random follow site site denote nonempty neighborhood markov collection subset satisfy neighborhood q respectively concept definition nonempty generalize neighborhood subset site nonempty set subset subset contain one empty b nonempty subset exclusive detection two part herein denote bad unknown part know part nonempty site associate site connect neighborhood connect site site order unique subset exclusive choose I prove neither site connect rather henceforth site site connect site
replace finite sample type paper rx toward directional complicated manifold intersect near neighbor direction let distance fx fx type edge depend manifold dominate operator laplace away volume approximately volume depend scaling near singular tend infinity effect fourier vanish precise ccc reflect somewhat simplify think boundary quite operator example directional type singular usual operator expect negative think difference
analog function assumption approximated spirit nonparametric estimation seek impose graph call estimate rich still encode precision force graphical thus relax normality restrict undirected graph figure summarize family model think model regression marginal copula estimate transform jointly normal univariate fully model covariance glasso refer log model back least iterative optimization package implement inverse level multivariate forest acyclic estimate light yield nonparametric family acyclic two marginal sparsity tractable family
slightly small homotopy procedure utilize number homotopy step homotopy homotopy algorithm value adaptively follow update every homotopy adjust change support toward elsewhere predefine homotopy instead iterative observe adaptive reconstruction assign adaptive encourage remain homotopy step require entire recently level shrinkage reduce adopt selection explicit exploit regularization homotopy homotopy build homotopy numerical compare propose accuracy old warm well homotopy solve path decrease homotopy lasso homotopy toward final piecewise path length direction segment sign sequence analyze kkt optimality zero critical easy homotopy homotopy critical value desire every homotopy direction move calculate matrix
weighted extension direct vertex new analytic decade application technique vast hope come decade fs fs height pt pt fs mid fs scale rgb ed signal processing laboratory california institute david naturally field graph theoretic concept harmonic overview way domain highlight incorporate structure processing signal review translation multiscale transform brief open possible extension form domain numerous weight connect either proportional vertex graph signal bar bar example engineering datum describe disease census datum human pattern g stock brain image connectivity connectivity graph thus image view represent vision text popular supervise help label method build image proximity process reference method account graph signal way either visually dimensional storage use operator digital process question field
minimization minimization problem enforce per instance desire relaxation sparsity last decade problem entry aim find entry intend sense cs linear namely relation numerical competitive area cs error compressive imaging recognition enforce attract increase attention year group block kx non
since singular exponentially add attack exponentially attack attack attack produce estimate follow since hold lp decode attack exponentially fraction attack bit estimator logistic column solve I attack produce solve theorem error scale privacy mechanism add noise least attack recover entry attack decode attack mechanism add private compare theorem reduce full establish distortion attribute decompose k multipli adversary adversary every define solve analysis use discussion loss xy every attribute let get adversary q zero loss bound away zero imply
address regard general exponential bad complexity form desirable incorporate strength association interesting partially fp reaction would thank anonymous suggestion science university institute advantage search score bayesian informative belief network possible causal pair arise among novel operator advantage belief skeleton advantage former naturally prior since node way belief path correspond relation network derive experimental stem observational amount exercise week occurrence experimental amount exercise belief cause incorporate belief strength
denote cardinality complement letter represent matrix hermitian hadamard j k denote pdf dirac delta begin combine bp mf approach briefly algorithm slight modification unified framework arbitrary pdf grouped index denote index part refer denote bp mf marginal pz
present notion multi armed bandit exponentially reward prove optimality suggest conclude machine base slot bandit problem tuple reward reward associate objective reward crucial face payoff payoff payoff machine
chance category forecast observation forecast index nf nf nf I range perfect score total forecast category discriminant range score except fraction forecast random chance unbiased forecast notice rand correct correct rand adjust pair expect maximum index r nf maximum number extract magnitude observation locate team al comprise density function display cluster
notably elegant relevant allow tight cm green grey target rotation angle cm cm cm cm da cm european
apply prove meet notice v representation support mail fr universit sciences f france counter recovery orthogonal pursuit omp ols noiseless omp ol good first iteration sufficient condition dictionary ols knowledge pursuit least take overcomplete particular noiseless usually solution support suboptimal tractable approach one basis algorithm mp pursuit omp raise question ensure paper novel answer omp ols widely recent analysis
way assess correction often partition adopt turn consider course group close adopt effect correction cluster addition consider absence center variation factor procedure factor center variation course factor may correct effect unobserved correction replicate dictionary sparse dictionary minimize jointly convex obtain minimizer discuss many iterative iteration relaxation restrict method think paper require rank strength application new try heuristic three benchmark choice discuss robustness hyperparameter choose replicate former use choose model model ridge acting choose positive care iterative counterpart estimator minimize correction small iterative use experiment author r interface gender find patient assess gender benchmark affected technical microarray brain patient involve study send university ann array share miss lead control gene identical use non iid dependent discussion could reasonable result process center array type line iteration dot line since gene affect sensitive cluster gender assess therefore try different correct retain center avoid plot retain keep panel clear variance main
trait eeg signal brain consist subject pattern approach literature feature occur pilot always new deal nmf modify cost function standard
cifar also op universit deep machine probabilistic date simultaneous joint largely encourage training deep machine model view
normal specify x set regular limit u degree lower replicate leave illustrate correct confidence range fix bias respectively adjust limit occur contrast correction procedure adjust limit necessary assumption hence correction correctly enough require correction vary error although eliminate increase limit correction replicate analysis p cp cb db cb db correct cp cb bootstrap correct cb double db replicate empirical probability interval correct
distribution class plot accuracy use rbf leave lin lin experiment include kernel lin unnormalized rbf normalize rbf validation perform rbf kernel repetition rbf rbf degree figure value function embed embed kernel tend impact level coincide depict equivalence image invariant consider handwritten equivalence class transformation parametrize factor along parametrize adopt distribution virtual sampling task digit digit
try match margin time run averaged choice margin need projection small follow times rs theory optimize input lose svms apply combine run fast twice increase projection two vertical ten ten fold ten matrix four individual genome individual fine population individual correspond nucleotide variation across genome allele allele miss population constitute tp projection classification four level notice vertical since run sample broad experiment population classification nearly identical method level close strongly support main fit combine projection region population task fast follow rs dense seem outperform number
eq need background reader pick subset class subgraph fs subgraph subgraph index converge supremum satisfy begin measurable subgraph function stationary polynomially ergodic uniformly let g page page let satisfy continuously since define finish initial use proposition result proceed polynomially ergodic denote respectively equation yield composition q e corollary q eq eq
learn kernel nlp k fold tune validation function smoothed fold cv aim address two method hyperparameter standard non different cv probabilistic ls need design loo cv base measure function classifier optimize usage loo cv propagation approximation actually however problem ep laplace approximation ep conduct criterion logarithm predictive nlp smoothed nlp ep loo cv probabilistic ls classifier ml use method l real benchmark show l quite competitive method ep optimize nlp optimize hyperparameter nlp optimize bias experimental inferior ep give brief gaussian ml criterion loo cv criteria illustrate loo cv optimization aspect specific
capability previously unseen combination unseen set repeat variance use create final either directly via actual perform predict make effect decompose procedure illustrate variance figure predict seven previously cd size space investigate model accuracy model uci repository feature alignment e adaboost provide action tendency variance prediction relatively dense variance variance apparent explanation suggest future whether would create model height quantify ten coefficient determination predict value progress curve monotonically rise reach exception segment correlation via except consistently prediction direct prediction accuracy attain full ten diverse seven determination predict great predictive treat variance huge normal therefore pair test confirm level deviation deviation predict height new one five
function perturbation exploit unlike development blind deconvolution pixel sparsity gibbs sample blind alternate blind illustrate real sparse principle atomic scale demonstrate first spatial less imaging map atomic spin formulation spin force reconstruct force rely wiener whereas least square however parameter physics probe probe external unfortunately tune circumstance image reconstruction suffer image partially usually deconvolution paper problem hierarchical bayesian
mechanic study collective behavior present well comprehensive utilize differential equation diffusion mechanic except opposite justification number accuracy point represent continuous available underlie represent since examine evolution density movement transformation identify backward shift total merge perfectly simple dimensional location cluster flow law hence dynamic yield law law dynamic provide analytical evolution
develop book variational area naive computational respect small instance gps reliable important rank gene great instead lack fitting model thousand impractical fix problematic constraint phenomenon early detailed derivation gp stochastic completely matrix multivariate
close take cut alone chain rule jacobian e g formula apply set likely estimator mx hx show various exact column agree say generate procedure
pass technical distribute receiver interference wireless within probabilistic share information predefine propose probabilistic message message determine overhead interference wireless potential additionally technique extremely design receiver architecture could beneficial propagation decoding assume channel joint decentralize
intelligence task intelligence user user work call self question filter six select user content independent feedback semantic comment generate worker ask article bundle feedback article bundle
random classification near mahalanobis depth moment estimate separate multi regression vector alternative current substantially feature extend I rather procedure yield rather stable solution procedure class two dd modify procedure base validation linearize x x I exponent feature equal discrimination feature bottom approach successively new basic subspace straight subspace discrimination discrimination separate dd classify discrimination depth h restriction imply straight discrimination calculate two coordinate pair characterize angle indicator minimize pair next geometrically value straight discrimination accord far angle minimum angle attain calculate etc stop basic angle
image handwritten digit screen two rejection discard feature e solver screening screen run solver screen equally spaced speedup solver screen lasso rule solver dpp solver solver solver solver mnist run second screen cancer obtain mass index ratio protein patient patient cancer therefore patient image face image pose trial image trial digit data grey handwritten testing dimension randomly digit get matrix randomly trial screen improvement discard inactive dpp compare observe improvement discard row show speedup moreover improvement discard inactive improvement ratio discard inactive result gain rule mnist speedup gain mnist screening solver substantial four family dpp rule compare screen computational cost family dpp compare art screening rule flexibility family dpp see dual problem share similarity look projection onto close entirely
separate say nu wu wu find vertex nu wu w wu z nu w z wu wu wu wu together practically instance hard problem feasible toward computational even many stable instance seek dependency corollary practically locally stable instance theorem remark counter theorem corollary conjecture computational traditionally quantify lead
use ari bic superior array collect interval remove max min reduce gene select gene start ari choose factor factor ari great choose bic choose lrr propose parsimonious gaussian mainly intend term rather penalize thereby likelihood mean absolute coefficient scenario specify mathematical require cross
base relevant pde seem complicated answer obtain heuristic jeffreys simple algebra function pde clear extent approximation exactly problem idea immediately care regularity exponential family requirement drive pde connection describe lose case equally fisher build already instance familiar thing like similarity dimension reduction feature immediately choice bad may conditional focus know statistic general differential drive function I analogue statistic carry challenging section home statistic important space turn notion framework auxiliary variable dimensional subspace observe I take modification denote accord association normalize simplification helpful corresponding density I predictive giving example q plausibility corollary proposition definition remark step inferential free focus unobserved efficient I feature observe case simultaneous fisher differential drive validity prove admit I flexible variance bayes set validity fisher view middle bayesian approach influence idea current frequentist example datum dependent prior promise new I observe free fundamental idea uncertainty fully
proposal distribute correspondingly define couple hasting kernel center consequence wasserstein mh proposal explicit give wasserstein hold ball value contraction proposal wasserstein hasting easily wasserstein hold mh proposal disadvantage second condition euler proposal positive exist restrictive often satisfied minimum state ball r enough coincide moreover sufficient hold q consequence proposition wasserstein metric implicit transition mr theorem exist path generally modify suitable neighborhood wasserstein base correspond two satisfied h w chain proposal joint wasserstein r probability ball base lyapunov function implicit quantitative implicit
matter likelihood inference provide numerical mean marginal various model analyse estimate base analytical function frequentist estimate counterpart uninformative implementation abc analyse tail slightly short posterior wider obtain approach choice original incorporate inclusion unlikely would result affect conclusion posterior spherical inclusion additionally normal sample transformation conditioning automatic unstable density numerically importance transformation
whether task constructive avoid view development task view propagate along neighbor natural neighbor mml limitation potential sensitivity choose iterative algorithm great chance view formulation thereby success demonstrate algorithms concern multiple setting task view multi multiple feature description motivation performance task multi performance correlate contrary understand propagation task successful propagation idea behind identify success two problem optimally close alternate mml optima provide function reconstruct dependent weight independently reduce function denote function description refer lagrange multipli enforce sum matrix parameter feature respectively control sparsity construct place notice empirically indicate indicate contribute indicate goal miss fix weight learning relax classify rewrite format carry regularizer operation row attempt optimal method closed form inversion prediction obtain form matrix inversion identity final prediction unlabele instance
add entry determine obviously gray multiplication importantly diagonal position nonzero adjacency matrix twice entry within stack row equation execute backward substitution extraction matrix nonzero experience computational scale forward substitution compute second nonzero suggest format truncation non even display pattern image size similar image coincide sketch work objective affine minimization path move initial gray determine minimize call first begin parameterization variable path equation drastically simplify nan standard denoise ode equation principle segment segment differential solution solve involve yield nonzero path close time z early straightforward model operation end cost path approximately comparable
level scalar operator term encourage sparsity split ad experimental literature support ad especially exhibit favorable structure instance bi subproblem close per convergence extensive test indeed formal appendix consensus q generate converge benefit sparsity minimization far call application scale traffic flow change service user source network failure service attack service anomaly engineering network traffic challenge however load flow pair flow link traffic flow connect carry traffic detail traffic carry link measure compactly clean traffic flow anomaly load indicate flow order link counterpart traffic flow occur addition flow suppose anomaly give measurement goal fashion primary define low cf sparse pursuit applicable adopt anomaly p readily simplify general residual
hierarchical argument occur special hierarchy favor hierarchy word main likely small corresponding word focus main effect final useful distinguish call coefficient practical restriction difference substantial hierarchical split run top misclassification versus practical lasso node nonzero edge take argument extreme hierarchy develop weak hierarchy name think strong impose principle multivariate spline vector produce vector response lasso optimization one control relative importance square extension apply column deviation predictor center centering notion strong hierarchy situation want exclude grow focus generalization induce q onto structured choose nest penalty specialized penalty suggest penalty generalize propose lasso strong constraint interpretable remark penalty literature admit demand sparsity pay attention
convert handle computer viewpoint reconstruction equally space elegant motivate research mathematic brief introduction mathematically evaluate evaluation continuous endowed arise area term hilbert space reproduce product shannon band rkhs reproducing search shannon formula reproduce frame formulae reproduce banach frame banach space semi formulae enable continuous go countable remarkable however countable infinite computer infinitely raise question shannon lead exponentially decay often come
correct ssc correct subspace however computationally extensive develop generalize foundation prove generalized theorem invariant norm norm mathematical completeness arise popularity norm closed rank regularize connection previous prove minimize yield reveal true invariant choose suitable remain provably obey classic factorization demonstrate devise art computationally use although generally frobenius norm transpose pseudo state unitary natural small know fan frobenius three belong
form variable point correspond corresponding definition subsample induction subsampling process measure poisson another measure normalize measure measure evy derivation move thus subsample decade distance base take superposition induce operation definition complete configuration jump impossible z k furthermore normalizing denote argument subsample write subsampling equivalent divide use patch nm bernoulli evy mm mm mm chen mm normalize mm national act act nan computer university theory instance dependent topic model introduce background measure process slice
far sequence star principle accommodate expect poor fractional error significant rely full benefit magnitude realize p insight ap svm overall ap estimation accuracy spectra achieve surveys survey ultimately method spectra processing snr spectra bp rp dispersion make rough snr snr bp rp much large cover wide range unlike dominate far ap averaging trend restrict range address science structure identify star analyse three case mix turn rather rather ap science sample select completeness galaxy sample separation mix magnitude select estimate effective true range completeness fraction false high completeness contamination achieve preferable contamination star truly star accuracy provide ap contamination star contamination science k star star build proceeding table star conversely cost arguably true ccccc svm model star concern select star disk thing star define star four list separate statistic
apply completely exist dependent random framework construct covariate motivated flexible model minimize structure datum bayesian model attention learning statistic community us component feature model exchangeable justify believe structure grow challenge assumption maintain property original nonparametric process nonparametric measure index space know process nonparametric find literature disjoint subset normalize gamma process ibp known beta binomial positive exponential completely represent space dependent operation offer great dependency conjugacy dependent beta normalize relationship aid understand development covariate
operator notation work note notation stem boolean algebra page de law de several theorem phrase restriction theorem prove easy hence hold phrase phrase prove induction rather important analogue system se se se surprising establish follow easily disjoint set symbol q operator prefer sort characteristic transform change bit cube contain refer set partial note sort operation every pair derivative xx sort restriction sometimes translate classical sort close observation edge sort eq sort system apply shift sort require make sort
clutter type short occur occur type clutter traversal obstacle mat ern clutter length occur short length traversal obstacle level short traversal length mat ern clutter obstacle traversal occur clutter type form length cluster clutter long regular clutter obstacle window locate traversal occur windows length obstacle length clutter obstacle clutter length decrease obstacle clutter tends decrease obstacle short occur occur length window length length length clutter occur type occur short length short traversal length occur mat ern clutter obstacle short traversal type traversal clutter shape obstacle traversal shorter cluster clutter regular trend tend flat obstacle level increase obstacle increase tend obstacle number decrease shape window v window shape obstacle length trend similar cluster clutter likewise regular clutter ern cluster length short length clutter clutter window occur clutter v window shortest clutter window shortest occur window length occur hc w average traversal length tend obstacle increase short traversal short traversal length occur mat ern clutter short traversal poisson clutter type obstacle traversal length clutter shape obstacle traversal shorter cluster type obstacle obstacle type w shape window tend flat obstacle trend similar clutter w clutter poisson w windows clutter window ern clutter close obstacle significant interaction clutter reasonable interaction background clutter effect obstacle type traversal background clutter obstacle find interaction obstacle clutter main effect obstacle type clutter obstacle clutter obstacle form traversal length type obstacle short traversal length treatment hc obstacle length clutter short mat ern traversal clutter tend long traversal sort shaped shaped obstacle clutter type short occur treatment occur hc traversal occur ern clutter traversal similar clutter tend clutter traversal sort shape v shaped mean traversal shaped obstacle length
parent parent combinatorial fast overhead b mb parent proper responsible requirement versus candidate parent complete score memory need score parent store share efficiently reduce keep parent pick score search efficiency correctness illustration high high score reduction move record right half example entry assign large involved keep track original reduction memory store current involve entry entry among obtain original high parent algorithm ghz intel e processor
e removal u graph observation independence exploit variable rank property satisfie u choice thus neighbor find order criterion neighbor variable sufficient correctly recover impose find eq mixture tree separate separate degree component overlap model include establish sequel complexity exponentially relax separation satisfy property include propose effective neighbor neighbor involve computational complexity number test perform node conditioning involve assumption make dimension neighboring eq relax strict separation local separation decay refer bind local rank condition great satisfie arrange n relate allow grow cardinality sample also enough consider
continuity infimum drift iii conclusion derive sufficient ergodicity asymmetric volatility ask kind ergodicity ergodicity skewed purpose work ms support capital grant remark section
partition indicator complete group aspect estimation enable treat scale combination surrogate truncate approximate tuning dc decomposition nonconvex say replace approximate subproblem restrict provide update iterate indicate local dc critical routine detail section propose estimator achieve presentation utilize entire feasible group consistent oracle
determine computing setup reason take hypothesis approach extend problem address dissimilarity datum issue section dissimilarity estimate dissimilarity measure dissimilarity distribution dissimilarity measure divergence method dissimilarity contribute decide measure include popular divergence unknown cope plug definition divergence plug reliable novel basic direct density learn density intuitive know two know know necessarily estimate substantially easy density promise representative demonstrate usefulness principle avoid solve recognition density principle
exploration phase one obtain unknown large regret diameter necessary markov chain treat I next memory section stationary ergodic case asymptotically sublinear acknowledgment project european fp grant author currently science ks regret set plausible mdps lemma random identically fix state recall transition color choose transition state hoeffding union hoeffding color note action pair color write
fx fx gx reasoning gx note p inequality iv prove vi follow similar finally vi analogous variable define use bind establishe define k bx pp pp assertion vi ii note lem assumption bx j claim sequence iii exactly iii whenever vi corollary example provide quantile root behave construct behave uniformly coverage tend
figure detect environmental compare environmental create environmental previously neutral simulated frequency slope include correlated environmental dataset set percentage false base lm glm neutral fp contrast produce low power reject neutral simulated return bayes rank association positive number negative score parameter assess measure receiver auc replicate whereas linear performance value factor analyze genomic throughout great forest snps express sequence est individual environmental component describe describe fall environmental score great environmental snp great confirm snps snps functional annotation discover protein protein stress stress
original labeling book probably relational som combine advantage som organization computation som handle describe performance project either categorical paris fr example pairwise dissimilarity variant generalize whereas som rough relational som suffer version provide result bad article line justify several
limit arbitrary template binary background white disk white contaminate significant snr db fig observe spin perfectly signal merely guarantee spin manifold incoherent informally incoherence template sufficiently spin spin manifold entry choose incoherent dimension b spin intrinsic matrix rip spin construction rip combine lemma serve cover manifold low probability measurement cardinality upper geometry correspondingly specify bound measurement preserve em spin tractable computation example manifold input return location match fourier transform operation tractable approximate projection spin weak noise shot acquisition fig spike undesirable sparse
reach result centralize u I readily assertion centralize denote iterate adapt process evolve adapt auxiliary eq hence hence conclude hypothesis exist let note readily w tt introduce adapt evolve reduce also fact complete proof note adapt lemma observe hence adapt action conclude construct satisfie process bound event exist functional contraction application yield conclude contradict complete lemma immediate consequence characterization achieve simulate behavior example agent cardinality control pair stage one stage sample state action uniform simulate near communication link place circle simulate trajectory trajectory trajectory sampling probability past illustrate path centralize centralized centralized randomly solid factor among action consensus
great correspond number pearson coefficient value value zero space pair one orthogonal diagonal engineering coefficient monotonic sampling rank multi orthogonality measure variable particular orthogonal exhaustive point hypercube possibility prescribe apply uncertainty divide range value centre randomly couple result discrete variable quite simplification optimisation mainly discrete domain simplify space significantly restriction sa restriction different value needed level equal homogeneity result appear alternatively author mixed programming generate design evaluation criterion way sa intensive reason develop recent work topic design far genetic optimisation bound design establish
infer prescribe structure grey description random direct graph favor prescribe structure translate convenient replace extract perfect compression necessary perfect always amount threshold sample show partition overlap correct different plant block observe threshold lie region
work simulated statistical employ sec remark two issue propose testing behavior l evy stable evy simulate stable pdf stable mention variance evy irrelevant straight correspond asymptotic evy pdf two empirical pdfs visible fourth moment simulate random fourth converge infinite fourth evy stand simple present indicate former constant behave
lm lm verify misclassification n lx induce nn next derive classifier classifier misclassification sure kernel b converge sure pi rf pi rf dominant pi rf belong deal derive misclassification unsupervise classification fix assumption misclassification plug tight rf pi pi verify upper
channel expression ss pixel nonlinear simply pixel difference pixel neighbor hyperspectral organize sample angle denote matrix spectral channel diagonal element vary band easy product unnormalize pixel yield could unnormalize pixel characterize unnormalized likelihood diagonal adapt transform computed series rotation final computing efficient pca adaptation artificial treat equilibrium map energy attractive energy distance constant function pairwise shape result attractive force formation force adaptive potential act field thereby nature point nature distance negligible map I map range force map behavioral distance appeal intuitive property differentiable differentiable pair map e tangent inverse approximate decay unnormalized generate grow map unnormalize bound spherical field shape harmonic planning center origin force field continuou unbounded invariant similarity unbounded field map coordinate pair equilibrium strong force field search configuration unnormalized yield multidimensional artificial field elastic relate motion illustration force force mostly effective short propose entail multidimensional
break imputation observe social friend reconstruct information explanatory social ht ci ci yes yes yes yes yes ht ci ci yes l c yes yes yes cm de sa e sa author email
poisson binomial dispersion poisson tail alternative allow dispersion normal allow dispersion allow binomial less law unable poisson
find cone dual cone cone follow cone cardinality regularize least square literature example thresholding recently zhang propose penalty solve well finding unconstrained least successfully cone particular propose constrain programming sequence generate minimizer method finding method iteration local propose penalty dynamically accumulation local minimizer outline paper
use select fan li choice scad mainly scad simulation come fan last two size simulation examine smoothing ps ps simulation random subroutine software choose wavelet spline code code local matlab knot select initial knot factor fourth stable tuning present scad fits cm ps ps eps spline slightly cubic spline also observe penalize spline
attain subset observation membership sub application model select bic classification membership ccc est run comprise know run e label select even bic group bic sub cluster among three group suggest cluster introduce parsimonious model cluster latent structure explanatory parsimonious version combine constraint constraint factor likelihood estimation algorithm sensitive maxima dimensional step hierarchical utilize likelihood word artificial regard latent
subgraph valid define reach realize abuse notation allow universal constant essentially bound xy lemma condition along chernoff relaxation remain include detailed suffice achieve appendix favorable give dependent direct value preserve special satisfied possible define universal constant conjunction localization similar employ remain conclusion remain valid bind excess prove concentration via analogous base aside possible arise specification entropy preserve validity substitution derive constant essentially appendix preserve validity brevity detail omit validity theorem arrive except replace preserve validity theorem omit brevity computable note precede imply reasoning argue find remain replace definition essentially brevity running algorithm main maintain showing sufficiently reduce guarantee state request component follow application algorithm value upon obtain complete round denote h g eq total mh furthermore event I sm j eq imply next event j u j value base inductive event trivially imply hypothesis imply eq plug particular always establish j complete imply particular keep least er suffice theorem statement event includes state furthermore event note bernoulli least prove er trivially otherwise classic
part equation lebesgue constant depend theorem follow easy prove minimize achieve adaptively slightly finite reason higher high begin result extend manifold depend eq positive form case embed hilbert space orthogonal complement describe tangent bend map mt tangent orthogonal complement crucially case dimension complement k high greatly well analysis slow order performance datum key region manifold point flat guarantee vanishing infinity approximated tracking tangent space fundamental region vanish diameter going infinity unless take root surface curvature arbitrary k domain type link tight manifold bad method easy quantity less restrictive region curvature cause result present care take throughout constant constant
rational quadratic additive combination considerably suitable scale grid otherwise grid divide six performance simulate real affect finally simulate credible single simulated choose unimodal tail challenging separate figure model narrow gamma mode gaussian middle la importance la integration density median line statistically difference first mode plot practical interactive figure show correspond credible follow set visually la la mcmc leave log cv performance across importance laplace galaxy datum improvement perform similar slightly density galaxy tail observation look mcmc
degree gs similarity regression preferable tune competitive one bind difficult bind benchmark specific allele testing outperform affinity machine yield training build predictor represent represent bind bind function output bind affinity real training method represent output give lose energy true bind regression fundamental draw accord predictor small access assume tell predictor risk suitably call predictor predictor measure accuracy complexity elaborate tell subspace span q change efficiently computable feature induce extremely large dimensionality running point kernel correspond space normally value normally mostly denote product product space protein kk fs minimum coincide gradient vanish kernel ridge semi inverse exist
inconsistent domain transfer target impact help conduct fine grain analysis different target domain show first rating tail really grain cf without fine grain usually preference user method respect long case instance well rating user historical rating benefit avoid none apply extremely compose record result table utilize domain type domain observe compare book part information interest movie preference wikipedia record graph directly relate still netflix source observe wikipedia record domain transfer method transfer compare source domain
insufficient autocorrelation allow principled outcome binary instead combine sign undesirable one deal use allow outcome maximization chain statistic develop alternate change e degenerate method preferred software validate posterior quantile estimate move network autocorrelation present parameter suggestion complete paper process influence channel member message widely model assume social interact realistic large connect people distance influence refinement approach outcome individual explicitly method simultaneous residual eq outcome
base use recursive formula adaptation possibility iteration mh pdf choice well section seem correct ergodicity property future degeneracy first iteration poor collect produce mh assign among mean update
let consider node collect information centralize optimal write operation centralize across impractical centralized architecture highly rely fusion fusion filter distribute algorithm distribute algorithm base classical neighbor estimate take diffusion denote node exchange weight I cardinality update adaptive I ta represent filter combination mainly combination rule laplacian ik ik first disadvantage sensitive statistic mean I algorithm focus rule nevertheless individual evolutionary predefine besides reveal distribute sequel graphical exist give distribute evolutionary biology differ property equilibrium widely process communication streaming spectrum approach dynamic equilibrium period strategy ess ess stand player player equilibrium ne stability moreover
trading summation appropriate us rate strongly regret lipschitz mirror without function bregman strongly norm regret regret stability require strong continuity lipschitz strongly square norm continuous diameter bregman generating function incur algorithm bound r use optimality strong use side forward formally previous note strong condition convexity need select r r q optimizing prove online objective
important arise approach ordering partially order significantly optimization final network demonstrate network census weather accurately dependencie domain graphical acyclic dag extensively wide variety instance gene medical machine vision behavior probability random
generally call exhibit practical important collection text sound broadly use develop step conditional variable derive stochastic empirical follow organize document simplex denote topic topic infinite collection associate distribution simplex simplex entry index draw topic proportion assignment relevant expectation dirichlet w v topic assume document exhibit proportion topic topic proportion draw model lda exhibit assume dirichlet exchangeable improve prior analyze amount capture describe degree exhibit assign explore large document central researcher develop method include markov monte propagation inference develop variational collection illustrate topic lda collection document corpus compare algorithms lda collection inference subset document collection variational well derive two represent categorical like assignment document assign th likewise word document abuse review distribution simplex one factorial dirichlet parameter derivative log gamma derive put dirichlet variable set document local variable proportion assignment hide topic complete distribution variational conditional conditional topic assignment approximation variational topic conditional th dirichlet assign conditional dirichlet variational dirichlet dirichlet document different proportion conditional context global th dirichlet topic hyperparameter topic global variable complete conditional
fit break unnecessary parent output depend causality fit choose additive linear aic order build spline gaussian automatically optimization residual series combination one correlation thm correlation insufficient fail run determine bandwidth median distance input heuristic fitting choice unobserved model one try discover causal whenever version repeat able remain exclude useful investigate principle remain mostly require fine
replacement might randomize careful accumulate increment select iteration many increment specialized case toy scalar apply initialize incremental stepsize replacement use sample square toy illustrate scalar consider stepsize perform replacement replacement inequality always close expectation sort toy example gaussian replacement subtract side multiply index multiply risk namely demonstrate without replacement respect noise sequence simplify thing take expectation
course incidence incidence diabetes relate adopting may continue year consideration simulate simulate carlo applicability analysis strictly separate pde mention input exploit predict accuracy structure inherent affine interpolation incidence work affine incidence diabetes age year
experiment change transformation purpose drop network present make network interesting order motivate natural change influence content network follow coherence network content begin examine coherence edge drop add involve change structure analysis present coherent connectivity view measure enhance overall graph affect connectivity protein belong use protein class network average number scatter measure original statistically significant connectivity plot class performance prediction explanation help prediction protein connectivity original connectivity original improvement auc prediction transformation functional connectivity class accuracy average particularly help improve connectivity prediction original show important transformation enhance connect early utilize score identify
nonlinear switching process generate weighted variable construct experiment successfully discriminate dynamic human extremely process occur person subtle pattern observe movement gender person remarkable learn generative dynamic environment brain infer propose generative continuous dynamic online accurately discriminate dynamic implement implement winner many switch active associate variable parametric equation movement primitive observation standard
interval assumption population four step apply value formula simulation various fr distinct three compare square summarize via significance er von ks pearson summarize table illustrate new compute coverage probability summarize illustrated sample shape
big outside determination confidence interval event allow confidence note bayesian determination frequentist
inference infeasible feasible numerical community individual similar overall diverse context protein human method know interact distinct community across community structure representation diverse analytical candidate partition depend display yet understand consist reveal reflect case procedure belief propagation perform classification node consist partition block belong block
pose low solution np current impose make amenable via exist alternate small represent capital letter letter column transpose default norm denote spectral trace represent pca sign prediction etc reason applicability memory flexible modeling c amenable parallelization despite largely heuristic approach problem b complete low provide show recovery minimization attractive additive
outside simplex ice write covariance control tradeoff good simplex simplex choice version worth note geometric either volume regularizer enforce abundance datum cloud regularizer fractional update introduce cca boundary cone concept cca start eigenvector large eigenvector basis belong cone convex minus imagine hyperspectral sensor type tree spectra almost spectra associated spectra pixel consist perhaps represent true nature vertex region interior design find hull unless fail happen rely design devise identify hyperspectral class convex piecewise approach design latter next rely fuzzy cluster assign one cluster allow every fuzzy assignment membership two point objective attempt spectra membership ice abundance analytic update formula membership multiplier fractional one still solve minima update formula version pure algorithm simulate data set illustrate pixel snr db follow pure simplex pure simplex distribute bottom mixed set produce good pure nmf produce degradation bottom leave still significant degradation pixel close remove finally produce pixel vertex set reach mixture mixed yield simplex statistical powerful high computational represent spectral formulate statistical case hyperspectral blind tool separation tool hyperspectral abundance hyperspectral datum abundance imply applicability hyperspectral signature ica
let vector exponential sample exp construct write functional q equivalently mode weight denote distribution anneal van level canonical thompson walk set energy energy evolutionary liu liu slice sampler uniformity contour exhibit global local example derivative difficulty additive slice sampler boltzmann chain example
exploration asynchronous helpful convergence case careful perhaps idea adapt tree store message inefficient many convex cover problematic message pass algorithm dominant arbitrary message cover finite cover lift minimize domain contain even objective use trick theorem force computation twice differentiable second partial derivative semidefinite demonstrate computation convex guarantee independent twice continuously condition convergence min scale extend subject future without diagonal break irreducible argument eigenvalue existence definition dominant cover scale positive symmetric cover definite loss value irreducible frobenius scalar radius nz ic consider c iy ty iy combine see imply word walk root let potential bound depth diagonal eigenvalue ii contain need ij far multiply potential multiplied possibility leaf
statistical work take structure evident successful conversely goal may successful configuration variable right column disadvantage pattern lose level relation branch view kernel mark node anchor west induction anchor anchor west anchor west rewrite node anchor fill white l l fill proof tree oppose sequence branch devote generic index lemma goal dissimilarity goal ml pg analyse case rewrite mul n big rewrite exp big seq odd move h move big odd odd square goal rewrite big fact big add next consideration pg extract pg goal discovery area machine irrespective feature extraction pattern tool limited exception rule proof extraction major respect allow goal pg choose property shape shape goal feature uniformly rich may see library number list library trivial rewrite trivial inside double five consecutive extraction pg name link step hypothesis inductive lemma track extraction goal shape still infer goal user treat goal step ml want might advantage apply library adaptation change library
evaluate horizon learn generalize outperform worth well report associate optimistic shortest find totally policy mean index score bad arm arm times decrease ucb reward play preferred suggest optimistic paradigm multi ucb horizon table give train horizon percentage count observe learn percentage real percentage better numerical core perform whole take hour hour symbolic case take minute bit less hour symbolic explain careful way obtain optimal symbolic algorithm cpu able rapidly reject strategy armed formulation
piece automatically extract result learn behavioral twitter occur network piece feature automatically extract behavioral role learn time collective twitter united conference importance role visualization clearly analysis twitter day role role find regularity pattern arise thus spike presence anomaly small spike observe type role step locate step pattern importantly towards relatively trend whereas interestingly pattern twitter involve conference drastically change conference come un conference form connect dynamical completely manner systematically behavioral world ip role membership individual role past behavior role measure formally role interpret membership measurement component coefficient degree contribution time contribution average evolve behavioral trace
assumption recover signal arrive sparse sa na induction conclusion trivially least imply u ss argument proof combine arrive l ls imply ss induction assumption ss ss note arrive step
orthogonal subsequently anomalous greedy anomalous link anomaly correlation link anomaly operator filter traffic anomaly filter choice wavelet fast fouri e accommodate change see detail comprehensive test estimator leverage sparsity cs plus decomposition fit incomplete cf l minimize pseudo unfortunately albeit natural typically nuclear singular adopt since one controlling appeal develop anomaly offer traffic subsequently task exploit spatio anomaly procedure ht roc curve leverage jointly via anomaly show blue move implementation generality miss sparse encountered pursuit also refer pca available aforementioned however superposition challenge cs paradigm recovery arguably dimensional adopt missing entry implement link traffic measurement use aggregation operational paradigm adopt limitation architecture information raw measurement communication translate miss raise concern central isolated failure reason motivate embed anomaly per computational task locally rely link link message anomaly attain centralized counterpart
baseline helpful preference map show classifier weighted vote diversity indexing claim appear appropriate prediction modality show competitive fusion incorporation order preserve constraint sometimes beyond pac theoretically indexing ranking etc corollary cm
go go vice small vice error small suggest connect threshold impose additional b whereas relationship show approximation easily equal cauchy conclude complete whereas nonlinear equal right similarly step representation analogously improve negligible specifically addition third condition type jt w depend derive nonlinear follow q relationship property select every proof mi e attain weighted e
gibbs partition drive proceed conference complex combinatorial nj formulae gibbs exchangeable appear analysis exchangeable gibbs triangles science approach estimation discover cm gibbs exchangeable exchangeable th statistic game volume cm brownian motion characterize
recommend quickly refer back let denote radius smoothed version small form hausdorff reason commonly strict distance analogous familiar subsection self intersection reach weak find reference gradient whenever define let ax contain aa bend circle slightly create straight reach corner reach set homotopy continuous map case smooth write nearly homotopy k extensive let ax jk f stacking length conversely vector length stacking think anti hadamard calculus jacobian q fx ff
approach result use method I I average along sub learn rule em extensive derive correlation benefit correlation modify deal application sparsity zero row mutually parameterize multivariate gaussian nonnegative hyperparameter control definite matrix correlation structure partition reader convenience quite intra inter inter intra inter correlation
create subset document represent difference market similar level look problem model technique represent document collection text represent vocabulary uninformative contrast description ir key naturally incorporate example create guide semantic meaningful despite suffer many bayesian incorporate perspective parallelization scalability remain aspect sparse concept incorporation side semantic feature improve ir capture derivation however discard employ sampler birth hasting sample gibbs gibbs j generality concept come well ibp exchangeability write factor dm call simplified word document word document explain supplementary material metropolis hasting flip value since exist let concept
sample form table du matrix singular pick fail choice discuss choice topic conditional information document satisfie unit return algorithm permutation illustrative appendix remark boost confidence convergence polynomially failure repeat sufficiently interval scale factor sample depend bound factor eigenvalue approach much broad view w k conditionally column specify else conditional hybrid depend style cm fill black sep hide name name style minimum inner style hide name name observe name h follow degeneracy general remark view dimension notational stick case mean develop estimation topic
rgb rgb blue fill blue shape blue shape draw red thick thick nlp view reduction purpose view supervise unlabeled datum rare article york time recent year easy classified finance need human feature supervise unsupervised view occur
time news influence account micro platform view message account follow novel message collection account major select abc france news multiple account retrieve account tweet focused tweet contain chemical intervention pick show hour snapshot source two account tweet relevant black line tweet short user content long period first tweet keyword tweet period tweet contain relevant black line tweet user diagram user tweet depend past past solid tweet box denote period star tweet establish ground news influence user tweet manually formal performance message arrival network user activity although continuously might twitter see figure without one delay minute period merge news source active binary tweet contain tweet contain send describe inactive period precede model past entropy logistic fit denote computed manner overfitte description penalty parametric entropy quantify conditioning within could tweet tweet keyword period example likely due fitting news user activity correspond negative remove criterion infer positive tp tn evaluate
identify cluster fourth lr sake completeness add column basis lr intermediate table group correspond trait describe ordinal multidimensional concern ask health scale develop compose equally item category minimum low loading aggregate suitable matrix accommodate item treat dim section optimal latent aim avoid employ comparison differ rely increase bic start est est est multi maximum number parameter bic lk bic lk bic basis adopt correspondence small fall repeat vary start possible high log estimate bic est concern good logit link logit index free difficulty completely multidimensional est multi est np lk bic logit second
vector proper get zero full proper joint spline write present simulation carlo nest approximation structure conditional straightforward conditional metropolis sampling technique adaptive sde unfortunately sde care next key hasting adaptive sde write possible conditional b c na b I order approximation approximation proposal conditional n sample package integrate laplace handle model compute estimate mcmc technique hierarchical observe likelihood marginal construct without
learned stage dictionary learn task task character among character induce since interested task tune validation task simulate overlap target value rr go code optical character miss employ capital character digit regard space interval level divide possible digit random pixel value image transfer vs vs atom dictionary learn tune show trace ridge worse sc apply pixel analyze
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characteristic apply person simultaneously pose part assign orientation detector body location identify pose detector precise many variation pose people model pose encourage diverse predict single person approximately pixel possible select frame people camera serious person annotate head location reference orientation angle value part pose dpp instead build factorize treat pose root head arm body part singleton pairwise quality derive model part tree expect pose control hyperparameter hold quality score train image every location accord encourage example arm near detail overlap pose diversity location reference space evenly ignore orientation reference pose reference align width pose hold compare baseline pose normalize pose choose pose incorporate select pose maximum encouraging diversity pose normalize independent pose overlap pose pose overlap cover pose pose max long pose achieve might perform poorly people overlap stand generate spatially diverse strong visual datum randomly test training score well baseline irrelevant baseline pose total precision measure predict part expert head leave correspondingly fraction predict encourage diversity expense score mean metric tight radius acceptance perhaps resolution diversity obtain significantly baseline curve tend model improve pose give acceptance radius part match interval recall circle illustration cloud color probability marginal pose main loop synthetic entirely pose similar marginal pose successive left extreme combinatorial present dual normalization marginalization even memory limit mainly reason move become big natural analog vector occurrence dimensionality propagation reduce diversity cardinality dpp dpp cardinality denote structure note sampling mainly eigenvalue require eq structured must label part empirically toy diverse diversity optimize plan problem structure city factor singleton stop base google city popular consecutive stop preferred order path assignment arrival assign diversity feature singleton city result appear path country path tend along east increase emphasize show draw leave correspond city distance project sample dimension relatively projection approximation dramatically projection theoretically empirically justify projection black variational project complete graph diverse salient depend citation paper thus period article experiment possibility discover trend social medium image video overview collection mean insight object instance people image noisy manual tool query relationship continue automate key efficiently effectively weight edge diverse salient tracking program lead like link detection text collection however address probabilistic tractable work tracking division slice generative available evolve sort document distinction compare retrieval extract document collection find cluster entity take input stream article topic seek extract sentence occur cluster document contrast extract entity sentence focus propose news set intersect map event content specify query likewise build individually assume collection perhaps input make document collection diverse cover aspect length setting finally efficient memory projection allow simultaneously goal direct graph node edge define weight control dynamic likewise feature case might convenient feature node application context diverse diversity independently identically dual dpp sampling citation computer paper news comprise large approximately paper citation node remove
suggest report bic polynomial polynomial degree bold highlight highlight accord bic estimate round scatter fit component c mm brevity sake consideration reference mixture place collection nine area order quality life place health care united seem due prominent overlap group aspect evaluate clear group regression ht std nominal suppose forget directly consider display bic correspondence good consideration summarize
datum regard tolerance treat tolerance x positivity failure respect calculate impose scenario inner evolution solver termination condition inner loop parameter set constraint loop constraint sum inner loop terminate cx produce constraint impose solution object cx maximize failure exercise protocol numerically replicate denote optimizer outer quickly search instead force search value optimizer report zero nn iteration equal tolerance subsection report implement california institute technology small impact brief form smoothness ball angle impact impact pass fully say topology opposite situation merely refer shorthand complete common use impact complicate experimental scope find relevant mathematic select scalar post impact cross area measure optical convention mean distinct triplet response protocol tolerance observe independent failure fall area sub agreement grey criterion plot markov bind feature agreement markov maximizer
manuscript training svms linear svms radial try map solve possible must solve conduct robustness heuristic parameter kernel inferior polynomial parameter linear rigorous alternative classifying predict simply essentially distance vector margin equal probability age old patient maximum classify negative sophisticated smoothing polynomial contiguous point smoothed consequently directly polynomial smoothed point unnecessary result average filter tend information mean derivative examining assume aid discrimination example fluctuation something nature employ throughout three ignore apart use replicate replicate investigate construction detect disease protein aggregation regular longitudinal construct evidence aggregation exist solely threshold aggregation decide upon optimally address svm besides issue stop process possibility detect incorporation additional patient disease duration construction disease machine rt current test detection marker reduce specificity associate term induced exploit brain patient induce shape aggregate reaction aggregated emission rt create longitudinal interpret curve occurrence national
avoid membership apply child reach leaf leaf reliable vs score assign report sample predict tr fr fr tr tr propose drawback label problem sample tr fr boost widely categorization evaluate big flat
solve mle run polytope algorithm maxima run objective ordering occurrence experiment begin work problem problem whether non negative among mle mle boundary none lie lie model description determine polynomial irreducible component closure component intersect ml far algebraic algebraic could ask exact description seek boolean inequality find resolve range van illustrate nest factorization q irreducible component algebraic boundary ideal height determinant define study look topology affine variety mle regard ambient simplex parameter product parameter parametrization identifiable semi algebraic dimension fact topology maximization method find exposition emphasize operate entirely
conclude image figure display residual reconstruct comparison basis basis comparison image construct patch image rather eigenvector patch eigenvector clearly explain eigenvector patch white every patch image patch lastly patch clean image pixel denote similarly nm coordinate real eq patch coordinate nonlinear thresholding coefficient denoise clean patch patch pixel ball radius overlap center patch correspond pixel word patch exponential weight patch center neighboring patch variance connection translation invariant original shift reconstruction translation denoise eq method denoise problem process define patch wrong explain heat kernel solution diffusion notice expansion reconstruct base similarity expansion understand encode amplitude exponentially fast geometry
pd symmetric vector submatrix satisfy collection conditional conditional fact factorization statement either implicitly joint variable capability mixed random variable ideal undirected graphical ideal method implement
angle correspond around big small reasoning apply pick lie plane angle function metric ranking angular angular equivalent distance preserve grateful
correlation subject multidimensional scaling generate embedding common canonical analysis optimize regard fidelity multidimensional scale fidelity canonical enforce canonical correlation formulate generalize space calculate subject different respective canonical space embedding lack datum lie investigate relation datum experiment generalization around million article pointing explain language article
reference come disease axis co disease duration represent axis subject line word line time parallel triple duration respectively disease life change henceforth illustrate subject show birth disease life disease life
model cell map feature lp pool activate lp pooling method pool c traffic sign house classification multi stage ms branching figure
indicate min inferior almost result lack present max performance affect average proportion effective covariate fix size parameter set value roc true less joint separate method benefit penalization former complicate compare max two comparable sensitive note point represent average sensitivity specificity replication select maximize conditional log dimension proportion covariate strength non vector position sign magnitude separate level eventually almost sensitivity specificity point separate extra uninformative covariate size
sum involve positive exist existence eq argument strictly convention minimizer exist unique since equal constraint attain proof detail regard present ridge equivalence integrable rkh continuous operator derive multiplier banach space suitable lagrange multiplier existence lagrangian context lagrangian fr lagrange multiplier n use f hx hx problem become
subset subset usually superior however well fitting likely superior include suppose sure well regression estimate develop desirable use improve least scad fan li adaptive mcp zhang ridge subset sure screening guarantee fortunately appeal monotonicity fitting improve circle contour square leave orthogonal solution right example arbitrary fix mx difference proposition orthogonal penalize square cx ci iteratively
important finance decision central financial market simple value period available variable distribution cdf monotone specification cdf iid specification lag l pl lead dynamic probit root represented lag extension reaction correction var discrete unobserved typical forecasting resource recursive typically behave existence consistency estimate estimate inconsistent diagnostic goodness
krige express covariance correct krige orthogonal interpretation calculate conditional knowing note prove eqs plug eq difference
discuss contraction moment interest quasi link identity canonical response canonical link lead acknowledge robust sense influence canonical convex binary response function quasi hazard decrease hazard quasi loss clearly constant compatibility constant normalize design denote column canonical correlation span combination combination geometric compatibility never eigenvalue compatibility eigenvalue
political facebook approach ignore known user gender compute consider alternatively construct capture user proportion live relational though decrease accuracy even political link derive relational compute user view label initially typically refine infer cc cc inference many gibbs propagation ica survey concrete facebook special emphasis cc transformation task set relationship classification anomalous resolution figure relational find powerful elegant link decompose interpretation former predict interpretation part construct node link task task primary graph illustration summarize organize large link new intuitively facebook user value feature share political link people increase improve accuracy collective similarity neighbor infinite random walk second type graph facebook link thus figure result weight strong identify probable link rather weight link link may assign kind discrete label link related event link label influence feature add kind link count occur number link labeling perhaps feature construction section technique link special emphasis link model user friend technique section additional link figure discover discover facebook might friend people relevant characteristic node associate link could way though away short length original new link propagate algorithm cc apply identification group apply separately finally link influential weight assign friend eigenvector g represent graph give estimate estimate relational via representation inference arbitrary value add instance facebook friend friend count friend would autocorrelation feature might friend conservative political user friend feature political identifying feature essential performance prominent prediction interpretation turn ij refer vector node contain summarize notation relational transform link label node remove seek instance objective completely specify improve transformation four prediction present link link task link partially transformation task simultaneously create produce new node feature represent label introduce notion relational node construct link without sometimes article intrinsic aid reader summarize often focus interested create modify motivated way incomplete interested goal discover represent predict whose common subsequent learn seek evolve connection might interested object spatially pair individual successful summarize prediction summary predict weight great original link step link step link often yield uniform show compute link section describe non topology base technique exploit link predictor exploit relational define create exceed pair high similarity object domain cosine similarity represent node use cosine link show link link compare leverage
optimality condition inactive bad optimality problem within comprehensive description assume g essentially identical viewpoint smooth suggest assessment iteration current assume set current complete zero hoc fail otherwise optimal follow compute algorithm option computation optimality gap duality follow gap figure elastic net short penalization duality gap bound rather fairly coarse unless solution reach gap assess rough good computationally tight optimality gap provided guess optimal step current complete complement minimize magnitude finally low solve duality
sf bind expression rhs rr r sf last j inequality chernoff hoeffding chernoff hoeffding fact inequality use combine sf e j f combine lemma dr dr I dr define history play martingale define first never happen martingale notation us time play l bind
discuss detect fouri paradigm correspondingly empirical cdf representation think work given find unit execute outlier remain portion incorporate unity create exponential family correspond smooth fourier spline overall proper pt circle general edge furthermore discrete density feature device spline try regard see parametric circular spline empirical preserve metric borel topological empirical histogram haar number assume exploratory descriptor absolutely major derivative haar topological etc pose look mapping topological exist haar situation integral integral indirect fashion major reason tool fast transformation fourier transform let function support finitely definite induce turn space shift representative unitary
pi begin generic namely independent algorithm specify although useful purpose mind analyse provide useful later vary admit target interest investigate desire multi level smc mix strong often kind simplify state generate proof find generalised always hold show time get estimator short variance hand long long proposition stop summary tradeoff convenient trying level particle tradeoff serve flexible vary use smc within multi smc within density density consider stop process sample k accept accept ki pi p n addition proposal irreducible modify notation algorithm pi generic form enhance add update backward
replace matrix process call employ cause generic solve load loading vector repeat describe formulation step burden produce load monotonically use fully albeit use purpose besides conceptual formulation theoretical certain iterate load depend state giving framework previously literature provide order variance algebra involve open implement parallelization strategy formulation architecture ii serial computer core serial benchmark measure speedup core gpu codes section parallelism purpose compute
middle summary summarie unchanged rating overall randomize amazon hypothesis topic summary currently summary estimate superior lda estimate interpretable plausible stable approach modeling explore obtain quantify regularization within poorly rare word usage topic expectation summary dominate regardless content across topic panel rate lda panel panel refer fit topic show assign occurrence score dominate order logit panels lda assign total occurrence contrast frequent pattern across result big work quantify well within fashion dimension content word alone stop space quantity estimate base occurrence bias difference strategy lda normalization usage dirichlet topic regularize word usage analysis interpretability output stop word presence summary versus bottom arguably one propose stop word topic become negligible word design induce sensible prior vary contextual corpus stop topic emphasis stop lead appearance issue propose stop post amazon coherent proxy evidence support summary interpretable
give equation comes obtain small sum arm obtain state relax mab number show repeatedly arm fractional bind recall mab problem indicate small upper summing apply lemma fractional highlight previous notation previously introduce variable fractional time arm within fractional note show equation equation equation produce behaviour budget mab within particular arm cost mab accord limited mab mab limit fractional within mab domain fractional practice counterpart cl cl cl cl
arbitrary part sequential tree location new location grow enable pool reduce size discard crucially analytic enable large retain leaf yield discard preserve property discard subsequent stay arrive demand would available new tree local nature subtree nearby loss tree affect single leaf take subscript l leave enter usual leaf fine way wish consider recursive equation like represent stand inverse assume xy keep memory statistic crucially dimension additional multinomial discard may indicator vector count obvious manner sensible detail unfold preserve distribution non posterior unchanged never discard update demand loss argue limited
strongly correlate redundant rate preferable f l l l spam lasso c c l l r spam c l l sub ar table ar lasso method dataset e p select red select redundant propose width must choose carry choose propose comparison lasso delta clearly outperform input genome since value
stable point show requirement translate show highlight fact hypothesis allow generalization bound hypothesis space capture double useful choose probability guarantee get application union yet q note confidence tell choose yield proceed via predictor lipschitz first variable satisfie invoke ss n respect value es gs n fact lipschitz function f
variable express denote two observable define likelihood latent marginal prior include express reveal estimation plug measure distribution kullback construction bayes form error learning estimation difficult parameter good performance successfully word assignment situation posterior distribution bayes go zero theorem decrease variable
predictor spline express term basis plus panel consequently lie near span spectra pure near informative cosine problem methodology encourage use decomposition penalty strongly various many two step example component regression project predictor basis modification spline towards functional extend longitudinal publish longitudinal functional predictor proceed truncate represent spline longitudinal mixed model incorporate random predictor decompose visit accordingly longitudinal principal longitudinal outcome coefficient operator inform estimate unbiased predictor precision longitudinal penalize bias obtain weak compare propose evaluate effect explore confidence band probability evaluate partial real section implement observe
te tc tu tu u together tm tm tm ty tc tc tc tc tc predictor vector predict learner e regret successfully contextual bandit payoff learner predictor dimensional rank accord consider contextual problem underlie predictor good aim contextual armed bandit e thompson ts heuristic around old numerous reinforcement family matching algorithm ts contextual follow consist reward pp function ts simple way achieve play arm design generalization thompson fit use mab irrespective derive thus ts ucb interpret
obtained unfortunately preprocesse difference neither word advantage induce word gram try hidden word hide activation probability last sentence hide activation validation superior embed cm crf meaningful primarily syntactic representation short window enforce local semantic syntactic group word maintain small distance week digit year year visualization describe design word representation sentiment probabilistic document semi supervise sentiment capture mostly semantic word encode combine feature
principal cart split biased categorical allow split ordinal split variable unique ordinal multivariate cart include guide avoid select set approach make guide fit article guide longitudinal variable briefly guide univariate longitudinal fix concrete section compare prediction deal occur predictor analyze extend longitudinal school compare accuracy idea two longitudinal stress child conclude remark univariate response fit idea sign fit variable residual versus exhibit sign depend piecewise residual cluster end center obvious contingency chi test group indicate dash form form table count chi value small chi split serve split example test lack least exhaustive search reduction side practically important search well recursively large
propose rank correct low completion overcome nuclear confirm improvement correction presence play nuclear least foundation structured correction sampling great extend also interesting acknowledgement efficiently density function say elsewhere eq n fu operator eq otherwise reduce detail spectral operator reader thesis constrain continuously lk give feasible constraint tangent linearity lin equivalently l constraint difficult verify problem b previous condition reduce accord condition reduce characterization optimality decomposition orthogonal projection directional derivative nuclear like find rademacher bernoulli probability less estimate event hoeffde far imply proof theorem let notice imply direct calculation yield b ms inequality choose constant arrive intermediate result derive estimation deviation sum random norm version control operator norm characterize tail matrix extension find exponential z f exponential meanwhile direct calculation q calculation remain
throughout frequently relation every existence vector hence trivially obtain case hence q may distinguished possibly atomic agreement reliability model either q part prove complete determine carefully meet reliability estimate precede even hold put precede value could spurious would modification long precede add spurious interval bad distribution category
square motivated achieve property presentation change unify arbitrary parametric another parametric high sparse additive segment bin acknowledgement gm partially support grant fa collect well text reader use vector denote index support set extension notational denote denote ij model input exist literature ignore
row illustrate movement bottom previous connect tp green distinguished node movement static support node movement reflect static table due lack centroid temporal cost compare although static temporal require static four time present significant addition base support website cost notice table mean centroid cost evolutionary spectral achieve temporal static encourage mit reality mining social mit access equip phone medium access nearby device five proximity construct week mit event participant income student university business rest due number display clutter encourage reader website idea network first week class group incoming student separate work building incoming student proximity remain expect student affect separation group clear subsequent separation switch see color correspond create discuss notice group separate group another group time separate participant evident
add latent belong add variational single rx horizontal expansion wish break conditionally variational due applied scenario arise output believe example event version form hierarchy recall come set automatic relevance ard describe extend present output gp assign ard vector index hope encode assign relevance idea level obtain network obvious ard neuron learn ht general architecture cascade depict simplification demonstrate mapping leave intermediate include leave graphical variant variational covariance prior couple
aware machine tu group institute mathematics fu institute tu factorization component demonstrate real world potentially gold avoid spurious canonical cp multidimensional singular decomposition science many name discover discover
report experimental verify demand recent problem demand fact fast availability raw speed efficient problem smooth determine respectively predictor function fit sense cost decomposable sa bfgs respectively fast currently available incorporate area curve rating speak perfect sequence datum batch acquire could day alternatively large batch sequentially notation subscript write th practice know effective algorithm suffer mention regard variety proceed scale understand
select criterion filter optimum subset search good subset black embed use learner recursive elimination framework score g perform node single tree feature selection however limited ensemble tree accurate extract ensemble recently ensemble feature select forest redundant concept iteration use embed framework building acceptable desirable selection method reduce type
ergodicity impossible contour geometrically ergodic decay exponentially regular contour stronger complicated require theorem whose mutually exclusive behaviour tail number central limit coincide application variance bound geometrically ergodic hasting exhibit behaviour inference large like argument proof although loose geometric chain geometrically ergodic sensitive change understand qualitative result prefer variety addition justify reversible desire lee centre thank research grateful helpful proof markov bound reversible markov kernel early variance invariant theorem minor state space omit suffice
observation assume covariance design write effect specify overview resp equation term mixed simultaneously good normality pdf random ml q alternatively positive preferred numerical bad condition variance matrix inverse solution matrix unbiased blue uk k l u u g special conventional normally distribute sr
marginal I fw generalise iw effectively technique another eq sampling introduce dependency fortunately generalise latent use metropolis hasting hamiltonian monte carlo rejection use logarithm please number cluster advance turn update update update update give w mh denote accept move w jk jk k flexible obvious counterpart lead construct finite dimensional base item atomic existence rate atomic atom locate mass representation simply subsequent previously pick none partial bias measure exchangeable may exist specify section process measure gamma normalise show auxiliary list sampler derive correspond sampler extend general measure extension heterogeneous capture heterogeneity preference mixture degenerate item ever mixture component structured section mixture college show student interpretable preference conclude proposal work college degree handle application college office study
segmentation potentially conjunction denote element wise division derive pearson pearson testing histogram estimate histogram symmetric take bin virtue harmonic lie remove use harmonic original goodness statistic freedom regularize gamma cumulative cdf unlikely happen one analogy base harmonic although enjoy analogy since compare intuitively distribution empirically kernel degree work similarly degree refer paper form sum represent give approximation repeatedly plug side eq geometric straightforwardly see rather trick approximation full rate slow
powerful signal chance detect maximum window compare roc scale average roc curve scale window scale raw scan fdr control aforementione slice fmri slice color color color study inspection datum appear right slice leave central slice middle top slice slice shape seem false fdr false rate region previous paragraph many likely false positive region cluster spatial scan cluster paragraph spatial scan reveal shape fmri panel slice shape
decrease value way effective implementation slight original allowed additionally attribute importance present benchmark achieve forest internal assessment several derivation classification attribute equally disjoint relaxed bayes formula strict practically combination large object impossible simple assume independence
b b n position say random ii iii rescaled parameter density probability proof theorem proof step first step iv integral bound argument nm v restrict integral n n n n convexity inequality expression zero turn previous b b b dnn n n eq q ii satisfied mean eq last valid quasi n conclude theorem assertion reduce lemma iii c normality proof satisfied exist conclusion former sequence truncation argument concentration give w u w n w u first z use right go zero w w inequality see take sufficiently value n j dx ng j n ns
correlation among contrast separately pattern bayesian reason different latent relation type capture interaction latent factor object relation binary matrix consider latent factor observe enhanced hyperparameter gibbs space initialization stability result miss link quality clearly relational confirm relation prediction conduct impact variation improve overfitte result tradeoff model efficiency factorization dataset trend choose number object connect interaction within distinct relation france
coefficient matlab expand toeplitz intra reflect empirically calculate average ensure calculate denote fm real intra tend positive far constrain intra correlation average reconstruct fm rule due constructive people
explicitly page importance nothing change evolve al approximated stream outside node equation utilize transition probabilistic introduce walk popularity capture external popularity implicit human dynamic namely human rapidly change topic human large update receive considerable tensor
regularity sde assume typically stationary g unique unique identifiability sde unique minimum remark restrict process result concern framework derivative sde time let mh mh w h w h identity q write define account er h g jt eq obtain conditional imply obtain lemma distribute zero estimator state approximate reduce reduce restriction discretization comment end converge construct order estimator approximation reason formula high first moment variant multiplicative adequate weak convergence performance conventional linearization scheme bias ordinary differential equation equation various
correct p small report conclusion bind site breast cancer repeat validation split fold optimize selection repeat note extra training set separately receiver curve auc auc repeat fold ideal would choose histogram resemble si please represent histogram signature exist
continuity combine computational three subspace change sufficiently perform accomplished grow prune tree split merge branch bound adaptive nonparametric time branch increase high merging reduce make consideration residual penalty splitting algorithm split odd bi form figure data node represent parent means perform minor use prescribed residual partitioning leaf virtual child similar possible initialize root virtual easier virtual thereby recursive square expect small historical evolve control residual tolerance vary underlie control usually detect change varie track produce sequence allow merge splitting track increase track residual track residual close gaussian adapt generalized ratio change generalized procedure detect residual assume normal mean formulate point post quasi survey expect
agnostic construction bound frobenius apply agnostic theorem constrain seek r b cr define positive present algorithm select construct eqn compute leave considerably improve ratio deterministic svd first step run algebraic spectral rescale opt opt opt conclude need I multiply employ greedy column convenient matrix set vector u symmetric dominate index eq achieve search need
highly non parameter unlikely perfectly describe set mlp unit layer patch refer filter connect unit mlp represent patch datum mlp interest attempt input activation conversely cause activation layer unit output last hide generator mlp architecture purpose mlp detector generator detector detector generator bottom separately detector generator detector similar generator detector classify detector small pixel dot feature detector scale shift dictionary denoise explain patch htbp detector detector sort feature detector low noisy detector normalization display detector mean dc component auto train tie force learn detector feature generator intuition reasonable might generator allow mlp dictionary output coding suggest hide ask activation hide b activation mlp mlp evaluate berkeley activation center random mlp activation mlp either function mostly activation later measure neuron entropy distribution plot entropy bin repeat mlp entropy absolute reverse true unit absolute mlp high mlp absolute also behavior
monte general introduction smc shall smc smc sampler particle close iid seem close iid particles mention little exact posterior iteratively simulate particle parametric generate modal smc sampler idea mode simulate bold determinant transpose trace preliminary scale may facilitate assign read one work likelihood likelihood function difficulty next either likelihood include involve symmetric second determinant evaluate fouri integral third describe section regard first cholesky low triangle obtain compute quickly substitution cholesky operation completeness mention solve conjugate gradient involve unfortunately alternative approach evaluation satisfactory use approach importantly front decomposition scientific give magnitude take author introduction importance sample evaluate integral quadrature poorly numerical sect yet
intuitive rough bit suggest subsequent favorable non differentially learner subgradient generally score asymptotically gradient secondly estimation efficiency estimation subgradient centralized argument thus also apply derive first view conditional saddle theoretic optimization result upper application efficient mirror technical defer complete outline technique remainder precise notion devote information observe unique section constructive trading notation definition set assume give distribution point denote element q distribution support lastly symbol denote formal protocol information notion focus statistical formally access subgradient intuitively communication follow three vector subgradient variable ball cover variety online optimization approximation allow perturbation live radius regular kullback leibler view adversary controlling say publicly available provide mutual saddle conditional meaning saddle direction formalize notion collection regular q saddle calculation supremum infimum formulate notion differentially private setting define privacy infimum regular private take limited consist suitable set sensible choose minimize mutual definition specify receive belong guarantee mechanism study effect method sample assess term risk describe obtain uniformly gradient gradient excess risk distribution randomness perturbation gradient mask regular sub set denote loss belong random minimax loss giving
axis next imply entail eq j f easy kk jx explicit expression depend coefficient call adaptive lasso allow lar recently penalty huber criterion enjoy oracle pairwise tend group exist regression dominate combine ridge penalty elastic net en en ridge penalty propose version elastic net en property elastic oracle propose version huber criterion call huber relatively small error contribute grow hybrid shrink coefficient provide optimize objective nothing show ordinary huber account heavy tailed enjoy huber addition encourage spirit create penalize group avoid individually en
rr optimization thus applicable could could note proportional subtle rr constraint employ rr coefficient sufficiently small ridge close optimization employ rr algorithm interesting examine approach density detailed derivation conditional model model solve model employ lasso lasso employ implement employ software http www negative posterior follow likelihood employ eq eq conditional follow algorithm negative bayesian gamma normal normal consider general support
nonnegative learn matrix position observation clean allow sparse solid theoretical justification solution knowledge work datum additive without position rest organize section propose algorithm iterative optimization method justification method column represent nmf product minimize multiplicative objective
distance measurement sum population eq q pick decrease element entire empirical covariance close continuity learn subspace large th show norm canonical angle learn interpretable robot result model nominal detail robot control evolution govern equation generalize denote denote observation noise translation rotation robot nice expect observation dynamic easily derive motion robot pose nonlinear inaccurate optima hypothesis multiple make ideal common difficulty lack guarantee formulate range relate state several need weak motion hope track optima formulation learn desirable optima bias general expand dimensionality identification focus localization
number hyperspectral core similar step project complement extract strictly identify vertex use function maximize randomly reason solve point b enough extract isolate make rather algorithm recursively try volume hull volume induce extract approximated algorithm assume locate therefore noiseless ill condition confirm synthetic set allow highlight separable follow matrix randomly function svd tw hence might way set zero different position column remain one average geometrically hull column column outside contain dirichlet choose dirichlet generate boundary column hyperspectral mean construct separable four table describe total ht exp exp exp compute percentage curve figure generate experiment exp
none directly instance equation resort perform rigorous contrast inherent hardness motivate applicability section space branch extend approximate examine extend distribute kernel pair correspond product induce whenever reduce method metric efficient reduction strict subsequently desirable applicability near candidate every kernel previous object cosine candidate still query notion characterize hardness sp dp dc sp sp expansion effectively surface hyper sphere expensive scan linear runtime
sufficiently joint successful methodology length burn establish burn iterate retain autocorrelation within outli occurrence cut detect recall ga variability belief time setup time informative variability process
apply derivative pricing infer triplet neutral observable datum triplet completely distributional option quite differently find evidence calibration day section sd explicit estimator self section interval assess simulation apply discuss conclude sample variance defer basic evy stochastically stationary increment evy decomposition write brownian drift denote drift jump jump throughout absolutely lebesgue jump component variation therefore evy evy determine call triplet reduce one well dimensional characteristic depend pricing throughout neutral martingale exist strong finite activity self decomposable increase decrease additionally sd satisfied k
generic matrix mind inferential interest potential uncertainty evaluate measure approach miss maximization adopt ml em complete likelihood function particular expectation maximization carry equation approach report remain solving form derivative adapt conclude paragraph worth potential spatial discuss issue address estimation bias covariate part variability even sample covariate algorithm list namely provide likelihood asymptotically multivariate case normally distribute operator replace confidence problem estimate run pattern simulate enough
framework brevity setup iid poisson logistic couple power penalty spectrum nearly penalty shown save detect sparse occurrence along path predictor perfectly predict response therefore largely combination except variability second logistic convex penalty short time penalty separation early figure display select glm define negative false negative select positive number rough glm convexity tend lead appear significantly improve although admit predictor overall non convex penalty square error mse parameter estimate empirical comparable penalty selector mind result simulation vary numerous signal etc hope tool facilitate comparative study numerical package material run penalize glm generic combination convex mild path gps connection ode approach track smoothly avoid choose bayesian empirical quick besides extend
two treatment group result informally index assign fix treatment subject treatment outcome row treatment affect outcome help make asymptotic outcome treatment assignment model population thus treatment estimate effect ol coefficient ol regression paper analogy observational give analysis regression indicator chapter surface leaf consume weight quickly leave small simple random mean unbiased estimator expect motivate adjustment way leaf weight adjustment measure covariate measure population help estimator vector b ib connection adjustment remain mention despite agnostic much adjustment observational literature simple
classification generation set optimization analyse separable show margin lead good oppose order analysis control unbounde early adaboost sensitive choice condition goal manuscript minimize element risk convex empirical I risk perhaps available example relevant risk place theorem input statistical however briefly losse exponential sample suppose attain infimum subgradient descent employ cone descent adaboost iteration suffice suboptimal mostly descent lastly manuscript theory times b goal allow odd task bounding characteristic map countable perfect albeit I mb exist cm black position anchor west fill fill node north fill text simple appear example determine fail agree convex statement fairly additional regularity
tx difference arise moment maximum dependence key final next estimate belong modification apply bandit unlike chapter convention arm interval realization random similarly chapter rewrite denote strategy unimodal necessarily monotone decrease unimodal general unimodal choose return perturb bound repeatedly however exist belong irrespective regret incur ready version section chapter chapter operation financial engineering usa edu di di armed bandit unit allocate payoff slot armed american allocation multi armed must repeatedly bandit basic instance sequential fundamental trade exploitation sequential player must action past give high payoff clinical trial must decide treatment next create play domain service target benefit adapt service sequence request concrete domain page website website deal design font image payoff desire behavior ad pool ad bandit change time ad choose path send payoff path computer playing many move bandit version bandit huge tree game focus promising play survey cover many variant basic arm correspond forecaster select arm receive arm focus regret pseudo sequence plain regret fluctuation hope contrary control use formula let armed exploitation intrinsic seem explore actually see exploration exploitation principle uncertainty many sequential decision uncertain environment forecaster accumulate decide act plausible concentration inequality favorable identify base favorable plausible armed armed bandit weighting bandit function example attack armed uncertainty construct choose arm need bandit obtains call ucb assume reward hoeffding turn ucb first three indeed imply bind use obtain straightforward conclude denote leibl
even construct representation combine much design elaborate feature recently encode demonstrate much aforementioned provide broad theoretical offer explanation success result call feature encoding cross achieve popular benchmark however feature especially appeal produce encoding multiply triangle threshold slight idea modify term square take constrain triangle eq version threshold validation soft threshold similarity triangle feature computer vision approach encode assignment van version idea quantization rather hard assignment quantization scheme success threshold feature setting encode applying unsupervise triangle compression version level chart threshold detection recognition respectively
preserve integer stand span sample datum eigen denote eigenvector assume dimensional supervised reduction joint sample independently ij j main randomize rather input particularly reflect biology finance intrinsic appeal explicit control computational efficiency rapid convergence achieved investigate statistical highlight input focus practical sample population subspace information population algorithmic subspace generalization describe power iteration estimate span increase factorization problem fewer large deviation effect randomization argue computational combine implement computationally efficient shrinkage capture large estimation randomize study literature simplify result science discover approximation algorithm propose extend randomization capture scenario capture project basis set random bi validation project compute onto tu u
choice cutoff theorem give discussion proof useful tool limitation supplementary throughout notation integer random scalar distinguish constant ratio bound cardinality scalar generality mild condition surely borel also subset small assume write linear functional k l u interval natural quantile regression function formulate quantile function go generate admit quantile restriction location obey quantile obeys quantile restriction design suppose eq denote restriction estimation principal x dt consist expansion principal score
b u b piece formula corollary statistic north university nc li modern produce structure digital imaging flow frequently measurement underlie address scientific arise take matrix form crucial hinge coefficient signal approximated structure low lasso highly scalable algorithm develop facilitate selection demonstrate real nesterov nuclear modern produce unit include digital imaging contain color consist intensity cell channel motivate example eeg two normal subject stimulus channel subject scientific association pattern glm predictor include covariate gender classical glm deal pose na turn large eeg sample inherently e channel correlate dimensionality complex regularization covariate dimensionality preserve variety year
happen region combine example explain list pos contribute alone n contain redundant execution eventually replace small easy search tree depth depth backtrack partition object take never twice backtracking step backtrack computation involve split operation take complexity significantly relate indicate far backtrack equation bad hence adopt design algorithm seem heuristic may currently subset select e unfortunately happen frequently test alone region given generally subset less prove give reverse sensitive
group quantization know quantization representative I indicate distortion two need highlight quantization application consuming mean converge stop small discuss real seem big quantization could code quantization result denote center lsh try guide matrix adjacent group pick random natural pick adjacent median adjacent hyperplane easily associate plane previous projection generate projection usual projection select
bad forecaster share hoeffding jensen convex hoeffding substitution step sketch multiply examine difference inequality update third sum fact I recall convention conclude case forecaster hand hand extend generalized sharing technique way loss adaptive tuning paradigm choose learner expert classification special case online design difference cumulative element criterion
bundle maximize subject constraint bundle type linearly value concave explain agent behavior help predict future choice agent pair budget example function p b p every distribution example every set observation say complexity bound polynomial complexity learn algorithm learn mechanism bundle equal bundle agent select section learn predict require bundle exactly receive rich learn observation probability produce eq additive typically normalize lie function particular provide upper function low immediate start characterize linear utility intuition price respectively denote
chinese restaurant process crp customer chinese restaurant infinite number stick break resort constructive way construction stick break
pa rr r h e x x put together since henceforth pilot kernel pilot symmetric variate density square integrable come optimal pilot pn show everywhere asymptotic minimizer relative lemma analyze begin taylor use expand eq make expansion third section selector substituting yield lemma h h pn remark multivariate region contain derivative estimation receive excellent practical generalization derivative due bandwidth multivariate derivative achieve advance analytic tractable derivative behaviour explore application drive technique combine shift along argue exist bandwidth parameter exist derivative introduce validation method optimality plug root domain recently smoothed selector automatic small univariate surprising suffer equally literature bandwidth recent
p check x last finally conclude proximity require fortunately proximity subproblem q solve implie ball extremely well study hard fortunately solve root unlike proximity I generate approximate mixed proximity ultimately motivated author build upon q matrix formula lemma norm old matrix know invoke conjugate triangle old latter
also entrie nonnegative preserve normalization nonetheless class infinitely see infinitely concatenation analyze na need definite infinitely infinitely infinitely definite infinitely link space q moreover notice definite way suggest infinitely formalize semi exist contain hilbert metric infinitely notice normalization matrix theorem hilbert infinitely negative depict gram employ entropy path normalize alternative negative definite establish function quantity interpret functional far attention entropy gram certain operator act converge entropy population entropy bilinear fx fx x xx normalization eq orthonormal schmidt compact case extensively
sdca non hinge plot horizontal divide data sdca sgd work non smooth hinge sdca hinge small test error sdca sgd smoothed hinge axis correspond sdca gap ex descent solve svm theoretical guarantee dual ascent software package good paper present dual sdca sgd analysis sdca application associate regularize minimization let scalar tell number convergence iteration large specify examine proof small close close dependency occur deal sub optimality objective solution primal solution primal
approach candidate criterion costly remain refer extra choosing infer include square hyperparameter tb distribution estimate gp minima local minimizer multi modal minimizer
follow extension nonnegative sparse pca sparse conventional pc especially environmental science biology follow element recursive strategy separate series small respect need convenience rewrite virtue nonnegative model differ sparse evaluate matlab pc ghz cpu gb memory os list well synthetic evaluate recover truth sparse extraction utilize benchmark generated create observable intrinsic utilizing contain apply extract pc perform truth effectiveness algorithm adopt toy pc covariance apply schmidt randomly first two set
likelihood estimate kalman many specifically display phenomenon variance approach new forward accordance degeneracy ess stay ess drop proposal consistently forward run backward filter smc estimate superior show smc explore smc calculate utilize rely rejection
suggest direct may competitive penalization inference come validity relationship imply graphical rely assumption conditional course
freedom input pass sigma delta follow diag specify list r know column coordinate column indicate similar omit fit implement algorithm call eps scalar specify contain specifie start parameter specify iteration specify termination user argument expect hence attempt search well termination criterion control loop terminate criterion reach current log log default argument option likelihood display fit termination set detail argument
output draw white normal independent reconstruction accuracy select bottom bottom functional alone toy marginal coordinate label investigate enforce rescale determine size degree fix optimal validation positive irrelevant root error select versus value respect variation smoothness lead term smoothness h present set experiment vary unchanged account training space space kernel corrupt regression depend white sample rescale determine ratio fix value relevant leave unchanged ratio always hypothese selection versus decrease visualize plot minimum necessary clear visual inspection high want evaluate hypothesis unchanged set exception vary step signal noise display versus different hypothesis decrease hypothesis chance error number want evaluate effect well parameter theory unchanged polynomial fix relevant regression randomly combination degree involve display feature indeed present aim method model multiple
form exactly pairwise independent obtain upper moment inequality nr symmetry brevity term use bound show bottom divide numerator rhs obtain pe n u n z part large distribution distribution side compute note show derivative attain equal use pe imply interval arbitrarily long rate choose theorem encoding decode refined codebook tail
embed minimize introduce pairwise measure widely move row node give equip node eq sensible measure arrival site start match precision coordinate large scale eigenvalue eigenvector trivial include eigenvector eigenvalue dominate scale structural approximate aim estimate become maximally likely trace node specifically sum possible assignment free functional procedure maximization maximize maximize ideally gaussians easily lie ahead field field start addition introduce energy
generalize enable rooted tree dynamically multivariate generalize exist modeling flexibility conduct hasting build around overlap group technique em model model stock multivariate generalize induce period volatility market fluctuation stock build conservative portfolio recent scenario large irrelevant appropriately literature extremely popular seminal lasso zhang priors
respectively incremental dct corresponding coefficient coefficient correspond compact dct reconstruct representation follow last fig car define final likelihood criterion sigmoid ability model confidence compute score candidate bound visualization normalize location fig map obvious modal peak observation discriminative filter particle update observation tp frame state normalize state solve maximum evaluate video compose image video factor lead appearance illumination rotation motion pose etc sequence experiment main goal verify propose situation evaluate adaptive capability appearance change matlab intel core tracking normalize computational state particle module particle patch scale image pixel factor set buffer setting remain sequence achieve track state compete tracking decomposition online adaboost incremental l state utilize strategy slide inference code l utilize adaboost classifier select
sample controller increment measure use velocity start prior generate gp track horizon r gp gp gp gp gp filter color explain latent even outperform commonly use ratio robust unstable analytic gps se function implementation gp linearization density mapping nonlinear incoherent smoothing gps uncertainty
model term solve age specific publish calculation age burden dividing derive random stochastically problematic composite increment choose small accordance assume increment distribute
dependence compound kernel hilbert schmidt intuitively test cross dependence group separate cross exist group norm satisfy relation sufficient nuclear appendix theorem furthermore mc know nuclear norm topology treat quantify dependence obviously relation indistinguishable question still relation zero perturbation keep h apply correct incorrect eq switch incorrect correct relation compare hidden perturbation correspondingly strong close indistinguishable lemma correct relation
label label optimal partitioning may partition cut could tool optimize mixed combinatorial optimization poorly unary potential permutation true uninformative force exception case label graph efficient perfect dual statistical computing ise explore map mrfs partitioning provide partition cc partitioning label weight consider merge follow cost q twice small cost approach relate provide approximate likely segmentation poorly real image segment come colored subroutine difficult problem tractable
parameter finally discuss exposition writing sequel relevant quantity know support covariate exactly completely set well know closeness sub design define sub gaussian high partially solve noise miss estimate discuss covariate assume either know true assumption seem plausible instrumental whose row know rigorous asymptotic covariate independence operate design section iterative although currently analytical sub accord outline follow simple strong cauchy schwarz generic noise depend hence result subsequently estimator study finite bound simple corresponding instrumental variable economic performance indeed central paper able obtain
thank value ml mb define eq upper different successively cauchy schwarz every plug yield let xy py conclude proposition lemma use addition get absolute hold true imply q ps mx proposition union remain bind bernstein variance sup assumption eq absolute obtain separately constant absolute mx xu mx u hence deterministic appear follow every q nf mu mf remark follow u supplementary organize variance penalization detailed exact computation analyse fold criterion useful concentration supplement every proposition variance variance increment computation appear definition eq appear pm q oracle inequality hold correct fold penalization fold fold cross like suggest increase overall explain common
series h p soon representation trivial p h multiply limit desire follow observe consider reversible function positive measure satisfy whenever convergent hold lemma full proof give sake b metropolis hasting define check I f ap p pf qx qx exist x ap contradiction shall shorthand notation nx g zero variation p place acceptance remarkable fact see product metropolis hasting qx marginally implement marginal assumption may see allow interest aim algorithm equilibrium respective abstract pseudo illustrate one simple assume simplicity borel consist lebesgue consider density computed suggest approximate importance probability unbiased estimator pseudo constant seminal various application particle filter
boolean request facebook many score well request I rating often request statistically sound concept request demonstrate request user application facebook platform party application market resource average review description help phone hardware call history via restrict application application camera need application order study examine small map dimensionality visualize application analysis category relate finding request minimization low generative probabilistic request identify et correlation weak correlation average availability website application publish expand analysis far date focused technique identify apply static string et al build
smoother think right figure comparison include reconstruction depict mention believe induce change correlation figure case convolution noise results vary signal reconstruct original scheme matrix aforementione major come figure right see observation smooth worse change describe much rich specify move geometry identity observation follow complex reconstruction prominent reconstruction capture layer reconstruction reconstruction parameter well close model course add
follow derive assume therefore concentration event embed space moreover interpolation give supremum arbitrary satisfy satisfy universal substitute eqs probability moreover eqs far q side take assumption density choice probability less function estimator rate gaussian learning estimator rely technique develop involve stem expectation investigate ensure paper exactly need
bind jensen inequality help jensen difficult concave logarithm inequality conclusion complete tight maximize fix maximize guess maximize whole kl eventually current calculate later call devise
cloud future precisely cloud cloud entity show naive parallelization scheme propose provide parallelization vector order resource intuitive parallelization sequential therefore improve
learn perform depict learn thus indicate axis gray scale learn indicate
predictive order suppose give analytically web want predict consider dynamic stream thus dependency however gaussian relationship variable follow denote distribution analytically tractable condition
measurable p parameterize easy difference expand q jensen denote coordinate apply existence jk ensures hence write update sf c since recursion see ode eq quasi ode bound p easily satisfy theorem ode also g sf martingale adapt fx bound almost asymptotically stable ode fx fx prove convergent easily measurable martingale martingale jensen inequality jensen inequality fact argument convergence error term term n case summation odd show third claim follow ode see stationary lie follow show iteration track ode sequence obtain sf converge neighborhood stable immediately update
reconstruct provable introduction unlabeled piece leverage liu grouping merge piece introduce estimate rv nu ig represent recursive latent rv empirical I along ni constructed initialize take union span latent iteratively neighborhood tree run employ latent propose divide latent building fidelity optimize bayesian information search iteration limit quick implement correctness flexibility cycle contain many cycle tune latent page leverage cycle cycle node two cycle sufficiently neighborhood depict denote close span local neighborhood observe surrogate relationship child merge select merge discover latent distance select nearby compute provide asymptotic ise succeed potential ise bound potential depend edge ise potential potential consider counterpart potential expectation class
methodology system parameter local mode subject go population measurement straight forward example effect mcmc something department centre college bt bayesian computationally demand applicability application markov jump valid model convergence rate allow fast mcmc chemical formal physical applicability model chemical interact describe population network terminology notation chemical methodology model reaction interact ordinary differential fluctuation analytic form simplify hasting perturbation use local refer unknown paper nature metropolis hasting strong behaviour auto tuning good mixing adjust langevin mala hamiltonian hmc also metropolis hasting ess rate several however mala mechanism see chemical
may scale span use theoretic l matrix order decrease expand expansion basis leading mode analogous q completeness x correlation function kernel type significantly capability rare event remain step need give word theoretic generally manifold application establish overfitte address behavior shape curve context frobenius evaluate via lead considerable choice monitoring change propose via spectral measure measure energy usual order set energy spectrum energy information theory show mode
relate probabilistic estimation devote determination conclude remark auto property function q orthogonality verify property extend pp give quite focus pg gr suffice notice give semi auto highly set center diagonal semi auto model arbitrary dd
replace low summation efficiently use variational calculate expect hmm hmms sequence expect transition state state hmm maximize appendix update tuple include covariance base derive average base derive h cluster hmms relate hmms hmm center compactly center hmms leverage cluster positively break small set learn portion standard intermediate value directly direct time store second evaluate short implementation effectively parallel require allow model updating assume model requires learning follow intermediate computationally intensive considerable computational hierarchical implicitly virtual compatible generalize lastly virtual intermediate make virtual relatively positively run achieve hmms product hmms hmms virtual distribution hmms affinity compute hmms produce novel center suboptimal hmm example implement spectral mean point embed fail capture novel center especially high hierarchy leverage one obtain hmm cluster hmm center limited hmms I optimally center cluster involve hybrid learn hmm obtain h summarize hmms sc cluster hmm h accurate learn likelihood second drawback matrix hmms see music distribution embed costly exact carry I py alternative sum probability py dy affinity approximate distribution real respectively work spectral similarity show superior hmms
adaptive mdp dynamic give observe agent act optimally belief statistic lead exploitation uncertain unfortunately mdp dynamic action action readily derive execute real constitute agent respect prior occur planning find decision equation bayes adaptive monte planning forward tailored monte carlo search effort branch state base estimate clarity adaptive ba apply base ba simple search necessary simulation base tree ensure root originally partially observable search current compose represent
question answer general point short operate regime limit function reveal way answer density special unweighted short unweighted limit geometry implication short path unweighted accept pass high illustration opposite achieve application unweighted lead huge estimate manifold see illustration area compute unlabeled crucial exploit region distance
p pf pp p derive sensitivity subspace fit shape shape total subspace fit fitting note result tuple instance coordinate total sensitivity function observe project technical shape shape subset distance shape set abuse denote fit shape shape focus fitting seek integer shape tuple affine underlie euclidean projective cluster square tuple projective shape projective specific projective polynomially
simply treat player belief field kind towards point start clear mean offline equilibrium payoff map good equilibrium field speedup acceleration present observe time start point point fix speedup htb speedup summarize acceleration base sequence acceleration small start repeat acceleration seem great acceleration speedup satisfactory remarkable result challenge interactive system aim payoff satisfactory interactive user good good type seek solution nash criterion satisfactory offer alternative closely user make decision strategy make meet response strategy specifically option satisfactory action space methodology basic see solution player satisfactory pose existence satisfactory feasibility vector action necessary satisfactory na w w jj jj jj w jj n player necessarily target user satisfactory satisfied situation examine course
class verification may function imply filter sequence separable hilbert lag terminology frequency filter course converge technical sequence possess satisfy analogue theorems proposition establish previous immediate proof filter hx ty dimensional require belong process possess spectral operator consequently readily verify density minimize minimize employ infer operator consequently proof turn convergence integral tend theorem fix subsequent notational dependence limit assumption l jensen triangular lebesgue continuity b imply suppose exist clearly side v contradiction remain let turn first lag consistency result define assume proof tend lemma term jointly consider space x product operator possess yx x hilbert schmidt orthonormal hilbert schmidt show orthonormal hilbert schmidt compact class hilbert schmidt operator inner product h hx h define trace hilbert schmidt corollary remark
distribution mean necessarily interaction whose agent utility policy let oracle policy stage payoff stage policy consider total utility stage reward oracle reward special however payoff mapping payoff stage game stage payoff agent stage complete policy terminate end terminal termination payoff satisfy agent goal termination horizon discount reward reinforcement opponent select payoff maintain
section defer theorem rely theorem real target column construct svd projection spectral frobenius zero residual nc frobenius construct compute adaptive sampling base idea proportion arbitrary algorithm residual decrease theorem adaptive nr mr nr expectation randomize solve present follow selection algorithm near optimal
conduct topology large centralized localize community link although algorithm suffer requirement like algorithm practice use neighbor discovery recent research team community detection burden output detect community briefly discuss knowledge centrality heuristic view preserve locality location community community network lot attention detection detection community differ structural usually stronger sensitive tp fp detect community share exclude correspond person follow twitter friend undirected relationship undirected social vertex shorthand denote iteratively observe notation local call observer information depend observer facilitate discussion edge illustration observer node direct friend observer level node edge link visible observer thin observer within
cart tree cart cart learn classifier space h train cpu empirical il restrict test cost test learner extraction use weak use free auxiliary let formulate cost norm scalar assign weak cost extract learner difficult non non mixed sum recover commonly encourage plug cost relaxation norm sensitive single linear expand balance path show tree derive
mapping factor iff denote set exist semantic simplify semantic substitute equip edge model circle black scale
indicator accord positive entry remove separately reason remove remove statistic datum preprocesse attribute affinity use randomly unconstrained fill correct relation label quality rand guarantee compare algorithm unconstraine sl affinity directly supervise perform new try encode grouping projection trick sl report propose sl constraint encode laplacian use uci dataset largely sl adjust rand range quality mean maximum ari fig report ratio constrain observation algorithm utilize spectral hand amount consistently outperform margin ari able satisfy constraint algorithm significantly random fig quickly consistently practice fig ef suffer free unconstrained introduce small modify sl identify cut easily identify minimize outperform sl instance document write translate list document
calculate base algorithm treat calculate newly duration case locate life point direction duration might diagram person diagram relevant simultaneously depend age subject duration disease
difficult focus carefully neighbor confirm well fulfil robustness adapt triplet generalize metric xu come increase instance iid denote example lf lf loss lf generalize generalize almost imply notion weak pair sequence example learn almost recall sample belong notion average testing sense weakly fix weakly training example robust testing obtain
relatively fail boost predictor one fdr value argue opinion tool gene matter gene boost gene gene gene boost gene gene gene gene fdr fdr fdr gene gene fdr gene gene gene gene pls gene accuracy pls gene readily see false discovery interact gene expect cancer fdr pls gene identify non essential size start picture method fdr pls pool consistently gene increase sample ten gene pick gene boost gene boost know false identify gene false boost seem boost weak show majority exception rank rank ten ten gene determine gene chart allow gene cut ht model polynomial show describe taylor boost couple method success disease x show fdr pls three disease ht equal fdr boost accuracy x accuracy boost x x x gene pls x accuracy gene pls fdr continue relevant classify interaction cancer certain
posterior turn regard inferential analytic sde analytic available value discretization euler high taylor comprehensive observational fine stepsize value jj tt nj abc computational make turn previously significantly simulate sde soon trivial verify switching calculation propose proposal density simulate sde acceleration speedup kernel return sim relative produce considerable computational acceleration follow uniform denominator probability practice acceptance nature check verify simulate sde always regardless value fail accept proposal reject code initialization choose simulate sim generate sim sim sim sim sim sim sim sim increment modification considerable benefit particularly algorithm moderately accept obtain simulate sde benefit idea early rejection knowledge early generate proposal
dyadic discrete similar research et al base alternative approach relational apply problem essentially yu al variate gaussian generalize interaction relational incorporate capture globally property discover structure show benefit base optimization transfer algorithm complexity dataset evaluate relational still automatically factor cluster would marginalization prove form auxiliary easy il trivial form observe china fr relational application recommender bioinformatic characteristic thus source relational many world ask efficiently
corpus train lda hyperparameter place document proportion word symmetric symmetric improve lda compare widely hyperparameter except place look less design explicitly discover topic pattern capture word topic correctly identify non lda find quite identify topic topic interesting capture topic word topic lda actually adjust lda informative token ng change corpora
wide sde drive hilbert difficult common involve inherently term derivation advance hmc avoid strong assumption justification project justification analytically far beyond scope paper application verify mix sampler operator measure measure discuss sequel suggest coincide theoretical development construct definition hilbert space greatly extend develop carry sense separable formally state introduction argument think energy eq define hamiltonian dynamic hamiltonian preserve proof solution volume motivation direction velocity target space force end become show solution preserve hard carefully equation solve accept reject correct invariance standard work give rise volume preserve property look hamiltonian synthesis hmc table due mention easy acceptance invariant h hmc give calculate go ii nx standard sde give increasingly would vanish keep fix employ brownian path wrong case mala advanced hmc avoid degeneracy gaussian hamiltonian hamiltonian equation equation analytically analytically define eq small define define step proposal operator absolute law absolutely
decrease solve keep decrease iteration recover channel fista tv operator tv avoid list conduct mr multi channel validate conduct ghz intel cpu matlab matrix measurement mix white deviation evaluation reconstruct power original multi aid clinical diagnosis mr water water appear water suit imaging intensity weight mr correlate mr forest suppose fourier formulate conventional forest multi mr htbp extract domain frequency mr image mask compare fista fista fista fista demonstrate comparison among could model
zero multivariate gaussian n kn ni function j v work use cumulative control identical assumption hyperparameter characterize assumption expression posterior factorize incremental training approximation f nm ip ii site active basis note site site differ basis selection site optimization give initialize initialize basis selection site briefly add basis set incremental calculation carry
exceed paper powerful unconditional situation unconditional power underlie constant consistent conclusion preferable unconditional compare statistic note property simulate var unconditional variance test bivariate var generate write variance exist causality variance angular exist causality almost modify detect note positive definite instance check property lag length know autoregressive commonly ol light study table c c asymptotic nominal sample c nominal seem reasonably compare nevertheless outcome generalize recall size confirm almost nan hypothesis size
three cost examine dataset disease dataset consider minimum bm bp datasets bm tie l bm svm heart eight namely examine breast diagnostic breast spam cs exhibit tp auc tn bp svm breast diabetes spam dataset svm bp svm cs cancer diabetes imbalance ratio evaluate svm could exhibit tp tn auc compare tie experiment bm breast svm breast survival sensitive dataset past non mail performance table ad hoc consist subsequent algorithm rule prune sensitive ed bp algorithms ed suggest two sensitive box convert insensitive resampling accord kernel ed bp p ed bp black bb ci bb example also sensitive generalization cs svm cost cost result avoid superior dataset
subsampling object multiscale context large decision parameter multiscale across allow capture extra train representation edge abstraction pixel provide hierarchy observation level object capability assess automatically produce spatially naturally precision base extension neural network transform share weight justify combination region region enforce convolutional shift impose pixel wise label dense system complex typical pixel million naive convolutional view context multiscale convolutional extend scale multiscale pyramid construct multiscale pyramid laplacian pyramid process neighborhood zero convolutional multiscale per scale output scale normalize
unify os optimization os reject refine mode mode proposal dominate represent optimize find line refer far rejection usual mark stop far trial return accept second refinement history accept value trial history last provide detail decide accept mode accept enough maximum mode accept base trial line trial reject refinement hmm word observe noisy depend state gram language define x optimization huge need explicitly context
neighborhood lem I either technical require go topology auxiliary set nest assume subset requirement loss generality satisfy extra statement subset valid main ss measurable algebra decrease arbitrary claim validity efficient nested theorem since validity I nest xu valid combine previous display eq text automatically confirm discrete keep algebra text closure affect function continuous particularly light either paragraph predictive new measurable finally follow suppose iff iff iff put iff claim k j u interpretation measurable similar important ii random validity enough I shall produce optimal intuitively value want want typical mind present goal goal frequentist observe datum parameter uncertainty statistical unclear uncertainty come interpret level measure depend produce probabilistic unknown effort inference specification fisher inference variant inference calibrate recent focused incorporate frequentist
prove proposition least ensure prove force mathematical aware outside classical field acknowledgment point source solve difficulty challenge tend empty three algorithm solve anneal method combinatorial result backtracking solve solution however backtrack student mathematical science simulated anneal share common enjoy applicability projection feasibility problem convex programming optimization optimization combinatorial much backtracking annealing alternate np
example obtain send classical formulate counting nonzero encourage hard one pursuit ds constitutes match mp pursuit hybrid stage algorithm least hybrid accuracy high
proxy define note relaxed som positive kernel indeed condition impose nevertheless range string heat som act compare via walk day become several simple mixing description snapshot change adapt artificial strategy tackle stationarity risk structure dissimilarity flexibility artificial paradigm artificial originally traditional set difficulty evolve minimize quasi newton leverage
estimate give rise case bayesian without use cell single point bar one target responsible high expect contribution amount come decrease perform low sample negligible formulate posterior mean weighted posterior estimate concrete gaussian generality behave generally hamiltonian
strategy train predict take vote classifier receive party baseline favorable point margin support continue reach separable communication equal size run round ref stop reach underlie report communication cost accuracy applicable communication report number average run standard report send another incur word word describe assuming send classifier incur cost base find within small one value dataset test incur assume round send node reach cost round first account player observe small round
expect ei ei point max algorithm batch ei dynamically condition select explain algorithm suppose algorithm query start gradually ei pick second unobserved formulate q q ei close light sample close pick sequential ei line second know outcome whereas select ei method know parameter around curvature zero due closeness corollary square mean estimation show optimize summarize hybrid optimization batch
subsequent r phase eq therefore conclude fix pick unit particular follow turn later bind express coordinate depict I ii first inequality term first term various bound therefore thm orthonormal eq note suppose call subsequent call eigenvalue return permutation correspond permutation actually base case ignore assertion observe therefore assertion fix assume inductive least permutation inductive precede least call therefore quantity must triangle side separate another first assertion inductive inductive j assertion inductive thus induction principle exist hold complete variant stop key difference explicitly iteration converge within high condition tensor sphere iteration update eigenvector denote frobenius order ai ai symmetric stop eigenvector show stop condition condition hold I eq claim claim I k intuitive mean rise population approach prohibitive complementary take newton start often asymptotically latent evident well match observe intractable system multivariate fortunately class rich structure moment amenable descent include certain
general differ limited acceptance event characteristic integration variable experimental usually integrate symmetric although problem solve appropriate information incorporate act adjust smoothness unfold pdfs unity convenience indicator often commonly b spline eq substituting fluctuation account histogram observe experimentally value covariance bin column response
begin state measure limitation expansion wide parameterize parametrized parameter expansion taylor etc intuition statistical cast dynamical state special observable system output formalize follow make use necessity apply system light filter measure filter complex available approach polynomial spline smoothing filter implementation
index furthermore proposition issue see consideration develop poisson mass q function unity introduce variable poisson sampling use connection rewrite generic inversion assertion side sense one side use decrease integer set nest I I notation assertion alternate calculation show c summarize side via give belief plausibility I plausibility fisher true describe I frequentist observe plausibility show type ignore randomization issue correspond singleton assertion like tie
since core benefit cardinality suffer additional would boolean clause cm ti add pseudo boolean concluding equation describe result clause clause solver previously encode frequent constraint enumeration find frequent note encode exponential growth enumeration strategy opt adopt expressive enumeration prune part transaction approach rely heavily frequency threshold center frequent enumeration compact anti direction rely enumeration strategy solver output finer relevant valid strategy concept frequent frequent maximal either frequent follow upper option example may redundant transaction add must exclude base subset frequent frequent largely reduce impractical growth clause clause significantly high clause encode clause previously q clause directly find dual problem medium threshold fashion item frequent choice adopt
expand applicability general international dedicated research learn output space object difficulty encounter structured usual case reproduce elegant overcome kernel structural information relate kernel kde kernel kde reformulate et project structure rkh scalar regression input feature structure output pre kde beginning reduce component base measure kde kernels space approach permit exhaustive pre encountered kde kde kde minimizing capture closeness output
corollary proposition markov hasting mh inference challenge rejection mh scheme density proposal reject crucial drawback adaptive never proposal happen lead never converge overcome limitation adaptive target distribution strategy simplify sequence proposal algorithm outperform mcmc hasting mh rejection metropolis arm bayesian implementation smc know filter active area carlo powerful tool numerical stochastic simulation rejection mh single mode despite pdf problem adaptive never add region even procedure construct proposal g region density depend good imply indeed adaptation structural solve support mh increase markov partially discuss arm arm bound approach incorporate chain direct modification inside region mh learn past except current furthermore construction pdfs reduce effort require almost everywhere guarantee improve whole point introduce
spectrum conventional curve event estimate trade true discovery significantly hand speed essentially unchanged comparison experiment duration unchanged evident show present extend event spread probability repeat spread time recall represent arrival single scan expect number arrive bin single scan henceforth assume arrive give cause observation span generalize observe event cause one span poisson variable impact observe distinct event event drop impact event term vanish subtle impact weight observation observation weight include probability event exist identify event one form log weight event calculate set weight union time zero weight write event
core statistic us compute cn explain argument bound follow large uniformly hold prove proceed argument lemma n rough denominator since probability lemma fisher compare bind appendix lower desire speak several complex see operator et calculus orient circle center radius open domain whose calculus operator formalism prove contour complex plane lemma sequence define upper bind applying obtain refer counterpart calculus look definition contour change event lie circle circle almost work come nj obtain turn sketch eq decomposition eq proof get last us acknowledgement research partly b like
binary code output way replace definition sub policy binary learn complexity carlo training example state label training provide sub set combined final policy alg estimate ht kt algorithm identical slight distinction keep usual multiclass line map onto binary learn replace original action correspond code global define accord note ensure stability
forecast horizon ls stage forecast forecast approach good forecast spurious coefficient var contain spurious presence forecast less reliable column l l l ss l ss test week week forecast horizon air application concentration four air year california air observation previously component different var correlation domain ar partial correlation air specify range exclude pair stage coincide result bic var partial graph concern spectrum ar graph never result var contain zero spurious limitation substantially fit limitation partial since usually execute exhaustive pattern spectrum reach feasible numerical google trends ar selection dash apply display modulus I well see imply recover pattern estimate figure display finding agree series mainly large reflect major role intensity co relatively discovery finding underlying interaction air reader model
match letter individual index file file indicator ia unobserved verification record linkage match assumption latent simplify mixture field extension ci reduce need accomplish one restriction probability match denominator experience record linkage informally select linkage suggest et experience datum record linkage site study census record whether match find census record linkage application characteristic population study varied area impact linkage across percentage record record match correspond across across site among however variability
conditioning level th quantile laplace ratio different monte estimate change model change reference ht ad ht less dependent particular occur highly dependent row compare stem specify change asymptotically possess strong dependence estimate occur variable conclude induce dependence htbp cc ad show
replace function right equation obtain equation report value parameter marginal first enumeration acyclic graph graphical graphical often estimate involved step depth explore detail structure structure graphical graphical stand among model topology represents otherwise naturally split call separate step step graph structure arc direction common element depend however mostly multinomial variable local univariate bayesian markov former associate wishart respectively bayesian derivation identify arc dag arc element correspondence graph probabilistic inferential derive
correctness assume depend interaction notation emphasize let kernel underlie brevity instead thank plan g restrict draw accord abuse I j k r r r notation linearity trace k r f optimize drawing algorithm write return consider parameterize prominent dimensionality parameterize covariance per x traditional algorithm interval g discretization space leave might perform discretization space number continuously parameterize introduce kernel result suffer minima programming address fast search however combine kernel continuously parameterize similar describe randomized continuously parameterize similar denote plan k k brevity general resort sample auxiliary denote kernel parameter auxiliary dirac
assumption pass change bind surely outline procedure computationally tractable risk aggregate preference advantage datum construct relevance score preference discuss selection web information allow rank overview structure attempt preference preference relevance vector fit rank aggregation show limit aggregation whether reflect population detail framework description alternate david aggregation construct relevance phase preference skew symmetric matrix entry encode item recommend derive observe preference hadamard natural select square objective yield problem unnormalized laplacian preference observe decomposition I matrix note continuity solution likewise lipschitz skew symmetric variety aggregate pairwise datum skew upon specifically comparison mean distribution structure skew matrix point aggregation preference let grant count method score count time particular rate give skew score skew result ranking suggest one budget pairwise preference win opponent la eigenvector begin form reciprocal pairwise encode multiplicative preference item idea ratio strength generate empirical
n cyclic problem proximity project warm inner initialize nature initialization empirically prove guarantee discuss minimize without expand variable variable work p write advantage rely regularization overlap unitary furthermore ht q replicate much proximity depend overlap numerical comparison present aim family apply replicate index consider group amount entry take exact solution compute cyclic dual project dual less tolerance mean computing repetition situation convenient projection tolerance outer tolerance outer convenient resort dual optimal always subsection propose proximity benchmark group
assess measurement valid unbiased estimator dispersion assertion question determination location parameter minimum unbiased estimator huge induce pdf location control normal parameter denote estimator example generally address aim measurement world world conditional pdf vertical bar
kind concern efficiency inference reach I proceed proceed specify association structural model describes implicitly drive write result association describe subset know good assumption support combine specifie dependent assertion concern subset belief evidence plausibility readily magnitude assertion clear definition plausibility frequentist procedure plausibility plausibility consequence region nominal assume
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duality whereas duality gap notice duality function result methodology duality gap provide notably lasso optimistic path homotopy require solution enjoy exploit piecewise precision initialization invertible approximate direction x jj j reduce cm great direction
approximate map net definition notion message infimum ki li il il ki ik li il il il ki ik ik il il ij achieve message ideal message construct transformation subsection laplacian width graph simple notion matrix pass tree laplacian subgraph induce understand edge share subgraph induce cluster split decomposition exist factorization pass go dominant respective sum dominant approximately satisfy inequality message element assume non ideal message lie range intuitively effective node electrical roughly speak node net subset transformation note row diagonal definition row sum obtain stay follow approximate give pp net next prove round message pass recall make follow addition take tell round q far straightforward going follow readily induction k integrate w go
approximated value hold component approximate ready infer analytic stationary fourier domain truncation introduce transfer matrix obtain fouri obtain eq previous derivation inverse fouri fractional integral psd fractional integral independent process point way aim bound integration introduce truncation evaluation partition integral correspond rectangular evaluation read eq accuracy formula increase accurate scheme field treatment mathematical applicable satisfactory important
generalize cluster study simulate cluster define compose explanatory response joint partition model convenient type joint py py section broad canonical specify particular second derivative link marginal hereafter logit inverse thus variable binomial distribution apply discrete reason mainly interesting cluster differently mass moreover mass estimate gaussian finite denote generic result fact marginal marginal group extension
neuron spike strictly positive neuron period spike output come neuron neuron environment neuron spike neuron neuron notation reverse procedure perform else include neuron neuron neuron let external environment arrival positive spike rate mean environment trivial nothing happen also otherwise inactive time
author pay training require svm multiple requirement place label low desirable notion cost previously recognize barrier scaling notion characterize new source add example increase linearly pairwise similarity function label prohibitive propose jointly motivate extensive scale match extensive synthetic er work zhang valuable comment usa serious almost entity er prohibitive cost similarity exist literature er challenge er uniqueness characteristic synthetic state reality must scale superior constitute contribution entity novel contribution previously motivated er statement elaborate run multiple thorough world direction define dimension
consecutive finish phase action similarly distribute plug call mdp reward horizon node two denote latter implie expect mdp accumulate start state accord policy action union pair yield correspond take policy induce exactly finish phase apply action go last horizon variable know hoeffding hoeffding q verify bind markov way maximize coefficient decrease set lemma exponentially fast sample turn induction induction thing correctness eqs verify eqs iteration exactly bound eqs substitution respective eqs stem hoeffde partial sum e ty recursion q leaf node constant decrease valid rely modify modification induction induction satisfied grow beneficial might exponent expense coefficient
frequency c consistent single assumption tumor evolution process tumor population vice versa neither c frequency frequency figure either rule frequency triplet tumor likely pair tumor error frequency read note rule frequency rule estimate frequency tumor describe read count read copy profile tumor read implement non generative attempt explain frequency model visualization alone sufficient implicitly incorporate assign read column column parent parent child order always true brief parameterization allow represent discussion represent evolutionary distinct ii input reconstruct take union frequency copy appear aa indicate copy copy copy population frequency
wise prior mass pa distribution duration parameter prior collection duration generating exposition dependency component omit focus transition kronecker delta otherwise duration length switch model choice duration illustrate
insight reason unknown use trivial evaluate assignment validity suggest need easy answer need discover approach incorporate cope incomplete representation axiom answer query axiom formula verify nevertheless suffice complete restrictive impose concept utilize true probability formula game concept fair game prove theorem limited decision invoke axiom essentially system simultaneously reject valid preserve proof limited running assignment formula time either evaluate note bit claim run correctness first valid interpretation sample consistent summary see sample formula least hoeffding formula axiom solve accept may notion somewhat hoc weak appendix computationally notion likely infeasible purpose possess close decision solve invoke integrate implicit reasoning semantic standard largely due resolution turn excellent system surprisingly effective remain attractive natural operate clause inference resolution clause contain one appear infer find derive clause know clause typically actually use empty derive resolution incorporate hypothesis formula syntactic intuitively restriction wish lemma derivation syntactic capturing proceed recall give clause clause acyclic corresponding clause corresponding clause derivation clause appear input therefore prove derivation dag dag dag root edge correspond syntactic intuitive early possess decision restriction first establish resolution restriction effect proof syntactic formula axiom correspond clause derive restriction
smooth question answer question show bound infinitely go follow find give residual entry prove fw complete wherein solution sub problem initialize compute condition rate need problem dual iteration round minimize n w exactly overcome modify feasible accord change guarantee objective f k strategy additional spend sentence time wherein number get infinitely close go adaptively call total improve iteratively warm accelerate sentence check stop objective go infinity go theorem decrease take th decrease objective f immediately follow get fw precision fw h fw round substituting infinity without fortunately converge iteration converge iteration assumption usually stop time spend sentence successively respectively since iteration precision cost th wherein iteration empirically summary wherein round form factor therefore method scalable nesterov
process classify dft ds base empirical classifying threshold analytical empirical sample accord band different receive filter specific besides exclude psd band filter detection class different process classifier result correct classification characterize probable existence differently correct
random nn excess sketch hull smoothed marginal know rademacher smoothed trace norm rademacher yield theorem give supplementary material weighted specifically choose excess term smoothed weight trace norm new max smoothed norm simplify suppose choose excess max excess max come cost error guarantee require rich rich test local trace smoothed noisy entry trial draw random sphere moderate
reduce hdp random geometric explore nb across count restrictive nb nb constructed explore share nb analytic posterior comparable conditional posterior process conditional infeasible represent address atom gibbs gibbs nb block nb brevity infinite problem address small small constant modify integral whole risk level avoid truncation slice dirichlet auxiliary slice sampling nb count mixture amenable impose slice leave study block sampler slice represent simulation sampler employ collapse inference atom collapse usually marginal usually collapse conjugacy prior word atom likelihood atom develop collapse nb collapse rule future variety crf hdp latent marginally distribute fix learn crf fair discrete atom iteration gamma initialization enough last uci toolbox restrict five document corpus select word jk v great work appropriately averaged testing partition
power modeling law kernel generalize laplacian distribution mechanic positive definite insight regard correspond hilbert rkhs significance economic field law different finance traffic definite kernel study literature information quantity define measure base interested turn laplacian cauchy machine us term inherent
combine variance decrease rapidly enable ascent order cardinality cardinality observe reduction pca easy pca reliably apply rigorous pre processing couple life typically exponentially decrease allow many feature eliminate introduce block
sample sgd form j suitably scalar definite assume sample around locally optima dimension problem separable ignore diagonal omit gradient rewrite sgd unit variance h derive rate minimize loss whenever direction learn overall usual close optimum anneal schedule automatic almost surely converge subsection trivially parameter separable appendix full useful component
condition reasonable assumption tie interested trivial example exist inductive ml appendix brief mathematical technical input output x x l definition convenience convenience compact eq set I probability often adaboost within boost condition rarely condition useful specific part regard relate adaboost example weight perform essential main weight zero state later implementation adaboost consistent implementation update stay away low weighted hypothesis hypothesis close function make mistake easy imply negative incorrectly classify weak think converse third perfect stop weight w implication condition presentation state list dataset input note part convenient associate representative adaboost margin set whole feature call extensively use error hypothesis mistake equivalently mistake say mistake mistake eliminate mistake removal sound dominate hypothesis select dominate weight adaboost call fact mistake mistake associate e turn convenient classifier adaboost similarly mistake equivalent state early replace matrix form row column construction vector mistake hypothesis differently encode incorrectly whenever learner hypothesis effectively reduce common scheme apply pruning procedure remove mistake contain repeat repeat unique representative say mistake mistake mistake repeat mistake select adaboost execution adaboost select adaboost weight remove mistake equivalently correspond call subscript state previous frame adaboost system fix element representative update sound dot w boost context say differently value weak learner want emphasize simply learn adaboost adaboost classifier optimal characteristic margin introduce make traditional initialization replace surely drive fundamental almost e denominator adaboost depend round numerator adaboost decide whenever error implementation mapping dataset selection implementation scheme introduce hypothesis matter presentation adaboost cycle learner behave example execution state early informally adaboost appendix case say implementation adaboost implicitly deterministic mistake mistake implicitly mistake mistake low exist proxy define adaboost adaboost implication preliminary definition useful mathematical example tx tx adaboost build classification correspond effective label converge infinity hold grow replace change another margin formally bound positive weight round expectation hypothesis normalize similarly tx tx converge state practically present training say something strong behave optimal adaboost classifier effectively converge input
naturally generalize covariate estimate historical sized bootstrap deterministic still variation tail binomial draw analytically complete calculation straightforward numerically give size draw tail see appendix bootstrap averaging observe sized guarantee step tend infinity interval f analytically typically describe bad issue partly effort alone evidence plausibility misspecification conclusion interval reflect inherent difficulty correct tail select compare comparison approach example fully description plausibility law fail likelihood statistically wish produce confidence aid decision averaging pose risk inherent tail inconsistent appropriate frequentist currently insight draw rare event law variance fluctuation tail event large generate synthetic particularly challenging great fluctuation fluctuation perform even sample less event decay small may deviation estimate percent approach historical observing event global database number highly
bfgs hessian fall optima quickly figure make quickly variance large large update huber sufficiently find result analytic gradient factorize px contrast analytical draw average analytically optimize hypercube solution proceed choice far rise specific quadratic go replace objective smooth optimize typically smooth version make smooth classical approach potentially constraint relaxed relaxation optimum combine relaxation variational
set proposition let rademacher uniformly collection rademacher average gaussian compose complexity conditional bound scale rademacher every continuous additionally conditional comparison function point relation due zero separable gaussian gaussian essentially say bound another first maximum analytically difficult easy order let iid accord overcomplete useful bound dictionary overcomplete construct difficulty cover encoder stability concern hold enough guarantee sample weak exhibit proof overcomplete learn depend learn brevity quantity lastly proposition empty rh whose stable code guarantee code strategy fact strategy al turn notion provide good visualization illustrate guarantee code sd code stability code margin
run draw reduction determine input generator hence generate obvious generate show generation randomize take input access uniform n c central proof follow random consider second describe draw permutation compute use walk start output reach end suffice first least second assume eigenvalue graph imply location reach conclude range study polynomial finding several remain outline direction goal wide class several class seem investigation intersection note generation counting intersection independent pac know integer tree improve standard tree fast time note assignment tree reduction rely crucially implicit distribution investigate free quasi branching program similar determined computational generation wide range threshold future goal result wide range understand intersection monotone note counting former latter result natural language quasi branch program gaussian intersection pac know another interesting match boolean satisfying approximate bipartite preliminary suggest range namely color approximate generation admit rise forward inverse generation markov mix rapidly cut partition component sparse cut small random space least example belong uniform since generation possibility give physics formal ise critical temperature temperature spin come reduction fix boundary bring mix existence cut chain know color cut state approximate uniform generation even chain generation mix interesting chain mix polynomial yet cut formally ise mix hand spin make acknowledgement helpful approximate suggest signature scheme helpful acknowledgement helpful theorem claim definition remark pt nsf award foundation grant california berkeley author berkeley fellowship university support nsf university generation focus assignment function inverse problem assignment belong formula goal output variation sort type uniform satisfy assignment inverse generation type call speak approximate uniform query inverse inverse signature certain negative literature formula intersection function formula forward generation hard certain type code hardness inverse combine construction plausible hardness easy generation computationally generation easy combinatorial research theoretical decade complexity know generation fundamental topic wide approximate generation perfect e linear threshold instance problem uniform describe briefly approximate generation generation combinatorial object object speak output must put every taking
equation finite one exist number cn c number initial k surely exist k c exponentially exist k l analogous summation fact random l decay coefficient cauchy real great integer stationary coefficient first elementary component eq mix proof impose immediately result think generalize original multidimensional justify expression eq stationarity covariance depend n k separately move representation summation index exceed cauchy schwarz decay c indice small thus sequence satisfied l normally normally zero fact notation proceed asymptotically normally asymptotic infinity lm step devote first stationary mix strong process condition device convergence multivariate l converge mean apply limit mixing obtain converge l absolutely note treat restrict one exponential reasoning show latter term observe cauchy schwarz step r limit chebyshev every proof
cluster outlier discard algorithm place partition traditional graph construction employ set statistic neighbor denote distance list number neighbor theoretically near lead randomly split datum calculate eq versa rank step time final low indicator smoothness pdf thm minimum average degree point cut typically rather successfully solve capture area cut ensure degree distant unbalanced cluster unbalanced
eq simple adaptation either ml em continuity argument use consistency adaptive describe condition set maximization addition addition continuous r conditional p hz py conditional true continuous require parameter log maximization function run dimension assumption parameter consider empirically order identifiable surely identifiable surely remarkably parameterize correct know non sufficiently set guarantee equation eventually enter set see space gaussian parameter context compressed consider lasso max many performance mmse main motivation gain compressed sense example
f fa f ff sa fa argue sa redundant finally follow supremum set bound f h combinatorial envelope redundant sa fa fa sa ia sa sa trivially picture rather combinatorial define whose face set symbolic sa f value combinatorial envelope positively decrease obviously restriction odd much relate actually norm different possible generalization group overlap connection exploit parameterize tuple precisely norm provide address immediately values propose abstract explicitly presentation norm support introduce view generalize w fa illustrated figure would induce pattern small support motivation call support fa view relaxation cover rigorous link rigorous envelope l w fa unit cube give combine fractional yield since low combinatorial envelope
weakly convergent attain limit pair x g v I integrable p g v integrable integrable apply lebesgue convergence h nk ng n k km l k k l h h n thin l v k hence mx h nx h mx weakly converge pick last hx assume measurable nf want b consider integrable n x apply theorem n nx integrable furthermore get dx k furthermore fine concentrated trivial bound think integral check integral lx restrict measure fulfil establish well state rigorous preliminary definition bc kb c mapping song define rather song eq song state theorem song smoothness hilbert schmidt boundedness although scope xx yx xx state exist n h schmidt vi use cauchy schwarz follow together lead exist integrable fail km z contradiction case justification derive
maximize response last third parameter expectation unique generalized maximization semi effect mean distribution parameter coefficient via book association et al one variation act response source effect definition multi effect know measure similarity effect structure unknown arbitrarily associate component th identifiability additional constraint restriction element covariance write array covariance rkh space
kkt condition work fix concavity value derivative decrease modify active method appear previous organize introduce path mcp penalty section detail regression poisson present lasso simultaneous selection estimation broadly section glm additional design matrix intercept coefficient find maximum exclude little abuse notation therefore minimize glm assume twice differentiable check kkt q easy condition q decrease poor predictive follow construct decrease logarithm kkt activate coordinate together formula calculate appendix logistic intercept penalize warm start keep mind outside additionally typically correction necessary get solution correction user
distribution heavily thus sensitive capture capture work fine synthetic multiscale dataset reflect explore effective affinity agglomerative u perform among robust affected link mnist c add image sc cost bold statistic show c c sc graph base cost fast sc link comparable image initial detect outlier color correspondence robust object vision integrate correspondence cluster range potential application recent agglomerative correspondence cluster acc graph gs respectively present acc gs outperform sm greatly
bar cd color mark option solid dash explicit mark error title xlabel ylabel cycle legend pos bar explicit mark option thick title ylabel black white legend pos north west blue mark thick bar explicit mark solid bar cd x cluster title xlabel edge ylabel list name black legend pos north west bar table mark mark option solid thick bar cd mark title xlabel ylabel list name black legend pos north west blue mark x thick cd mark color mark option thick bar cd mark expect two number ten complexity
concentrate early learner improvement mn l remark supplement x concentrate constraint need since usually seem complicated show learn provably subtle simply bottom cube integer quite join follow measure measure cube eq lebesgue measure notice component define
center let column entry specify dispersion parameter independent wishart determine stage diagonal linear ij ij g ij estimate generalize linear pooling represent variability determine parametrization predictor x subject covariate I equal specify old reflect response underlie correspond partially circle undirecte depend circle variable circle hyperparameter repetition corner specification linear mod example identity link linear mixed lead vb importance assess center parametrization center efficient gaussian partially response express offset approximate bernoulli response specification response quasi keep posterior beginning variational pass machine vb factorize conjugate density exponential without derive bernoulli assume belong pass
hard inequality hold problem special appear considerably hard concentration iff ph thereby iff ar clearly obtain estimate measure begin serve introduce quantization exist compute extend produce sec focus two measure prove sec mean intermediate optimal quantization broad exist
bayesian formulae formulae equivalence frequentist bayes eq find several formulae q jeffreys
detector practice might reverse initialization aspect heuristic appearance improvement yet almost par detector multiple scene suggest wide detection cc split split template spatially flexible template appearance characterize connect learning involve step box part disjoint possibly interpretable profile vision critical discriminative reflect correspond really experimentally might actually reverse aspect show
series multiple physical taylor expansion determine avoid physics device illustrate procedure calibration information present effort include apply g scale level reduce computational explore quantification compute research partly department nuclear security award de university support also thank prediction provide network method multi physics integrate adopt calibration uncertainty extension various scenario natural uncertainty due uncertainty calibration physics investigate calibration form interval determination parameter expression physics model multi experimental resource limited expansion calibration due structure know save taylor series expansion base discussion likelihood complex physics calibration aforementioned calibration modeling device interact dynamic computer model quantity actual quantity distinction represent specify deterministic contrast
explain vector discrete probability dirichlet index different base multinomial label w component unity zero nm r label multinomial classification straightforward observe class posterior classification derive na bayes address hide represent mi joint
forecast forecast high forecast greatly reduction sample mse mixed generalize often high want use eqs process fully stochastic researcher predictive simulate realization simulation condition eqs initialize pt go simulation simulate spatio temporal simulate realization fig couple column influence start soon similar right trace show solely fig pattern qualitative mixed reconstruct spatio
need use common function extra etc commonly method implement package primarily literature reasonable preferable bound g second namely ij conclusion term accuracy make focused attention sample simulation table ahead motivation study section simulation finite performance group least estimator select group sl penalize estimator apply select estimator call estimator minimizer penalty theoretically implement cubic spline evenly distribute knot penalty q certainly option choose cross aic intuitive implement pl estimate automatically
phenomena number call curse especially np hard present section latter use protein investigation probably important limit phenomenon molecular model exploratory former intensive select heavily available prior distribution size size arise model encode reality context informative prior posterior undesirable regard advantage flexible specify assumption less dominate non informative well understand posterior ease system biology monte
lee xu tree scan get traditional refine term alternatively extensively compare approach optimize fit reading structure ultrametric think read observation hierarchical base bin bin bin search operation endowed operation case constant time ultrametric dendrogram carry logarithmic ultrametric address big massive data
stack tensor tensor minimize eq form term second independent describe let j kb ti ib ib ib ib ib ib ib z ib ib ib ib ib ib order generalize cycle sequentially minimize index minimization cf consequence somewhat consider minimization yield yield r ax yu ib ib ib ib ib ib ib analogous expression hold complete provide user rate test attribute rate close return explore slightly rule introduce conclude rating
divergence enough chapter supremum sup give conditional guarantee band roughly speak require desire efficiency measure band minimax finite coverage efficiency standard regularity drive resembles confidence band regression inference trivial band exist hand band mild distinct band nonparametric band nonparametric interval bootstrappe constant include bootstrappe none furthermore produce fact aware provide free band optimality marginal validity predictor introduce notion efficiency section band selection conclude remark band extension region
rely assumption marker measurement longitudinal roc datum suppose marker patient suppose repeatedly measure marker marker marker marker u auc auc sensitivity define real function derivative statistic compare two longitudinal marker linear broad roc auc evaluating marker auc auc statistic allow patient marker marker time two marker point linear high marker positive bound
neighborhood svd pmf variation item average item nmf tend less variation two neighborhood cf perform considerably significantly user slope insensitive count pmf variation item count mae rating fix user display near mae bottom follow baseline item remarkably perform seem svd cf dependency density level dependencie density difference prediction item slope one pmf three nmf relatively slope pmf significantly baseline previously trend fix quantity arbitrary examine dependency prediction count density mae item fit multivariate dependency mae count figure contour mae axis panel cf panel
u du let choice resolution lemma sm sd du ax true eq odd eq coefficient main map prove u u du u u notation du b du coordinate expand du nu dt pn du du du dt k substitution du u pn du pn du du du pn eq schwarz lastly eq function complete contain sd u sm complete
around maxima maxima quite small advantage pdf well suit optimisation intrinsic multiplier sa difficulty find lie
establish hypothesis enable distance apply lipschitz boundedness evaluate hessian log hessian suggest approximation ij define parameterized lemma appendix vector write choose give eq conclude consider equation bound lemma recursively relate constant
grow least see also theoretical algorithm smc approximate pseudo likelihood incremental weight observation subsequent transition nu adopt drawback algorithm grows typically maintain must advanced strategy investigate rejection note differ smc particle shown every realize versus mention smc term smc article modification notation static markov carlo operate standard distribution interest marginal metropolis q concentrate presentation abc procedure step resample also proceed depend store p k n accept reject probability accept ki ni ki ni ki approximation estimate smc extend include summation extend
fix drift establish situation transition share point transition example communication although never situation lead whenever inequality hence condition require drift satisfy markov equation theorem set visit infinitely define family lyapunov w satisfied indeed obtain note loss c vx suggest possibility precisely characterize return form drift know cc v x convenient prove exist hold cm w satisfied c follow indeed consider deduce x consequently existence exist deduce c existence x upper vx vx role play control practical relevance possess desire control driven approximation recursion describe root
square form ordinary obtain term maxima random apply norm q event uniqueness concentration let projection subspace entry mean sub universal second absolutely continuous unique suppose hold linear strictly minimizer turn cone z solution concern second trivial conversely assertion third fact lebesgue probability use state uniqueness follow position recall conversely hence hold contradict show treat event less set state readily fact optimality problem yield result immediate reasoning indeed scheme appendix bind establish proof span let span negative sequel threshold note contain event span corresponding fix positive conditional controlling usual exceed assertion establish coefficient kkt optimality satisfie employ minimizer low handle back system maximal component cf fulfil indicate negative lasso suggest literature minimum coefficient obey probability shall build previous paragraph whose gram constant denominator convenience constant well compute order give active j trivially conversely bind second latter scale optimality imply bind side similarly active set unique slight modification proof back substitution apart modification place eigenvector eigenvalue turn step active kkt follow inequality eq back
multi goal regularizer edge kt regularization gaussian regularizer drop graphical lasso term originally minimize original smooth upper provide multi multi task lead problem multi penalty non uniqueness boundedness maximization regularizer smooth task learning dual norm dual virtue swap min furthermore note solution get kt n k find optimum primal dual zero k block descent problem contain regularizer method descent rarely converge point correspond zeros precision minima among primal graphical simplify
column column single row general goal understand fundamentally hard version competitive side refine classification dual price key ingredient together lp bound geometry classifier pack input priori either reveal coarse input probability full lead yield management inherent accelerate development arguably study program application include ad management indeed uniform constraint online understand reveal much develop packing unknown require value column offline scale generality packing non ignore satisfy property column randomly approximation allocation ad
single correlate protein uncorrelated simulation block protein simulate protein control jointly patient change covariance mutation patient usual discovery method see outperform group moderate induce suffer problematic panel protein patient plot effect procedure grow become increasingly difficult gene interaction interaction exhaustive exhaustive run section gene dataset patient patient patient microarray along gene use gene measure multiple find interaction cutoff find significant find entirely small roughly nan hypothese true small regression surprisingly drop however unfortunately marginal interaction
unbalanced naive cart boost forest method approximately response significantly unbalanced show estimation main present classifier build structure tree classifier unbalanced occurrence minimum support sup minimum four estimation performance ht sup specificity auc classifier area curve learn min sup covariate consist aggregation bagging error error correlate take area mine large database express potential valuable relationship viewpoint weak classifier derive already relevance real variable categorical nominal elementary notation pattern categorical variable hence event pattern probability occurrence coverage index
supervise reduction pca bring designing framework change calculate assume incomplete namely unlabele search set self look subspace give increase separability absence detail calculate sample word lb self equivalent c influence control way derive call include xy xx yy yy please section summarize discussion extension calculate design another dimensionality categorization audio index appropriate three appropriate correlation find class introduce cca
hand side side go associate shorthand associate contain associate shorthand associate association edge associate edge associate measurement edge edge go associate measurement flip associate either edge associate go associate convenient adopt shorthand measurement associate refer association index understand refer go prove hand side big edge associate suppose index happen repeat z time measurement additional I z I z z rt proof similarly associate one remain due association assume former similar prof conclude piece observe continuity strict
add package namely first author ep lemma department computer uk ac continue method binary prediction problem predictor calibrate independently property introduce simplified predictor chapter property form property complicate whereas version behind way validity call part introduce technique uncertainty probabilistic facilitate various loss interested measurable z n ni measurable equivalence output denominator predict predictor hull sometimes refer variable take perfect get predictor satisfy equality say calibrate small
enable readily interesting show correct paper discussion author sparse popularity apply readily
infinity ultimately strategy theorem ucb value optimal well future prevent grow ucb bandit one comment contrary armed horizon small able interesting elegant derive request good mass minimize request give answer request number request oracle close assumption ucb probability least informally lag oracle restrictive assumption expert variation next another clear good ucb ucb use recall ucb attain expert note get fact event use request expert uniquely see
average either average finding capture cf model scenario herein apply gaussian example example label use comprehensive recommended index ari use ari evaluate herein present datum chemical region physical chemical r fit setting well bic ignore bic bic classification associate respective respective classification four merging probability slight ari go average inferior ari good probability window discuss seven model ari three model component average merge good ari compare average window good model window bic good bank available six old bank bic use
prove quite cluster cluster important classification system classification major release computing figure google grow retrieve title body increase decade closely term figure cluster post detail decrease relate historical follow overview computer science school development lead google justification albeit cluster lead science replace entirely principle general automate service start year publish team carefully page york city company
undirected node edge recommendation user follow acknowledgement describe inspire make recommendation represent web link arrive movie movie movie movie direct node item information profile weight specify rating item tag profile category weight relationship mutual twitter graph collaborative
control evolution say know literature mdps treat transition matrix observe build assume identically gaussian inferential convenience process model interpret two quantify expression right controller error thus characterize mdp mix r rx rx ax tx separation state exist mdp observe maker entirely determined also implicitly infer govern criterion observer collect apply controller inefficient observer observer controller control transition unknown approach case however controller inference henceforth may policy behavior system person omit choice intractable integral henceforth invariant scalar eq assume selection scaling observe outcome alternative function utility interpret covariate exist posterior improper parameter identifiability example zero suitable remain zero sec probability
topic document nonparametric hierarchy together inferential topic model include collapse gibbs complementary relax exchangeability correlation idea technique work tag meta importantly multiple widely vision image hierarchy augment spatial spatially none generation multi previous factored sharing even compact factored diverse multidimensional cluster modeling topic build upon classic whose flexible user rate dirichlet number group little place
neighbor ising despite simplicity modern image reference parameter active research validate control boundary condition unlike mrf particular mrf compare different compute chain ghz intel matlab ease visual display logarithmic observe agreement
maximum variant instead dataset examine general depend rank advantage gp contrast neural specification hyperparameter handle principled automate determine hyperparameter probability reward observe logarithm plug optimize solver conjugate able evaluate partial incorporate regard indicator generalization capability measure equal flexible effective achieve fall quickly rank sr approximation sr possible penalize frequentist definite parameterize scalar parameter square exponential
address theoretical algorithm separability penalty unique coordinate part hence second directional exist close relate nonconvex model algorithm solution step satisfy limit stationary simple initialize criterion although graphical
tr tr network I tr k compare strategy profile description use kronecker always stochastic doubly cf eq conclude hermitian compatible dimension profile conclude relation list conclusion diffusion combination matrix construction leave doubly need construction couple select construction originally symbol refer likewise symbol refer second laplacian matrix selection another degree choice assign average combination rule whereby type l rule construction weight largely dependent degree connection influence weight neighbor appropriate weighting ignore across sufficient solely connectivity node design combination aware network devise strategy able adapt statistical adjusting combination assign relevance network formulate combination adaptive become adaptive perform reasonably illustrate consider generally block already view network appear conclude justify relate expression series scale property independent instead minimize alternative minimizing use express combination follow q node associate noise measure amount trace covariance shall noise solution refer intermediate neighbor meaningful large rule appear eq stochastic covariance across size variance across network simple averaging table table evaluate neighbor term know realization learn product particular happen neighborhood neighborhood procedure product recursive motivate recursion write eq side use steady consideration fact hand depend product fourth desire enable let denote one recursion one become product optimal provide adaptive construction diffusion rely neighborhood construction select combination motivate adaptive manner aware variation profile also perturbation exchange neighboring factor channel study degradation noisy straightforward proceed subsequently presence operation exchange account node indicate noisy node source flow mind receive neighbor vector noise scalar e send refer variable noise measurement ki exist locally model receive spatially zero noise process perturb adaptive perturb quantity ki original quantity quantity subject exchange interested examine whose recursion largely shall emphasize main deviation presence may diffusion case noise exchange account additional show general begin aggregate represent exchange combination expression aggregate exchange noise covariance neighborhood scale covariance block follow scalar eq absence exchange datum I aggregate noisy note perturb hold exchange datum reduce introduce exchange see version denote receive scalar noisy node mean contrast mean follow study general zero shall limit discussion exchange henceforth exchange estimate block diagonal block repeat lead arrive recursion repeat recursion link replace arise influence driving appear involve reflect exchange weight involve account exchange involve account involve account exchange exchange weight recursion simplify expectation accord recursion recursion encounter
layer pre compare na I classifier nb principal analysis word paragraph context speech conduct automatically resolve language lexical ambiguity extraction level learn addition typical engineering natural language coverage feature similar deep learning behavior belief conduct compare english lexical lee evaluate various claim vector machine svm classifier kernel analysis kernel feature result show na I
study auc consistency loss introduce calibration prove calibration necessary insufficient minimize finding distance consistent auc addition derive regret bound surrogate loss equivalence loss auc exponential find auc surrogate auc surrogate consistent surrogate loss proper adaboost infinite use medical exhibit well theoretically parametric semi non parametric assumption auc apply mining rank incoherent aggregated auc regard rank especially bipartite ranking prediction auc
inference incorrect make scalable area key acknowledgement national security engineering technology dr influence call confidence interact denote actor position position actor actor actor differ actor change specify roughly actor keep percent position percent original actor interaction appeal underlie parameter partition consensus eventually consensus consensus large common point partial consensus regime value exactly actor attract asymptotic actor partition location toward empty sense adaptation analytic replace take dependent yield focus apparent actor become latent second high dropping term q right side eq side summary
universal adapt play formulate online inefficient relaxation meta relaxation devote complexity localize particular convex localization play adaptive integral turn provide idea define penalization localize analysis play localization give shrinking need unlikely rise develop illustrate example emphasize localization idea place set set cumulative minimizer empirical minimizer henceforth notation time th block inductive bind game rather ix history fast idea form trick notice player history meta previous play game localize adaptive size successive ix localize ix might empirical define radius set enforce bound exhibit choose parameter initialize block local sequential complexity possibly rademacher meta rademacher property replace localize upper instance replace rademacher bound integral bound set instance latter loose pp p trick course advance idea appendix admissible notice grow linearly block quite objective close strong relaxation ball localize remark spirit proof function demonstrate localization case guarantee take without advance certain call imagine expert gain clearly winner minimizer game since else work
take robust consequence polynomially combine theorem robustness linearization linearization mistake logarithmic dependence make decrease tolerance achieve well corollary labeling small polynomially mistake unweighted star mistake irrespective specifically irrespective assign central star actual ready operates weight shorthand notation q span spanning weighted graph mistake satisfie run span ratio mistake relationship similar linear corollary quantifie similar corollary phase linearize discuss subsection span unweighted weight graph complex outside scope paper naive run memory tree describe phase linearize line initially terminal traversal make node leave construct add leaf usual maintain whose label reveal list figure constant implementation node adjacent bottom reveal grey doubly depict traversal predicting stop soon mark traversal begin keep go determined determine leave locate respect go traversal mark connect start go use mark find goes reach right via node within determined quantify key observation internal gets visit trial visit mark first second visit mark mark preprocesse operation show time trial linear trial instance first trial section comparison real weight different
euclidean distance kernel careful svms rbf show near neighbor predict amongst neighborhood
proportion vote entire crowd indicate consistently high baseline indicate effectively member crowd identify pick crowd baseline straight note separately unweighted performance value baseline demonstrate random yield vote make informed member crowd template initial subsequent added step scheme affect selection vote test effect q high latter additional select three remove capability whereas make limited rather tend third component seed random without quality combine voting weighted vote
indistinguishable quadratic base fail reject hypothesis performance integer cause occur detect sample right outperform well frequency reasonable practice yield test rkh distance distance coincide testing parameter energy rkhs machine researcher associate testing chi inference particular problem reproduce one e statement term directly term valid reproduce x product product start expand em xy x xx yy x xy xx yy yy yy xy xy xx yy expansion
euclidean space state apply respective member definite kernel readily definite ari light graph al positive definite structure review relevant highly forecast form probability future retain class measure hausdorff quantify probabilistic
nu nu break nu k false n alpha beta alpha crp factor lambda new I value home paper n net home paper true code home exp home paper home paper true home seq shape nu lambda lambda n sim net home paper sim home sim sim sim sim cn sim home sim home value sim home paper nf
estimator define distribution pool define argue x fy comparison density interval du density estimate distribution density choose distribution density start whose goodness test du du du yield gx interpret transformation du goodness
achievable rate parameter performance limit infinite asymptotically recognition error completely complete side exploit side difference achievable margin since improvement achievable margin model viterbi algorithm recognition limit baseline information error text simulation indicate recognition test label reach gain baseline efficacy incorporate access able provide gain
condition depend database dependency computation query considerable subspace framework recognition constraint point notation close generate rv repetition projection repeat candidate scan return main choose close nr q iid cauchy nonzero constant notational convenience write first minimizer comment several perhaps gap close subspace subspace suggest reduction proportional notice weak dimensionality reduction impossible state overall implication guarantee constant probability easily take failure drop exponentially suffice generate candidate subspace candidate nearest second resource reason especially small query desire amongst query subspace retain project basically query reciprocal nominal gap near negligible fix nr r great random provide gap clear enter current extremely bind issue lead cauchy iid cauchy iid
effectively independent empirical suggest approximate theoretical designing topic model statistical recovery parameter algorithm provably meet separability assumption separability require anchor probability topic contain anchor guarantee proceed step anchor anchor second moment word occurrence provable assume correlated choice evidence economic economic run sample empirical program anchor reduce anchor inversion unstable generate topic three contribution replace programming combinatorial anchor selection separability prove presence second present interpretation
back result ode proposition recently ordinary relate incidence equation incidence incidence ordinary ode describe relate incidence article incidence interpret inverse pose inverse
variance bins correspond group bar center minimum bin demonstrate actual observation prominent model algebraic characterize extensive necessary combinatorial observe compute transition explain explanation relate one technique aspect study aspect present identifiability largely independent matrix completion apply demonstrate argue exploit problem beneficial practical algebraic combinatorial statement practical ignore conversely algebra various interesting therefore involve field valuable rt new european european framework carry fellowship ex european research framework agreement author universit completion intrinsic approach apart introduce combinatorial rank whether complete complete interpret version extend exposition hoc way statement algorithmic implication quickly contain principled application read reconstruct matrix position occur practically collaborative link prediction netflix correspond rate movie reduce complete arbitrarily partial single adapt combinatorial observation reconstruction full validate algorithm identify low allow study via g include answer main question
value order quantity alternative formulae solve base recursive backward next explain backward probability hmms evidence evidence hmms formulae quantity convention backward forward quantity observation prove apply rule independence establish classical inference forward quantity hold backward straightforward consequence corollary sample recursively markov sake completeness present hmms evidence formulae obtain modify formulae backward hmms evidence formulae impose add formulae quantity convention si forward quantity sx rr precede constrain evidence b extend formulae
parameter blockmodel restriction asymptotic mle blockmodel blocks edge bernoulli edge partition membership specifie matrix observe undirected loop sbm evidence community small grow slowly two ensure edge necessary control otherwise dense prevent become grow tight connection slowly grow network allow probability necessary quickly quickly high sbm restriction equal let interval small high tractable large propose recall denote partition arbitrary
identify predictor improvement promise consideration critical office center center support office environmental office science random uncertainty replicate relevant estimate df df nr nr clarity outline model standard simple training group split effect avg avg lar lar yes lar lar selector selector yes rr rr yes regression ridge yes rr correspond ridge assignment avg describe lars ds average divide highlight significance previously section section paper lead basis pursuit
element chain maximum decay partition need relevant scale explore decay suggest attain require obtain sample dag dag partition reach short sample construction much chain recall standard take hybrid reasonably advantage chain possibility short example disadvantage potentially acceptance operation probability chance get acceptance move application dag example restrict simply produce graph meet requirement complicate restrict density speed uniformity remain graph enumeration graph markov restriction incorporate enumeration example impose dags pair record arc also reverse order arcs make check result slow dags fraction dag increase possibilitie grow modification remove enumeration apart possibly stage enumeration dag number incoming arc receive simple allow connection avoid dag arc change restriction incorporate dag easily reconstruct dags arc binomial arc separately limit previous dag limit arc one complicated arc newly add mean
predict acc classifiers tie evaluation perfectly match acc predict rd tied rd predict early see metric system validation bad blind table predict metric metric acc pre predict match predict identify fairly consistent depend interest investigation blind crowd aggregate pseudo gold classifier average crowd exploit rank developed crowdsource track measure measure vs alternative classifier rank accordingly rank finally compare correlation determine achieve great correlation evaluate learner expensive represent bottleneck retrieval ir expert foundation evaluating paradigm enable evaluation ir scenario collection ad hoc item arrive accurately show ir operational increasingly manually traditional least insufficient label shared ir particularly sample technique stable labeling bottleneck
subject figure nevertheless ml indeed denote substantial incorrect consistent score part plot well represent total albeit think assess sd quantile normal monotone ill pose variation case bandwidth score even represent indicate reach around score translate subject get correct kernel total commonly shape assumption strong negative skewness note hard moreover differ two subject form gender differential literature occur subject belong group different choose recent strategy illustrate datum come efficacy conduct center california study concern effectiveness useful part via bank ask agree agree agree statement indicate favorable preliminary directly
exhaustive explore adjust behavior table need efficient estimation procedure minimal determine order residual remain outli subset contingency table fr partially contingency table outlier poisson subset cell count minimal pattern likelihood estimate thereby expect criterion minimal position couple performance develop outli identification method keyword contingency robustness outlier outli omp cell outli minimal pattern result minimal pattern independence method discuss conclusion comment poisson may model structural column design vector give notion far
protein bioinformatics matching view return target node label relation role type information store learn edge bipartite ordinal label rank ranking nothing occur second relation represent namely differently relation kernel cover multi respectively ignore condition object know light explain ranking object define total conditioning fourth straightforwardly reciprocal meaningful domain main setting extend regularize square rank iterative exploit kronecker graph million label relation incorporate consider reciprocal relation via corresponding product namely study reciprocal kernel prove edge direction machine also reciprocal kronecker completely generalization show basic ranking task expressive function indicate likely generalize rank learning set scalability considerably large state solver fast organize formal theoretic review edge allow model two reciprocal base derive algorithm connection difference relate learning emphasis section present promise world term computational introduce label provide represent underlie certain relation take real straightforwardly appropriate vice versa notation kernel consider parameter datum cardinality input consider select hypothesis algorithm function parameter theorem minimizer admit dual associate rkhs feature triplet correctly aim minimize follow
rank collection reduce theoretic contribution yahoo rating well comparison fisher rank another schedule game cluster within division poor ranking conference scheduling experimental synthesis connectivity ranking variety alternative address inherent inconsistent consequence rank comparison old recent theory ranking pose alternative dataset comparison refer oppose comparison pairwise preference comparison express q data subscript rank satisfy collect expect rank tradeoff ranking follow question collect additional pair maximally propose algorithm rank design community lead choice include constraint specify collect rank complete vertex alternative arcs arc yield find weight desire spectral design second theoretic question primary contribution previous ranking amount briefly receive recent paper os assessment crowdsource experiment without ranking like collection preference propose datum construct example algebraic os
change exist analogous procedure likelihood every maximum lead q nan maximize nan following unbounded treat section whether change occur change stop detect receive repeat compute propose occur contain monitor discard observation become determine monitoring false alarm constitute change occur ideally would equal generally compute usual expensive procedure carry real corresponding store arise example involve conditional variable sort fashion small order threshold sequence
within copula model copula compute mean function basis appropriate optimize copula conditional effect procedure resort pt training model predictive density hoeffding mean elaborate volatility financial examine efficacy covariance model daily return index include exchange rate asset matlab application first scenario return daily price series business daily composite uk sp third business day daily effective sp month series benchmark assess model train contain daily return scenario training datum vector shift ahead series contain index remain
present present present large replace follow differ hence expense factor sample probability nice norm gaussian exponential tail column factor argue probability row factor follow optimize regression give construct minimize probability least entire dimension extend point subspace dimension note analog solve singular hyperplane pass take regression sometimes solve subspace multiple become optimal separate regression problem recover appendix reveal vector continue hold matrix continue hold imply desire shrink additional union input summarize fast construct solution run problem vector regression save factor regression approximation essentially overhead regression matrix separate regression regression regression svd separate regression regression problem take make norm problem reduce regression onto regression regression onto replace vector subspace replace
convex assume strong analogous exist imply guess know average stop know decide schedule maintain average time maintain returning iterate eq provably alternatively easily maintain iterate suboptimal average give parameterize average computed accept publication scheme slightly tailor strongly merely simplicity
none solution initialize meet constraint update result see return solution close knowledge percentage signal gap median convexity undesirable might positive portfolio study euclidean onto replace hyperplane motivation address quantum portfolio illustrate provable quadratic subject assume operator simplicity project
node state paper generalization operational wireless highly nontrivial special plot qualitatively change affect behavior small accuracy intermediate accuracy overfitting improve rate sample state accuracy operational accuracy parameterize fig balance classification communication mechanism behavior transfer broad classifier access protocol partly way complicated remain certain maximize may solution preliminary fig plot solution sensor signal know
conventional innovation period well know g drift estimator whole drift conventional reason innovation decrease value histogram limit nh z I innovation error value average decrease illustrate innovation state go order h u correspond one decrease e finding precisely summarize average exact innovation adaptive improve innovation large sampling innovation far illustrate l l l limit c ccc ccc average exact innovation conventional order ccc r r r r r difference average innovation conventional l c c iv exact innovation estimator conventional figure histogram limit innovation available conventional
flow financial economic market excess volatility recent attempt huge business ii extract trading activity remove identify hierarchy news differ dimension investigate news market one news aggregate manner account impact total news record period study release economic serious limitation significant could short news day temporal partition address simultaneous relevant financial trading activity raw text million news record thompson examine impact trading activity stock list stock period determine piece information explain activity regression news event volume explain news good correspondence evolution trading measure daily volume correspondence
approximate bad vx minimization modular correspondingly algorithm every get strongly minimization note bad complexity much practice h try cost strong amongst find choose capture relevance subset na bayes applicable meaning express submodular subsection regularize mutual observed laplace help without experiment find representative subset formulation entropy factorization formulation factor factored simple iteratively factor factored mutual information lastly na nb heuristic modular modular submodular submodular generally outperform procedure quite thus
form maximum greedy rank notion check greedy cast flow vertex towards capacity hypergraph maximal clique decomposable see converse naturally polytope constraint optimization form replace convex remain integral remain primal relaxed clique feasible integral relaxed convex relaxation primal constraint convex minimize polynomial constraint complex introduce lagrange multiplier maximize em cover constraint clique constraint therefore relate primal dual variable cost
large action short virtue exploration behaviour identify towards particularly demonstrate line fashion theoretic influence environment previously transition assume main relax calculation vector application exploration long assume precise transition interact environment address state advance computational point vector space action discretize however naive discretization soon become infeasible work partly agent artificial laboratory grant science foundation n research european part project www grow fp angle measure axis angular velocity nm describe q angular velocity mean physical symbol dynamic simulation keep scalar action discretize continuous domain angle angular artificial virtual physical environment fully various automatically example dynamic programming specifie explicitly goal want application perfectly reasonable sometimes subtle human sensible investigate coupling prefer resort task research idea organization goal drive agent universal rule optimize define reward kind behavior artificial inherently interesting step
randomness hold exchangeability family second among carry information training calibration proportion standard ideal proportion prediction calibration become confident variance cf
well remark admm scheme adopt quite different cc admm e e compare synthetic create proximal specialized code http mit edu code comparison sparsity denote percentage nonzero always small zero cpu times second report r dim cpu e e e produce comparable objective compare fast table ten sometimes four need hour minute
walk two group walk label start reach measure unlabeled show walk probability walk step define bag develop section suffer thus path entry bag path put term graph could framework tackle network et suggest avoid compute computing input small community obtain approach suffer hyperparameter community node compute reduce investigate work technique span tree laplacian compute enable address graph liu investigate relational latent dimension social twitter example approach promising label define last walk force outperform group centrality
ask probability note equally analysis fail vertex find least reality loop actual isotropic implie find sample vertex vertex input map simplex simplex isometry q sphere variation distance distribution simplex set let simplex orthogonal orthogonal decomposition approximate maximum eq q study maxima moment direction geometry maxima minima optimization like although isotropic n complete maxima third homogeneous symmetric direction conclude consider simplex identify via na v equivalent embed canonical hyperplane accord constant maxima condition equation say coordinate put order necessary eliminate list candidate scale projection vector maxima global continuous maxima lagrangian semidefinite restrict tangent square coordinate elsewhere
acceptance momentum flip tr relevant represent label transition net
beyond absolute position benefit increase possible vary region improvement tree recommend seem forest need point yet c balance diabetes breast scale diabetes incorporate position metric distance example extension consistency metric learn propagation generate specific unlabeled well ten uci measure cross constraint construct explicitly method inherently method efficiency specific approach time comparable position dependence allow
cluster extend gradually merge agglomerative dendrogram depend distance merging procedure linkage linkage first statistical noticed dealing cluster generally surely mean group seek center certain notion clustering assume parametric modeling g bayes partitioning regard motivation view see instance clearly component underlie mixture certainly way dense arbitrarily coincide several component need might well unimodal suggesting shape cluster motivate example helpful cluster separate density maxima mode modal understand provide precise goal explore next population cluster fully researcher lead parametric attempt modal level define
movie see long describe opinion influence opinion formation influence individual rating preference vector influence influence user attribute default also change member case normally assume individual necessarily decision social influence motivation apply group assume describe affect evolution describe certain social influence disagreement consideration member regime rating recommendation beginning may ap equation directly initially assume recommendation different factor
separate backward computationally exponential recalling seek induce input unfortunately find much hard output output method output long allow let approximate output extend let nan tw normalise h probable probable probable remove trivially return sort appear good shorter technique speech observe output iteratively store running prediction memory network store
find region ec reader circle cone black ec ec field surprising convex geometric argument come point ignore either expand essentially precede q ec density ec illustrate rejection likelihood threshold cut ec cone ec representation case easy coefficient cone center origin sphere geodesic probability content subset beta necessary requirement must intersect similar establish convex rejection region provide euclidean radius sphere radius spherical geodesic radius near contribute pointed ignore root
state enable model brief relevant inference extend hdp thorough parameter stick distribution parameterize notation hdp role transition dp draw link dp discrete typical state towards chinese restaurant assignment infinite hdp context hdp individually state reduce hdp purpose note reduce hdp transition hdp hdp hmm maintain mix correlation thus provide new alternative exist hdp hmm hmm towards transition provide encourage process elsewhere hdp allow duration duration distribution duration explicit duration remainder component complex factorial self transition emphasize allow transition transition simplify clear allow explicit duration allow transition duration challenge particular hdp model tool duration fit make bayesian semi treat parameter good bayesian later allow compare hdp case state duration index
infeasible differ support introduce use induce approach maximize provide clear set variational gps generalize group auxiliary induce task sense induce portion detail follow introduction group specify parameter n j hyper parameter likelihood observation low j j posterior variational variation completely free entirely additional derive simplified predictive ready integrate leave th jj g low achievable jensen equality variational
history coin infinitely identical coin minimize coin coin minimal monotonically increase coin log optimal description coin coin decrease coin coin constant coin give repeatedly coin log coin starts coin coin terminate likelihood coin log state get heavy coin respectively coin get get heavy coin get heavy coin coin eq heavy
sampling introduce estimate independence node rw strong adjustment applicable evaluate technique wide topology facebook art facebook guarantee available idea estimator sample node rather sample combine together challenge rw unchanged rw sampling se finally claim gray many network walk rw estimating property global property type node exist inefficient rw sampling address induce study participant field cost measurement concrete budget desire feasible rw rw use www networks offline derive sec estimate
necessary equivalence raw statistic equivalence justify identical correspond nan identically distribute bernoulli hypothesis I therefore infimum quantity nan statistic contain equivalence monotonically statistic contain statement lemma number decrease give intend microarray cutoff point equivalent observe equivalence great q equivalent cutoff equivalent cutoff positive recover gene hypothesis true hypothesis therefore q gene cutoff
condition modern area increasingly high dimensional challenge mean definite covariance arise rank throughout finance sciences matrix invertible inverse covariance powerful allow discovery multivariate encode conditional draw normal distribution covariance matrix surely dense high setting perform sparse penalize
tx fx kt satisfy tp gx k proximal many include available problem efficiently subproblem framework bootstrap previously solver loss suggest newton outer higher nd confirm empirically newton gradient expensive quadratic subproblem newton approximation bfgs bfgs sr statistic heavily generally set algorithm warm start find cost fitting solution kkt mixed opposed interest prediction feature build conditional full form choose pairwise pairwise independent potential r present experimental conditional sample true figure theoretical
arrival evy strictly transform arrival evy ie dd atom evy obtain gamma poisson stable often prior survival measure construct normalize refer g q great use application image directly instead exchangeable infinitely exchangeable sequence de infinitely sequence distribution sequence chinese restaurant de measure exchangeable exchangeable process likelihood integrate beta exchangeable ibp subset nonparametric version conjugacy distribution analytically chinese restaurant parameter represent analogy customer restaurant number table single table one customer customer restaurant table customer table choose measure define measure discrete day formal technical report criterion collection measure support distinct reasonably easy either computationally familiar converge posterior converge specification typically assume metric however hierarchical exchangeable hierarchical model collection discuss covariate index dependent nonparametric dirichlet frequently nonparametric also collection partition
extra content notice content penalty introduce operate completion admit explicit parameterization kind paradigm need content reader apparent simulation fact cf think widely accept model certainly simulate would favor approach understand interesting informative question suitable model consider content issue focus collaborative filter methodology impose alignment shrink attribute regression attribute produce novel insight recommendation mean thorough theoretical mention section bring opposed approach plausible generative rating outline idea practitioner easy manner support natural science engineering research project create associate constructive especially grateful em recommender recommendation either content matrix approach rate rate recommend
lead solve mode bottleneck factor matrix surprisingly show full column correctly efficiently use decomposition substitute rewrite ki exactly entry arrange order define unfold replace product way unfold unfold call unfold tensor tensor number unfold mode simplicity refer merged tensor transpose unfold arbitrarily product tensor mode reduce several rao product component immediately uniquely permutation respectively rao recover rao solve problem decompose apply component optimally rao set proper pre problem comprehensive simplicity subset linearly obviously useful appendix jj rao mild factor matrix mode likely full rank cause mode key
template optical flow template reference motion template event training sample sample ann knn coding feature feature location six sa knn achieve future extent additional image addition feature extraction like fast certainly achieve improvement feed grant plan science innovation lt city science technology unit theorem college computer information classify diverse video challenging ignore past realistic annotate six location classify consist four phase
annotation noisy search keyword unsupervise annotation raw tag feature purpose model text describe quantitative qualitative evaluation present mean raw use th feature eigenvector equivalent normalize unit cluster row nc hard obtain normalize way nc factorize nonnegative normalize assignment highest nc soft cluster document topic document use posterior lead hard map cluster posterior investigate kernel baseline motivate consideration large number image want annotation method tag truth paper unfortunately standard use namely tc view tag propose modal retrieval internet wikipedia labels annotation million imagenet choose collect publicly available query category cat tag dictionary average tag per image keyword tag retrieval collect ten imagenet result tag national manually annotate city etc important wide list tag ground truth annotation concept tag web category include concept mark relevant query dataset contain concept second view text web page tag appear stop list document tag dictionary also tag characteristic class whose annotation whose come image comparative implementation wide manually annotate come collect image inconsistent quality large vocabulary view visual
univariate multivariate margin unify however characterization describe term unit sphere statistically comprehensive max describe device propose generate family max stable distribution already vary multivariate concept generator development nonparametric parametric latter frequentist multivariate asymptotic independence accurately refine seminal behaviour maxima structure dependence related highlight aspect multivariate sample collect
procedure implement recommend complementary multiply carry measure data set section monte present p value section reporting make remark concern interpretation significance reasonably model homogeneity aside consideration multiple testing statistic homogeneity reject question
ibp consist plus learn latent ibp ibp ibp truncate allow iteration reconstruction incorporate reconstruction pt ibp ibp extensions data ibp absence word vocabulary ibp text statistic model corpora ibp ibp tailed corpora collection graphic hold replicate experiment perform original three parameter ibp negative distribution
chain operate representation space chain define advantage stem move abstract auto encoder good auto side variant variant mm embed kernel hessian embed visualization hidden representation well preserve neighborhood property local density neighborhood sample covariance empirical machine manifold window achieve test mean neighboring local basis eigenvalue predictor neighbor predict objective discover manifold
achieve propose algorithm conduct learn overlap employ whole pc comparable five common precision learn scale averaging framework reconstruction complete community serve meaningful protein gene well li department science china email engineering institute east china normal china email edu cn science china email cn motivation apply average currently model average super novel network principle divide comprise first perform robust community community merge contribution robust segment much small accord allocate depend denote closeness challenge pearson one introduce overcome robust small sub community current perform dependency intra community make intra unbiased credible analyze primary
simulation r r c random knot knot knot full parent spc spc spc spc parent spc stationary square interval time simulate random miss location simulation study two trend evident well spc knot knot knot even close parent wrong bad score region assume covariance parametric examine performance unlikely conduct study sample mat ern simulation study mat ern covariance simulate add spatially study knot knot fix knot spc spc spc sec parent spc covariance observe random
brownian motion bm epidemic decide adopt model bm basis fig sm bm interesting alternative criterion likelihood iterate filter subject particle filter estimate expect uncertainty credible interval month early stage sensitivity explore robustness concentrate
inter instead time average paper inter cell interference interference noise interference fractional beyond though traffic ix analogous active bs ix feasible bs eventually bss indicator exploit controller know bs switching strategy last knowledge detail section minimize overall bss shown consumption bs linearly coverage energy consumption bss consumption stay bs traffic vary bs traffic generalize summarize besides consumption bs consumption bs order avoid quality service delay formulate specifically queue flow system little active association consumption ensure balancing reflect consumption mdp tuple traffic controller mode otherwise bs correspondingly metric bs cell strength traffic traffic load transform determine volume bss meanwhile immediate cost environment feed bs controller discount state
constraint take programming problem example handle slack article particularly svm lee preserve incorporate address norm svms obtain classification svm mainly motivated adopt slightly exploit solve learn easily explain drawback depend image computation convenient derive wolfe multiplier lagrangian tucker plug q dual qp equal otherwise entirely example considerably small original introduce algorithm classifier point formulate computation radius image mapping dot dot wolfe solution formula square subset concept going explore immediately deep presence objective mild kernel ignore within I implement constant thus equivalence construction dense qp expensive matter kind one take account approximate solution within modify priori try specify definition tolerance ball algorithms scale bc iterations r kb exploit idea support reduction include expression moreover look k I evaluation lack reduce
north rectangle south east enforce effect center constant model gibbs covariate index possible knot observation region covariate posterior take informative penalty entire spline wider credible spline global responsible thin fix knot place quantile precision spline correctly gibbs sampler determinant posterior avoid thus penalty zero eq spline control smoothing shift lack smoothing smoothness gamma spline hasting inside sampler credible calculate quickly sample credible interval report coefficient spline credible effect interest credible interpretable forecast manner forecast precede take advantage day one hour come hour training week forecast hour advance value forecast forecast treat imputation week forecasting error iteration eq forecast sample asymptotically section chain check unimodal draw posterior even ci spline marginally normally interested contain credible informative look spline say zero posterior
effort call rbf encoder center mean step simplify basis center experiment value coordinate elastic give similarity parameter normalize mac alternate slow parallel mac qp achieve nearly processor reconstruction elastic run mac qp manifold loop mac c indicate marker speedup experiment approximately reach processor rbf autoencoder architecture decoder try search define weight include regularization selection know aic omit error training parameter decoder layer output dimension encoder equivalently output dimension center autoencoder lm rbf autoencoder network constrain point mac qp complex e previous optimize mac qp model net good potentially nest function plus indicate marker change annotated architecture move impose parameter follow minor architecture continuous final total incur large training function optimization result achieve approximately speedup mac drastically runtime nest jointly exist
present theoretic epidemic cascade prior super graph cascade cascade reliable outcome I epidemic bind variable find likely cascade node algorithm computation need infect learn tree node greedy explain cause large infected remove theoretic cascade approximate recovery notion sir corollary super specify factor cascade final sir epidemic cascade direct graph infection allow illustrate role compact emphasize thus merely thus system probability cascade regime corollaries union basis thus recover neighbor super neighborhood find neighborhood entire neighborhood neighborhood super graph epidemic cascade cascade
minimize difficulty non feature interaction redundancy combination optimization since completely redundancy interaction forward select search deal redundant detect presence start iteratively instead potential interact go problematic seem easier discrete consider feature interaction feature table however reveal dependency nonetheless unclear mi capable detect nonlinear cause property instead permit efficiently compute past exhaustive impractical univariate combine feature good strategy combine ccccc forward interaction dependency needed efficiency excellent forward backward exhaustive mi variation exist impractical optimization consist zero learn
order require hold therefore distribute unfortunately therefore remark knowledge still ht finding verify text categorization news article evenly topic perform topic label wish total number news article otherwise wish function step matrix pool satisfy communication subgradient consensus consensus consensus gradient proximal weight converge neighborhood achieve highlight
amount alignment somewhat alignment statistic reflect weakly correlate alignment statistic statistic influence influence quantity stability statistic relatively theoretical paper describe meet table fix assume page solution overview stability statistic theorem within away stability statistic adversarial structure outlier type note follow subspace angle subspace imagine advance pose oracle regression intractable l theorem even outlier compare standard formulation pca search minimize subspace minimize sum residual tends sensitive consequence contribute short section contain page argument prove type primal technical detail hard working dominate paper elaborate analysis idea replace nearby objective take exact result paragraph classic optimization see believe idea relaxation data function technical use target function perturb contain subspace perturb coincide technical result lemma show residual perturb minimizer close perturbation objective residual define perturb move feasible ascent lemma subtract second indeed feasible feasible apply lemma right reach finish argument eq second
graph n ns follow appear necessarily small q plug expression shall j asymptotically probability weak treatment valid immediately asymptotic proceed fact condition condition proceed end similarly numerical experiment dependent restrict bivariate autoregressive ar let bivariate set normal recursively bivariate version autoregressive previously procedure bivariate copula copula result since copula bivariate frank copula define dependent multipli sequence test move initially detail convolution choose median choice table monte
formula logical conjunction follow protein logic define hold redundant logic logic note every property describe static boolean asynchronous study functional aspect need guarantee loop describe logical name compatible satisfy property logical note enforce cause within logical fix look truth evaluate condition behind change steady compatible loop existence steady consider well work
coherence atom however still enforce atom r trace q coherence take partial derivative derivative minimizer objective run iteration successively I partial evaluate gradient computing dictionary ccc singular spectra self coherence coherent flat correspond svd lie sequentially coherence atom
reach change uncertainty stop fall point coarse variation explore adjust bellman update uncertainty keep track visit gp experiment rl algorithm fine show gp short thing learn optimal high sample always exception direct fit iteration method hoc domain examine explore gp perform initially explain generalization capability gps propagation car acceleration function transition iteration modify due dependent hyperparameter selection region transition car indeed position estimate gps
respective currently confirm subsection learn view secondary channel pattern vector sense channel slot channel attempt take interference message send discard receiver slot experience throughput interference sense throughput secondary slot configuration slot choose employ channel usage slot channel sum equal employ share slot channel selection complete selection describe round access throughput slot average expect throughput take prevent mechanism derive pattern channel thus interference cause show usage sharing channel selection error mechanism loss cause interference allocate channel scheme interference among high throughput high scenario throughput function number address spectrum secondary network formulate
deal time experiment effectiveness code publicly bayesian well hyperparameter fast human procedure almost set level significantly performance usually specify easily weight desirable another automate view tune black invoke expensive involve primary completion desirable making seek elegant approach function typically sample gaussian process posterior make pick hyperparameter next experiment optimize expect ei ei ucb
vary pair fix probability fix centrality variety description various algorithm shall score randomness outcome comparison comparison connect walk irreducible add centrality transition centrality regularize prior beta give regularization maximize solve wise logistic regression provide regularize centrality regularize mle regularize generalized count method si aggregate ranking widely break pairwise apply count rank count ranking generalize normalize centrality rank lead eigenvector matrix different base lead prominent top eigenvector rank mc mc mc
fold reduce additional challenge demonstrate generative rank optimal divide training feature remaining perform vary vs attain portion j accuracie bold maximum summarize poor svm polynomial rbf hand perform accuracy multiclass marginally generative discrimination text build way eliminate work multi interesting theorem exist paper large nature curse employ perform enforce density use jeffreys discriminate extend multi
convergence robust regression major minimize I response residual eigenvalue eigenvalue interestingly statistic california continuously differentiable onto separate set nontrivial project intersection newton interior medium modern application statistic potentially thousand instance optimization quasi newton illustrate several statistic formulate instance continuously convex closed include find iterate iterate generate drive strict unless c n pair lead penalization positive weight loss sum penalty differently useful constraint equally consequently uniform notational exposition distance require simultaneous point present ascent projection method exhibit ascent start c acceleration
characterization separation prove useful elementary omit brevity dag remove oriented form edge vertex calculus method interested detect strength parent instrumental absence let dag vertex separate pa share equal px px strongly correlate become compatible markov exist
line gaussian depend norm universal upper induce average sub reach much bound characterization learn rich term upper dependent lower bind hold rich feature adapt calculate distribution denote suffice distribution dimension every rich margin excess algorithm bind complexity establish margin regularize rigorously sample regularization discriminative sample bound decide label rule margin eigenvalue vector provide extend discuss sample complexity notation provide necessary margin dimension upper general must matrix implication proof prove set require type bind
update data enkf variant exist construct draw perturb equivalently kalman treat draw update distribution normal member update separately center member center datum actual ensemble combination distribution gray dot right frame see gray represent dot right ensemble draw produce give mean variance member enkf though first second two enkf representation appear right skewed enkf posterior enkf stage easily generalize stage enkf break information two twice twice enkf twice produce produce ensemble enkf twice gray right frame representation dot right frame see frame relationship cover somewhat evaluation
datum prior locally normalise would easy nevertheless least extent prior generalise qualitative shape normalise isolate early figure vertex figure contribution normalise isolate dotted learn dominate learn pass learn globally normalise although plot generalised random scope comparison student teacher leave case gp student globally normalise teacher normalise r enyi match figure show unity probabilistic similar case generalised figure locally learn teacher globally case two extend graph component qualitative globally normalise bayes expect uncertain vertex intermediate variance become receive kernel local exactly enyi law graph low peak normalise display variance dominant regard character package color conjunction terminal explanation graphic explanation terminal graphic ltb lt lt lt lt ltb lt lt lt lt lt lt ltb ltb ltb teacher normalise kernel power generalise exponent cutoff terminal option explanation load package package graphic explanation terminal macro ltb lt lt lt lt lt ltb lt lt lt lt ltb ltb ltb enyi teacher globally generalise exponent cutoff package conjunction
expectation come quadratic bilinear dt jt dt v jt dt eq w w dt proceed pp w w piece kp kp dt dt st dt w st first since turn whose adjoint project complement correspond st st dt st w st dt successively adjoint w st dt dt dt dt exploit norm put together upper dt p kt tp proof start introduce linearity trace diagonal reasoning follow hence give conclusion triangle definition sub hold z uniqueness assume j kt dt exploit dt j unique advantage exploit desire proceed dt p noiseless surely bound light lemma first follow apply desire proposition equivalent q exist
readily find q energy dx x p fr random clearly integer value fr event occurs see respectively imply odd infinitely often superior mean therefore inferior empty mean highlight fr mean set fail inner need exhibit property include true sequence realization subsequence infimum event occur follow approximated formulae clearly represent visit infinitely hence arithmetic expect every exist reach conclusion disagreement classical convergence
crp take unlike dp family result application exchangeable dd crp distribution represent dynamic evolution base user couple process dirichlet hdp introduce coupling use extend allocation hdp equivalently extension crp chinese restaurant crf crf couple coupling rich depend relationship evolution address parametric static time amenable evolution context recurrent crf scalability recurrent crf static problem medium focus around interest pattern medium use topic focus account social medium rich etc site recommendation assess interest activitie study user al model account popularity various relate deal preference attempt understand pattern twitter apart temporal dynamic study factor explore social model wide network factor social medium build chinese restaurant evolution application basic associate individual movie
function say enumeration close enumeration close equivalent valid relax proposition appear include term form condition continuity axiom general interpretation probability separate interpretation preserve proposition number individual axiom ns practice natural axiom type proposition definition property suggest strongly part valid principle advance sure idea inductive weak turn useful strongly development major alphabet sentence condition necessary sufficient discuss detail alphabet countable free enumeration hence universal condition fail statement read iff assume assumption get separate case separate say separate term separate prove prove set interpretation interpretation borel algebra topology need define alphabet sentence closed intersection topology borel form equivalently term countable algebra algebra interpretation finitely additive additive countable collection set hold ia always countable equivalent continuity algebra probability sentence interpretation finitely suppose finite interpretation useful countable sentence algebra finitely claim suppose contrary contradict thus prove decrease alphabet countable interpretations borel let suppose sentence valid finitely collection disjoint finitely proposition alphabet countable countable borel algebra borel interpretation intend intend member separate interpretation separate interpretation alphabet borel algebra need denote close finite intersection algebra consider
cca version modal discriminative pca lda handle supervise meet generalize class framework recover framework define propose generalize alignment author domain event detection corpora tv entity corpora image versus start significant attention computer covariate long new one survey adaptation theory thorough survey relate field mention introduction label fundamental devoted aspect fashion benefit
subsequent criterion deterministic criterion view field generalize training net optimize limited datum conditioning set work answer train time rather data descent
landscape evolve toward parallelism mcmc identify end extreme independent parallel nothing chain limitation non version extreme alternatively chain intermediate use parallelism share information mix na I grain execution nevertheless parallelism possible objective core low slice sampling elliptical elliptical generalize elliptical slice parallelism dynamically desire evaluate method x use wish sometimes objective carlo formulate classical example simulate operator produce sample use compute expectation typically unnormalized notation somewhat chain carlo adaptive rely coordinate accomplished slice insight conditional uniform slice potentially easier uniformly among typically define leave slice describe interval expand necessary acceptable procedure phase efficient first direction consider implementation later detail unclear
relevance exhibit mixture original gaussian contrary principal figure hyperparameter isotropic review classifier hyperparameter classification moderate dimension dimensional reader reference framework tune optimize margin svm penalization optimization hyperparameter leave radius margin optimal hyperparameter objective follow later gradient class classifier indeed problem must vs prefer require optimal similar since gradient enter gradient derivative conventional framework hyperparameter detail use methodology consecutive
show ad value thus satisfy ad achieve candidate efficiency iterative suppose ad ad subset correspond self calibrate prediction different ad raw candidate ad randomly bid efficiency maximize query mechanism offline maximize np hard even minimum feedback arc player rank minimize player minimize number time player player player generality show ad winner ad maximize raw exactly also observe np construction perfect efficiency calibrate illustrate follow four ad tuple c guarantee show also need already query
median univariate variable empirical curve point median versus curve minute geometric quantile generalization unlike median homogeneous coordinate coordinate appropriately compute argument favor g functional fact datum exactly median median central instant consumption record week week compute wise measurement consumption wise median notice coordinate median week situation median fact take account time point compute instant instant use fact fulfil point median note define nice direction towards unchanged obvious extend geometry index quantile way interpret solve zhang effort measurement work subset henceforth realization inclusion contain inclusion variety direct element design
conditional deviation give second moment unit respectively require limit follow continuous measurable mapping nh hx tb initial drift weakly distribution continuous step half length point x condition usual acceptance indicator denote embed discrete variable variable come mh acceptance original benchmark keep
result key good linkage record firstly j step e compose equal remain equation replace log estimate maximum frequency count usual stop assess measure consecutive start algorithm take great restriction linkage em clear greater refer individual agree information record tuple necessarily probability information high parameter easy determine constraint take determine start partition fine low criterion note procedure use direct partition branch partition identify constraint whenever among probability diagram link size small reasonable starting determine maximum element maximum reasonable starting determine minus notice rather merely guide start em maxima take hold marginal observe left side set
space ds exclude ds ds svm forecaster forecaster ds filter quantity slight improvement locate accurately precisely rmse filtering dominate incorrectly via forecaster capture forecaster significant gain historical set interesting feature forecast observation time forecasting performance excellent attempt state forecast find prediction svm forecaster additionally laplace fit likely variance able deviation gaussian actual observation influential result ds ds misspecification svm historical dominate filter stochastic firstly svm forecaster training set relative system significantly initially filter
margin optima peak normal middle shape symmetric well margin respectively well margin behave margin situation change remarkable kernel zero value top normal margin locate evolution proceed zero margin capture good exhibit introduction correlation seems find many reliably critical population two achieve population capable multivariate information fail optimize margin successful though say aspect marginal role follow aspect report combine copula apply truncation
rejection period local exist local consecutive update period cyclic loop matrix local period cyclic property transition ergodic condition ergodic transition state integer define ergodicity ergodicity markov condition considered choose way choose actual however choice short rate sequential small uniformly far ergodicity sequential even simple algorithm ergodicity ise practically check compare sequential assess lattice fast convergence short parameter conventional autocorrelation nearly
te coupling coefficient strength depend internal explain measure coupling define entropy second term first strength depend belong parent analytical numerical advantage mutual mi parent mi input depend coefficient process condition te though three depend interaction external formula mit solely depend couple coefficient autoregressive generally additive mit depends prove coupling strength theorem section variance parent separable mix entropy part entropy could gray couple rather thick nonlinear mit numerically constraint full coupling reach next formalize strength multivariate sect couple link follow derivation lag link define mean dependence source process parent I
choice covariance one x diagonal hyperparameter zero elsewhere apply ern kernel know choose produce already decide denote arbitrary property gps jointly gaussian x complement eq lattice simplex size set number optimization b bp assume variance construct combination optimize optimize sufficient include probability improvement thompson refer detail main function search densely shrink surrogate intuition function discard
yy yy yy yy throughout derivation therefore last proof follow tail bind gaussian covariance dimensional jj inequality probability least simplify probability complete convenience notation cone yy yy function convexity loss equivalently eq argument contradict proof figure support use f score norm present table glasso glasso norm glasso glasso glasso c c glasso glasso glasso glasso glasso theorem corollary remark pt section department statistic study partial convex establish competitive empirical exist various synthetic real copy unknown covariance
label assign instance find categorization organize problem categorization together brief explanation confirm second use alternative experiment server recently generalize claim extensive section manual multiple exhaustive think occur topic match abstract description category example medical deal mesh keyword work label admit belong music possible third label second occurrence label third song degree label phenomenon initially capture binary third look contribution label category combination binary account capture label crucial give rich instance every define assume binary content instance represent label
compute assignment business attribute compare user business user business user business role compute agreement assignment sign penalize alternative would pair set function equally sized set conceptually differ attribute see enable share standard new framework new role cost hybrid anneal convert term assignment eq terms business assign simplify auxiliary apparent assign contribution role assignment substitute assignment ik k eq directly compare business cost compute substituting procedure arise recursion turn tn ik gibbs multiple repeatedly find cost business information role attribute optimizer user role mining role group together irrelevant attribute infer without attribute theoretic relevance random assignment variable business generic job business quantity entropy mutual quantifie assign entropy business attribute mutual gain knowledge attribute compare entropy small reduce relevance bit knowledge care observation relevance business high imagine fair na differ considerably one compute attribute observation user highly relevant compute value sufficiently measure entropy different le access job unit attribute code index kind contract initially would user business task compute result depict histogram attribute relevance reduction much little role experiment time
proposition u proceed continue terminate inequality hence rewrite solution rule like relevant know square result regularization also cross validation motivated vector independently sparsity recall select vector ease elastic parameter varied necessarily
equivalent recursion start subgradient end adaptive may boundedness line convergence argument gap extend denote maximizer recursion convexity smoothness g note equality classical equation desire proposition consider optimize iteration extend search recursion proposition proof assume constant one v
rule year extent language order relationship partially rule another matter cm remark question university universe analysis
degree remove eventually interesting arise conduct ratio limit fundamentally consider hypothesis constrain penalize ny q endow hilbert reproduce trivial bivariate z g z basis limit distribution calculation fr g w g g g g g g ds w fr derivative additional rate verify assumption satisfied g main local present likelihood satisfy nz pn restriction local proposition explicitly reproduce uniquely rkh certain facilitate calculation example explicitly specify remark depend bias specify estimation fortunately smoothness true satisfied fourier coefficient satisfy phenomenon type since counterpart practice section counterpart simultaneous band originally density estimation method require relaxed translate
filter covariance subject wide range also datum subject trade optimization initialize use leave manner subject training use allow method cluster interpretation perform analysis whether subject kullback leibler kl kl matrix distance matrix user large difference may test small subject accuracy may degree stationarity performance significance description covariance test matrix subject train subject fig analyse user square angle subspace tu exist equality principal angle amount preserve project thus indicate subspace angle picture relation restrict angle tends extract type discriminative filter prominent direction measure similarity discriminative random
lc lc lc lc lc lc cx grey posterior mat ern field square krige see panel leave show ni lc lc lc lc lc lc marginal b lc lc lc lc lc lc lc lc lc lc right krige definition see carlo satisfying correct nothing distribution approximation ni handling cl bc bc tc ct ct ct ct ct ct cl cl bc bc tc ct ct ct ct green configuration configuration mcmc simulation use posterior ni twice estimate sample curves green ni ni method setting twice color show two carlo perform finitely posterior discrete indeed error full configuration ni cl cl bc bc tc tc ct ct ct ct ct ct cl bc tc tc ct ct blue green
multipli procedure extensive carry nine absolutely f multiplier routine moment likelihood imply classical regularity van appear q information equivalently respect unknown implementation gradient package detail subsection numerical multivariate degree freedom f dimension define statistic transform fit realization q size parameter continuous namely arise distribution g continue generation determine minimizer scale fix w shape parametrization use random size family multipli goodness carry von statistic kolmogorov rate column
bottom optical flow move backward mainly bottom identify due resolution sample optical flow one region prediction forward backward stop encode colour axis represent optical problem describe turn joint making prediction decide low state conditionally fairly greatly complexity memory incremental multivariate gmm arrive sensor learn incremental g around likelihood explain threshold component covariance optical
weather induce traffic speed conclude road traffic weather trend confirm quantitative conduct devote follow analyze draw speed forecast weather section road link road link classify well road class importance connectivity demand concern c attribute major road road secondary road road road importance road road minor road gps device position sensor sensor car traffic
flow term box constraint switch hour controller take one second pick iterative process low air moderate level process start repeat separate general denote interval usage describe effect relationship average factor xt nonparametric relationship scatter plot energy represents scatter energy hour xt allow probability simplicity quantitie
runtime total runtime prox well primal descent prox sdca derive conjugate sub whenever sign sign therefore q zero q sdca minimize accuracy prox sdca accord prox sdca mirror online therein closely average sdca
try mention theorem run version available page help look conference seek conference format format certain specify instance body copy live area center top cm leave
relaxed setting suggest learn time choose mnist fall completely top tune fall top training propagate interesting competitive energy pooling network stack joint layer capable distinct capture transform auto issue negativity form minima inference different training contribution context convolutional sparse learn manner simplicity integration pool amenable channel produce sparse feature minimize hyper map form give solution pseudo two operation produce large influence small neighborhood constrain make compute weight neighborhood map max except work treat paper neighborhood overlap overlap brevity operation
slightly careful counting index collection size n feature distribution satisfie n assume consistent ordered term uniform among old order derive ibp n final poisson number time draw exist formulation particularly provide infer partition partition specify block characterize survey vast clustering indicate height observe generative dominant specie th region indicate overall weight height region measure block guarantee uniqueness posterior rule carlo mcmc desire posterior prove true equilibrium gibbs sequentially sequence note exchangeable remain z nz nz rule full crp cover worth note exchangeable rule programming induce integer label parameter partition assignment separately lead similarly follow generative feature block let appearance belong parameter parameter review motivate example customer describe economics science describe book science book cardinality great customer book set book sale book picture train picture contain element directly might serve appearance remain ny sampler ibp encoding parameter every requirement argument satisfy consider symmetric argument argument proportion fraction stick
also mechanism template merge learn new template fusion pattern error take final rank system majority vote realize show interest fusion recognition visible color convolutional dynamic seem dynamic demonstrate dynamic rate capture sum powerful literature fusion powerful yet term rate architecture fusion interesting stop otherwise capture reach rejection acceptance score mechanism present verification modality necessary face acquisition kind architecture hierarchical multiple layer mathematical sum min
criterion coefficient index generative reformulate q lie element signal represent overcomplete joint signal simply overcomplete combine optimal atom index non row alternatively indicate solution actually overcomplete use benefit dictionary iterative complexity dimension large least guarantee boundedness origin r kp shortest distance kp kp let nan space I minimum singular value induce support index boundedness need unbounde x
throughout cover exclude special day validation set day keep exclude day correspond public day access forecast period omit unit consumption time hour explain operational constraint concern half operational forecasting lc day minute median graphical representation performance right symbol represent similarity definition several possibly benchmark procedure serve comparison benchmark compound expert size j category model get expert varied behavior expert expert nonlinear approach load effect process weather add forecast seven change gradient temperature short lead group generalize implement software idea adapt consumption parametric priori behavior expert account economic drastically expert univariate weather functional forecast day similarity expert depict bar represent sort value expert scatter frequency expert operational variant correction expert inactive prediction part consumption generate operational expert active period practitioner period length period always active report
gmm color patch gmm patch patch finite mixture al equivalently soft bregman soft exponential family mathematically hard clustered cluster interpret isotropic al mean et mathematical likelihood exponential mixture rate distortion expectation special von duality bregman divergence assign local update convergence monotonically discuss initialization strategy describe basic notion bregman divergence bregman divergence study base generic mle describe section discuss proximity assignment mle mle finally conclude paper discuss parametric distribution q density sufficient denote inner matrix q natural order family order
avoid eps b eps eps nearly method sample explicitly include calculate constraint I base reliable biased hand I large simulation imply I exclude physics mechanism assume former put desirable aim optimal useful information determine maximize bayesian method suitable model one situation previous generalize play robust estimation theory core entropy obtain introduce quantity quantity principle quantity adopt theory advantage prior bias exclude effectiveness
covariance cc correspond separate appendix ht bottom symbol configuration simulate size hmm spherical emission evaluate mse mse conditional evaluate criterion mse indicator classification rate posteriori rule hmm merging regard provide bad satisfying overlap explain merging suit ht well provide good estimation group far show
exponential family map neighbor neighbor two I kt writing form acknowledge advanced probabilistic traditionally exploit width graphical exact approximate possibility highly connect context relational probabilistic clear formally symmetry exploiting mean formulation typically preserve mathematical group concept graphical notion
propose combination mask reservoir store delay use later step nonlinear previous delay hardware use paragraph give brief system represent leave operate constant power nm device encode level optical act delay reach convert scheme experimental setup reservoir reservoir analog layer green part generator amp scope ni acquisition multiply mask level generator correspond add producing digital intensity state computer
logarithmic batch corrupt mini batch online lipschitz bound exchange message nesterov accelerate achieve unconstraine problem characterize present assume objective function lipschitz convex topology message introduce extend gradient present result experiment paper function sequentially receive loss datum subgradient unknown
des est des est centre de en des es dans la est la les la une la la de la est par la ne les les pour ce en
ph generated control hamiltonian reinforcement learn stability knowledge actor near drawback generate actor find balance actor physical setup due boundedness learn currently present freedom rgb university technology cd com g r reinforcement base system intuitive way achieve passive storage consideration calculate complex partial differential pde issue balance loop energy preserve pde matching specification hamiltonian control control law actor enable near policy close loop landscape near standard make perspective incorporate learning thank parameterization
ideal estimate vote house separate political color measure political fit house correctly vote correctly vote big predict hill r vote understand point cut side issue hill order top connect ideal row panel remain segment color still vote hill vote consistently within area education vote political spectrum consistently vote united position hill health care position put odd many particularly ideal hill position classic spectrum certain vote financial policy education take position traditional traditional model goal develop model usual political illustrate kind top panel point various political position bottom position model representative representative posterior give voting available ideal roll classical popularity point encode discuss ideal issue code topic model detail popularity ideal issue landscape represent issue value one describe change example leave left issue vote issue adjustment adjust ideal point vote use find inference find sparsity adjust issue issue probabilistic issue ideal point draw issue
f g hx quadratic compare alm optimization practice guarantee complex admm use default alm parameter need close easy fact five dataset alm admm slow alm mainly alm fy proper possibly non convex gradient method proximity operation main develop max possess advantage term gap aim narrow paper
justify along way carefully acquisition design feature train profile neighbor gaussian error suggest benefit aim future stroke feature computer affect identify resolve design decision use modality modality image front camera usage pattern research support intel science foundation center artificial university user interact phone behavioral raw demonstrate behavioral interact phone e learn user reject current achieve inter test week experimental mechanism long implement part modal user computer device typically user face challenge input dominate perspective interact device usage mobile device short activity email read simple weak increase device completely recent attack break entry successfully quite record eight reading text phone geometric discriminate apparent difference come stroke pressure cover extent mobile device computer implicit would passive user device ideally device complement monitoring accuracy substitute grow body
biological sample center scaling procedure simple normalization normalization independent sparse create penalty option preprocesse example multinomial sparse group sparse give model class cross subset approximately estimate expect generalization simply differently compare non estimator block fit one fit set order compare cancer datum estimate misclassification non right approximately expression cancer divide tumor cell select tumor sample unbalanced see lasso run evident group group however number zero
create loss unobserve datum low payment triangle jointly estimate make assumption feature illustrate mixture however specification framework explore augment assumption specification scale distribution incur loss comprise year variate p log incur joint loss denote year specifie low tail frank member source family give consider distribution payment development year mean prior distribution year portion correspond triangle payment loss incur loss independent precise particular auxiliary factor also augmentation predict payment remark incorporation payment incur loss copula independent model conditional evaluation analytic evaluate equivalent model analytically posterior distribution mcmc comprise gibbs update copula resort augmentation scheme able perform via mcmc hasting section mcmc static mass acceptance likely improve iteration progress autocorrelation estimator integral functional several adaptive distinguish feature algorithm standard utilize kernel past variant metropolis adapt history chain present ergodicity simple ergodic appropriate condition ensure ergodicity two strategy kernel stationary correspond static parameter start condition sampler produce draw distribution iterate tend infinity sufficient converge tv jj pd
estimate causal genetic account event restrict baseline project several attempt call analysis baseline imputation work however account clinical trial variable determine randomization present causal problem encounter clinical allocate follow graph affect analysis outcome explain intend include participant trial per participant include illustrate individual design control age control create structure probability individual depend graphical presentation draw individual case incoming index condition health status dependence take account estimate population
perfect loading possess perfect structure error datum n respectively rotation via loading eq six correctly example rotation yield dense maximum often enhance sparsity employ likelihood estimate rotation take loading give theorem express enhance modify control fitness solution penalty control amount maximum rotation technique near penalization dense model paper apply nonconvex achieve penalty scad nonconvex sparsity consideration
corner fill gaussian mixture nonlinear trade statistical linearization depend dynamically split quantify induce linearization gaussian linearize direction specific linearization performance estimation nonlinear statistical linearization filter bayesian estimation n measurement measurement actual measurement vector process white assume describe mixture assume framework namely step q step transition dynamic dirac delta determine system acquire k likelihood nonlinear exist density component substitute mixture give rise vector replace
obtain survey gradient estimate difference paper problem estimation base result describe related problem fact whole depict figure successively may possibility understand variable difficult obtain completion analytically apply normally monte notice simulation another arise reliability system engineering element keep fail lose work one working maximize expect relation element directly retain sample expect system illustrate depict important may solve activity network network
observe control chart play significant role though difficult uniformly depth range fit gamma distribution fit except lead lack due resort bootstrappe computationally save time fit point pt pt pt
orient thin corner high distant thick play player corner corner corner corner distant present compact reality large always shape move omit pattern entirely interpretation unless context observer course match pca correlate uncorrelated one significant pattern old corner approach pca low corner high modern pattern game perhaps play experiment sample corner immediately obvious visual significant player shape sequence obviously center orient thick play somewhat certainly correlation large correspondence pattern table coefficient pca limited reliability pattern feature carry c significant however extend normalization significant dimension elsewhere accurate contrast emphasis occur frequently require extended normalization notably specific
estimator consistency hold fix supplement consistent generality assume clarity sufficient stand x case impact handle limit distribution direction main corollary corollary usually typical random moment assumption x x bn c bn n n n n r n b n suppose tell theorem estimator enjoy meet magnitude incorrectly estimate effect penalize estimator limit one addition suboptimal estimator whose corresponding consistency normality n assumption n third condition two improve step challenge main region degree freedom asymptotic confidence region keep asymptotic confidence level property condition interest last tail supplement first
node mix graph analog jj jj ij laplacian nothing use w give auc give along hyperparameter paper principle probabilistic mixed propose highlight show usefulness author anonymous section usa provide tend connect label edge connect node ir relational neighbor graph deal mixed well world descriptor I pattern evaluation graph relational unlabele assumption violate depend underlying relational pair
usefulness model education exercise identify relevant covariate party exclude party origin exclude consider summarize htb gender g substantially opinion employ inferential statistic adequate drawback quantify base statistical law use support inferential attempt party outcome covariate model parametric limitation concern specific functional link outcome limitation aspect category covariate assume easy
share share connectivity number chemical perturbation cell line response enhance pathway interaction wide condition highlight cell biological activation introduce characterization response genome scale search modeling reveal gene utilize interaction limit guide signature functional unit network gene involve simultaneous activation signature vary precise whereas observe potentially alternative activate condition since response detection efficient proxy identify state task detect characterize measurement state index signature gene association observation underlie lead n triple w define shape fluctuation frequency gene signature feasibility assumption predefine difference expression predefine channel array use
theoretic rely concept approximate behavior mathematical avoid stable outcome steady behavior alternative diffusion social network see reference possible interested instance paper particular concentrate mention variety problem research computational relatively little graphical address success practical begin availability datum collect result complex individual people individual group system g trading internet individual customer device demand future considerable technical assume availability payoff availability arguably availability observation motivate theoretic formal player concentrate call e steady pure stable come behavioral lead potentially goal aim give graphical deal relative broad behavioral address arbitrary emphasis game introduce theoretic generative game behavioral seem capture validate argue exist considerable amount evidence capture learn illustration please constitute parametric address player help bring increase infer game behavioral player observe player application vote game refer reader far show learn logistic characteristic influence highlight influence member party member opposite party dash line unite tight time current vice display party bottom corner bottom leave opposite third new york arcs influence prominent explanation term allow focus influence business prominent member summarize pursuit framework behavioral game deal player hundred objective line sense steady joint making behavioral might end steady attempt effort behavioral state appendix far strictly behavioral action take require payoff agnostic player unit office company etc like un vote record behavioral entity single computationally provable
outperform standard probe level expression involve thousand rapidly collection reasonably annotate assess set collection type disease public microarray group class singleton annotation describe retrieve scalable preprocesse availability term sufficient available include origin batch approach set retrieve date file header tag laboratory day assess specifically array preprocesse addition probe set probe mapping probe use follow sd replicate change percentile gene change obtain slope slope concentration versus intensity medium intensity high intensity slope fold nominal log change nominal fold change nominal roc positive standardize intensity
loss bin large require reliably increase decision principle working power distribution world quantity set scientific domain include organization power merge branch specie interval volume ice power ten stay within year number plus bin spanning day stay omit wind united state accord enhance ef wind speed united states interval human census california measure amplitude area km per disease associate power fit indicate statistically plausible text denote next std ccccc arbitrary wind ef scale max wind knot knot population city logarithmic km disease gene logarithmic naturally bin raw analysis conclusion analyze primary law plot fit several case include bin boundary entire supplementary conduct claim shape inspection suggest claim summarize include statistical hypothesis law good fit power law behavior law still confirm heavy tailed law stay wind disease quantitie law law quantity fit multiplicative mechanism normal exponential
value addition notice side monotonic computed h equation self strictly sided investigate work regularity algebraic asymptotic relation conservative q strictly usual I characterize proportion reject distribution obtain hand evidence value hypothesis specify asymptotic dimension adopt threshold relation cut actual logical arise leave namely evidence value replace equivalent confidence state axiom seen satisfy axiom axiom nan propose analyze light abstract calculus abc propose abc symbolic know nonempty interval characteristic subset disjoint subset basically calculus property axiom
time es chains change variation geometrically study evolution strategy search es child add gaussian parent call good covariance adapt iteration investigate cumulative adapt adaptation evolution cumulative introduce make exponentially sphere dynamical problem optimum randomly
previous know netflix rating movie recovery impossible study iterative project svd incomplete surrogate lead parallel replacement completion since long minimization quadratic order norm matrix small recursive spectrum lasso enjoy
less investigate pressure genome association diabetes control phenotype think relate c pressure course body mass treatment individual various observe phenotype conceptual status understand genetic marker valuable suitable statistical analysis limit phenotype analyze perform snp phenotype analyze methodology determine associate conceptual truly phenotype variable phenotype longitudinal point suggestion phenotype treatment longitudinal significant little use treat remain replace person therefore methodology suitable snp associate estimate phenotype probability consistently suggest relate snp rs covering suggest minor allele snp latent part
recover normalize view rather suppose stress size svd algorithm subtle straightforward derive b invertible claim say canonical positive return subset canonical proof hmms eigenvector hmm reduction view reduce view find integer sum average document second third moment empirical moment singular vector matrix large singular reconstruct z remark exact succeed run primarily x lda follow sample permutation permutation column normalize accuracy alternative small element smallest relate obtaining currently accuracy probability sample
summation entropy joint product set n py variable obviously negativity information use theorem entropy identical empty conditional define follow mutual inequality distribution special transfer entropy n extension
arm auto intuition like behavior arise force trying already especially stack rbms deep belief net dimensional representation good obtained walk vector modify train reconstruction minus unnormalized small poor concerned try deal certain parameterization represent density trivially everywhere potential find look conceptually argue eq equivalent parameterization denoise yield tie obtain weight demonstrate energy tie lead integral parameterization denoise auto encoder flexibility well connection involve score denoise exist denoise auto particular family denoise yield stochastically case minimize square perform score matching x px width corruption reconstruction eq criterion say function per derivative denoise smooth I appear desirable undesirable since score
maximize optimization arise trajectory seed monotone modular successfully scenario submodular success library become submodular unless note effective people previous orientation clutter also position relative submodular crucially environment maintain property always algorithm contextual control library regressor I n figure diagram train environment row feature attribute environment denote loss slot example environment particular beneficial within classifier input environment ease understanding walk
figure three previously quantity test exact expectation fit progress cm procedure kernel cm together product expectation assign set fast apply exact expectation minima exhaustive search superiority min search slow seem boundary marginal polytope small convexity constant cm obtain mean match without candidate compute namely setup cm consider know
trace strictly well corruption bind corruption max ideal tight relaxation max suggest cluster spirit compress sense guarantee subject input class cluster objective clustering disagreement objective locality ratio cut try globally typically np hard ahead monotone principal component affinity transform advance algorithm might change dramatically incidence iff otherwise belong incidence e block write phrase correlation objective certainly cluster relax
candidate use undirected encode symmetric adjacency entry derivation avoid whose explain quantify community possibly overlap gd entry zero positive scale define encode community interest specify individual basis binary connection member community individual essentially two row conditionally treat individual variable restriction way addition imply edge clique clique information inferential computing involve estimation matrix basis element
seek follow absence light past I time previous play time spike neuron differ neuron base apply circuit spike motivated specific circuit probabilistic formulation employ link see approach pattern repeat count spike align model randomly ensure poisson bernoulli suggest however design setting behavior stimulus spike blue short vertical bar neuron fire stimulus nonlinear polynomial model review spline point view sparse sophisticated procedure overcome challenge theory simple add additional randomly process control consequence pattern pattern cover possible could point process create similar
phase notation arm phase occur one type slightly accept arm algorithm would reject event look arm whose least n type stage arm accept contradiction occur symmetric bad occur arm gap leave accepted arm active
family euler lagrange must form iterative parameter example take develop guarantee optimize updating euler lagrange equation make deviation algorithm parallel hardware factor runtime regime enable parallel update partially kl depend function implement conjugate gradient descent perform
probability surely conclude sum assumption pair standard note update expert ball dual simplify lemma estimate argument upper get inequality jensen simplify last corollary update expert unit dual scaling conclude conclude bandit pointwise also regret inequality arm pointwise maximal loss conclude strategy show rademacher give choice minimax expression compact triangle introduce ball let event coordinate leave far equal event q write attain prove lemma rademacher pick admissible rademacher pick strategy respectively rademacher pick tr admissible relaxation mention equation regret play statement provide vertex
two unsupervise external internal outside validity internal cluster cluster cluster separation tell separate combine since explicitly external include datum cluster external measure give cluster describe fall entropy set cluster compare set set compare paper consist wide web handle etc result matrix run value
life approximately month states disease disease present interpretation state absence backward require disease chain markov admit limit chain gibbs sampler cardinality longitudinal analyse presence specie therefore briefly section indicator presence absence choice correspondence analyse main counterpart analysis matrix correspond associate analogous correspond mle zero set nine target partition six sub scheme give walk metropolis rwm rwm heavily efficiency correctly curvature algorithm sub block current markov scale adaptation scale equilibrium conditional use forward algorithm acceptance conditional disease backward disease particular conjugacy dirichlet conditional posterior
aid economic represent direction comparison performance guarantee serve measure predictive event block iid data increase separation widely spaced block nearly expectation theorem prediction var linear collection sequence ar truncate error truncate need truncate notice analogously risk need row stationarity eq supremum class publicly economic necessary require create four description unit availability service consumption business hour population individually fit smoothing convention series show maximize filter surface rough information parameter constrain lie plausible normal strict estimate low upper corollary theorem definition assumption economic support grant r thank david valuable institute bound forecast many autoregressive move space compete high probability well make assumption generate motivate standard economic financial
proceed close suitable regularization type admit hilbert admit minimizer conversely contain uniquely minimize family admit first uniquely decompose eq minimizer observe setting belong
follow complete proof adjacency example remove direction match relation bind term equal contribute upper ji dependence recall show equality symmetry ij complete eq logical rhs average independent bounding key one example bind start independent noting lemma n bt lemma imply pick pseudo fast long history success statistic theoretical convergence theory multinomial value two area national security name detect physics literature review review partition cut modularity focus probabilistic overlap community perhaps study node define adjacency
trajectory generate simulation agent interact start state tuple action accord tuple proceed incremental solution initial proceed instead weight solve equation solution difference list converge deterministic state know condition violate nevertheless treat derivation implementation deterministic solve discussion regularization equivalent output one consider candidate function search possible candidate parameter weight quadratic solve consider mean regressor one choose approximate denote give submatrix tm point om operation relevant term distinguished unsupervise unsupervised incomplete cholesky
measure fourier interpret borel corresponding section estimate compute accord simply enforce unity level set support consider regularize energy summarize contribution estimator energy observable estimator reasonably balance reduction nonlinearity continue definition key concept energy system well measure support nonlinear theory space behave leverage connection control relationship previously certain forced system reverse drive invariant sde possible object
resort field variation inference recover variable context generative reverse inference variation factor visible observation latent variable think real variable element function unit interact interpret form visible bias respectively energy fully specify energy include factored representation yet highly dependent three binary spike adopt interaction group local define encoding block use binary effectively subspace characterize imagine dimension like edge detector
important develop theoretical develop achieve theoretical programming improve conservative practice develop offer empirical analyze program bilinear term empirical build bilinear programming optimize conservative norm compute solution main challenge mdp consume impractical return parameterized function address issue idea easy optimize small domain maximize minimize policy reason policy bind return base frequency represent fraction spend frequency
random eq generality oracle reliability oracle relate suppose variance comparative outcome distribution side shape realize motivate subject pair aid tune familiar query novel pairwise finite bandit comparison work work algorithm evaluation nearly match prove function empty intersection merely lipschitz bind noiseless bind relevant problem convergence match aim gap understand spirit simplify aid proof respect randomization every sufficiently infimum
set rich propose algorithm sampler vb super sparse interpolation bic bp camera house unless sr patch hyper standard uninformative e truncation image use house apply use interpolation color layer cb cr kind bp compare base interpolation gold sr near interpolation bilinear base representation also image fair change noise variance edge make image interpolation might boost remove beta construction covariate close factor assignment removal bp super improvement bp learning patch dictionary dictionary consist algorithm gibbs sampler train near c c bic bp
curve overlap empty bin become large curve overlap derive mat verify theoretical curve shape empty essentially h curve e mat overlap experimental theoretically mat mse confirm ii accurate hashing without replacement mat curves approximation confirm unbiased somewhat empirical validate hashing add hashing overlap validate expect bin occur goal hashing reduction expect nevertheless strategie empty bin integrated r dataset hashing procedure sample deal simply replace original code almost numerator drawback course strategy couple strategy review detail sec encode take strategy strategy essentially
structure tuning stability estimate adapt selection consider smoothing amount sufficient upper bound empty search problem range goodness fit replacement denote many frequency stable parameter link variable exchangeable factorial model choose take illustrate tuning consider scheme structure four represent column second g mainly fp tp negative precision matrix link element positively estimate tp tn false discovery discover whereas look estimate element covariance fact indicate scenario multivariate time presence absence tune efficient solver infer propose graphical tool analyze relationship universal
combine complete assume q q similar wherein satisfie eigenvector normalize sample scatter xt numerator normalization n recall numerator surely pattern exist showing generalization ability determine training reasonably conclusion size quadratic sample population covariance linear important pattern recognition
lemma b prove bernstein
f compare hence counterpart replace view large increase select also reduce convenient ready u pe pe tune lemma upper achieve optimal explain selection tuning toeplitz toeplitz diagonal exact toeplitz twice except gram first adjacent origin hand compare apply adjacent subscript long theorem b hamming diagram diagram visualize successful impossible density characterize require call interestingly pp approximately number signal rare large hamming procedure recover phase diagram rate hard numerically display diagram b fig toeplitz model view special gram convenient extreme ill pose hand exploit take proper work extreme small eigenvalue carefully detail retain different construct estimate I pe pe j k continue order node behind introduce component split small set separately patch performance cp stand change choose tuning parameter hamming satisfy p tuning exponent phase minimax hamming dominate hamming distinguish isolate hamming distance distinguish change triplet triplet change right hard boundary
experimental split evenly default univariate kernel compare test ol independently apply output outperform leave identity optimize output mass learn scalar ols gm tradeoff numerical class categorization literature target author object discrimination main experiment allow subproblem respectively note negligible subproblem update comparison effectively unconstraine psd cone hour insufficient progress make subproblem precision efficient implementation per iteration solver appropriate rapid progress accuracy fact three classification accuracy highly report
inequality first feasibility admm review comprehensive separately turn field datum machine intensive practical apply method linearize accelerate however classic assume value reality ignore
candidate form h multi map mutual enough set largely risk empirical risk different small hold regularization tr important normalization explain least average operator eq second task task allow fix limit dimensional subspace standard learning support already bring benefit sphere constrain
cone deal simple background gamble contain reduce condition bernstein discuss interesting elsewhere consider framework exchange possibility gamble preference fit impose open favorable gamble possibility gamble fit right discuss gamble gamble favorable gamble correspond strictly desirable gamble axiom look axiom formulate desirable gamble gamble factor recall consider trivially finally combination closure exchangeability reject repeat finite exchangeability illustrative trivial actually form intersection situation theoretic find towards model agent form accept reject one reject framework expression natural extension gamble background status assessment acceptable gamble regular regular side north east inner sep ex ex sep ap rp xshift rectangle xshift south west yshift ex south rectangle yshift ex north east xshift south west right xshift north east ex yshift east rp illustration nine work assessment draw reject gamble acceptable gamble acceptable gamble gamble reject status acceptable gamble cone accept cf actually framework brief overview axiom axiom framework acceptable background conditional constant gamble background ef desirable gamble contain gamble background closure axiom really gamble background necessarily finite closure imply elsewhere axiom say slightly look show correspondence axiom ignore call set gamble gamble factor homogeneity avoid partial sure scaling combination closure turn identical acceptable net use mathematical finance specifically see scale low risk closure requirement fall scope framework deal gamble way via expectation operator background argue restrict attention reference turn connect focus coherent gamble constant gamble identify notational convenience theory real fair first mind agent see exchange either second mind acceptable gamble agent higher translate cf real partition ph ph ff linearity convex segment
try behavioral pattern ip may action attack device business run email spam ip enter email service problem far important repeat rapid reaction email record length historical create informative record window multiple attribute aggregate predict behavior future prediction base property current aggregate transaction transaction estimate base transaction allow send unique property speed improve depict reference ip record g historical window length small historical integer exponentially grow historical benefit record size window window carefully choose smallest cover fs ip aggregate historical record contain feature set fs ip fs ip fs include email actual depend service email record take address error window change behavior spam vice versa play role ip ip change spam versa goal change stay long small value alternate back indeed realistic single behavior ip record train learn classifier window attribute relational
centralized order go observation kullback leibler kl every furthermore remainder describe decentralized detection literature classify systematically fusion second combine fusion center version communication alphabet integer threshold center additionally stationary independent increment detection rule fusion threshold choose alarm recursion multiply unit acquire rate stand rule study take sensor time easy optimize threshold threshold optimal sensor threshold way fusion detect support bit message policy fusion center sensor min shoot asymptotically alternative alarm suggest walk ki since determine decentralized contrary possible
importantly error contribute eqs acquire rate cardinality mind situation n c h eqs experimental model correct object context difference co occurrence context regardless target purely observational value select object select episode eq occurrence regardless refer reader ref mechanic mapping limitation allow analytical extremely bar word correspond storage capability make performance subject control experiment sophisticated capture human observational interestingly value law minimal report child map limitation impose frequency acknowledgment
observation measurement series random target target I evolves denote definition newly target detect target detected finally define measurement detect th mapping target measurement target observation target indicate detect target target detect time target name style name x name name name name x death target observation write mass lebesgue law main aim computing mle also static treat evaluate metropolis hasting move present online important iterative calculate follow step step repeat converge show calculate statistic arise omit notation sufficient statistic estimate tracking relation make explicit later expectation w solution depend matrix target velocity plane direction record outside interval detection reference pf
kernel task task kernel study latent model linear idea allow arbitrary task due similarity neither used perform refers water meta attribute task powerful present shoot theoretical investigation zero shot attribute label attribute internet source attribute paper kind linguistic source google yahoo evaluation approach attribute generalize task category equip category attribute help boost categorization heavily rely share part generalization maximum analysis find share latent modality attribute
eq basis vector quantitie overall partition use connect label unary probability necessary mapping invertible binary previously lead suppose vertex homogeneous unary
addition kde addition also continuity inequality find ne ff f countable k np approach suffer curse dimensionality tradeoff misspecification bias fall outside conversely parametric approximate moderately combine place mass family derive constraint match impose around example probability mass concentrate around result exponential contain added penalty encourage sparsity making within statistic vanish true family approach unknown
agglomerative compression minimize first group program encode membership underlie subspace infer cluster combination point c solution writing combination point expressive combination point representation fact number great rise infinitely nonzero refer combination direction ideally datum nonzero infinitely minimize objective n ij decrease infinity toward solution increase hard problem count number efficiently find consider prefer also rewrite program matrix ni ideally correspond next infer guarantee recover solve ideally infer segmentation build weight nn weight similarity ideal ideally correspond subspace point combination may necessarily particular make sure get post describe approach ideally row whose eigenvector symmetric graph step similarity normalize sparse norm point select euclidean value euclidean point nonzero put emphasis keep stronger sure ssc edge subspace weak ssc connection subspace subspace find lie program corrupt normalize spectral corrupt entry due hoc datum perfectly motion segmentation trajectory corrupt noise entry large
report reflect see sbm model perform one network however network structure homogeneou model show single improvement sbm membership perform single membership membership membership higher less accurate mixed membership show supervise times blockmodel network class mix membership
note momentum represent variational boltzmann weight eventually stability clear dynamical either discrete eq time several neural converge start corresponding point globally unit q specify lyapunov eq case follow
obvious prove number wide web internet tool life activity education virtual online handle numerous piece hardware software together internet evolution user interact evolve behaviour give privacy medium life social facebook boost web usage increase overall store expansion web current facebook million million massive stream digital text attract social communication rely access perhaps equally access although situation web weather long notice reference previous paragraph fairly start piece finding present weather point resource particular make weather uk location way automatic dealing amount interpretation become human feasible form lie notion text world define inference occur simplify amount simple conclusion apply artificial mining field improvement framework relevant context effort purpose project already sub constrain aim goal might seem address proper justification hypothesis course amount subset practice early stage ph project finding initial answer project dedicate develop derive conclusion inference form extend model retrieval processing massive amount aim reduce consider event pattern statistical perform case quantify fraction life measure reaction opinion supervise learn abstract discovery project aim discover specific applicable could carefully depend aforementioned question aim text inference extraction play important several work thesis attract health news medium research public shape perspective twitter mit ball new social medium detect daily release link paper review study forecasting article media observer pp impact research chapter thesis review ph three list provide infer chapter european conference machine principle knowledge database tool detect uk exploit twitter describe improvement publication chapter transaction methodology explore web two uk chapter social network social medium news www france associate score real life amongst evidence connect give normalise document raw term term normalise hold retrieve normalise document ij nm calculate document appear document retrieve inverse frequency use formula scheme nm ij text text basis variant merge something help reduce vocabulary index text approach use removal one make list research research researcher research remove keep character create distinction comparative since automate rule might addition desirable semantic mostly quite article proposition english text stop list generate frequent english manual word automatically maximum word spam decide set minimum frequency depend threshold frequency word behind remove stop word dimensionality increase behind appendix notion scientific artificial intelligence machine scenario input datum regressor show nonparametric bootstrap lasso likewise bag address incorporate whereas give consistent algebraic form entire removing help dimensionality feature statistical reduce mm two first project explain value method project collect impossible derive web website suppose update content resource article video type evolution complementary tool web twitter characteristic character post experimental derivation thesis collect use twitter create new account per twitter allow message see maximum character default private tweet publish account tweet interface type account public account people people distinction public one private sided tweet character user also reproduce tweet incorporate topic tweet start hash something topic twitter refer justify twitter content twitter social comprise sub dense one close friend latter influential great twitter identify vast twitter human short quantity connection direction united uniform population represent activity seek user connect interaction collaborative tweet work environment several aspect twitter encourage interaction student great applicability twitter news persistent nature enable break twitter communication political real life extremely limit twitter rich opinion mining sentiment track intelligence source easily regard political opinion tv start combine particular twitter temporal reaction political twitter preliminary approach due author tweet valuable medium connect twitter build tweet real life occurrence come conclusion degree importance twitter side relationship mechanism allow rapid spread conduct online make behaviour stage work work reason force collect store essential describe reference library really feed extensive language content manner recent development format support twitter use atom content retrieve atom match atom feed retrieve twitter actual tweet feed tag explanatory example date tweet language prefer orient framework establish interface useful already tweet collect study tweet tweet request
lead less problem show experiment achieve report strength joint near neighbor distance mutual reduce effect curse dimensionality effect orthogonality follow review detailed state art wide family section relate description cite principal project direction maximum whose usually difficult determine computed dataset arise threshold automatically
give weight mse assign situation truth ground selection fully gain compete
extend ei ed em hypothese enumeration infinitely often output make collection hypothesis restrict fail symmetric list hypothese output extension hypothesis must
heavily unbalanced supervise classifier curve supervise assign object roc positive roc curve gradually false negative specify classification proportion threshold mainly relative misclassification area roc widely curve lie strictly perform area admit several interpretation randomly member
regularizer realize neighbor classical tree group mini batch neighbor iteration apply parameter split rating similarly choose randomly optimization correction remain testing estimate rmse mae score provide neighborhood affect relation column align neighborhood increase rmse rmse result validation surface fig illustration mini size summarize follow neighborhood surface fig imply validation surface good indicator
eq anti differentiable square necessarily symmetric property directional derivative symmetric value separable possibly smooth non manuscript attention selection denoise rank matrix close
stability yield fix eq compute apply positively homogeneous jacobian first local belief propagation coherent usual bp convergence connectivity theorem impose decrease mutual explain sparse variable prop prop prop prop express field
subject censor considerable back censor also presence independent censor idea paper censor covariate replace rather strong assumption survival censor time iterative base principle mass survival subject censor exploit underlie martingale mechanism quantile simultaneously impact covariate work cope censor employ mass idea cite common rather involved rely heavily important far present paper sensible weak certain kind derive discuss remainder deal quantile yield substantial penalization identify predictor regression considerable context censor quantile develop penalty unconditional survival censor smoothing fix sparse context censor independence censor assumption many inefficient name concerned distribution example predictor
hold choice get maximizer relate phenomenon initial partition row update cluster lin keep make long locally row label profile likelihood iteratively perform converge optimum perform practice likelihood order conventional sort operation comparison algorithmic cost spectral computing indirect proportion misclassifie pl simulation poisson rate pl pl misclassification mean km sim profile likelihood partition separately initialization simulation block row column row column membership block q sparse present size deviation scenario profile criterion method sign although sim
prove existence jt minimize know np fortunately build reasonable suboptimal assign arbitrarily jt existence least possibility text depth ex var var thick hmm build jt cluster name instead result jt height ex var node var var label edge edge edge thick c edge thick edge fig jt jt cover also intersection less illustrative
dark background noise version intensity area reciprocal big pre contrast means variance image di perfect lee whole test summarize form show situation analyze
conjecture strength role determine minimax really around minima motivating convexity strength direct dimension reduce task stochastically dimensional sign exposition interest unique corrupted hence boundary switch point fx g probability linearly ie hence exponent provide obey like active classification query label excess symmetric difference modification proof probably show use see appendix x x originally prove provide nice
forest rapidly forest situation show band look image city area deal apply image manually four manually case degree polynomial employ statistical parameter detect usual shape etc detect four label heterogeneous homogeneous tend heterogeneous approach image spline contour vary degree specify area coarse calculate segment performance term image acceptable addition edge quantify estimator intensity component consistently nine model detail robust estimator contour
standard assume consider th induction least apply choice ks classic imply large project direction fall interval put k constant passive exist hypothesis find focus isotropic non isotropic pass whitening ultimately algorithm output consistent prove prove ks consistent classifier example include whose fall length project density bound constant need chernoff bind draw example fail member enough total number constant factor complete proof point
matlab n inf double os windows inf inf inf inf inf inf inf inf inf inf inf inf inf inf inf inf inf inf mac os inf inf inf inf inf inf inf matlab windows inf inf inf inf inf inf inf inf inf inf inf inf mac os inf inf inf inf inf inf inf windows inf inf inf
probit scale control bayes hold far py pp commonly make e normalize expand taylor binary linear mix binary trait scaling scale true correction back ht ht ht c keyword reference double mixture ridge mixture contain distribution normal distribution scale respectively scale respectively combine random capture sample structure normal distribution list table include refer different effect distribution keyword list fitting height tc obtain error height standard calculate iteration replicate intra family split phenotype computation perform core intel ghz comprehensive many suffice provide rough guide computational burden simulation snps causal snps cd trait height tc cd ghz cpu million deviation calculate replicate replicate intra split table ht error standard posterior ccccc cd il usa statistics il usa mail edu edu abstract linear mixed model widely recently genome study two situation assumption arise specification hyper monte phenotype explain phenotype value combine advantage sparse regression phenotype outperform large previously suggest implement available summary genetic variation characteristic quantitative trait human production improvement help relationship ultimately lead clinical practice present help tackle challenge mix
discard parent additional evaluation reflect trial r pt pt pt pt pt trial pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt
model experiment deeply crucial achieving introduce sharing share neighbor overlap within convolutional filter input filter set work much big image sampling set remove overlap comparison architecture neighboring offset diagonal considerably make set filter offset keep constant filter simple extend follow bottom straightforward stack top express eq discuss due factorial another high state aggregated aggregated posterior sampling consist top
begin ensure penalty focus coarse grid infer apply hierarchy argument satisfy number sake element large combining contradict satisfy penalty never minimizer oracle right equip restrict attention choose class select carefully reason present complete theorem necessarily smallest bind bound turn satisfie substitute assumption assumption verify remark establish sufficient look observe irrespective term due form minimize regularization fast rate computationally inferential modify motivating budget bound exponentially exponential rate use assumption fast hierarchy dimension loss square boost assumption integral consideration identical conjunction conclude budget expect argument heavily estimator process positively correlate constrained set difficult relate previous early perform
multiply obtain bayes discuss validation correspond conduct independently rule observe essentially probability observe individual factor multiply assumption logarithmic scale factor bayes threshold bayes cost decision reject hypothesis appropriate help reduce assume correct decision wrong probability true result bayes collect validation may evidence reject may reasonable alternative probability hypothesis exercise pdf pdfs model calculation helps correct bayes different metric hypothesis test start pdf quantity hypothesis denominator affect numerator standard variable negligible base eqs substitute statistic combine eqs obtain bayes cdf similarly cdf freedom choose letting threshold hypothesis
change likely detector observation likely drift control positive generate false positive roughly stream conclude approach concept pass overhead control although extended streaming predict object whether correct stream incorrect correspond detect drift beyond use error stream treat underlie black neural machine modular drift detection drift detector discriminant scheme detect propose however either change adapt weighted chart recently present general chart stream explain situation present describe chart drift name compare recently move originally detect increase variable observe change stream proceed estimator essentially form control old independent deviation occur away
condition study effort aim efficiently observation respect set backward hmms lead quadratic recursive formulae illustration apply time temperature interest influence height ex node var x var thick thick edge thick edge dash dash start hidden sequence heterogeneous
bootstrappe network enough bootstrappe quite slow fairly tight present classic nuisance present nonetheless derive limit sparse graph simulation confirm information criterion former derivation aic asymptotic tool theory suggest fold global network good way set miss positive inference present principled block ratio block degree network excellent correction point selection network divide naturally module community connect discover modeling range
nod fuzzy memory forecasting forecasting shift memory represent long exploit month value previous see fig fuzzy consistently low forecasting number increase shift continue improve fuzzy step fuzzy shift node numerically qualitative extract memory system successfully learn month information short correlation sufficient essential scale fig order represent fuzzy future begin use forecast distant predict prediction treat memory representation point end predict series begin shift show generate node continue cycle fuzzy memory represent sufficiently capture two shift signal describe overfitte regression extract coefficient require span forecasting perform row red shift ensure split extract coefficient fig compare plot red goal within shift although contain self evolve resource associate complete shift evolve moment subsample spaced extract corresponding series turn forecast sufficient red fig extend temporal memory outperform low come long scale rather pick help overfitte fuzzy information forecasting generalization environment stimulus
take rate regularize matrix affect seem method measure seem converge pairwise identify causal outperform ad hoc left study well world project recently devote causal connection among vector variable exploit present recent compare
adjoint sum may prohibitive particularly surrogate introduce section address expansion either admissible indicate pl expansion former logarithm obtain complete derivation input affine basis typically uncertain main perturbation analysis derive gradient incorporates show analytical estimate gradient experimental stochastic approximation deterministic physics base partial differential numerical able impact extract sample prototype iterative resemble appear rate necessarily well focus special discrete variable development estimator practical setting largely assessment conduct design practical suggest sa sensitivity step choice class comparable substantially sa design approach difference algorithm selection property introduce underlying present bfgs challenge evaluate expansion numerical bfgs sensor involve conclusion relative strength choose sequential allow experiment next paper continuously parameterize space infer indirect observation perform continuous random density evidence assume lead
metric distance similar keep apart ordinal rating label object likely employ label compute set implementation aim maximize average optimization experience local optimum ascent much consequently penalty detail use rating euclidean metric observe feature object object feature underlie metric
produce appear smooth trial trial produce recursive minimization objective indicate level give provide collection uniquely smoothness etc function give kernel smooth hyperparameter determine pt range e relatively large begin specification trajectory pt eq q range let define restricted partition I encourage smoothness partition unbalanced tree equal parent encouraging level less robustness child locally think level encode fine denote covariance notational simplicity inherent conjugacy one hierarchy gps marginally gp encodes define covariance two location contain distance level fall example determine band influential restrict evaluate specific
naive truth high model experiment section problem pick tb l international book com www com co store com com book books book usa model author recall estimate l round round voting compare result book author good accuracy reliability ranking compare accuracy truth value additional source source dependent assign tie reliability accuracy peak gradually
close strength hard optimize energy unary pair ij ji strength pair wise strong energy average energy negative energy poorly bind among used motivate focus swap sim scale character use partially multiscale result run anneal gibbs anneal cc show energy relative energy baseline multiscale combine consistently energy variance b experiment baseline percent strictly energie pair energy define energy perform chinese represent pattern represent energy efficiently test energy unified act primal energy dual energy addition visualization chinese character energy highlight submodular energy make multiscale expand result percent art fraction energy address frame video co
threshold start motivation understand md analysis separability separable case online sample erm erm batch understanding whether rather guarantee arise argument erm algorithm one online erm gap analysis sense mirror provide well possible desirable understand mirror contribution provide online improve regret use agnostic hinge even online guarantee ensure also erm focus choose minimizer rather match low logarithmic require
test homogeneity homogeneity testing homogeneity I dependent attempt contribution research reduce strongly mix mix homogeneity sequence say error immediate type pd n also growth choose result probability condition theorem let generate x least strict linkage target strict generate pair total number statement obvious homogeneity
arc galaxy arc big curvature curve linearity p make arbitrarily long short evident short curvature identical across straight cloud point point nan cloud fact value attempt cloud additional galaxy global projection distance curve notion physical origin scatter remain red galaxy highly property galaxy importantly fact shape depend correlated arc track separate early galaxy tell track material produce recent series galaxy green find formation galaxy separate star star form prominent color emission star formation galaxy population appear pca green density red color fig average keep arc increase except green stay decrease continue decrease monotonically formation emission equivalent drop move emission last pure form region branch separate pure star arc length basically nan activity agreement activity cause formation classify roughly lf come dark matter central galaxy galaxy galaxy power lf finite end galaxy hand shape function particular high mass central whose
far method nystr om transformation need number sample point force theoretically point double rapid bottleneck employ learn fast implementation number old take gain accuracy dataset set iteration take second average second medium test gaussian however tight instead seeks encourage overfitte gaussian solution repeat constraint repeat lie medium dataset median min randomly select scale cross dataset get exist work
within coefficient within consider turn symmetric process nine control chart lr chart apply chart chart chart depend list attain four depend reference bold comparison bad behave control chart sr chart lr chart small sr chart five scheme value lr chart behave good show assumption behave chart change generalize scheme chart chart behave reference chart chart lr sr chart choice magnitude advance chart ar chart structure chart
analysis however important group bi van zhang zhang zhang introduce concept superior standard lasso sparsity sparse recently conduct analysis lasso inequality restrict show bound class group sparse vector error enjoy excellent prediction general satisfy model zhang fan point tend coefficient one selection lin lasso minimize prediction similarly group select false false false construct base without penalty functions bridge frank dt fan li mcp dt zhang penalty estimator least assume bridge fan li penalty obtain bridge scad mcp penalty zhang problem mcp invariance multipli ensure amount regularization one mcp penalty scad adequate attention regression
inference provably even choice take variational bag comparable choose vb objective corpus use corpus trained lda test criterion convergence tp comparison depict vb much take less however vb often overall vb quickly significantly heavily vb suggest vb tp sparse infer proportion fraction infer latent depict inference always sparse topic surprisingly achieve topic reason multiplication solution vb provably theoretical experiment row goodness inference appear
constant give constant determine finite polynomial large simply request build strong classic exact sequence boolean instance small concept test definition shannon term check read boolean e must definition checking test target require test distinguish obtain individual lx generalized basis exist generally simply projection individual checking check test compare alternative read exist bit stand boolean permutation
divided subset site q keyword generate transaction begin mapping site send remain transaction site transaction request site candidate fail generally coincide site class transaction termination home transaction start next choice find user inactive tweet furthermore rarely user
road valuable flow planning road due cost dedicated trace equip match road real store later complex task address discover part road conduct understand movement well reason network trajectory briefly
especially extend neighborhood graph total span tree construction belong enyi entropy simulation number knn knn make neighbor value rp distribute individual group precision estimation accord estimation precision illustrate fig estimation notice estimation rp moderately estimator challenge deal dataset modality filter version
categorical dimensionality thereby share dimension new dimension use encode hyperparameter benefit categorical mathematical lose enable clean software engineering keep track space similar compute combine kernel straightforward use k iff pair input continuous include ten categorical heuristic value sort enable deal decay elimination clause discretize eight base continuous different exclude optimality gap preprocesse categorical categorical categorical simplex half categorical barrier parameter mostly configuration restrict cutoff second run default parameter instance benchmark instance amongst machine model prediction configuration run per node ghz intel bit processor ram cutoff second take requirement cpu year run take second year sound method large yield surprisingly prediction base datum section vary one distribution benchmark amongst one configuration ten machine fold previously unseen rr sp rt rf rr pp rt rf pp rf rr sp nn rt rf additional correlation log likelihood ccccc forest quantify solver runtime varied observe forest
infimum inequality union least discrepancy guarantee combination hypothesis combination weight favorable guarantee lead determining follow regularization standard qp available software practice require bound also hold close discrepancy spam filter union additionally case remain unchanged period cycle task spam political sentiment problem major
classification system produce range I lastly validate anomaly learn classifier transaction anomaly interesting trend classifier effective anomaly anomaly preferable rate anomaly index classifier found probably offer six apply variation distinct classification gm classifier across test rank auc rank rank result construct multivariate show dataset anomaly support anomaly base high anomaly curve reflect effort learn effectiveness additionally converge amount current contain transaction derive transaction divide subset transaction next partition sized construct finally result improve increase improved achieve monotonic dataset different gm reach classifier examine domain univariate approach anomaly outli outli anomaly unsupervised anomaly none instance apply one paradigm output algorithm calculate factor instance high local instance local suitable purpose outli acceptable widely repository type two prominent similar
process minimal maximal know degree structure process well series reveal periodic maximal randomness possess degree physical quantify statistical measure effective measure distribution detect
even summarize used algebra operation code segment efficient high additional execution high gpu processor cost use two cpu implementation matlab cpu call implementation precision cholesky hypercube scheme implementation power recent release relax allow section challenge effectively gp repeatedly solution lead suppose generate
nc edges superiority ht connectivity model see nc nc ad graph simultaneously multiple maximize log necessary condition block screening develop efficient multiple employ multiple scheme propose demonstrate efficiency screening explore use direction shrink method warm find initial improve possibility section theorem remark multiple graphical simultaneously fuse penalty encourage graph structure motivate brain network brain control brain patient cognitive ad nc identical brain formulation contribution condition decomposable key property screening present subgraph allow subgraphs cost demonstrate effectiveness efficiency undirected graphical explore relationship vision analysis
dataset trajectory evaluate importance drop vice versa compare segment analogy trajectory similarity graph similarity graph weight graph represent e e contain trajectory loose devise strict similarity edge e segment mentioned similarity graph discover road segment cluster analogy segment generator briefly direct along unique loose graph edge seven road across region
reduce twice ten cifar convolutional pooling follow layer neighborhood pooling pool layer layer unit softmax layer class convolutional cifar dropout strong regularization fourth pooling layer layer layer filter dropout softmax input weight imagenet dropout cnn horizontal augmentation help seven layer convolutional pooling layer layer pool convolutional layer pixel neighboring neuron bank filter input input pool response output group connect channel convolutional layer image filter pixel two max nonlinearity fourth convolutional layer one normalization third filter connect subset channel pool convolutional layer group unique channel produce layer globally neuron layer
negligible hour subsample large collection expense parallel convergence training invoke friend user single power scale facebook shall analyse sm multiple modality arguably single metric sm sm hold collection interest context capture user interest task easy latent like allocation lda visualization normally improve classification large amount train bad bag word stem dimensionality reduction necessity picture aspect link label small collection way use sm sm avoid dimensionality text come play statistically plain baseline setup ground truth interest sm predict vector user exploit support classifier input cross four collection take hour train normalize word document summarize
ep via factor approximate factorize distribution natural property exponential interpret factor exact link message repeat minimize I information incorporate update w kl projection datum clean kl minimization relaxed propagation factor describe relaxation iw I much relaxation relaxation unnormalized replacement adaptively handle outli accurately I
easy acyclic relaxation coincide polytope subgraph keep exposition cover g detail preserve idea involve decomposition straightforwardly subgraph contain grid graph energy energy uv represent way function q smooth subgradient I computable constitute subgradient one primal kind smooth g soft approximate acyclic vector marginal I q subgraph consider special positive tree reweighte free energy show e special acyclic depend probability gibbs maximization primal estimate iterate vanish entropy gibbs associate subgraph entropy combination free separable r
structural network combine acquisition strong similarity applicability recommender simple regularize practical beneficial knowledge challenge path various differ hand variety processing decade measurement suffer high cost low accuracy large infeasible two contribution rating network completion quantifying path investigate rating ordinal large advantageous metric rating carry streaming
edge hybrid feature combine neighbourhood explore great two language l exploit report implementation cl k n n set accuracy propose adopt approach row mae five obtain statistically test set method report result predict adopt distribution report probability rating number rating ccccc e error report mae adopting distribution unbalanced svd c c c
model follow l l preliminary positive risk multivariate student shrinkage estimator sign stein hypothesis classification however implication exist however pearson multivariate student multivariate cauchy distribution concern particular mathematical al et estimation parameter belong sub stein preliminary belonging describe
suit situation different consist random size let population assign sample perform detection aforementioned nonparametric rank empirical candidate straight maxima note identify section simulation assessment edge run computer intel operating programming language version graphic produce shall parametric situation rectangular window distribution window compose use span variety encounter practice range smoothed look situation edge algorithm differ texture situation window detect describe absolute difference locate store array
lin undirected dependence concentration fitting matrix empirical variant depend want element unobserved jointly effect source instance observe stock price price two term marginalization variable second proportional since trace
achieve rate source distortion design matrix varied trade distortion length space per distortion distortion decay sparse achieve distortion computationally encode square distortion compress sense compression shannon rate long goal information storage codebook complexity channel superposition essential terminal code build computationally theory develop computationally compression show function source small near length rapidly distortion encoding source alphabet code rate decode excess scheme slower survey paper technique quantization code attain
control remainder organize apply inclusion principle analytic poisson wide application address problem binomial variable rule possible effort eliminate confidence limit stop result appear conference throughout notation integer denote notation proceed propose analytic stop random specify confidence level estimate virtue sample number sample define absolute integer n sn rule nn n mean termination unnecessary rule purpose sequence n n eventually stop termination sampling procedure stage integer sampling sn appendix stop
suppose fire remove table partial fire let manifold equation small converge h ise model experiment compute seed identically distribute sample fig kl kl expand ise seed computation learn retrieve structure define follow stock
serve binary certainly gaussian clique ise deal interaction essentially uniquely table number node graph graph bernoulli ise multivariate member predictor eq covariate build model predictor valid log likelihood q generalize regression thus logistic coefficient vector square implement nevertheless hessian fisher pt negative multivariate logistic two coefficient hessian negative
cr award american google supported award fellowship nonnegative row one factorization permutation write offer strong coefficient row suppose great reach line I number tell representation row distance great I else introduce extra monotonically extend hold row list may well nonnegative assertion contradiction minimize construct satisfy representation c kk follow occur constraint conclude sum fall factor separable
particular point expression moment follow moment triangular set equation hold thus formulae direct gr basis elimination term generator basis generator parameter map substitute previous alternatively check substitution compute expression considerably simple probability gr basis execute coordinate factor define simple except denominator reflect meaning factor iff iid coin unlikely one iff hide subsequent model flip subsequent two model identically factor occur equilibrium define equilibrium hide restrict yield parametrization parametrization newly occur vanish lie meaningful matrix computation gain specification affine minute ideal take memory gb exhibit parametrization trace version formula generator integer marginalization
hellinger read parameter large useful actual parameter know appendix b paper subsequent repetition estimate though improve increase testing draw seem feasible probability distribution discrete focus testing assumption latter application display fit significance mind section simply consistency trying decide likely false try alternative handling nuisance always exactly repetition extreme substantial nevertheless amount associate seem experimental enforce repeat integer value value value value combination possibility know proper priori permutation specify bin entail sort frequency whether simulate many freedom permutation compute assess consistency simulation draw obtain root square run simulation conduct follow distribution experimental generate step calculate calculated empirical
assumption meet aware standard gibbs reversible reversible irreducible irreducible undirecte binary symmetric indicate fast sampler graphical part sampler state probability state standard gibbs directly gibbs sampler move standard fact analytically fast show symmetry fast several stochastic variation chain rather argument simplification method theorem value metric couple markov markov chain numerous application considerable amount devoted couple insight coupling graph
experiment fig confirm nonzero unit nonzero normalize specify averaged setup dynamic figure dynamic reason outline section experiment report magnitude three see thresholding almost performance comprehensive thresholding result enhance understanding approach linear response future extend fundamental bad coherence model orthonormal banach banach space value
number loss quantity positive table paper result density q formula joint variable evaluate visualize investigate copula degree copula precise loss continue marginal distribution display copula leave skewed density multimodal become distinct skewness readily write skewed mixture skewed mixture unimodal gamma modal influence figure display quantile numerical integration root solver setting dot line indicate claim number independence monotone association equal gauss frank copula grey solid indicate expected policy
brain prove brain region recover challenge recovery ill address univariate predictive domain fit use logistic lr multiclass problem quantity amplitude fmri linear hold estimation suffer particularly signal
use q depend complete pack proof quadratic previous bound number perturbation sphere complicated argument aim proof hausdorff hausdorff v follow relate hausdorff distance hausdorff convex every pair clearly gb gb fx gx
real number finitely arise finite latter exact arithmetic mathematic solve thus quickly approximate answer lead notion us truncation actually algorithm exact error tell run moreover forward forward bound backward expect condition much optimization scientific seen develop algorithm well pose unstable turn much partial history recursion calculus machine lead belief formally capture qualitative highlight role logic study turn rich many algorithm run lead notion np np hardness np completeness lead qualitative question turn problem intractable np sense sort provide possible several way prove algorithm relaxation relaxed program combinatorial convex duality implicit refer meaningful interested want learner sensitive quality interest provide meaningful ill pose solution meaningful noise one function specify geometric exactly parameter solution quality regularization manner lead natural trade optimize fitting though often hard
bound similarity learn norm term inequality advance secondly function specifically loss square frobenius bias zero minimizer xx equation z f combine together imply theorem c consequently imply yield consistency true function application aspect equally notion natural
correspond tend theorem clique clique disjoint contain node therefore recover plant clique size clique size clique clique plant clique model agree exist minimum plant clique provide match guarantee consequence exact bipartite subset subgraph induce bipartite disjoint subgraph set subgraph divide root equal disjoint disjoint density subgraph want characteristic sum disjoint clique pose nonconvex quadratic program let w relax semidefinite program block index nonconvex clique instance disjoint graph know weight edge complement vertex construct sample two plant define feasible objective describe plant semidefinite disjoint subgraph complete graph feasible sample plant scalar scalar maximum density disjoint tend exponentially tends plant comprise modification accommodate
component sufficiently incoherent compression restrict isometry hard subject henceforth admissible sparse draw certain random satisfy random collection restriction amplitude sufficiently formalize exposition henceforth accord element row q high practice specify take whose value recover coincide follow bound rate bernoulli addition incoherence constant generality inside special case hence expect incoherence subspace eliminate theorem section compression give rise fall column space admissible take account vice versa could condition deal benefit incoherent price e follow cauchy likewise obtains follow incoherent generate form uniformly collection partial independent incoherent cf clear incoherent suffice small close column unity attain single element aforementioned compression consist unitary entry subsample select ensure require
marginal specify numerically maximize routine conjugate gradient optimize type ii alternatively equation prediction computational phase hyperparameter marginal expensive involve input cholesky decomposition computation computational part depend hyperparameter evaluate ccccc storage variance complexity reduce g subset away induce plus nearby iterative speed give fully conditional recommend improve fast transform amount ignore complexity full nature paper gp gp apply size description possible alternative randomly cluster point complexity suitable
suitably scale convex hull norm carry kind induce matrix imagine take matrix hull general relaxation tight trace norm ball obtain trace bring paper condition raise obtain oracle poorly well effort algebraic nonetheless oracle estimator rank tractable viewpoint
ik definition approximately control theoretically improved note design must know prove test take perform compare control make design point mostly concentrate point stage stage empirically give trade design median fine scale include coefficient scale threshold plot access set family wavelet basis implement moment dot rough recover shape rough improve point level visually large take plotted run median design outperform consistent compare function level together present difficult design block significant still conclude spatially acknowledgement anonymous valuable
detection add anchor ii projection square obtain face cone current normal normalize lie hyperplane tb algorithm separable nmf separable ray point criterion cone residual identify extreme lemma column cone extreme cone form lagrangian tr lagrange project current extreme current kkt use lagrange hence ti right side ji I regard correctness add maximizer extreme ray select iteration identify extreme current diag strictly positive inverse j strictly positive I jj least remark maximum generate extreme add anchor point
step ise ise seem cutoff critical approximate question good sample output control approximation make close repeat median building calculate ise unbiased variation w
ise discrete mrfs slice image mrfs investigate slice field particularly suited segmentation generally intractable grow exponentially hyperparameter homogeneity image large contrary homogeneous region interesting know drawing achieve use introduce correlation may hide deep markov field gauss adapt memory resource introduce pixel number priori kf associate mainly depend application explain fix image noisy set produce bayesian propose assign coefficient work unnecessary priori theorem follow proportional unfortunately mmse unknown think receive alternative use sample
q sparse notion subgradient restriction minimal need acknowledge existence restrict imply convexity obviously subgradient determine invoke axiom subgradient drop context slight abuse terminology restrict linearization sparse function define align say linearization form strong rsc restrict smoothness condition quantify bregman divergence similar rsc convert bregman divergence rsc vice various rsc important difference bound bound globally unlike rsc require vector subtle difference invoke rsc neighborhood parameter require curvature main result furthermore suppose ii magnitude indicate estimate close magnitude analogous guarantee noisy accuracy obviously properly imply statistical datum target rather
iteration linearly sg iteration cost suit sg optimize iteration train uniformly among yield assumption suitably decrease step size sublinear take write sag iteration momentum sag recent momentum lead decrease momentum gradient sag gradient dual lead size rather author sg value asymptotic efficiency newton rate step size combine iterate display appeal epoch sg average objective remain sublinear various option accelerate smooth accelerated method conjugate hessian free rate deterministic discuss sg decrease convergence part sg iteration iteration basic sg size relationship work converge additional accelerate sg related advantage
furthermore point gram form pass alternative although reasonably sized prohibitive application realistic system however schmidt obtain advantageous pre computation e dynamic carlo dynamic integral evaluate analytically operator act state show integral term act operator efficient perspective path carlo aim estimation allow maintain
process intend enhance noise scene result wishart trace distance divergence neighbor present region come produce reject reject filter
stationary fundamental compression world assumption implicit rather modify stationary promise meta automatically generalize exist suited allow low begin terminology sequential source alphabet finite empty symbol string concatenation source define satisfy compatibility whenever define familiar rule j notation describe partition segment say overlap another exist segment notation piecewise stationary source partition source
mean know estimator regret raw moment median bind moment good trade estimator computational indeed constant update median linear question mean good moment requirement variant heavily rely unclear similar tail bandit focus attention concentration work decision nonparametric monte
trust significance reason automatic new remarkable cosine trend phase fact time perspective clear part cosine model properly course hoc remarkable high order small improvement clear quantitative qualitatively essentially decision could high statistic coupling deterministic trend synchronization relation model mean consequence highly make begin hypothesis hypothesis try find usually situation neither decision whether exclude almost daily cycle exclude long phase time project synchronization support university grant member mathematical image joint phase synchronization grateful extensive discussion synchronization view plot plot phase deterministic term phase process close straight call stochastically nonzero exist couple trend deterministic trend couple relation case trend equilibrium involve correction equilibrium reference international journal potential international journal co correction university synchronization circuit physical phase synchronization extend system synchronization report l dynamical ed mathematical series digital process david large synchronization international journal r detect synchronization autoregressive ratio statistic root make var review white forecast biological neuron k co representation synchronization regard brain region abstract sense equation system one include
model post processing rule validate example curve algorithm complex controlling different try obtain validate performance base benchmark median variable validate accuracy display rule couple htbp accuracy tree soft soft previous input soft rule input see use marker marker code
candidate set ei simulator output illustrate example candidate maximizer ei grow design summarize simulator simulator output select candidate ei calculate ei ei simulator condition meet go initial run aid assess space step step gp ei integration stop tie remainder evaluation simulator demonstrate dimensional approach present global average performance base stationary gp performance design shoot simulator output estimate increase result present section I design ei sequential ei ei ei instead ei ei ei outline operational setting section setting discuss ei follow
go able category bias go recommend method address case guess absence strong estimate directly use fdr assign biological category reliably estimate biological category local discovery identify promise maximum find mle perform order go ideal mle category component thus order representation application method recommend category discovery
offline e vast volume learn kalman filter model online model time step spectral model sophisticated incremental make unclear operate architecture demonstrate parallel robot scheme overhead paper learn share behaviour robot gradient td hundred policy unique develop online base bellman computational architecture correspondence traditional mean dramatically scope life immediate work learn learn control predictive employ e g feature robot learn useful al thousand accurately mobile great
look co two gene multinomial moderate large collect hyper parameter combination attractive might certain exploratory em mcmc model surface drastically capabilitie simultaneous protein per cell sort surface homogeneous cell characterize technology array single device enable cell hundred classic cell intrinsic nature result expression heterogeneity carry important expression population heterogeneity provide snapshot system multiple simultaneously heterogeneous whole monitoring cell intensity typically thresholde subset boolean combination positivity biological technology single cell thresholding cell subset boolean large
eigenvector define easy frobenius bind n specify nystr om om frobenius besides intrinsic low cause nystr om measure frobenius large spectrum examine nystr theory paper structure nystr
standard ht homogeneous h eq standard observation eq equivalently ij
consequently situation justify possibility b rule attempt next stage ir scan attempt detect identify case person stage unnecessary indeed similarly diagnosis disease stage sense begin early treatment stage cascade measurement stage complex region complex measurement classifier illustrate advantage scheme sense modality centralize reject utilize nd sensor achieve centralized htb approach call sensor sense modality derive reduce cost acquisition sequential reject measurement classifier cast dp stage stage scenario produce consequently unlike estimate adopt parametric utilize go discrete instead go function formulate utilize wise evaluate decomposition formulate erm stage decision transform reject
paper portion final find describe section graph throughout paper distinct edge sum edge unweighted roughly speak boundary number weight edge vertex add create remove create strictly boundary complicated remove strictly decrease satisfy outside every edge outside great edge outside outside illustrate graph half place cut
interaction trace specialized call web usage mining mining user behavior website place context extract file web usage trace become critical website management need build site help retain current cross sale track leave logical service traffic flow etc account entire usage trace
million sensitive visually variation patch image instead reduce color effective efficiency color ambiguity randomly raw effective image size virtual generative many object recognition raw element color retrieve
political structure even usually refer configuration agent network height network measurement agent small send parent agent message message throughout make information aggregate information convergence star criterion converge exponentially exponent star tree unbounde height binary leaf distance minimum leave large message unit likelihood total probability convergence tree combination gives learn social elaborate rate follow rule break odd rule optimal however majority achieve strategy show majority question close rule differ way fusion become next consider learn agent level generate message send parent intermediate receive message send place decision make information agent aggregate maker
reduce parameter vocabulary reduction sample singular matrix small become increasingly hard number dimension point write u likewise u range u product make substitution j noting substitution get mn proof three suppose take pr pr less trivial iid hmm lastly bind square assumption algebra mn minimum bound
weak equilibria obtain hardness corollary utilize tailor game outcome utilize weak hull nash equilibria equilibrium calibration calibration tx td nash follow weak return ne constitute randomize arrange lemma
convex beyond paper big multiclass become increasingly machine algorithm establish much understand multiclass collection binary base relaxation relax convex statistical quantify incur derive explicitly relate excess misclassification excess exhaustive presentation generalize class several indeed interpret relaxation linear offer speed work due function quantitative similar restriction notably classification might lead consistent restriction
minimize norm risk contour risk evenly risk happen end begin high take respect parameter risk risk sensible solution place emphasis task strategy particularly appropriate learner relative amount observation minimize risk subset compositional relaxation minimax case exhibit situation small fraction fundamentally hard remain task reduce level minimize maximum minimize soft intuitively formalize
last term note assume constant confidence almost happen space choose tend ball small close almost domain f e square weak heavy tailed bad optimal investigation algorithm paper pair form construct rank functional define measurable barrier verify shannon minimizer totally fx barrier difficulty analysis overcome full feature large first
heterogeneous various entity mass fusion aspect improve one integration quantity model gp objective gps gaussian process dependent gps extend handle multiple technique exploit spatial correlation perform simulated need equation represent regression uncertainty subject modification value individual model location east north depth datum may demonstrate stationary auto set covariance set gps consideration respectively convolution detail evaluate gps likewise concentration equation incorporate output gp noise hyperparameter auto heterogeneous output main cross auto function gps address cite kernel isotropic relationship covariance smoothing may develop model smoothing smoothing transform stationary nn kernel cross two correlation respective mathematical formalism mathematically white gp smooth white show stationary covariance covariance equation smoothing ix square e suggest covariance convolution two covariance
dimension probabilistic produce solution problem algorithm offer representation exploit minima result provide minima outline regression problem examine serious analyze understand
far apply inference need hierarchical bayesian undirected discuss trivial either unconstraine change set parameter resort procedure could solve method focus whole theory deal factor include likelihood function general trivial constrained see example present theoretical use concrete develop restrict linear operator banach space ap inference fp subsequent statement tool primal relationship optimization general space result duality banach space conjugate duality banach function bound satisfy either supremum dual banach duality apply density summarize constraint mapping product banach function observation kl relax kl regularity condition difficult check hold examine depend correspond duality posterior compute though representation operator feature convex optimum multiplying term prior distribution coefficient optimization problem remark put depend model extra contribute prior constrain feature otherwise regularization theorem prior widely use bayes implicit bayes rule define could integral implicit flexible model constraint systematically domain automatically piece knowledge allow incorporation knowledge knowledge especially expert normally low behave skewed worth although generic usually make assumption primal solve apply along development model inconsistent present leave systematic consistency future example appendix first classification x put use
classify active inactive divergence instability solution perturbation break replica influence instability discuss later first second former provide indicate success information extraction guide marker inverse classify depend infinity keep around plant locally otherwise along form
step step sa stable sa step path imply imply everything omit sa stable even omit stable doubly proposal iteration implement walk metropolis hasting v target pdf distribution panel diagonal equal artificial aim switch compatible adaptive illustrate behaviour multimodal two weight represent version particular distribution solution near coordinate recover marginal make mcmc discard burn assess look restriction efficiency autocorrelation sample histogram marginal quite well figure unimodal number enough explore original seed hand broad seed indicate autocorrelation reference function mcmc ht pdf panel depict thin line indicate appropriate correspond background
turn predictor occurrence report recent spike manner even signal potential could combine discrete train continuous scenario conceptual analogy spike basically sum exponentially spike spike train apart decay calculation local corner spike spike rather elementary regard complementary spike spike lag sensitive instantaneous instantaneous lag eliminate pt spike train spike directly address analyze dataset contain train spike bivariate average distance identical train dissimilarity profile profile come linear small instantaneous spike train equal rate reduction level detail instantaneous spike train result spike train movie material two possible spike train dissimilarity whole whereas average lead overall spike train interval application mind appropriate whole onto single might high train could intermediate useful high local averaging spike instantaneous exception spike spike global picture fact different resolution slide window spike dependence spike account spike individual spike among spike contrast measure pool histogram value regardless spike train qualitatively reliability difference ratio train computer memory concern profile successive time absence train instant depend
economic recent blockmodel connection respective variable blockmodel network form fitting blockmodel block connection probability histogram nonparametric array term co article co misspecification generative separate exchangeability significantly generalize blockmodel generate definition vertex parameter affinity connect network observation identifiable hence indistinguishable preserve many model recognize present constant co blockmodel specifies approximation blockmodel exchangeable follow sm tn ij give blockmodel parameter loss blockmodel latent vector class membership index blockmodel class separately exchangeable blockmodel task equivalent fix co task estimator categorical blockmodel contain triple whose subset contain measure construction cluster array partition likewise leibl estimator
perform analysis stage aic standard regression summary ridge may attribute robust handling adjustment illustrate use summary top panel bottom panel infinite visual clarity open figure square region almost increase rejection relative unlike term statistic adjustment indicate efficient may concern model production occurrence term block largely large inclusion extreme whereby intersect point inclusion distribution inclusion generalize shape extreme consist number total create exploit dimension technique consider spherical shaped focus priori subset six consist empirical quantile inclusion complete due value sensitive precise quantile quantile close maximum subset substantial alg none pt number pls regularization integration cross moderate regularization ridge weak solution method include square pls ridge ridge neural network introduction summary statistic adjustment column good general offer provide slight improvement substantially along statistic consider computationally exhaustive enumeration neural adjustment
convexity valid display association q display minimax theory q note measurable test direct upper bound explicit address match small compute symmetric diagonal adjacency undirecte find whether clique np semidefinite lambda semidefinite programming scalar program canonical use point relaxation nonconvex program prove major relaxation day though change yield z source relaxation semidefinite trace schwarz drop original optimization linear canonical aforementioned relaxation building early somewhat first inequality remain stay use dual duality direct functional property variant detection deviation st ij follow element bounding lemma taking probability event yield
expect good reward reward empirically minimize regret principle rational maintain find arm compare arm high mean effect single estimate
norm frobenius row index variate draw q unbiased I well know inverse sample either good introduction liu estimator constant decompose estimator obtain put column possess desirable liu rate drawback adaptive tuning need drawback minimization drive adaptive variability motivation p sample eq major upper hard hereafter entry denote step modify procedure take account individual note jj jj side relaxed estimate conditional estimate variability individual dependent
expect capable demonstrate extensive computational great deal many study extend exist study conclude remark component denote diagonal form index set index row index addition semidefinite operator low minimizer nonsmooth approximate threshold play crucial role section point derive nonzero stationary minimizer order similar general stationary statement hold hold yield multiply twice recently chen derive interesting nonzero minimizer special next also minimizer stationary twice
trust solution relaxation lemma therefore equality calculation omit f x denote x x r f write ty poorly area strategy favorable environment actually adversarial guarantee speak dispersion visit overcome problem generalize deterministic environment finite space work determine bad trajectory lipschitz trivial compatible function search expression worst possible return lower previous depend dispersion trajectory conservative investigate np stage np exactly preserve nature generalization lead guarantee drop problem polynomial trust lagrangian quadratic prove relaxation
quasi analysis necessary answer devote induce connect underlie theorem like admit well exponential type nr sx field n sx I dt concern every exist essential role random index large estimate estimator order convergence dimensional wiener theorem generalize n sx handle I quasi likelihood analysis type estimator moreover type latter practical application pursuit field question h discuss involve set let c n ij pp fact exist c kn u pn kn p u relation c write number cn j complete c k fulfil fx jx kx f admit expansion jx jx fx x x near compact admit support support one dimensional jx j onto easy fx
outperform recovery considerably sequence compressed comparison task mt robust cs computationally efficient affine cs admm robust cs good fista robust considerable huber despite fact slow need recover compressed term speed triangle inequality immediate mm mm assumption lemma compress cs cope corruption cs model combine robust cs recovery newly formulate cs solve rather inefficient cs limitation cs cs efficient advance non furthermore formulation norm though fista cs fista additionally solve cs formulation efficiently differ fista update extend number affine norm powerful cs cope extension cs include fista provably admm cs formulation
validation fit publication com component careful standard cross fail pca characteristic present pca idea predict auto many extract manner locally project som describe curve subspace replace manifold nonlinear mapping focus auto technique technique approach
regard final assumption smoothness constant z region cube sake brevity investigate kernel gaussian precise result satisfied nx long defer decay fx random quantity fx nx n fx nx band width ill hand grow system fix wise indirect price uniformity easily function band extend time define df ty imaging function correspond model statement add fourier consideration fourier transform transformation nx tt db g k h tn additional asymptotic require kernel assume mn mn mn mb k moment satisfy mb mh ta h
cluster apply become heuristic picture state desirable large depend geometry example room state seek partition directly worth coarse goal macro state fine coarse coarse average path extent capture fine sense markov determine hierarchy shortest particular coarse coarse generality rich transfer hierarchical framework overview current relate concerned discover specific possible transfer share constitute correspondence reward task instance option transfer abstraction identify carry kind transfer set common formalism specify mind immediately within also eigenfunction domain trajectory eigenfunction function define varied since coarse fine within mdps expand describe derive either transition result value may store library multiscale mdps paper contrast procedure support transfer coarse scale handle framework policy transfer challenging scale consideration refinement representation always mdps scale present knowledge transfer mdps multiscale center hierarchical coarse problem mdps local fine coarse scale argue multiscale efficiently solve transfer localize solution consider localize scale across would expect effort subsection generalization compression access pre model involve average need relaxed entirely multiscale compression include completely set bottleneck detection regime initially local heat evolve exploration starting need estimate coarse transition reward process detect bottleneck proceeding picture successively add compressed simulator could make reach state multiscale may approach discrete representative state discretization general dense prescribe geometry largely multiscale translation laplacian build trajectory coarse mdp handle continuous discrete continuous fine include policy factor slightly usual action reward discount adopt take collect transition mapping allow later policy thought satisfy action deterministic placing unit action explicitly track avoid matter constraint need assign policy discuss policy restrict feasible suitably markov action accord tensor hadamard finally denote take continuous space haar measure volume given define assign discount horizon take sequence action bring expectation ps action discount criterion choice commonly function compute dynamic ss stationary policy achieve policy whenever policy primarily mdp determination usual approach conditioning apply markov stochastic transition discount see one green analogy markov govern restriction subset unchanged restriction sub definition outside operation expectation respect restrict respect truncation action although efficient fact define quantity locally interest quantity restriction lastly vector scalar follow individually detail connect bottleneck whose certain policy compress mdps mdps fine mdps multiscale construction perfectly next fine compression enjoy scale scale fine scale high thought level abstraction original novel mdps yield significant roughly mdp compute across section establish mdps solution problem problem transfer devoted subsection subsection overview concern implement latter read big picture proof subsection appendix detection partitioning mdp c induce equivalence yield cluster equivalence plus bottleneck state denote associate markov policy consist reward additional reach say discussion make embedding cluster visualize reference graph
power allocation pa vectors simplex work action power amount level author point performance efficiency achieve substantially report contribution level proportion mean rely local nash ne organized sec wireless formalize sec present formal analytical property iteration ne well fraction ne validate analysis c sub
ascent may expensive attack optimal efficiently terminate predefined validation follow section method dimensional evaluate effectiveness mnist handwritten digit recognition first generation covariance assign figure otherwise consist class experiment blue serve start refine attack termination attack plot rbf background surface explicitly hinge loss
panel green line interval great show load terminal graphic ltb lt lt lt lt ltb lt lt lt lt bp highly noise quantity power panel package load package package graphic explanation terminal need graphic macro ltb lt lt ltb lt lt lt lt lt lt reconstruction quantity describe panel great log spectrum row logarithmic version dark reconstruction right bottom bottom uncertainty without white theory linearity simplicity effect observational mask uncorrelate one sec first discuss sphere periodic discretized case normalization power spectrum show fig highly variance panel reconstruction panel reconstruction apparent former homogeneity since impact higher demand region well represent uncertainty high linearity signal variance nonlinear high level reconstruction sec reconstruct usage partly prevent scale drop dominate still directly affect uncertainty lack prominent signal dominate reconstruct dominate signal around pixel linearity
compute similar u j implementation size cluster represent update concatenation pixel alternative consist whereby sparsity impose propose context specifically ensure fit pca provide coefficient coefficient eq soft investigate alternate method multiplier algorithm adapt poisson though practice strategy factorization extract fine consist avoid cluster improve partition approach result redundancy region perform patch similarly enforce inside patch size matrix factorize author cluster drive patch increase load improvement particularly noise paper compare similar adapt clustering
tail thompson close ucb thank bernoulli thompson sampling apply bandit therefore ucb complex setting computable efficiently use encouraging conjugate suggest pose challenge often heavily dependent sample give work support national project european community grant pt pt claim corollary remark proof question optimality thompson open positively bernoulli comparison
figure vs sample value cost ucb sensitive h cost minimum show exploration greedy well exploration sampling prominent h aware empirically greedy figure arm sampling occur monte carlo mdps improvement numerous application argue perspective cr scheme attempt regret inspire ideally ab well speed aware
dominate state fix let yield truncate appeal theorem provide kf fact already nevertheless arbitrarily large smoothness form manner formalize rise approach normal analogously discuss gradually achieve imply arrive immediately binomial converge converge marginally unity far q admit smooth serve slightly rapidly kn mixed distribution formally regime theorem attain coincide obtain shrink toward rate agree corollary result general poisson variate dispersion evaluate linearly relative poisson limit regime increasingly decay sampling limit degree realize govern characterize variation achievable complementary role play specification law network pairwise bernoulli trial achieve thorough understanding population parameterize smooth realistic summarize conclusion implication practitioner crucially highlight
backtrack line start repeatedly multiple rare multipli theorem use combine version well balancing demonstrate series generate improvement controller performance aforementione common approach control forecast future optimize horizon forecast realize first forecasting repeat develop regression focus attention dynamic evolve discrete action minimize cost quadratic gx illustrate balance cart cart move along axis linearization absence force
occurrence estimation respective pr innovation standard theoretical spectrum fit residual summary adapt deviation respectively computation autoregressive handle quickly moderate summary std pr summary furthermore versus deviation hessian estimation illustrate table hessian est est hessian show variance deviation integration implement integrate involved integral however limit ii iii
non mle somewhat ht interval observation solid df bandwidth ht precede estimator status mle smooth integrate w discrete plug seem attain distribution somewhat whereas natural dimensional respectively see bivariate bias take direction usual bandwidth lead play seem attain local bivariate order theorem numerator
ef mle p inverse profile exponential x reverse profile make help understanding adopt alternative alternative jx j know analytically extract distribution mean preferred score constant treat along case model intractable integral log latent mc e location family shape write model agent noise alternative focus eq density concave concave accord log
focus belong consider membership membership feature convenience intercept logistic parameter presence feature attribute graph latent affinity tendency assignment tendency link
greedy reach htbp leave one segment segment therefore good basis control optimization segment start segment basis quite spectrum mapping object offer characteristic
scatter deduce cost zero reduce zero question besides measurement anonymous review comment quality national foundation china project excellent team support liu grant leave side coherence side term denote transpose refer real value dimensional small compressive sense community measurement formulate reconstruct unknown interest
location object scene bin capture typical reduce normalize random run cross pair notably std fold accuracy scene dataset category difficult scene vary foreground category image extract dimensionality preserve incorporate spatial information location normalize show enyi accuracy attain previous method decrease though applicability tb science scale simulation phenomenon occur exploratory experiment data flow calculate contiguous sub phenomenon stationary parallel plane velocity center latter label negative kernel hellinger distance base outcome parameter achieve hellinger hellinger evaluate slice resolution slice result velocity classification slightly region little canonical somewhat instance explore large phenomenon distribution region fig pick
describe decrease guarantee play role aspect future empirically unique see figure relaxed grid side c impossible identifiability distribution uniform optimal node assume initially dag associate delay graph network access even necessarily dag regular network add let associate unit access
txt color solid triangle index header comma include dataset lambda l densely triangle mark option solid index header sep comma match lambda l header sep comma lambda color densely dot mark mark solid col sep comma include lambda l header col sep confidence txt header col sep comma txt densely mark mark index header col comma include lambda txt index header col comma lambda densely style thick index index header col comma lambda include match avg txt solid repeat header sep comma include match lambda header sep comma lambda gray header col sep comma lambda txt xlabel pass ylabel area log pos east col comma lambda txt solid thick square repeat header comma dataset lambda index header col sep comma lambda product confidence thick mark repeat header true col comma l densely dash style header comma match lambda txt index header col comma lambda txt color densely mark mark option solid header col sep comma match lambda product ls txt table header comma lambda txt color solid mark table header col sep comma dataset txt densely dash thick mark table header sep comma lambda txt col comma include lambda color densely style header true col comma lambda txt index sep comma lambda mark header col comma lambda avg txt header col sep match lambda confidence txt solid style mark table header col comma lambda txt xlabel effective pass ylabel style legend true col comma product solid mark header true col sep comma include datum lambda txt comma matching product txt none triangle mark repeat true col comma dataset matching lambda l txt densely dash thick mark triangle repeat option col comma lambda l opt header sep comma lambda product l densely dot mark mark option header col sep comma include match txt col sep comma lambda green solid thick repeat table sep comma data lambda txt densely mark table col sep comma lambda txt index col comma include lambda confidence color densely dot mark table comma lambda txt table index header true col sep comma lambda confidence txt color black solid repeat index header col comma include dataset lambda true comma lambda txt color gray style thick index comma datum dataset lambda txt see regularization amongst method figure cut give keep well see far scheme mit nlp initialize decrease pass multiply evaluation oracle count extra pass appear plot method initialize set dual option hamming sequence match task gold hamming error supplementary randomize classic frank wolfe separable despite show duality frank subgradient however unlike subgradient frank wolfe allow duality guarantee outperform solver amongst interest solver tailor applicability svms svms graph combinatorial object due difficulty deal constraint rate single subgradient structural svms practitioner sensitive terminate iteration solve structural frank wolfe recent signal svms key iterate use subgradient cutting thus frank wolfe wide applicability method apply bregman projection space svms also subgradient cut frank wolfe exist solver like cut plane technique see frank wolfe unfortunately subgradient prohibitive reduce randomize frank
unlabele audio six convert gray shift gradient extract image segment frame frame second audio sample reduce dimension result feature process window constitute audio location digit elsewhere audio mention design regression yield solely solely goal domain predictor recognition accuracy two domain present restrict boltzmann machine corpus digit appear within perform poorly row nevertheless well hard bad single domain success predictor train rbm audio audio analyze problem domain domain setting domain derive result use expression domain showed present synthesis neutral face context demonstrate despite method design though audio sentence corpus subspace orthogonality satisfy every consequently every q estimator fix respectively complete claim orthogonality address problem domain
non conjugate guarantee present fitting convergent furthermore new speed importance paper explore connection obvious approximation mc approximate directly available analytically wish understand tradeoff case bias variance first expansion spread everything recover make randomness variance mc terms taylor equal analytic suggest analytic derivation term first estimator term approximate numerator variance three analytic typically beneficial worth mse give contribution actually obvious family bias variance exactly case interest suggest family capable even possible exact sufficient vanish variance mean replace use distribution recover normalize regression really correspond exponential true predict mc benefit gain related h proposition david stanford edu approximate minimize kullback divergence intractable provide close exponential mixture distribution make precise several tractable quantity marginal monte approximation kullback approximate rely analytic conjugacy variable conditional analytically member exponential applicable equivalent sufficient unnormalized log
training training test set methodology li segmentation initial pool around segment segment extract two bag dense gray scale descriptor contour foreground descriptor descriptor pyramid locally texture vector score balance fair fourier feature gd want segment gd table set accuracy average class scales discuss able hyper train attribute minute cc
belong correct add generation treat generate block orient degree correct risk reader direct shorthand computing appear difficult except approximate assume eq however determine challenge say law cutoff essence treat impose hyperparameter open structure fit network growth power law cutoff specifically direct instance nod useful network various way exponent eq vertex network undirecte block block average degree degree power exponent bind degree poisson describe upper normalize know block force infer fully
formalism set equation numerically formalism set contribute link read read related lagrangian label node instead label satisfy two fraction link notice several link link b rescale version c dependency solution clarity scheme case close network working node
select fit set tend interior lead addition particular pattern derive slight proposition end verification lasso assumption
pg tc tc tc tc tc multiplication bc kx bc generated figure illustrate solve homotopy four axis cumulative proximal homotopy method vertical segment indicate homotopy reflect objective gap solve regularization parameter demonstrate slow sublinear rate last slow convergence pg dense several fast end contrast maintain iterate stage homotopy iteration gap clearly monotonically pg however sublinear maintain plot pg explain pg stay sublinear become automatically exploit convexity pg behind discussion homotopy strategy improves compare still slow perform homotopy homotopy method inner precision take reach precision number optimality take inner stage lack strong multiplication one evaluate gradient cost proximal pg need require three multiplication method cost eight multiplication confirm
signal guarantee exist measure ratio rule observe mass become posteriori map locally minimize strategy test underlie wrong go interested failure decision learn underlying presence channel recall channel equal symbol occurrence channel recall node observe immediate decision two exist p j error immediate private use strategy complement law total generalize symbol q j away size node exist decision measurement equivalent assumption observe immediate previous unbounded infinity probability result exist prove network let note satisfy guarantee consider
simplify make assign sequence change affect examine asymptotic delay since kl functional notation regard furthermore consequence lower proof nonnegative furthermore polynomial moment theorem polynomial notion notation sequence exist nonnegative side constant could bind algebra statement statement b nb na na nc na n n nc na n log inequality statement inequality n condition n n event similarly probability union p polynomial assertion accord omit implicitly state allow equivalence statement regard prior constant introduce occur density note use convention
draw visualize huge graph transform preprocesse mention price recent variable absence edge node
outlier final error make examine functional base gain make real synthetic datum water equivalent ten near three directly classify conditional discrimination discrimination estimate quickly fast estimate less accurate estimate absence classification direct knn offset speed like feature extraction surface full non yet variable range neural inverse possibility become resolution acknowledgement thank manuscript education research project
elaborate procedure type boltzmann deep boltzmann boltzmann machine eq unnormalized involve sum rewrite partition follow eq notice compute analytical sequence transition one draw draw mm evolve denote transition alternate ratio partition anneal result datum eq time produce estimate demonstrate center deep boltzmann machine section deep boltzmann discriminative
show small well suffice section sparsity eigenvector moderately able select significant produce use eigenvector determination difficult issue section describe regularity assume ba hold maximize impose hyperparameter observe convenient trivial increasingly estimate describe risk estimator implication c estimate c replace compare also establish preliminary eigenvector c become establish conduct eigenvector thresholde additional technical certain circumstance slightly risk property seem well describe behavior somewhat significance notice space imply ba condition satisfy important emphasize asymptotic sense whose total prescribe hyperparameter apart supremum usual inspection ba one weak condition eigenvector require look geometry subproblem final
office research twice continuously ii lipschitz continuous sequence descent theorem apply proximal newton exact newton newton focus minimize proximal newton tailor newton include newton analysis inexact method e subproblem newton newton inexact proximal rate adaptive stopping exactly demonstrate benefit rich literature monotone newton unify close continuous necessarily nonsmooth assume attain proximal mapping projection convex entry proximal nonsmooth minimize iterate first include accelerate method simple composite gradient composite neither subgradient subgradient
general ica entropy yield decrease excellent relation frequency property importance temporal introduce technique ica temporal search minimize entropy iterative analytic solution optimum lag signal detect nonparametric macro pca signal classification stationary hold ty unit variance last plug proposition axiom criterion theorem exercise theorem presentation remark
process context topic document global possible relate variational stochastic variational inference factorize searching leibl parameter conjugate conjugate exponential simple follow stochastic inference sample vector conjugate conjugacy two motivate maximize gradient conjugate exponential immediately statistic calculate form iteration optimize direction index subset step among observation select optimize expectation subset equally probable w proceed optimize update convergence similar exception sum though batch exponential ignore select matrix inverse simplify old index subsample document subsample subtree optimize distribution subtree word allocation eqs collect eqs inference stochastic entail optimize local parameter document follow variational table case select latent word wish consider lda hdp allocation consume additionally mean translate subtree entire subtree reduce activate
dynamical include present use basic interact feasible political spectra influence international involved opinion opinion formation model collective locate endow spin could total state network name existence feedback opinion meaning evolve time link opinion change modify due topological structure evolution numerical continuous formation communication agent topology division feature opinion system model close reality additional apart opinion birth death division opinion social
quantitie piece compare reduction th data reduction median approach th I mean predictor high approach correspondence sharp vary find begin end smooth also end regard band problematic lead interval true plot covariance mean smoothly even order maintain separation suggest property quantile quantile I quantile th spline compute true standardize I overall superior standardized residual choose minimize lack closeness summary confirm even highlight investigate well improvement locally consequence show select vary propose clearly induce mean proposal accommodate tail characteristic financial algorithm new generate subsequent maintain continuity ji black mean mean green black simulate accord subsection smoother drive posterior show mixing assess
set micro day second parameter memory usage seven collect computed main concept micro summarize anomalous interested discover trend change day behavior final informative curve positively skewed differ mostly curve interpret representative give curve spread median characterize value day case detect observe box opposite depict box vary trend show
vary measure root true state estimate meaningful number compute figure show bar state plot pf serial implementation la la divergence la significant well scatter multiple illustrate accuracy latter attain much pf variant la parallel follow transition execution figure respectively perform pf la similar draw computation yield table run monte approximation respective perform show
shorthand vector hope minimize provable pruning loss goal pruning assign answer score become decompose transition form hmm model pruning segment prune
use frequently rest coordinate comment step setup initially block nesterov weighted structure subspace permutation identity enable coordinate notation work block decomposition pick one criterion natural write uniqueness view write definition separability respect block simplicity sometimes block block block space denote let far iw inner weight l uniformly fx standard block group block convex strongly q study regularize monotonicity variation choose block point randomly block comment begin set set set value precisely brevity refer set property get chance update doubly give descent composite loose introduction develop map comment nx introduction besides separability lead block stay ready technical expectation involve name develop complexity motivate concept section devote property choose subset subset assume choose degree separability sampling situation analyze monotonicity guarantee directly inclusion arise encode block
value strategy assume uniquely condition compare represent gaussian deviation kl divergence distribution finally obtain applie deal showing imply oppose error distributional armed bandit imply derivative bandit randomized strategy coincide show average general convex rely insight specifically thm lead thm turn function provide non stochastic section low look easy derivative fix randomized function euclidean attempt later intuition case optima pick close optimum order therefore far get easy lead
eq notice sufficiently constant dynamically monotone generally outperform counterpart propose lipschitz constant proceed variant exact parameter go step go step simultaneously b c violate em outer outer inner terminate outer convex modulus follow q
feasible since sum column solve another program see optimal entry separable state characteristic spread simplex x say nonnegative whose column column column matrix perturb perturb belong word order robustness permutation construct satisfy focus prove robustness fact attention admit separable identify permutation deal margin isolate topic context impractical near easy measuring say relate whose equivalent differ multiplicative
index nan old without step coordinate basis th row old column outcome multiply vector remain event basis augment sum complete set column column matrix index cc denote block dependent row top half row denote independent among differ row row dependent bottom require suppose zero result remove least eigenvalue symmetric contain linearly long introduce extending require
desire convexity machine example compute gradient pool suitably formalize even generalize well sparsity corollary result shorthand state condition rsc row parameter vector setup definition modify logistic finite pool suppose satisfied sparse universal constant q purpose et objective term strongly section regression previously ease least cost need assumption brevity shorthand characterize identically state use shorthand epoch universal number dominant stochastic draw iteration match sparse corollary approximately corollary early corollaries lipschitz follow cardinality weak universal observe corollary analogous obtain involve replace rsc reader implement manner achieve length unless nesterov address propose epoch also additionally strong constant fix epoch set budget version length bad past work function technical ease presentation state fix epoch parameter recall cardinality logarithmic optimally concern prove case least square turn characterize
depend explicitly facilitate presentation notation belong due paper constant assumption ensure eigenvector depending ensure spectral close concentrate effectively dimensional relatively space simplify q interested alternatively bound pca every eq
correspond potential maximize potential obtain collective prediction individual atom independently set exploit meta predicate signature effectively expressive relational stack graphical instance instance set stack present yield work margin offer dimensional appropriate subgraph act interpretation procedure section mode expand predict biological help drug kernel test perspective consist interpretation binary g molecular genetic half start molecular molecular radius ccccc frame atom element element h atom atom type atom type link specification domain predicate atom chemical functional group functional group via signature serve purpose simplify atom replace list atom signature twice atom play chemical tuple directional first syntactic extra domain learn employ list permutation atom tuple unnecessary list term whole kernel regularization report composition surprise similar expand molecular specification trivially background atom summarize table language bad atom presence group unfortunately unable refine hypothesis atom almost coarse grain optimum ahead expensive bagging boost bias bag atom bagging x code repeating time fold root rmse absolute fold outperform cart rmse atom extensively match publish fold run second core ghz disadvantage powerful thank kernel additionally due powerful feature set radius respectively signature analogy empirical recall signature dramatically start complement create predictor stack obtain binary fed stage predict
minimize discrimination bias consistent reasoning formally appropriate normalize normalize lagrange multiplier enforce restriction result nonlinear mean marginal rewrite compact quantity two way multiply
easy optima optimum present variational essentially locally identifiable possibly noisy signal identifiability identifiability global context learn require come closely tie tv neighboring component rank operator nan modify projection span suboptimal solution project subgradient analysis algorithm program example unfortunately change difficult face synthesis v technique seek local approximation keep repeat operator subproblem line line solve although convex solve matrix noiseless project subgradient operator subproblem subgradient value subgradient subgradient long needs project unfortunately find attempt differential projection projection row norm norm row row normalise random due normalised sphere onto tight frame calculate linear method practically work needs project
separately subject european pass evidence procedure snps additionally exclude snps allele miss snp allele miss snp snp mapping procedure pathway provide gene map functional database classify number cell current relate reaction start list gene least pathway assign within gene pathway illustrate ad human map base pair question snps map gene snp half snps pass within pathway use canonical pathway gene molecular signature map many gene around know pathway snps pathway map pathway exclude large pathway map snps highly redundant pathway subset pathway pathway pathway snps overlap pathway snp range pathway pathway distribution snps snp distribution snps pathways gene pathway pathway pathway include gene highlight review pathway snps map gene pathway number snps study list gene pathway snp hand one
adaptive seem scheme reason error statistical number trial suitable use good next large change p three invariant projective match scheme around small two scheme routine figure adaptive I sphere match adaptive bad scheme htbp dependence average green sequences calculation carlo integration radius two scheme appearance peak column right column run three right column lower behaviour cluster state great direction cluster around function update hilbert schmidt former mention true state measurement
balance space repeat follow namely complementary j py px line ensemble evolve procedure outcome code computationally inner loop performance autocorrelation serial move generic extremely draw complementary q px huge associate method autocorrelation autocorrelation especially affine acceptance fraction agreement acceptance propose step chain independent target conversely perform walk regard
follow strong compare dependent find sample almost thm indicate take classic subgradient stepsize strong stepsize sgd testing suggest argue proper yield rate stepsize compare nesterov behavior theoretically accord prop nonsmooth part nonsmooth apply nonsmooth inferior due run plot deviation bar first subgradient result assume take value vs
encode iff least sized partition terminate consistent terminate sized need able equivalence class state disjoint minimum grouping obtain one initially explain way splitting probability match matching figure split split exactly one think grouping split probability put put stochastically split put match split imply support consider relate concerned therefore want relate distribution grouping way grouping motivation partition tuple non state iff iff stochastic mention grouping give group parent say tree go start parent argument notational point say every partition exist relation partition analogous sketch partition define problem find consistent finding sized
equation since generality magnitude among constant w bm integer magnitude every entry c next require except syntactic sake clarity give complete hypothesis prove lie hyperplane prove dx n dx x create vector round coordinate near multiple rounding lie moreover take must value n w recalling dx relate community view represent th voting social choice correspond design final researcher efficiently uniform attention david give learn al show accurate theoretically suffice specify within theoretic theory generalization recent threshold predicate hardness approximation considerable interest community provably fairly recently give parameter fx sufficiently precise output integer fx efficient polynomial dependence magnitude see doubly exponential implication agnostic uniform pac run quasi polynomial elaborate integer weight magnitude bound approximate representation weight integer must get away small approximated subsequently improve tool useful context hardness construction object approach
periodic infinite generator use implement acceptable variant variant elsewhere random generator necessary resort device mathematical string principle long random sense string generate express web page www computational simulation sample population interested build section resource mathematics generator binary string test concept notion number concept randomness randomness section devote present objective preliminary specification accept complete notion bit introduce understanding present instance generate digit digit could ideal
leave range anneal accept replace previous marginal ip p sufficiently converge specifie topology example topology symmetric difference edge sp optimal framework consensus alignment intractable proposal summarize numerical illustrate calculation substitution aa computing equation compute f f v artificial site gap f character equation v use paper compute macro percent percent proposition corollary department division university california berkeley usa address inference molecular process generally marginal homology extensive integrate derivation difficulty probability play role accuracy overall
depict backward allow simple execute step number htp case ii path zero coupling meet hyperparameter perfect carlo randomly perfect estimate paper forward metropolis dependent collect burn burn period seem arbitrarily draw mean histogram show htp histogram draw histogram backward coupling exception make much
geodesic multinomial support high fisher arbitrarily singular therefore nice multinomial width proportional fisher information show inspection see positive spectrum come consequence denote except whose example chapter comprise distinct spectrum comprise satisfy typically replicate central importance exponentially fisher matrix vanish bin symmetry eigenvalue resemble mode dominant omit asymmetric potentially natural look behave issue typical simplex call face bins positive count face face span concave face strictly decrease normal unobserved face zero count flat log lack strict concavity immediate constant fixing b affine let v mix decompose direct appendix closure play computational geometry concept furthermore example boundary simplex
verify q q q apply prove suppose minimax identity leave hand saddle side asymptotically tight connection chernoff dr dp dp saddle never big converse intermediate three possibility proceed impossible give follow contradict empty think letter channel space either infinite whenever implicitly assume topological space borel algebra close set generalization finite channel capacity finite uniform channel geometrically interpret respect enyi divergence order ar extend channel capacity extend plausible extend minimax argument convex subset topological compact convex continuous quasi quasi verify convexity minimax letting never converse space sample space redundancy always continuous redundancy achieve supremum continuous compact attain convexity simplex semi achieve q r may center inequality must channel capacity redundancy particular channel shannon zero
already act noise noise fall class estimation problem posteriori map development measurement well proceed statistical bring ignore develop formulation nonlinear regression variance new exploit composite special subproblem solve implement series preserve smooth numerical smooth capability approach conclusion kalman know definite state
achieve notion consistency function question incomplete framework whether constructed show policy cost randomized randomization consistent must feasible able mean third rarely affect average idea adapt feasibility start definition I kn denote mean refer determined vector policie q population equivalence period coincide history force purpose force population
representation look eq normalization positivity think g normalization write eq inequality concrete complete transformation dot putting rhs ensure negative second large program polynomial feasibility equality obviously concrete upper factor determine lp representation thousand variable provide formulation mixture probability avoid write equation hold transition transition say exchangeable clearly occur extend joint string impose like
band bandwidth overhead gs genome wide small evaluation vector residual term hamming criterion optimality optimality mathematically demand yet paradigm gs hamming especially signal rare weak provide subset optimality advantage gs gs sufficiently adequate tie signal strength tune challenging subsampling scheme gs insensitive mis attribute important screen property screening pick threshold retain fraction hamming negligible mean except subgraph exceed original conclude generality throughout gs gs computation threshold proof lemma similar second cost step goal retain signal try minimize controlling threshold tradeoff high may signal screen computational burden characterize gs sure setting theorem gs formally depend setting screen r convenient negligible many component moderate reduce regression define notation obvious regression involve letting g j contain negligible negligible dimensional finally gs gs magnitude procedure low result inference constraint optimal optimal another tune different step discussion notational end retain gs subgraph p component summation subgraph probability event claim connect subgraph support event possibility connect subset hold I gs gs I least op op w w proof need probability control claim definition em numerical screening subset comparison hard experiment tie spirit refine iterate screening refinement behave poorly find
profile economic given see second density bandwidth useful estimator privacy utility beyond merely underlie goal release differentially private draw release differentially basis analysis original datum exploratory example previous however histogram suboptimal converge assumption true smooth preferred statistic lead statistical example regression task develop outline privacy section give value theory broad space discuss recall input database row database say write whenever differ element exist database adjacent whenever element may characterize private
forecaster player choose possibly action player simultaneously player adversary incur goal player internal total delay objective represent start problem optimization dimension paper consider website set ad select bipartite choose gain click ad easily example span communication well understand paper feedback adversarial arm regret online short path problem derive suboptimal term dependency derive discuss detail still survey overview bandit
unable evolution hide feature deep architecture address stack rbms deep stack rbms side train temporal autoencoder simple dynamic allow stimulus interaction hide layer different energy autoregressive rbm train rbm sample hide connection structure fashion visible layer activation delay corrupt visible
di author term exclude word question offer elegant dominate parametric author analogously
nontrivial problem tb problem multiclass laplacian neighbor first eigenvector laplacian clustering display follow fidelity determine select point fidelity point procedure fidelity fidelity evolve develop fidelity seed region configuration form nearly decrease run display average error per technique ghz ratio relaxed involve prior balance even calculation
cox ii meet n slightly self contain despite integer case bi bc l equation ad b b ni triangular q triangular support part
compatible matrix component e individual component present seek matrix extensively decade simply ordinary large create stack detailed sub unique since arbitrary proper qr decomposition hereafter generality purpose implicitly assume common solve cause additional hence implicitly indeed restriction worth ordinary equivalently matrix stacking matrix partition distinguished involve part restriction part unknown compute equation compute truncate estimating play role pseudo inverse need qr decomposition svd respectively fix norm threshold find otherwise
vector onto eq et rbf function detail category prediction actual function detail general least classic advantage predict example attribute probability eq attribute lr omit role accordance probability concentration fix relatively give totally among five dataset h ccc record notation interval give
worth vary typical give rise question discuss concentration adjoint adjoint refer adjoint jensen integrable value cm risk class put contribution rich scalar criterion thank concentration upper bayes pac derive elegant risk majority derive achieve art multiclass formulation framework majority
density verify calculation carry complete calculate numerator denominator denominator numerator divide denominator since correspond corresponding term evy process locate w iw gamma ease update update give tw w poisson appropriately rescale sampling q mh moves irreducible go probability use recursion backward forward recursion eq
package include illustrative population believe parametric diagnosis examine data package discrete discrete package discrete variant bandwidth fix select di mail age adopt form table life table raw extensively situation age counterpart present change agree characterize opinion form closely neighbor relationship progress smoothly
exist see nuisance rely simplify composite hypothesis alternative nuisance sophisticated eliminate interference cause formal justification theory see power limitation refer reader therein issue simple problem versus test base proof v statistical generalized reduce tie via reader notice connection cut program first cut within cut know np tree past decade notably approach semi relaxation cut metric minimum suggest second relaxation
emphasis paper self barrier perturbation rate careful slightly sense elementary point hand result online bandit feedback clear playing instead incur convex thus thank eq mirror efficient strategy specific precisely former improve result latter linear optimization choose randomize compact player adversary incur forecaster
predictor lag enable superiority lag variable lag lag package regard number coefficient predictor error snr diagonal diagonal set snr snr evaluate algorithm train testing pass training sparsity fold set inverse standardize quite summarize dense lag stay inverse choose cross lag large cause lag discover l lag snr error testing training error error testing testing section argue possible explanation uniformity adopt choice lag
n correction th subsequently correct obtain eq literature estimator overview estimator extend multivariate arise nonparametric result exponent satisfy inequality place incorporate estimation estimate exponent incorporate inequality independence constrain constrain denote turn
drb drb drb drb drb set triple bind drb ic score allele name sequence item appear allele pair affinity keep handle algorithm insufficient denominator tuple contain allele pair bind normalize suggest select show compare allele auc drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb drb suggest substitution describe drb identify analyze detail classify drb drb substitution bind ray use linkage proximity order element proximity define choose weighting build leave linkage instead linkage allele treat cluster successively union merge leave separate family assign cluster cluster describe cluster two close identical st st st st st ten drb drb st drb extensive medical call oriented disease gene human ray sufficiently assign allele allele drb gene drb drb drb drb gene drb human assign allele g drb digits allele protein family first type
graphical prior posterior expect nonetheless extreme observation prior almost curve difference impact exception bs estimation mcmc computation option little least range extreme sensitivity posterior simplicity work multivariate take fit distance otherwise fit ab unit reason exclude individual daily class belong nonetheless posterior predictive one answer nine minor problem negligible people minor daily task unit properly summary value fraction observation appear problematic look find unit gender individual use site journal conservative replicate analyze chain spline thank thin dramatically effect connect spline characterize trace
see alternative regularity f satisfy asymptotic calculation show asymptotic schwarz obtain hand iff alternative small formula explain third perfect agreement finding hypothesis
image technology compatibility sentence edge compatibility define demonstrate assignment bipartite become cover sentence assignment constrain node aspect bipartite use goal sentence explain aspect rating aspect figure constraint review sentence constraint add additional unique aspect dataset separable discard discard aspect difficult constraint absolutely semi supervise unsupervised condition initialize datum choose option statistic labeling compare level agreement corpus monotonic minimize loss maximize loss diversity technique diversity introduce require bipartite objective match image framework minimize upper rating rating overall rating review rate review miss aspect rating understand user user rating na I learn aspect rating fully rate review review user mixed aspect appear review make apart appeal
minimal column vector bold capital letter denote indicate denote weight unknown space occur naturally audio video code mt state focus discount mdps learning choose receive environment change transition stochastic action discount reward start act define bellman develop value policy reward iteratively apply guess optimal bellman operator estimate transition square difference derivation state extensive computational space induce random gradient proof offline
backward backward pass conceptually explicit pass notably truncation backward approximation less require intermediate backward use inference believe respectively h sl university california berkeley automatic control novel particle backward considerably improve instead achieve forward framework challenge markovian state principle backward trace trajectory seems promise markovian flexibility must kernel path degeneracy sampler add yield considerably particle unfortunately simulation
eq iteration second denote j last il therefore next give know identity therefore distribution j g j concrete complete reformulate z include convenience decomposed view belong group recall j multivariate distribution fact formulate develop close remain note I expectation expectation z g use update decompose depend eq separately two subsection optimize optimize find q interval j j denote matrix ii fine accordingly optimization since immediately find j j optimal series shift shift shown inverse fourier transform wise conjugate equation respect reformulate derivative eq obviously decrease optimization form rbf amount find therefore
decision employ cost depend precision bipartite give round precision player cost fix denote cost log growth decentralize decision monotone precision bipartite matching nm unknown choose arbitrarily slowly arbitrarily close result affect limitation optimal play j total play I jt jt jt mm bipartite account skip similar I least follow chernoff false get put regret theorem l bind linear time monotonically illustrative establish follow computation imagine decentralize base could unfortunately player decentralize grow number use different growth decentralized arm reward arm
inequality error express dft pmf monotonically certainly work eventually process meet know eq rewritten imply monotonicity condition method move broad monotonicity integer positive kn pt type might first exist fit proof next calculate treatment unit store holding take
find computationally remain promise principal subspace analogously principal instead one manifold low upper row highlight existence condition statement complicate multiple pt subspace minor nontrivial similarly violate entire necessarily occurrence selection refer subspace search active bind interpretation sparsity regime generally second pca problem low sparsity mild become involve ball row sparse generalize combination outside sparse dense regime remain upper analyzing case state simple upper row row maximizer certain radius principal nontrivial serve prototype involve main upper specific state involve involve process mild dimensional since extent qualitatively analogous condition enough
equation kullback leibler neighborhood prior marginal hausdorff metric special dimensional distribution resp kp n divergence divergence specification p g kp g p c particularly usually violate conclusion worth depend remove result weak hausdorff wasserstein wasserstein distance subset pt tend degenerate leibl nx alternatively view kullback coupling pt kp kp follow kp kp kp n nk summation take pt c inequality due kp kp element nh nh combine display obtain kk p w assume nc appear essential exchangeable kp g gp j distribution definition kullback prove main assumption support bind say density kp pp j k q note similar
total additive column reconstruction additive noise ht reconstruction projection measurement reconstruction measurement technique compressive measurement via tv sense interest exceed threshold identification level
variant regard extension extend reduce hull minimum shall illustrate sample decompose uncertainty hull hold loss subset index define function uncertainty set present problem q solve optimization eq z need first function decision appropriate term rate hard separate step simplify analysis study figure derive conjugate dual late duality property primal notation equal infimum subscript drop clear clearly regularization may depend size introduce assumption metric set continuous respect deterministic training function condition satisfy negativity e subdifferential e inequality condition hold q satisfied strictly inequality shall sufficient existence assumption gap empty
snr level reconstruction rapidly tie l competitive signal model lag examine snr lasso roughly tie l limit large quantization small figure examine snr fidelity well snr first confidence deterministic reconstruction degradation optimum sparsity confidence high l performance confidence evident snr try evident performance measurement quantization advantage lower specify varied lasso performance mark highly
conduct maximization bfgs variational shall evaluate like david discussion way nonparametric introduce infinite random location break dependent measure hyperparameter beta stick author propose stick impose probable vector associate explicitly position capable regard
different ol ol criterion internal validation computational setting implement strategy exploit randomly iteratively vertex connect connected sparsity exploit arc iteratively exploit sparsity pattern path sort code encourage entry square one penalty penalty pair link arc encourage pattern number use implementation algorithm use approximately penalty group six left vertex link arc legend present top art often image patch natural approach introduce consist extract predefine induce regularization clean patch patch average reconstruct full set good choose orthogonal cosine transform dct organize order frequency dag neighbor go low high frequency large million difficult since exactly solution admits close soft tool proximal image step exploit arc large arc approach equivalently quality image optimize keep denoise table standard image answer observed computation factor include patch second cpu core I penalty require solve quadratic minimum allowing patch three fast process move significant instance indeed penalty superiority penalty denoise scheme base overlap evaluate quality individual opposite draw section thesis error without table penalty possibly penalty helpful denoise patch art address question choose successively gaussian bm course
expectation outcome product coincide experiment advance direction advance particular send particle coin choose measurement direction observe correlation run fair coin imagine actually outcome outcome four I avoid hoeffding inequality already imagine copy different locality cs cm angle radius cs cm prediction quantum certain selection vector lie plane angle choice difference equal positively correlated degree correspond anti correlation fourth ab ab absolute mechanics notable paris later coin aspect direction signal light consider start experiment available online ab statistical delta calculation check binomial count roughly locality chance ab ab b describe system spin truth concern picture repeat time advance leave many pair actual individual time outcome detector still particle fail could measure neither detect advance definite measurement outcome setting rapidly value occur close belong event event return whether picture particle separately really point quantum mechanic principle seem reason could equal equal experimentally recover situation first
mechanism martingale exchangeability construct sequence p small design simple mixture strategy apply stream martingale suggest use well aim martingale exchangeability present construction exchangeability previously experimental result test life exchangeability plug definition take joint infinite random exchangeable tool martingale exchangeability keep conditional
behaviour see fig right polynomial assume vanish improper prior approximate piecewise polynomial near assume piecewise coincide ordinary interpolation e near ordinary interpolation position grid interpolation four interpolation cell grid corner grid analogous high grid spline nd value position away posterior away sufficiently smooth grow distribution prior conjecture argument interpolation scheme expansion adapt terminology scheme estimator choose whose parameter et motivation rather approximation constraint make usage scheme assume uncorrelated easily adapt q obvious improper prior wang et approach constrain obtain unconstraine take limit wang et
c r ic ik subtract z al z km predictive liu computer validation lin h tu involve dynamic multi model new york discussion imputation york variance trial g carlo maximization family space spatial lee h lee h experiment high dimensional west university computer branch krige gibbs krige hypercube design et computer help response response help affect curve turn residual influence life distortion consider set residual different profile tool
evaluation bilinear text estimate error replicate times square square bilinear advantage square important difference bilinear save function correction support dms index definition introduce generalize make efficient interest sum bilinear allow reduced evaluation alternative bilinear paper extend correction sum subset sensitivity introduction index systematic quantity property systematic new reduce early work strategy mean due appear review avoid index value interpretability bilinear include index
training run dictionary long increase significantly far conduct experiment activity video densely smooth max pooling video representation approach comprehensive indicate higher report cite ssc code code subsequent combine max ratio variance discrimination realize histogram code fisher code improve code sift descriptor patch code video avoid processing extraction previous recognition video densely extract sift c fuse sc ssc lasso
decade work introduce setup property simply ms aggregation space dictionary value goal estimate observation literature available ease q dictionary certain boundedness function except strong design rule suboptimal paper propose star remarkably parameter rule design sample idea subsequently similar study enjoy star solution greedy enjoy optimal deviation extend various direction aggregate aggregate projection first put weight prior weight appear derive convex optimization approximately newly propose formulation produce deviation model
path denote vertex graph slot choose reveal slot point regret selection extra incur compare select minimum end end policy path cost policy present path tail heavy cost tail cost tail moment heavy tail tailed denote chernoff
chain estimator depend estimator think lebesgue variance eq term rewrite df hx df hx indicate ideal take select upper term valid efficiency formulate term large infinity embed index setup formalize random non negative integrable markov probability section rigorously formulate select eq mcmc sampler find gibbs rare criterion formulate one estimator bias could infinite estimator introduce walk exceed high identically distribute distribution probability convenient
value minimize substitute multiply minimization respect minimization minimize eq increase function negative tn tn
award research innovation shift distribution exploit relationship general approach mathematically elegant straightforward novel demonstrate real illustrate classification mixture shift asymmetric laplace perform mark model well finite mixture naturally cluster arise finite write density proportion density density usually type often gaussian density mixture popularity mathematical flexibility capture density herein
observation exist either linearization particle moment matching incorporation introduce additional tracking lead track refinement ep distribution improve decision expectation widely deterministic inference ep factor specify pass algorithm approximate q minimize ep provably message invariant invertible compete approximate tb dynamical three type message respectively compute I factor ti derivative update alg node factor three message denote
c inner furthermore rational alternatively run emphasize answer purely need encode nonnegative system immediate implication nearly exponential show exponential somewhat surprising nonnegative reasoning polynomial inequality plausible property matrix submatrix play instance nonnegative rank nonnegative infeasible geometry subset inequality hope infeasible infeasible subset yet case nonnegative al crucial nonnegative row map point map intermediate intermediate simplex triangle case instance construct explicit restrict point modification property simplex write triangle triangle triangle
product b column product unbiased matrix replace ab
reach matrix could ever reach transition long calculate equation eq know look operation zero diagonal diagonal calculate rest process hazard basic hazard hazard function first quantity next certain viewpoint provide visit consider reliability probable cumulative version probable exclusive event add cumulative presenting time intuitive interpretation minus cdf state cdf must reach reach time quantity useful
track lc dl propose recognition suppose face x cn cc treat overall query identify code minimization scalar achieve recognition even act open code sample dictionary discrimination adaptive overcomplete pool linearly signal conventional dictionary learning learn
entry kk kk k inverse symmetry either alone storage initialize yield organize practice enough path follow solve ode successive ode notation f p jacobian ode differential ode calculation ode jacobian hessian df store light whole matrix incur cost direction usually constrain far application especially briefly proof proposition change equality program hessian strict convexity alternatively constraint represent p quadratic path segment satisfy ode non non cost derivative small expensive compute path qr either one leave specific function convex
high posterior meaning dirac mass final trace mode condition start proposal acceptance probability approximately effect empirical proposal acceptance adaptive accordingly small state explore big jump reject often small figure choice possibility express rather advantage possibility small static chain show rise ergodic accept initial distribution skewed proposal choice roughly comparable multimodal posterior advantageous mix area proposal purpose nontrivial application function potentially use volume explore severe meaning wave pde mean explain restrictive prior recover log pressure standard structure dimensional random covariance draw gaussian gibbs reader correspond posterior comparison algorithm forward mod mesh bandwidth solver operation page draw keep allow ms draw ei value trace prior acceptance around clear generate algorithm converge demonstrate importance assumption solution subsection image idea observational conjugate parameter setup correspond closed noisy datum
dense focus type mrf integrate extensively direct graphical model normalization apply low robustness related spike mrfs modelling organize hierarchical bayesian mrf hyper parameter experiment set mrf widely use mrfs log parametrization potential associate potential clique graphical equivalently remove learning allow consider subject infer connectivity two commonly correspond penalty optimization base sparse special approximate like one sparse mrfs spike point mass spike dirac uninformative component mild shrinkage
randomly kronecker delta use top storage algorithm bottom retrieval explain introduction explain try overcome world rare mathematically translate capability recognize short pattern person white
invertible hence cast observation form namely matrix induce question multinomial factorization independence translate nonetheless might aspect context possibility research piece come novel approach combine
note taylor add subtract inverse f multiply side expansion write simply roughly expansion first hope decrease sufficiently fast formalize assumption convert order control expansion term decrease quickly recall event goal move lemma prove constant respectively immediate occur recall union imply constant let assumption matrix define norm begin jensen inequality assumption cauchy inequality thereby obtain immediately indeed event cauchy schwarz lemma obtain complete inequality event occur know eq shorthand bit q triangle q recalling remainder high leave application couple series jensen schwarz proof guarantee c lemma inequality note without take decomposition begin inequality theorem definition average vector minimizer minimizer empirical eq parallel condition conjunction lemma couple indeed apply lemma statement remainder establish lemma provide auxiliary exist throughout notational shorthand big also shorthand optimization error recall lemma constant prove moment expansion lead
conditional distribution determine add result figure empirical predictor approximation w optimality poor actual bad affect concern performance discard layer comparison bind dataset layer deep optimistic thus final intuition upper architecture layer rbms upper generative use rbm result confirm likelihood rbm dataset alignment thus compare final vs validation likelihood training vs discuss early regularization could optimistic validation training upper training upper mention figure log difficult training almost optimal picture likelihood attribute increase upper still hyper training validation predictor notably lack optimistic maximize inference parameter display really ok replace well color really ok idea future validation still auto validation network compare upper reasonable well compare high might validation upper perform selection validation vs log experimental set small summing however exact layer situation mode distribution mode obtain validation since display sample dataset display true nice compare figure show bind optimistic successful layer
omp add backward statistically work difference backward object add significant extra gain ensure correctness specify loss target building modify kind singleton element entire row remove backward step whose removal kind compare forward correctness immediate removal understand reward step inclusion appropriate
resolution refine approximation static give accurate annotated boolean formulae membership equivalence respectively main loop choose approximation use query query teacher give query abstract presence answer teacher answer another practice nevertheless loop infer empirical evidence example dramatically technique predicate demand abstraction parsimonious invariant algorithm terminate loop invariant predicate example atomic predicate drawback loop invariant predicate essential loop atomic predicate otherwise formulae teacher never answer address predicate framework essential inconsistent free formula expressive predicate interpolation atomic predicate refine abstraction initial atomic predicate loop atomic predicate refine abstraction answer abstraction refine abstraction answer throughout sequence happen
mobile phone macro mobile phone activity section homogeneous performing extract though previous extensive city make attractive forest approach describe test lead forest useful ability time make time number efficiently forest moreover introduce frequently feature exploit later large share location denote blue bar forest vote count vote forest obtain input series implement vote vote denote vote count vote vote prediction input series calculation matlab implementation forest feature location hour mobile phone hour average feature location activity day phone activity test validation forest fraction
corpus corpus partition give fold call test testing performance mention validate fully nest fold partition fold block repeat produce fold validate train give validate fold outer validate transform fold rank gain perform accord lda suggest estimate matrix fact diagonal lda task run major report similar
combine natural fix gradient number convert unique natural suitable parameter write sensible metric treat likelihood x imagine suggest uninformative input gaussian distribution well generate metric
signal compress measurement matrix gaussian sparsity vary datum every pareto iteration reflect new extensive simulation approach compress
approach formulate finally important limitation automatically tune project foundation grant st decomposable efficiency guide optimum building sampling candidate incorporate run solve similar reliability much area automate technique previous basic idea problem strength statistic statistic bias instance strongly solve study metric practically decomposable class key technique
expression interpret small exploratory view structure possible vast automate human closely transformation automate interpretability transformation combine way view weight visual interpretability scheme investigation automate visual semantic exploratory present visualize axis city new york usa york usa pose practical challenge visual analysis increase dimension irrelevant redundant costly collect store exploratory prevent user explore visually extraction seek suitable help challenge create feature amongst extraction visual various exploratory pursuit multidimensional scaling evolutionary constructive reduction design intuitive measure study match human consider
audio separation law device utilize device device load device collect correspond periodic device background superposition component show segment simulation periodic two exponentially decay circuit circuit state prototype subject offset device speech overall raw audio combination noise depict ideal
ci result propose ci correct accelerate skewness particularly converge ci ci hypothesis reject significance ci dissimilarity reject similarity spaced range perturbation time hypothesis reject percentage time reject average repetition use base bound show fine perturbation need estimate bound reject similarity ability similarity distributional fr near neighbor knn mmd measure curve query observation number query knn mmd rbf choose video unique video decrease decrease drop add noise frame lastly rgb
relative profile carlo sensitive fix comparison different unity fix plausibility acceptable table interval classical approximation third order accurate plausibility marginal plausibility shorter confidence mean mean shape source lambda pl sim lambda cccc method first marginal plausibility interval plausibility describe allow construction exact frequentist parametric simulate variation carefully lee method compare many give answer correlation example example outperform model distribute portion come sample nuisance hierarchical hierarchical independent write likelihood free suffice carlo step readily plausibility parametric tend difficulty plausibility near sided bootstrap true get close increase particular shall coverage probability
show work take instead issue project project constraint conduct fix jointly happen similar dimension degenerate case positive iff condition convexity region term suppose trivial prove minimizer henceforth take place domain slack variable thank schwarz basic cauchy schwarz hand complete optimization nonsmooth nuclear norm spirit literature parameterized differentiable envelope distance norm differentiable multivariate center set datum nonlinearity make contribution laplacian term highlight phenomenon book history major predict cross sale volume aggregated test book aim predict sale book predict sale co product graph one market binary item graph co graph product co
eigenvector energy choose sample privacy privacy pca capture whereas gap become quite ht un pca subspace scale utility perform however comparable projection indicate point show optimal differentially private investigation future differentially additional assumption verify truly effect ideal privacy total variation distance differential yet analytical sampler empirically account bring specifically result related capture approximate pca low possible packing utility bound perform space classical dimensionality component rank vast amount collect user behavior survey subject contain sensitive typically therefore discover take study guarantee differential privacy gain significant community privacy privacy risk output database differential sensitivity data proportional sensitive change change smoothed set sub query
order accuracy bit bit hashing reduction online platform online platform accuracy batch package face training exceed common repeatedly model coefficient bottleneck load many disk os hash high another loading feature time follow notational online sgd training svm q replace stochastic popular time point update sgd code initially careful every observe epoch dataset sensitive increase h th epoch hashing present accuracy regularization achieve original perhaps epoch reach accuracy hashing hashing reflect load data loading dominate sgd converted format oppose batch report loading time helpful communication
db db db db ccccc bm db db db db db db db db db db db db db db db db db db db hill db house db db db db db db db db db ccccc db db db db db db db c db db db db hill db db house db db db db db db db db db db db htbp ccccc bm mlp db db db db db c db db db couple db db db db hill db db db house db db db db db db db db db db db db db db db compare bm outperform ten image bm image achieve result bm house method outperform significantly well bm well outperform table suggest prior method perform method bm high db htbp compare mlp bm bm average image improvement whereas improvement middle compare achieve mlp bm dataset berkeley db bm compare achieve mlp bm bm
indicate discover underlie sparse thresholding entry might expect constraint stop early constraint compatibility provide justification series accuracy dual value layer enumeration link bottom center variable cycle greedy start first fundamental cycle cycle stop degree cycle dual factor reduce contribute estimation level contribution primal coupling result quality cycle satisfactory beyond percent size temperature value cm ef error bipartite layer fundamental number dual plot value plot primal cycle basis propagation independent cycle propagation bp bethe bp bethe propagation reflect principle absence improvement temperature direct bethe probably possibly insufficient away future width ab curves modal bp ising turn original motivation calibrate model inference complete historical plot ise mrf associate obtain map cumulative inference perform index ise study variety real alternative mrf mrf happen high wish ensure algorithm propagation real inference concern mainly follow purely approach rely constraint analytical coupling picture correlate modal possibility relevant bethe possibly correction modal kind pseudo moment explain modal like relevant mentioned surprisingly perform determine sparse estimate inverse direct amenable
separate precisely estimate modify gauss explicitly definite ignore see form subproblem constant proxy natural obtain parameter datum conduct place reflect surface naturally cast square framework transform record discretized receiver location denote column approach discretize locate hz typically ratio situation bfgs method modify
conversely syntactic emphasize language dl distinguish framework table language dl dl cl clause clause dl dl safe tight dl safe head dl dl dl role dl dl dl reasoning calculus boolean yes implementation yes semantic dl part cl separate connect minimal interface kind coupling illustrate derive previous external evaluation implement part couple achieve failure picture dl hybrid dl cl read comprehensive effect precisely dl answer answer occur rule reduce answer ip argument recursive answer combination logic role must non dl propose reasoning dl complete answer dl occur
series third parametric assumption topic lda overview tensor decomposition lda topic mixture hide approach impose constraint degeneracy paper furthermore parametric topic sub idea paper topic employ matrix recall establish topic anchor uniquely anchor degree requirement presence word matrix variable mix idea dictionary learn ask pair et study full square noiseless knowing enjoy sparsity state clearly recover explain I idea leverage counterpart assume intuitive arrive parametric consider case framework sparse work refer structural collection associate progress identifiability identifiable moreover gaussian combination identifiable et al gaussian normally work deal latent contrast identifiability linear learn object intensive investigation past community vast biology economic learn extensively year tree tractable likelihood approach test score direct undirected relaxation variable challenge latent undirected latent model
exclude characteristic cluster exact achievable spectral bring consider ht cccc cluster vary child vary vary vary vary none total analysis look e child interact child interaction thus point consideration work data effectiveness teacher run category simplicity ground use answer cluster measure spectral second although visually inaccurate secondly figure take potentially less well display odd run entirely unstable set spectral offer important business address separate cluster similar become datum mining range bioinformatic dependent medical imaging aid vision utilize surveillance represent population attribute leave company
perhaps reconstruct note follow claim still see express ignore incorrectly express uncertainty irrelevant play express divide characteristic variance resp write contrary situation real background equal eq equal sc x p se mp se mp se mp n p sn put part yield analogue odd roughly evidence population highly expert working see interpret rewrite odd eq sp sp sp I retrieve likelihood depend prior quantity note however quantity prior quantity take obtain likelihood course thing prefer care likelihood useful rather complicated weight evidence evidence turn prefer without frequency effect weight population contain evidence role frequency weight suppose belong long
idea collect verification return candidate extension fraction begin candidate solve trial verification oracle order exist good polynomial shall address condition extension rank easy height rational k counting determine height polynomially linear counting problem exist fully randomized approximation problem count approximation number two get algorithm incorporate extra eq approximate compute k ji k good whole complete verification return depend position fraction remains preference sort information give order preference compute matching algorithm make verification u stable return notice verification totally iii time stable matching verification oracle algorithm current discussion preference list analysis cost trial considerable room improvement unnecessary way almost even unbounded computational power create initially create verification direct path verification note gets make query uniformly match strictly positive probability instance consistent far verification oracle sufficient go could prefer previous could happen preference preference preference randomly preferred averaging prefer put together probability strictly need large comment stable preference bind refer trial formula clause unknown u formula solution verification return false first computation satisfy assignment exist oracle target also search standard clause n terminate oracle assignment ask verification confirm terminate program return idea elimination extensively pac formula proceed propose clause return
first use optimal parameterization within hamiltonian learn uncertainty clear experimental hamiltonian illustrate advantage tractable involve presence result show error experiment irrespective rao frequency characterize unknown perhaps importantly unknown end show region accord perform term experimental conversely fail cost time vary advanced optimization search advance resample technique substantial reduce experiment extension control system united acknowledge question apply infer quantum system parameter trade resource avoid storage post processing learn change process rao quantum processor first require accurate characterization dynamic device code tool implement quantum carry quantum computer unable simulate call analog quantum validity depend simulation dynamical simulator present output expect characterization generation quantum discuss really know nothing highly unlikely typically insight form hamiltonian process additional knowledge process
successfully recover wavelet full wavelet note problem nonconvex q eq give easily modify objective basically generalization solve stationary
coarse energy energy procedure construct pyramid principled algebraic look matrix coarse act coarse label yield end equation one key pyramid straightforward coarse energy multiscale heavily aggregate fine assignment
put emphasis plot different effective iteration divide size performance comparison trend worst prove strategy still pass iterate worst average second typically among
ff precision error statistical adjust regularization proof adopt thm fp eq triangle cauchy schwarz q imply thus union n argument rate remain iteration front change error constant w eq covariance conjecture present thorough convergence graphical kronecker implement penalty enforce lin covariances kronecker generalize flip factor iterates converge local maximum ff establish fast ff glasso analysis root target al dimensional slight glasso diagonal matrix penalty glasso et report analysis significantly rate infinity kronecker show outperform ff glasso rate ff glasso achieve hold
bring much enable hierarchical approach even configuration significantly reach hierarchical database database dataset show satisfactory world number analyst work exploit cm cm cm universit le du france
exploration random discuss detail regret unlike benefit look close see exploration budget exploration lead exploration similarly start explore imagine dark room room little little whole exactly speak hence point far previous I choose select search exploitation close optimizer baseline like unknown costly classic apply frequently
gaussian signal purely model evolve accord random distribution g walk generative calculate datum identify fit uncertainty infer goodness fit extend measure time standard true event event event estimate uncertainty event partition set signal logarithm base reflect gaussian strictly quantity e gamma shape appendix em gamma rarely sufficient specify procedure canonical prior mean determined inspection curve shape scale fig
jk calculate held word word compare actual rate mixture nb count appropriate hdp base marked performance follow nb crf beta outperform nb percentage somewhat intuitive showing share dispersion document nb hold various lda nb crf hdp mark nb active topic word model document learn algorithm framework distinct mechanism document weakly couple indicate topic non topic sharp either close zero control n jk count usually indicate rarely observe use smooth gradually control jk modeled allow confirm
cost fitness evolution capability learn fitness code sequence give agent code fitness landscape bit distance ham landscape fitness fitness landscape fitness bit zero environment evaluation fitness exercise evaluation fitness function grow fitness landscape capability grow subset evolutionary slow evolutionary hill growth growth
merely hence trace simulate produce trace see figure part trace property make important making parameter zero iteration reduce significance unseen zero divide parameter alternative fraction strategy unconstraine progress converge use course motivate second checking platform human genome project vast biological pathway biological chemical abstract demonstrate model check handle space biological model efficacy dynamical comprise five reaction decay mass
skeleton direct orientation equivalent skeleton keep edge complete introduce essential one complete let least isolate undirected subgraph isolate observational direction undirecte perform intervention measure complexity chain graph study equivalence complete play role construct complete transition modify completed carry graphical six edge edge direct direct edge complete modify edge operator represent undirecte complete except modify section supplementary pe na introduce several modify operator chain move happen complete stay order complete modify characterization introduce alternative exploiting complete might complete also valid valid valid direct acyclic dag accord belong unique operator complete valid define operator modify occur complete operator guarantee operator condition result complete condition imply valid complete skeleton use complete dag every undirecte terminate undirected alternative operator modify material include generate modify creates complete consistent describe supplementary material constructing complete operator complete construct subset complete valid operator operator complete apply markov markov start arbitrary complete repeat complete result complete stay transition
converge exact bound perturbation argument require plug polynomial method moment introduce speak estimator find mixture observe moment parameter system challenge early undesirable moment reliable year restriction distance polynomial complexity estimator base require resource exponential mixing dimension degenerate multi view contrast base separation condition degenerate degeneracy condition discuss degeneracy condition weak separation parameter close degenerate sample polynomially natural quantity measure closeness degeneracy condition
tensor q explicit proportional decrease carry choose form parameter full rotation matrix obtain step flow equation hold define final point vanish pair vanish result matrix degenerate possibly problematic tensor operator expansion
respectively illustration nature carry exclude mcp scad eliminate nonzero covariate model mcp make smooth scad penalty make transition g scad change mcp representative path mcp group group straightforward fit scad mcp need replace multivariate thresholding update update method list f f recall note expression possess every respect see iteration scad penalty furthermore lasso minimize convex minimum mcp scad convex nonconvex similar interestingly show even update sequence converge orthonormal update group initial produce multiple converge considerable lasso mcp group scad algorithm logistic recall simple possible iteratively reweighte typically equation diagonal mcp lack simple context depend hessian let positive logistic bounding consist
norm penalty element clear generic minimize sparse dictionary coordinate minimization follow many represent code apply every overlap patch neighboring patch pixel shift thus redundant introduce sparse formulation convolutional convolutional coding provide function predictor function convnet element predictor consider predictor final unsupervised energy input follow dictionary filter step fix inference carry sparse code fista propose iterative shrinkage thresholding improve size momentum domain adopt
review part hold get validation computationally demand though lot exploit parallelization practice usually employ speed heuristic may local minima test see quality local heuristic cross datum reflect large finish effective heuristic require error subset minimum converge process subset reveal way take candidate automate propose size substantial testing control roughly speak number dropping criterion speed result method less significant loss certain optimality yet availability vast guide region take learner section discuss section fast testing state synthetic real world set conclude skip self hoeffding test evaluate remain confidence lie outside perform drop similar approach pair false devise pac emphasis application domain kind suit costly train evaluation direct would one learn half half maximal model procedure utilize remain benefit necessary evaluation whether belong top extend upper procedure nearly infinite domain evolutionary configuration concept boost bernstein extend algorithm concept evaluate increase probable wide domain reinforcement relevance topic hyper model hierarchical previously observe performance candidate configuration historical apply deep belief potential second lead learn toolbox auto search like concept combine armed bandit seem way armed bandit identify large choose
pose necessarily hypothesis might machine insufficient limit number regard treat technique sensitive reliable allow establish sample single voxel frequency voxel radius frequency increase irrespective voxel optimistic manner data regularity voxel voxel member voxel voxel simultaneously contain voxel voxel member voxel voxel equation voxel member center member non voxel contain prove diameter voxel less equal central voxel spherical voxel great exist second voxel voxel contrary assume contain necessarily vice versa lemma voxel imply reasoning every every voxel voxel case contradict requirement numerically
penalty n x generalize freedom correction adjust covariate iy wang adopt idea evaluate perturbation derivative wang liu perturbation quantile function time computationally surface show path fix explore show construct respect piecewise linear bi optima computation time figure display convergence propose multiclass method measure convergence
would still appropriate importance sparse carry implicit instance weakly correlate spatially order variable apart time little assumption example variable association possibly reduce notion large clearly degenerate matrix small perhaps property reduce largely bound estimation jump finite eigenvector eigenvalue establish norm hold appropriately small continue pt pn r similar introduce rank detail arrange arrange henceforth number plot calculate section justify exist detect jump jump gap level definition estimation aic criterion special structure jump spectra jump decay treat offer jump counterpart less study exist eigenvalue via back asymptotic analysis grow mostly
x h ranking regret analyze classifier rank reduced subset show equal classification calibrate yy every proper therefore calibrate let r logistic therefore calibrate term associate balanced loss q balanced loss distribution analysis logistic loss f f combining suggest balanced regret optimize justification situation usual exponential minimize balanced loss rank
employ admm estimation problem derive optimal base effectiveness future formal sparse oracle convex application sparse namely arbitrary diagonal consist u imply matrix pair additive easily verify clutter denote context trace norm lemma derive matrix summarize lemma complete bind location set entry denote entry entry
increase focus acceptance rate obtain unnormalized rate proposal improve exactly cdf inside technique term e methodology target unbounded case transformation px ease exposition generality illustrate rv know unnormalized class previous limit necessary sufficient ensuring limit equal us bound unnormalized transform htb unbounde suitable increase unnormalized pdf vertical restriction combine two subsection obtain pdf unbounded unbounded pdf higher vertical denote section complementary analyze suitable unnormalized rv belong monotonic inside mean invertible interest domain transform rv unnormalized pdf assume without loss pdfs case pdfs support ensure monotonic decrease may difficult define sake monotonic pdfs pdfs unbounded pdfs case pdfs case imply assume normalization unnormalize unbounded support unnormalize obtain monotonic inside unnormalized sufficient condition sufficient equal complementary case unbounded monotonically decrease vertical unnormalized bound support rv monotonic unnormalized inverse rv unnormalized term pdf unbounded second term transform finite vertical decrease imply therefore high order unbounded subsection demonstrate rv inverse target pdf set summarize subsection transformation target pdf either c vertical transformation pdf pdf none monotonic monotonic fast monotonically decrease
signal separation literature like propose elegant approximate propose hessian directional characteristic hessian provide complete noisy ica sign case since extract square definite fourth point relate specifically refer tensor fourth tensor simultaneous view single scalar similar expansion fourth expand x algebraic share real value cross know r c ix
combination perform publish global ensemble analysis combine explicitly combine know correlation consistently make complex object one simplify analysis tool complex effort go task complement analysis satisfactory team later aim flexible well development scientific new physics significance observation often limit observation common determination estimate credible represent compatible hypothesis hypothesis fit quantify describe statistical procedure
order instability desirable replace version like identify like perturbation serve always satisfy zero although area cs terminology derive pt pt version non cauchy schwarz relation equality attain opposite continuous identically continuous sensible illustrate essence sort plot triangle axis profile see geometric way panel plot level sx align axis effective cross select number iteration level tuning solution erm differ soft sparsity angle cite method theoretical
category ai db ec ir net graph cut directional community explore direct define weakly reach regardless meanwhile direction call e satisfie se v te te propose type edge alternate backward call connectivity sequence e htb direct adjacency connectivity concept analyze centrality web page world connectivity regardless role develop hold alternate connectivity two terminal next source terminal set maximal call source part directional directional directional edge terminal direct belong directional tb asymmetric decomposition directional directional box membership terminal directional source terminal directional show figure way partition adjacency exhibit directional source terminal node role terminal require except terminal surprising work find directional achieve simple directional terminal appendix algorithm direct directional prior search one drawback search directional real network directional one expect absolutely community order community external edge cut directional community directional community adjacency matrix notice cut emphasize cut cut community
p w ard p jk iw linear covariance ard obtain go go general choose whether nearly gp model prior modify hyperparameter variable residual plus minus residual base distribution hyperparameter unfortunately latent overview classic metropolis easy implement dimensional distribution proposal efficient appropriate neither big generally unknown generally
aggregation mean differ differ advantage induce kernel poorly example kernel hand perturbation improvement compare kernel offer free improvement wider observe induce advantageous differ dimension kernel fine characteristic benchmark independent ica angle gaussian finally pass variable bandwidth able case indistinguishable sensitive quadratic base fail addition increase frequency make small right kernel bandwidth type independence occur high set heuristic practice test informally exponent play gaussian dependency small median marginal capture point dependency establish notion embedding definite interpret mmd respect type readily use product class investigate show family choice dimensional particularly learn statistical side testing
amount line orientation document character probabilistic capture part identify however removal level character character vs though pattern consist character represent e therefore least mechanism character discriminate reconstruction possible solely single page factor corruption character character character instance character character type discrimination character become especially strongly learn character character character performance contain instance character character consist alphabet representation feature absence individual pattern variational learn
fill height width corner em thick fill draw black thick fill black nash game algorithm measurement perturbation perturbation payoff perturbation perturbation modify vanish perturbation sure equilibrium extend nash non perturbation function perturbation nash previous general payoff write markovian section case I estimate stochastic perturbation helpful node try follow action propose perturbation game node perturbation method discrete state converge locally nash equilibrium vanish payoff concave find even provide discrete ode remainder provide stochastic algorithm ode example provide paper proof summarize notation h symbol space payoff decision
explore gaussian compute miss computationally expensive span tree observe fill value em imputation apply problem database contain miss value survey database simple deal discard however technique suit valuable value imputation note extensive imputation scope propose lin focus show value imputation prefer preserve machine piece take account instance indicate consideration mind address presence describe assess discuss gaussian
eq decomposition problem feasible least assumption pd method nice pd become numerous trivial pd reformulate eq ready pd penalty arbitrary apply find subproblem perform k go step set x k observe thus reasonable terminate method practical termination relative terminate iteration method enhance one execute outer find th outer start kx k repeat terminate outer iteration next b notational index iterate block choose solve x local minimizer vector observe affine together minimizer generate accumulation saddle exist affine statement eq accumulation exist subsequence q use continuity imply together saddle
sn example definition see fa di identity lx kx obtain eq generator see theorem align apply sign apply statement lemma appear tv obtain state proof observe sum rewrite side term sum belong partition apply complete department mathematics department mathematics university article volume title definition david obtain appear generator derivative copulas copula due copula inner
concern half however purpose relevant express function certain associate explicit relate property asymptotic decay rapidly explicit expression product converge slowly accelerate give show compute give computation comment generalise characteristic limit sequence concept merely applie character specify
reach challenging figure input match node learn trial slow base mutation slow macro initially grow reach around see average static receive achievable form combination action conduct select extended action exploration action modify purely running ga mutation report great fuzzy ga fuzzy element accommodate modification range way action update ga execute usual good compare criterion output illustrate trial fast none drawback exploration conduct high predict online furthermore weight include last discard parent evaluation rapidly converge traditionally encode environmental associate beyond number fuzzy artificial behaviour kind artificial biological paper dynamical within term genetic
huge try box quite robust huge scaling mh possibility design symmetry pdfs improve note study proposal find analytic usually within appear target good note prefer numerical computationally calculation weight indeed impossible proposal clearly choice help portion mh section pdfs proposal tackle distribution opinion promise use chain select function consideration adaptive independent pdfs fit already important proposal freedom specifically confirm work literature candidate generate sequentially whereas condition
fail case permutation field h h site order permutation recommend type field series ensure simulate c site make one explain simulate marginal h h field h advantageous herein simulate field site nearby mean space construct nearby site priori within known assume assign maintain desire include site connect site site portion sx x rhs recursively sx sx probability marginal use smoothed state undesirable replace use portion picture special case zero term h space I different covariance site connect notion neighbor markov accumulate neighbor state one assertion one match collection probability covariance correlate ease future sx sx reduce sx sx auxiliary sx x form
infinite obtain rate analyze infinite laplacian small given define tend infinity integral embed subtle manifold tangent one hence inverse intersection projection onto union tangent however still restrict piece tangent convert tangent follow let tangent jacobian point resp sufficiently analyze simple smooth give near neighbor boundary normal sufficiently boundary interior hence laplacian near interior point oppose difference plane
approach outline article marginal acyclic nonparametric little explore vertex edge edge exclude encode sum clique fully nonparametric specified possibility remain assumption discuss possibility conclude two exist joint jacobian function make identifiable demand variance jx jx independence encode respect obeys x center form thus let matrix rewrite covariance hence diagonal zero identical example density three different family transform concavity density much rich normal lead dimensional estimate enable distribution
change eq take argument size cause inactive enter small size become support sign update update step homotopy every come compute solve compute know construction cost application element homotopy step instead explicitly product add recursively addition rank often suffer issue become close number close stable cholesky factorization qr support cholesky involve homotopy support section present homotopy iterative change wish new weight incorporate change replace propose homotopy homotopy change phase old one homotopy take also homotopy computationally initialize optimality column small diagonal respectively move
laplacian often localize precisely identify graph spectral notion study trade focus extract static ability localize transform lie euclidean space aforementioned processing vertex classical major obstacle processing technique dependency fundamental significantly challenging translate analog change translate option vertex useful generalized translation fact graph shift translation graph signal multiply translation analogous trivial define intuitively create capture structural addition previously application order process localize operation process weight localize transform leverage year process research develop implementation localize transform extract high graph address algebraic concept harmonic literature spectral graph e therein focus opposed signal finally localize technique manifold pass operation enhance decomposition level continue signal processing overview next way encode graph classical frequency survey operator filter translation
relaxation verification acknowledgment research nf nf nsf view conclusion document represent either laboratory reproduce herein lemma remark property conjecture berkeley edu imaging md compressive accurately original class lagrangian minimization relaxation expression enforce exploit interpretability example group block different entry coefficient reconstruct since
mle mle mle logistic set vector goal noise release simplicity q estimator obtain solve adversary could logistic reduce use noisy overcome mle section require condition function hessian distortion low estimator linear represent nx account discrepancy predict outcome give known likelihood represent total design transform odd see concatenation require lipschitz lipschitz log bound xx ba matrix hence equation adversary mle construct write attack operate construct attack attack similar logistic operate logistic attribute private attack achieve attribute exist every logistic every private run
swap operator phenomenon give belief pair advantage score address causal improve identify orientation edge identify skeleton factor cause network causal cause belief inherently one believe path separately marginal infer wish joint give belief belief axiom compute belief uninformative incoherent coherent induce joint efficiently compute take advantage learn causal relation facilitate learn skeleton propose operator quality several regard causal variable absence belief prior network argue
symbol interference frequency channel probabilistic pdf pdf coefficients p account prior pmf bit conditionally conditionally independent capture derive iterative receiver inference fig receiver beliefs pilot symbol channel probabilistic variable next assume run bp obtain message symbol message pass nx nx
subsection consider anomaly anomaly analyze ad introduce product order anomaly type exist machine u anomaly type refer type denote expect tu u suboptimal kt show anomaly exponentially decrease exponential
km occur terminate km south gray cluster way measure add diagnostic tool highly finding consider extend include delay model high area also behavior identify classify help risk occurrence great acknowledgement thank comment suggestion connection useful suggestion help mm mm mm mm http project mm z via diffusion ann university verification weather forecast mm
trick apply allow work space give da evaluate toy inter svm iterative illustrate difficulty minimization source action
maximum p residual take eq hypothesis exist recovery omp omp namely omp omp stagewise omp ols literature often analyse ol remain rare reason ol consume omp therefore ol compressive ols atom dictionary atom close ol omp contrary correlate inverse significantly differ ol argument motivate omp although imply atom weak exhibit behavior omp ols general analysis omp specifically
center contain uv arise batch technical heterogeneity lead spurious association differentially expression could cause partially cluster new uv strong effect end predict sample begin end well patient accept combine study heterogeneous merging sample remove uv difficult consider uv batch essentially factor gene lead variance bayes use replicate estimate batch replicate center remove replicate batch factor factor also influence uv interact poor factor difficult estimate uv factor effect equally lead conclusion variation unobserve covariance gene matrix gene address interest intuitively good long uv factor variant recent propose uv microarray affect variation analysis state method recently effect variation interest factor jointly induce penalty add relax constraint jointly variation effect ice iterate factor maximization yield ice finally factor observe difficult indeed want study cancer batch cluster one use large set sample identify subtle uv know advance interest person start become uv use uv interest axis principal variance snp uv correlate observe variation project along batch factor
incorporate eeg exponential valuable information nmf addition capable devise group nonnegative update use competition apply frequency bin find kind
would acknowledge popular restrict boltzmann however scheme activation activation interpretation sort globally coherent inference translate employ success entirely fulfil
scale parameter also value interval perform ridge adjustment abc improve assume credible large coverage interval even credible central credible compare precise abc estimator true location abc care suggest suggest determination hoc practically intuitively sensible determine confident produce consistent cc cc cc width width cc width adjust coverage probability frequentist credible possess frequentist coverage procedure parameter correction asymptotically unbiased central correct
rkhs induce function rkh dense universal e universal every every induce linearity every analogous nonlinear treat feature map detail standard use define long entirely effort practical notion distribution work function derive function certain recently rbf despite success classifier therefore class subsection svms amount expect kernel I alternatively k empirical sample respectively may
original margin margin original hold margin accurately optimization use dual datum ratio random worth random technique extensively compress theorem knowledge soft deal curse rr project pairwise preserve perturb distance argue projection recently construct hadamard short hadamard matrix power matrix product matrix column construction factor project important prior preserve orthogonality transform take advantage recent projection generalized embed use run sparsity integer appropriately construction start let let proceed create I concatenation stacking row matrix I matrix finally diagonal construct stacking namely
iteration rejection sampler quantile quantile table run replication realize chain empirical truth interval ccccc draw ccccc sbm ccc sbm replication assess estimate quantile setting quantile subsampling simulation result substantially effort block estimate run markov chain iteration median three bm sbm take second time rs sense often substantial barrier approach alternatively bm sbm software implement let stationary probability space fs ts elementary k sn n mixing
bayesian integrate hyperparameter expression available usually estimate optimize mention early integral calculate analytically gp noise certain laplace approximation two insight laplace approximation comprehensive comparison exact monte binary cost prohibitive learn information linear technique ep algorithm apply classification find approximate distribution q diag tf nf ip ep site site cavity tractable cavity f ii identity mean variance cavity site site py tf z moment ep ep z expression conjugate technique give various loo cv optimization criterion give optimize distribution py th computation discuss average logarithm nlp loo generic probabilistic
bias behave differently behaviour variety arise merge apply dataset merge independent examine range unseen classifier estimate upper proceed bias experimental methodology conclusion clarity paper author assume instance predict possible instance cd create explicit induce misclassification predict formally misclassification classifier create likelihood course approach bootstrappe hold error low estimation present behaviour classifier follow rate decay bayes combination less dataset periodic fit predict point value variability significant deviation law extension significance permutation fit result fit bound pac begin iid division case empirically measure error eq vc dimension algorithm probability effectively make inherently classifier think arrange higher relate maximum loose development similar exploit
iteration reject inefficient move slow probable space method inefficient metropolis acc acc acc parameter acc acc acc evaluate monte acc acc acc acc burn acc upper tune instrumental end burn consequently transition fix ergodicity stationary conservative also describe direct parameter difficult nonlinearity slow evaluation candidate sampling calculate linearize convolution also correspondingly replace similar adopt deconvolution convolution spread partially deviation
multiple must independent structure shrink cluster variance mean tt contraction direction heterogeneous contraction pattern past mean intuitively appeal able unless successively instability diffusion dimensional establish dimension order instability examine use energy lyapunov widely dynamical equation energy discussion follow normal entropy lyapunov functional first order dt I lyapunov non decrease complete law thus shrink physical unstable density
variance square smooth often simple give mat ern modify length vary range slowly extremely treat essentially fit gps synthetic normally gp square exponential ml square variance lead surface estimate generative indicate
well behave choice section possible appear lemma calculate right hand numerically jacobian family find must prove composed step also solve newton problem continuous allow optimization define shape define therefore generalize fairly
message strategy base perfect knowledge interference channel formulate ability exchange infer bit inference bp field distribute channel precision decode pass message connect determine link complement write letter vector respectively hermitian hadamard covariance system information receiver
distribute identically document exchangeability basic document topic word distribution topic suggest give topic recursively topic formally mathematical lda vocabulary topic select select select join opposite document topic
sized accord table nd scale cauchy cauchy contaminate contaminate location sample exponential location asymmetric dd nine classifier discriminant neighbor mahalanobis ms mh dd dm correspondingly motivation package mass leave cross validation relatively calculation depth exactly calculation done circular avoid choose dd degree function dd never phase distribution evaluate pair classifier perform box figure depth arise dd assign throughout kind bad assignment rule advantage lda dm figure leave behave perform since cope dd perform like dd performance dd cauchy location alternative close attribute classifier picture contaminate leave dd dd slightly
dual dual transform kkt duality kkt corollary remark regression technique find sparse dimension feature sample extremely solve remain problem safe rule inactive predictor inactive reduce inactive inactive constraint dual polytope projection mainly uniqueness inactive group currently screen synthetic inactive lasso various scale daily life extract datum build interpretable desirable popular explanatory widely square regression appeal equivalent feature follow success wide dimension large solving memory aim optimization usage exist roughly divided safe screening heuristic guarantee discard discard sis association application correctness strong check kkt repeat screen safe screening discard safe screening safe base key search effective safe dual result discard inactive efficient screening rule screen rule safe discard indicate dual formulate dual onto polytope elegant accurately g
observation cut maximal observation cut recursively decrease easy warm yield efficient solve simplification find vertex maximal clearly observation conclude cut consequently cut cut instance observation cut define cut exist case empty matching matching observation leave area practically part vast address issue major practice view
form application analytical derive ig element estimate towards label unobserve factor variance matrix update use detail family stage stage convergence extensive detail derive penalize maximize weak number bic intuitively traditional detail criterion asymptotic identifiability general parameter identifiability mixture criterion sufficiently close cluster parameter
scalar recall pde pde powerful indeed ordinary du c eq introduce sort location say thing concrete write projection pde pde solution besides pde rarely pde issue decomposition pde depend difficult depend address computation pde stand alone follow pde statistic development respect integrable integral sign baseline density eq u ix e pde couple technique readily easy see somewhat calculation fisher ignore term therefore scale similar I case simple auxiliary variable association pde likewise example highlight observable dependent variable auxiliary variable shall case association solve approach seek map orthogonal column easy satisfie differential construct conditional I build important unnecessary dimensional quantity detail I characterize inference translate briefly construction many iid efficiency auxiliary though typically variable fully unobserved characteristic motivate demonstrate reduction development dimension reduction combine counterpart call validity establish property I familiar sufficient dimension I work say two classical problem work differential technique reduction beyond ordinary besides development framework notion idea statistic valid
proposition r control dependence immediately obtain formula simplify fix sequel iw v v dt ik jt vector omit w vx vx vx vx vx vx hx vx vx vx w vx obtain w h hx h assertion apply derive individual hold vx vx vx vx x x vx eq prove vx vx dt euclidean obtain vx vx vx x tx ty dt w therefore vx vx vx correspondingly vx vx fix estimate lemma euler proposal direct assertion proposal
summary statistic examine estimate indeed joint r package fit transform mixture component spherical perform bic care dimension tend bayes selection histogram posterior specification top panel marginal derive spherical quantile plot automatic statistic marginal estimate although perfect improve considerably summary regression regression work even high obtain flexible modelling place compare early
use expert guide constrain selection color image success width task instance fact computation inversion require much denominator fraction practice local recover equation add identity small value show simple work ignore classify function ignore possibly incorrect label assume suffer serious relax replace inconsistent namely allow partial expand fill location relax inference inconsistent penalty penalty convex unconstraine rewrite accordingly label inference optimization classify mathematically formulate let know equivalent consequently locate mml world multi setting empirical mml supervised ranking perform principled component pca reduction optimization harmonic multi similar assume function come fix rbf ik mml approach determine importance task measure mml evaluation micro average measure micro real belong synthesis description extract sequence natural scene image belong category fall text mining consist text
equality type parameterization gradient programming equality exponential parameterization necessary condition program strictly convex nf claim prove strict guarantee uniqueness continuity path turn imply uniqueness continuity original dy concrete check span generate hull constrain geometric program trajectory solid contour contour ode solver matlab evaluate derivative seven solid contour dash contour semidefinite linear minimize complicated involve nonconvex way enforce semidefinite convex simplify assume unique triangular equal ji kl express denote formula I kk kl lk term triangular path problematic surrogate add decrease possess minimum identity deduce accommodate notation
approach flow anomaly protocol collect anomaly noise common temporal pattern flow traffic difficulty flow utilize section comprise assess spike select entry traffic internet anomaly depict anomaly specification matrix take previous centralized centralized datum centralize scheme fig depict relative control trend convergent behavior small zero almost end enforce internet application however delay infeasible one end delay dependency delay low nearby end bottleneck link distribute processing motivate delay test internet reveal dominant demonstrate entry drop delay relative compressed completion principal pursuit capture traffic anomaly trace large spatially distribute cognitive cost stationary point solve identity q equivalence advantage follow alternative norm follow explore lagrangian constraint notational accordance structure matrix
formulation relate directly effect negative characterization demand regularization interpretable quantity hierarchy regularization freedom effect force hierarchy another sparsity aim develop focus problem interaction model scope address specifically tool problem previous interaction selection fall three specify hierarchy requirement relate paper proposal regularize implication impose framework real datum distinction draw hierarchy typically ny also include ridge discuss may involve notational pair produces guarantee motivation proposal hierarchy th symmetry add hierarchy undesirable relaxation replace solve add give establish relaxation david cox total interaction effect additional advantage relaxation restrictive good make hierarchy interaction relative main effect desirable fraction tuning modification net supplementary material write similarity group penalty
thm thm thm thm science china grant zhang china grant office development extract possible one motivated study point subspace result transform search keyword sampling reproduce describe phenomenon reconstruction long way rkh approximation wiener question consider point practical ask strategy pursuit extract number question next subspace span known extend operator transform significantly computational section lead search sampling commonly space
occur even though easily accomplish refer reader many factorization component analysis agglomerative recent subspace ssc provably norms fix minimum frobenius iff frobenius generalize norm remarkable svd moreover remain optimal invariant frobenius product block decomposable easy work frobenius norm much thin svd complement ap ap minimum frobenius solution unique iff large largely adapt frobenius decomposable question whether frobenius norm unfortunately put
total jump unobserved jump observe become normalise hx remainder evy e hx px reveal constant use substitute simplify substitute variable let simplify get term term second fourth equal reject subsampling superposition separate take define process evy random measure evy evy merging impact evy jump evy argument detail first divide overlap use nm proof adapt measure mean measure introduce dependency finally dependency composition normalize measure randomize model random kind hierarchical model dependency get correlate treat stochastic model many
even contrast rely model candidate solution spectra affect degeneracy forward accurate bias pdf degeneracy g degeneracy algorithm magnitude however residual combination poor four model separately rich degeneracy bias relate temperature nm spectrum star star deeply assume make otherwise star spectrum incorrectly star rich star poor star might expect systematic two presence residual reveal weak degeneracy probably either effective temperature explain colour versa degeneracy degeneracy method unlike degeneracy reduce bottom panel star mean absolute bottom row star svm open circle green closed circle vertical line spectral f change panel g star especially star star sensitive form star probably consequence increase spectra h degeneracy higher would degeneracy competitive particular star low star row show mean star symbol circle red circle vertical indicate green surface generally responsible svm although degradation considerable dominate well magnitude helps constrain turn infer star accurately star star
topic well however base suggest construct beta achieve model decade hold decade decade hold find decade maximize dynamic model static version train static predict decade hold choose baseline select decade table increase par substantial take force nuclear activation table word peak united topic peak prominent address address show able multi modal topic framework prior retain exist conjugacy dependent literature specific effectiveness exchangeable latent time vary model superior dependent poisson dependent topic model predictive
di intersection hyperplane independent set prove linear independence exist regular q proper affine proper hyperplane discuss follow statement equivalent cell cell whose closure agree prove first follow useful proved induction non entry exist affine since affine translate enough since exist hyperplane let prof let statement regular strongly accord independent solution q strongly particular therefore hc contradiction item corollary hyperplane statement h follow let hyperplane intersect hyperplane intersect easy cell namely euclidean hyperplane hyperplane dimension intersection remain hyperplane operator
clutter obstacle continuous traversal perform minimize length continuous placing maximize traversal know clutter call brevity clutter motivation variant information require sensor technology necessarily nonetheless still clutter instance research know clutter set adjacency integer unit start vertex target intersect disk obstacle perform either disk devise length traversal exploitation capability call discretization statistical reader history fall rd algorithm continuous high provably instance suboptimal computable adapt discretized function q indicator expression disk probability disk obstacle ard algorithm would short walk disk might course traversal final trajectory walk since enter disk disk remove disk center obstacle repeat reach clutter obstacle clutter side obstacle disk center clutter obstacle disk important consider know clutter clutter example locate place disk unit obstacle clutter show circle solid circle truly equip sensor respective disk obstacle obstacle gray scale reflect disk true obstacle sensor traversal color probability show lattice discretization traversal ard obstacle clutter pass avoid point two actual status obstacle clutter side circle middle obstacle mark disk color obstacle lattice discretization traversal experiment disk radius unit disk center ensure clutter mark sample mark particular setup possess characteristic call extensive realization mainly identify category inter interaction tend pattern point category section describe six process background clutter disk spatial find brief spatial excellent intensity region pattern plane intensity call process four two disjoint independent problem possible scenario clutter start toward vice versa line toward simulate modification intensity location two process intensity intensity plane independently cluster cluster context existence point specific probability give star galaxy clutter formation encounter poisson randomize case cox mat
major subroutine give order new scoring different propose order define eq equal maximal combination parent consistent mathematically score scoring subroutine part graph function score consistent order operation avoid consume logarithm score lead incorrect generate max globally well consistent globally need construct thus avoid reduce decrease compatible parent indicate bit slow experiment bit huge order precede therefore vector parent notice compare generate parent range see increase limit parent number speedup parent
tree routine individual briefly mixture guarantee outline l l c product mixture separate random matrix isolate b h w j l jk let b r c triplet computation k h u find pairwise tree subgraph coincide tree liu exact marginal routine pair compute span tree complete complexity remark also briefly roughly conditional estimate decomposition selecting separate suitable threshold statistic similar graphical obtain analysis vector empirical vector I bind concentration bound denote norm frobenius norm sample result success exact ia h r q computed operator assumption observable u particular root polynomial invertible result imply isolate
well geometrically ergodic irreducible product probability lebesgue measure integer exist eq obtain eq convolution q prove prove every borel proceed repeatedly ergodicity
theorem grow exist candidate candidate incorporate considerable redundant group kind however deviation applicable feature agree constant regularization hellinger mainly specify sub mean continue mention critical dc lasso two modification focus often accelerate commonly fast solve sparse convenience follow depend supplement introduce whose involve become lasso preliminary
plain indeed difference regard stability robustness still maintain possess pe experimentally demonstrate detection even plain experimentally investigate artificial human activity sense twitter sensitivity issue algorithm red asymmetric green divergence change pe detection show p divergence behave imply forward point illustrate propose implementation median sample combination grid validation experiment contain manually sample e change mean number origin change step matrix change significant early
mix choose accordingly mdp learn colored bandit eliminate mdps parameter mdp learn mdp maximal transition translation knowledge know translation function color color diameter bound analogously additional assigning aggregate structured diameter mdp upper mix mix mix arm close reach sample reach necessary reach trial recall prove mix mix mix prove support upper mix j recall horizon confidence achieve
root bootstrap counterpart denote p p n give quite kernel degree proceed useful arbitrary kernel kernel arbitrary whenever notation define kernel arise theorem restriction theorem establish conclusion pointwise asymptotically valid theorem appendix fx gx g iii gx gx g gx fx gx nominal uniformly behave uniformly size greater strong convergence asymptotic approximation coverage probability poor
activity nucleotide bind activity act movement bind activation bind dna bind bind stress bind protein bind activity nucleotide bind activity activity heart protein development pathway activity bind bind bind activity bind activity pathway activation protein positive cell cell development anti cell communication negative dna pathway cell protein bind act incorporation molecular incorporation specific dna bind bind bind bind activity gene gene protein bind activity bind bind bind rna bind nucleotide development dna bind bind bind activity rna ii activity rna bind nucleotide bind binding via bind
classify apart node label cluster part figure book political orientation book political grid neutral book subtle categorical data som world data relation common dissimilarity categorical graph aspect knowledge mining prototype recent overview method tackle neural particular extension map
add b k k hence b g substitute k b b k b k g last equal far expand g k k rip eq term lemma gm x x use x rearrange recurrence relation b small rearrange bind choose mechanism omit detail brevity observe merely condition signal recovery constant could signal describe applicable wide attract considerable discuss instance utilize instance indicate kind gain offer practice signal overcomplete union orthonormal spin solve orthonormal basis expansion basis form algorithm solve efficiently depth basis commonly refer
develop comment constant proof continue arbitrary exposition instant laplacian require satisfy emphasize greatly analytical n ti devote eventually obtain accurate update may rewrite let successive iterate ni section generic possibly follow value value adapt processes contraction quantify behavior integer consensus subspace admits adapt satisfie depend constant weight state adapt deterministic sequence constant self sequence conclude assertion complete property inequality adapt hypothesis constant make recall stopping time constant stop deterministic give uk fall proposition event follow zero proof involve requirement trivially laplacian dependent dynamic drop evolves process hypothesis last inequality use fact lemma immediately preserve readily operator
usage sa like criterion seem optimisation process criterion complex annealing criterion differ involve movement position latter choose one drive metropolis criterion stand reduce multiplicative iteration reach evaluation guarantee achieve nevertheless frequent also decide result deep search reliable criterion originally develop achieve result optimal bad property estimate mutual criterion comparison attain variable design reduce randomness optimisation optimisation design evaluate plot figure devoted optimize criterion rectangle result criterion repeat figure give easily ease criterion recall rectangle eight plot individual order ml cn plot correspond restriction apply design plot
color correspond production material minimize well establish monte carlo hasting na implementation membership find drastically improve local partially structure propose label choose neighbor subsequent case minimization strategy belong propagation bp large approach
describe matlab using combine ad test alpha evy evy propose tends check exceed maximum model l evy ad exceed alpha via approach exhibit test evy confidence follow evy parameter depict htb illustrate section last ad last contain c yes evy stability obtain constitute l evy evy stable distribution behave ad indicate
analyze classifier classifier adopt classifier learn employ previous measure focus classifier unlabele contrast unsupervise nearest nn classifier unlabele classification derive misclassification generalization nn focus average classification measure misclassification error comprise plug encourage boundary separation weight induce laplacian region plug
vector nature base graph pairwise similarity scale memory usage objective probable pairwise observation usage pixel square memory usage method decomposition instability capable difficulty usage complexity concept rank greedy lead eigenvalue kernel efficient local develop directional scheme computational indicate complexity prohibitive efficient solution hyperspectral challenge embed overlap patch patch coordinate pixel reference merge result hard pixel contain cover resolution sensor poor overlap signature cover incorporate characteristic model challenge appeal numerous maxima formation explain observe create local still need neighbor maxima show weak map close convergence maxima handle short distance hope increase magnitude local report design algorithmic insight new dimensionality introduce kernel capability capture hyperspectral disjoint encode transform field demonstrate general applicability problem idea motion motivate short range map force act pixel iterative gradient energy configuration propose acquire propose desirable preserve topology induce strong structure disjoint spatially motivated cluster formulation popular assumption work visualization trajectory demonstrate overcome
past odd ci participant year response datum firstly test participant run ii estimator naive estimator effect regression ignore observed evidence dependence still replacement bias issue issue sampling reach spread dependence response take
survey proportional replacement rgb rgb corollary section rgb pt title derive algorithm develop case population assume
hard constrain decade gain numerous area example compressed finding system project method regularize programming cone conclude nonempty ball present technical close smooth whose arbitrarily respect projection map onto show statement gx lx lx next project arbitrary set project use gradient strongly modulus convexity continuity imply let let monotonicity immediately exist
fan fan scad regression scad penalty spline pose minimize fan initial minimizer scad singular origin step quadratic q eq compute third numerically fan li drawback stay also burden non regression knot generally spline knot error region hence value value place knot binomial numerical
likelihood among selection analysis analysis mainly conduct actually adjust agreement original rand index rand comparison observation group pair ari agreement negative indicate present parsimonious write datum set consist matrix detail ht c c datum note remain bic model model large good component latent estimate parameter sort number select horizontal hereafter factor bic
vc subgraph mention hull interest explicit condition relate excess risk express term convenient simplify inequality implicit convenient state distance excess risk express introduce g condition existence often refer satisfying condition often formulate abstract result weak replace add condition use place surrogate loss function explore condition satisfy satisfied function particular state condition classification nonnegative thus inverse nonnegative continuous easily type extensively thesis wang linear specifically disagreement coefficient class b wang reader refer discussion coefficient lead pg measurable van van distribution minimal pseudo van triangle inequality universal satisfie satisfy h h h p take set denote cover q since universal satisfie need monotonicity satisfy definition q arise abstract set vc gx pseudo statement convention say subgraph van interested concern function monotone vc subgraph van vc vc q apply vc arrive relaxed h use pf r xy xy h xy risk vc class imply apply log condition calibrate let erm er em note special loss value simplify xy bf vc minimax passive turn theorem xy j vc bound arrive regard vc class universal calibrate letting
necessarily see positive definite relate final imply lemma separable second fundamental possibly ambient complement although countable patch function always minimum let geodesic prove follow patch appropriate geodesic triangle neighborhood geodesic small extension mr orthogonal patch convex curve smoothness clearly establish closeness geodesic neighborhood hold minimum smoothness restrictive geodesic neighborhood eqs use region adapt old dx x manifold aside control interest q choose balance paper eight page long exceed nine page review consider presentation conference automatic line generation file version specific paper please number file version since early year blind file author
carlo draw multivariate predictive need map hyperparameter step compare case quadratic well present rejection enforce tail zero improve additional importance multivariate exact posterior posterior scale negative separately adaptively skewness distribution split observe need speed first principal axis take second sampling compute computational demand experiment code importance number sample use minute method gps speed grid large form exploit kronecker covariance covariance evenly spaced point form covariance th product evaluate unfortunately kronecker multiplication exploit reduce correspond eigenvector construct eigenvector
subsection kernel positive definite therefore feature word hyper value transform gs fact gs semi provide say alphabet symmetric convolution b kernel string alphabet ai lx x x symmetric semi definite positive gs except specialized exponential rbf obtain convolution eq equation distribution one still convolution omit paper gs cope pre stage pre computation stage complexity gs tuple number pre complexity form see gs follow position string q computation involve operation complexity programming recurrence therefore operation consequently complexity gs close approximation gs post kernel require inverse guarantee stage indeed positive greatly speed prediction far similar kernel vanish exponentially rapidly distance approximate gs
show iteration build weak domain base variance mis predict decrease ensemble hypothesis rest section mean absolute mae define also mae facilitate optimization previous confident consistency scalar compose hypothesis learner consist additive weak learner build additive subsequently derive scalar derivation omit index represent equivalently loss transformation zero updating rule also adopt similar perform level selection likely task boost weighting proportion single notation instance music rating book rating movie rating netflix wikipedia log netflix record
decision adequate accommodate two equivalence explain diffusion person influence neighbor network equivalence model person influence neighbor considerable model real explain effect autocorrelation ambiguity network choice reflect regime attribute consideration actor opinion take present plausible autocorrelation logistic speed method quick conditionally biased direct directional distinguished counterpart analyze autocorrelation outcome accommodate network component coefficient sign must redundant significant network
state symmetric error estimate correlation contiguous adaptation namely mh hence mh average location proposal htb average proposal target configuration sake simplicity target pdfs specifically pdf gaussians different gaussian mean
focus ability gene group issue eventually formulate network strategy fitness interact update incorporate treat network pure strategy update neighbor regard updating summarize correspondence player player combine neighbor equilibrium convergence state player strategy update fitness player locally interaction player baseline fitness example fitness interpret quality utility determine predefined utility intensity relative fitness strong fitness equal death death birth db I proportional fitness birth replace death process strategy death strategy proportional birth process proportional fitness kind match kind update filter network degree include node strategy proportional fitness term adopt information calculate case rule update node I update
conceptually minor difference usage proximal nonsmooth sparsity notion forward stability convert stable forward exist prove regret simplify exist quantity algorithm online consecutive iterate far closely uniform define stable well uniformly stable interestingly stability forward incur access next go make attain q counterpart bound online concept besides right give bound relate third show stability
publicly capture check knowledge study include heuristic lack deal problem investigate decentralized clustering author thank discussion theorem theorem united ct usa university bayesian dag capture structure particular challenging bayesian arguably
want infinite forget far estimate try way maintain acknowledgment david nsf grateful comment calculate hidden show break exponential family depend second local proceeding imply focus write natural context yield noisy exactly context optimize ascent wang wang david wang scalable technique class allocation collection article stochastic handle small bayesian counterpart apply massive massive raw million book store want text book build datum website million description item want recommend item user continuously collect want classifier sequences million people make hypothesis trait illustrate challenge high structure observe finally never stream address challenge elegant visual language observation graphical explore organization book trait recommendation model powerful connect strategy quantity scalable address massive set store require computer specialize hardware topic collection document topic collection exhibit probabilistic model structure infer structure collection illustrate algorithm probabilistic vocabulary article analyze massive collection like traditional posterior article article behind study topic build variational transform traditionally iterate analyze inefficient stochastic noisy use variational inference show give subsample call variational graphical stochastic variational dirichlet
differ instantaneous ik x mu k tx instantaneous instantaneous instantaneous ii differ instantaneous instantaneous together introduce observation assumption bs inference infer causal functional call noise causality exploit additive instantaneous effect instantaneous propose algorithm insufficient datum mostly avoid answer code upon request finitely series whether causal influence review inference iid apply instantaneous describe causality instantaneous common
replacement provide elegant markovian incremental algorithm replacement probabilistic tool compare replacement scheme conclude conjecture arithmetic guarantee without replacement another spirit randomize algorithm exact solve projection medical device consist incremental nn replacement expectation average order tuple similarly replacement define inequality geometric inequality hold replacement implementation return pass iteration document arithmetic variety setting establish tool
reflect incidence widely know calculate solve cauchy euclidean figure incidence year year expect reveal factor work incidence stationary independence net applicability care health limitation need duration disease duration response disease disease duration play major role age disease diabete cause
binary pre unweighted addition numerator absence reward case similar set term avoid bias measure protein interaction majority finally generalization network conservative estimation weight much wherein original see common protein evaluate measure framework graph protein methodology evaluation first measure strength protein input operate higher interaction higher structured constitute thresholding performance next run function go simple neighborhood inspire score protein annotate protein relevant cross roc curve score separately section study subsequent explain trend interaction protein transform detail transform describe number connect ensure network bias variation protein primarily score edge weakly measure transform create network
preliminary flexible dynamic representation small along limit flexibility need could parametric discriminate potentially model smoothly combine kalman track multimodal impact switch delay attribute limitation discrimination perturbation actual change particular response cf even perturbation stable improved distribution increase unnecessary dynamic person active transition time mostly discrete principle dynamic kalman filter discrimination process illustrate simple motion human
cdf q parametric associate unbiased n distribution condition show pareto hence cover thus situation quantile establish quantile well hill tail denote statistic eq author standard stand convergence situation real necessarily tail
bayes flat coverage mean probably principle confidence big determination principle frequentist coincide whereas
definition usa paris france e microsoft centre france multiple type label stochastic label correspond focus scenario reconstruct correctly transition belief insensitive threshold know correlate absolutely fully graph underlying recover belief phenomenon implication lose detection sensitivity belief noise second locally resemble ability propagation retrieve structure
appropriately main idea e section th expression get irrespective precise error term let rip see finish theorem rank theorem v v present essentially decrease bit iterate iterate particular replace iterate remain lemma imply span exactly modify also iterate version state suppose span subsequent span
rely type linear scale light material occur fact resolution fine enough material completely mixed measuring depict material detector measure band spectrum weight spectra weight conversely physical interaction light material occur light reflect object hyperspectral illustrative derivation model borel show product sufficient bilinear material consist material mixing level average individual lead illustrate leave material layer devoted mechanism real scenario nonlinear back year find material scene infer signature pose rely scene hard spectra later aim kernel algorithm mixture design sufficiently nonlinearity degree radial polynomial expansion inspire successively e occur scene term differ rely firstly multiple vice secondly signature signature extraction achieve strategy neural reduce parameter fashion reference therein seem sufficiently cover wide however assume prior extraction attempt conduct introduce algorithmic n simplex volume compute geodesic distance approximate short path distance nearest even recently geodesic datum fully mixture cite one assume either present accurately mix parameter great evaluate usefulness beginning expand past decade discuss rest processing hyperspectral dimensionality classical determination inversion sparse determination inversion brief light correction intrinsic property stress span spectra band identify subspace
j slice markov b g gray robust optimization parallel slice discussion r rate metropolis sampler application van j wang b university reconstruct landscape carlo j theorem axiom exercise remark summary via slice sampling south u edu edu work school business
variance equal definition precisely gauss unfortunately neither algorithm asynchronous message sufficiently ij notice min converge demonstrate reweighte pass compare typical min min sum illustrate asynchronous reweighted message asynchronous message asynchronous always converge min converge behavior apparent asynchronous definite increase decrease reweighted mark legend entry legend pos north east xlabel ylabel style number cd x cp plot mark axis width entry style pos north east xlabel ylabel format cd plot plot asynchronous entire positive region asynchronous require graph sufficiently matrix observation many leverage end standard approach correctness duality ascent tree reweighte fail walk variance matrix show exist monotone convergence open future open performance reweighte algorithm definite conjecture large force min
pg branching encode globally convention first branch store column indicate branch highlight bold sum procedure explain pg obtain ml pg invoke level feature input pg number ml pg pg assign numerical lemma provide blind assign consecutive library cell concatenation element numerical sum odd collect ml pg proof design prove concept interaction several process separately explain ml pg detail pg pg user familiar pg useful pg machine choice make tree explain extraction demand user pg extraction disadvantage engine ml pg datum interactive extraction three proof library send data mining engine extraction development pg ready pg build modular complete environment format hash table name lemma second sensible engine concrete format necessary engine equally popular machine matlab file case file extend learn extraction proof ml pg flexible use invoke engine ml connect machine mechanism connect ml pg ml ml pg choice matlab ml pg advance
problem whereas policy prior kind thus simulate resp bandit episode resp compose armed bernoulli expectation draw bandit draw two bernoulli arm expectation reward distribution change interval standard rejection obtain policy policy introduce policy previously design bernoulli hyper tune policy refer detailed policy policy knowledge kind fair choice careful policy default use nothing enforce constraint automatically identify constraint symbolic maximal lead apply described formula among strategy result million candidate strategy formula equivalence distinct algorithm report
compact behavioral representation additional large explore structural formal structural dynamic report framework exploratory conclude remark direction graph automatically behavioral sequence flexible learn representative search instead choose implicitly component structural network time edge inactive node make edge unweighte instantaneous duration node temporal snapshot active order snapshot represent record ex ex set feature matrix ex ex ex tt discover role large discover representative degree unweighted weighted aggregate exist node sum mean aggregation retain extract formally evolve extract feature active feature snapshot matrix property network graph discover structural nmf minimum nonnegative positive intuitively
general case solve induction conclusion suppose l note eq continue assumption set conclusion follow subtract side side equal iterate side arrive use l relation use x x combine continue accord induction provide
volume carry common flow dl scheme attractive flexibility utilize dl motivate far suppose predicting dictionary supervise form count regularity link graph gram count flow flow use correspond snapshot incomplete count suitable l ls count regularizer norm give laplacian regularization explicitly write w b encourage count accord typically adopt regularization encourage lie approximate count relate cost solver well available readily apparent imputation depend dl historical describe canonical form seek dictionary apply learn use count structure abstraction similar operational b unbounde smoothness regularization validate although convex descent solver update fashion dl enhance sequentially train operational prediction phase summarize auto thick thick circle inner sep minimum mm fill draw blue minimum draw corner thin color height corner thin bottom color height cm l b c node red line width south start
histogram library rbf tuned fusion experiment svm linear fusion margin sum unweighted vote q series rbf layer represent kernel see compare stack tune result computational performance pairwise resolution base fold cv cm france
conclude probability finally multiple hypothesis substantially complex consider elsewhere presentation air office fa grant nf foundation grant dms university california department mathematics corollary proposition section pc almost optimal almost sequential test composite pt pt minus sequentially nan composite hypothesis density sequential likelihood ratio sequential test first weight minimize term appropriate specify minimize negligible minimax leibler divergence expansion operating characteristic lead
discover ann exchangeable development theory volume page institute mathematical statistic cm brownian bridge uniform partition science pages institute california stochastic et mathematics ann di able provide conjecture central central number local
onto interpret whose entry bx bx g ok g eq q begin ridge together verify part get small unique r ok homotopy explain part hence notice dominant dominant eigenvalue require theorem get follow denoise shift go great detail approximate path mesh move along toward maxima consider kernel collection trajectory trajectory gradient path conversely mesh trajectory shift replace advantage log constrain shift mesh repeat hx decompose b default bandwidth select mesh remove mesh mesh provide
impulse structure non lie get turn turn potentially asynchronous manner useful pass sense cs change vary several change exploit enhanced ability benefit limit signal condition algorithm capable understand role discuss limit category focus category combinatorial paper support support denote index non zero
describe figure place normal analytically inference flexibility form feature flexibility toy learn concept base image rely corpora explicit image semantic choose co occurrence language within english vocabulary three sec infer word concept word probability co occurrence create vocabulary expand differ translation see rank word rank direction english vice pair rather word dimensional model combine corpus concept semantic performance use incorporate several coherence concept primarily server white concept order perform monte sampling update primarily concept document end conjugacy exist throughout performance force long another infinite
team intersect people car email repository routine book file lose start person change fan book fourth graphic pass never need explicitly independence u eq b yield care handle exploit view go via eigenvector respective eigenvalue give though essential difference perturbation argument defer appendix event eq event union bind therefore ii p lemma distinct eigenvalue j jj define bound k j orthogonality make establish subspace perturbation orthonormal singular similarly x x claim assumption claim column orthonormal orthogonal complement fourth claim claim
hmm dirichlet allocation language multi cca guarantee datum conditioning model feature provide vector work well lot nlp throughout use cca hide non state cca despite still end estimator look example view generate similar view previous solve
information feedback quantify achievable rate source noiseless control relevance testing portfolio theory pair message content relationship message parametric direct information independently information complexity wu sample mutual plug empirical analogous graphical bayesian network node causality autoregressive independence causality identify instance conditionally network direct preliminary proved address uniqueness estimator graph although explicitly pairwise condition extend instantaneous direct network use wu al algorithm parent whether autoregressive topology investigate liu autoregressive lastly direct correctness denote alphabet alphabet collection process alphabet easily denote give similarity condition remove clear remark consider distribution x kullback kl condition follow p I mutual however quantify correlation sense quantify justify direct formulation causality note general conditioning conditioning channel although use condition entail relationship examine base deterministic characterized influence state evolve induce analogous system fully condition notation condition e mn causal positivity avoid degenerate case purely argue strict causality relevant strict causality essential factorization strictly generative strictly form variable causal
model configuration response configuration maximize respect em implement describe denote unobserved configuration complete denote free removing adopt decompose subject column element conditional maximize respect expect substitute x expected denote explicit maximum fisher illustrate follow add respect expression matrix canonical add quantity compute cf suggest initialize deterministic start strategy class multidimensional lc depend parameterization term link iv item aspect suitable criterion suggest mostly rely lr criterion lose lr prefer satisfie regularity large sample size criterion bic parsimonious class logit function difficulty parameterization basis lr lr test may trait collapse
penalize basis complicate derivative fit spline possess attractive stability boundary connection spline markovian flexibility author smoothing spline generalize spline adaptive spline spatially spline also spline smoothing impose spline ordinary counterpart p knot smoothing window regression mixture smoothing bayesian equip adaptive smoothing stochastic differential brief form notation nonzero conditional neighbor zero relate allow numerical sparse major inferential enhance link nest integrate fast accurate inference connection
different involve sparse g therein gradient bt ingredient proximity form proximity operator projection ball minimization function otherwise minimization require problem operator see base independently generic follow whose replacement equal prescribe task sphere new seven code task report learn sc discussion quantity generate new regularization parameter repeat time average plot method figure clearly indicate outperform experiment approach
statement
diverse diverse game reasonably time make kernel advantage though component kind frequently good field case allow factor computation possibility yet structure bad doubly exponential subset exponential assign pose challenge need technique structure dpp dpp share message pass min sum replace special compute quantity factor normalize structured variety new application diverse instance good translation high perhaps aid code rna possibility representative world pose pose tend overlap salient citation paper want number major corpus text article article cover development structural assumption make give rise message sometimes demonstrate technique projection dramatically maintain yield improve baseline remain modern computer number item roughly particle universe course type perform probability subset jump ground dpp dpp implicitly part instance position element might sentence give leverage doubly exponential possibility think part finite possibility player position player discretize position th challenge long define diversity decomposition present recall nonnegative diversity feature specify structure structure build factorization analogous clique field part part factor factorization score decompose argue quite tracking prefer high traffic allow enforce path part clique mrf define multiplicative graph thus mrfs binary mrfs structure model diverse diversity function could coarse position player enforce instead norm bias towards every bias seem practice diversity balance high diverse structured contribution especially significant imagine unlikely remain go roughly position unique dpp serious combinatorial introduction dramatically increase item subtle might factor quality likely structure probable mrf nearly identical sample likely contain truly diverse set describe technical develop intuition study result motion track task follow simplified example motion assume particle discretized trajectory model trajectory involve depend sensor simplicity determine quality trajectory position measure smoothness primary mode depict blue curve leave quality maximize particle move essence path start position smoothly time trajectory similar trajectory pass near trajectory nearby position feature diversity feature diversity effect quality visualization expect trajectory draw model row panel trajectory independently probabilitie evident due preference diverse tend still smooth start near fig particle trajectory top quality particle make usual intractable dpp small feature diversity could marginal linearity add contribution however sum number expensive substitute factorization turn pass order statistic standard pass special product eq decompose structure message graph pass graph factor graph bipartite graph type nod structure model associate similarly node distinct edge graph connect factor relationship auto f minimum cm f minimum circle minimum draw f edge dot dot tracking node circular factor depend appear factor allow assign configuration whenever graph merge go forward tree convert factor bound factor describe introduce weight goal belief propagation large structure structure assignment node factor every assignment nod single indicate assign recursive function pass along depend whether versa factor variable message come message local propagation pass phase call forward pass backward pass root pass edge prove inductive potential note pass message edge
precise mean classification herein ij ij reflect uncertainty component bic numerical mainly conduct cluster classification adjust rand rand rand pairwise divide observation plus group pair assume value pairwise true membership perfect interpret ari allow possibility correctly classify ari would relate continue artificial real datum algorithm repetition initialization polynomial artificial limit case application regression moreover posterior membership arise
feasible change even change redundant constraint bind near first redundant secondly suffice total short hold differ geometrically check vertex cube along adjacent form redundant redundant redundant observation constraint put way figure isolate relevant redundant furthermore every far away redundant point close feasible respect isolate length neighbourhood demonstrating give redundant say x euclidean hull compact face axis corner redundant redundant feasible contradict feasibility similarly contradiction complete closure rectangular close see heuristic redundant hold enough content redundancy concern feasible constraint set z say eq observation say bind extreme feasible say extreme redundant bind bind relevant converse implication extreme point redundant black dot show dot bind unchanged constraint non bind provide result maximizer q bind g maximizer observation q maximizer feasible claim note converse introduction infeasible still
difference profile diagnostic tool obtained plot age examine curve profile present pattern combination ht patient plot clear correlation measure enough confirm aid detect confirm separability specificity datum reliable test robustness reliable however well cross fit robustness derivative hour inferior measure maximum increase exploit patient age hour way effective high hour show classify remove reduce considerably change wish read simplicity threshold suggest svm recover find hour discriminate calibration secondary likely test reading overlap value confirm distinguish neither conjunction classifying limitation manuscript suggestion extension surveillance rt compare use marker protein confirm year sd year patient year diagnosis classify clinical rt every minute hour longitudinal patient f year lp date date duration death datum rt technique specificity rigorous suggest exploit reliably superior protein increase specificity potential issue address figure replicate obvious positive henceforth note tend remain constant notably
classifier root leaf reliability formula reject categorization information manually code three category topic topic code category organize hierarchy basis business economic political document category depend investigate assignment categorization assign category also split level region tc
verify solution verify select distribution generate verify critical computed homotopy instance instance user berkeley code analogue matrix interesting critical point considerably explain rational entry algebraic solution thus algebraic extension rank symmetric abstract group symmetric group case six write imply explicitly complex critical analogous identity hold namely likelihood variety topological ml version theorem true develop global negative practitioner restrict non comprise diagonal row algebraic relaxation closure see tensor format distribution random negative column transform form identify far inclusion maximum estimation use compute point function
bottom topology graph use clean operator manifold perturb weight access try reconstruct graph topology associate image filter patch unless study energy eigenvector first scale amount image denoise compute eigenvector associate add scale intermediate image denoise eigenvector image display eigenvector visually pc build red il function radial leave fig display radial frequency radial coincide supervise construct classifier improve practice pass denoise fig study number use two stage patch introduce notice visually eigenvector another patch patch patch becoming perturb eigenvector patch minimize patch eigenvector require texture aware limitation patch size patch ambient compare effect patch image pass noise mm number neighbor patch
graph check problem graph variable many standard interaction connectivity vertex factorization conditional independence statement factor consideration discrete precede paragraph algebraic geometry distribution independence subset
dissimilarity angle respectively angular distance scenario deduce ff consider extend explicit angle transform leave transform angle invariant triangular apply sign reverse
distance distance experimental datum experiment demonstrate utility relation learning new point label relation learn point run leave carlo calculate describe generate kind test choose space e english dissimilarity dissimilarity dissimilarity figure document base indexing frequency inverse figure relation axis get solid learn manifold matching
individual subject subject several thousand simulation time diagnosis function occur approximated integration ex minus ex address minus event death diagnosis disease death disease article move closely diagram history detail current
good cm pooling lp pool perform insight vary single max pooling yield validation htb stack convnet convnet ms small convnet convnet convnet
tp mutation status status gene rank p q q propose ise covariate covariate strength association smoothly covariate strength link motivate limitation covariate interpretation discover covariate focus extend categorical mix involve continuous another condition neighborhood regression versa frequently understand principle research support grant dms dms ar support nsf dms gm omit separate literature hold version population also proposition j satisfied assume moreover uniqueness
record hz measure movement acquisition delay appear movement measure signal compute sum band filter movement keep channel segment label movement compose movement amplitude movement signal task functional response learn label recognize recognize recognize movement movement operator polynomial hilbert multiplication operator computable operator inverse integral eigenfunction regularization
sis fs fs angle fan sis popularity screening number select control denote sis fs sis sis initial new dimensional well fs estimator repetition criteria eight coverage percentage repetition respectively table local well particular local around sis significantly improve note inconsistent fs small rule predictor improve cr ii sis fitting many basic fs fs iii
misspecification dynamic misspecification affect marginal alone test inconsistent continuous case know alternative part continuous f f z fy usual assumption automatically therefore proposition check satisfied equal r noting exist function k b f f cm theorem assumption diagnostic effect bootstrap dynamic feature dynamic entail method tailor
conditional variance would part covariance correct batch formulae adopt n z
depend moreover level therein condition small already play derivation least avoid case large improve bound actually indeed want require condition hold mean loss drop depend practice apply treat estimate absolute deviation condition strictly result assume element note theorem detect word proportion nonzero sufficiently absolute nonzero eq positive quasi similar imply positive
find relational count predict link link apply structure feature topic link traditionally use document infer mixture latent topic document inter similarity metric link however topic capable perform discussion neighborhood global predict link summarize metric simple metric approach local neighborhood devise measure topology similarity link show metric neighbor share strategy normalization nine common neighbor perform well nine metric new use normalization yield average two case assign highlight importance metric specific dataset investigation evaluate random coefficient type network base perform relationship hierarchical perform good link find propose walk large I walk variant show base sophisticated computation node count short path demonstrate even fairly metric effective anomalous prediction statistically initial measure metric require walk node label relational unlabeled accuracy add probability walk roughly heart engine metric propose interestingly metric base required walk start walk efficiently recursive set supervise combine meta discuss arbitrary technique decomposition global modify structure reduce approach spurious suggest score node unseen meta global metric future neighbor surprisingly learn combine instance metric feature classifier naive approach simple link metric supervise probabilistic propose method network report simple metric global perform combine metric promise discuss sensitive biology costly incomplete removal issue analyze propose different domain web analysis citation community many subsection perform attribute key kind feature mix relational use domain study web page find relational page similarly short predict link link must consistent rank wider select important training diverse new combine kind network network feature construct use relational feature yield combine new pairwise protein interaction tensor linear useful domain might user preference recommender system propose relate euclidean simple diversity relevant link systematically generate search learn link logistic link equality continue add long information score improve find search co citation search avoid challenge second link prediction supervise relational topology describe apply possible link flat link make prediction involved link influence nearby newly predict node use propagation compare intensive note get belief propagation significant simpler involve repeatedly predict node link label prediction logistic approach possibility walk directly target social link study likely recommend node behavior initial link probability walk arrive walk process argue reduce graph base show outperform final unsupervised yield new reveal project low existence kind adjacency
level solver compute elastic net grid qualitatively regard display top middle represent ratio dark region indicate time region fast run reach large gain active penalty reach net soft algorithm behave evaluation stand publicly package choose package implement substitute implementation resolution bad package publicly available web available benchmark package elastic net reference use since solve problem rely solution penalty value empty model sensible return large regularization due numerical instability encounter extreme square mostly rely spurious case restrict henceforth along objective
observe suboptimal play arm arm multiple help binomial thank chernoff hoeffding equal let chernoff hoeffding bind variable common na trial happen dx inequality define play outcome trial chernoff ix k inequality chernoff bound refer I let denote play sum j
